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{{Other uses of|p-box|P-box (disambiguation){{!}}P-box}}
[[Image:continuous p-box gamma(2,interval(2,4)),steps=500.png|thumb|300px|alt=A continuous p-box depicted as a graph with abscissa labeled X and ordinate labeled Probability|A p-box (probability box).]]
 
A '''probability box''' (or '''p-box''') is a characterization of an [[uncertain number]] consisting of both [[uncertainty quantification|aleatoric and epistemic uncertainties]] that is often used in [[risk analysis]] or quantitative [[uncertainty]] modeling where numerical calculations must be performed. [[Probability bounds analysis]] is used to make arithmetic and logical calculations with p-boxes.
 
An example p-box is shown in the figure at right for an uncertain number ''x'' consisting of a left (upper) bound and a right (lower) bound on the probability distribution for ''x''.  The bounds are coincident for values of ''x'' below 0 and above 24.  The bounds may have almost any shapes, including step functions, so long as they are monotonically increasing and do not cross each other.  A p-box is used to express simultaneously incertitude (epistemic uncertainty), which is represented by the breadth between the left and right edges of the p-box, and variability (aleatory uncertainty), which is represented by the overall slant of the p-box.
 
==Interpretation==
{{multiple image
  | direction = vertical
  | width    = 160
  | footer    = '''Dual interpretation of p-boxes'''
  | image1    = Continuous p-box showing interval probability interval(0.4, 0.36) that x is 2.5 or less.png
  | alt1      = p-box with dotted lines showing probability interval associated with an ''x''-value
  | caption1  = Probability that ''x'' is 2.5 or less is between 4% and 36%
  | image2    = Continuous p-box showning interval 95th percentile interval(9,16).png
  | alt2      = P-box with dotted lines showing interval 95th percentile
  | caption2  = 95th percentile is between 9 and 16
  }}
There are dual interpretations of a p-box.  It can be understood as [[Upper and lower bounds|bounds]] on the cumulative probability associated with any ''x''-value.  For instance, in the p-box depicted at right, the probability that the value will be 2.5 or less is between 4% and 36%.  A p-box can also be understood as bounds on the ''x''-value at any particular probability level.  In the example, the 95th percentile is sure to be between 9 and 16.
 
If the left and right bounds of a p-box are sure to enclose the unknown distribution, the bounds are said to be ''rigorous'', or absolute.  The bounds may also be the tightest possible such bounds on the distribution function given the available information about it, in which case the bounds are therefore said to be ''best-possible''.  It may commonly be the case, however, that not every distribution that lies within these bounds is a possible distribution for the uncertain number, even when the bounds are rigorous and best-possible.
 
==Mathematical definition==
P-boxes are specified by left and right bounds on the [[cumulative distribution function|cumulative probability distribution function]] (or, equivalently, the [[survival function]]) of a quantity and, optionally, additional information about the quantity’s [[mean]], [[variance]] and distributional shape (family, unimodality, symmetry, etc.).  A p-box represents a class of probability distributions consistent with these constraints.
 
Let {{Unicode|&#x1D53B;}} denote the space of distribution functions on the [[real number]]s {{Unicode|ℝ}}, i.e., {{Unicode|&#x1D53B;}} = {''D'' | ''D'' : {{Unicode|ℝ}} → [0,1], ''D''(''x'') ≤ ''D''(''y'') whenever ''x'' < ''y'', for all ''x'', ''y'' [[Naive_set_theory#Sets.2C_membership_and_equality|&isin;]] {{Unicode|ℝ}}}, and let {{Unicode|&#x1D540;}} denote the set of real [[Interval (mathematics)|intervals]], i.e., {{Unicode|&#x1D540;}} = {''i'' | ''i'' = [''i''<sub>1</sub>, ''i''<sub>2</sub>], ''i''<sub>1</sub> ≤ ''i''<sub>2</sub>, ''i''<sub>1</sub>, ''i''<sub>2</sub> ∈ {{Unicode|ℝ}}}.  Then a p-box is a quintuple {''{{overbar|F}}'', <u>''F''</u>, ''m'', ''v'', '''F'''}, where ''{{overbar|F}}'', <u>''F''</u> ∈ {{Unicode|&#x1D53B;}}, while ''m'', ''v'' ∈ {{Unicode|&#x1D540;}}, and '''F''' ⊆ {{Unicode|&#x1D53B;}}.  This quintuple denotes the set of distribution functions ''F'' ∈ {{Unicode|&#x1D53B;}} matching the following constraints: <!--are the formulas only good for non-negative random variables?-->
 
:<u>''F''</u> (''x'') &le; ''F''(''x'') &le; ''{{overbar|F}}''(''x''),
:<big>&int; </big>{{su|p=&infin;|b=-&infin;}} ''x'' [[Riemann-Stieltjes integral|d''F''(''x'')]] [[Naive_set_theory#Sets.2C_membership_and_equality|&isin;]]  ''m'',
:<big>&int;</big>{{su|p=&infin;|b=-&infin;}} ''x''<sup>2</sup>d''F''(''x'') &ndash; <big>(&int; </big>{{su|p=&infin;|b=-&infin;}} x d''F''(''x'')<big>)</big><sup>2</sup> [[Naive_set_theory#Sets.2C_membership_and_equality|&isin;]] ''v'', and
:''F'' [[Naive_set_theory#Sets.2C_membership_and_equality|&isin;]] '''F'''.
 
Thus, the constraints are that the distribution function ''F'' falls within prescribed bounds, the mean of the distribution (given by the [[Riemann-Stieltjes integral]]) is in the interval ''m'', the variance of the distribution is in the interval ''v'', and the distribution is within some admissible class of distributions '''F'''.
The Riemann–Stieltjes integrals do not depend on the differentiability of ''F''.
 
P-boxes serve the same role for [[random variables]] that [[upper and lower probabilities]] serve for [[event (probability theory)|events]].  In [[robust Bayes analysis]]<ref>Berger, J.O. (1984). The robust Bayesian viewpoint. Pages 63–144 in ''Robustness of Bayesian analyses'', edited by J.B. Kadane, Elsevier Science.</ref> a p-box is also known as a [[distribution band]].<ref>Basu, S. (1994). [http://www.jstor.org/pss/25050991 Variations of posterior expectations for symmetric unimodal priors in a distribution band]. ''[[Sankhya (journal)|Sankhyā: The Indian Journal of Statistics]], Series A'' '''56''': 320–334.</ref><ref>Basu, S., and A. DasGupta (1995). [http://www.math.niu.edu/~basu/Research/dband.ps Robust Bayesian analysis with distribution bands]. ''Statistics and Decisions'' '''13''': 333–349.</ref>  A p-box can be constructed as a closed neighborhood of a distribution ''F'' ∈ {{Unicode|&#x1D53B;}} under the [[Kolmogorov–Smirnov test|Kolmogorov]], [[Lévy metric|Lévy]] or [[Wasserstein metric]].  A p-box is a crude but computationally convenient kind of [[credal set]].  Whereas a credal set is defined solely in terms of the constraint '''F''' as a convex set of distributions (which automatically determine ''{{overbar|F}}'', <u>''F''</u>, ''m'', and ''v'', but are often very difficult to compute with), a p-box usually has a loosely constraining specification of '''F''', or even no constraint so that '''F '''= {{Unicode|&#x1D53B;}}.  Calculations with p-boxes, unlike credal sets, are often quite efficient, and algorithms for all standard mathematical functions are known.
 
A p-box is minimally specified by its left and right bounds, in which case the other constraints are understood to be vacuous as {''{{overbar|F}}'', <u>''F''</u>, [&ndash; <small><math>\infty</math></small>, <math>\infty</math>], [0, <math>\infty</math>], {{Unicode|&#x1D53B;}}}.  Even when these ancillary constraints are vacuous, there may still be nontrivial bounds on the mean and variance that can be inferred from the left and right edges of the p-box.
 
==Where p-boxes come from==
P-boxes may arise from a variety of kinds of incomplete information about a quantity, and there are several ways to obtain p-boxes from data and analytical judgment.
 
===Distributional p-boxes===
When a probability distribution is known to have a particular shape (e.g., normal, uniform, beta, Weibull, etc.) but its parameters can only be specified imprecisely as intervals, the result is called a distributional p-box, or sometimes a parametric p-box.  Such a p-box is usually easy to obtain by enveloping extreme distributions given the possible parameters.  For instance, if a quantity is known to be normal with mean somewhere in the interval [7,8] and standard deviation within the interval [1,2], the left and right edges of the p-box can be found by enveloping the distribution functions of four probability distributions, namely, normal(7,1), normal(8,1), normal(7,2), and normal(8,2), where normal(μ,σ) represents a normal distribution with mean μ and standard deviation σ.  All probability distributions that are normal and have means and standard deviations inside these respective intervals will have distribution functions that fall entirely within this p-box.  The left and right bounds enclose many non-normal distributions, but these would be excluded from the p-box by specifying normality as the distribution family.
 
===Distribution-free p-boxes===
Even if the parameters such as mean and variance of a distribution are known precisely, the distribution cannot be specified precisely if the distribution family is unknown.  In such situations, envelopes of all distributions matching given moments can be constructed from inequalities such as those due to [[Markov's inequality|Markov]], [[Chebyshev's inequality|Chebyshev]], [[Chebyshev's_inequality#Variant:_One-sided_Chebyshev_inequality|Cantelli]], or Rowe<ref>Rowe, N.C. (1988). [http://faculty.nps.edu/ncrowe/unary2.htm Absolute bounds on the mean and standard deviation of transformed data for constant-sign-derivative transformations]. ''SIAM Journal of Scientific and Statistical Computing'' '''9''': 1098–1113.</ref><ref>Smith, J.E. (1995). Generalized Chebychev inequalities: theory and applications in decision analysis. ''Operations Research'' '''43''': 807–825.</ref> that enclose all distribution functions having specified parameters.  These define distribution-free p-boxes because they make no assumption whatever about the family or shape of the uncertain distribution.  When qualitative information is available, such as that the distribution is [[Unimodal#Unimodal_probability_distribution|unimodal]], the p-boxes can often be tightened substantially.<ref>
Zhang, J. and D. Berleant (2005). Arithmetic on random variables: squeezing the envelopes with new joint distribution constraints. Pages 416–422 in ''Proceedings of the Fourth International Symposium On Imprecise Probabilities and Their Applications (ISIPTA ’05)'', Carnegie Mellon University, Pittsburgh, July 20–23, 2005.</ref>
 
===P-boxes from imprecise measurements===
When all members of a population can be measured, or when random sample data are abundant, analysts often use an [[empirical distribution function|empirical distribution]] to summarize the values.  When those data have non-negligible [[measurement uncertainty]] represented by interval ranges about each sample value, an empirical distribution may be generalized to a p-box.<ref name=Ferson-etal-2007>Ferson, S., V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg (2007). [http://www.ramas.com/intstats.pdf ''Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty'']. Sandia National Laboratories, SAND 2007-0939, Albuquerque, NM.</ref>  Such a p-box can be specified by cumulating the lower endpoints of all the interval measurements into a cumulative distribution forming the left edge of the p-box, and cumulating the upper endpoints to form the right edge.  The broader the measurement uncertainty, the wider the resulting p-box.
 
Interval measurements can also be used to generalize distributional estimates based on the [[method of moments (statistics)|method of matching moments]] or [[maximum likelihood]], that make shape assumptions such as normality or lognormality, etc.<ref name=Ferson-etal-2007 /><ref>Xiang, G., V. Kreinovich and S. Ferson, (2007). Fitting a normal distribution to interval and fuzzy data. Pages 560–565 in ''Proceedings of the 26th International Conference of the North American Fuzzy Information Processing Society NAFIPS'2007'', M. Reformat and M.R. Berthold (eds.).</ref>  Although the measurement uncertainty can be treated rigorously, the resulting distributional p-box generally will not be rigorous when it is a sample estimate based on only a subsample of the possible values. But, because these calculations take account of the dependence between the parameters of the distribution, they will often yield tighter p-boxes than could be obtained by treating the interval estimates of the parameters as unrelated as is done for distributional p-boxes.
 
===Confidence bands===
There may be uncertainty about the shape of a probability distribution because the sample size of the empirical data characterizing it is small.  Several methods in traditional statistics have been proposed to account for this sampling uncertainty about the distribution shape, including [[Kolmogorov%E2%80%93Smirnov_test#Setting_confidence_limits_for_the_shape_of_a_distribution_function|Kolmogorov–Smirnov]]<ref>Kolmogorov, A. (1941). Confidence Limits for an Unknown Distribution Function. ''Annals of Mathematical Statistics'' '''12''': 461–463.</ref> and similar<ref>Owen, A.B. (1995). Nonparametric likelihood confidence bands for a distribution function. ''Journal of the American Statistical Association'' '''90''': 516–521.</ref> [[confidence band]]s, which are [[distribution-free]] in the sense that they make no assumption about the shape of the underlying distribution.  There are related confidence-band methods that do make assumptions about the shape or family of the underlying distribution, which can often result in tighter confidence bands.<ref>Cheng, R.C.H., and T.C. Iles (1983). Confidence bands for cumulative distribution functions of continuous random variables. ''Technometrics'' '''25''': 77–86.</ref><ref>Cheng, R.C.H., B.E. Evans and J.E. Williams (1988). Confidence band estimations for distributions used in probabilistic design. ''International Journal of Mechanical Sciences'' '''30''': 835–845.</ref><ref name=murphy95>Murphy, S.A. (1995). [http://www.stat.lsa.umich.edu/~samurphy/papers/conf_int.pdf Likelihood ratio-based confidence intervals in survival analysis]. ''Journal of the American Statistical Association'' '''90''': 1399–1405.</ref>  Constructing confidence bands requires one to select the probability defining the confidence level, which usually must be less than 100% for the result to be non-vacuous.  Confidence bands at the (1&nbsp;&minus;&nbsp;α)% confidence level are defined such that, (1&nbsp;&minus;&nbsp;α)% of the time they are constructed, they will completely enclose the distribution from which the data were randomly sampled.  A confidence band about a distribution function is sometimes used as a p-box even though it represents statistical rather than rigorous or sure bounds.  This use implicitly assumes that the true distribution, whatever it is, is inside the p-box.
 
An analogous Bayesian structure is called a Bayesian p-box,<ref>Montgomery, V. (2009). [http://maths.dur.ac.uk/stats/people/fc/thesis-VM.pdf ''New Statistical Methods in Risk Assessment by Probability Bounds'']. Ph.D. dissertation, Durham University, UK.</ref> which encloses all distributions having parameters within a subset of parameter space corresponding to some specified probability level from a Bayesian analysis of the data.  This subset is the [[credible interval|credible region]] for the parameters given the data, which could be defined as the highest posterior probability density region, or the lowest posterior loss region, or in some other suitable way.  To construct a Bayesian p-box one must select a prior distribution, in addition to specifying the credibility level (analogous to a confidence level).
 
===Envelopes of possible distributions===
When there are multiple possible probability distributions that might describe a variable, and an analyst cannot discount any of them based on available information, a p-box can be constructed as the envelope of the various cumulative distributions.<ref name=Fersonetal03>Ferson, S., V. Kreinovich, L. Ginzburg, D.S. Myers, and K. Sentz (2003). [http://www.ramas.com/unabridged.zip ''Constructing Probability Boxes and Dempster–Shafer Structures'']. Sandia National Laboratories, SAND2002-4015, Albuquerque, NM.</ref><ref>Fu, G., D. Butler, S.-T. Khu, and S. Sun (2011). Imprecise probabilistic evaluation of sewer flooding in urban drainage systems using random set theory. ''Water Resources Research'' '''47''': W02534.</ref>  It is also possible to account for the uncertainty about which distribution is the correct one with a sensitivity study, but such studies studies become more complex as the number of possible distributions grows, and combinatorially more complex as the number of variables about which there could be multiple distributions increases.  An enveloping approach is more conservative about this uncertainty than various alternative approaches to handle the uncertainty which average together distributions in [[stochastic mixture]] models or [[Bayesian model average]]s.  The unknown true distribution is likely to be within the class of distributions encompassed by the p-box.  In contrast, assuming the true distribution is one of the distributions being averaged, the average distribution is sure to be unlike the unknown true distribution.
 
===P-boxes from calculation results===
P-boxes can arise from computations involving probability distributions, or involving both a probability distribution and an interval, or involving other p-boxes.  For example, the sum of a quantity represented by a probability distribution and a quantity represented by an interval will generally be characterized by a p-box.<ref>Ferson, S., and L.R. Ginzburg (1996). Different methods are needed to propagate ignorance and variability. ''Reliability Engineering and Systems Safety'' '''54''': 133–144.</ref>  The sum of two random variables characterized by well-specified probability distributions is another precise probability distribution typically only when the [[copula (statistics)|copula]] (dependence function) between the two summands is completely specified.  When their dependence is unknown or only partially specified, the sum will be more appropriately represented by a p-box because different dependence relations lead to many different distributions for the sum.  [[Kolmogorov]] originally asked what bounds could be placed about the distribution of a sum when nothing is known about the dependence between the distributions of the addends.<ref name=Frank-etal-1987>Frank, M.J., R.B. Nelsen and B. Schweizer (1987). Best-possible bounds for the distribution of a sum—a problem of Kolmogorov. ''Probability Theory and Related Fields'' '''74''': 199–211.</ref>  The question was only answered in the early 1980s.  Since that time, formulas and algorithms for sums have been generalized and extended to differences, products, quotients and other binary and unary functions under various dependence assumptions.<ref name=Frank-etal-1987 /><ref>Yager, R.R. (1986). Arithmetic and other operations on Dempster–Shafer structures. ''International Journal of Man-machine Studies'' '''25''': 357–366.</ref><ref>Williamson, R.C., and T. Downs (1990). Probabilistic arithmetic I: Numerical methods for calculating convolutions and dependency bounds. ''International Journal of Approximate Reasoning'' '''4''': 89–158.</ref><ref>Berleant, D. (1993). Automatically verified reasoning with both intervals and probability density functions. ''Interval Computations'' '''1993 (2) ''': 48–70.</ref><ref>Berleant, D., G. Anderson, and C. Goodman-Strauss (2008). Arithmetic on bounded families of distributions: a DEnv algorithm tutorial. Pages 183–210 in ''Knowledge Processing with Interval and Soft Computing'', edited by C. Hu, R.B. Kearfott, A. de Korvin and V. Kreinovich, Springer (ISBN 978-1-84800-325-5).</ref><ref>Berleant, D., and C. Goodman-Strauss (1998). Bounding the results of arithmetic operations on random variables of unknown dependency using intervals. ''Reliable Computing'' '''4''': 147–165.</ref><ref>Ferson, S., R. Nelsen, J. Hajagos, D. Berleant, J. Zhang, W.T. Tucker, L. Ginzburg and W.L. Oberkampf (2004). [http://www.ramas.com/depend.pdf ''Dependence in Probabilistic Modeling, Dempster–Shafer Theory, and Probability Bounds Analysis'']. Sandia National Laboratories, SAND2004-3072, Albuquerque, NM.</ref>
 
These methods, collectively called [[probability bounds analysis]], provide algorithms to evaluate mathematical expressions when there is uncertainty about the input values, their dependencies, or even the form of mathematical expression itself.  The calculations yield results that are guaranteed to enclose all possible distributions of the output variable if the input p-boxes were also sure to enclose their respective distributions.  In some cases, a calculated p-box will also be best-possible in the sense that ''only'' possible distributions are within the p-box, but this is not always guaranteed. 
For instance, the set of probability distributions that could result from adding random values without the independence assumption from two (precise) distributions is generally a proper [[subset]] of all the distributions admitted by the computed p-box.  That is, there are distributions within the output p-box that could not arise under any dependence between the two input distributions.  The output p-box will, however, always contain all distributions that are possible, so long as the input p-boxes were sure to enclose their respective underlying distributions.  This property often suffices for use in [[risk analysis]].
 
==Special cases==
Precise [[probability distribution#Probability_distributions_of_real-valued_random_variables|probability distributions]] and [[interval (mathematics)|interval]]s are special cases of p-boxes, as are [[real number|real values]] and [[integer]]s. Because a probability distribution expresses variability and lacks incertitude, the left and right bounds of its p-box are coincident for all ''x''-values at the value of the cumulative distribution function (which is a non-decreasing function from zero to one).  Mathematically, a probability distribution ''F'' is the degenerate p-box {''F'', ''F'', E(''F''), V(''F''), ''F''}, where E and V denote the expectation and variance operators.  An interval expresses only incertitude.  Its p-box looks like a rectangular box whose upper and lower bounds jump from zero to one at the endpoints of the interval.  Mathematically, an interval [''a'', ''b''] corresponds to the degenerate p-box {H(''a''), H(''b''), [''a'', ''b''], [0, (''b''&ndash;''a'')<sup>2</sup>/4], {{Unicode|&#x1D53B;}}}, where H denotes the [[Heaviside step function]].  A precise scalar number ''c'' lacks both kinds of uncertainty.  Its p-box is just a step function from 0 to 1 at the value ''c''; mathematically this is {H(''c''), H(''c''), ''c'', 0, H(''c'')}.
 
==Applications==
{{:Applications of p-boxes and probability bounds analysis}}
 
==Criticisms==
''No internal structure''.  Because a p-box retains little information about any internal structure within the bounds, it does not elucidate which distributions within the p-box are most likely, nor whether the edges represent very unlikely or distinctly likely scenarios.  This could complicate decisions in some cases if an edge of a p-box encloses a decision threshold.
 
''Loses information''.  To achieve computational efficiency, p-boxes lose information compared to more complex [[Dempster-Shafer_theory#Formal_definition|Dempster-Shafer structures]] or [[credal set]]s.<ref name=Fersonetal03 />  In particular, p-boxes lose information about the mode (most probable value) of a quantity. This information could be useful to keep, especially in situations where the quantity is an unknown but fixed value.
 
''Traditional probability sufficient''.  Some critics of p-boxes argue that precisely specified probability distributions are sufficient to characterize uncertainty of all kinds.  For instance, [[Dennis Lindley|Lindley]] has asserted, "Whatever way uncertainty is approached, probability is the ''only'' sound way to think about it."<ref>Page 71 in Lindley, D.V. (2006). ''Understanding Uncertainty''. John Wiley & Sons, Hoboken, New Jersey.</ref><ref>http://en.wikiquote.org/wiki/Dennis_Lindley</ref><!--ref><<O’Hagan>> < / ref -->  These critics argue that it is meaningless to talk about ‘uncertainty about probability’ and that traditional [[probability theory|probability]] is a complete theory that is sufficient to characterize all forms of uncertainty.  Under this criticism, users of p-boxes have simply not made the requisite effort to identify the appropriate precisely specified distribution functions.
 
''Possibility theory can do better''.  Some critics contend that it makes sense in some cases to work with a [[possibility theory|possibility]] distribution rather than working separately with the left and right edges of p-boxes.  They argue that the set of probability distributions [[Possibility_theory#Possibility_theory_as_an_imprecise_probability_theory|induced]] by a possibility distribution is a subset of those enclosed by an analogous p-box's edges.<ref>Baudrit, C., D. Dubois, H. Fargier (2003). Représentation de la connaissance probabiliste incomplète. Pages 65–72 in ''Actes Rencontres Francophones sur la Logique Floue et ses Applications (LFA'03), Tours, France''. Cépaduès-Éditions.</ref><ref>Baudrit, C. (2005.) [http://www.univ-orleans.fr/mapmo/membres/baudrit/manuscrit_these.pdf ''Représentation et propagation de connaissances imprécises et incertaines : Application à l'évaluation des risques liés aux sites et aux sols pollués'']. Ph.D. dissertation, Université Paul Sabatier, Toulouse III.</ref>  Others make a counterargument that one cannot do better with a possibility distribution than with a p-box.<ref>Troffaes, M.C.M., and S. Destercke (2011).  [http://dx.doi.org/10.1016/j.ijar.2011.02.001 Probability boxes on totally preordered spaces for multivariate modelling]. ''International Journal of Approximate Reasoning'' (in press).</ref>
 
==See also==
* [[uncertain number]]
* [[interval (mathematics)|interval]]
* [[cumulative distribution function|cumulative probability distribution]]
* [[upper and lower probabilities]]
* [[credal set]]
* [[risk analysis]]
* [[uncertainty propagation]]
* [[probability bounds analysis]]
* [[Dempster–Shafer theory]] and the section on [[Dempster–Shafer theory#Formal_definition|Dempster–Shafer structure]]
* [[imprecise probability]]
* [[Confidence_band#Pointwise_and_simultaneous_confidence_bands|simultaneous confidence bands]] on distribution and [[survival function]]s using likelihood ratios<ref name=murphy95 />
* [[Confidence_band#Pointwise_and_simultaneous_confidence_bands|pointwise]] binomial confidence intervals for ''F''(''X'') for a given ''X''<ref>Meeker, W.Q., and L.A. Escobar (1998). ''Statistical Methods for Reliability Data'', John Wiley and Sons, New York.</ref>
* [[list of uncertainty propagation software|uncertainty propagation software]]
 
==References==
{{Reflist|2}}
 
==Additional references==
* Baudrit, C., and D. Dubois (2006). [http://www.univ-orleans.fr/mapmo/membres/baudrit/csda06.pdf Practical representations of incomplete probabilistic knowledge]. ''Computational Statistics & Data Analysis'' '''51''': 86–108.
 
* Baudrit, C., D. Dubois, D. Guyonnet (2006). [http://www.univ-orleans.fr/mapmo/membres/baudrit/ieee06.pdf Joint propagation and exploitation of probabilistic and possibilistic information in risk assessment]. ''IEEE Transactions on Fuzzy Systems'' '''14''': 593–608.
 
* Bernardini, A., and F. Tonon (2009). [http://www.caee.utexas.edu/prof/tonon/Publications/Papers/Paper%20P%202008-02%20Extreme%20probability%20distributions%20of.pdf Extreme probability distributions of random/fuzzy sets and p-boxes]. ''International Journal of Reliability and Safety'' '''3''': 57-78. [http://savannah.gatech.edu/workshop/rec08/documents/REC08_Paper_Bernardini.pdf (alternative link)]
 
* Destercke, S., D. Dubois and E. Chojnacki (2008). [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V07-4T2M5W9-1&_user=10&_coverDate=11%2F30%2F2008&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=5e765b0819db1b4f43f21cd7e90fe509&searchtype=a Unifying practical uncertainty representations – I: Generalized p-boxes].  ''International Journal of Approximate Reasoning'' '''49''': 649–663 .
 
* Dubois, D. (2010). (Commentary) Representation, propagation, and decision issues in risk analysis under incomplete probabilistic information. ''Risk Analysis'' '''30''': 361–368. {{doi|10.1111/j.1539-6924.2010.01359.x}}.
 
* Dubois, D., and D. Guyonnet (2011). Risk-informed decision-making in the presence of epistemic uncertainty. ''International Journal of General Systems'' '''40''': 145–167.
 
* Guyonnet, D., F. Blanchard, C. Harpet, Y. Ménard, B. Côme and C. Baudrit (2005). [http://www.brgm.fr/publication/pubDetailRapportSP.jsp?id=RSP-BRGM/RP-54099-FR Projet IREA—Traitement des incertitudes en évaluation des risques d'exposition]. Rapport BRGM/RP-54099-FR, Bureau de Recherches Géologiques et Minières, France.
 
{{DEFAULTSORT:Probability Box}}
[[Category:Probability bounds analysis]]
[[Category:Risk analysis]]
[[Category:Uncertainty of numbers]]

Latest revision as of 03:39, 18 January 2014

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Since Ground to Flooring peak is at all times > Flooring to Ceiling heights, some developers prefer to quote the floor to flooring top. Again, we must always examine between developments utilizing ground to ceiling heights. Core Central Region (CCR) > Central (or RCR, Rest of central region) > Remainder of Singapore (or OCR, Outside Central Area) Since students that dwell inside 1km or 2km to a school gets entry precedence to that school, proximity to good faculties adds to the worth of the event. Information to the shopping for process Guide me by way of the process of shopping for accomplished properties (rental/property already built liao). I need to have a minimum of an idea of it! The lawyer's fees don't embrace stamp charges or disbursements correctly incurred bythe lawyer. Promoting Your Home or Flat

One other distinguished actual property developer is Keppel Land Restricted. It contributes to Singapore's skyline with landmark developments such as the Ocean Monetary Centre and Marina Bay Financial Centre within the Central Enterprise District. The company is also Asia's premier dwelling developer with world-class iconic waterfront properties at Keppel Bay and Marina Bay. It has a land financial institution of about a hundred thirty million sq. toes with a pipeline of about 70,000 homes across Asia and the Center East to faucet into the demand for quality housing pushed by residence ownership aspirations.

Philip Ng, who holds degrees in civil and geotechnical engineering in addition to metropolis planning, has over the previous decade or so spruced up the corporate in Singapore and hired many professionals. The group has developed many award-successful buildings. United Abroad Financial institution Group chairman Wee Cho Yaw described Mr Ng as 'an previous pal of more than 50 years' who had 'an intuitive aptitude for reading property cycles'. Even the biggest property corporations in Hong Kong take their hats off to his firm in Hong Kong, where it enjoys a excessive standing,' Mr Liew added. Ng Teng Fong, one in every of Asia's richest tycoons and the founder and chairman of Singapore's largest unlisted property group, Far East Group, died Tuesday aged 82, the developer said. SB Improvement Pte Ltd

Relying whether you might be buying the property for own keep or funding, location performs an important function. Properties in prime districts retain their value very nicely and they usually have the very best property developers in singapore capital achieve in a bullish property market. Properties in the suburbs are lower in value and could also be more suitable for own keep than funding. In case you can buying the property for investment, the properties in prime districts like district 09, 10, eleven or the Central Business District are the most secure purchase. Properties with sea view on the East Coast are also great for a resort residence or investment.

Only Singapore citizens and accredited persons can purchase Landed 'residential property'as outlined within the Residential Properties Act. Foreigners are eligible to purchaseunits in condominiums or residences which aren't landed dwelling homes. Foreignerswho wish to purchase landed property in Singapore should first search the approval ofthe Controller of Residential Property. Your Lawyer's Role in a Property Purchase Pending completion of your buy, your lawyer will lodge a caveat towards the title to the property - this serves to notify the public (and any third occasion fascinated in the property) that you've a legitimate interest or claim to the title of the property arising from the contract for the sale and purchase. Property was all the time on his thoughts Property

The primary week makes or breaks a developer. A perceived excessive take up price within the first week of launch helps gross sales tremendously. Nothing helps gross sales greater than a headline title of "80% of launch property bought over the weekend". That keeps enthusiasm for the mission for the remainder of the week when folks flock to see the show flat subsequently. Ascendas Pte Ltd - This property improvement firm in Singapore presents actual property services that embrace mission management, renovation, estate and property management. Along with this, it also manages non-public real estate funds. What's non-restricted residential property underneath the Residential Property Act Sale and lease of non-residential properties are subject to GST. Financing your Property EC web site attracts S$a hundred and seventy.1m bid

Mr Brooke, for one, expects prices in urban areas to rise 15 per cent within the subsequent two years, while he sees prices in outlying areas taking one other three-four years to get well. - 2002 April sixteen by Audrey Tan Singapore Business Instances Does anyone have any info /expertise of the property market in Singapore? My partner and I want to transfer there soon and have seen a great place we like nonetheless we actually do not know if the market has reached its peak or whether it's imagined to proceed to rise. Keppel Land is Asia's premier house developer with world-class iconic waterfront residences at Keppel Bay and Marina Bay in Singapore. Hedges Park, a brand new condominium by Hong Leong Group with impressive full facilities. Shaw Properties (1997) Pte Ltd Yeo Hiap Seng Limited

A continuous p-box depicted as a graph with abscissa labeled X and ordinate labeled Probability
A p-box (probability box).

A probability box (or p-box) is a characterization of an uncertain number consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be performed. Probability bounds analysis is used to make arithmetic and logical calculations with p-boxes.

An example p-box is shown in the figure at right for an uncertain number x consisting of a left (upper) bound and a right (lower) bound on the probability distribution for x. The bounds are coincident for values of x below 0 and above 24. The bounds may have almost any shapes, including step functions, so long as they are monotonically increasing and do not cross each other. A p-box is used to express simultaneously incertitude (epistemic uncertainty), which is represented by the breadth between the left and right edges of the p-box, and variability (aleatory uncertainty), which is represented by the overall slant of the p-box.

Interpretation

People are nonetheless rushing to buy properties in Singapore. A new residential condominium mission in Bedok had a long queue three days before its official launch and bought more than 350 units on the very first day. Nonetheless, there is a ‘development' that had emerged in 2011 that the majority have possibly overlooked.

In August, more than a hundred individuals across 16 states reportedly had been sickened with salmonella after consuming mangos In September, the Meals and Drug Administration detained mango imports from a Mexican packing house after the fruits had been linked to the sickenings Save Cash On Your Online Purchasing Make saving money when purchasing on-line a breeze with Bodge's favourite app, Invisible Hands. This helpful program does the price-checking analysis for you, and will warn you when a unique site has a better value - or let you realize that you have found the most effective deal. Watch Posted by Edison Foo September 29, 2013 HEARTH SALE!!! From only $9xx,000! We hope this listing will help launch your search for new properties in Singapore. It's unlawful to purchase ice cream after 6 pm, in Newark.

for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed

After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant

Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013

6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!

BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill There are dual interpretations of a p-box. It can be understood as bounds on the cumulative probability associated with any x-value. For instance, in the p-box depicted at right, the probability that the value will be 2.5 or less is between 4% and 36%. A p-box can also be understood as bounds on the x-value at any particular probability level. In the example, the 95th percentile is sure to be between 9 and 16.

If the left and right bounds of a p-box are sure to enclose the unknown distribution, the bounds are said to be rigorous, or absolute. The bounds may also be the tightest possible such bounds on the distribution function given the available information about it, in which case the bounds are therefore said to be best-possible. It may commonly be the case, however, that not every distribution that lies within these bounds is a possible distribution for the uncertain number, even when the bounds are rigorous and best-possible.

Mathematical definition

P-boxes are specified by left and right bounds on the cumulative probability distribution function (or, equivalently, the survival function) of a quantity and, optionally, additional information about the quantity’s mean, variance and distributional shape (family, unimodality, symmetry, etc.). A p-box represents a class of probability distributions consistent with these constraints.

Let PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER denote the space of distribution functions on the real numbers PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER, i.e., PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER = {D | D : PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER → [0,1], D(x) ≤ D(y) whenever x < y, for all x, y ∈ PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER}, and let PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER denote the set of real intervals, i.e., PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER = {i | i = [i1, i2], i1 ≤ i2, i1, i2 ∈ PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER}. Then a p-box is a quintuple {Template:Overbar, F, m, v, F}, where Template:Overbar, F ∈ PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER, while m, v ∈ PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER, and F ⊆ PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER. This quintuple denotes the set of distribution functions F ∈ PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER matching the following constraints:

F (x) ≤ F(x) ≤ Template:Overbar(x),
∫ Template:Su x dF(x) ∈ m,
∫Template:Su x2dF(x) – (∫ Template:Su x dF(x))2 ∈ v, and
F ∈ F.

Thus, the constraints are that the distribution function F falls within prescribed bounds, the mean of the distribution (given by the Riemann-Stieltjes integral) is in the interval m, the variance of the distribution is in the interval v, and the distribution is within some admissible class of distributions F. The Riemann–Stieltjes integrals do not depend on the differentiability of F.

P-boxes serve the same role for random variables that upper and lower probabilities serve for events. In robust Bayes analysis[1] a p-box is also known as a distribution band.[2][3] A p-box can be constructed as a closed neighborhood of a distribution F ∈ PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER under the Kolmogorov, Lévy or Wasserstein metric. A p-box is a crude but computationally convenient kind of credal set. Whereas a credal set is defined solely in terms of the constraint F as a convex set of distributions (which automatically determine Template:Overbar, F, m, and v, but are often very difficult to compute with), a p-box usually has a loosely constraining specification of F, or even no constraint so that F = PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER. Calculations with p-boxes, unlike credal sets, are often quite efficient, and algorithms for all standard mathematical functions are known.

A p-box is minimally specified by its left and right bounds, in which case the other constraints are understood to be vacuous as {Template:Overbar, F, [– ∞, ∞], [0, ∞], PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

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Where p-boxes come from

P-boxes may arise from a variety of kinds of incomplete information about a quantity, and there are several ways to obtain p-boxes from data and analytical judgment.

Distributional p-boxes

When a probability distribution is known to have a particular shape (e.g., normal, uniform, beta, Weibull, etc.) but its parameters can only be specified imprecisely as intervals, the result is called a distributional p-box, or sometimes a parametric p-box. Such a p-box is usually easy to obtain by enveloping extreme distributions given the possible parameters. For instance, if a quantity is known to be normal with mean somewhere in the interval [7,8] and standard deviation within the interval [1,2], the left and right edges of the p-box can be found by enveloping the distribution functions of four probability distributions, namely, normal(7,1), normal(8,1), normal(7,2), and normal(8,2), where normal(μ,σ) represents a normal distribution with mean μ and standard deviation σ. All probability distributions that are normal and have means and standard deviations inside these respective intervals will have distribution functions that fall entirely within this p-box. The left and right bounds enclose many non-normal distributions, but these would be excluded from the p-box by specifying normality as the distribution family.

Distribution-free p-boxes

Even if the parameters such as mean and variance of a distribution are known precisely, the distribution cannot be specified precisely if the distribution family is unknown. In such situations, envelopes of all distributions matching given moments can be constructed from inequalities such as those due to Markov, Chebyshev, Cantelli, or Rowe[4][5] that enclose all distribution functions having specified parameters. These define distribution-free p-boxes because they make no assumption whatever about the family or shape of the uncertain distribution. When qualitative information is available, such as that the distribution is unimodal, the p-boxes can often be tightened substantially.[6]

P-boxes from imprecise measurements

When all members of a population can be measured, or when random sample data are abundant, analysts often use an empirical distribution to summarize the values. When those data have non-negligible measurement uncertainty represented by interval ranges about each sample value, an empirical distribution may be generalized to a p-box.[7] Such a p-box can be specified by cumulating the lower endpoints of all the interval measurements into a cumulative distribution forming the left edge of the p-box, and cumulating the upper endpoints to form the right edge. The broader the measurement uncertainty, the wider the resulting p-box.

Interval measurements can also be used to generalize distributional estimates based on the method of matching moments or maximum likelihood, that make shape assumptions such as normality or lognormality, etc.[7][8] Although the measurement uncertainty can be treated rigorously, the resulting distributional p-box generally will not be rigorous when it is a sample estimate based on only a subsample of the possible values. But, because these calculations take account of the dependence between the parameters of the distribution, they will often yield tighter p-boxes than could be obtained by treating the interval estimates of the parameters as unrelated as is done for distributional p-boxes.

Confidence bands

There may be uncertainty about the shape of a probability distribution because the sample size of the empirical data characterizing it is small. Several methods in traditional statistics have been proposed to account for this sampling uncertainty about the distribution shape, including Kolmogorov–Smirnov[9] and similar[10] confidence bands, which are distribution-free in the sense that they make no assumption about the shape of the underlying distribution. There are related confidence-band methods that do make assumptions about the shape or family of the underlying distribution, which can often result in tighter confidence bands.[11][12][13] Constructing confidence bands requires one to select the probability defining the confidence level, which usually must be less than 100% for the result to be non-vacuous. Confidence bands at the (1 − α)% confidence level are defined such that, (1 − α)% of the time they are constructed, they will completely enclose the distribution from which the data were randomly sampled. A confidence band about a distribution function is sometimes used as a p-box even though it represents statistical rather than rigorous or sure bounds. This use implicitly assumes that the true distribution, whatever it is, is inside the p-box.

An analogous Bayesian structure is called a Bayesian p-box,[14] which encloses all distributions having parameters within a subset of parameter space corresponding to some specified probability level from a Bayesian analysis of the data. This subset is the credible region for the parameters given the data, which could be defined as the highest posterior probability density region, or the lowest posterior loss region, or in some other suitable way. To construct a Bayesian p-box one must select a prior distribution, in addition to specifying the credibility level (analogous to a confidence level).

Envelopes of possible distributions

When there are multiple possible probability distributions that might describe a variable, and an analyst cannot discount any of them based on available information, a p-box can be constructed as the envelope of the various cumulative distributions.[15][16] It is also possible to account for the uncertainty about which distribution is the correct one with a sensitivity study, but such studies studies become more complex as the number of possible distributions grows, and combinatorially more complex as the number of variables about which there could be multiple distributions increases. An enveloping approach is more conservative about this uncertainty than various alternative approaches to handle the uncertainty which average together distributions in stochastic mixture models or Bayesian model averages. The unknown true distribution is likely to be within the class of distributions encompassed by the p-box. In contrast, assuming the true distribution is one of the distributions being averaged, the average distribution is sure to be unlike the unknown true distribution.

P-boxes from calculation results

P-boxes can arise from computations involving probability distributions, or involving both a probability distribution and an interval, or involving other p-boxes. For example, the sum of a quantity represented by a probability distribution and a quantity represented by an interval will generally be characterized by a p-box.[17] The sum of two random variables characterized by well-specified probability distributions is another precise probability distribution typically only when the copula (dependence function) between the two summands is completely specified. When their dependence is unknown or only partially specified, the sum will be more appropriately represented by a p-box because different dependence relations lead to many different distributions for the sum. Kolmogorov originally asked what bounds could be placed about the distribution of a sum when nothing is known about the dependence between the distributions of the addends.[18] The question was only answered in the early 1980s. Since that time, formulas and algorithms for sums have been generalized and extended to differences, products, quotients and other binary and unary functions under various dependence assumptions.[18][19][20][21][22][23][24]

These methods, collectively called probability bounds analysis, provide algorithms to evaluate mathematical expressions when there is uncertainty about the input values, their dependencies, or even the form of mathematical expression itself. The calculations yield results that are guaranteed to enclose all possible distributions of the output variable if the input p-boxes were also sure to enclose their respective distributions. In some cases, a calculated p-box will also be best-possible in the sense that only possible distributions are within the p-box, but this is not always guaranteed. For instance, the set of probability distributions that could result from adding random values without the independence assumption from two (precise) distributions is generally a proper subset of all the distributions admitted by the computed p-box. That is, there are distributions within the output p-box that could not arise under any dependence between the two input distributions. The output p-box will, however, always contain all distributions that are possible, so long as the input p-boxes were sure to enclose their respective underlying distributions. This property often suffices for use in risk analysis.

Special cases

Precise probability distributions and intervals are special cases of p-boxes, as are real values and integers. Because a probability distribution expresses variability and lacks incertitude, the left and right bounds of its p-box are coincident for all x-values at the value of the cumulative distribution function (which is a non-decreasing function from zero to one). Mathematically, a probability distribution F is the degenerate p-box {F, F, E(F), V(F), F}, where E and V denote the expectation and variance operators. An interval expresses only incertitude. Its p-box looks like a rectangular box whose upper and lower bounds jump from zero to one at the endpoints of the interval. Mathematically, an interval [a, b] corresponds to the degenerate p-box {H(a), H(b), [a, b], [0, (b–a)2/4], PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

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Applications

Applications of p-boxes and probability bounds analysis

Criticisms

No internal structure. Because a p-box retains little information about any internal structure within the bounds, it does not elucidate which distributions within the p-box are most likely, nor whether the edges represent very unlikely or distinctly likely scenarios. This could complicate decisions in some cases if an edge of a p-box encloses a decision threshold.

Loses information. To achieve computational efficiency, p-boxes lose information compared to more complex Dempster-Shafer structures or credal sets.[15] In particular, p-boxes lose information about the mode (most probable value) of a quantity. This information could be useful to keep, especially in situations where the quantity is an unknown but fixed value.

Traditional probability sufficient. Some critics of p-boxes argue that precisely specified probability distributions are sufficient to characterize uncertainty of all kinds. For instance, Lindley has asserted, "Whatever way uncertainty is approached, probability is the only sound way to think about it."[25][26] These critics argue that it is meaningless to talk about ‘uncertainty about probability’ and that traditional probability is a complete theory that is sufficient to characterize all forms of uncertainty. Under this criticism, users of p-boxes have simply not made the requisite effort to identify the appropriate precisely specified distribution functions.

Possibility theory can do better. Some critics contend that it makes sense in some cases to work with a possibility distribution rather than working separately with the left and right edges of p-boxes. They argue that the set of probability distributions induced by a possibility distribution is a subset of those enclosed by an analogous p-box's edges.[27][28] Others make a counterargument that one cannot do better with a possibility distribution than with a p-box.[29]

See also

References

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Additional references

  • Dubois, D. (2010). (Commentary) Representation, propagation, and decision issues in risk analysis under incomplete probabilistic information. Risk Analysis 30: 361–368. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park..
  • Dubois, D., and D. Guyonnet (2011). Risk-informed decision-making in the presence of epistemic uncertainty. International Journal of General Systems 40: 145–167.
  1. ↑ Berger, J.O. (1984). The robust Bayesian viewpoint. Pages 63–144 in Robustness of Bayesian analyses, edited by J.B. Kadane, Elsevier Science.
  2. ↑ Basu, S. (1994). Variations of posterior expectations for symmetric unimodal priors in a distribution band. Sankhyā: The Indian Journal of Statistics, Series A 56: 320–334.
  3. ↑ Basu, S., and A. DasGupta (1995). Robust Bayesian analysis with distribution bands. Statistics and Decisions 13: 333–349.
  4. ↑ Rowe, N.C. (1988). Absolute bounds on the mean and standard deviation of transformed data for constant-sign-derivative transformations. SIAM Journal of Scientific and Statistical Computing 9: 1098–1113.
  5. ↑ Smith, J.E. (1995). Generalized Chebychev inequalities: theory and applications in decision analysis. Operations Research 43: 807–825.
  6. ↑ Zhang, J. and D. Berleant (2005). Arithmetic on random variables: squeezing the envelopes with new joint distribution constraints. Pages 416–422 in Proceedings of the Fourth International Symposium On Imprecise Probabilities and Their Applications (ISIPTA ’05), Carnegie Mellon University, Pittsburgh, July 20–23, 2005.
  7. ↑ 7.0 7.1 Ferson, S., V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg (2007). Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty. Sandia National Laboratories, SAND 2007-0939, Albuquerque, NM.
  8. ↑ Xiang, G., V. Kreinovich and S. Ferson, (2007). Fitting a normal distribution to interval and fuzzy data. Pages 560–565 in Proceedings of the 26th International Conference of the North American Fuzzy Information Processing Society NAFIPS'2007, M. Reformat and M.R. Berthold (eds.).
  9. ↑ Kolmogorov, A. (1941). Confidence Limits for an Unknown Distribution Function. Annals of Mathematical Statistics 12: 461–463.
  10. ↑ Owen, A.B. (1995). Nonparametric likelihood confidence bands for a distribution function. Journal of the American Statistical Association 90: 516–521.
  11. ↑ Cheng, R.C.H., and T.C. Iles (1983). Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25: 77–86.
  12. ↑ Cheng, R.C.H., B.E. Evans and J.E. Williams (1988). Confidence band estimations for distributions used in probabilistic design. International Journal of Mechanical Sciences 30: 835–845.
  13. ↑ 13.0 13.1 Murphy, S.A. (1995). Likelihood ratio-based confidence intervals in survival analysis. Journal of the American Statistical Association 90: 1399–1405.
  14. ↑ Montgomery, V. (2009). New Statistical Methods in Risk Assessment by Probability Bounds. Ph.D. dissertation, Durham University, UK.
  15. ↑ 15.0 15.1 Ferson, S., V. Kreinovich, L. Ginzburg, D.S. Myers, and K. Sentz (2003). Constructing Probability Boxes and Dempster–Shafer Structures. Sandia National Laboratories, SAND2002-4015, Albuquerque, NM.
  16. ↑ Fu, G., D. Butler, S.-T. Khu, and S. Sun (2011). Imprecise probabilistic evaluation of sewer flooding in urban drainage systems using random set theory. Water Resources Research 47: W02534.
  17. ↑ Ferson, S., and L.R. Ginzburg (1996). Different methods are needed to propagate ignorance and variability. Reliability Engineering and Systems Safety 54: 133–144.
  18. ↑ 18.0 18.1 Frank, M.J., R.B. Nelsen and B. Schweizer (1987). Best-possible bounds for the distribution of a sum—a problem of Kolmogorov. Probability Theory and Related Fields 74: 199–211.
  19. ↑ Yager, R.R. (1986). Arithmetic and other operations on Dempster–Shafer structures. International Journal of Man-machine Studies 25: 357–366.
  20. ↑ Williamson, R.C., and T. Downs (1990). Probabilistic arithmetic I: Numerical methods for calculating convolutions and dependency bounds. International Journal of Approximate Reasoning 4: 89–158.
  21. ↑ Berleant, D. (1993). Automatically verified reasoning with both intervals and probability density functions. Interval Computations 1993 (2) : 48–70.
  22. ↑ Berleant, D., G. Anderson, and C. Goodman-Strauss (2008). Arithmetic on bounded families of distributions: a DEnv algorithm tutorial. Pages 183–210 in Knowledge Processing with Interval and Soft Computing, edited by C. Hu, R.B. Kearfott, A. de Korvin and V. Kreinovich, Springer (ISBN 978-1-84800-325-5).
  23. ↑ Berleant, D., and C. Goodman-Strauss (1998). Bounding the results of arithmetic operations on random variables of unknown dependency using intervals. Reliable Computing 4: 147–165.
  24. ↑ Ferson, S., R. Nelsen, J. Hajagos, D. Berleant, J. Zhang, W.T. Tucker, L. Ginzburg and W.L. Oberkampf (2004). Dependence in Probabilistic Modeling, Dempster–Shafer Theory, and Probability Bounds Analysis. Sandia National Laboratories, SAND2004-3072, Albuquerque, NM.
  25. ↑ Page 71 in Lindley, D.V. (2006). Understanding Uncertainty. John Wiley & Sons, Hoboken, New Jersey.
  26. ↑ http://en.wikiquote.org/wiki/Dennis_Lindley
  27. ↑ Baudrit, C., D. Dubois, H. Fargier (2003). Représentation de la connaissance probabiliste incomplète. Pages 65–72 in Actes Rencontres Francophones sur la Logique Floue et ses Applications (LFA'03), Tours, France. Cépaduès-Éditions.
  28. ↑ Baudrit, C. (2005.) Représentation et propagation de connaissances imprécises et incertaines : Application à l'évaluation des risques liés aux sites et aux sols pollués. Ph.D. dissertation, Université Paul Sabatier, Toulouse III.
  29. ↑ Troffaes, M.C.M., and S. Destercke (2011). Probability boxes on totally preordered spaces for multivariate modelling. International Journal of Approximate Reasoning (in press).
  30. ↑ Meeker, W.Q., and L.A. Escobar (1998). Statistical Methods for Reliability Data, John Wiley and Sons, New York.
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