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'''''H''<sub>∞</sub>''' (i.e. '''"''H''-infinity"''') '''methods''' are used in [[control theory]] to synthesize controllers achieving robust performance or stabilization.  To use ''H''<sub>∞</sub> methods, a control designer expresses the control problem as a [[mathematical optimization]] problem and then finds the controller that solves this.  ''H''<sub>∞</sub> techniques have the advantage over classical control techniques in that they are readily applicable to problems involving multivariable systems with cross-coupling between channels; disadvantages of ''H''<sub>∞</sub> techniques include the level of mathematical understanding needed to apply them successfully and the need for a reasonably good model of the system to be controlled.  Problem formulation is important, since any controller synthesized will only be 'optimal' in the formulated sense: optimizing the wrong thing often makes things worse rather than better. Also, non-linear constraints such as saturation are generally not well-handled.
'''Pythagorean expectation''' is a formula invented by [[Bill James]] to estimate how many games a [[baseball]] team "should" have won based on the number of [[run (baseball)|runs]] they scored and allowed.  Comparing a team's actual and Pythagorean winning percentage can be used to evaluate how lucky that team was (by examining the variation between the two winning percentages).  The name comes from the formula's resemblance to the [[Pythagorean theorem]].<ref>[http://thegamedesigner.blogspot.com/2012/05/pythagoras-explained.html The Game Designer|title=Pythagoras Explained]</ref>


The term '''''H''<sub>∞</sub>''' comes from the name of the mathematical space over which the optimization takes place: ''H''<sub>∞</sub> is the space of [[matrix (mathematics)|matrix]]-valued functions that are [[analytic function|analytic]] and bounded in the open right-half of the [[complex plane]] defined by Re(''s'')&nbsp;>&nbsp;0; the ''H''<sub>∞</sub> norm is the maximum singular value of the function over that space.  (This can be interpreted as a maximum gain in any direction and at any frequency; for [[Single-Input and Single-Output|SISO]] systems, this is effectively the maximum magnitude of the frequency response.)  ''H''<sub>∞</sub> techniques can be used to minimize the closed loop impact of a perturbation: depending on the problem formulation, the impact will either be measured in terms of stabilization or performance. 
The basic formula is:


Simultaneously optimizing robust performance and robust stabilization is difficult.  One method that comes close to achieving this is [[H-infinity loop-shaping|''H''<sub>∞</sub> loop-shaping]], which allows the control designer to apply classical loop-shaping concepts to the multivariable frequency response to get good robust performance, and then optimizes the response near the system bandwidth to achieve good robust stabilization.
:<math>\mathrm{Win} = \frac{\text{runs scored}^2}{\text{runs scored}^2 + \text{runs allowed}^2} = \frac{1}{1+(\text{runs allowed}/\text{runs scored})^2}</math>


Commercial software is available to support ''H''<sub>∞</sub> controller synthesis.
where Win is the winning ratio generated by the formula. The expected number of wins would be the expected winning ratio multiplied by the number of games played.


== Problem formulation ==
==Empirical origin==
Empirically, this formula correlates fairly well with how baseball teams actually perform. However, statisticians since the invention of this formula found it to have a fairly routine error, generally about 3 games off. For example, in 2002, the New York Yankees scored 897 runs, allowing 697 runs. According to James' original formula, the Yankees should have won 62.35% of their games.
:<math>\mathrm{Win} = \frac{\text{897}^{2}}{\text{897}^{2} + \text{697}^{2}} = 0.623525865</math>
Based on a 162 game season, the Yankees should have won 101.07 games. The 2002 Yankees actually went 103-58.<ref>[http://www.baseball-reference.com/teams/NYY/2002.shtml Baseball-reference.com | 2002 NY Yankees Statistics<!-- Bot generated title -->]</ref>


First, the process has to be represented according to the following standard configuration:
In efforts to fix this error, statisticians have performed numerous searches to find the ideal exponent.


[[Image:H-infty plant representation.png]]
If using a single number exponent, 1.83 is the most accurate, and the one used by baseball-reference.com, the premier website for baseball statistics across teams and time.<ref>[http://www.baseball-reference.com/about/faq.shtml#pyth Baseball-reference.com | What is pythagorean winning percentage?<!-- Bot generated title -->]</ref> The updated formula therefore reads as follows:


The plant ''P'' has two inputs, the exogenous input ''w'', that includes reference signal and disturbances, and the manipulated variables ''u''. There are two outputs, the error signals ''z'' that we want to minimize, and the measured variables ''v'', that we use to control the system. ''v'' is used in ''K'' to calculate the manipulated variable ''u''. Notice that all these are generally [[vector (geometry)|vectors]], whereas '''P''' and '''K''' are [[matrix (mathematics)|matrices]].
:<math>\mathrm{Win} = \frac{\text{runs scored}^{1.83}}{\text{runs scored}^{1.83} + \text{runs allowed}^{1.83}} = \frac{1}{1+(\text{runs allowed}/\text{runs scored})^{1.83}}</math>


In formulae, the system is:
The most widely known is the Pythagenport formula<ref>[http://www.baseballprospectus.com/article.php?articleid=342 Baseball Prospectus | Articles | Revisiting the Pythagorean Theorem<!-- Bot generated title -->]</ref> developed by [[Clay Davenport]] of [[Baseball Prospectus]]:
:<math>\mathrm{Exponent} = 1.50 * log(\frac{R+RA}G) +0.45</math>


:<math>\begin{bmatrix} z\\ v \end{bmatrix} = \mathbf{P}(s)\, \begin{bmatrix} w\\ u\end{bmatrix} = \begin{bmatrix}P_{11}(s) & P_{12}(s)\\P_{21}(s) & P_{22}(s)\end{bmatrix} \, \begin{bmatrix} w\\ u\end{bmatrix}</math>
He concluded that the exponent should be calculated from a given team based on the team's runs scored (R), runs allowed (RA), and games (G). By not reducing the exponent to a single number for teams in any season, Davenport was able to report a 3.9911 root-mean-square error as opposed to a 4.126 root-mean-square error for an exponent of 2.<ref>[http://www.baseballprospectus.com/article.php?articleid=342 Baseball Prospectus | Articles | Revisiting the Pythagorean Theorem<!-- Bot generated title -->]</ref>


:<math>u = \mathbf{K}(s) \, v</math>
Less well known but equally (if not more) effective is the [[Pythagenpat]] formula, developed by David Smyth.<ref>[http://gosu02.tripod.com/id69.html W% Estimators<!-- Bot generated title -->]</ref>
:<math>\mathrm{Exponent} = (\frac{R+RA}G)^{.287} </math>


It is therefore possible to express the dependency of ''z'' on ''w'' as:
Davenport expressed his support for this formula, saying: <blockquote>
After further review, I (Clay) have come to the conclusion that the so-called Smyth/Patriot method, aka Pythagenpat, is a better fit. In that, ''X''&nbsp;=&nbsp;((''rs''&nbsp;+&nbsp;''ra'')/''g'')<sup>0.285</sup>, although there is some wiggle room for disagreement in the exponent. Anyway, that equation is simpler, more elegant, and gets the better answer over a wider range of runs scored than Pythagenport, including the mandatory value of 1 at 1&nbsp;rpg.<ref>[http://baseballprospectus.com/glossary/index.php?mode=viewstat&stat=136 Baseball Prospectus | Glossary<!-- Bot generated title -->]</ref>
</blockquote>


:<math>z=F_\ell(\mathbf{P},\mathbf{K})\,w</math>
These formulas are only necessary when dealing with extreme situations in which the average number of runs scored per game is either very high or very low. For most situations, simply squaring each variable yields accurate results.


Called the ''lower linear fractional transformation'', <math>F_\ell</math> is defined (the subscript comes from ''lower''):
There are some systematic statistical deviations between actual winning percentage and expected winning percentage, which include [[bullpen]] quality and luck.  In addition, the formula tends to [[Regression toward the mean|regress toward the mean]], as teams that win a lot of games tend to be underrepresented by the formula (meaning they "should" have won fewer games), and teams that lose a lot of games tend to be overrepresented (they "should" have won more).


:<math>F_\ell(\mathbf{P},\mathbf{K}) = P_{11} + P_{12}\,\mathbf{K}\,(I-P_{22}\,\mathbf{K})^{-1}\,P_{21}</math>
=="Second-order" and "third-order" wins==
In their Adjusted Standings Report,<ref>[http://www.baseballprospectus.com/statistics/standings.php Adjusted Standings Report]</ref> [[Baseball Prospectus]] refers to different "orders" of wins for a team. The basic order of wins is simply the number of games they have won. However, because a team's record may not reflect its true talent due to luck, different measures of a team's talent were developed.


Therefore, the objective of <math>\mathcal{H}_\infty</math> control design is to find a controller <math>\mathbf{K}</math> such that <math>F_\ell(\mathbf{P},\mathbf{K})</math> is minimised according to the <math>\mathcal{H}_\infty</math> norm. The same definition applies to <math>\mathcal{H}_2</math> control design. The infinity norm of the transfer function matrix <math>F_\ell(\mathbf{P},\mathbf{K})</math> is defined as:
First-order wins, based on pure run differential, are the number of expected wins generated by the "pythagenport" formula (see above). In addition, to further filter out the distortions of luck, [[sabermetrician]]s can also calculate a team's ''expected'' runs scored and allowed via a [[runs created]]-type equation (the most accurate at the team level being [[Base Runs]]). These formulas result in the team's expected number of runs given their offensive and defensive stats (total singles, doubles, walks, etc.), which helps to eliminate the luck factor of the order in which the team's hits and walks came within an inning. Using these stats, sabermetricians can calculate how many runs a team "should" have scored or allowed.


:<math>||F_\ell(\mathbf{P},\mathbf{K})||_\infty = \sup_\omega \bar{\sigma}(F_\ell(\mathbf{P},\mathbf{K})(j\omega))</math>
By plugging these expected runs scored and allowed into the pythagorean formula, one can generate second-order wins, the number of wins a team deserves based on the number of runs they should have scored and allowed given their component offensive and defensive statistics. Third-order wins are second-order wins that have been adjusted for strength of schedule (the quality of the opponent's pitching and hitting). Second- and third-order winning percentage has been shown to predict future actual team winning percentage better than both actual winning percentage and first-order winning percentage.


where <math>\bar{\sigma}</math> is the maximum [[singular value]] of the matrix <math>F_\ell(\mathbf{P},\mathbf{K})(j\omega)</math>.
==Theoretical explanation==
Initially the correlation between the formula and actual winning percentage was simply an experimental observation.  In 2003, Hein Hundal provided an inexact derivation of the formula and showed that the Pythagorean exponent was approximately 2/(''σ''√''π'') where ''σ'' was the standard deviation of runs scored by all teams divided by the average number of runs scored.<ref>{{cite news|first=Hein|last=Hundal|title=Derivation of James Pythagorean Formula (Long)|url=http://groups.google.com/group/rec.puzzles/browse_thread/thread/3be0e6ad49631ddb/bfb52d16b12955ac?q=hein+hundal+pythagorean&fwc=1}}</ref>  In 2006, Professor Steven J. Miller provided a statistical derivation of the formula<ref>{{cite journal |author1=Miller |journal=Chance Magazine 20 (2007), no. 1, 40–48 |pages=9698 |title=A Derivation of the Pythagorean Won-Loss Formula in Baseball |year=2005 |arxiv=math/0509698 |bibcode=2005math......9698M}}</ref> under some assumptions about baseball games: if runs for each team follow a [[Weibull distribution]] and the runs scored and allowed per game are statistically independent, then the formula gives the probability of winning.<ref>{{cite journal|first=Steven J|last=Miller|title=A Derivation of the Pythagorean Won-Loss Formula in Baseball|journal=Chance Magazine 20 (2007), no. 1, 40--48|pages=9698|year=2005|arxiv=math/0509698|bibcode=2005math......9698M }}</ref>


The achievable ''H''<sub>∞</sub> norm of the closed loop system is mainly given through the matrix ''D''<sub>11</sub> (when the system ''P'' is given in the form (''A'', ''B''<sub>1</sub>, ''B''<sub>2</sub>, ''C''<sub>1</sub>, ''C''<sub>2</sub>, ''D''<sub>11</sub>, ''D''<sub>12</sub>, ''D''<sub>22</sub>, ''D''<sub>21</sub>)). There are several ways to come to an ''H''<sub>∞</sub> controller:
==Use in basketball==
* A [[Youla-Kucera parametrization]] of the closed loop often leads to very high-order controller.
American sports executive [[Daryl Morey]] was the first to adapt James' Pythagorean expectation to professional basketball while a researcher at [[STATS, Inc.]]. He found that using 13.91 for the exponents provided an acceptable model for predicting won-lost percentages:
* [[Riccati equation|Riccati]]-based approaches solve 2 [[Riccati equation]]s to find the controller, but require several simplifying assumptions.
* An optimization-based reformulation of the Riccati equation uses [[Linear matrix inequality|Linear matrix inequalities]] and requires fewer assumptions.


== See also ==
:<math>\mathrm{Win} = \frac{\text{points for}^{13.91}}{\text{points for}^{13.91} + \text{points against}^{13.91}}.</math>
*[[Hardy space]]
*[[H square]]
*[[H-infinity loop-shaping]]
*[[Linear-quadratic-Gaussian control]] (LQG)


== References ==
Daryl's "Modified Pythagorean Theorem" was first published in ''[http://morey.org/pythbook.gif STATS Basketball Scoreboard, 1993-94]''.<ref>{{cite book |last= Dewan |first= John |coauthors= Don Zminda, STATS, Inc. Staff |title= [[STATS Basketball Scoreboard, 1993-94]] |publisher= [[STATS, Inc.]] |year= 1993 |month= October |isbn= 0-06-273035-5|page= 17}}</ref>
{{refbegin}}
* {{citation|last1=Skogestad|first1=Sigurd|last2=Postlethwaite|first2=Ian|title=Multivariable Feedback Control: Analysis and Design|year=1996|publisher=Wiley|isbn=0-471-94277-4}}
* {{citation|last1=Skogestad|first1=Sigurd|last2=Postlethwaite|first2=Ian|title=Multivariable Feedback Control: Analysis and Design|year=2005|publisher=Wiley|isbn=0-470-01167-X|edition=2nd|url=http://www.nt.ntnu.no/users/skoge/book/}}
* {{citation|last=Simon|first=Dan|title=Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches|year=2006|publisher=Wiley|url=http://academic.csuohio.edu/simond/estimation/}}
* {{citation|last1=Green|first1=M.|last2=Limebeer|first2=D.|title=Linear Robust Control|year=1995|publisher=Prentice Hall|url=http://www3.imperial.ac.uk/portal/pls/portallive/docs/1/7287085.PDF}}
* {{Citation
| author = V. Barbu and S. S. Sritharan
| year = 1998
| title = H-infinity Control of Fluid Dynamics
| journal = Proceedings of the Royal Royal Society of London, Ser. A.
| volume = 545
|pages= 3009–3033
|url= http://www.nps.edu/Academics/Schools/GSEAS/SRI/R19.pdf
| postscript = .
}}


Noted basketball analyst [[Dean Oliver (statistician)|Dean Oliver]] also applied James' Pythagorean theory to professional basketball. The result was similar.


{{refend}}
Another noted basketball statistician, [[John Hollinger]], uses a similar Pythagorean formula except with 16.5 as the exponent.


[[Category:Control theory]]
==Use in pro football==
[[Category:Hardy spaces]]
The formula has also been used in [[American football|pro football]] by football stat website and publisher [[Football Outsiders]], where it is known as '''Pythagorean projection'''. The 2011 edition of ''Football Outsiders Almanac''<ref>''Football Outsiders Almanac 2011'' (ISBN 1466246133), p.xviii</ref> states, "From 1988 through 2004, 11 of 16 [[Super Bowl]]s were won by the team that led the [[National Football League|NFL]] in Pythagorean wins, while only seven were won by the team with the most actual victories. Super Bowl champions that led the league in Pythagorean wins but not actual wins include the [[2004 New England Patriots season|2004 Patriots]], [[2000 Baltimore Ravens season|2000 Ravens]], [[1999 St. Louis Rams season|1999 Rams]] and [[1997 Denver Broncos season|1997 Broncos]]."


[[de:H-unendlich-Regelung]]
Although ''Football Outsiders Almanac'' acknowledges that the formula had been less-successful in picking Super Bowl participants from 2005–2008, it reasserted itself in 2009 and 2010. Furthermore, "[t]he Pythagorean projection is also still a valuable predictor of year-to-year improvement. Teams that win a minimum of one full game more than their Pythagorean projection tend to regress the following year; teams that win a minimum of one full game less than their Pythagoerean projection tend to improve the following year, particularly if they were at or above .500 despite their underachieving. For example, the [[2008 New Orleans Saints season|2008 New Orleans Saints]] went 8-8 despite 9.5 Pythagorean wins, hinting at the improvement that came with the [[2009 New Orleans Saints season|next year's]] [[2009 NFL season|championship season]]."
[[ar:تحكم إتش إنفينتي]]
 
[[fr:Hinfini]]
==See also==
[[ja:H∞制御理論]]
* [[Baseball statistics]]
[[pl:H-nieskończoność]]
* [[Sabermetrics]]
[[ru:H∞-управление]]
* [[Football Outsiders]]
[[uk:H∞-керування]]
 
==Notes==
{{reflist|2}}
 
==External links==
*{{cite journal|author1=Miller|journal=Chance Magazine 20 (2007), no. 1, 40--48|pages=9698|title=A Derivation of the Pythagorean Won-Loss Formula in Baseball|year=2005|arxiv=math.ST/0509698|bibcode=2005math......9698M}}
*[http://www.sportility.net/cgi-bin/pointspread.cgi?league=mlb Current Major League Baseball Pythagorean expectation.]
*[http://www.pro-football-reference.com/blog/?p=337 Adjusting football’s Pythagorean Theorem]
 
[[Category:Bill James]]
[[Category:Baseball terminology]]
[[Category:Baseball statistics]]
[[Category:Sports records and statistics]]

Revision as of 08:45, 11 August 2014

Pythagorean expectation is a formula invented by Bill James to estimate how many games a baseball team "should" have won based on the number of runs they scored and allowed. Comparing a team's actual and Pythagorean winning percentage can be used to evaluate how lucky that team was (by examining the variation between the two winning percentages). The name comes from the formula's resemblance to the Pythagorean theorem.[1]

The basic formula is:

Win=runs scored2runs scored2+runs allowed2=11+(runs allowed/runs scored)2

where Win is the winning ratio generated by the formula. The expected number of wins would be the expected winning ratio multiplied by the number of games played.

Empirical origin

Empirically, this formula correlates fairly well with how baseball teams actually perform. However, statisticians since the invention of this formula found it to have a fairly routine error, generally about 3 games off. For example, in 2002, the New York Yankees scored 897 runs, allowing 697 runs. According to James' original formula, the Yankees should have won 62.35% of their games.

Win=89728972+6972=0.623525865

Based on a 162 game season, the Yankees should have won 101.07 games. The 2002 Yankees actually went 103-58.[2]

In efforts to fix this error, statisticians have performed numerous searches to find the ideal exponent.

If using a single number exponent, 1.83 is the most accurate, and the one used by baseball-reference.com, the premier website for baseball statistics across teams and time.[3] The updated formula therefore reads as follows:

Win=runs scored1.83runs scored1.83+runs allowed1.83=11+(runs allowed/runs scored)1.83

The most widely known is the Pythagenport formula[4] developed by Clay Davenport of Baseball Prospectus:

Exponent=1.50*log(R+RAG)+0.45

He concluded that the exponent should be calculated from a given team based on the team's runs scored (R), runs allowed (RA), and games (G). By not reducing the exponent to a single number for teams in any season, Davenport was able to report a 3.9911 root-mean-square error as opposed to a 4.126 root-mean-square error for an exponent of 2.[5]

Less well known but equally (if not more) effective is the Pythagenpat formula, developed by David Smyth.[6]

Exponent=(R+RAG).287

Davenport expressed his support for this formula, saying:

After further review, I (Clay) have come to the conclusion that the so-called Smyth/Patriot method, aka Pythagenpat, is a better fit. In that, X = ((rs + ra)/g)0.285, although there is some wiggle room for disagreement in the exponent. Anyway, that equation is simpler, more elegant, and gets the better answer over a wider range of runs scored than Pythagenport, including the mandatory value of 1 at 1 rpg.[7]

These formulas are only necessary when dealing with extreme situations in which the average number of runs scored per game is either very high or very low. For most situations, simply squaring each variable yields accurate results.

There are some systematic statistical deviations between actual winning percentage and expected winning percentage, which include bullpen quality and luck. In addition, the formula tends to regress toward the mean, as teams that win a lot of games tend to be underrepresented by the formula (meaning they "should" have won fewer games), and teams that lose a lot of games tend to be overrepresented (they "should" have won more).

"Second-order" and "third-order" wins

In their Adjusted Standings Report,[8] Baseball Prospectus refers to different "orders" of wins for a team. The basic order of wins is simply the number of games they have won. However, because a team's record may not reflect its true talent due to luck, different measures of a team's talent were developed.

First-order wins, based on pure run differential, are the number of expected wins generated by the "pythagenport" formula (see above). In addition, to further filter out the distortions of luck, sabermetricians can also calculate a team's expected runs scored and allowed via a runs created-type equation (the most accurate at the team level being Base Runs). These formulas result in the team's expected number of runs given their offensive and defensive stats (total singles, doubles, walks, etc.), which helps to eliminate the luck factor of the order in which the team's hits and walks came within an inning. Using these stats, sabermetricians can calculate how many runs a team "should" have scored or allowed.

By plugging these expected runs scored and allowed into the pythagorean formula, one can generate second-order wins, the number of wins a team deserves based on the number of runs they should have scored and allowed given their component offensive and defensive statistics. Third-order wins are second-order wins that have been adjusted for strength of schedule (the quality of the opponent's pitching and hitting). Second- and third-order winning percentage has been shown to predict future actual team winning percentage better than both actual winning percentage and first-order winning percentage.

Theoretical explanation

Initially the correlation between the formula and actual winning percentage was simply an experimental observation. In 2003, Hein Hundal provided an inexact derivation of the formula and showed that the Pythagorean exponent was approximately 2/(σ√π) where σ was the standard deviation of runs scored by all teams divided by the average number of runs scored.[9] In 2006, Professor Steven J. Miller provided a statistical derivation of the formula[10] under some assumptions about baseball games: if runs for each team follow a Weibull distribution and the runs scored and allowed per game are statistically independent, then the formula gives the probability of winning.[11]

Use in basketball

American sports executive Daryl Morey was the first to adapt James' Pythagorean expectation to professional basketball while a researcher at STATS, Inc.. He found that using 13.91 for the exponents provided an acceptable model for predicting won-lost percentages:

Win=points for13.91points for13.91+points against13.91.

Daryl's "Modified Pythagorean Theorem" was first published in STATS Basketball Scoreboard, 1993-94.[12]

Noted basketball analyst Dean Oliver also applied James' Pythagorean theory to professional basketball. The result was similar.

Another noted basketball statistician, John Hollinger, uses a similar Pythagorean formula except with 16.5 as the exponent.

Use in pro football

The formula has also been used in pro football by football stat website and publisher Football Outsiders, where it is known as Pythagorean projection. The 2011 edition of Football Outsiders Almanac[13] states, "From 1988 through 2004, 11 of 16 Super Bowls were won by the team that led the NFL in Pythagorean wins, while only seven were won by the team with the most actual victories. Super Bowl champions that led the league in Pythagorean wins but not actual wins include the 2004 Patriots, 2000 Ravens, 1999 Rams and 1997 Broncos."

Although Football Outsiders Almanac acknowledges that the formula had been less-successful in picking Super Bowl participants from 2005–2008, it reasserted itself in 2009 and 2010. Furthermore, "[t]he Pythagorean projection is also still a valuable predictor of year-to-year improvement. Teams that win a minimum of one full game more than their Pythagorean projection tend to regress the following year; teams that win a minimum of one full game less than their Pythagoerean projection tend to improve the following year, particularly if they were at or above .500 despite their underachieving. For example, the 2008 New Orleans Saints went 8-8 despite 9.5 Pythagorean wins, hinting at the improvement that came with the next year's championship season."

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

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  • Current Major League Baseball Pythagorean expectation.
  • Adjusting football’s Pythagorean Theorem
  1. ↑ The Game Designer|title=Pythagoras Explained
  2. ↑ Baseball-reference.com | 2002 NY Yankees Statistics
  3. ↑ Baseball-reference.com | What is pythagorean winning percentage?
  4. ↑ Baseball Prospectus | Articles | Revisiting the Pythagorean Theorem
  5. ↑ Baseball Prospectus | Articles | Revisiting the Pythagorean Theorem
  6. ↑ W% Estimators
  7. ↑ Baseball Prospectus | Glossary
  8. ↑ Adjusted Standings Report
  9. ↑ Template:Cite news
  10. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  11. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  12. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  13. ↑ Football Outsiders Almanac 2011 (ISBN 1466246133), p.xviii
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