Freundlich equation

<multifractal>...</multifractal>A Strange Attractor that exhibits multifractal scaling

Example of a multifractal electronic eigenstate at the Anderson localization transition in a system with 1367631 atoms.

A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed.

Multifractal systems are common in nature, especially geophysics. They include fully developed turbulence, stock market time series, real world scenes, the Sun’s magnetic field time series, heartbeat dynamics, human gait, and natural luminosity time series. Models have been proposed in various contexts ranging from turbulence in fluid dynamics to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more. The origin of multifractality in sequential (time series) data has been attributed, to mathematical convergence effects related to the central limit theorem that have as foci of convergence the family of statistical distributions known as the Tweedie exponential dispersion models as well as the geometric Tweedie models. The first convergence effect yields monofractal sequences and the second convergence effect is responsible for variation in the fractal dimension of the monofractal sequences.

From a practical perspective, multifractal analysis uses the mathematical basis of multifractal theory to investigate datasets, often in conjunction with other methods of fractal analysis and lacunarity analysis. The technique entails distorting datasets extracted from patterns to generate multifractal spectra that illustrate how scaling varies over the dataset. The techniques of multifractal analysis have been applied in a variety of practical situations such as predicting earthquakes and interpreting medical images.

Definition

In a multifractal system $s$, the behavior around any point is described by a local power law:

$s({\vec {x}}+{\vec {a}})-s({\vec {x}})\sim a^{h({\vec {x}})}.$

The exponent $h({\vec {x}})$ is called the singularity exponent, as it describes the local degree of singularity or regularityTemplate:Disambiguation needed around the point ${\vec {x}}$.

The ensemble formed by all the points that share the same singularity exponent is called the singularity manifold of exponent h, and is a fractal set of fractal dimension D(h). The curve D(h) versus h is called the singularity spectrum and fully describes the (statistical) distribution of the variable $s$.

In practice, the multifractal behaviour of a physical system $X$ is not directly characterized by its singularity spectrum D(h). Data analysis rather gives access to the multiscaling exponents $\zeta (q),\ q\in {\mathbb {R} }$. Indeed, multifractal signals generally obey a scale invariance property which yields power law behaviours for multiresolution quantities depending on their scale $a$. Depending on the object under study, these multiresolution quantities, denoted by $T_{X}(a)$ in the following, can be local averages in boxes of size $a$, gradients over distance $a$, wavelet coefficients at scale $a$... For multifractal objects, one usually observes a global power law scaling of the form:

$\langle T_{X}(a)^{q}\rangle \sim a^{\zeta (q)}\$

at least in some range of scales and for some range of orders $q$. When such a behaviour is observed, one talks of scale invariance, self-similarity or multiscaling.

Estimation

Using the so-called multifractal formalism, it can be shown that, under some well-suited assumptions, there exists a correspondence between the singularity spectrum $D(h)$ and the multiscaling exponents $\zeta (q)$ through a Legendre transform. While the determination of $D(h)$ calls for some exhaustive local analysis of the data, which would result in difficult and numerically unstable calculations, the estimation of the $\zeta (q)$ relies on the use of statistical averages and linear regressions in log-log diagrams. Once the $\zeta (q)$ are known, one can deduce an estimate of $D(h)$ thanks to a simple Legendre transform.

Multifractal systems are often modeled by stochastic processes such as multiplicative cascades. Interestingly, the $\zeta (q)$ receives some statistical interpretation as they characterize the evolution of the distributions of the $T_{X}(a)$ as $a$ goes from larger to smaller scales. This evolution is often called statistical intermittency and betrays a departure from Gaussian models.

Modelling as a multiplicative cascade also leads to estimation of multifractal properties for relatively small datasets (Template:Harvnb). A maximum likelihood fit of a multiplicative cascade to the dataset not only estimates the complete spectrum, but also gives reasonable estimates of the errors (see the web service [1]).

<techniques>...</techniques>

Practical application of multifractal spectra

Multifractal analysis is analogous to viewing a dataset through a series of distorting lenses to home in on differences in scaling. The pattern shown is a Hénon map

<distort>...</distort>Multifractal analysis has been used in several fields in science to characterize various types of datasets. In essence, multifractal analysis applies a distorting factor to datasets extracted from patterns, to compare how the data behave at each distortion. This is done using graphs known as multifractal spectra that illustrate how the distortions affect the data, analogous to viewing the dataset through a "distorting lens" as shown in the illustration. Several types of multifractal spectra are used in practise.

D_Q vs Q

<dqvsq>...</dqvsq>

D_Q vs Q spectra for a non-fractal circle (empirical box counting dimension = 1.0), mono-fractal Quadric Cross (empirical box counting dimension = 1.49), and multifractal Hénon map (empirical box counting dimension = 1.29).

<dimensional ordering>...</dimensional ordering>One practical multifractal spectrum is the graph of D_Q vs Q, where D_Q is the generalized dimension for a dataset and Q is an arbitrary set of exponents. The expression generalized dimension thus refers to a set of dimensions for a dataset (detailed calculations for determining the generalized dimension using box counting are described below).

Dimensional ordering

The general pattern of the graph of D_Q vs Q can be used to assess the scaling in a pattern. The graph is generally decreasing, sigmoidal around Q=0, where D_(Q=0) ≥ D_(Q=1) ≥ D_(Q=2). As illustrated in the figure, variation in this graphical spectrum can help distinguish patterns. The image shows D_(Q) spectra from a multifractal analysis of binary images of non-, mono-, and multi-fractal sets. As is the case in the sample images, non- and mono-fractals tend to have flatter D_(Q) spectra than multifractals.

The generalized dimension also offers some important specific information. D_(Q=0) is equal to the Capacity Dimension, which in the analysis shown in the figures here is the box counting dimension. D_(Q=1) is equal to the Information Dimension, and D_(Q=2) to the Correlation Dimension. This relates to the "multi" in multifractal whereby multifractals have multiple dimensions in the D_(Q) vs Q spectra but monofractals stay rather flat in that area.

$f(\alpha )$ vs $\alpha$

Another useful multifractal spectrum is the graph of $f(\alpha )$ vs $\alpha$ (see calculations). These graphs generally rise to a maximum that approximates the fractal dimension at Q=0, and then fall. Like D_Q vs Q spectra, they also show typical patterns useful for comparing non-, mono-, and multi-fractal patterns. In particular, for these spectra, non- and mono-fractals converge on certain values, whereas the spectra from multifractal patterns are typically humped over a broader extent.

Estimating multifractal scaling from box counting

<calculations>...</calculations> Multifractal spectra can be determined from box counting on digital images. First, a box counting scan is done to determine how the pixels are distributed; then, this "mass distribution" becomes the basis for a series of calculations. The chief idea is that for multifractals, the probability, $P$, of a number of pixels, $m$, appearing in a box, $i$, varies as box size, $\textstyle \epsilon$, to some exponent, $\textstyle \alpha$, which changes over the image, as in Template:EquationNote. NB: For monofractals, in contrast, the exponent does not change meaningfully over the set. $P$ is calculated from the box counting pixel distribution as in Template:EquationNote.

$\epsilon$ = an arbitrary scale (box size in box counting) at which the set is examined

$i$ = the index for each box laid over the set for an $\epsilon$

$m_{[i,\epsilon ]}$ = the number of pixels or mass in any box, $i$, at size $\epsilon$

$N_{\epsilon }$ = the total boxes that contained more than 0 pixels, for each $\epsilon$

$P$ is used to observe how the pixel distribution behaves when distorted in certain ways as in Template:EquationNote and Template:EquationNote:

$Q$ = an arbitrary range of values to use as exponents for distorting the data set

• When $Q=1$, Template:EquationNote equals 1, the usual sum of all probabilities, and when $Q=0$, every term is equal to 1, so the sum is equal to the number of boxes counted, $N_{\epsilon }$.

These distorting equations are further used to address how the set behaves when scaled or resolved or cut up into a series of $\epsilon$-sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following:

• An important feature of Template:EquationNote is that it can also be seen to vary according to scale raised to the exponent $\textstyle \tau$ in Template:EquationNote:

Thus, a series of values for $\tau _{(Q)}$ can be found from the slopes of the regression line for the log of Template:EquationNote vs the log of $\epsilon$ for each $Q$, based on Template:EquationNote:

• <generalized dimension>...</generalized dimension>For the generalized dimension:

• $\textstyle \alpha _{(Q)}$ is estimated as the slope of the regression line for Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
  
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  Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
  
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  This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
  
  A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
  
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  Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. where:

• Then $f_{\left(\alpha _{(Q)}\right)}$ is found from Template:EquationNote.

• The mean $\textstyle \tau _{(Q)}$ is estimated as the slope of the log-log regression line for $\textstyle \tau _{{(Q)}_{[\epsilon ]}}$ vs $\textstyle \epsilon$, where:

In practise, the probability distribution depends on how the dataset is sampled, so optimizing algorithms have been developed to ensure adequate sampling.

See also

• Multifractal Model of Asset Returns (MMAR)
• Multifractal Random Walk model (MRW)
• Fractional Brownian motion
• Mandelbrot cascade, continuous cascade and lognormal cascade
• Detrended fluctuation analysis
• Tweedie distributions

References

11. Veneziano, D. and Essiam, A.K. (2003). Flow through porous media with multifractal hydraulic conductivity.

Water Resources Research 39: doi: 10.1029/2001WR001018. issn: 0043-1397.

External links

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  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
  
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  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

• Movies of visualizations of multifractals