Rename (relational algebra)

In statistics, the power transform is from a family of functions that are applied to create a rank-preserving transformation of data using power functions. This is a useful data transformation technique used to stabilize variance, make the data more normal distribution-like, improve the validity of measures of association such as the Pearson correlation between variables and for other data stabilization procedures.

Definition

The power transformation is defined as a continuously varying function, with respect to the power parameter λ, in a piece-wise function form that makes it continuous at the point of singularity (λ = 0). For data vectors (y₁,..., y_n) in which each y_i > 0, the power transform is

$y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda (\operatorname {GM} (y))^{\lambda -1}}},&{\text{if }}\lambda \neq 0\\[12pt]\operatorname {GM} (y)\log {y_{i}},&{\text{if }}\lambda =0\end{cases}}$

where

$\operatorname {GM} (y)=(y_{1}\cdots y_{n})^{1/n}\,$

is the geometric mean of the observations y₁, ..., y_n. The logarithm log here is the natural base-e logarithm, not the base-10 logarithm.

The inclusion of the (λ − 1)th power of the geometric mean in the denominator implies that the units of measurement do not change as λ changes. That makes it possible to compare sums of squares of residuals and choose the value of λ that minimizes that sum.

The value at Y = 1 for any λ is 0, and the derivative with respect to Y there is 1 for any λ. Sometimes Y is a version of some other variable scaled to give Y = 1 at some sort of average value.

The transformation is a power transformation, but done in such a way as to make it continuous with the parameter λ at λ = 0. It has proved popular in regression analysis, including econometrics.

Box and Cox also proposed a more general form of the transformation that incorporates a shift parameter.

$\tau (y_{i};\lambda ,\alpha )={\begin{cases}{\dfrac {(y_{i}+\alpha )^{\lambda }-1}{\lambda (\operatorname {GM} (y))^{\lambda -1}}}&{\text{if }}\lambda \neq 0,\\\\\operatorname {GM} (y)\ln(y_{i}+\alpha )&{\text{if }}\lambda =0,\end{cases}}$

which holds if y_i + α > 0 for all i. If τ(Y, λ, α) follows a truncated normal distribution, then Y is said to follow a Box–Cox distribution.

Box–Cox transformation

The power transformation above is used by Box–Cox for calculating the profile likelihood function for λ. However, the Box–Cox transformation itself is defined as

$y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda }},&{\text{if }}\lambda \neq 0,\\[8pt]\log {(y_{i})},&{\text{if }}\lambda =0.\end{cases}}$

The transformation above does not require that $y>0$.

Use of the power transform

• Power transforms are ubiquitously used in various fields. For example, multi-resolution and wavelet analysis, statistical data analysis, medical research, modeling of physical processes, geochemical data analysis, epidemiology and many other clinical, environmental and social research areas.

Example

The BUPA liver data set contains data on liver enzymes ALT and γGT. Suppose we are interested in using log(γGT) to predict ALT. A plot of the data appears in panel (a) of the figure. There appears to be non-constant variance, and a Box–Cox transformation might help.

The log-likelihood of the power parameter appears in panel (b). The horizontal reference line is at a distance of χ₁²/2 from the maximum and can be used to read off an approximate 95% confidence interval for λ. It appears as though a value close to zero would be good, so we take logs.

Possibly, the transformation could be improved by adding a shift parameter to the log transformation. Panel (c) of the figure shows the log-likelihood. In this case, the maximum of the likelihood is close to zero suggesting that a shift parameter is not needed. The final panel shows the transformed data with a superimposed regression line.

Note that although Box–Cox transformations can make big improvements in model fit, there are some issues that the transformation cannot help with. In the current example, the data are rather heavy-tailed so that the assumption of normality is not realistic and a robust regression approach leads to a more precise model.

Econometric application

Economists often characterize production relationships by some variant of the Box–Cox transformation.

Consider a common representation of production Q as dependent on services provided by a capital stock K and by labor hours N:

$\tau (Q)=\alpha \tau (K)+(1-\alpha )\tau (N).\,$

Solving for Q by inverting the Box–Cox transformation we find

$Q={\big (}\alpha K^{\lambda }+(1-\alpha )N^{\lambda }{\big )}^{1/\lambda },\,$

which is known as the constant elasticity of substitution (CES) production function.

The CES production function is a homogeneous function of degree one.

When λ = 1, this produces the linear production function:

$Q=\alpha K+(1-\alpha )N.\,$

When λ → 0 this produces the famous Cobb–Douglas production function:

$Q=K^{\alpha }N^{1-\alpha }.\,$

Activities and demonstrations

The SOCR resource pages contain a number of hands-on interactive activities demonstrating the Box–Cox (Power) Transformation using Java applets and charts. These directly illustrate the effects of this transform on Q-Q plots, X-Y scatterplots, time-series plots and histograms.

Notes

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References

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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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• Carroll, RJ and Ruppert, D. On prediction and the power transformation family. Biometrika 68: 609–615.
• 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
  
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  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
• Handelsman, DJ. Optimal Power Transformations for Analysis of Sperm Concentration and Other Semen Variables. Journal of Andrology, Vol. 23, No. 5, September/October 2002.
• Gluzman, S and Yukalov, VI. Self-similar power transforms in extrapolation problems. Journal of Mathematical Chemistry, Volume 39, Number 1 / January, 2006, 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., 47–56.
• Howarth, RJ and Earle, SAM. Application of a generalized power transformation to geochemical data Journal Mathematical Geology, Volume 11, Number 1 / February, 1979, DOI 10.1007/BF01043245, pages 45–62.

External links

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