Mathematical re-expression, often nonlinear, of data values. Data are often transformed either to meet the assumptions of a statistical model or to make the results of an analysis more interpretable.

Data transformation is the mathematical re-expression, often nonlinear, of data values. Data are often transformed either to meet the assumptions of a statistical model or to make the results of an analysis more interpretable. Specific types of transformations are covered by the and and tags on Cross Validated.

  1. Linear transformation. A linear transformation preserves linear relationships between variables. Therefore, the correlation between x and y would be unchanged after a linear transformation. Examples of a linear transformation to variable x would be multiplying x by a constant, dividing x by a constant, or adding a constant to x. The most common linear transformations are centering and standardizing.

  2. Nonlinear transformation. A nonlinear transformation changes (strengthens or weakens) linear relationships between variables and, thus, changes the correlation between variables. Examples of a nonlinear transformation of variable x would be taking the square root of x or the reciprocal of x.

    In regression, nonlinear transformations of the response variable are sometimes used to achieve homoscedasticity and / or make the distribution of the residuals more normal. For example, here are a few methods of transforming variables sometimes used in regression settings:

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refers to scaling all numeric variables in the range [0,1], such as using the formula: $$x_{new}=\frac{x-x_{min}}{x_{max}-x_{min}}$$

refers to a transform to the data set to have zero mean and unit variance, for example using the equation: $$x_{new}=\frac{x-\overline{x}}{s}$$

refers to the use of power transformations developed by statisticians George E. P. Box and David Cox in 1964 to identify an appropriate exponent, $\lambda$ (i.e. $Y^{\lambda}$), on the range of $[-5,5]$. For $\lambda=0$, the data is transformed by $\log_{10}$. Therefore, for $\lambda=2$ the data $Y$ is transformed by the equation $Y^2$

Reference: stattrek.com

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