Main Content


Box-Cox transformation



[transdat,lambda] = boxcox(data) transforms the data vector data using the Box-Cox transformation method into transdat. It also estimates the transformation parameter λ.


transdat = boxcox(lambda,data) transform the data using a certain specified λ for the Box-Cox transformation. This syntax does not find the optimum λ that maximizes the LLF.


collapse all

Use boxcox to transform the data series contained in a vector of data into another set of data series with relatively normal distributions.

Load the SimulatedStock.mat data file.

load SimulatedStock.mat

Transform the nonnormally distributed filled data series TMW_CLOSE into a normally distributed one using Box-Cox transformation.

 [Xbc, lambdabc] = boxcox(TMW_CLOSE)
Xbc = 1000×1


lambdabc = 0.2151

Compare the result of the TMW_CLOSE data series with a normal (Gaussian) probability distribution function and the nonnormally distributed TMW_CLOSE.

subplot(2, 1, 1);
grid; title('Nonnormally Distributed Data');
subplot(2, 1, 2);
grid; title('Box-Cox Transformed Data');

The bar chart on the top represents the probability distribution function of the data series, TMW_CLOSE, which is the original data series. The distribution is skewed toward the left (not normally distributed). The bar chart on the bottom is less skewed to the left. If you plot a Gaussian probability distribution function (PDF) with similar mean and standard deviation, the distribution of the transformed data is close to normal (Gaussian). When you examine the contents of the resulting object Xbc, you find an identical object to the original object TMW_CLOSE but the contents are the transformed data series.

Input Arguments

collapse all

Data, specified as a positive column vector.

Data Types: double

Lambda, specified as a scalar numeric or structure.

If the input data is a vector, lambda is a scalar. If the input is a financial time series object (tsobj), lambda is a structure with fields similar to the components of the object. For example, if tsobj contains series names Open and Close, lambda has fields lambda.Open and lambda.Close.

Data Types: double | struct

Output Arguments

collapse all

Data Box-Cox transformation, returned as a vector.

Lambda transformation parameter, returned as a numeric.

More About

collapse all

Box Cox Transformation

boxcox transforms nonnormally distributed data to a set of data that has approximately normal distribution. The Box-Cox transformation is a family of power transformations.

If λ is not = 0, then


If λ is = 0, then


The logarithm is the natural logarithm (log base e). The algorithm calls for finding the λ value that maximizes the Log-Likelihood Function (LLF). The search is conducted using fminsearch.

Version History

Introduced before R2006a

expand all

See Also