# Principal Component Regression (PCR)

Principal Component Regression (PCR) combines PCA and OLS. It has advantages compared to classic regression. Available in Excel with the XLSTAT software.

## What is Principal Component Regression

PCR (Principal Components Regression) is a regression method that can be divided into three steps:

- The first step is to run a
**PCA**(Principal Components Analysis) on the table of the explanatory variables, - Then run an
**Ordinary Least Squares regression**(OLS regression) also called linear regression on the selected components, - Finally
**compute the parameters**of the model that correspond to the input variables.

## Principal Component Regression models

PCA allows to transform an X table with n observations described by variables into an S table with n scores described by q components, where q is lower or equal to p and such that (S’S) is invertible. An additional selection can be applied on the components so that only the r components that are the most correlated with the Y variable are kept for the OLS regression step. We then obtain the R table.

The OLS regression is performed on the Y and R tables. In order to circumvent the interpretation problem with the parameters obtained from the regression, XLSTAT transforms the results back into the initial space to obtain the parameters and the confidence intervals that correspond to the input variables.

PCR results: Correlation and observations charts and biplots

As PCR is build on PCA, a great advantage of PCR regression over classical regression is the available charts that describe the data structure. Thanks to the correlation and loading plots it is easy to study the relationship among the variables. It can be relationships among the explanatory variables, as well as between explanatory and dependent variables. The score plot gives information about sample proximity and dataset structure. The biplot gather all these information in one chart.

## Prediction with Principal Component Regression

Principal Componenet Regression is also used to build predictive models. XLSTAT enable you to predict new samples' values.

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