# Multivariate Regression

# Multivariate Regression

Multivariate regression is a distinct type of analysis.

Multivariate methods are tools to deal with high-dimensional data sets. In general these large data sets are challenges for data processing, for example visualization and modeling. One problem is that not all dimensions of a high-dimensial model are useful for a good model. One of the goals of multivariate regression models is to reduce the dimensions and at the same moment leave the contained information the same.

If multivariate regression is the best tool for analyzing depends on the question that should be answered. It can be useful if it is a study at differen times and only if huge data should be analysed because of finding complex structures. It is also important that the dependent variables should correlate in some way. And the data set should be outlier free.

An example to use multivariate regression is data-mining, to find unknown structures and dependencies between coefficients.

The results of a multivariate regression is one equation for each dependent variable. Which should be the same result as from the individual analysis. The advantage of the multivariate regression is that multivariate regression gives information about the corellation between dependent variables. If the depdendent variables are dependent from the independent variables or in which combination the are dependent from each other.

To differentiate between regression models the models are described often depending on the outcome variable. For example a linear regression is continuous. From the statistical side is a multivariate analysis a statistical model which has two or more dependent outcome variables.

It is important to not mix multivariate and multivariable regression. In comparison to the multivariate regression is the multivariable analysis a statistical model which has multiple independent response variables.

## Multivariate Linear Regression Model:

The general form of a multivariate linear regression model is:

$y = a + b_1 \cdot x_1 + … + b_n \cdot x_n \epsilon_i$

where a is the regression constant, $b_i$ is the gain coefficient of the i-th feature, $x_i$ is the expression of the i-th feature and the error term $\epsilon$, the deviation of the data from the regression line.

# Related

- [notebook] Multiple Regression in Python
- [slides] Multiple and multivariate regression