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Correlation and Variance-Covariance Matrices

Variance-covariance and correlation matrices are among the most important quantitative measures of a data set that characterize statistical relationships involving dependence.

Specifically, the covariance measures the extent to which variables “fluctuate together” (that is, co-vary). The correlation is the covariance normalized to be between -1 and +1. A positive correlation indicates the extent to which variables increase or decrease simultaneously. A negative correlation indicates the extent to which one variable increases while the other one decreases. Values close to +1 and -1 indicate a high degree of linear dependence between variables.

Details

Given a set $X$ of $n$ feature vectors $x_1=(x_{11},\dots ,x_{1p}),\dots ,x_n=(x_{n1},\dots ,x_{np})$ of dimension $p$, the problem is to compute the sample means and variance-covariance matrix or correlation matrix:

Correlation and Variance-Covariance Matrices

Statistic	Definition
Means	$M=(m(1),\dots ,m(p))$, where $m\left(j\right)=\frac{1}{n}\sum _i{x}_{ij}$
Variance-covariance matrix	$Cov=(v_{ij})$, where $v_{ij}=\frac{1}{n-1}\sum _{k=1}^n(x_{ki}-m(i))(x_{kj}-m(j))$, $i=\overline{1,p}$, $j=\overline{1,p}$
Correlation matrix	$Cor=(c_{ij})$, where $c_{ij}=\frac{v_{ij}}{\sqrt{v_{ii}\cdot v_{jj}}}$, $i=\overline{1,p}$, $j=\overline{1,p}$

Computation

The following computation modes are available:

• Batch Processing
• Online Processing
• Distributed Processing

Examples

C++ (CPU)Python*

Batch Processing:

• cov_dense_batch.cpp

• cov_csr_batch.cpp

Performance Considerations

To get the best overall performance when computing correlation or variance-covariance matrices:

• If input data is homogeneous, provide the input data and store results in homogeneous numeric tables of the same type as specified in the algorithmFPType class template parameter.

• If input data is non-homogeneous, use AOS layout rather than SOA layout.

Product and Performance Information
Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex​. Notice revision #20201201

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