second moment

The second moment refers to the expected value of the square of a random variable.

$$\mathbb{E}[X^2]$$

Second Moment Matrix

The second moment matrix represents the pairwise relationships between different samples in a dataset. For a data matrix $X$ where each row is a sample, it is computed as:

$$\text{Second Moment Matrix}=X\cdot X^T$$

For $n$ samples with $d$ features:

• $X$ is an $n\times d$ matrix
• The resulting second moment matrix is $n\times n$
• Entry $(i,j)$ represents the dot product between samples $i$ and $j$, and the diagonal entries $(i,i)$ represent the second moment of each sample.
• This is just the uncentered version of the covariance matrix.

Computing Sample-Sample Second Moment

Consider two samples:
Sample 1: $[1,2,3]$
Sample 2: $[2,4,6]$

The sample-sample second moment matrix is computed as:

$$\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\end{array}\right]\cdot \left[\begin{array}{cc}1& 2\\ 2& 4\\ 3& 6\end{array}\right]=\left[\begin{array}{cc}14& 28\\ 28& 56\end{array}\right]$$