Question $4$: Non$-$negative Matrix Factorization $($NMF$)$

The non$-$negative matrix factorization is a dimension reduction technique that is often applied to image data. It is similar to PCA except that is only applicable for strictly positive data. We can apply the NMF to vectorized versions of the shot surface. This is useful because we can convert the observed matrix of shot surfaces into:

Bases: Identifying modes of shooting style $($number of modes is determined by n$\_$components argument to NMF function below$)$

Coefficients: How each players shooting style could be expressed as a $($positive$)$ linear combination of these bases

The NMF solves the following problem: given some matrix

is

matrix, NMF computes the following factorization:

where

is

matrix and

is

matrix.

In this homework, we have the following:

The data matrix

is of dimension

$=\{$number of players$\}$ and

$=\{$number of total square bins on the court$\}.$ Each column corresponds to a player, with entries corresponding to a "flattened" or "vectorized" version of the $2$d histograms plotted in part $4$b$.$

Bases matrix:

Columns

contain the shot "bases". First, we will try it with

bins in $5$a$,$ and then with

bins in $5$d$.$

Coefficient matrix: H

Each column of

gives a coefficient for each of the bases vectors in

$,$ and there are

columns for each player.

The sklearn library is one of the main Python machine learning libraries. It has a built in NMF function for us$.$ The function below runs this function and normalizes the basis surfaces to sum to $1.$