In my previous post I developed some basic laws for doing unit analysis with linear algebra. My goal has been to build up a capability for applying proper unit analysis to numeric computing, in particular machine learning and data science.

The key test of success for a unit analysis of linear algebra is whether it can be applied to real methods in data science. As an initial demonstration, in this post I will use the ideas from my previous post to do a unit analysis of one of the oldest methods in data science - Linear Regression.

To review, the linear regression model predicts some dependent scalar variable \(y\) from a vector of independent variables \(x = [ x_1 \dots x_m ]\). The parameters of the model are a vector \(\beta = [ b_1 \dots b_m ]\) such that the model estimates \(\hat y = \beta \cdot x\).

There are many variations on the algorithms for fitting parameters \(\beta\) to training data. In this post I’ll be working with the classic least squares estimation, which is given by the matrix formula:

\[\large \hat \beta = \left( X^T X \right) ^ {-1} X^T Y\]

Recall that in this formula, \(X\) is a table of \(n\) data samples, and \(Y\) is a column of corresponding dependent value measurements:

\[\large X = \begin{bmatrix} x_{11} & x_{12} & \dots & x_{1m} \\ x_{21} & x_{22} & \dots & x_{2m} \\ \vdots & & \ddots \\ x_{n1} & x_{n2} & \dots & x_{nm} \\ \end{bmatrix} \quad \quad Y = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \\ \end{bmatrix}\]

Consider the structure of \(X\) and \(Y\). Each of its columns holds \(n\) samples of the values of one kind of measurement, or feature. All the values in this column, therefore, can be described as having some particular unit. The same is true of our dependent values in \(Y\). For example, if we were trying to learn a crude model for predicting a person’s weight from their height and age, our data might look like so:

\[\large X = \begin{bmatrix} 190\ cm & 21\ yr \\ 175\ cm & 35\ yr \\ \vdots \\ 200\ cm & 51\ yr \\ \end{bmatrix} \quad \quad Y = \begin{bmatrix} 80\ kg \\ 55\ kg \\ \vdots \\ 91\ kg \\ \end{bmatrix}\]

If we apply the concept of a unit signature to the example above, then we have:

\[\large \Upsilon X = \begin{bmatrix} cm & yr \\ cm & yr \\ \vdots \\ cm & yr \\ \end{bmatrix} \quad \quad \Upsilon Y = \begin{bmatrix} kg \\ kg \\ \vdots \\ kg \\ \end{bmatrix}\]

Both \(X\) and \(Y\) are examples of tabular matrices. In other words, their unit signatures are of the general form:

\[\large \Upsilon X = \begin{bmatrix} u_{1} & u_{2} & \dots & u_{m} \\ u_{1} & u_{2} & \dots & u_{m} \\ \vdots & & \ddots \\ u_{1} & u_{2} & \dots & u_{m} \\ \end{bmatrix} \quad \quad \Upsilon Y = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \\ \end{bmatrix}\]

In my previous post, I developed the formula for the unit signature of tabular products and we can apply that directly to get:

\[\large \Upsilon X^T X = \begin{bmatrix} u_1 ^ 2 & u_1 u_2 & \dots & u_1 u_m \\ u_1 u_2 & u_2 ^ 2 & \dots & u_2 u_m \\ \vdots & & \ddots \\ u_1 u_m & u_2 u_m & \dots & u_m ^ 2 \\ \end{bmatrix} \quad \quad \Upsilon X^T Y = \begin{bmatrix} u_1 v \\ u_2 v \\ \vdots \\ u_m v \\ \end{bmatrix}\]

Furthermore, we have the formula for the unit signature of a tabular inverse:

\[\large \Upsilon (X^T X)^{-1} = \begin{bmatrix} (u_1 u_1)^{-1} & (u_2 u_1)^{-1} & \dots & (u_m u_1)^{-1} \\ (u_1 u_2)^{-1} & (u_2 u_2)^{-1} & \dots & (u_m u_2)^{-1} \\ \vdots & & \ddots & \\ (u_1 u_m)^{-1} & (u_2 u_m)^{-1} & \dots & (u_m u_m)^{-1} \\ \end{bmatrix}\]

Putting it all together, we arrive at the unit signature for \(\hat \beta\):

\[\large \begin{aligned} \Upsilon \hat \beta & = \Upsilon \left( \left( X^T X \right) ^ {-1} X^T Y \right) \\ & = \Upsilon \left( X^T X \right) ^ {-1} \quad \Upsilon X^T Y \\ & = \begin{bmatrix} (u_1 u_1)^{-1} & (u_2 u_1)^{-1} & \dots & (u_m u_1)^{-1} \\ (u_1 u_2)^{-1} & (u_2 u_2)^{-1} & \dots & (u_m u_2)^{-1} \\ \vdots & & \ddots & \\ (u_1 u_m)^{-1} & (u_2 u_m)^{-1} & \dots & (u_m u_m)^{-1} \\ \end{bmatrix} \begin{bmatrix} u_1 v \\ u_2 v \\ \vdots \\ u_m v \\ \end{bmatrix} \\ &= \begin{bmatrix} v / u_1 \\ v / u_2 \\ \vdots \\ v / u_m \\ \end{bmatrix} \end{aligned}\]

To help make this concrete, applying these forms to our earlier example looks like this:

\[\large \Upsilon \hat \beta = \begin{bmatrix} (cm\ cm)^{-1} & (yr\ cm)^{-1} \\ (cm\ yr)^{-1} & (yr\ yr)^{-1} \\ \end{bmatrix} \begin{bmatrix} cm\ kg \\ yr\ kg \\ \end{bmatrix} = \begin{bmatrix} kg / cm \\ kg / yr \\ \end{bmatrix}\]

The point of a unit analysis is to sanity check whether our units make sense. Recall that we apply our model like so: \(\hat y = \hat \beta \cdot x\). We can check the corresponding unit signatures to see if they are consistent, and the law for unit signatures of inner products shows that they are:

\[\large \begin{aligned} \Upsilon \hat y & = \Upsilon \hat \beta \cdot x \\ v & = [ v / u_1, v / u_2 \dots v / u_m ] \cdot [ u_1, u_2 \dots u_m ] \\ v & = v \\ \end{aligned}\]

Going back to our example, we would have:

\[\large kg = [ kg / cm, kg / yr] \cdot [cm, yr]\]

There is a lot of work to do, to discover how widely these unit analysis techniques can be applied in data science mathematics. However, I’m encouraged that they yield a proper unit analysis on a real world algorithm like Linear Regression.