Multiple R-squared

While calculating R² previously, we saw how it was the amount of variance explained by the model:

Since the variance is the mean squared error, we can multiply both the var(ε) and var(y) terms by the sample size and arrive at the following alternative equation for R2:

This is simply the sum of squared residuals over the sum of squared differences from the mean. Incanter contains the incanter.core/sum-of-squares function that makes this very simple to express:

(defn r-squared [coefs x y]
   (let [fitted      (i/mmult x coefs)
         residuals   (i/minus y fitted)
         differences (i/minus y (s/mean y))
         rss         (i/sum-of-squares residuals)
         ess         (i/sum-of-squares differences)]
     (- 1 (/ rss ess))))

We use the variable names rss for residual sum of squares and ess for explained sum of squares. We can calculate the matrix R² for our new model as follows:

(defn ex-3-21 []
  (let [data (swimmer-data)
        x (->> (feature-matrix ["Height...