How do I calculate r-squared using Python and Numpy?

Question

I'm using Python and Numpy to calculate a best fit polynomial of arbitrary degree. I pass a list of x values, y values, and the degree of the polynomial I want to fit (linear, quadratic, etc.).

This much works, but I also want to calculate r (coefficient of correlation) and r-squared(coefficient of determination). I am comparing my results with Excel's best-fit trendline capability, and the r-squared value it calculates. Using this, I know I am calculating r-squared correctly for linear best-fit (degree equals 1). However, my function does not work for polynomials with degree greater than 1.

Excel is able to do this. How do I calculate r-squared for higher-order polynomials using Numpy?

Here's my function:

import numpy

# Polynomial Regression
def polyfit(x, y, degree):
    results = {}

    coeffs = numpy.polyfit(x, y, degree)
     # Polynomial Coefficients
    results['polynomial'] = coeffs.tolist()

    correlation = numpy.corrcoef(x, y)[0,1]

     # r
    results['correlation'] = correlation
     # r-squared
    results['determination'] = correlation**2

    return results

Answer 1

From the numpy.polyfit documentation, it is fitting linear regression. Specifically, numpy.polyfit with degree 'd' fits a linear regression with the mean function

E(y|x) = p_d * x**d + p_{d-1} * x **(d-1) + ... + p_1 * x + p_0

So you just need to calculate the R-squared for that fit. The wikipedia page on linear regression gives full details. You are interested in R^2 which you can calculate in a couple of ways, the easisest probably being

SST = Sum(i=1..n) (y_i - y_bar)^2
SSReg = Sum(i=1..n) (y_ihat - y_bar)^2
Rsquared = SSReg/SST

Where I use 'y_bar' for the mean of the y's, and 'y_ihat' to be the fit value for each point.

I'm not terribly familiar with numpy (I usually work in R), so there is probably a tidier way to calculate your R-squared, but the following should be correct

import numpy

# Polynomial Regression
def polyfit(x, y, degree):
    results = {}

    coeffs = numpy.polyfit(x, y, degree)

     # Polynomial Coefficients
    results['polynomial'] = coeffs.tolist()

    # r-squared
    p = numpy.poly1d(coeffs)
    # fit values, and mean
    yhat = p(x)                         # or [p(z) for z in x]
    ybar = numpy.sum(y)/len(y)          # or sum(y)/len(y)
    ssreg = numpy.sum((yhat-ybar)**2)   # or sum([ (yihat - ybar)**2 for yihat in yhat])
    sstot = numpy.sum((y - ybar)**2)    # or sum([ (yi - ybar)**2 for yi in y])
    results['determination'] = ssreg / sstot

    return results