Numy v0.1.5 Numy.Fit.SimpleLinear View Source

Simple Linear Regression.

In y = α + βx, one approach to estimating the unknowns α and β is to consider the sum of squared residuals function, or SSR.

∑rᵢ² = ∑(yᵢ - α - βxᵢ)²

It is a fact that among all possible α and β, the following values minimize the SSR:

β = cov(x,y) / var(x)

α = ȳ - βx̄

Example

iex(34)> x = Numy.Lapack.Vector.new(0..9)
#Vector<size=10, [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]>
iex(35)> y = Vc.scale(x,2) |> Vcm.offset!(-3.0) # make slope=2 and intercept=-3
#Vector<size=10, [-3.0, -1.0, 1.0, 3.0, 5.0, 7.0, 9.0, 11.0, 13.0, 15.0]>
iex(36)> err = Numy.Lapack.Vector.new(10) |> Vc.assign_random |> Vcm.offset!(-0.5) |> Vcm.scale!(0.1)
iex(37)> Vcm.add!(y,err) # add errors to the ideal line
iex(38)> line = Numy.Fit.SimpleLinear.fit(x,y)
{-2.9939933270609496, 1.9966330251198818} # got intercept=-3 and slope=2 as expected
iex(40)> Numy.Fit.SimpleLinear.predict(0,line)
-2.9939933270609496
iex(41)> Numy.Fit.SimpleLinear.predict(10,line)
16.97233692413787
iex(9)> predicted = Numy.Lapack.Vector.new(Numy.Fit.SimpleLinear.predict(Enum.to_list(0..9),line))
iex(10)> Numy.Fit.SimpleLinear.pearson_correlation(predicted, y)
0.9999884096405113

Link to this section Summary

Functions

covariance(x, y)

In statistics, covariance is a measure of the joint variability of 2 random variables.

covariance_over_variance(x, y)

Calculate cov/var in one step.

fit(x, y)

Find slope and intercept of the line that fits the input data.

pearson_correlation(x, y)

Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations.

predict(x, arg)

Calculate y by x using slope and intercept.

variance(x)

In statistics, variance is the expectation of the squared deviation of a random variable from its mean.

Link to this section Functions