Efficient online linear regression algorithm in python

I've found solution from this paper: updating simple linear regression. The implementation is as below:

def lr(x_avg,y_avg,Sxy,Sx,n,new_x,new_y):
    """
    x_avg: average of previous x, if no previous sample, set to 0
    y_avg: average of previous y, if no previous sample, set to 0
    Sxy: covariance of previous x and y, if no previous sample, set to 0
    Sx: variance of previous x, if no previous sample, set to 0
    n: number of previous samples
    new_x: new incoming 1-D numpy array x
    new_y: new incoming 1-D numpy array x
    """
    new_n = n + len(new_x)

    new_x_avg = (x_avg*n + np.sum(new_x))/new_n
    new_y_avg = (y_avg*n + np.sum(new_y))/new_n

    if n > 0:
        x_star = (x_avg*np.sqrt(n) + new_x_avg*np.sqrt(new_n))/(np.sqrt(n)+np.sqrt(new_n))
        y_star = (y_avg*np.sqrt(n) + new_y_avg*np.sqrt(new_n))/(np.sqrt(n)+np.sqrt(new_n))
    elif n == 0:
        x_star = new_x_avg
        y_star = new_y_avg
    else:
        raise ValueError

    new_Sx = Sx + np.sum((new_x-x_star)**2)
    new_Sxy = Sxy + np.sum((new_x-x_star).reshape(-1) * (new_y-y_star).reshape(-1))

    beta = new_Sxy/new_Sx
    alpha = new_y_avg - beta * new_x_avg
    return new_Sxy, new_Sx, new_n, alpha, beta, new_x_avg, new_y_avg

Performance comparison:

Scikit learn version that calculate 10k samples altogether.

from sklearn.linear_model import LinearRegression
x = np.arange(10000).reshape(-1,1)
y = np.arange(10000)+100*np.random.random_sample((10000,))
regr = LinearRegression()
%timeit regr.fit(x,y)
# 419 µs ± 14.6 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

My version assume 9k sample is already calculated:

Sxy, Sx, n, alpha, beta, new_x_avg, new_y_avg = lr(0, 0, 0, 0, 0, x.reshape(-1,1)[:9000], y[:9000])
new_x, new_y = x.reshape(-1,1)[9000:], y[9000:]
%timeit lr(new_x_avg, new_y_avg, Sxy,Sx,n,new_x, new_y)
# 38.7 µs ± 1.31 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

10 times faster, which is expected.

Nice! Thanks for sharing your findings :) Here is an equivalent implementation of this solution written with dot products:

class SimpleLinearRegressor(object):
    def __init__(self):
        self.dots = np.zeros(5)
        self.intercept = None
        self.slope = None

    def update(self, x: np.ndarray, y: np.ndarray):
        self.dots += np.array(
            [
                x.shape[0],
                x.sum(),
                y.sum(),
                np.dot(x, x),
                np.dot(x, y),
            ]
        )
        size, sum_x, sum_y, sum_xx, sum_xy = self.dots
        det = size * sum_xx - sum_x ** 2
        if det > 1e-10:  # determinant may be zero initially
            self.intercept = (sum_xx * sum_y - sum_xy * sum_x) / det
            self.slope = (sum_xy * size - sum_x * sum_y) / det

When working with time series data, we can extend this idea to do sliding window regression with a soft (EMA-like) window.

You can use accelerated libraries that implement faster algorithms - particularly https://github.com/intel/scikit-learn-intelex

For linear regression you would get much better performance

First install package

pip install scikit-learn-intelex

And then add in your python script

from sklearnex import patch_sklearn
patch_sklearn()