how to vectorize forecasting when using polynomials in numpy

Question

essentially would like to find the most efficient solution (numpy) that essentially allows me to extend np.poly1d to K dimensions.

test case:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt


class Polyfit:

    @staticmethod
    def from_fit_to_forecast(df, forecast_values, dates_forward, x_data, y_data, order=2):
        # nice vectorized params estimation
        all_params = np.polyfit(x_data, y_data, order)

        # terrible fit of data as I loop over them
        new_df = pd.DataFrame([np.poly1d(i)(x_data) for i in all_params.T], columns=df.index, index=df.columns).T
        forecast_df_second = pd.DataFrame(
            [np.poly1d(i)(forecast_values) for i in all_params.T], columns=dates_forward, index=df.columns).T

        return new_df, forecast_df_second

    @staticmethod
    def gen_data(k_steps):
        data = 1 + np.random.rand(100, 4) / 300 - (np.random.rand(100, 4) / 10) ** 2
        dates = pd.date_range('2010-1-1', freq='D', periods=100)
        dates_forward = pd.date_range(max(dates) + pd.Timedelta(1, unit='d'), freq='D', periods=k_steps)
        return pd.DataFrame(data, columns=list('ABCD'), index=dates).cumprod(), dates_forward

    def __init__(self, k_steps_forward=20):
        self.original_data, dates_forward = self.gen_data(k_steps_forward)

        x_data = list(range(len(self.original_data.index)))
        max_x_data = max(x_data)
        forecast_values = list(range(max_x_data + 1, max_x_data + 1 + k_steps_forward, 1))
        y_data = self.original_data.values

        self.fit_df_2, self.forecast_2 = self.from_fit_to_forecast(
            self.original_data, forecast_values, dates_forward, x_data, y_data, order=2)

cls = Polyfit(k_steps_forward=30)

print(cls.fit_df_2)
print(cls.forecast_2)

the critical point is in the from_fit_to_forecast where I do this:

[np.poly1d(i)(forecast_values) for i in all_params.T]

which slows things down considerably. Also, since I would also be using the 2nd order polynomial, I tried playing around with np.dot or other things that work with matrixes but no avail.

any suggestions?

Answer 1

So you got a bunch of polynomial coefficients from

all_params = np.polyfit(x_data, y_data, order)

(where y_data is a 2D array) and you want to evaluate all of them at the points x_data. A vectorized way to do this, as explained below, is:

(x_data.reshape(-1, 1)**np.arange(order, -1, -1)).dot(all_params)

Here is a small example where the fit is perfect (2nd degree polys through three points), so you can see that the evaluation is correct

x_data = np.array([1, 2, 3])
y_data = np.array([[5, 6,], [9, 8], [7, 4]])
order = 2
all_params = np.polyfit(x_data, y_data, order)
(x_data.reshape(-1, 1)**np.arange(order, -1, -1)).dot(all_params)

outputs

array([[ 5.,  6.],
       [ 9.,  8.],
       [ 7.,  4.]])

Explanation

x_data.reshape(-1, 1)**np.arange(order, -1, -1) creates a matrix of powers of x_data points, starting from highest, e.g.,

x1**2 x1**1 x1**0
x2**2 x2**1 x2**0

This matrix gets multiplied, by way of matrix multiplication, with the coefficients of quadratic ax**2 + bx + c, which looks like

a1 a2
b1 b2
c1 c2

The result is exactly the values of polynomials.