Autoregressive Forecast

Fit an autoregressive model AR(p) by least squares and use it to forecast.

An AR(p) model predicts each value from the p values before it:

x_t = c + phi_1 * x_{t-1} + phi_2 * x_{t-2} + ... + phi_p * x_{t-p}

Fitting it is ordinary linear regression. For every t from p to n - 1, make one row [1, x_{t-1}, ..., x_{t-p}] with target x_t. Solve for (c, phi_1, ..., phi_p) by least squares, for example with np.linalg.lstsq.

To forecast steps values, predict the next value, append it to the series, and repeat. Later forecasts use earlier forecasts as inputs.

Function signature

def ar_forecast(x, p, steps):

Example

Input

ar_forecast(np.array([1.0, 2.0, 4.0, 8.0, 16.0, 32.0]), p=1, steps=2)

Output

array([ 64., 128.])

Reasoning

Every value is twice the one before, so least squares finds c = 0 and phi_1 = 2 with zero error. The next value is 2 * 32 = 64, and the one after it is 2 * 64 = 128, built from the first forecast.

Constraints and notes

• len(x) > 2 * p, so there are more rows than unknowns.
• The coefficient for lag 1 multiplies the most recent value.
• Return an array of shape (steps,).