Time Series Convolution (Supplementary Materials)

A convolutional kernel approach for reinforcing the modeling of time series trends and interpreting temporal patterns, allowing one to leverage Fourier transforms and learn sparse representations.

In this post, we intend to explain the essential ideas of our latent research work:

I. Appendix for Imputing Time Series

A. Circulant Matrix Nuclear Norm Minimization

Below is the Python code for reproducing the circulant matrix nuclear norm minimization on partially observed traffic volume data in Portland, USA.

import numpy as np

def compute_mape(var, var_hat):
    return np.sum(np.abs(var - var_hat) / var) / var.shape[0]

def compute_rmse(var, var_hat):
    return np.sqrt(np.sum((var - var_hat) ** 2) / var.shape[0])

def circ_opt(y_true, y, lmbda, gamma, maxiter = 50, show_iter = 10):
    T = y.shape
    pos_train = np.where(y != 0)
    y_train = y[pos_train]
    pos_test = np.where((y_true != 0) & (y == 0))
    y_test = y_true[pos_test]
    z = y.copy()
    w = y.copy()
    def y_true, y
    for it in range(maxiter):
        x = update_x(z, w, lmbda)
        z = update_z(y_train, pos_train, x, w, lmbda, gamma)
        w = w + lmbda * (x - z)
        if (it + 1) % show_iter == 0:
            print('Iter: {}'.format(it + 1))
            print('MAPE: {:.6}'.format(compute_mape(y_test, x[pos_test])))
            print('RMSE: {:.6}'.format(compute_rmse(y_test, x[pos_test])))
            print()
    return x