How to solve several independent time series at the same time using scikit linear regression model

Question

I try to predict multiple independent time series simultaneously using sklearn linear regression model, but I seem not be able to get it right.

My data is organised

Answer 1

@ali_m I don't think this is a duplicate question, but they are partly related. And of course it's possible to apply and predict time series simultaneously using a linear regression model similar to sklearn:

I created a new class LinearRegression_Multi:

class LinearRegression_Multi:
    def stacked_lstsq(self, L, b, rcond=1e-10):
        """
        Solve L x = b, via SVD least squares cutting of small singular values
        L is an array of shape (..., M, N) and b of shape (..., M).
        Returns x of shape (..., N)
        """
        u, s, v = np.linalg.svd(L, full_matrices=False)
        s_max = s.max(axis=-1, keepdims=True)
        s_min = rcond*s_max
        inv_s = np.zeros_like(s)
        inv_s[s >= s_min] = 1/s[s>=s_min]
        x = np.einsum('...ji,...j->...i', v,
                      inv_s * np.einsum('...ji,...j->...i', u, b.conj()))
        return np.conj(x, x)    

    def center_data(self, X, y):
        """ Centers data to have mean zero along axis 0. 
        """
        # center X        
        X_mean = np.average(X,axis=1)
        X_std = np.ones(X.shape[0::2])
        X = X - X_mean[:,None,:] 
        # center y
        y_mean = np.average(y,axis=1)
        y = y - y_mean[:,None]
        return X, y, X_mean, y_mean, X_std

    def set_intercept(self, X_mean, y_mean, X_std):
        """ Calculate the intercept_
        """
        self.coef_ = self.coef_ / X_std # not really necessary
        self.intercept_ = y_mean - np.einsum('ij,ij->i',X_mean,self.coef_)

    def scores(self, y_pred, y_true ):
        """ 
        The coefficient R^2 is defined as (1 - u/v), where u is the regression
        sum of squares ((y_true - y_pred) ** 2).sum() and v is the residual
        sum of squares ((y_true - y_true.mean()) ** 2).sum().        
        """        
        u = ((y_true - y_pred) ** 2).sum(axis=-1)
        v = ((y_true - y_true.mean(axis=-1)[None].T) ** 2).sum(axis=-1)
        r_2 = 1 - u/v
        return r_2

    def fit(self,X, y):
        """ Fit linear model.        
        """        
        # get coefficients by applying linear regression on stack
        X_, y, X_mean, y_mean, X_std = self.center_data(X, y)
        self.coef_ = self.stacked_lstsq(X_, y)
        self.set_intercept(X_mean, y_mean, X_std)

    def predict(self, X):
        """Predict using the linear model
        """
        return np.einsum('ijx,ix->ij',X,self.coef_) + self.intercept_[None].T

Which can be applied as follow, using the same declared variables as in the question:

LR_Multi = LinearRegression_Multi()
LR_Multi.fit(X_stack[:,:half], y_stack[:,:half])
y_stack_pred = LR_Multi.predict(X_stack[:,half:])
R2 = LR_Multi.scores(y_stack_pred, y_stack[:,half:])

Where the R^2 for the multiple time series are:

array([ 0.91262442,  0.67247516])

Which is indeed similar to the prediction method of the standard sklearn linear regression:

from sklearn.linear_model import LinearRegression

LR = LinearRegression()
LR.fit(X1[:half], y1[:half])
R2_1 = LR.score(X1[half:],y1[half:])

LR.fit(X2[:half], y2[:half])
R2_2 = LR.score(X2[half:],y2[half:])
print R2_1, R2_2
0.912624422097 0.67247516054