# Source code for tensorly.metrics.regression

```from .. import backend as T

# Author: Jean Kossaifi <jean.kossaifi+tensors@gmail.com>

[docs]
def MSE(y_true, y_pred, axis=None):
"""Returns the mean squared error between the two predictions

Parameters
----------
y_true : array of shape (n_samples, )
Ground truth (correct) target values.
y_pred : array of shape (n_samples, )
Estimated target values.

Returns
-------
float
"""
return T.mean((y_true - y_pred) ** 2, axis=axis)

[docs]
def RMSE(y_true, y_pred, axis=None):
"""Returns the regularised mean squared error between the two predictions
(the square-root is applied to the mean_squared_error)

Parameters
----------
y_true : array of shape (n_samples, )
Ground truth (correct) target values.
y_pred : array of shape (n_samples, )
Estimated target values.

Returns
-------
float
"""
return T.sqrt(MSE(y_true, y_pred, axis=axis))

def R2_score(X_original, X_predicted):
"""Returns the R^2 (coefficient of determination) regression score function.
Best possible score is 1.0 and it can be negative (because prediction can be
arbitrarily worse).

Parameters
----------
X_original: array
The original array
X_predicted: array
Thre predicted array.

Returns
-------
float
"""
return 1 - T.norm(X_predicted - X_original) ** 2.0 / T.norm(X_original) ** 2.0

def reflective_correlation_coefficient(y_true, y_pred, axis=None):
"""Reflective variant of Pearson's product moment correlation coefficient
where the predictions are not centered around their mean values.

Parameters
----------
y_true : array of shape (n_samples, )
Ground truth (correct) target values.
y_pred : array of shape (n_samples, )
Estimated target values.

Returns
-------
float: reflective correlation coefficient
"""
return T.sum(y_true * y_pred, axis=axis) / T.sqrt(
T.sum(y_true**2, axis=axis) * T.sum(y_pred**2, axis=axis)
)

def covariance(y_true, y_pred, axis=None):
centered_true = T.mean(y_true, axis=axis)
centered_pred = T.mean(y_pred, axis=axis)

if axis is not None:
# TODO: write a function to do this..
shape = list(T.shape(y_true))
shape[axis] = 1
centered_true = T.reshape(centered_true, shape)
shape = list(T.shape(y_pred))
shape[axis] = 1
centered_pred = T.reshape(centered_pred, shape)

return T.mean((y_true - centered_true) * (y_pred - centered_pred), axis=axis)

def variance(y, axis=None):
return covariance(y, y, axis=axis)

def standard_deviation(y, axis=None):
return T.sqrt(variance(y, axis=axis))

def correlation(y_true, y_pred, axis=None):
"""Pearson's product moment correlation coefficient"""
return covariance(y_true, y_pred, axis=axis) / T.sqrt(
variance(y_true, axis) * variance(y_pred, axis)
)
```