numpy.py

# AUTOGENERATED! DO NOT EDIT! File to edit: ../../nbs/losses.numpy.ipynb.

# %% auto 0
__all__ = ['mae', 'mse', 'rmse', 'mape', 'smape', 'mase', 'rmae', 'quantile_loss', 'mqloss']

# %% ../../nbs/losses.numpy.ipynb 4
from typing import Optional, Union

import numpy as np

# %% ../../nbs/losses.numpy.ipynb 7
def _divide_no_nan(a: np.ndarray, b: np.ndarray) -> np.ndarray:
    """
    Auxiliary funtion to handle divide by 0
    """
    div = a / b
    div[div != div] = 0.0
    div[div == float("inf")] = 0.0
    return div

# %% ../../nbs/losses.numpy.ipynb 8
def _metric_protections(
    y: np.ndarray, y_hat: np.ndarray, weights: Optional[np.ndarray]
) -> None:
    assert (weights is None) or (np.sum(weights) > 0), "Sum of weights cannot be 0"
    assert (weights is None) or (
        weights.shape == y.shape
    ), f"Wrong weight dimension weights.shape {weights.shape}, y.shape {y.shape}"

# %% ../../nbs/losses.numpy.ipynb 11
def mae(
    y: np.ndarray,
    y_hat: np.ndarray,
    weights: Optional[np.ndarray] = None,
    axis: Optional[int] = None,
) -> Union[float, np.ndarray]:
    """Mean Absolute Error

    Calculates Mean Absolute Error between
    `y` and `y_hat`. MAE measures the relative prediction
    accuracy of a forecasting method by calculating the
    deviation of the prediction and the true
    value at a given time and averages these devations
    over the length of the series.

    $$ \mathrm{MAE}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}_{\\tau}) = \\frac{1}{H} \\sum^{t+H}_{\\tau=t+1} |y_{\\tau} - \hat{y}_{\\tau}| $$

    **Parameters:**<br>
    `y`: numpy array, Actual values.<br>
    `y_hat`: numpy array, Predicted values.<br>
    `mask`: numpy array, Specifies date stamps per serie to consider in loss.<br>

    **Returns:**<br>
    `mae`: numpy array, (single value).
    """
    _metric_protections(y, y_hat, weights)

    delta_y = np.abs(y - y_hat)
    if weights is not None:
        mae = np.average(
            delta_y[~np.isnan(delta_y)], weights=weights[~np.isnan(delta_y)], axis=axis
        )
    else:
        mae = np.nanmean(delta_y, axis=axis)

    return mae

# %% ../../nbs/losses.numpy.ipynb 15
def mse(
    y: np.ndarray,
    y_hat: np.ndarray,
    weights: Optional[np.ndarray] = None,
    axis: Optional[int] = None,
) -> Union[float, np.ndarray]:
    """Mean Squared Error

    Calculates Mean Squared Error between
    `y` and `y_hat`. MSE measures the relative prediction
    accuracy of a forecasting method by calculating the
    squared deviation of the prediction and the true
    value at a given time, and averages these devations
    over the length of the series.

    $$ \mathrm{MSE}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}_{\\tau}) = \\frac{1}{H} \\sum^{t+H}_{\\tau=t+1} (y_{\\tau} - \hat{y}_{\\tau})^{2} $$

    **Parameters:**<br>
    `y`: numpy array, Actual values.<br>
    `y_hat`: numpy array, Predicted values.<br>
    `mask`: numpy array, Specifies date stamps per serie to consider in loss.<br>

    **Returns:**<br>
    `mse`: numpy array, (single value).
    """
    _metric_protections(y, y_hat, weights)

    delta_y = np.square(y - y_hat)
    if weights is not None:
        mse = np.average(
            delta_y[~np.isnan(delta_y)], weights=weights[~np.isnan(delta_y)], axis=axis
        )
    else:
        mse = np.nanmean(delta_y, axis=axis)

    return mse

# %% ../../nbs/losses.numpy.ipynb 19
def rmse(
    y: np.ndarray,
    y_hat: np.ndarray,
    weights: Optional[np.ndarray] = None,
    axis: Optional[int] = None,
) -> Union[float, np.ndarray]:
    """Root Mean Squared Error

    Calculates Root Mean Squared Error between
    `y` and `y_hat`. RMSE measures the relative prediction
    accuracy of a forecasting method by calculating the squared deviation
    of the prediction and the observed value at a given time and
    averages these devations over the length of the series.
    Finally the RMSE will be in the same scale
    as the original time series so its comparison with other
    series is possible only if they share a common scale.
    RMSE has a direct connection to the L2 norm.

    $$ \mathrm{RMSE}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}_{\\tau}) = \\sqrt{\\frac{1}{H} \\sum^{t+H}_{\\tau=t+1} (y_{\\tau} - \hat{y}_{\\tau})^{2}} $$

    **Parameters:**<br>
    `y`: numpy array, Actual values.<br>
    `y_hat`: numpy array, Predicted values.<br>
    `mask`: numpy array, Specifies date stamps per serie to consider in loss.<br>

    **Returns:**<br>
    `rmse`: numpy array, (single value).
    """
    return np.sqrt(mse(y, y_hat, weights, axis))

# %% ../../nbs/losses.numpy.ipynb 24
def mape(
    y: np.ndarray,
    y_hat: np.ndarray,
    weights: Optional[np.ndarray] = None,
    axis: Optional[int] = None,
) -> Union[float, np.ndarray]:
    """Mean Absolute Percentage Error

    Calculates Mean Absolute Percentage Error  between
    `y` and `y_hat`. MAPE measures the relative prediction
    accuracy of a forecasting method by calculating the percentual deviation
    of the prediction and the observed value at a given time and
    averages these devations over the length of the series.
    The closer to zero an observed value is, the higher penalty MAPE loss
    assigns to the corresponding error.

    $$ \mathrm{MAPE}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}_{\\tau}) = \\frac{1}{H} \\sum^{t+H}_{\\tau=t+1} \\frac{|y_{\\tau}-\hat{y}_{\\tau}|}{|y_{\\tau}|} $$

    **Parameters:**<br>
    `y`: numpy array, Actual values.<br>
    `y_hat`: numpy array, Predicted values.<br>
    `mask`: numpy array, Specifies date stamps per serie to consider in loss.<br>

    **Returns:**<br>
    `mape`: numpy array, (single value).
    """
    _metric_protections(y, y_hat, weights)

    delta_y = np.abs(y - y_hat)
    scale = np.abs(y)
    mape = _divide_no_nan(delta_y, scale)
    mape = np.average(mape, weights=weights, axis=axis)

    return mape

# %% ../../nbs/losses.numpy.ipynb 28
def smape(
    y: np.ndarray,
    y_hat: np.ndarray,
    weights: Optional[np.ndarray] = None,
    axis: Optional[int] = None,
) -> Union[float, np.ndarray]:
    """Symmetric Mean Absolute Percentage Error

    Calculates Symmetric Mean Absolute Percentage Error between
    `y` and `y_hat`. SMAPE measures the relative prediction
    accuracy of a forecasting method by calculating the relative deviation
    of the prediction and the observed value scaled by the sum of the
    absolute values for the prediction and observed value at a
    given time, then averages these devations over the length
    of the series. This allows the SMAPE to have bounds between
    0% and 200% which is desirable compared to normal MAPE that
    may be undetermined when the target is zero.

    $$ \mathrm{sMAPE}_{2}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}_{\\tau}) = \\frac{1}{H} \\sum^{t+H}_{\\tau=t+1} \\frac{|y_{\\tau}-\hat{y}_{\\tau}|}{|y_{\\tau}|+|\hat{y}_{\\tau}|} $$

    **Parameters:**<br>
    `y`: numpy array, Actual values.<br>
    `y_hat`: numpy array, Predicted values.<br>
    `mask`: numpy array, Specifies date stamps per serie to consider in loss.<br>

    **Returns:**<br>
    `smape`: numpy array, (single value).

    **References:**<br>
    [Makridakis S., "Accuracy measures: theoretical and practical concerns".](https://www.sciencedirect.com/science/article/pii/0169207093900793)
    """
    _metric_protections(y, y_hat, weights)

    delta_y = np.abs(y - y_hat)
    scale = np.abs(y) + np.abs(y_hat)
    smape = _divide_no_nan(delta_y, scale)
    smape = 2 * np.average(smape, weights=weights, axis=axis)

    if isinstance(smape, float):
        assert smape <= 2, "SMAPE should be lower than 200"
    else:
        assert all(smape <= 2), "SMAPE should be lower than 200"

    return smape

# %% ../../nbs/losses.numpy.ipynb 32
def mase(
    y: np.ndarray,
    y_hat: np.ndarray,
    y_train: np.ndarray,
    seasonality: int,
    weights: Optional[np.ndarray] = None,
    axis: Optional[int] = None,
) -> Union[float, np.ndarray]:
    """Mean Absolute Scaled Error
    Calculates the Mean Absolute Scaled Error between
    `y` and `y_hat`. MASE measures the relative prediction
    accuracy of a forecasting method by comparinng the mean absolute errors
    of the prediction and the observed value against the mean
    absolute errors of the seasonal naive model.
    The MASE partially composed the Overall Weighted Average (OWA),
    used in the M4 Competition.

    $$ \mathrm{MASE}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}_{\\tau}, \\mathbf{\hat{y}}^{season}_{\\tau}) = \\frac{1}{H} \sum^{t+H}_{\\tau=t+1} \\frac{|y_{\\tau}-\hat{y}_{\\tau}|}{\mathrm{MAE}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}^{season}_{\\tau})} $$

    **Parameters:**<br>
    `y`: numpy array, (batch_size, output_size), Actual values.<br>
    `y_hat`: numpy array, (batch_size, output_size)), Predicted values.<br>
    `y_insample`: numpy array, (batch_size, input_size), Actual insample Seasonal Naive predictions.<br>
    `seasonality`: int. Main frequency of the time series; Hourly 24,  Daily 7, Weekly 52, Monthly 12, Quarterly 4, Yearly 1.
    `mask`: numpy array, Specifies date stamps per serie to consider in loss.<br>

    **Returns:**<br>
    `mase`: numpy array, (single value).

    **References:**<br>
    [Rob J. Hyndman, & Koehler, A. B. "Another look at measures of forecast accuracy".](https://www.sciencedirect.com/science/article/pii/S0169207006000239)<br>
    [Spyros Makridakis, Evangelos Spiliotis, Vassilios Assimakopoulos, "The M4 Competition: 100,000 time series and 61 forecasting methods".](https://www.sciencedirect.com/science/article/pii/S0169207019301128)
    """
    delta_y = np.abs(y - y_hat)
    delta_y = np.average(delta_y, weights=weights, axis=axis)

    scale = np.abs(y_train[:-seasonality] - y_train[seasonality:])
    scale = np.average(scale, axis=axis)

    mase = delta_y / scale

    return mase

# %% ../../nbs/losses.numpy.ipynb 36
def rmae(
    y: np.ndarray,
    y_hat1: np.ndarray,
    y_hat2: np.ndarray,
    weights: Optional[np.ndarray] = None,
    axis: Optional[int] = None,
) -> Union[float, np.ndarray]:
    """RMAE

    Calculates Relative Mean Absolute Error (RMAE) between
    two sets of forecasts (from two different forecasting methods).
    A number smaller than one implies that the forecast in the
    numerator is better than the forecast in the denominator.

    $$ \mathrm{rMAE}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}_{\\tau}, \\mathbf{\hat{y}}^{base}_{\\tau}) = \\frac{1}{H} \sum^{t+H}_{\\tau=t+1} \\frac{|y_{\\tau}-\hat{y}_{\\tau}|}{\mathrm{MAE}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}^{base}_{\\tau})} $$

    **Parameters:**<br>
    `y`: numpy array, observed values.<br>
    `y_hat1`: numpy array. Predicted values of first model.<br>
    `y_hat2`: numpy array. Predicted values of baseline model.<br>
    `weights`: numpy array, optional. Weights for weighted average.<br>
    `axis`: None or int, optional.Axis or axes along which to average a.<br>
        The default, axis=None, will average over all of the elements of
        the input array.

    **Returns:**<br>
    `rmae`: numpy array or double.

    **References:**<br>
    [Rob J. Hyndman, & Koehler, A. B. "Another look at measures of forecast accuracy".](https://www.sciencedirect.com/science/article/pii/S0169207006000239)
    """
    numerator = mae(y=y, y_hat=y_hat1, weights=weights, axis=axis)
    denominator = mae(y=y, y_hat=y_hat2, weights=weights, axis=axis)
    rmae = numerator / denominator

    return rmae

# %% ../../nbs/losses.numpy.ipynb 41
def quantile_loss(
    y: np.ndarray,
    y_hat: np.ndarray,
    q: float = 0.5,
    weights: Optional[np.ndarray] = None,
    axis: Optional[int] = None,
) -> Union[float, np.ndarray]:
    """Quantile Loss

    Computes the quantile loss between `y` and `y_hat`.
    QL measures the deviation of a quantile forecast.
    By weighting the absolute deviation in a non symmetric way, the
    loss pays more attention to under or over estimation.
    A common value for q is 0.5 for the deviation from the median (Pinball loss).

    $$ \mathrm{QL}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}^{(q)}_{\\tau}) = \\frac{1}{H} \\sum^{t+H}_{\\tau=t+1} \Big( (1-q)\,( \hat{y}^{(q)}_{\\tau} - y_{\\tau} )_{+} + q\,( y_{\\tau} - \hat{y}^{(q)}_{\\tau} )_{+} \Big) $$

    **Parameters:**<br>
    `y`: numpy array, Actual values.<br>
    `y_hat`: numpy array, Predicted values.<br>
    `q`: float, between 0 and 1. The slope of the quantile loss, in the context of quantile regression, the q determines the conditional quantile level.<br>
    `mask`: numpy array, Specifies date stamps per serie to consider in loss.<br>

    **Returns:**<br>
    `quantile_loss`: numpy array, (single value).

    **References:**<br>
    [Roger Koenker and Gilbert Bassett, Jr., "Regression Quantiles".](https://www.jstor.org/stable/1913643)
    """
    _metric_protections(y, y_hat, weights)

    delta_y = y - y_hat
    loss = np.maximum(q * delta_y, (q - 1) * delta_y)

    if weights is not None:
        quantile_loss = np.average(
            loss[~np.isnan(loss)], weights=weights[~np.isnan(loss)], axis=axis
        )
    else:
        quantile_loss = np.nanmean(loss, axis=axis)

    return quantile_loss

# %% ../../nbs/losses.numpy.ipynb 45
def mqloss(
    y: np.ndarray,
    y_hat: np.ndarray,
    quantiles: np.ndarray,
    weights: Optional[np.ndarray] = None,
    axis: Optional[int] = None,
) -> Union[float, np.ndarray]:
    """Multi-Quantile loss

    Calculates the Multi-Quantile loss (MQL) between `y` and `y_hat`.
    MQL calculates the average multi-quantile Loss for
    a given set of quantiles, based on the absolute
    difference between predicted quantiles and observed values.

    $$ \mathrm{MQL}(\\mathbf{y}_{\\tau},[\\mathbf{\hat{y}}^{(q_{1})}_{\\tau}, ... ,\hat{y}^{(q_{n})}_{\\tau}]) = \\frac{1}{n} \\sum_{q_{i}} \mathrm{QL}(\\mathbf{y}_{\\tau}, \\mathbf{\hat{y}}^{(q_{i})}_{\\tau}) $$

    The limit behavior of MQL allows to measure the accuracy
    of a full predictive distribution $\mathbf{\hat{F}}_{\\tau}$ with
    the continuous ranked probability score (CRPS). This can be achieved
    through a numerical integration technique, that discretizes the quantiles
    and treats the CRPS integral with a left Riemann approximation, averaging over
    uniformly distanced quantiles.

    $$ \mathrm{CRPS}(y_{\\tau}, \mathbf{\hat{F}}_{\\tau}) = \int^{1}_{0} \mathrm{QL}(y_{\\tau}, \hat{y}^{(q)}_{\\tau}) dq $$

    **Parameters:**<br>
    `y`: numpy array, Actual values.<br>
    `y_hat`: numpy array, Predicted values.<br>
    `quantiles`: numpy array,(n_quantiles). Quantiles to estimate from the distribution of y.<br>
    `mask`: numpy array, Specifies date stamps per serie to consider in loss.<br>

    **Returns:**<br>
    `mqloss`: numpy array, (single value).

    **References:**<br>
    [Roger Koenker and Gilbert Bassett, Jr., "Regression Quantiles".](https://www.jstor.org/stable/1913643)<br>
    [James E. Matheson and Robert L. Winkler, "Scoring Rules for Continuous Probability Distributions".](https://www.jstor.org/stable/2629907)
    """
    if weights is None:
        weights = np.ones(y.shape)

    _metric_protections(y, y_hat, weights)
    n_q = len(quantiles)

    y_rep = np.expand_dims(y, axis=-1)
    error = y_hat - y_rep
    sq = np.maximum(-error, np.zeros_like(error))
    s1_q = np.maximum(error, np.zeros_like(error))
    mqloss = quantiles * sq + (1 - quantiles) * s1_q

    # Match y/weights dimensions and compute weighted average
    weights = np.repeat(np.expand_dims(weights, axis=-1), repeats=n_q, axis=-1)
    mqloss = np.average(mqloss, weights=weights, axis=axis)

    return mqloss