Losses - Nixtla

The most important train signal is the forecast error, which is the difference between the observed value $y_{\tau}$ and the prediction $\hat{y}_{\tau}$ , at time $y_{\tau}$ :

e_{\tau} = y_{\tau}-\hat{y}_{\tau} \qquad \qquad \tau \in \{t+1,\dots,t+H \}

The train loss summarizes the forecast errors in different evaluation metrics.

from utilsforecast.data import generate_series

models = ['model0', 'model1']
series = generate_series(10, n_models=2, level=[80])
series_pl = generate_series(10, n_models=2, level=[80], engine='polars')

1. Scale-dependent Errors

Mean Absolute Error (MAE)

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

source

mae

 mae (df:~DFType, models:List[str], id_col:str='unique_id',
      target_col:str='y')

*Mean Absolute Error (MAE)

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.*

	Type	Default	Details
df	DFType		Input dataframe with id, actual values and predictions.
models	List		Columns that identify the models predictions.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

def pd_vs_pl(pd_df, pl_df, models):
    np.testing.assert_allclose(
        pd_df[models].to_numpy(),
        pl_df.sort('unique_id').select(models).to_numpy(),
    )

pd_vs_pl(
    mae(series, models),
    mae(series_pl, models),
    models,
)

Mean Squared Error

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

source

mse

 mse (df:~DFType, models:List[str], id_col:str='unique_id',
      target_col:str='y')

*Mean Squared Error (MSE)

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.*

	Type	Default	Details
df	DFType		Input dataframe with id, actual values and predictions.
models	List		Columns that identify the models predictions.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    mse(series, models),
    mse(series_pl, models),
    models,
)

Root Mean Squared Error

\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}}

source

rmse

 rmse (df:~DFType, models:List[str], id_col:str='unique_id',
       target_col:str='y')

*Root Mean Squared Error (RMSE)

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.*

	Type	Default	Details
df	DFType		Input dataframe with id, actual values and predictions.
models	List		Columns that identify the models predictions.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    rmse(series, models),
    rmse(series_pl, models),
    models,
)

source

bias

 bias (df:~DFType, models:List[str], id_col:str='unique_id',
       target_col:str='y')

*Forecast estimator bias.

Defined as prediction - actual*

	Type	Default	Details
df	DFType		Input dataframe with id, actual values and predictions.
models	List		Columns that identify the models predictions.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    bias(series, models),
    bias(series_pl, models),
    models,
)

2. Percentage Errors

Mean Absolute Percentage 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}|}

source

mape

 mape (df:~DFType, models:List[str], id_col:str='unique_id',
       target_col:str='y')

*Mean Absolute Percentage Error (MAPE)

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.*

	Type	Default	Details
df	DFType		Input dataframe with id, actual values and predictions.
models	List		Columns that identify the models predictions.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    mape(series, models),
    mape(series_pl, models),
    models,
)

Symmetric Mean Absolute Percentage Error

\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}|}

source

smape

 smape (df:~DFType, models:List[str], id_col:str='unique_id',
        target_col:str='y')

*Symmetric Mean Absolute Percentage Error (SMAPE)

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 100% which is desirable compared to normal MAPE that may be undetermined when the target is zero.*

	Type	Default	Details
df	DFType		Input dataframe with id, actual values and predictions.
models	List		Columns that identify the models predictions.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    smape(series, models),
    smape(series_pl, models),
    models,
)

3. Scale-independent Errors

Mean Absolute Scaled Error

\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})}

source

mase

 mase (df:~DFType, models:List[str], seasonality:int, train_df:~DFType,
       id_col:str='unique_id', target_col:str='y')

*Mean Absolute Scaled Error (MASE)

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.*

	Type	Default	Details
df	DFType		Input dataframe with id, actuals and predictions.
models	List		Columns that identify the models predictions.
seasonality	int		Main frequency of the time series; Hourly 24, Daily 7, Weekly 52, Monthly 12, Quarterly 4, Yearly 1.
train_df	DFType		Training dataframe with id and actual values. Must be sorted by time.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    mase(series, models, 7, series),
    mase(series_pl, models, 7, series_pl),
    models,
)

Relative Mean Absolute Error

\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})}

source

rmae

 rmae (df:~DFType, models:List[str], baseline:str, id_col:str='unique_id',
       target_col:str='y')

*Relative Mean Absolute Error (RMAE)

Calculates the RAME 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.*

	Type	Default	Details
df	DFType		Input dataframe with id, times, actuals and predictions.
models	List		Columns that identify the models predictions.
baseline	str		Column that identifies the baseline model predictions.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    rmae(series, models, models[0]),
    rmae(series_pl, models, models[0]),
    models,
)

4. Probabilistic Errors

Quantile 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)

source

quantile_loss

 quantile_loss (df:~DFType, models:Dict[str,str], q:float=0.5,
                id_col:str='unique_id', target_col:str='y')

*Quantile Loss (QL)

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.*

	Type	Default	Details
df	DFType		Input dataframe with id, times, actuals and predictions.
models	Dict		Mapping from model name to the model predictions for the specified quantile.
q	float	0.5	Quantile for the predictions’ comparison.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

Multi-Quantile Loss

\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})

source

mqloss

 mqloss (df:~DFType, models:Dict[str,List[str]], quantiles:numpy.ndarray,
         id_col:str='unique_id', target_col:str='y')

*Multi-Quantile loss (MQL)

MQL calculates the average multi-quantile Loss for a given set of quantiles, based on the absolute difference between predicted quantiles and observed values.

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.*

	Type	Default	Details
df	DFType		Input dataframe with id, times, actuals and predictions.
models	Dict		Mapping from model name to the model predictions for each quantile.
quantiles	ndarray		Quantiles to compare against.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    mqloss(series, mq_models, quantiles=quantiles),
    mqloss(series_pl, mq_models, quantiles=quantiles),
    models,
)

Coverage

source

coverage

 coverage (df:~DFType, models:List[str], level:int,
           id_col:str='unique_id', target_col:str='y')

Coverage of y with y_hat_lo and y_hat_hi.

	Type	Default	Details
df	DFType		Input dataframe with id, times, actuals and predictions.
models	List		Columns that identify the models predictions.
level	int		Confidence level used for intervals.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    coverage(series, models, 80),
    coverage(series_pl, models, 80),
    models,
)

Calibration

source

calibration

 calibration (df:~DFType, models:Dict[str,str], id_col:str='unique_id',
              target_col:str='y')

Fraction of y that is lower than the model’s predictions.

	Type	Default	Details
df	DFType		Input dataframe with id, times, actuals and predictions.
models	Dict		Mapping from model name to the model predictions.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    calibration(series, q_models[0.1]),
    calibration(series_pl, q_models[0.1]),
    models,
)

CRPS

\mathrm{sCRPS}(\hat{F}_{\tau}, \mathbf{y}_{\tau}) = \frac{2}{N} \sum_{i} \int^{1}_{0} \frac{\mathrm{QL}(\hat{F}_{i,\tau}, y_{i,\tau})_{q}}{\sum_{i} | y_{i,\tau} |} dq

Where $\hat{F}_{\tau}$ is the an estimated multivariate distribution, and $y_{i,\tau}$ are its realizations.

source

scaled_crps

 scaled_crps (df:~DFType, models:Dict[str,List[str]],
              quantiles:numpy.ndarray, id_col:str='unique_id',
              target_col:str='y')

*Scaled Continues Ranked Probability Score

Calculates a scaled variation of the CRPS, as proposed by Rangapuram (2021), to measure the accuracy of predicted quantiles y_hat compared to the observation y. This metric averages percentual weighted absolute deviations as defined by the quantile losses.*

	Type	Default	Details
df	DFType		Input dataframe with id, times, actuals and predictions.
models	Dict		Mapping from model name to the model predictions for each quantile.
quantiles	ndarray		Quantiles to compare against.
id_col	str	unique_id	Column that identifies each serie.
target_col	str	y	Column that contains the target.
Returns	DFType		dataframe with one row per id and one column per model.

pd_vs_pl(
    scaled_crps(series, mq_models, quantiles),
    scaled_crps(series_pl, mq_models, quantiles),
    models,
)

API Reference

​1. Scale-dependent Errors

​Mean Absolute Error (MAE)

​mae

​Mean Squared Error

​mse

​Root Mean Squared Error

​rmse

​bias

​2. Percentage Errors

​Mean Absolute Percentage Error

​mape

​Symmetric Mean Absolute Percentage Error

​smape

​3. Scale-independent Errors

​Mean Absolute Scaled Error

​mase

​Relative Mean Absolute Error

​rmae

​4. Probabilistic Errors

​Quantile Loss

​quantile_loss

​Multi-Quantile Loss

​mqloss

​Coverage

​coverage

​Calibration

​calibration

​CRPS

​scaled_crps

1. Scale-dependent Errors

Mean Absolute Error (MAE)

mae

Mean Squared Error

mse

Root Mean Squared Error

rmse

bias

2. Percentage Errors

Mean Absolute Percentage Error

mape

Symmetric Mean Absolute Percentage Error

smape

3. Scale-independent Errors

Mean Absolute Scaled Error

mase

Relative Mean Absolute Error

rmae

4. Probabilistic Errors

Quantile Loss

quantile_loss

Multi-Quantile Loss

mqloss

Coverage

coverage

Calibration

calibration

CRPS

scaled_crps