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RegressionMetrics

Bases: ABC

A collection of regression metrics.

Methods:

Name Description
coefficient_of_determination

Compute the coefficient of determination (R²) on the given data.
mean_absolute_error

Compute the mean absolute error (MAE) on the given data.
mean_directional_accuracy

Compute the mean directional accuracy (MDA) on the given data.
mean_squared_error

Compute the mean squared error (MSE) on the given data.
median_absolute_deviation

Compute the median absolute deviation (MAD) on the given data.
summarize

Summarize regression metrics on the given data.
Source code in src/safeds/ml/metrics/_regression_metrics.py 
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class RegressionMetrics(ABC):
    """A collection of regression metrics."""

    @abstractmethod
    def __init__(self) -> None: ...

    @staticmethod
    def summarize(
        predicted: Column | TabularDataset | TimeSeriesDataset,
        expected: Column | TabularDataset | TimeSeriesDataset,
    ) -> Table:
        """
        Summarize regression metrics on the given data.

        Parameters
        ----------
        predicted:
            The predicted target values produced by the regressor.
        expected:
            The expected target values.

        Returns
        -------
        metrics:
            A table containing the regression metrics.
        """
        expected = _extract_target(expected)
        predicted = _extract_target(predicted)
        _check_equal_length(predicted, expected)

        coefficient_of_determination = RegressionMetrics.coefficient_of_determination(expected, predicted)
        mean_absolute_error = RegressionMetrics.mean_absolute_error(expected, predicted)
        mean_squared_error = RegressionMetrics.mean_squared_error(expected, predicted)
        median_absolute_deviation = RegressionMetrics.median_absolute_deviation(expected, predicted)

        return Table(
            {
                "metric": [
                    "coefficient_of_determination",
                    "mean_absolute_error",
                    "mean_squared_error",
                    "median_absolute_deviation",
                ],
                "value": [
                    coefficient_of_determination,
                    mean_absolute_error,
                    mean_squared_error,
                    median_absolute_deviation,
                ],
            },
        )

    @staticmethod
    def coefficient_of_determination(
        predicted: Column | TabularDataset | TimeSeriesDataset,
        expected: Column | TabularDataset | TimeSeriesDataset,
    ) -> float:
        """
        Compute the coefficient of determination (R²) on the given data.

        The coefficient of determination compares the regressor's predictions to another model that always predicts the
        mean of the target values. It is a measure of how well the regressor explains the variance in the target values.

        The **higher** the coefficient of determination, the better the regressor. Results range from negative infinity
        to 1.0. You can interpret the coefficient of determination as follows:

        | R²         | Interpretation                                                                             |
        | ---------- | ------------------------------------------------------------------------------------------ |
        | 1.0        | The model perfectly predicts the target values. Did you overfit?                           |
        | (0.0, 1.0) | The model is better than predicting the mean of the target values. You should be here.     |
        | 0.0        | The model is as good as predicting the mean of the target values. Try something else.      |
        | (-∞, 0.0)  | The model is worse than predicting the mean of the target values. Something is very wrong. |

        **Note:** Some other libraries call this metric `r2_score`.

        Parameters
        ----------
        predicted:
            The predicted target values produced by the regressor.
        expected:
            The expected target values.

        Returns
        -------
        coefficient_of_determination:
            The calculated coefficient of determination.
        """
        expected = _extract_target(expected)
        predicted = _extract_target(predicted)
        _check_equal_length(predicted, expected)

        # For TimeSeries Predictions, where the output is a list of values.
        # Expected results are internally converted to a column containing multiple Columns for each prediction window
        # Currently only used in fit_by_exhaustive_search, where prediction metrics have to be calculated internally.
        if isinstance(expected.get_value(0), Column):
            sum_of_coefficient_of_determination = 0.0
            for i in range(expected.row_count):
                predicted_row_as_col: Column = Column("predicted", predicted[i])
                expected_row_as_col = expected.get_value(i)
                sum_of_coefficient_of_determination += RegressionMetrics.coefficient_of_determination(
                    predicted_row_as_col,
                    expected_row_as_col,
                )
            return sum_of_coefficient_of_determination / expected.row_count

        residual_sum_of_squares = (expected._series - predicted._series).pow(2).sum()
        total_sum_of_squares = (expected._series - expected._series.mean()).pow(2).sum()

        if total_sum_of_squares == 0:
            if residual_sum_of_squares == 0:
                return 1.0  # Everything was predicted correctly
            else:
                return 0.0  # Model could not even predict constant data

        return 1 - residual_sum_of_squares / total_sum_of_squares

    @staticmethod
    def mean_absolute_error(
        predicted: Column | TabularDataset | TimeSeriesDataset,
        expected: Column | TabularDataset | TimeSeriesDataset,
    ) -> float:
        """
        Compute the mean absolute error (MAE) on the given data.

        The mean absolute error is the average of the absolute differences between the predicted and expected target
        values. The **lower** the mean absolute error, the better the regressor. Results range from 0.0 to positive
        infinity.

        Parameters
        ----------
        predicted:
            The predicted target values produced by the regressor.
        expected:
            The expected target values.

        Returns
        -------
        mean_absolute_error:
            The calculated mean absolute error.
        """
        expected = _extract_target(expected)
        predicted = _extract_target(predicted)
        _check_equal_length(predicted, expected)

        if expected.row_count == 0:
            return 0.0  # Everything was predicted correctly (since there is nothing to predict)

        # For TimeSeries Predictions, where the output is a list of values.
        # Expected results are internally converted to a column containing multiple Columns for each prediction window
        # Currently only used in fit_by_exhaustive_search, where prediction metrics have to be calculated internally.
        if isinstance(expected.get_value(0), Column):
            sum_of_mean_absolute_errors = 0.0
            for i in range(expected.row_count):
                predicted_row_as_col: Column = Column("predicted", predicted[i])
                expected_row_as_col = expected.get_value(i)
                sum_of_mean_absolute_errors += RegressionMetrics.mean_absolute_error(
                    predicted_row_as_col,
                    expected_row_as_col,
                )
            return sum_of_mean_absolute_errors / expected.row_count

        return (expected._series - predicted._series).abs().mean()

    @staticmethod
    def mean_directional_accuracy(
        predicted: Column | TabularDataset | TimeSeriesDataset,
        expected: Column | TabularDataset | TimeSeriesDataset,
    ) -> float:
        """
        Compute the mean directional accuracy (MDA) on the given data.

        This metric compares two consecutive target values and checks if the predicted direction (down/unchanged/up)
        matches the expected direction. The mean directional accuracy is the proportion of correctly predicted
        directions. The **higher** the mean directional accuracy, the better the regressor. Results range from 0.0 to
        1.0.

        This metric is useful for time series data, where the order of the target values has a meaning. It is not useful
        for other types of data. Because of this, it is not included in the `summarize` method.

        Parameters
        ----------
        predicted:
            The predicted target values produced by the regressor.
        expected:
            The expected target values.

        Returns
        -------
        mean_directional_accuracy:
            The calculated mean directional accuracy.
        """
        expected = _extract_target(expected)
        predicted = _extract_target(predicted)
        _check_equal_length(predicted, expected)

        if expected.row_count == 0:
            return 1.0

        # Calculate the differences between the target values
        predicted_directions = predicted._series.diff().sign()
        expected_directions = expected._series.diff().sign()

        return predicted_directions.eq(expected_directions).mean()

    @staticmethod
    def mean_squared_error(
        predicted: Column | TabularDataset | TimeSeriesDataset,
        expected: Column | TabularDataset | TimeSeriesDataset,
    ) -> float:
        """
        Compute the mean squared error (MSE) on the given data.

        The mean squared error is the average of the squared differences between the predicted and expected target
        values. The **lower** the mean squared error, the better the regressor. Results range from 0.0 to positive
        infinity.

        **Note:** To get the root mean squared error (RMSE), take the square root of the result.

        Parameters
        ----------
        predicted:
            The predicted target values produced by the regressor.
        expected:
            The expected target values.

        Returns
        -------
        mean_squared_error:
            The calculated mean squared error.
        """
        expected = _extract_target(expected)
        predicted = _extract_target(predicted)
        _check_equal_length(predicted, expected)

        if expected.row_count == 0:
            return 0.0  # Everything was predicted correctly (since there is nothing to predict)

        # For TimeSeries Predictions, where the output is a list of values.
        # Expected results are internally converted to a column containing multiple Columns for each prediction window
        # Currently only used in fit_by_exhaustive_search, where prediction metrics have to be calculated internally.
        if isinstance(expected.get_value(0), Column):
            sum_of_mean_squared_errors = 0.0
            for i in range(expected.row_count):
                predicted_row_as_col: Column = Column("predicted", predicted[i])
                expected_row_as_col = expected.get_value(i)
                sum_of_mean_squared_errors += RegressionMetrics.mean_squared_error(
                    predicted_row_as_col,
                    expected_row_as_col,
                )
            return sum_of_mean_squared_errors / expected.row_count

        return (expected._series - predicted._series).pow(2).mean()

    @staticmethod
    def median_absolute_deviation(
        predicted: Column | TabularDataset | TimeSeriesDataset,
        expected: Column | TabularDataset | TimeSeriesDataset,
    ) -> float:
        """
        Compute the median absolute deviation (MAD) on the given data.

        The median absolute deviation is the median of the absolute differences between the predicted and expected
        target values. The **lower** the median absolute deviation, the better the regressor. Results range from 0.0 to
        positive infinity.

        Parameters
        ----------
        predicted:
            The predicted target values produced by the regressor.
        expected:
            The expected target values.

        Returns
        -------
        median_absolute_deviation:
            The calculated median absolute deviation.
        """
        expected = _extract_target(expected)
        predicted = _extract_target(predicted)
        _check_equal_length(predicted, expected)

        if expected.row_count == 0:
            return 0.0

        # For TimeSeries Predictions, where the output is a list of values.
        # Expected results are internally converted to a column containing multiple Columns for each prediction window
        # Currently only used in fit_by_exhaustive_search, where prediction metrics have to be calculated internally.
        if isinstance(expected.get_value(0), Column):
            sum_of_median_absolute_deviation = 0.0
            for i in range(expected.row_count):
                predicted_row_as_col: Column = Column("predicted", predicted[i])
                expected_row_as_col = expected.get_value(i)
                sum_of_median_absolute_deviation += RegressionMetrics.median_absolute_deviation(
                    predicted_row_as_col,
                    expected_row_as_col,
                )
            return sum_of_median_absolute_deviation / expected.row_count

        return (expected._series - predicted._series).abs().median()

coefficient_of_determination

Compute the coefficient of determination (R²) on the given data.

The coefficient of determination compares the regressor's predictions to another model that always predicts the mean of the target values. It is a measure of how well the regressor explains the variance in the target values.

The higher the coefficient of determination, the better the regressor. Results range from negative infinity to 1.0. You can interpret the coefficient of determination as follows:

R² Interpretation
1.0 The model perfectly predicts the target values. Did you overfit?
(0.0, 1.0) The model is better than predicting the mean of the target values. You should be here.
0.0 The model is as good as predicting the mean of the target values. Try something else.
(-∞, 0.0) The model is worse than predicting the mean of the target values. Something is very wrong.

Note: Some other libraries call this metric r2_score.

Parameters:

Name Type Description Default
predicted Column | TabularDataset | TimeSeriesDataset

The predicted target values produced by the regressor.

 required
expected Column | TabularDataset | TimeSeriesDataset

The expected target values.

 required

Returns:

Name Type Description
coefficient_of_determination float

The calculated coefficient of determination.
Source code in src/safeds/ml/metrics/_regression_metrics.py 
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@staticmethod
def coefficient_of_determination(
    predicted: Column | TabularDataset | TimeSeriesDataset,
    expected: Column | TabularDataset | TimeSeriesDataset,
) -> float:
    """
    Compute the coefficient of determination (R²) on the given data.

    The coefficient of determination compares the regressor's predictions to another model that always predicts the
    mean of the target values. It is a measure of how well the regressor explains the variance in the target values.

    The **higher** the coefficient of determination, the better the regressor. Results range from negative infinity
    to 1.0. You can interpret the coefficient of determination as follows:

    | R²         | Interpretation                                                                             |
    | ---------- | ------------------------------------------------------------------------------------------ |
    | 1.0        | The model perfectly predicts the target values. Did you overfit?                           |
    | (0.0, 1.0) | The model is better than predicting the mean of the target values. You should be here.     |
    | 0.0        | The model is as good as predicting the mean of the target values. Try something else.      |
    | (-∞, 0.0)  | The model is worse than predicting the mean of the target values. Something is very wrong. |

    **Note:** Some other libraries call this metric `r2_score`.

    Parameters
    ----------
    predicted:
        The predicted target values produced by the regressor.
    expected:
        The expected target values.

    Returns
    -------
    coefficient_of_determination:
        The calculated coefficient of determination.
    """
    expected = _extract_target(expected)
    predicted = _extract_target(predicted)
    _check_equal_length(predicted, expected)

    # For TimeSeries Predictions, where the output is a list of values.
    # Expected results are internally converted to a column containing multiple Columns for each prediction window
    # Currently only used in fit_by_exhaustive_search, where prediction metrics have to be calculated internally.
    if isinstance(expected.get_value(0), Column):
        sum_of_coefficient_of_determination = 0.0
        for i in range(expected.row_count):
            predicted_row_as_col: Column = Column("predicted", predicted[i])
            expected_row_as_col = expected.get_value(i)
            sum_of_coefficient_of_determination += RegressionMetrics.coefficient_of_determination(
                predicted_row_as_col,
                expected_row_as_col,
            )
        return sum_of_coefficient_of_determination / expected.row_count

    residual_sum_of_squares = (expected._series - predicted._series).pow(2).sum()
    total_sum_of_squares = (expected._series - expected._series.mean()).pow(2).sum()

    if total_sum_of_squares == 0:
        if residual_sum_of_squares == 0:
            return 1.0  # Everything was predicted correctly
        else:
            return 0.0  # Model could not even predict constant data

    return 1 - residual_sum_of_squares / total_sum_of_squares

mean_absolute_error

Compute the mean absolute error (MAE) on the given data.

The mean absolute error is the average of the absolute differences between the predicted and expected target values. The lower the mean absolute error, the better the regressor. Results range from 0.0 to positive infinity.

Parameters:

Name Type Description Default
predicted Column | TabularDataset | TimeSeriesDataset

The predicted target values produced by the regressor.

 required
expected Column | TabularDataset | TimeSeriesDataset

The expected target values.

 required

Returns:

Name Type Description
mean_absolute_error float

The calculated mean absolute error.
Source code in src/safeds/ml/metrics/_regression_metrics.py 
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@staticmethod
def mean_absolute_error(
    predicted: Column | TabularDataset | TimeSeriesDataset,
    expected: Column | TabularDataset | TimeSeriesDataset,
) -> float:
    """
    Compute the mean absolute error (MAE) on the given data.

    The mean absolute error is the average of the absolute differences between the predicted and expected target
    values. The **lower** the mean absolute error, the better the regressor. Results range from 0.0 to positive
    infinity.

    Parameters
    ----------
    predicted:
        The predicted target values produced by the regressor.
    expected:
        The expected target values.

    Returns
    -------
    mean_absolute_error:
        The calculated mean absolute error.
    """
    expected = _extract_target(expected)
    predicted = _extract_target(predicted)
    _check_equal_length(predicted, expected)

    if expected.row_count == 0:
        return 0.0  # Everything was predicted correctly (since there is nothing to predict)

    # For TimeSeries Predictions, where the output is a list of values.
    # Expected results are internally converted to a column containing multiple Columns for each prediction window
    # Currently only used in fit_by_exhaustive_search, where prediction metrics have to be calculated internally.
    if isinstance(expected.get_value(0), Column):
        sum_of_mean_absolute_errors = 0.0
        for i in range(expected.row_count):
            predicted_row_as_col: Column = Column("predicted", predicted[i])
            expected_row_as_col = expected.get_value(i)
            sum_of_mean_absolute_errors += RegressionMetrics.mean_absolute_error(
                predicted_row_as_col,
                expected_row_as_col,
            )
        return sum_of_mean_absolute_errors / expected.row_count

    return (expected._series - predicted._series).abs().mean()

mean_directional_accuracy

Compute the mean directional accuracy (MDA) on the given data.

This metric compares two consecutive target values and checks if the predicted direction (down/unchanged/up) matches the expected direction. The mean directional accuracy is the proportion of correctly predicted directions. The higher the mean directional accuracy, the better the regressor. Results range from 0.0 to 1.0.

This metric is useful for time series data, where the order of the target values has a meaning. It is not useful for other types of data. Because of this, it is not included in the summarize method.

Parameters:

Name Type Description Default
predicted Column | TabularDataset | TimeSeriesDataset

The predicted target values produced by the regressor.

 required
expected Column | TabularDataset | TimeSeriesDataset

The expected target values.

 required

Returns:

Name Type Description
mean_directional_accuracy float

The calculated mean directional accuracy.
Source code in src/safeds/ml/metrics/_regression_metrics.py 
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@staticmethod
def mean_directional_accuracy(
    predicted: Column | TabularDataset | TimeSeriesDataset,
    expected: Column | TabularDataset | TimeSeriesDataset,
) -> float:
    """
    Compute the mean directional accuracy (MDA) on the given data.

    This metric compares two consecutive target values and checks if the predicted direction (down/unchanged/up)
    matches the expected direction. The mean directional accuracy is the proportion of correctly predicted
    directions. The **higher** the mean directional accuracy, the better the regressor. Results range from 0.0 to
    1.0.

    This metric is useful for time series data, where the order of the target values has a meaning. It is not useful
    for other types of data. Because of this, it is not included in the `summarize` method.

    Parameters
    ----------
    predicted:
        The predicted target values produced by the regressor.
    expected:
        The expected target values.

    Returns
    -------
    mean_directional_accuracy:
        The calculated mean directional accuracy.
    """
    expected = _extract_target(expected)
    predicted = _extract_target(predicted)
    _check_equal_length(predicted, expected)

    if expected.row_count == 0:
        return 1.0

    # Calculate the differences between the target values
    predicted_directions = predicted._series.diff().sign()
    expected_directions = expected._series.diff().sign()

    return predicted_directions.eq(expected_directions).mean()

mean_squared_error

Compute the mean squared error (MSE) on the given data.

The mean squared error is the average of the squared differences between the predicted and expected target values. The lower the mean squared error, the better the regressor. Results range from 0.0 to positive infinity.

Note: To get the root mean squared error (RMSE), take the square root of the result.

Parameters:

Name Type Description Default
predicted Column | TabularDataset | TimeSeriesDataset

The predicted target values produced by the regressor.

 required
expected Column | TabularDataset | TimeSeriesDataset

The expected target values.

 required

Returns:

Name Type Description
mean_squared_error float

The calculated mean squared error.
Source code in src/safeds/ml/metrics/_regression_metrics.py 
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@staticmethod
def mean_squared_error(
    predicted: Column | TabularDataset | TimeSeriesDataset,
    expected: Column | TabularDataset | TimeSeriesDataset,
) -> float:
    """
    Compute the mean squared error (MSE) on the given data.

    The mean squared error is the average of the squared differences between the predicted and expected target
    values. The **lower** the mean squared error, the better the regressor. Results range from 0.0 to positive
    infinity.

    **Note:** To get the root mean squared error (RMSE), take the square root of the result.

    Parameters
    ----------
    predicted:
        The predicted target values produced by the regressor.
    expected:
        The expected target values.

    Returns
    -------
    mean_squared_error:
        The calculated mean squared error.
    """
    expected = _extract_target(expected)
    predicted = _extract_target(predicted)
    _check_equal_length(predicted, expected)

    if expected.row_count == 0:
        return 0.0  # Everything was predicted correctly (since there is nothing to predict)

    # For TimeSeries Predictions, where the output is a list of values.
    # Expected results are internally converted to a column containing multiple Columns for each prediction window
    # Currently only used in fit_by_exhaustive_search, where prediction metrics have to be calculated internally.
    if isinstance(expected.get_value(0), Column):
        sum_of_mean_squared_errors = 0.0
        for i in range(expected.row_count):
            predicted_row_as_col: Column = Column("predicted", predicted[i])
            expected_row_as_col = expected.get_value(i)
            sum_of_mean_squared_errors += RegressionMetrics.mean_squared_error(
                predicted_row_as_col,
                expected_row_as_col,
            )
        return sum_of_mean_squared_errors / expected.row_count

    return (expected._series - predicted._series).pow(2).mean()

median_absolute_deviation

Compute the median absolute deviation (MAD) on the given data.

The median absolute deviation is the median of the absolute differences between the predicted and expected target values. The lower the median absolute deviation, the better the regressor. Results range from 0.0 to positive infinity.

Parameters:

Name Type Description Default
predicted Column | TabularDataset | TimeSeriesDataset

The predicted target values produced by the regressor.

 required
expected Column | TabularDataset | TimeSeriesDataset

The expected target values.

 required

Returns:

Name Type Description
median_absolute_deviation float

The calculated median absolute deviation.
Source code in src/safeds/ml/metrics/_regression_metrics.py 
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@staticmethod
def median_absolute_deviation(
    predicted: Column | TabularDataset | TimeSeriesDataset,
    expected: Column | TabularDataset | TimeSeriesDataset,
) -> float:
    """
    Compute the median absolute deviation (MAD) on the given data.

    The median absolute deviation is the median of the absolute differences between the predicted and expected
    target values. The **lower** the median absolute deviation, the better the regressor. Results range from 0.0 to
    positive infinity.

    Parameters
    ----------
    predicted:
        The predicted target values produced by the regressor.
    expected:
        The expected target values.

    Returns
    -------
    median_absolute_deviation:
        The calculated median absolute deviation.
    """
    expected = _extract_target(expected)
    predicted = _extract_target(predicted)
    _check_equal_length(predicted, expected)

    if expected.row_count == 0:
        return 0.0

    # For TimeSeries Predictions, where the output is a list of values.
    # Expected results are internally converted to a column containing multiple Columns for each prediction window
    # Currently only used in fit_by_exhaustive_search, where prediction metrics have to be calculated internally.
    if isinstance(expected.get_value(0), Column):
        sum_of_median_absolute_deviation = 0.0
        for i in range(expected.row_count):
            predicted_row_as_col: Column = Column("predicted", predicted[i])
            expected_row_as_col = expected.get_value(i)
            sum_of_median_absolute_deviation += RegressionMetrics.median_absolute_deviation(
                predicted_row_as_col,
                expected_row_as_col,
            )
        return sum_of_median_absolute_deviation / expected.row_count

    return (expected._series - predicted._series).abs().median()

summarize

Summarize regression metrics on the given data.

Parameters:

Name Type Description Default
predicted Column | TabularDataset | TimeSeriesDataset

The predicted target values produced by the regressor.

 required
expected Column | TabularDataset | TimeSeriesDataset

The expected target values.

 required

Returns:

Name Type Description
metrics Table

A table containing the regression metrics.
Source code in src/safeds/ml/metrics/_regression_metrics.py 
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@staticmethod
def summarize(
    predicted: Column | TabularDataset | TimeSeriesDataset,
    expected: Column | TabularDataset | TimeSeriesDataset,
) -> Table:
    """
    Summarize regression metrics on the given data.

    Parameters
    ----------
    predicted:
        The predicted target values produced by the regressor.
    expected:
        The expected target values.

    Returns
    -------
    metrics:
        A table containing the regression metrics.
    """
    expected = _extract_target(expected)
    predicted = _extract_target(predicted)
    _check_equal_length(predicted, expected)

    coefficient_of_determination = RegressionMetrics.coefficient_of_determination(expected, predicted)
    mean_absolute_error = RegressionMetrics.mean_absolute_error(expected, predicted)
    mean_squared_error = RegressionMetrics.mean_squared_error(expected, predicted)
    median_absolute_deviation = RegressionMetrics.median_absolute_deviation(expected, predicted)

    return Table(
        {
            "metric": [
                "coefficient_of_determination",
                "mean_absolute_error",
                "mean_squared_error",
                "median_absolute_deviation",
            ],
            "value": [
                coefficient_of_determination,
                mean_absolute_error,
                mean_squared_error,
                median_absolute_deviation,
            ],
        },
    )