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Multivariate

uncertainty_flow.multivariate

Multivariate uncertainty modeling.

BaseCopula

Base class for copula families.

Source code in uncertainty_flow/multivariate/copula.py 
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class BaseCopula:
    """Base class for copula families."""

    name: ClassVar[str]
    has_lower_tail: ClassVar[bool]
    has_upper_tail: ClassVar[bool]

    def __init__(self):
        self.theta_: float | None = None
        self.fitted_ = False

    def _validate_fitted(self) -> None:
        """
        Ensure model is fitted before accessing parameters.

        Raises:
            ModelNotFittedError: If the model has not been fitted yet.
        """
        if not self.fitted_ or self.theta_ is None:
            raise ModelNotFittedError(self.__class__.__name__)

    @abstractmethod
    def fit(self, residuals: np.ndarray) -> "BaseCopula":
        """Fit copula on residual matrix."""
        ...

    @abstractmethod
    def sample(
        self,
        marginals: np.ndarray,
        n_samples: int = 1000,
        quantile_levels: np.ndarray | None = None,
        random_state: int | np.random.Generator | None = None,
    ) -> np.ndarray:
        """Generate joint samples from copula."""
        ...

    def _compute_bic(self, log_likelihood: float, n_params: int, n_samples: int) -> float:
        """Compute BIC for model selection."""
        return float(float(n_params) * np.log(n_samples) - 2 * log_likelihood)

    def log_likelihood(self, residuals: np.ndarray) -> float:
        """
        Compute log-likelihood for BIC calculation.

        Default implementation returns -inf. Subclasses should override
        with proper implementation for accurate BIC calculation.
        """
        return -np.inf

    def _to_copula_space(self, residuals: np.ndarray) -> np.ndarray:
        """Transform residuals to uniform copula space via empirical CDF."""
        n_samples, n_targets = residuals.shape
        uniform = np.zeros_like(residuals)
        for t in range(n_targets):
            sorted_vals = np.sort(residuals[:, t])
            ranks = np.searchsorted(sorted_vals, residuals[:, t])
            uniform[:, t] = (ranks + 1) / (n_samples + 1)
        return uniform

fit(residuals) abstractmethod

Fit copula on residual matrix.

Source code in uncertainty_flow/multivariate/copula.py 
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@abstractmethod
def fit(self, residuals: np.ndarray) -> "BaseCopula":
    """Fit copula on residual matrix."""
    ...

sample(marginals, n_samples=1000, quantile_levels=None, random_state=None) abstractmethod

Generate joint samples from copula.

Source code in uncertainty_flow/multivariate/copula.py 
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@abstractmethod
def sample(
    self,
    marginals: np.ndarray,
    n_samples: int = 1000,
    quantile_levels: np.ndarray | None = None,
    random_state: int | np.random.Generator | None = None,
) -> np.ndarray:
    """Generate joint samples from copula."""
    ...

log_likelihood(residuals)

Compute log-likelihood for BIC calculation.

Default implementation returns -inf. Subclasses should override with proper implementation for accurate BIC calculation.

Source code in uncertainty_flow/multivariate/copula.py 
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def log_likelihood(self, residuals: np.ndarray) -> float:
    """
    Compute log-likelihood for BIC calculation.

    Default implementation returns -inf. Subclasses should override
    with proper implementation for accurate BIC calculation.
    """
    return -np.inf

ClaytonCopula

Bases: BaseCopula

Clayton copula for modeling lower tail dependence.

Lower tail dependence: extreme low values tend to co-occur. Ideal for modeling simultaneous extreme low values (e.g., market crashes).

Examples:

>>> import numpy as np
>>> residuals = np.array([
...     [-1, -2],
...     [-0.5, -1],
...     [0, 0],
...     [0.5, 1],
...     [1, 2],
... ])
>>> copula = ClaytonCopula()
>>> copula.fit(residuals)
>>> copula.theta_ > 0  # theta > 0 indicates lower tail dependence
True
Source code in uncertainty_flow/multivariate/copula.py 
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class ClaytonCopula(BaseCopula):
    """
    Clayton copula for modeling lower tail dependence.

    Lower tail dependence: extreme low values tend to co-occur.
    Ideal for modeling simultaneous extreme low values (e.g., market crashes).

    Examples:
        >>> import numpy as np
        >>> residuals = np.array([
        ...     [-1, -2],
        ...     [-0.5, -1],
        ...     [0, 0],
        ...     [0.5, 1],
        ...     [1, 2],
        ... ])
        >>> copula = ClaytonCopula()
        >>> copula.fit(residuals)
        >>> copula.theta_ > 0  # theta > 0 indicates lower tail dependence
        True
    """

    name = "clayton"
    has_lower_tail = True
    has_upper_tail = False

    def fit(self, residuals: np.ndarray) -> "ClaytonCopula":
        """
        Fit Clayton copula via maximum likelihood.

        Args:
            residuals: Residual matrix shape (n_samples, 2) — bivariate only

        Returns:
            self (for method chaining)
        """
        if residuals.ndim != 2:
            raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")

        if residuals.shape[1] != 2:
            raise InvalidDataError(
                f"ClaytonCopula supports bivariate only, got {residuals.shape[1]} dimensions"
            )

        uniform = self._to_copula_space(residuals)
        u, v = uniform[:, 0], uniform[:, 1]

        def neg_log_likelihood(theta: float) -> float:
            if theta <= 0:
                return 1e10
            try:
                h = (u ** (-theta) + v ** (-theta) - 1) ** (-1 / theta)
                ll = (
                    -1 / theta * np.log(u ** (-theta) + v ** (-theta) - 1)
                    + (-theta - 1) * (np.log(u) + np.log(v))
                    - np.log(h)
                )
                if not np.isfinite(ll).all():
                    return 1e10
                return float(-np.sum(ll))
            except (ValueError, OverflowError, ZeroDivisionError):
                return 1e10

        from scipy.optimize import minimize_scalar

        result = minimize_scalar(neg_log_likelihood, bounds=(0.01, 10), method="bounded")
        self.theta_ = float(result.x)
        self.fitted_ = True

        return self

    def log_likelihood(self, residuals: np.ndarray) -> float:
        """Compute log-likelihood for BIC calculation."""
        self._validate_fitted()

        uniform = self._to_copula_space(residuals)
        u, v = uniform[:, 0], uniform[:, 1]
        theta = self.theta_
        assert theta is not None

        try:
            h = (u ** (-theta) + v ** (-theta) - 1) ** (-1 / theta)
            ll = (
                -1 / theta * np.log(u ** (-theta) + v ** (-theta) - 1)
                + (-theta - 1) * (np.log(u) + np.log(v))
                - np.log(h)
            )
            return float(np.sum(ll))
        except (ValueError, OverflowError, ZeroDivisionError):
            return -np.inf

    def sample(
        self,
        marginals: np.ndarray,
        n_samples: int = 1000,
        quantile_levels: np.ndarray | None = None,
        random_state: int | np.random.Generator | None = None,
    ) -> np.ndarray:
        """
        Generate joint samples from Clayton copula.

        Args:
            marginals: Quantile predictions for each target
                      shape (n_samples_input, n_targets, n_quantiles)
            n_samples: Number of Monte Carlo samples to generate

        Returns:
            Joint samples shape (n_samples, n_targets)
        """
        self._validate_fitted()

        if marginals.shape[1] != 2:
            raise InvalidDataError(
                f"ClaytonCopula supports bivariate only, got {marginals.shape[1]} dimensions"
            )

        n_samples_input, n_targets, _ = marginals.shape
        theta = self.theta_
        assert theta is not None
        rng = _resolve_rng(random_state)

        s1 = rng.uniform(0, 1, size=(n_samples_input, n_samples))
        s2 = rng.uniform(0, 1, size=(n_samples_input, n_samples))

        u = s1
        v = (s1 ** (-theta) * (s2 ** (-theta / (theta + 1)) - 1) + 1) ** (-1 / theta)

        uniform_samples = np.stack([u, v], axis=-1)
        return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

    def __repr__(self) -> str:
        if self.fitted_:
            return f"ClaytonCopula(theta={self.theta_:.3f}, fitted=True)"
        return "ClaytonCopula(fitted=False)"

fit(residuals)

Fit Clayton copula via maximum likelihood.

Parameters:

Name Type Description Default
residuals ndarray

Residual matrix shape (n_samples, 2) — bivariate only

 required

Returns:

Type Description
'ClaytonCopula'

self (for method chaining)
Source code in uncertainty_flow/multivariate/copula.py 
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def fit(self, residuals: np.ndarray) -> "ClaytonCopula":
    """
    Fit Clayton copula via maximum likelihood.

    Args:
        residuals: Residual matrix shape (n_samples, 2) — bivariate only

    Returns:
        self (for method chaining)
    """
    if residuals.ndim != 2:
        raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")

    if residuals.shape[1] != 2:
        raise InvalidDataError(
            f"ClaytonCopula supports bivariate only, got {residuals.shape[1]} dimensions"
        )

    uniform = self._to_copula_space(residuals)
    u, v = uniform[:, 0], uniform[:, 1]

    def neg_log_likelihood(theta: float) -> float:
        if theta <= 0:
            return 1e10
        try:
            h = (u ** (-theta) + v ** (-theta) - 1) ** (-1 / theta)
            ll = (
                -1 / theta * np.log(u ** (-theta) + v ** (-theta) - 1)
                + (-theta - 1) * (np.log(u) + np.log(v))
                - np.log(h)
            )
            if not np.isfinite(ll).all():
                return 1e10
            return float(-np.sum(ll))
        except (ValueError, OverflowError, ZeroDivisionError):
            return 1e10

    from scipy.optimize import minimize_scalar

    result = minimize_scalar(neg_log_likelihood, bounds=(0.01, 10), method="bounded")
    self.theta_ = float(result.x)
    self.fitted_ = True

    return self

log_likelihood(residuals)

Compute log-likelihood for BIC calculation.

Source code in uncertainty_flow/multivariate/copula.py 
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def log_likelihood(self, residuals: np.ndarray) -> float:
    """Compute log-likelihood for BIC calculation."""
    self._validate_fitted()

    uniform = self._to_copula_space(residuals)
    u, v = uniform[:, 0], uniform[:, 1]
    theta = self.theta_
    assert theta is not None

    try:
        h = (u ** (-theta) + v ** (-theta) - 1) ** (-1 / theta)
        ll = (
            -1 / theta * np.log(u ** (-theta) + v ** (-theta) - 1)
            + (-theta - 1) * (np.log(u) + np.log(v))
            - np.log(h)
        )
        return float(np.sum(ll))
    except (ValueError, OverflowError, ZeroDivisionError):
        return -np.inf

sample(marginals, n_samples=1000, quantile_levels=None, random_state=None)

Generate joint samples from Clayton copula.

Parameters:

Name Type Description Default
marginals ndarray

Quantile predictions for each target shape (n_samples_input, n_targets, n_quantiles)

 required
n_samples int

Number of Monte Carlo samples to generate

 1000

Returns:

Type Description
ndarray

Joint samples shape (n_samples, n_targets)
Source code in uncertainty_flow/multivariate/copula.py 
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def sample(
    self,
    marginals: np.ndarray,
    n_samples: int = 1000,
    quantile_levels: np.ndarray | None = None,
    random_state: int | np.random.Generator | None = None,
) -> np.ndarray:
    """
    Generate joint samples from Clayton copula.

    Args:
        marginals: Quantile predictions for each target
                  shape (n_samples_input, n_targets, n_quantiles)
        n_samples: Number of Monte Carlo samples to generate

    Returns:
        Joint samples shape (n_samples, n_targets)
    """
    self._validate_fitted()

    if marginals.shape[1] != 2:
        raise InvalidDataError(
            f"ClaytonCopula supports bivariate only, got {marginals.shape[1]} dimensions"
        )

    n_samples_input, n_targets, _ = marginals.shape
    theta = self.theta_
    assert theta is not None
    rng = _resolve_rng(random_state)

    s1 = rng.uniform(0, 1, size=(n_samples_input, n_samples))
    s2 = rng.uniform(0, 1, size=(n_samples_input, n_samples))

    u = s1
    v = (s1 ** (-theta) * (s2 ** (-theta / (theta + 1)) - 1) + 1) ** (-1 / theta)

    uniform_samples = np.stack([u, v], axis=-1)
    return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

FrankCopula

Bases: BaseCopula

Frank copula for symmetric dependence.

Features symmetric tail dependence (technically zero tail dependence in the limit). Used when extreme events are not more likely in one tail than the other.

Examples:

>>> import numpy as np
>>> residuals = np.array([
...     [-1, -2],
...     [-0.5, -1],
...     [0, 0],
...     [0.5, 1],
...     [1, 2],
... ])
>>> copula = FrankCopula()
>>> copula.fit(residuals)
>>> copula.theta_ != 0  # theta != 0 indicates dependence
True
Source code in uncertainty_flow/multivariate/copula.py 
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class FrankCopula(BaseCopula):
    """
    Frank copula for symmetric dependence.

    Features symmetric tail dependence (technically zero tail dependence in the limit).
    Used when extreme events are not more likely in one tail than the other.

    Examples:
        >>> import numpy as np
        >>> residuals = np.array([
        ...     [-1, -2],
        ...     [-0.5, -1],
        ...     [0, 0],
        ...     [0.5, 1],
        ...     [1, 2],
        ... ])
        >>> copula = FrankCopula()
        >>> copula.fit(residuals)
        >>> copula.theta_ != 0  # theta != 0 indicates dependence
        True
    """

    name = "frank"
    has_lower_tail = False
    has_upper_tail = False

    def fit(self, residuals: np.ndarray) -> "FrankCopula":
        """
        Fit Frank copula via maximum likelihood.

        Args:
            residuals: Residual matrix shape (n_samples, 2) — bivariate only

        Returns:
            self (for method chaining)
        """
        if residuals.ndim != 2:
            raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")

        if residuals.shape[1] != 2:
            raise InvalidDataError(
                f"FrankCopula supports bivariate only, got {residuals.shape[1]} dimensions"
            )

        uniform = self._to_copula_space(residuals)
        u, v = uniform[:, 0], uniform[:, 1]

        def neg_log_likelihood(theta: float) -> float:
            if theta == 0:
                return 1e10
            try:
                with np.errstate(divide="ignore", invalid="ignore"):
                    exp_neg_theta_u = np.exp(-theta * u)
                    exp_neg_theta_v = np.exp(-theta * v)
                    exp_neg_theta = np.exp(-theta)
                    numerator = (exp_neg_theta_u - 1) * (exp_neg_theta_v - 1)
                    h = -np.log(numerator / (exp_neg_theta - 1) + 1) / theta
                    ll = (
                        theta * (u + v - 1)
                        - np.log(exp_neg_theta_u - 1)
                        - np.log(exp_neg_theta_v - 1)
                        - np.log(exp_neg_theta - 1)
                        + theta * h
                    )
                    if not np.isfinite(ll).all():
                        return 1e10
                    return float(-np.sum(ll))
            except (ValueError, OverflowError, ZeroDivisionError):
                return 1e10

        from scipy.optimize import minimize_scalar

        result = minimize_scalar(neg_log_likelihood, bounds=(-30, 30), method="bounded")
        self.theta_ = float(result.x)
        self.fitted_ = True

        return self

    def log_likelihood(self, residuals: np.ndarray) -> float:
        """Compute log-likelihood for BIC calculation."""
        self._validate_fitted()

        uniform = self._to_copula_space(residuals)
        u, v = uniform[:, 0], uniform[:, 1]
        theta = self.theta_
        assert theta is not None

        try:
            h = (
                -np.log(
                    (np.exp(-theta * u) - 1) * (np.exp(-theta * v) - 1) / (np.exp(-theta) - 1) + 1
                )
                / theta
            )
            ll = (
                theta * (u + v - 1)
                - np.log(np.exp(-theta * u) - 1)
                - np.log(np.exp(-theta * v) - 1)
                - np.log(np.exp(-theta) - 1)
                + theta * h
            )
            return float(np.sum(ll))
        except (ValueError, OverflowError, ZeroDivisionError):
            return -np.inf

    def sample(
        self,
        marginals: np.ndarray,
        n_samples: int = 1000,
        quantile_levels: np.ndarray | None = None,
        random_state: int | np.random.Generator | None = None,
    ) -> np.ndarray:
        """
        Generate joint samples from Frank copula.

        Args:
            marginals: Quantile predictions for each target
                      shape (n_samples_input, n_targets, n_quantiles)
            n_samples: Number of Monte Carlo samples to generate

        Returns:
            Joint samples shape (n_samples, n_targets)
        """
        self._validate_fitted()

        if marginals.shape[1] != 2:
            raise InvalidDataError(
                f"FrankCopula supports bivariate only, got {marginals.shape[1]} dimensions"
            )

        n_samples_input, n_targets, _ = marginals.shape
        theta = self.theta_
        assert theta is not None
        rng = _resolve_rng(random_state)

        s1 = rng.uniform(0, 1, size=(n_samples_input, n_samples))
        s2 = rng.uniform(0, 1, size=(n_samples_input, n_samples))

        u = s1
        denom = np.exp(-theta * u) * (1 - s2) + s2
        v = -np.log(1 + s2 * (np.exp(-theta) - 1) / denom) / theta

        uniform_samples = np.stack([u, v], axis=-1)
        return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

    def __repr__(self) -> str:
        if self.fitted_:
            return f"FrankCopula(theta={self.theta_:.3f}, fitted=True)"
        return "FrankCopula(fitted=False)"

fit(residuals)

Fit Frank copula via maximum likelihood.

Parameters:

Name Type Description Default
residuals ndarray

Residual matrix shape (n_samples, 2) — bivariate only

 required

Returns:

Type Description
'FrankCopula'

self (for method chaining)
Source code in uncertainty_flow/multivariate/copula.py 
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def fit(self, residuals: np.ndarray) -> "FrankCopula":
    """
    Fit Frank copula via maximum likelihood.

    Args:
        residuals: Residual matrix shape (n_samples, 2) — bivariate only

    Returns:
        self (for method chaining)
    """
    if residuals.ndim != 2:
        raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")

    if residuals.shape[1] != 2:
        raise InvalidDataError(
            f"FrankCopula supports bivariate only, got {residuals.shape[1]} dimensions"
        )

    uniform = self._to_copula_space(residuals)
    u, v = uniform[:, 0], uniform[:, 1]

    def neg_log_likelihood(theta: float) -> float:
        if theta == 0:
            return 1e10
        try:
            with np.errstate(divide="ignore", invalid="ignore"):
                exp_neg_theta_u = np.exp(-theta * u)
                exp_neg_theta_v = np.exp(-theta * v)
                exp_neg_theta = np.exp(-theta)
                numerator = (exp_neg_theta_u - 1) * (exp_neg_theta_v - 1)
                h = -np.log(numerator / (exp_neg_theta - 1) + 1) / theta
                ll = (
                    theta * (u + v - 1)
                    - np.log(exp_neg_theta_u - 1)
                    - np.log(exp_neg_theta_v - 1)
                    - np.log(exp_neg_theta - 1)
                    + theta * h
                )
                if not np.isfinite(ll).all():
                    return 1e10
                return float(-np.sum(ll))
        except (ValueError, OverflowError, ZeroDivisionError):
            return 1e10

    from scipy.optimize import minimize_scalar

    result = minimize_scalar(neg_log_likelihood, bounds=(-30, 30), method="bounded")
    self.theta_ = float(result.x)
    self.fitted_ = True

    return self

log_likelihood(residuals)

Compute log-likelihood for BIC calculation.

Source code in uncertainty_flow/multivariate/copula.py 
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def log_likelihood(self, residuals: np.ndarray) -> float:
    """Compute log-likelihood for BIC calculation."""
    self._validate_fitted()

    uniform = self._to_copula_space(residuals)
    u, v = uniform[:, 0], uniform[:, 1]
    theta = self.theta_
    assert theta is not None

    try:
        h = (
            -np.log(
                (np.exp(-theta * u) - 1) * (np.exp(-theta * v) - 1) / (np.exp(-theta) - 1) + 1
            )
            / theta
        )
        ll = (
            theta * (u + v - 1)
            - np.log(np.exp(-theta * u) - 1)
            - np.log(np.exp(-theta * v) - 1)
            - np.log(np.exp(-theta) - 1)
            + theta * h
        )
        return float(np.sum(ll))
    except (ValueError, OverflowError, ZeroDivisionError):
        return -np.inf

sample(marginals, n_samples=1000, quantile_levels=None, random_state=None)

Generate joint samples from Frank copula.

Parameters:

Name Type Description Default
marginals ndarray

Quantile predictions for each target shape (n_samples_input, n_targets, n_quantiles)

 required
n_samples int

Number of Monte Carlo samples to generate

 1000

Returns:

Type Description
ndarray

Joint samples shape (n_samples, n_targets)
Source code in uncertainty_flow/multivariate/copula.py 
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def sample(
    self,
    marginals: np.ndarray,
    n_samples: int = 1000,
    quantile_levels: np.ndarray | None = None,
    random_state: int | np.random.Generator | None = None,
) -> np.ndarray:
    """
    Generate joint samples from Frank copula.

    Args:
        marginals: Quantile predictions for each target
                  shape (n_samples_input, n_targets, n_quantiles)
        n_samples: Number of Monte Carlo samples to generate

    Returns:
        Joint samples shape (n_samples, n_targets)
    """
    self._validate_fitted()

    if marginals.shape[1] != 2:
        raise InvalidDataError(
            f"FrankCopula supports bivariate only, got {marginals.shape[1]} dimensions"
        )

    n_samples_input, n_targets, _ = marginals.shape
    theta = self.theta_
    assert theta is not None
    rng = _resolve_rng(random_state)

    s1 = rng.uniform(0, 1, size=(n_samples_input, n_samples))
    s2 = rng.uniform(0, 1, size=(n_samples_input, n_samples))

    u = s1
    denom = np.exp(-theta * u) * (1 - s2) + s2
    v = -np.log(1 + s2 * (np.exp(-theta) - 1) / denom) / theta

    uniform_samples = np.stack([u, v], axis=-1)
    return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

GaussianCopula

Bases: BaseCopula

Fit a Gaussian copula on residuals to model inter-target correlation.

Captures linear dependence between targets.

Examples:

>>> import numpy as np
>>> residuals = np.array([
...     [1, 10],
...     [2, 20],
...     [3, 30],
... ])
>>> copula = GaussianCopula()
>>> copula.fit(residuals)
>>> print(copula.correlation_matrix_)
[[1. 1.]
 [1. 1.]]
Source code in uncertainty_flow/multivariate/copula.py 
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class GaussianCopula(BaseCopula):
    """
    Fit a Gaussian copula on residuals to model inter-target correlation.

    Captures linear dependence between targets.

    Examples:
        >>> import numpy as np
        >>> residuals = np.array([
        ...     [1, 10],
        ...     [2, 20],
        ...     [3, 30],
        ... ])
        >>> copula = GaussianCopula()
        >>> copula.fit(residuals)
        >>> print(copula.correlation_matrix_)
        [[1. 1.]
         [1. 1.]]
    """

    name = "gaussian"
    has_lower_tail = False
    has_upper_tail = False

    def __init__(self):
        super().__init__()
        self.correlation_matrix_: np.ndarray | None = None

    def fit(self, residuals: np.ndarray) -> "GaussianCopula":
        """
        Fit copula on residual matrix.

        Args:
            residuals: Residual matrix shape (n_samples, n_targets)

        Returns:
            self (for method chaining)
        """
        if residuals.ndim != 2:
            raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")

        n_samples, n_targets = residuals.shape

        # Check for constant (zero-variance) columns before computing correlation
        constant_cols = []
        for t in range(n_targets):
            if np.all(residuals[:, t] == residuals[0, t]):
                constant_cols.append(t)

        if len(constant_cols) > 0:
            raise InvalidDataError(
                f"Target columns at indices {constant_cols} are constant (zero variance). "
                "Cannot compute correlation matrix. Check if target values vary across samples."
            )

        self.correlation_matrix_ = np.corrcoef(residuals.T)

        # Condition correlation matrix for numerical stability
        min_eigval = 1e-6
        try:
            eigenvals = np.linalg.eigvals(self.correlation_matrix_)
            if np.any(np.isnan(eigenvals)):
                raise InvalidDataError(
                    "Correlation matrix contains NaN values. "
                    "This may indicate zero-variance columns or invalid residuals."
                )
            if np.any(eigenvals < min_eigval):
                deficit = min_eigval - eigenvals[eigenvals < min_eigval].min()
                conditioning = np.eye(n_targets) * deficit
                self.correlation_matrix_ = self.correlation_matrix_ + conditioning

                try:
                    np.linalg.cholesky(self.correlation_matrix_)
                except np.linalg.LinAlgError:
                    raise InvalidDataError(
                        "Correlation matrix is too ill-conditioned to condition. "
                        "This may indicate very high correlation between targets."
                    )
                eigenvals = np.linalg.eigvals(self.correlation_matrix_)

            if np.any(eigenvals < min_eigval):
                raise InvalidDataError(
                    f"Correlation matrix is too ill-conditioned. "
                    f"Minimum eigenvalue: {np.min(eigenvals):.2e}, "
                    f"Threshold: {min_eigval:.2e}. "
                    f"This may indicate very high correlation between targets."
                )
        except np.linalg.LinAlgError as e:
            raise InvalidDataError(
                f"Failed to validate correlation matrix: {e}. "
                f"Check for linear dependencies in your data."
            )

        self.fitted_ = True
        self.theta_ = 0.0

        return self

    def log_likelihood(self, residuals: np.ndarray) -> float:
        """Compute log-likelihood for BIC calculation."""
        self._validate_fitted()

        uniform = self._to_copula_space(residuals)

        try:
            cov = self.correlation_matrix_
            if cov is None:
                return -np.inf
            mvn = stats.multivariate_normal(cov=cov)
            ll = np.sum(mvn.logpdf(stats.norm.ppf(uniform)))
        except (np.linalg.LinAlgError, ValueError):
            ll = -np.inf

        return float(ll)

    def sample(
        self,
        marginals: np.ndarray,
        n_samples: int = 1000,
        quantile_levels: np.ndarray | None = None,
        random_state: int | np.random.Generator | None = None,
    ) -> np.ndarray:
        """
        Generate joint samples from copula.

        Args:
            marginals: Quantile predictions for each target
                      shape (n_samples_input, n_targets, n_quantiles)
            n_samples: Number of Monte Carlo samples to generate

        Returns:
            Joint samples shape (n_samples, n_targets)
        """
        self._validate_fitted()

        n_samples_input, n_targets, _ = marginals.shape
        rng = _resolve_rng(random_state)

        try:
            cov = self.correlation_matrix_
            if cov is None:
                raise ModelNotFittedError("GaussianCopula")
            normal_samples = stats.multivariate_normal.rvs(
                mean=np.zeros(n_targets),
                cov=cov,
                size=n_samples_input * n_samples,
                random_state=rng,
            )
        except (np.linalg.LinAlgError, ValueError):
            # Fallback for singular correlation matrices
            cov = self.correlation_matrix_
            if cov is None:
                raise ModelNotFittedError("GaussianCopula")
            assert cov is not None
            normal_samples = rng.multivariate_normal(
                mean=np.zeros(n_targets),
                cov=cov,
                size=n_samples_input * n_samples,
            )

        normal_samples = np.asarray(normal_samples).reshape(n_samples_input, n_samples, n_targets)
        uniform_samples = stats.norm.cdf(normal_samples)
        return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

    def __repr__(self) -> str:
        if self.fitted_:
            assert self.correlation_matrix_ is not None
            n_targets = self.correlation_matrix_.shape[0]
            return f"GaussianCopula(n_targets={n_targets}, fitted=True)"
        return "GaussianCopula(fitted=False)"

fit(residuals)

Fit copula on residual matrix.

Parameters:

Name Type Description Default
residuals ndarray

Residual matrix shape (n_samples, n_targets)

 required

Returns:

Type Description
'GaussianCopula'

self (for method chaining)
Source code in uncertainty_flow/multivariate/copula.py 
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def fit(self, residuals: np.ndarray) -> "GaussianCopula":
    """
    Fit copula on residual matrix.

    Args:
        residuals: Residual matrix shape (n_samples, n_targets)

    Returns:
        self (for method chaining)
    """
    if residuals.ndim != 2:
        raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")

    n_samples, n_targets = residuals.shape

    # Check for constant (zero-variance) columns before computing correlation
    constant_cols = []
    for t in range(n_targets):
        if np.all(residuals[:, t] == residuals[0, t]):
            constant_cols.append(t)

    if len(constant_cols) > 0:
        raise InvalidDataError(
            f"Target columns at indices {constant_cols} are constant (zero variance). "
            "Cannot compute correlation matrix. Check if target values vary across samples."
        )

    self.correlation_matrix_ = np.corrcoef(residuals.T)

    # Condition correlation matrix for numerical stability
    min_eigval = 1e-6
    try:
        eigenvals = np.linalg.eigvals(self.correlation_matrix_)
        if np.any(np.isnan(eigenvals)):
            raise InvalidDataError(
                "Correlation matrix contains NaN values. "
                "This may indicate zero-variance columns or invalid residuals."
            )
        if np.any(eigenvals < min_eigval):
            deficit = min_eigval - eigenvals[eigenvals < min_eigval].min()
            conditioning = np.eye(n_targets) * deficit
            self.correlation_matrix_ = self.correlation_matrix_ + conditioning

            try:
                np.linalg.cholesky(self.correlation_matrix_)
            except np.linalg.LinAlgError:
                raise InvalidDataError(
                    "Correlation matrix is too ill-conditioned to condition. "
                    "This may indicate very high correlation between targets."
                )
            eigenvals = np.linalg.eigvals(self.correlation_matrix_)

        if np.any(eigenvals < min_eigval):
            raise InvalidDataError(
                f"Correlation matrix is too ill-conditioned. "
                f"Minimum eigenvalue: {np.min(eigenvals):.2e}, "
                f"Threshold: {min_eigval:.2e}. "
                f"This may indicate very high correlation between targets."
            )
    except np.linalg.LinAlgError as e:
        raise InvalidDataError(
            f"Failed to validate correlation matrix: {e}. "
            f"Check for linear dependencies in your data."
        )

    self.fitted_ = True
    self.theta_ = 0.0

    return self

log_likelihood(residuals)

Compute log-likelihood for BIC calculation.

Source code in uncertainty_flow/multivariate/copula.py 
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def log_likelihood(self, residuals: np.ndarray) -> float:
    """Compute log-likelihood for BIC calculation."""
    self._validate_fitted()

    uniform = self._to_copula_space(residuals)

    try:
        cov = self.correlation_matrix_
        if cov is None:
            return -np.inf
        mvn = stats.multivariate_normal(cov=cov)
        ll = np.sum(mvn.logpdf(stats.norm.ppf(uniform)))
    except (np.linalg.LinAlgError, ValueError):
        ll = -np.inf

    return float(ll)

sample(marginals, n_samples=1000, quantile_levels=None, random_state=None)

Generate joint samples from copula.

Parameters:

Name Type Description Default
marginals ndarray

Quantile predictions for each target shape (n_samples_input, n_targets, n_quantiles)

 required
n_samples int

Number of Monte Carlo samples to generate

 1000

Returns:

Type Description
ndarray

Joint samples shape (n_samples, n_targets)
Source code in uncertainty_flow/multivariate/copula.py 
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def sample(
    self,
    marginals: np.ndarray,
    n_samples: int = 1000,
    quantile_levels: np.ndarray | None = None,
    random_state: int | np.random.Generator | None = None,
) -> np.ndarray:
    """
    Generate joint samples from copula.

    Args:
        marginals: Quantile predictions for each target
                  shape (n_samples_input, n_targets, n_quantiles)
        n_samples: Number of Monte Carlo samples to generate

    Returns:
        Joint samples shape (n_samples, n_targets)
    """
    self._validate_fitted()

    n_samples_input, n_targets, _ = marginals.shape
    rng = _resolve_rng(random_state)

    try:
        cov = self.correlation_matrix_
        if cov is None:
            raise ModelNotFittedError("GaussianCopula")
        normal_samples = stats.multivariate_normal.rvs(
            mean=np.zeros(n_targets),
            cov=cov,
            size=n_samples_input * n_samples,
            random_state=rng,
        )
    except (np.linalg.LinAlgError, ValueError):
        # Fallback for singular correlation matrices
        cov = self.correlation_matrix_
        if cov is None:
            raise ModelNotFittedError("GaussianCopula")
        assert cov is not None
        normal_samples = rng.multivariate_normal(
            mean=np.zeros(n_targets),
            cov=cov,
            size=n_samples_input * n_samples,
        )

    normal_samples = np.asarray(normal_samples).reshape(n_samples_input, n_samples, n_targets)
    uniform_samples = stats.norm.cdf(normal_samples)
    return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

GumbelCopula

Bases: BaseCopula

Gumbel copula for modeling upper tail dependence.

Upper tail dependence: extreme high values tend to co-occur. Suited for simultaneous extreme high values (e.g., extreme rainfall).

Examples:

>>> import numpy as np
>>> residuals = np.array([
...     [-1, -2],
...     [-0.5, -1],
...     [0, 0],
...     [0.5, 1],
...     [1, 2],
... ])
>>> copula = GumbelCopula()
>>> copula.fit(residuals)
>>> copula.theta_ >= 1  # theta >= 1 indicates upper tail dependence
True
Source code in uncertainty_flow/multivariate/copula.py 
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class GumbelCopula(BaseCopula):
    """
    Gumbel copula for modeling upper tail dependence.

    Upper tail dependence: extreme high values tend to co-occur.
    Suited for simultaneous extreme high values (e.g., extreme rainfall).

    Examples:
        >>> import numpy as np
        >>> residuals = np.array([
        ...     [-1, -2],
        ...     [-0.5, -1],
        ...     [0, 0],
        ...     [0.5, 1],
        ...     [1, 2],
        ... ])
        >>> copula = GumbelCopula()
        >>> copula.fit(residuals)
        >>> copula.theta_ >= 1  # theta >= 1 indicates upper tail dependence
        True
    """

    name = "gumbel"
    has_lower_tail = False
    has_upper_tail = True

    def fit(self, residuals: np.ndarray) -> "GumbelCopula":
        """
        Fit Gumbel copula via maximum likelihood.

        Args:
            residuals: Residual matrix shape (n_samples, 2) — bivariate only

        Returns:
            self (for method chaining)
        """
        if residuals.ndim != 2:
            raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")

        if residuals.shape[1] != 2:
            raise InvalidDataError(
                f"GumbelCopula supports bivariate only, got {residuals.shape[1]} dimensions"
            )

        uniform = self._to_copula_space(residuals)
        u, v = uniform[:, 0], uniform[:, 1]

        def neg_log_likelihood(theta: float) -> float:
            if theta < 1:
                return 1e10
            try:
                t = (-np.log(u)) ** theta + (-np.log(v)) ** theta
                h = np.exp(-(t ** (1 / theta)))
                psi = -np.log(-np.log(u)) - np.log(-np.log(v))
                ll = t ** (1 / theta) + psi - np.log(h)
                if not np.isfinite(ll).all():
                    return 1e10
                return float(-np.sum(ll))
            except (ValueError, OverflowError, ZeroDivisionError):
                return 1e10

        from scipy.optimize import minimize_scalar

        result = minimize_scalar(neg_log_likelihood, bounds=(1.0, 10), method="bounded")
        self.theta_ = float(result.x)
        self.fitted_ = True

        return self

    def log_likelihood(self, residuals: np.ndarray) -> float:
        """Compute log-likelihood for BIC calculation."""
        self._validate_fitted()

        uniform = self._to_copula_space(residuals)
        u, v = uniform[:, 0], uniform[:, 1]
        theta = self.theta_
        assert theta is not None

        try:
            t = (-np.log(u)) ** theta + (-np.log(v)) ** theta
            h = np.exp(-(t ** (1 / theta)))
            psi = -np.log(-np.log(u)) - np.log(-np.log(v))
            ll = t ** (1 / theta) + psi - np.log(h)
            return float(np.sum(ll))
        except (ValueError, OverflowError, ZeroDivisionError):
            return -np.inf

    def sample(
        self,
        marginals: np.ndarray,
        n_samples: int = 1000,
        quantile_levels: np.ndarray | None = None,
        random_state: int | np.random.Generator | None = None,
    ) -> np.ndarray:
        """
        Generate joint samples from Gumbel copula.

        Args:
            marginals: Quantile predictions for each target
                      shape (n_samples_input, n_targets, n_quantiles)
            n_samples: Number of Monte Carlo samples to generate

        Returns:
            Joint samples shape (n_samples, n_targets)
        """
        self._validate_fitted()

        if marginals.shape[1] != 2:
            raise InvalidDataError(
                f"GumbelCopula supports bivariate only, got {marginals.shape[1]} dimensions"
            )

        n_samples_input, n_targets, _ = marginals.shape
        theta = self.theta_
        assert theta is not None
        rng = _resolve_rng(random_state)

        s1 = rng.uniform(0, 1, size=(n_samples_input, n_samples))
        s2 = rng.uniform(0, 1, size=(n_samples_input, n_samples))

        u = s1
        v = _solve_gumbel_conditional(s1, s2, theta)

        uniform_samples = np.stack([u, v], axis=-1)
        return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

    def __repr__(self) -> str:
        if self.fitted_:
            return f"GumbelCopula(theta={self.theta_:.3f}, fitted=True)"
        return "GumbelCopula(fitted=False)"

fit(residuals)

Fit Gumbel copula via maximum likelihood.

Parameters:

Name Type Description Default
residuals ndarray

Residual matrix shape (n_samples, 2) — bivariate only

 required

Returns:

Type Description
'GumbelCopula'

self (for method chaining)
Source code in uncertainty_flow/multivariate/copula.py 
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def fit(self, residuals: np.ndarray) -> "GumbelCopula":
    """
    Fit Gumbel copula via maximum likelihood.

    Args:
        residuals: Residual matrix shape (n_samples, 2) — bivariate only

    Returns:
        self (for method chaining)
    """
    if residuals.ndim != 2:
        raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")

    if residuals.shape[1] != 2:
        raise InvalidDataError(
            f"GumbelCopula supports bivariate only, got {residuals.shape[1]} dimensions"
        )

    uniform = self._to_copula_space(residuals)
    u, v = uniform[:, 0], uniform[:, 1]

    def neg_log_likelihood(theta: float) -> float:
        if theta < 1:
            return 1e10
        try:
            t = (-np.log(u)) ** theta + (-np.log(v)) ** theta
            h = np.exp(-(t ** (1 / theta)))
            psi = -np.log(-np.log(u)) - np.log(-np.log(v))
            ll = t ** (1 / theta) + psi - np.log(h)
            if not np.isfinite(ll).all():
                return 1e10
            return float(-np.sum(ll))
        except (ValueError, OverflowError, ZeroDivisionError):
            return 1e10

    from scipy.optimize import minimize_scalar

    result = minimize_scalar(neg_log_likelihood, bounds=(1.0, 10), method="bounded")
    self.theta_ = float(result.x)
    self.fitted_ = True

    return self

log_likelihood(residuals)

Compute log-likelihood for BIC calculation.

Source code in uncertainty_flow/multivariate/copula.py 
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def log_likelihood(self, residuals: np.ndarray) -> float:
    """Compute log-likelihood for BIC calculation."""
    self._validate_fitted()

    uniform = self._to_copula_space(residuals)
    u, v = uniform[:, 0], uniform[:, 1]
    theta = self.theta_
    assert theta is not None

    try:
        t = (-np.log(u)) ** theta + (-np.log(v)) ** theta
        h = np.exp(-(t ** (1 / theta)))
        psi = -np.log(-np.log(u)) - np.log(-np.log(v))
        ll = t ** (1 / theta) + psi - np.log(h)
        return float(np.sum(ll))
    except (ValueError, OverflowError, ZeroDivisionError):
        return -np.inf

sample(marginals, n_samples=1000, quantile_levels=None, random_state=None)

Generate joint samples from Gumbel copula.

Parameters:

Name Type Description Default
marginals ndarray

Quantile predictions for each target shape (n_samples_input, n_targets, n_quantiles)

 required
n_samples int

Number of Monte Carlo samples to generate

 1000

Returns:

Type Description
ndarray

Joint samples shape (n_samples, n_targets)
Source code in uncertainty_flow/multivariate/copula.py 
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def sample(
    self,
    marginals: np.ndarray,
    n_samples: int = 1000,
    quantile_levels: np.ndarray | None = None,
    random_state: int | np.random.Generator | None = None,
) -> np.ndarray:
    """
    Generate joint samples from Gumbel copula.

    Args:
        marginals: Quantile predictions for each target
                  shape (n_samples_input, n_targets, n_quantiles)
        n_samples: Number of Monte Carlo samples to generate

    Returns:
        Joint samples shape (n_samples, n_targets)
    """
    self._validate_fitted()

    if marginals.shape[1] != 2:
        raise InvalidDataError(
            f"GumbelCopula supports bivariate only, got {marginals.shape[1]} dimensions"
        )

    n_samples_input, n_targets, _ = marginals.shape
    theta = self.theta_
    assert theta is not None
    rng = _resolve_rng(random_state)

    s1 = rng.uniform(0, 1, size=(n_samples_input, n_samples))
    s2 = rng.uniform(0, 1, size=(n_samples_input, n_samples))

    u = s1
    v = _solve_gumbel_conditional(s1, s2, theta)

    uniform_samples = np.stack([u, v], axis=-1)
    return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

PairwiseChainCopula

Bases: BaseCopula

Chain copula for d-dimensional dependence (d >= 2).

Decomposes a d-dimensional joint into d-1 bivariate copulas arranged in a chain (a simplified vine). Each pair captures dependence between consecutive targets. Supports tail dependence via Archimedean families.

Examples:

>>> import numpy as np
>>> residuals = np.column_stack([
...     np.random.randn(50),
...     np.random.randn(50) + 0.5 * np.random.randn(50),
...     np.random.randn(50) + 0.3 * np.random.randn(50),
... ])
>>> copula = PairwiseChainCopula()
>>> copula.fit(residuals)
>>> copula.fitted_
True
Source code in uncertainty_flow/multivariate/copula.py 
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class PairwiseChainCopula(BaseCopula):
    """
    Chain copula for d-dimensional dependence (d >= 2).

    Decomposes a d-dimensional joint into d-1 bivariate copulas arranged in a
    chain (a simplified vine).  Each pair captures dependence between
    consecutive targets.  Supports tail dependence via Archimedean families.

    Examples:
        >>> import numpy as np
        >>> residuals = np.column_stack([
        ...     np.random.randn(50),
        ...     np.random.randn(50) + 0.5 * np.random.randn(50),
        ...     np.random.randn(50) + 0.3 * np.random.randn(50),
        ... ])
        >>> copula = PairwiseChainCopula()
        >>> copula.fit(residuals)
        >>> copula.fitted_
        True
    """

    name = "pairwise_chain"
    has_lower_tail = False
    has_upper_tail = False

    def __init__(self):
        super().__init__()
        self._pair_copulas: list[BaseCopula] = []
        self._n_targets: int = 0

    def fit(self, residuals: np.ndarray) -> PairwiseChainCopula:
        """
        Fit the chain by learning a bivariate copula for each consecutive pair.

        Args:
            residuals: (n_samples, d) residual matrix with d >= 2.

        Returns:
            self
        """
        if residuals.ndim != 2:
            raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")
        if residuals.shape[1] < 2:
            raise InvalidDataError("PairwiseChainCopula requires at least 2 targets")

        n_targets = residuals.shape[1]
        self._n_targets = n_targets
        self._pair_copulas = []

        for i in range(n_targets - 1):
            pair_data = residuals[:, [i, i + 1]]
            bivariate_families = [f for f in COPULA_FAMILIES if f != "pairwise_chain"]
            best_family = auto_select_copula(pair_data, families=bivariate_families)
            pair_copula = COPULA_FAMILIES[best_family]()
            pair_copula.fit(pair_data)
            self._pair_copulas.append(pair_copula)

            if pair_copula.has_lower_tail:
                self.has_lower_tail = True
            if pair_copula.has_upper_tail:
                self.has_upper_tail = True

        self.theta_ = 0.0
        self.fitted_ = True
        return self

    def log_likelihood(self, residuals: np.ndarray) -> float:
        """Sum of log-likelihoods across all pairs."""
        self._validate_fitted()
        total = 0.0
        for i, pc in enumerate(self._pair_copulas):
            pair_data = residuals[:, [i, i + 1]]
            total += pc.log_likelihood(pair_data)
        return total

    def sample(
        self,
        marginals: np.ndarray,
        n_samples: int = 1000,
        quantile_levels: np.ndarray | None = None,
        random_state: int | np.random.Generator | None = None,
    ) -> np.ndarray:
        """
        Sequential conditional sampling through the chain.

        Draws u_1 ~ Uniform, then for each subsequent target draws
        u_{i+1} conditional on u_i using the i-th pair copula.

        Args:
            marginals: (n_input_rows, d, n_quantiles) quantile predictions.
            n_samples: Monte Carlo samples per input row.
            quantile_levels: Quantile levels.
            random_state: Random seed.

        Returns:
            Mapped samples via _inverse_from_marginals.
        """
        self._validate_fitted()

        n_rows, d, n_q = marginals.shape
        rng = _resolve_rng(random_state)

        uniform_samples = np.empty((n_rows, n_samples, d))
        uniform_samples[:, :, 0] = rng.uniform(0, 1, size=(n_rows, n_samples))

        for i in range(d - 1):
            pair_copula = self._pair_copulas[i]
            prev_u = uniform_samples[:, :, i]
            w = rng.uniform(0, 1, size=(n_rows, n_samples))

            if isinstance(pair_copula, GaussianCopula):
                corr = pair_copula.correlation_matrix_
                if corr is not None:
                    rho = corr[0, 1]
                else:
                    rho = 0.0
                z_prev = stats.norm.ppf(np.clip(prev_u, 1e-10, 1 - 1e-10))
                z_w = stats.norm.ppf(np.clip(w, 1e-10, 1 - 1e-10))
                z_cond = rho * z_prev + np.sqrt(max(1 - rho**2, 1e-10)) * z_w
                uniform_samples[:, :, i + 1] = stats.norm.cdf(z_cond)
            elif isinstance(pair_copula, ClaytonCopula):
                theta = pair_copula.theta_ if pair_copula.theta_ is not None else 0.01
                a = prev_u ** (-theta) + w ** (-theta / (theta + 1)) - 1
                uniform_samples[:, :, i + 1] = np.where(a > 0, a ** (-1 / theta), 1e-10)
            elif isinstance(pair_copula, GumbelCopula):
                prev_clipped = np.clip(prev_u, 1e-10, 1 - 1e-10)
                theta_val = pair_copula.theta_ if pair_copula.theta_ is not None else 1.01
                uniform_samples[:, :, i + 1] = _solve_gumbel_conditional(
                    prev_clipped,
                    w,
                    theta_val,
                )
            elif isinstance(pair_copula, FrankCopula):
                theta = pair_copula.theta_ if pair_copula.theta_ is not None else 0.01
                u_clipped = np.clip(prev_u, 1e-10, 1 - 1e-10)
                denom = np.exp(-theta * u_clipped) * (1 - w) + w
                uniform_samples[:, :, i + 1] = -np.log(1 + w * (np.exp(-theta) - 1) / denom) / theta
            else:
                uniform_samples[:, :, i + 1] = w

        return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

    def __repr__(self) -> str:
        if self.fitted_:
            families = [type(pc).__name__ for pc in self._pair_copulas]
            return f"PairwiseChainCopula(pairs={families}, fitted=True)"
        return "PairwiseChainCopula(fitted=False)"

fit(residuals)

Fit the chain by learning a bivariate copula for each consecutive pair.

Parameters:

Name Type Description Default
residuals ndarray

(n_samples, d) residual matrix with d >= 2.

 required

Returns:

Type Description
PairwiseChainCopula

self
Source code in uncertainty_flow/multivariate/copula.py 
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def fit(self, residuals: np.ndarray) -> PairwiseChainCopula:
    """
    Fit the chain by learning a bivariate copula for each consecutive pair.

    Args:
        residuals: (n_samples, d) residual matrix with d >= 2.

    Returns:
        self
    """
    if residuals.ndim != 2:
        raise InvalidDataError(f"residuals must be 2D, got shape {residuals.shape}")
    if residuals.shape[1] < 2:
        raise InvalidDataError("PairwiseChainCopula requires at least 2 targets")

    n_targets = residuals.shape[1]
    self._n_targets = n_targets
    self._pair_copulas = []

    for i in range(n_targets - 1):
        pair_data = residuals[:, [i, i + 1]]
        bivariate_families = [f for f in COPULA_FAMILIES if f != "pairwise_chain"]
        best_family = auto_select_copula(pair_data, families=bivariate_families)
        pair_copula = COPULA_FAMILIES[best_family]()
        pair_copula.fit(pair_data)
        self._pair_copulas.append(pair_copula)

        if pair_copula.has_lower_tail:
            self.has_lower_tail = True
        if pair_copula.has_upper_tail:
            self.has_upper_tail = True

    self.theta_ = 0.0
    self.fitted_ = True
    return self

log_likelihood(residuals)

Sum of log-likelihoods across all pairs.

Source code in uncertainty_flow/multivariate/copula.py 
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def log_likelihood(self, residuals: np.ndarray) -> float:
    """Sum of log-likelihoods across all pairs."""
    self._validate_fitted()
    total = 0.0
    for i, pc in enumerate(self._pair_copulas):
        pair_data = residuals[:, [i, i + 1]]
        total += pc.log_likelihood(pair_data)
    return total

sample(marginals, n_samples=1000, quantile_levels=None, random_state=None)

Sequential conditional sampling through the chain.

Draws u_1 ~ Uniform, then for each subsequent target draws u_{i+1} conditional on u_i using the i-th pair copula.

Parameters:

Name Type Description Default
marginals ndarray

(n_input_rows, d, n_quantiles) quantile predictions.

 required
n_samples int

Monte Carlo samples per input row.

 1000
quantile_levels ndarray | None

Quantile levels.

 None
random_state int | Generator | None

Random seed.

 None

Returns:

Type Description
ndarray

Mapped samples via _inverse_from_marginals.
Source code in uncertainty_flow/multivariate/copula.py 
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def sample(
    self,
    marginals: np.ndarray,
    n_samples: int = 1000,
    quantile_levels: np.ndarray | None = None,
    random_state: int | np.random.Generator | None = None,
) -> np.ndarray:
    """
    Sequential conditional sampling through the chain.

    Draws u_1 ~ Uniform, then for each subsequent target draws
    u_{i+1} conditional on u_i using the i-th pair copula.

    Args:
        marginals: (n_input_rows, d, n_quantiles) quantile predictions.
        n_samples: Monte Carlo samples per input row.
        quantile_levels: Quantile levels.
        random_state: Random seed.

    Returns:
        Mapped samples via _inverse_from_marginals.
    """
    self._validate_fitted()

    n_rows, d, n_q = marginals.shape
    rng = _resolve_rng(random_state)

    uniform_samples = np.empty((n_rows, n_samples, d))
    uniform_samples[:, :, 0] = rng.uniform(0, 1, size=(n_rows, n_samples))

    for i in range(d - 1):
        pair_copula = self._pair_copulas[i]
        prev_u = uniform_samples[:, :, i]
        w = rng.uniform(0, 1, size=(n_rows, n_samples))

        if isinstance(pair_copula, GaussianCopula):
            corr = pair_copula.correlation_matrix_
            if corr is not None:
                rho = corr[0, 1]
            else:
                rho = 0.0
            z_prev = stats.norm.ppf(np.clip(prev_u, 1e-10, 1 - 1e-10))
            z_w = stats.norm.ppf(np.clip(w, 1e-10, 1 - 1e-10))
            z_cond = rho * z_prev + np.sqrt(max(1 - rho**2, 1e-10)) * z_w
            uniform_samples[:, :, i + 1] = stats.norm.cdf(z_cond)
        elif isinstance(pair_copula, ClaytonCopula):
            theta = pair_copula.theta_ if pair_copula.theta_ is not None else 0.01
            a = prev_u ** (-theta) + w ** (-theta / (theta + 1)) - 1
            uniform_samples[:, :, i + 1] = np.where(a > 0, a ** (-1 / theta), 1e-10)
        elif isinstance(pair_copula, GumbelCopula):
            prev_clipped = np.clip(prev_u, 1e-10, 1 - 1e-10)
            theta_val = pair_copula.theta_ if pair_copula.theta_ is not None else 1.01
            uniform_samples[:, :, i + 1] = _solve_gumbel_conditional(
                prev_clipped,
                w,
                theta_val,
            )
        elif isinstance(pair_copula, FrankCopula):
            theta = pair_copula.theta_ if pair_copula.theta_ is not None else 0.01
            u_clipped = np.clip(prev_u, 1e-10, 1 - 1e-10)
            denom = np.exp(-theta * u_clipped) * (1 - w) + w
            uniform_samples[:, :, i + 1] = -np.log(1 + w * (np.exp(-theta) - 1) / denom) / theta
        else:
            uniform_samples[:, :, i + 1] = w

    return _inverse_from_marginals(marginals, uniform_samples, quantile_levels)

auto_select_copula(residuals, families=None)

Select best copula family via BIC.

Parameters:

Name Type Description Default
residuals ndarray

Residual matrix shape (n_samples, n_targets)

 required
families list[str] | None

List of copula families to consider. Defaults to all available families.

 None

Returns:

Type Description
str

Name of selected copula family

Examples:

>>> import numpy as np
>>> residuals = np.array([
...     [1, 10],
...     [2, 20],
...     [3, 30],
... ])
>>> selected = auto_select_copula(residuals)
>>> selected in ["gaussian", "clayton", "gumbel", "frank"]
True
Source code in uncertainty_flow/multivariate/copula.py 
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def auto_select_copula(
    residuals: np.ndarray,
    families: list[str] | None = None,
) -> str:
    """
    Select best copula family via BIC.

    Args:
        residuals: Residual matrix shape (n_samples, n_targets)
        families: List of copula families to consider.
                  Defaults to all available families.

    Returns:
        Name of selected copula family

    Examples:
        >>> import numpy as np
        >>> residuals = np.array([
        ...     [1, 10],
        ...     [2, 20],
        ...     [3, 30],
        ... ])
        >>> selected = auto_select_copula(residuals)
        >>> selected in ["gaussian", "clayton", "gumbel", "frank"]
        True
    """
    if families is None:
        families = list(COPULA_FAMILIES.keys())

    n_samples = residuals.shape[0]
    n_targets = residuals.shape[1]
    best_bic = np.inf
    best_family = "gaussian"

    if n_targets > 2:
        warnings.warn(
            f"Auto-selecting copula for {n_targets}D data. "
            f"Only Gaussian copula supports dimensions > 2. [UF-W006]",
            UncertaintyFlowWarning,
            stacklevel=3,
        )

    for family_name in families:
        copula_cls = COPULA_FAMILIES[family_name]

        if n_targets > 2 and family_name in ("clayton", "gumbel", "frank"):
            continue

        try:
            copula = copula_cls()
            copula.fit(residuals)
            ll = copula.log_likelihood(residuals)

            if family_name == "gaussian":
                n_params = n_targets * (n_targets - 1) // 2
            elif family_name == "pairwise_chain":
                n_params = n_targets - 1
            else:
                n_params = 1
            bic = n_params * np.log(n_samples) - 2 * ll

            if bic < best_bic:
                best_bic = bic
                best_family = family_name

        except (ValueError, OverflowError, np.linalg.LinAlgError):
            continue

    return best_family