Student T

systematica.models.arbitrage_index.student_t

StudentT

StudentT(
    kendall_tau: float,
    rotation: str | systematica.models.arbitrage_index.utils.BaseCopulaRotation = BaseCopulaRotation.R0,
    start: float = 0.001,
    stop: float = 0.999,
    num: int = 100,
)

Bivariate Student’s t copula estimator.

The copula is defined as C_(u, v) = T_(t_v^(u), t_v^(v)), where t_v^ is the univariate t-quantile and T_ is the bivariate t CDF. For numerical stability we constrain the degrees of freedom v to [2, 50]: values below 2 produce extremely heavy tails, while values above 50 are effectively Gaussian.

Notes:

rho: Fitted correlation parameter (in (-1, 1)). ddof: Fitted degrees of freedom (in [2, 50]).

See Also:

Elliptical copulas (Gaussian, Student-t) are symmetric and do not require rotation variants.

Method generated by attrs for class StudentT.

Ancestors

• systematica.models.arbitrage_index.base.BaseCopula
• abc.ABC

Instance variables

• lower_tail_dependence: float: Theoretical lower tail dependence coefficient.

• upper_tail_dependence: float: Theoretical upper tail dependence coefficient.

• ddof: float: Degree of freedom. If set to auto, applies best fitted degree of freedom. Defaults to auto.

• ddof_: float: Best fitted degree of freedom applied to the copula model.

Methods

set_ddof

set_ddof(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
) ‑> float

Set degree of freedom.

Returns:

Type	Description
float	Degree of freedom value.

Raises:

Type	Description
AssertionError	Degree of freedom must be between 1 and 50.

density

density(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
) ‑> numpy.ndarray

Calculate log probability density of the bivariate copula: $P(U=u, V=v)$.

Parameters:

Name	Type	Default	Description
u	tp.Array1d	--	First uniform marginal.
v	tp.Array1d	--	Second uniform marginal.

Returns:

Type	Description
tp.Array1d	Log probability density.

cumulative_density

cumulative_density(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
) ‑> numpy.ndarray

Calculate cumulative density of the bivariate copula: $P(U<=u, V<=v)$.

Parameters:

Name	Type	Default	Description
u	tp.Array1d	--	First uniform marginal.
v	tp.Array1d	--	Second uniform marginal.

Returns:

Type	Description
tp.Array1d	Cumulative probability density.

arbitrage

arbitrage(
    self,
    ui: float,
    vi: float,
) ‑> Tuple[float, float]

Compute the h-function (partial derivative) for the bivariate Clayton copula, a.k.a. the mispricing index, for every time step in the trading period using the estimated copula.

Parameters:

Name	Type	Default	Description
ui	float	--	The first component of the bivariate input.
vi	float	--	The second component of the bivariate input.

Returns:

Type	Description
tp.Tuple[float, float]:	The mispricing indices for the copula, a.k.a. arbitrage.

partial_derivative

partial_derivative(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
) ‑> numpy.ndarray

Compute the h-function (partial derivative) for the bivariate Clayton copula, a.k.a. the mispricing index, for every time step in the trading period using the estimated copula.

Parameters:

Name	Type	Default	Description
u	tp.Array1d	--	The first component of the bivariate input.
v	tp.Array1d	--	The second component of the bivariate input.

Returns:

Type	Description
tp.Array2d	The mispricing indices for the copula, a.k.a. arbitrage.

Raises:

Type	Description
AssertionError	u and v must have the same shape.

score

score(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
) ‑> numpy.ndarray

Compute the log-likelihood score of each sample (log-pdf) under the model.

u and v are bivariate inputs (u, v) where each row represents a bivariate observation. Both u and v must be in the interval [0, 1], having been transformed to uniform marginals.

Parameters:

Name	Type	Default	Description
u	tp.Array1d	--	The first component of the bivariate input.
v	tp.Array1d	--	The second component of the bivariate input.

Returns:

Type	Description
tp.Array1d	The log-likelihood score of each sample under the fitted copula.

Raises:

Type	Description
AssertionError	u and v must have the same shape.