Skew Normal¶

This model is a univariate skewed normal distribution motivated by Azzalini et al. (2003), “Graphical models for skew-normal variates” and is defined by

\[x \in N(0, 0.5)\]

\[\mu = a*x+b\]

\[\sigma = c*x^2+d\]

\[\alpha = skew_{high} + sigmoid(x) * (skew_{high}-skew_{low})\]

\[y = 2*PDF(x;\mu,\sigma^2)*CDF(x\alpha)\]

While PDF and CDF referring to the probability density and cumulative distribution function of a normally distributed random variable with \(skew_{high} = -4\) and \(skew_{low}=-0.1\).

class cde.density_simulation.SkewNormal(random_seed=None)[source]¶

This model represents a univariate skewed normal distribution.

cdf(X, Y)[source]¶

Conditional cumulated probability density function P(Y < y | x) of the underlying probability model

Parameters

X – x to be conditioned on - numpy array of shape (n_points, ndim_x)
Y – y target values for witch the cdf shall be evaluated - numpy array of shape (n_points, ndim_y)

Returns

P(Y < y | x) cumulated density values for the provided X and Y - numpy array of shape (n_points, )

conditional_value_at_risk(x_cond, alpha=0.01, n_samples=1000000)¶

Computes the Conditional Value-at-Risk (CVaR) / Expected Shortfall of the fitted distribution. Only if ndim_y = 1

Parameters

x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)
alpha – quantile percentage of the distribution
n_samples – number of samples for monte carlo model_fitting

Returns

CVaR values for each x to condition on - numpy array of shape (n_values)

covariance(x_cond, n_samples=1000000)¶

Covariance of the fitted distribution conditioned on x_cond

Parameters

x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)
n_samples – number of samples for monte carlo model_fitting

Returns

Covariances Cov[y|x] corresponding to x_cond - numpy array of shape (n_values, ndim_y, ndim_y)

get_params(deep=True)¶

Get parameters for this estimator.

Parameters: deep (boolean, optional) – If True, will return the parameters for this estimator and contained subobjects that are estimators.
Returns: params – Parameter names mapped to their values.
Return type: mapping of string to any

kurtosis(x_cond, n_samples=1000000)¶

Kurtosis of the fitted distribution conditioned on x_cond

Parameters: x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)
Returns: Kurtosis Kurt[y|x] corresponding to x_cond - numpy array of shape (n_values, ndim_y, ndim_y)

log_pdf(X, Y)¶

Conditional log-probability log p(y|x). Requires the model to be fitted.

Parameters

X – numpy array to be conditioned on - shape: (n_samples, n_dim_x)
Y – numpy array of y targets - shape: (n_samples, n_dim_y)

Returns

conditional log-probability log p(y|x) - numpy array of shape (n_query_samples, )

mean_(x_cond, n_samples=None)[source]¶

Conditional mean of the distribution :param x_cond: different x values to condition on - numpy array of shape (n_values, ndim_x)

Returns: Means E[y|x] corresponding to x_cond - numpy array of shape (n_values, ndim_y)

pdf(X, Y)[source]¶

Conditional probability density function p(y|x) of the underlying probability model (

Args:
X: x to be conditioned on - numpy array of shape (n_points, ndim_x) Y: y target values for witch the pdf shall be evaluated - numpy array of shape (n_points, ndim_y)

Returns:
p(X|Y) conditional density values for the provided X and Y - numpy array of shape (n_points, )

plot(xlim=(-5, 5), ylim=(-5, 5), resolution=100, mode='pdf', show=False, numpyfig=False)¶

Plots the distribution specified in mode if x and y are 1-dimensional each

Parameters

xlim – 2-tuple specifying the x axis limits
ylim – 2-tuple specifying the y axis limits
resolution – integer specifying the resolution of plot
mode – spefify which dist to plot [“pdf”, “cdf”, “joint_pdf”]

plot2d(x_cond=[0, 1, 2], ylim=(-8, 8), resolution=100, mode='pdf', show=True, prefix='', numpyfig=False)¶

Generates a 3d surface plot of the fitted conditional distribution if x and y are 1-dimensional each

Parameters

xlim – 2-tuple specifying the x axis limits
ylim – 2-tuple specifying the y axis limits
resolution – integer specifying the resolution of plot

plot3d(xlim=(-5, 5), ylim=(-8, 8), resolution=100, show=False, numpyfig=False)¶

Generates a 3d surface plot of the fitted conditional distribution if x and y are 1-dimensional each

Parameters

xlim – 2-tuple specifying the x axis limits
ylim – 2-tuple specifying the y axis limits
resolution – integer specifying the resolution of plot

set_params(**params)¶

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Returns
Return type: self

simulate(n_samples=1000)[source]¶

Draws random samples from the unconditional distribution p(x,y)

Parameters: n_samples – (int) number of samples to be drawn from the conditional distribution
Returns: (X,Y) - random samples drawn from p(x,y) - numpy arrays of shape (n_samples, ndim_x) and (n_samples, ndim_y)

simulate_conditional(X)[source]¶

Draws random samples from the conditional distribution

Parameters: X – x to be conditioned on when drawing a sample from y ~ p(y|x) - numpy array of shape (n_samples, ndim_x)
Returns: Conditional random samples y drawn from p(y|x) - numpy array of shape (n_samples, ndim_y)

skewness(x_cond, n_samples=1000000)¶

Skewness of the fitted distribution conditioned on x_cond

Parameters: x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)
Returns: Skewness Skew[y|x] corresponding to x_cond - numpy array of shape (n_values, ndim_y, ndim_y)

std_(x_cond, n_samples=1000000)¶

Standard deviation of the fitted distribution conditioned on x_cond

Parameters: x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)
Returns: Standard deviations sqrt(Var[y|x]) corresponding to x_cond - numpy array of shape (n_values, ndim_y)

tail_risk_measures(x_cond, alpha=0.01, n_samples=1000000)¶

Computes the Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)

Parameters

x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)
alpha – quantile percentage of the distribution
n_samples – number of samples for monte carlo model_fitting

Returns

VaR values for each x to condition on - numpy array of shape (n_values)
CVaR values for each x to condition on - numpy array of shape (n_values)

value_at_risk(x_cond, alpha=0.01, n_samples=1000000)¶

Computes the Value-at-Risk (VaR) of the fitted distribution. Only if ndim_y = 1

Parameters

x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)
alpha – quantile percentage of the distribution
n_samples – number of samples for monte carlo model_fitting

Returns

VaR values for each x to condition on - numpy array of shape (n_values)