LeastSquares Density Ratio Estimation¶
Implementation of LeastSquares Density Ratio Estimation (LSCDE) method introduced in [SUG2010] with some extra features.
This approach estimates the conditional density of multidimensional inputs/outputs by expressing the conditional density in terms of the ratio of unconditional densities r(x,y):
Instead of estimating both unconditional densities separately, the density ratio function r(x,y) is directly estimated from samples. The density ratio function is modelled by the following linear model:
where \(\alpha=(\alpha_1, \alpha_2,...,\alpha_b)^T\) are the parameters learned from samples and \(\phi(x,y) = (\phi_{1}(x, y),\phi_{2}(x,y),...,\phi_{b}(x,y))^T\) are kernel functions such that \(\phi_{l}(x,y) \geq 0\) for all \((x,y)\in D_{X} \times D_{Y}\) and \(l = 1, ..., b\).
The parameters \(\alpha\) are learned by minimizing the a integrated squared error.
After having obtained \(\widehat{\alpha} = argmin_{\alpha} ~ J(\alpha)\) through training, the conditional density can be computed as follows:
[SUG2010] propose to use a Gaussian kernel with width \(\sigma\) (bandwidth parameter), which is also the choice for this implementation:
where \(\{(u_{l},v_{l})\}_{l=1}^b\) are center points that are chosen from the training data set. By using Gaussian kernels the optimization problem \(argmin_{\alpha} ~ J(\alpha)\) can be solved analytically. Also, the denominator in (1) is traceable and can be computed analytically. The fact that training does not require numerical optimization and the solution can be computed fully analytically is the key advantage of LSCDE.
While [SUG2010] propose to select center points for the kernel functions randomly from the training set, our implementation offers further center sampling methods:
all: use all data points in the train set as kernel centers
random: randomly selects k points as kernel centers
k_means: uses kmeans clustering to determine k kernel centers
agglomorative: uses agglomorative clustering to determine k kernel centers

class
cde.density_estimator.
LSConditionalDensityEstimation
(name='LSCDE', ndim_x=None, ndim_y=None, center_sampling_method='k_means', bandwidth=0.5, n_centers=500, regularization=1.0, keep_edges=True, n_jobs=1, random_seed=None)[source]¶ LeastSquares Density Ratio Estimator
http://proceedings.mlr.press/v9/sugiyama10a.html
 Parameters
name – (str) name / identifier of estimator
ndim_x – (int) dimensionality of x variable
ndim_y – (int) dimensionality of y variable
center_sampling_method – String that describes the method to use for finding kernel centers. Allowed values [all, random, distance, k_means, agglomerative]
bandwidth – scale / bandwith of the gaussian kernels
n_centers – Number of kernels to use in the output
regularization – regularization / damping parameter for solving the leastsquares problem
keep_edges – if set to True, the extreme y values as centers are kept (for expressiveness)
n_jobs – (int) number of jobs to launch for calls with large batch sizes
random_seed – (optional) seed (int) of the random number generators used

conditional_value_at_risk
(x_cond, alpha=0.01, n_samples=1000000)¶ Computes the Conditional ValueatRisk (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
 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)
 Returns
Covariances Cov[yx] corresponding to x_cond  numpy array of shape (n_values, ndim_y, ndim_y)

eval_by_cv
(X, Y, n_splits=5, verbose=True)¶ Fits the conditional density model with crossvalidation by using the score function of the BaseDensityEstimator for scoring the various splits.
 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)
n_splits – number of crossvalidation folds (positive integer)
verbose – the verbosity level

fit
(X, Y, **kwargs)[source]¶ Fits the conditional density model with provided data
 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)

fit_by_cv
(X, Y, n_folds=3, param_grid=None, verbose=True, n_jobs=1)¶ Fits the conditional density model with hyperparameter search and crossvalidation.  Determines the best hyperparameter configuration from a predefined set using crossvalidation. Thereby,
the conditional loglikelihood is used for simulation_eval.
Fits the model with the previously selected hyperparameter configuration
 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)
n_folds – number of crossvalidation folds (positive integer)
param_grid –
(optional) a dictionary with the hyperparameters of the model as key and and a list of respective parametrizations as value. The hyperparameter search is performed over the cartesian product of the provided lists. Example: {“n_centers”: [20, 50, 100, 200],
”center_sampling_method”: [“agglomerative”, “k_means”, “random”], “keep_edges”: [True, False]
}

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

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[yx] corresponding to x_cond  numpy array of shape (n_values, ndim_y, ndim_y)

log_pdf
(X, Y)[source]¶ Predicts the conditional logprobability log p(yx). 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 logprobability density log p(yx)  numpy array of shape (n_query_samples, )

mean_
(x_cond, n_samples=1000000)¶ Mean of the fitted distribution conditioned on x_cond :param x_cond: different x values to condition on  numpy array of shape (n_values, ndim_x)
 Returns
Means E[yx] corresponding to x_cond  numpy array of shape (n_values, ndim_y)

mean_std
(x_cond, n_samples=1000000)¶  Computes Mean and Covariance of the fitted distribution conditioned on x_cond.
Computationally more efficient than calling mean and covariance computatio separately
 Parameters
x_cond – different x values to condition on  numpy array of shape (n_values, ndim_x)
 Returns
Means E[yx] and Covariances Cov[yx]

pdf
(X, Y)[source]¶ Predicts the conditional density p(yx). 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 probability density p(yx)  numpy array of shape (n_query_samples, )

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 1dimensional each
 Parameters
xlim – 2tuple specifying the x axis limits
ylim – 2tuple 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 1dimensional each
 Parameters
xlim – 2tuple specifying the x axis limits
ylim – 2tuple specifying the y axis limits
resolution – integer specifying the resolution of plot

predict_density
(X, Y=None, resolution=50)¶ Computes conditional density p(yx) over a predefined grid of y target values
 Parameters
X – values/vectors to be conditioned on  shape: (n_instances, n_dim_x)
Y – (optional) y values to be evaluated from p(yx)  if not set, Y will be a grid with with specified resolution
resulution – integer specifying the resolution of simulation_eval grid
 Returns: tuple (P, Y)
P  density p(yx)  shape (n_instances, resolution**n_dim_y)
Y  grid with with specified resolution  shape (resolution**n_dim_y, n_dim_y) or a copy of Y in case it was provided as argument

sample
(X)[source]¶ sample from the conditional mixture distributions  requires the model to be fitted
 Parameters
X – values to be conditioned on when sampling  numpy array of shape (n_instances, n_dim_x)
 Returns: tuple (X, Y)
X  the values to conditioned on that were provided as argument  numpy array of shape (n_samples, ndim_x)
Y  conditional samples from the model p(yx)  numpy array of shape (n_samples, ndim_y)

score
(X, Y)¶ Computes the mean conditional loglikelihood of the provided data (X, Y)
 Parameters
X – numpy array to be conditioned on  shape: (n_query_samples, n_dim_x)
Y – numpy array of y targets  shape: (n_query_samples, n_dim_y)
 Returns
average log likelihood of data

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

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[yx] 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[yx]) 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 ValueatRisk (VaR) and Conditional ValueatRisk (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 ValueatRisk (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
 Returns
VaR values for each x to condition on  numpy array of shape (n_values)
 SUG2010(1,2,3)
Sugiyama et al. (2010). Conditional Density Estimation via LeastSquares Density Ratio Estimation, in PMLR 9:781788 (http://proceedings.mlr.press/v9/sugiyama10a.html)