Mixture Density Network¶
The Mixture Density Network (MDN) [BISHOP1994] combines a conventional neural network (in our implementation specified as \(estimator\)) with a mixture density model for modeling conditional probability distributions \(p(tx)\). Given a sufficiently flexible network and considering a parametric mixture model, the parameters of the distribution \(t\) can be determined by the outputs of the neural network provided the input to the network is \(x\) (in our implementation specified as X_ph (placeholder) and X). This approach therefore constitutes as a framework capable of approximating arbitrary conditional distributions.
The following example develops a model for Gaussian components with isotropic component covariances, while \(K\) being the number of components of a single mixture (our model currently allows to choose an arbitrary number of (global) mixture components, see parameter L below) and \(\pi(x)\) denoting the mixing coefficients:
It is both feasible to replace the components by components of other distributions and extending the MDN to arbitrary covariance matrices. Although the later is generally much more difficult, it has been shown by [TANSEY2016] that, for example one can have the MDN output the lower triangular entries in the Cholesky decomposition.
Using \(x\) as input, the mixing coefficients \(\pi_{k}(x)\), the means \(\mu_{k}(x)\), and the variances \(\sigma_{k}^2(x)\) can be governed by the outputs of neural network. Assuming the mixture model has L mixture components (in our implementation specified as n_centers), the total number of network outputs is given by \((K+2)L\).
The mixing coefficients are computed as a set of \(L\) softmax outputs, where \(a_k^{\pi}\) determine the mixing coefficients emitted by the network:
ensuring the constraint that \(\pi_k(x)\) over \(K\) must sum to 1. Similarly, the variances must me larger or equal to zero. Due to isotropy we have \(L\) kernel widths \(\sigma_k(x)\) which are determined by the network output \(a_k^{\sigma}\) and can be represented as exponentials:
For the \(K \times L\) means we directly use the network outputs: \(\mu_k(x) = a_{kj}^{\sigma}\).
The weights and biases \(w\) of the neural network are learned by minimizing the negative logarithm of the likelihood (maximum likelihood) over \(N\) data points:
This can be executed via the standard backpropagation algorithm, given that suitable expressions for the derivations can be obtained.

class
cde.density_estimator.
MixtureDensityNetwork
(name, ndim_x, ndim_y, n_centers=10, hidden_sizes=(16, 16), hidden_nonlinearity=<function tanh>, n_training_epochs=1000, x_noise_std=None, y_noise_std=None, entropy_reg_coef=0.0, weight_decay=0.0, weight_normalization=True, data_normalization=True, dropout=0.0, random_seed=None)[source]¶ Mixture Density Network Estimator
See “Mixture Density networks”, Bishop 1994
 Parameters
name – (str) name space of MDN (should be unique in code, otherwise tensorflow namespace collitions may arise)
ndim_x – (int) dimensionality of x variable
ndim_y – (int) dimensionality of y variable
n_centers – Number of Gaussian mixture components
hidden_sizes – (tuple of int) sizes of the hidden layers of the neural network
hidden_nonlinearity – (tf function) nonlinearity of the hidden layers
n_training_epochs – Number of epochs for training
x_noise_std – (optional) standard deviation of Gaussian noise over the the training data X > regularization through noise
y_noise_std – (optional) standard deviation of Gaussian noise over the the training data Y > regularization through noise
entropy_reg_coef – (optional) scalar float coefficient for shannon entropy penalty on the mixture component weight distribution
weight_decay – (float) the amount of decoupled (http://arxiv.org/abs/1711.05101) weight decay to apply
weight_normalization – (boolean) whether weight normalization shall be used
data_normalization – (boolean) whether to normalize the data (X and Y) to exhibit zeromean and std
dropout – (float) the probability of switching off nodes during training
random_seed – (optional) seed (int) of the random number generators used

cdf
(X, Y)¶ Predicts the conditional cumulative probability p(Y<=yX=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 cumulative probability p(Y<=yX=x)  numpy array of shape (n_query_samples, )

conditional_value_at_risk
(x_cond, alpha=0.01, n_samples=10000000)¶ Computes the Conditional ValueatRisk (CVaR) / Expected Shortfall of a GMM. Only if ndim_y = 1
Based on formulas from section 2.3.2 in “Expected shortfall for distributions in finance”, Simon A. Broda, Marc S. Paolella, 2011
 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=None)¶ 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, random_seed=None, verbose=True, eval_set=None, **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)
eval_set – (tuple) eval/test set  tuple (X_test, Y_test)
verbose – (boolean) controls the verbosity (console output)

fit_by_cv
(X, Y, n_folds=3, param_grid=None, random_state=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] }
random_state – (int) seed used by the random number generator for shuffeling the data

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

get_params_internal
(**tags)¶ Internal method to be implemented which does not perform caching

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)¶ 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
onditional logprobability log p(yx)  numpy array of shape (n_query_samples, )

mean_
(x_cond, n_samples=None)¶ 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=None)¶  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)¶ Predicts the conditional probability 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 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=100)¶ 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
resolution –
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

reset_fit
()¶ resets all tensorflow objects and :return:

sample
(X)¶ 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=10000000)¶ 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)
 BISHOP1994
Bishop (1994). Mixture Density Networks, Technical Report, Aston University (http://publications.aston.ac.uk/373/)
 TANSEY2016
Tansey et al. (2016). Better Conditional Density Estimation for Neural Networks (https://arxiv.org/abs/1606.02321)