PlnDiag

The PlnDiag model is a Pln model where the covariance is not assumed full but only diagonal.

PlnDiag Documentation

class pyPLNmodels.PlnDiag(endog, *, exog=None, offsets=None, compute_offsets_method='zero', add_const=True)[source]

PLN model with diagonal covariance. Inference is faster, as well as memory requirements.

Examples

>>> from pyPLNmodels import PlnDiag, load_scrna
>>> data = load_scrna()
>>> pln = PlnDiag(data["endog"])
>>> pln.fit()
>>> print(pln)
>>> pln.viz(colors=data["labels"])

>>> from pyPLNmodels import PlnDiag, load_scrna
>>> data = load_scrna()
>>> pln = PlnDiag.from_formula("endog ~ 1 + labels", data=data)
>>> pln.fit()
>>> print(pln)
>>> pln.viz(colors=data["labels"])

Parameters:

endog (Tensor | ndarray | DataFrame)
exog (Tensor | ndarray | DataFrame | Series | None)
offsets (Tensor | ndarray | DataFrame | None)
compute_offsets_method ({'logsum', 'zero'})
add_const (bool)

__init__(endog, *, exog=None, offsets=None, compute_offsets_method='zero', add_const=True)[source]

Initializes the model class.

Parameters:

endog (Union[torch.Tensor, np.ndarray, pd.DataFrame]) – The count data.
exog (Union[torch.Tensor, np.ndarray, pd.DataFrame], optional(keyword-only)) – The covariate data. Defaults to None.
offsets (Union[torch.Tensor, np.ndarray, pd.DataFrame], optional(keyword-only)) – The offsets data. Defaults to None.
compute_offsets_method (str, optional(keyword-only)) –
Method to compute offsets if not provided. Options are:
- ”zero” that will set the offsets to zero.
- ”logsum” that will take the logarithm of the sum (per line) of the counts.
Overridden (useless) if offsets is not None.
add_const (bool, optional(keyword-only)) – Whether to add a column of one in the exog. Defaults to True.

Return type:

Pln

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data)
>>> pln.fit()
>>> print(pln)

compute_elbo()[source]

Compute the elbo of the current parameters.

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data)
>>> pln.fit()
>>> elbo = pln.compute_elbo()
>>> print(elbo)

get_variance_coef()[source]

Calculate the variance of the regression coefficients using the sandwich estimator. Returns None if there are no exogenous variables in the model.

Returns:: Variance of the regression coefficients.
Return type:: torch.Tensor
Raises:: ValueError – If the number of samples is less than the product of the number of covariates and dimensions.

Examples

>>> from pyPLNmodels import PlnDiag, load_scrna
>>> rna_data = load_scrna(dim = 10)
>>> pln = PlnDiag(rna_data["endog"], exog=rna_data["labels_1hot"], add_const=False)
>>> pln.fit()
>>> variance = pln.get_variance_coef()
>>> print('variance', variance)

get_confidence_interval_coef(alpha=0.05)[source]

Calculate the confidence intervals for the regression coefficients. Returns None if there are no exogenous variables in the model.

Parameters:: alpha (float (optional)) – Significance level for the confidence intervals. Defaults to 0.05.
Returns:: interval_low, interval_high – Lower and upper bounds of the confidence intervals for the coefficients.
Return type:: Tuple(torch.Tensor, torch.Tensor)

Examples

>>> from pyPLNmodels import PlnDiag, load_scrna
>>> rna_data = load_scrna(dim = 10)
>>> pln = PlnDiag(rna_data["endog"], exog=rna_data["labels_1hot"], add_const=False)
>>> pln.fit()
>>> interval_low, interval_high = pln.get_confidence_interval_coef()

>>> import torch
>>> from pyPLNmodels import PlnDiag, PlnDiagSampler
>>>
>>> sampler = PlnDiagSampler(n_samples=1500, add_const=False, nb_cov=4)
>>> endog = sampler.sample() # Sample PlnDiag data.
>>>
>>> pln = PlnDiag(endog, exog=sampler.exog, add_const=False)
>>> pln.fit()
>>> interval_low, interval_high = pln.get_confidence_interval_coef(alpha=0.05)
>>> true_coef = sampler.coef
>>> inside_interval = (true_coef > interval_low) & (true_coef < interval_high)
>>> print('Should be around 0.95:', torch.mean(inside_interval.float()).item())

get_coef_p_values()[source]

Calculate the p-values for the regression coefficients. Returns None if there are no exogenous variables in the model.

Returns:: p_values – P-values for the regression coefficients.
Return type:: torch.Tensor

Examples

>>> from pyPLNmodels import PlnDiag, load_scrna
>>> rna_data = load_scrna(dim = 10)
>>> pln = PlnDiag(rna_data["endog"], exog=rna_data["labels_1hot"], add_const=False)
>>> pln.fit()
>>> p_values = pln.get_coef_p_values()
>>> print('P-values: ', p_values)

summary()[source]

Print a summary of the regression coefficients and their p-values for each dimension. Returns None if there are no exogenous variabes in the model.

Examples

>>> from pyPLNmodels import PlnDiag, load_scrna
>>> rna_data = load_scrna(dim = 10)
>>> pln = PlnDiag(rna_data["endog"], exog = rna_data["labels_1hot"], add_const = False)
>>> pln.fit()
>>> pln.summary()

property number_of_parameters: Returns the number of parameters of the model.

property entropy: Entropy of the latent variables.

property AIC: Akaike Information Criterion (AIC).

property BIC: Bayesian Information Criterion (BIC) of the model.

property ICL: Integrated Completed Likelihood criterion.

biplot(column_names, *, column_index=None, colors=None, title='')

Visualizes variables using the correlation circle along with the pca transformed samples. If the endog has been given as a pd.DataFrame, the column_names have been stored and may be indicated with the column_names argument. Else, one should provide the indices of variables.

Parameters:

column_names (List[str]) – A list of variable names to visualize. If column_index is None, the variables plotted are the ones in column_names. If column_index is not None, this only serves as a legend. Check the attribute column_names_endog.
column_index (Optional[List[int]], optional keyword-only) – A list of indices corresponding to the variables that should be plotted. If None, the indices are determined based on column_names_endog given the column_names, by default None. If not None, should have the same length as column_names.
title (str optional, keyword-only) – An additional title for the plot.
colors (list, optional, keyword-only) – The labels to color the samples, of size n_samples.

Raises:

ValueError – If column_index is None and column_names_endog is not set, that has been set if the model has been initialized with a pd.DataFrame as endog.
ValueError – If the length of column_index is different from the length of column_names.

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data=data)
>>> pln.fit()
>>> pln.biplot(column_names=["MALAT1", "ACTB"])
>>> pln.biplot(column_names=["A", "B"], column_index=[0, 4], colors=data["labels"])

property coef

Property representing the regression coefficients of size (nb_cov, dim). If no exogenous (exog) is available, returns None.

Returns:: The coefficients or None if no coefficients are given in the model.
Return type:: torch.Tensor or None

property covariance

Property representing the covariance of the model.

Returns:: The covariance.
Return type:: torch.Tensor

property dict_latent_parameters: The latent parameters of the model.

property dict_model_parameters: The parameters of the model.

property dim: Number of dimensions (i.e. variables) of the dataset.

property elbo: Returns the last elbo computed.

property endog

Property representing the endogenous variables (counts).

Returns:: The endogenous variables.
Return type:: torch.Tensor

property exog

Property representing the exogenous variables (covariates).

Returns:: The exogenous variables or None if no covariates are given in the model.
Return type:: torch.Tensor or None

fit(*, maxiter=400, lr=0.01, tol=1e-06, verbose=False)

Fit the model using variational inference. The lower the tol (tolerance), the more accurate the model.

Parameters:

maxiter (int, optional) – The maximum number of iterations to be done. Defaults to 400.
lr (float, optional(keyword-only)) – The learning rate. Defaults to 0.01.
tol (float, optional(keyword-only)) – The tolerance for convergence. Defaults to 1e-6.
verbose (bool, optional(keyword-only)) – Whether to print training progress. Defaults to False.

Raises:

ValueError – If maxiter is not an int.

Return type:

Pln object

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data)
>>> pln.fit()
>>> print(pln)

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data)
>>> pln.fit(maxiter=500, verbose=True)
>>> print(pln)

classmethod from_formula(formula, data, *, compute_offsets_method='zero')

Create an instance from a formula and data.

Parameters:

formula (str) – The formula.
data (dict) – The data dictionary. Each value can be either a torch.Tensor, np.ndarray, pd.DataFrame or pd.Series. The categorical exogenous variables should be 1-dimensional.
compute_offsets_method (str, optional(keyword-only)) –
Method to compute offsets if not provided. Options are:
- ”zero” that will set the offsets to zero.
- ”logsum” that will take the logarithm of the sum (per line) of the counts.
Overridden (useless) if data[“offsets”] is not None.

Return type:

Pln

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data=data)

property latent_mean

Property representing the latent mean conditionally on the observed counts, i.e. the conditional mean of the latent variable of each sample.

Returns:: The latent mean.
Return type:: torch.Tensor

property latent_parameters: Alias for dict_latent_parameters.

property latent_positions

The (conditional) mean of the latent variables with the effect of covariates removed.

See also

pyPLNmodels.Pln.latent_variables()

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data)
>>> pln.fit()
>>> print("Shape latent positions", pln.latent_positions.shape)
>>> pln.viz(remove_exog_effect=True) # Visualize the latent positions

property latent_sqrt_variance

Property representing the latent square root variance conditionally on the observed counts, i.e. the square root variance of the latent variable of each sample.

Returns:: The square root of the latent variance.
Return type:: torch.Tensor

property latent_variables

The (conditional) mean of the latent variables. This is the best approximation of latent variables. This variable is supposed to be more meaningful than the counts (endog).

See also

pyPLNmodels.Pln.latent_positions()

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data)
>>> pln.fit()
>>> print(pln.latent_variables.shape)
>>> pln.viz() # Visualize the latent variables

property latent_variance: Property representing the latent variance conditionally on the observed counts, i.e. the conditional variance of the latent variable of each sample.

property list_of_parameters_needing_gradient: The list of all the parameters of the model that needs to be updated at each iteration.

property loglike: Alias for elbo.

property marginal_mean: The marginal mean of the model, i.e. the mean of the gaussian latent variable.

property model_parameters: Alias for dict_model_parameters.

property n_samples: Number of samples in the dataset.

property nb_cov: int: The number of exogenous variables.

property offsets

Property representing the offsets.

Returns:: The offsets.
Return type:: torch.Tensor

property optim_details

Property representing the optimization details.

Returns:: The dictionary of optimization details.
Return type:: dict

pca_pairplot(n_components=3, colors=None)

Generates a scatter matrix plot based on Principal Component Analysis (PCA) on the latent variables.

Parameters:

(int (n_components) – Defaults to 3. It Cannot be greater than 6.
optional) (The number of components to consider for plotting.) – Defaults to 3. It Cannot be greater than 6.
(np.ndarray) (colors) – sample in the endog property of the object. Defaults to None.
n_components (bool)

Raises:

ValueError – If the number of components requested is greater: than the number of variables in the dataset.

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data=data)
>>> pln.fit()
>>> pln.pca_pairplot(n_components=5)
>>> pln.pca_pairplot(n_components=5, colors=data["labels"])

plot_correlation_circle(column_names, column_index=None, title='')

Visualizes variables using PCA and plots a correlation circle. If the endog has been given as a pd.DataFrame, the column_names have been stored and may be indicated with the column_names argument. Else, one should provide the indices of variables.

Parameters:

column_names (List[str]) – A list of variable names to visualize. If column_index is None, the variables plotted are the ones in column_names. If column_index is not None, this only serves as a legend. Check the attribute column_names_endog.
column_index (Optional[List[int]], optional) – A list of indices corresponding to the variables that should be plotted. If None, the indices are determined based on column_names_endog given the column_names, by default None. If not None, should have the same length as column_names.
title (str) – An additional title for the plot.

Raises:

ValueError – If column_index is None and column_names_endog is not set, that has been set if the model has been initialized with a pd.DataFrame as endog.
ValueError – If the length of column_index is different from the length of column_names.

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data=data)
>>> pln.fit()
>>> pln.plot_correlation_circle(column_names=["MALAT1", "ACTB"])
>>> pln.plot_correlation_circle(column_names=["A", "B"], column_index=[0, 4])

plot_expected_vs_true(ax=None, colors=None)

Plot the predicted value of the endog against the endog.

Parameters:

ax (Optional[matplotlib.axes.Axes], optional) – The matplotlib axis to use. If None, the current axis is used, by default None.
colors (Optional[Any], optional) – The labels to color the samples, of size n_samples. By default None (no colors).

Returns:

The matplotlib axis.

Return type:

matplotlib.axes.Axes

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln(data["endog"])
>>> pln.fit()
>>> pln.plot_expected_vs_true()
>>> pln.plot_expected_vs_true(colors=data["labels"])

plot_regression_forest(alpha=0.05, figsize=(10, 10))

Creates a forest plot for regression coefficients with confidence intervals (5%).

Parameters:

alpha (float) – The confidence parameter.
figsize (tuple) – The size of the figure.

property precision

Property representing the precision of the model, that is the inverse covariance matrix.

Returns:: The precision matrix of size (dim, dim).
Return type:: torch.Tensor

predict(array_like=None)

projected_latent_variables(rank=2, remove_exog_effect=False)

Perform PCA on latent variables and return the projected variables.

Parameters:

rank (int, optional) – The number of principal components to compute, by default 2.
remove_exog_effect (bool, optional) – Whether to remove or not the effect of exogenous variables. Default to False.

Returns:

The projected variables.

Return type:

numpy.ndarray

remove_zero_columns = True

show(savefig=False, name_file='', figsize=(10, 10))

Display the model parameters, norm evolution of the parameters and the criterion.

Parameters:

savefig (bool, optional) – If True, the figure will be saved to a file. Default is False.
name_file (str, optional) – The name of the file to save the figure. Only used if savefig is True. Default is an empty string.
figsize (tuple of two positive floats.) – Size of the figure that will be created. By default (10,10)

sigma(): Covariance of the model.

transform(remove_exog_effect=False)

Returns the latent variables. Can be seen as a normalization of the counts given.

Parameters:: remove_exog_effect (bool (optional)) – Whether to remove or not the mean induced by the exogenous variables. Default is False.
Returns:: The transformed endogenous variables (latent variables of the model).
Return type:: torch.Tensor

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1", data=data)
>>> pln.fit()
>>> transformed_endog = pln.transform()
>>> print(transformed_endog.shape)
>>> pln.viz()
>>> transformed_no_exog = pln.transform(remove_exog_effect=True)

viz(*, ax=None, colors=None, show_cov=False, remove_exog_effect=False)

Visualize the latent variables. One can remove the effect of exogenous variables with the remove_exog_effect boolean variable.

Parameters:

ax (matplotlib.axes.Axes, optional) – The axes on which to plot, by default None.
colors (list, optional) – The labels to color the samples, of size n_samples.
show_cov (bool, optional) – Whether to show covariances, by default False.
remove_exog_effect (bool, optional) – Whether to remove or not the effect of exogenous variables. Default to False.

Examples

>>> from pyPLNmodels import Pln, load_scrna
>>> data = load_scrna()
>>> pln = Pln.from_formula("endog ~ 1 + labels", data=data)
>>> pln.fit()
>>> pln.viz()
>>> pln.viz(colors=data["labels"])
>>> pln.viz(show_cov=True)
>>> pln.viz(remove_exog_effect=True, colors=data["labels"])

optim: torch.optim.Optimizer

List of methods and attributes

Public Data Attributes:

`number_of_parameters`	Returns the number of parameters of the model.
`entropy`	Entropy of the latent variables.
`optim`

Inherited from Pln

`list_of_parameters_needing_gradient`	The list of all the parameters of the model that needs to be updated at each iteration.
`dict_model_parameters`	The parameters of the model.
`dict_latent_parameters`	The latent parameters of the model.
`latent_variables`	The (conditional) mean of the latent variables.
`latent_positions`	The (conditional) mean of the latent variables with the effect of covariates removed.
`number_of_parameters`	Returns the number of parameters of the model.
`entropy`	Entropy of the latent variables.
`optim`

Inherited from BaseModel

`remove_zero_columns`
`list_of_parameters_needing_gradient`	The list of all the parameters of the model that needs to be updated at each iteration.
`dict_model_parameters`	The parameters of the model.
`model_parameters`	Alias for dict_model_parameters.
`dict_latent_parameters`	The latent parameters of the model.
`latent_parameters`	Alias for dict_latent_parameters.
`n_samples`	Number of samples in the dataset.
`dim`	Number of dimensions (i.e. variables) of the dataset.
`endog`	Property representing the endogenous variables (counts).
`exog`	Property representing the exogenous variables (covariates).
`nb_cov`	The number of exogenous variables.
`offsets`	Property representing the offsets.
`latent_mean`	Property representing the latent mean conditionally on the observed counts, i.e. the conditional mean of the latent variable of each sample.
`latent_variance`	Property representing the latent variance conditionally on the observed counts, i.e. the conditional variance of the latent variable of each sample.
`latent_sqrt_variance`	Property representing the latent square root variance conditionally on the observed counts, i.e. the square root variance of the latent variable of each sample.
`coef`	Property representing the regression coefficients of size (nb_cov, dim).
`covariance`	Property representing the covariance of the model.
`precision`	Property representing the precision of the model, that is the inverse covariance matrix.
`marginal_mean`	The marginal mean of the model, i.e. the mean of the gaussian latent variable.
`latent_variables`	The (conditional) mean of the latent variables.
`latent_positions`	The (conditional) mean of the latent variables with the effect of covariates removed.
`elbo`	Returns the last elbo computed.
`loglike`	Alias for elbo.
`BIC`	Bayesian Information Criterion (BIC) of the model.
`ICL`	Integrated Completed Likelihood criterion.
`AIC`	Akaike Information Criterion (AIC).
`number_of_parameters`	Returns the number of parameters of the model.
`entropy`	Entropy of the latent variables.
`optim_details`	Property representing the optimization details.
`optim`

Public Methods:

`__init__`(endog, *[, exog, offsets, ...])	Initializes the model class.
`compute_elbo`()	Compute the elbo of the current parameters.
`get_variance_coef`()	Calculate the variance of the regression coefficients using the sandwich estimator.
`get_confidence_interval_coef`([alpha])	Calculate the confidence intervals for the regression coefficients.
`get_coef_p_values`()	Calculate the p-values for the regression coefficients.
`summary`()	Print a summary of the regression coefficients and their p-values for each dimension.

Inherited from Pln

`__init__`(endog, *[, exog, offsets, ...])	Initializes the model class.
`from_formula`(formula, data, *[, ...])	Create an instance from a formula and data.
`fit`(*[, maxiter, lr, tol, verbose])	Fit the model using variational inference.
`compute_elbo`()	Compute the elbo of the current parameters.
`plot_correlation_circle`(column_names[, ...])	Visualizes variables using PCA and plots a correlation circle.
`biplot`(column_names, *[, column_index, ...])	Visualizes variables using the correlation circle along with the pca transformed samples.
`pca_pairplot`([n_components, colors])	Generates a scatter matrix plot based on Principal Component Analysis (PCA) on the latent variables.
`transform`([remove_exog_effect])	Returns the latent variables.
`plot_expected_vs_true`([ax, colors])	Plot the predicted value of the endog against the endog.
`viz`(*[, ax, colors, show_cov, ...])	Visualize the latent variables.
`get_variance_coef`()	Calculate the variance of the regression coefficients using the sandwich estimator.
`get_confidence_interval_coef`([alpha])	Calculate the confidence intervals for the regression coefficients.
`get_coef_p_values`()	Calculate the p-values for the regression coefficients.
`summary`()	Print a summary of the regression coefficients and their p-values for each dimension.
`plot_regression_forest`([alpha, figsize])	Creates a forest plot for regression coefficients with confidence intervals (5%).

Inherited from BaseModel

`__init__`(endog, *[, exog, offsets, ...])	Initializes the model class.
`from_formula`(formula, data, *[, ...])	Create an instance from a formula and data.
`fit`(*[, maxiter, lr, tol, verbose])	Fit the model using variational inference.
`show`([savefig, name_file, figsize])	Display the model parameters, norm evolution of the parameters and the criterion.
`plot_correlation_circle`(column_names[, ...])	Visualizes variables using PCA and plots a correlation circle.
`biplot`(column_names, *[, column_index, ...])	Visualizes variables using the correlation circle along with the pca transformed samples.
`compute_elbo`()	Compute the elbo of the current parameters.
`projected_latent_variables`([rank, ...])	Perform PCA on latent variables and return the projected variables.
`transform`([remove_exog_effect])	Returns the latent variables.
`viz`(*[, ax, colors, show_cov, ...])	Visualize the latent variables.
`__repr__`()	Generate the string representation of the model.
`predict`([array_like])
`sigma`()	Covariance of the model.
`pca_pairplot`([n_components, colors])	Generates a scatter matrix plot based on Principal Component Analysis (PCA) on the latent variables.
`plot_expected_vs_true`([ax, colors])	Plot the predicted value of the endog against the endog.

Private Data Attributes:

`_covariance`
`_precision`
`_abc_impl`
`_time_recorder`
`_dict_list_mse`
`_latent_mean`
`_latent_sqrt_variance`
`_coef`

Inherited from Pln

`_description`	Description of the model.
`_coef`
`_covariance`
`_additional_attributes_list`	The attributes that are specific to this model.
`_additional_methods_list`	The methods that are specific to this model.
`_endog_predictions`	Abstract method the predict the endog variables.
`_abc_impl`
`_time_recorder`
`_dict_list_mse`
`_latent_mean`
`_latent_sqrt_variance`

Inherited from BaseModel

`_name`
`_description`	Description of the model.
`_default_dict_model_parameters`
`_default_dict_latent_parameters`
`_precision`
`_marginal_mean`
`_useful_methods_list`
`_useful_attributes_list`
`_additional_attributes_list`	The attributes that are specific to this model.
`_additional_methods_list`	The methods that are specific to this model.
`_dict_for_printing`	Property representing the dictionary for printing.
`_endog_predictions`	Abstract method the predict the endog variables.
`_latent_dim`
`_abc_impl`
`_time_recorder`
`_dict_list_mse`
`_latent_mean`
`_latent_sqrt_variance`
`_coef`
`_covariance`

Inherited from ABC

_abc_impl

Private Methods:

Inherited from Pln

`_init_model_parameters`()	The model parameters are profiled in the ELBO, no need to initialize them.
`_init_latent_parameters`()	Initialization of latent parameters.
`_get_two_dim_latent_variances`(sklearn_components)	Computes the covariance when the latent variables are embedded in a lower dimensional space (often 2) with sklearn_components.

Inherited from BaseModel

`_get_model_viz`()
`_trainstep`()	Compute the elbo and do a gradient step.
`_compute_loss`(elbo)
`_initialize_timing`()
`_print_beginning_message`()
`_print_end_of_fitting_message`(...)
`_init_parameters`()
`_print_start_init`()
`_print_end_init`()
`_init_model_parameters`()	Initialization of model parameters.
`_init_latent_parameters`()	Initialization of latent parameters.
`_set_requiring_grad_true`()	Move parameters to the GPU device if present.
`_handle_optimizer`(lr)
`_fitting_initialization`(lr, maxiter)
`_project_parameters`()	Project some parameters such as probabilities.
`_update_closed_forms`()	Update some parameters.
`_track_mse`()
`_print_stats`(iterdone, maxiter, tol)	Print the training statistics.
`_pca_projected_latent_variables_with_covariances`([...])	Perform PCA on latent variables and return the projected variables along with their covariances in the two dimensional space.
`_get_two_dim_latent_variances`(sklearn_components)	Computes the covariance when the latent variables are embedded in a lower dimensional space (often 2) with sklearn_components.