Pln

Pln Documentation

class pyPLNmodels.Pln(endog, *, exog=None, offsets=None, compute_offsets_method='zero', add_const=True)

Simplest model, that is the original PLN model from Aitchison, J., and C. H. Ho. “The Multivariate Poisson-Log Normal Distribution.” Biometrika. Variance estimation of regression coefficients are available, thanks to: “Evaluating Parameter Uncertainty in the Poisson Lognormal Model with Corrected Variational Estimators” from Batardière, B., Chiquet, J., Mariadassou, M.

.

Examples

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

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

See also

pyPLNmodels.PlnDiag, pyPLNmodels.PlnPCA, pyPLNmodels.PlnAR, pyPLNmodels.sandwich.SandwichPln, pyPLNmodels.Pln.__init__(), pyPLNmodels.Pln.from_formula()

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)

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

See also

pyPLNmodels.Pln.from_formula(), pyPLNmodels.PlnPCA, pyPLNmodels.PlnMixture

Examples

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

See also

pyPLNmodels.Pln, pyPLNmodels.Pln.__init__()

Examples

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

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)

property list_of_parameters_needing_gradient

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

compute_elbo()

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)

property dict_model_parameters

The parameters of the model.

property dict_latent_parameters

The latent parameters of the model.

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_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 number_of_parameters

Returns the number of parameters of the model.

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])

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"])

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"])

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)

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

See also

pyPLNmodels.Pln.pca_pairplot(), pyPLNmodels.PlnPCA.pca_pairplot(), pyPLNmodels.Pln.biplot(), pyPLNmodels.PlnPCA.biplot()

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"])

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"])

get_variance_coef()

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 Pln, load_scrna
>>> rna_data = load_scrna()
>>> pln = Pln(rna_data["endog"], exog=rna_data["labels_1hot"], add_const=False)
>>> pln.fit()
>>> variance = pln.get_variance_coef()
>>> print('variance', variance)

See also

pyPLNmodels.Pln.summary(), pyPLNmodels.Pln.get_coef_p_values(), pyPLNmodels.Pln.get_confidence_interval_coef()

get_confidence_interval_coef(alpha=0.05)

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 Pln, load_scrna
>>> rna_data = load_scrna()
>>> pln = Pln(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 Pln, PlnSampler
>>>
>>> sampler = PlnSampler(n_samples=1500, add_const=False, nb_cov=4)
>>> endog = sampler.sample() # Sample Pln data.
>>>
>>> pln = Pln(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())

See also

pyPLNmodels.Pln.summary(), pyPLNmodels.Pln.get_coef_p_values()

get_coef_p_values()

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 Pln, load_scrna
>>> rna_data = load_scrna()
>>> pln = Pln(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)

See also

pyPLNmodels.Pln.summary(), pyPLNmodels.Pln.get_confidence_interval_coef()

summary()

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 Pln, load_scrna
>>> rna_data = load_scrna()
>>> pln = Pln(rna_data["endog"], exog = rna_data["labels_1hot"], add_const = False)
>>> pln.fit()
>>> pln.summary()

See also

pyPLNmodels.Pln.get_confidence_interval_coef()

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 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.

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 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

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_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_variance

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

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

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.

optim: torch.optim.Optimizer

List of methods and attributes

Public Data Attributes:

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.
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:

_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:

_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.

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