nemos.observation_models.GaussianObservations#

class nemos.observation_models.GaussianObservations[source]#

Bases: Observations

Model observations as Gaussian random variables.

The GaussianObservations is designed to model data based on a Gaussian distribution with a given mean (predicted rate) and variance (scale). It provides methods for computing the negative log-likelihood, generating samples, and computing the residual deviance for the given spike count data.

Attributes

`default_inverse_link_function`	Identity link function for Gaussian observations.
`scale`	Getter for the scale parameter of the model.

__init__()[source]#

Methods

`__init__`()
`deviance`(neural_activity, predicted_rate[, ...])	Compute the residual deviance for a Gaussian model.
`estimate_scale`(y, predicted_rate, dof_resid)	Estimate the scale of the model based on the GLM residuals.
`get_params`([deep])	From scikit-learn, get parameters by inspecting init.
`likelihood`(y, predicted_rate[, scale, ...])	Compute the observation model likelihood.
`log_likelihood`(y, predicted_rate[, scale, ...])	Compute the Gaussian negative log-likelihood.
`pseudo_r2`(y, predicted_rate[, score_type, ...])	Pseudo-\(R^2\) calculation for a GLM.
`sample_generator`(key, predicted_rate[, scale])	Sample from the Gaussian distribution.
`set_params`(**params)	Set the parameters of this estimator.

classmethod __init_subclass__(**kwargs)#

Set the set_{method}_request methods.

This uses PEP-487 [1] to set the set_{method}_request methods. It looks for the information available in the set default values which are set using __metadata_request__* class attributes, or inferred from method signatures.

The __metadata_request__* class attributes are used when a method does not explicitly accept a metadata through its arguments or if the developer would like to specify a request value for those metadata which are different from the default None.

References

property default_inverse_link_function#: Identity link function for Gaussian observations.

deviance(neural_activity, predicted_rate, scale=1.0)[source]#

Compute the residual deviance for a Gaussian model.

Parameters:

neural_activity (Array) – The spike count activity. Shape (n_time_bins, ) or (n_time_bins, n_neurons) for population models.
predicted_rate (Array) – The predicted firing rates. Shape (n_time_bins, ) or (n_time_bins, n_neurons) for population models.
scale (Union[float, Array]) – Scale parameter of the model.

Return type:

Array

Returns:

The residual deviance of the model.

Notes

The deviance is a measure of the goodness of fit of a statistical model. For a Gaussian model, the residual deviance is computed as:

\[D(y_{tn}, \hat{y}_{tn}) = \frac{(y_{tn} - \hat{y}_{tn})^2}{\sigma^2}\]

where \(y\) is the observed data, \(\hat{y}\) is the predicted data, and \(\sigma^2\) is the scale (variance) of the model. Lower values of deviance indicate a better fit.

estimate_scale(y, predicted_rate, dof_resid)[source]#

Estimate the scale of the model based on the GLM residuals.

For \(y \sim \mathcal{N}(\mu, \sigma^2)\) the scale is equal to,

\[\hat{\phi} = \frac{1}{n - p} \sum_{i=1}^n (y_i - \hat{\mu}_i)^2.\]

with \(n\) the number of samples and \(p\) the number of parameters in the model.

Therefore, the scale can be estimated as the variance of the residuals.

Parameters:

y (Array) – Observed spike counts.
predicted_rate (Array) – The predicted rate values.
dof_resid (Union[float, Array]) – The DOF of the residuals.

Return type:

Union[float, Array]

Returns:

The scale parameter. If predicted_rate is (n_samples, n_neurons), this method will return a scale for each neuron.

get_metadata_routing()#

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:: routing – A MetadataRequest encapsulating routing information.
Return type:: MetadataRequest

get_params(deep=True)#

From scikit-learn, get parameters by inspecting init.

Parameters:: deep – If True, will return the parameters for this estimator and contained subobjects that are estimators.
Return type:: dict
Returns:: A dictionary containing the parameters. Key is the parameter name, value is the parameter value.

likelihood(y, predicted_rate, scale=1.0, aggregate_sample_scores=<function Observations.<lambda>>)#

Compute the observation model likelihood.

This computes the likelihood of the predicted rates for the observed neural activity including the normalization constant.

Parameters:

y (Array) – The target activity to compare against. Shape (n_time_bins, ), or (n_time_bins, n_neurons).
predicted_rate (Array) – The predicted rate of the current model. Shape (n_time_bins, ), or (n_time_bins, n_neurons).
scale (Union[float, Array]) – The scale parameter of the model
aggregate_sample_scores (Callable) – Function that aggregates the log-likelihood of each sample.

Returns:

The likelihood. Shape (1,).

log_likelihood(y, predicted_rate, scale=1.0, aggregate_sample_scores=<function GaussianObservations.<lambda>>)[source]#

Compute the Gaussian negative log-likelihood.

Compute the Gaussian log-likelihood of the predicted rates for the observed neural activity up to a constant.

The formula for the Gaussian log-likelihood is given by:

\[\ell(\mu,\sigma^2) = -\frac{n}{2}\log(2\pi\sigma^2) - \frac{1}{2\sigma^2}\sum_{i=1}^{n} (y_i - \mu_i)^2.\]

where \(\mu\) is the predicted mean, \(\sigma^2\) is the variance (scale), and \(y\) is the observed data.

Parameters:

y (Array) – The target activity to compare against. Shape (n_time_bins, ), or (n_time_bins, n_neurons).
predicted_rate (Array) – The predicted rate of the current model. Shape (n_time_bins, ) or (n_time_bins, n_neurons).
scale (Union[float, Array]) – The scale parameter of the model.
aggregate_sample_scores (Callable) – Function that aggregates the log-likelihood of each sample.

Returns:

The Gaussian log-likelihood. Shape (1,).

pseudo_r2(y, predicted_rate, score_type='pseudo-r2-McFadden', scale=1.0, aggregate_sample_scores=<function Observations.<lambda>>)#

Pseudo-\(R^2\) calculation for a GLM.

Compute the pseudo-\(R^2\) metric for the GLM, as defined by McFadden et al. [2] or by Cohen et al. [3].

This metric evaluates the goodness-of-fit of the model relative to a null (baseline) model that assumes a constant mean for the observations. While the pseudo-\(R^2\) is bounded between 0 and 1 for the training set, it can yield negative values on out-of-sample data, indicating potential over-fitting.

Parameters:

y (Array) – The neural activity. Expected shape: (n_time_bins, )
predicted_rate (Array) – The mean neural activity. Expected shape: (n_time_bins, )
score_type (Literal['pseudo-r2-McFadden', 'pseudo-r2-Cohen']) – The pseudo-\(R^2\) type.
scale (Union[float, Array, ndarray[tuple[Any, ...], dtype[TypeVar(_ScalarT, bound= generic)]]]) – The scale parameter of the model.
aggregate_sample_scores (Callable)

Return type:

Array

Returns:

The pseudo-\(R^2\) of the model. A value closer to 1 indicates a better model fit, whereas a value closer to 0 suggests that the model doesn’t improve much over the null model.

Notes

The McFadden pseudo-\(R^2\) is given by:

\[R^2_{\text{mcf}} = 1 - \frac{\log(L_{M})}{\log(L_0)}.\]

Equivalent to statsmodels GLMResults.pseudo_rsquared(kind=’mcf’) .
The Cohen pseudo-\(R^2\) is given by:

\[\begin{split}\begin{aligned} R^2_{\text{Cohen}} &= \frac{D_0 - D_M}{D_0} \\\ &= 1 - \frac{\log(L_s) - \log(L_M)}{\log(L_s)-\log(L_0)}, \end{aligned}\end{split}\]

where \(L_M\), \(L_0\) and \(L_s\) are the likelihood of the fitted model, the null model (a model with only the intercept term), and the saturated model (a model with one parameter per sample, i.e. the maximum value that the likelihood could possibly achieve). \(D_M\) and \(D_0\) are the model and the null deviance, \(D_i = -2 \left[ \log(L_s) - \log(L_i) \right]\) for \(i=M,0\).

References

sample_generator(key, predicted_rate, scale=1.0)[source]#

Sample from the Gaussian distribution.

This method generates random numbers from a Gaussian distribution based on the given predicted_rate and scale.

Parameters:

key (Array) – Random key used for the generation of random numbers in JAX.
predicted_rate (Array) – Expected rate (mean) of the Gaussian distribution. Shape (n_time_bins, ) or (n_time_bins, n_neurons).
scale (Union[float, Array]) – The scale parameter (variance) for the distribution.

Returns:

Random numbers generated from the Gaussian distribution based on the predicted_rate and the scale.

Return type:

Array

property scale#: Getter for the scale parameter of the model.

set_params(**params)#

Set the parameters of this estimator.

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

Parameters:: **params (Any) – Estimator parameters.
Returns:: self – Estimator instance.
Return type:: estimator instance