nemos.observation_models.GammaObservations#

class nemos.observation_models.GammaObservations(inverse_link_function=<function GammaObservations.<lambda>>)[source]#

Bases: Observations

Model observations as Gamma random variables.

The GammaObservations is designed to model the observed spike counts based on a Gamma distribution with a given rate. It provides methods for computing the negative log-likelihood, generating samples, and computing the residual deviance for the given spike count data.

inverse_link_function#: A function that maps the predicted rate to the domain of the Poisson parameter. Defaults to jnp.exp.

Attributes

`inverse_link_function`	Getter for the inverse link function for the model.
`scale`	Getter for the scale parameter of the model.

__init__(inverse_link_function=<function GammaObservations.<lambda>>)[source]#

Methods

`__init__`([inverse_link_function])
`check_inverse_link_function`(...)	Check if the provided inverse_link_function is usable.
`deviance`(neural_activity, predicted_rate[, ...])	Compute the residual deviance for a Gamma 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.
`log_likelihood`(y, predicted_rate[, scale, ...])	Compute the Gamma 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 Gamma distribution.
`set_params`(**params)	Set the parameters of this estimator.

static check_inverse_link_function(inverse_link_function)#

Check if the provided inverse_link_function is usable.

This function verifies if the inverse link function:

Is callable
Returns a jax.numpy.ndarray
Is differentiable (via jax)

Parameters:: inverse_link_function (Callable) – The function to be checked.
Raises:: TypeError – If the function is not callable, does not return a jax.numpy.ndarray, or is not differentiable.

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

Compute the residual deviance for a Gamma model.

Parameters:

neural_activity (Array) – The spike coun 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 Gamma model, the residual deviance is computed as:

\[\begin{split}\begin{aligned} D(y_{tn}, \hat{y}_{tn}) &= 2 \left[ -\log \frac{ y_{tn}}{\hat{y}_{tn}} + \frac{y_{tn} - \hat{y}_{tn}}{\hat{y}_{tn}}\right]\\\ &= 2 \left( \text{LL}\left(y_{tn} | y_{tn}\right) - \text{LL}\left(y_{tn} | \hat{y}_{tn}\right) \right) \end{aligned}\end{split}\]

where \(y\) is the observed data, \(\hat{y}\) is the predicted data, and \(\text{LL}\) is the model log-likelihood. 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 \(y \sim \Gamma\) the scale is equal to,

\[\Phi = \frac{\text{Var(y)}}{V(\mu)}\]

with \(V(\mu) = \mu^2\).

Therefore, the scale can be estimated as the ratio of the sample variance to the squared rate.

Parameters:

y (Array) – Observed neural activity.
predicted_rate (Array) – The predicted rate values. This is not used in the Poisson model for estimating scale, but is retained for compatibility with the abstract method signature.
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_params(deep=True)#

From scikit-learn, get parameters by inspecting init.

Parameters:

deep

Return type:

dict

Returns:

out:: A dictionary containing the parameters. Key is the parameter name, value is the parameter value.

property inverse_link_function#: Getter for the inverse link function for the model.

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

Compute the Gamma negative log-likelihood.

This computes the Gamma negative log-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 Gamma negative log-likelihood. Shape (1,).

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

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

Compute the pseudo-\(R^2\) metric for the GLM, as defined by McFadden et al. [1] or by Cohen et al. [2].

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[Any, dtype[TypeVar(_ScalarType_co, bound= generic, covariant=True)]]]) – 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 Gamma distribution.

This method generates random numbers from a Gamma 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 (lambda) of the Poisson distribution. Shape (n_time_bins, ), or (n_time_bins, n_neurons)..
scale (Union[float, Array]) – The scale parameter for the distribution.

Returns:

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

Return type:

jnp.ndarray

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 (dict) – Estimator parameters.
Returns:: self – Estimator instance.
Return type:: estimator instance

nemos.observation_models.GammaObservations#

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