nemos.observation_models.PoissonObservations#

class nemos.observation_models.PoissonObservations[source]#

Bases: Observations

Model observations as Poisson random variables.

The PoissonObservations is designed to model the observed spike counts based on a Poisson 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.

Attributes

`default_inverse_link_function`
`scale`	Getter for the scale parameter of the model.

__init__()[source]#

Methods

`__init__`()
`deviance`(spike_counts, predicted_rate[, scale])	Compute the residual deviance for a Poisson model.
`estimate_scale`(y, predicted_rate, dof_resid)	Assign 1 to the scale parameter of the Poisson model.
`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 Poisson 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 Poisson 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#

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

Compute the residual deviance for a Poisson model.

Parameters:

spike_counts (Array) – The spike counts. 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 Poisson model, the residual deviance is computed as:

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

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]#

Assign 1 to the scale parameter of the Poisson model.

For the Poisson exponential family distribution, the scale parameter \(\phi\) is always 1. This property is consistent with the fact that the variance equals the mean in a Poisson distribution. As given in the general exponential family expression:

\[\text{var}(Y) = \frac{V(\mu)}{a(\phi)},\]

for the Poisson family, it simplifies to \(\text{var}(Y) = \mu\) since \(a(\phi) = 1\) and \(V(\mu) = \mu\).

Parameters:

y (Array) – Observed spike counts.
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]

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 PoissonObservations.<lambda>>)[source]#

Compute the Poisson negative log-likelihood.

This computes the Poisson negative log-likelihood of the predicted rates for the observed spike counts up to a constant.

Parameters:

y (Array) – The target spikes 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 Poisson negative log-likehood. Shape (1,).

Notes

The formula for the Poisson mean log-likelihood is the following,

\[\begin{split}\begin{aligned} \text{LL}(\hat{\lambda} | y) &= \frac{1}{T \cdot N} \sum_{n=1}^{N} \sum_{t=1}^{T} [y_{tn} \log(\hat{\lambda}_{tn}) - \hat{\lambda}_{tn} - \log({y_{tn}!})] \\\ &= \frac{1}{T \cdot N} \sum_{n=1}^{N} \sum_{t=1}^{T} [y_{tn} \log(\hat{\lambda}_{tn}) - \hat{\lambda}_{tn} - \Gamma({y_{tn}+1})] \\\ &= \frac{1}{T \cdot N} \sum_{n=1}^{N} \sum_{t=1}^{T} [y_{tn} \log(\hat{\lambda}_{tn}) - \hat{\lambda}_{tn}] + \text{const} \end{aligned}\end{split}\]

Because \(\Gamma(k+1)=k!\), see wikipedia for explanation.

The \(\log({y_{tn}!})\) term is not a function of the parameters and can be disregarded when computing the loss-function. This is why we incorporated it into the const term.

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 Poisson distribution.

This method generates random numbers from a Poisson distribution based on the given predicted_rate.

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]) – Scale parameter. For Poisson should be equal to 1.

Returns:

Random numbers generated from the Poisson distribution based on the predicted_rate.

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