neighbayes.models.SLX¶

class neighbayes.models.SLX(formula=None, data=None, y=None, X=None, W=None, priors=None, logdet_method=None, robust=False, w_vars=None, logdet_refit=True, logdet_refit_pad_sd=10.0, logdet_aaa_check=True, logdet_probe_check=True)[source]¶

Bayesian SLX (Spatial Lag X) model.

Adds spatial lags of the regressors \(X\) to a standard linear model, without any spatial dependence in \(y\):

\[y = X\beta + WX\theta + \varepsilon, \quad \varepsilon \sim N(0, \sigma^2 I).\]

The sampled coefficient vector stacks the local and lagged-regressor blocks as \([\beta, \theta]\). There is no Jacobian term because there is no spatial lag of \(y\).

Parameters:¶

formula : str, optional¶

Wilkinson-style formula, e.g. "y ~ x1 + x2". Requires data. Intercept is included by default; suppress with "y ~ x - 1".

data : pandas.DataFrame or geopandas.GeoDataFrame, optional¶

Data source for formula mode.

y : array-like, optional¶

Dependent variable of shape (n,). Required in matrix mode.

X : array-like or pandas.DataFrame, optional¶

Design matrix. Required in matrix mode. DataFrame columns are preserved as feature names.

W : libpysal.graph.Graph or scipy.sparse matrix¶

Spatial weights of shape (n, n). Used to construct the WX block. Accepts a libpysal.graph.Graph or any scipy.sparse matrix; legacy libpysal.weights.W is not accepted (use w.sparse or libpysal.graph.Graph.from_W(w)). Should be row-standardized.

priors : dict, optional¶

Override default priors. Supported keys:

beta_mu (array or float, optional): Normal prior mean for coefficients \([eta, heta]\). Default is Gelman et al. (2008) weakly-informative mean (mean(y) for intercept/constant columns, 0 for slopes).
beta_sigma (array or float, optional): Normal prior std for coefficients. Default is Gelman et al. (2008) weakly-informative scale: 2.5 * sd(y) for intercept/constant columns and 2.5 * sd(y) / sd(x_j) for each slope.
sigma2_alpha (float, default 2.0): Shape of the InverseGamma prior on \(\sigma^2\).
sigma2_beta (float, default Var(y)): Scale of the InverseGamma prior on \(\sigma^2\).
nu (float, default 4.0): Fixed Student-t degrees of freedom (only used when robust=True).

logdet_method : str, optional¶

Accepted for API consistency with the spatial-lag/error models but unused (SLX has no spatial Jacobian).

robust : bool, default False¶

If True, replace the Normal error with Student-t. See Robust regression below.

w_vars : list of str, optional¶

Names of X columns to spatially lag. By default all non-constant columns are lagged. Pass a subset to restrict which variables receive a spatial lag, e.g. w_vars=["income", "density"]. SLX requires at least one WX column; if filtering eliminates all of them a ValueError is raised.

Notes

Direct effects equal \(\beta\) and indirect effects equal \(\theta\). There is no global spillover multiplier because \(y\) has no spatial autoregression.

Robust regression

When robust=True, the error distribution is changed from Normal to Student-t:

\[\varepsilon \sim t_\nu(0, \sigma^2 I)\]

where \(\nu\) is a fixed hyperparameter set by priors={"nu": value} (default 4, LeSage’s rval); larger values approach the Normal. Values must exceed 2 so the variance exists.

__init__(formula=None, data=None, y=None, X=None, W=None, priors=None, logdet_method=None, robust=False, w_vars=None, logdet_refit=True, logdet_refit_pad_sd=10.0, logdet_aaa_check=True, logdet_probe_check=True)[source]¶

Methods

`__init__`([formula, data, y, X, W, priors, ...])
`fit`([draws, tune, chains, target_accept, ...])	Draw samples from the posterior.
`fitted_values`()	Return fitted values at posterior mean parameters.
`residuals`()	Return residuals `y - fitted_values`.
`spatial_diagnostics`()	Run Bayesian LM specification tests and return a summary table.
`spatial_diagnostics_decision`([alpha, format])	Return a model-selection decision from Bayesian LM test results.
`spatial_effects`([return_posterior_samples])	Compute Bayesian inference for direct, indirect, and total impacts.
`summary`([var_names])	Return posterior summary table.

Attributes

`inference_data`	Return the ArviZ InferenceData from the most recent fit.
`pymc_model`	Return the PyMC model object built for the most recent fit.

fit(draws=2000, tune=1000, chains=4, target_accept=None, random_seed=None, progressbar=True, sampler=None, gibbs_backend='auto', thin=1, n_jobs=-1, idata_kwargs=None, **sample_kwargs)[source]¶

Draw samples from the posterior.

Dispatches to this model’s Gibbs sampler (sampler="gibbs") or NUTS (sampler="nuts"). When sampler is None (default), Gibbs is used if the model has a registered Gibbs sampler, otherwise NUTS.

Parameters:¶

draws : int¶: Post-warmup draws, warmup steps, and number of chains.
tune : int¶: Post-warmup draws, warmup steps, and number of chains.
chains : int¶: Post-warmup draws, warmup steps, and number of chains.
target_accept : float, optional¶: Target acceptance rate for NUTS. NUTS-only: passing it with the Gibbs sampler raises TypeError. Defaults to 0.9 for NUTS.
random_seed : int, optional¶: Seed for reproducibility.
progressbar : bool, default True¶: Show progress bar(s) during sampling.
sampler : {"gibbs", "nuts", None}, default None¶: Sampling method. None auto-selects Gibbs when this model has one, else NUTS.
gibbs_backend : {"auto", "jax", "numpy"}, default "auto"¶: Execution backend for the Gibbs sampler. "auto" uses JAX when installed and supported by the family, else NumPy. Ignored for NUTS.
thin : int, default 1¶: Keep every thin-th post-warmup Gibbs draw (Gibbs only).
n_jobs : int, default -1¶: Parallel workers for the NumPy Gibbs path (Gibbs only).
idata_kwargs : dict, optional¶: {"log_likelihood": True} stores the complete Jacobian-corrected pointwise log-likelihood that az.loo / az.waic / az.compare need, for Gibbs and NUTS alike. Off by default, as in PyMC: it holds one value per draw, chain, and observation (16 GB at n = 250,000 with 4 × 2,000 draws). For NUTS the dict is also passed to pm.sample.
**sample_kwargs¶: For NUTS, forwarded to pm.sample (nuts_sampler=...); for Gibbs, the family’s declared options (an unsupported key raises).

Return type:¶

arviz.InferenceData

fitted_values()[source]¶

Return fitted values at posterior mean parameters.

Returns:¶: Posterior-mean fitted values (on the model’s native scale; fixed-effects-transformed for panel models).
Return type:¶: np.ndarray

property inference_data : arviz.data.inference_data.InferenceData | None[source]¶

Return the ArviZ InferenceData from the most recent fit.

Returns:¶: The inference data object, or None if the model has not been fit yet.
Return type:¶: arviz.InferenceData or None

property pymc_model : pymc.model.core.Model | None[source]¶

Return the PyMC model object built for the most recent fit.

For Gibbs-fitted models the PyMC model is not constructed during sampling; it is built lazily on first access so that downstream consumers (e.g. bridge sampling for marginal likelihoods) can evaluate logp and the prior under the same model definition used by the NUTS path.

Returns:¶: The model object used by fit(), or None if the instance has not been fit yet.
Return type:¶: pymc.Model or None

residuals()[source]¶

Return residuals y - fitted_values.

Returns:¶: Residual vector y - fitted_values on the same scale as fitted_values().
Return type:¶: np.ndarray

spatial_diagnostics()[source]¶

Run Bayesian LM specification tests and return a summary table.

Looks up the diagnostic suite registered for this model class and calls each test function on this fitted model, collecting the results into a tidy DataFrame. The set of tests depends on the model type — for example, an OLS model runs LM-Lag, LM-Error, LM-SDM-Joint, and LM-SLX-Error-Joint, while an SAR model runs LM-Error, LM-WX, and Robust-LM-WX. Panel models run the Panel--prefixed analogues (e.g. Panel-LM-Lag).

Requires the model to have been fit (.fit() called) and a spatial weights matrix W to have been supplied at construction time.

Returns:¶

DataFrame indexed by test name with columns:

Column	Description
statistic	Posterior mean of the LM statistic
median	Posterior median of the LM statistic
df	Degrees of freedom for the \(\chi^2\) reference
p_value	Bayesian p-value: `1 - chi2.cdf(mean, df)`
ci_lower	Lower bound of 95% credible interval (2.5%)
ci_upper	Upper bound of 95% credible interval (97.5%)

The DataFrame has attrs["model_type"] (class name) and attrs["n_draws"] (total posterior draws) metadata.

Return type:¶

pandas.DataFrame

Raises:¶

RuntimeError – If the model has not been fit yet.
ValueError – If no spatial weights matrix W was supplied.

See also

spatial_diagnostics_decision: Model-selection decision based on the test results.
spatial_effects: Posterior inference for direct/indirect/total impacts.

Examples

>>> ols = OLS(formula="price ~ income + crime", data=df, W=w)
>>> ols.fit()
>>> ols.spatial_diagnostics()
                 statistic  median  df  p_value  ci_lower  ci_upper
LM-Lag                3.21    2.98   1    0.073      0.12      8.54
LM-Error              5.67    5.34   1    0.017      0.34     12.10
LM-SDM-Joint          7.89    7.12   4    0.096      1.23     18.32
LM-SLX-Error-Joint    6.45    5.98   4    0.168      0.89     15.67

spatial_diagnostics_decision(alpha=0.05, format='graphviz')[source]¶

Return a model-selection decision from Bayesian LM test results.

Implements the decision tree from Koley and Bera [2024] (the Bayesian analogue of the classical stge_kb procedure in Anselin et al. [1996]). Panel models use the Panel--prefixed test analogues and the panel decision specs, following Elhorst [2014]. The decision logic depends on the current model type and the pattern of significant tests:

From OLS (6-test decision tree):

If only LM-Lag is significant → SAR.
If only LM-Error is significant → SEM.
If both are significant → use the Anselin–Florax / Koley–Bera robust pair: Robust-LM-Lag → SAR, Robust-LM-Error → SEM, both → SARAR. If neither robust test is significant, fall back to the lower raw p-value.
If neither naive test is significant → OLS.

From SAR (3-test decision tree):

LM-Error significant → SARAR; LM-WX significant → SDM; Robust-LM-WX significant → SDM.

From SEM (2-test decision tree):

LM-Lag significant → SARAR; LM-WX significant → SDEM.

From SLX (4-test decision tree):

Robust-LM-Lag-SDM significant → SDM; Robust-LM-Error-SDEM significant → SDEM; both → MANSAR; neither → SLX.

From SDM: LM-Error-SDM significant → MANSAR; else SDM.

From SDEM: LM-Lag-SDEM significant → MANSAR; else SDEM.

Parameters:¶

alpha : float, default 0.05¶: Significance level for the Bayesian p-values.
format : {"graphviz", "ascii", "model"}, default "graphviz"¶: Output format. "model" returns the recommended-model name string. "ascii" returns an indented box-drawing rendering of the full decision tree with the chosen path highlighted. "graphviz" returns a graphviz.Digraph object that renders inline in Jupyter; if the optional graphviz package is not installed a UserWarning is issued and the ASCII rendering is returned instead.

Returns:¶

Recommended model name when format="model", an ASCII tree string when format="ascii", or a graphviz.Digraph when format="graphviz" (with ASCII fallback on missing dep).

Return type:¶

str or graphviz.Digraph