neighbayes.models.SDM

class neighbayes.models.SDM(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 Spatial Durbin Model.

Combines a spatial lag of \(y\) with spatial lags of the regressors \(X\):

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

The sampled coefficient vector stacks the local and lagged-regressor blocks as \([\beta, \theta]\). The likelihood includes the spatial Jacobian \(\log|I - \rho W|\).

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). Accepts a libpysal.graph.Graph or any scipy.sparse matrix. The legacy libpysal.weights.W object is not accepted; pass w.sparse or libpysal.graph.Graph.from_W(w). Should be row-standardized; a UserWarning is raised otherwise.

priors : dict, optional

Override default priors. Supported keys:

  • rho_lower (float, default -1.0): Lower bound of the Uniform prior on \(\rho\).

  • rho_upper (float, default 1.0): Upper bound of the Uniform prior on \(\rho\).

  • beta_mu (float, default 0.0): Normal prior mean for \([\beta, \theta]\).

  • beta_sigma (float, default 1e6): Normal prior std for \([\beta, \theta]\).

  • 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

How to compute \(\log|I - \rho W|\). None (default) auto-selects by size: "eigenvalue" for n <= 500; for 500 < n <= 60000, "cheb_cholesky" (exact, sparse Cholesky at Chebyshev nodes) when W is symmetric else "aaa" (AAA rational approximation); "cheb_stochastic" for n > 60000. Explicit opt-ins: "chebyshev" (Barry-Pace) and "slq" (stochastic Lanczos quadrature).

logdet_refit : bool, default True

Rebuild the log-determinant interpolant halfway through warmup, on the range the chains have found rather than the interval implied by the prior. A warmup posterior is typically one to two orders of magnitude narrower than the prior, which needs far fewer interpolation nodes and drives the approximation error over the posterior’s support down to the factorization’s roundoff floor. Applies to "cheb_cholesky", "lu_cheb", "aaa" and "chol_aaa"; ignored otherwise.

The interpolant is only valid on its interval, so the refit window becomes the sampler’s support. The window is padded by logdet_refit_pad_sd warmup standard deviations, recorded in idata.attrs["logdet_refit_window"], and a warning is raised if the retained draws ever reach an edge the refit introduced.

logdet_refit_pad_sd : float, default 10.0

Padding for the refit window, in warmup posterior standard deviations. At the default the truncated tail is ~1e-23 under normality, and the padding costs a node or two at most.

logdet_aaa_check : bool, default True

For the AAA methods ("aaa", "chol_aaa"), set the number of exact factorizations from where the posterior lies. Warmup starts on a 14-node fit; halfway through, nodes are added only if the warmup posterior lies closer to a singularity of the Jacobian than four times the distance at which the fit’s poles resolve it. The count used is recorded in idata.attrs["logdet_aaa_nodes"]. Off, or on a path without a warmup midpoint, the count is fixed by the prior interval: 14 nodes within |ρ| ≤ 0.9, 18 otherwise.

logdet_probe_check : bool, default True

For "cheb_stochastic", set the number of Hutchinson probes from where the posterior lies. Warmup starts on 50 probes; halfway through, the probes’ own spread prices the bias they leave in the posterior mean of the spatial parameter, and the pool grows, to at most 200, until that bias is expected to stay under 0.0225 posterior sd. The count used is recorded in idata.attrs["logdet_probes"], with a warning if the cap is reached first. Probes cost setup time only; the cost of each draw does not depend on how many there are.

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"]. SDM requires at least one WX column; if filtering eliminates all of them a ValueError is raised.

Notes

Direct, indirect and total effects of \(X\) on \(y\) incorporate both the local and lagged-X blocks via the spatial multiplier \((I - \rho W)^{-1}\) and are reported by spatial_effects().

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):

  1. If only LM-Lag is significant → SAR.

  2. If only LM-Error is significant → SEM.

  3. 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.

  4. 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

See also

spatial_diagnostics

Compute the Bayesian LM test statistics.

References

Koley and Bera [2024], Anselin et al. [1996], Elhorst [2014]

spatial_effects(return_posterior_samples=False)[source]

Compute Bayesian inference for direct, indirect, and total impacts.

Computes impact measures for each posterior draw, then summarizes the posterior distribution with means, 95% credible intervals, and Bayesian p-values. This is the fully Bayesian analog of the simulation-based approach in LeSage and Pace [2009] and the asymptotic variance formulas in Arbia et al. [2020].

Models without a spatial lag on y do not exhibit global feedback propagation through \((I-\\rho W)^{-1}\). However, models with spatially lagged covariates (SLX, SDEM) can still have non-zero neighbor spillovers captured in the indirect term.

Parameters:
return_posterior_samples : bool, optional

If True, return a (DataFrame, dict) tuple where the dict contains the full posterior draws under keys "direct", "indirect", and "total". Default False.

Returns:

If return_posterior_samples is False (default), returns a DataFrame indexed by feature names with columns for posterior means, credible-interval bounds, and Bayesian p-values.

If return_posterior_samples is True, returns (DataFrame, dict) where the dict has keys "direct", "indirect", "total", each mapping to a (G, k) array of posterior draws.

Return type:

pd.DataFrame or tuple of (pd.DataFrame, dict)

summary(var_names=None, **kwargs)[source]

Return posterior summary table.

Parameters:
var_names : list, optional

Variable names to include in the summary.

**kwargs

Additional arguments passed to arviz.summary().

Returns:

Posterior summary statistics.

Return type:

pandas.DataFrame