bayespecon.models.flow.SEMFlow¶

class bayespecon.models.flow.SEMFlow(y, G, X, **kwargs)[source]¶

Bayesian spatial-error flow model with three free spatial parameters.

\[y = X\beta + u, \qquad B u = \varepsilon, \qquad B = I_N - \lambda_d W_d - \lambda_o W_o - \lambda_w W_w, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I_N)\]

where \(W_d = I_n \otimes W\), \(W_o = W \otimes I_n\), \(W_w = W \otimes W\). The Kronecker spatial structure is identical to SARFlow, but the spatial filter acts on the disturbance rather than the dependent variable. Equivalently the model implies a Gaussian likelihood with covariance \(\sigma^2 (B^\top B)^{-1}\).

Marginal mean is \(\mathbb{E}[y] = X\beta\), so there are no \(X\)-mediated spatial spillovers — the LeSage / Thomas-Agnan decomposition reduces to the closed-form expressions used by OLSFlow (direct effect equals \(\beta\), network effect equals zero). Use SARFlow if spillovers from observed covariates are of interest.

Parameters:¶

y : array-like, shape (n, n) or (N,)¶

Observed origin-destination flow matrix or its vec-form.

G : libpysal.graph.Graph¶

Row-standardised spatial graph on n units.

X : np.ndarray or pandas.DataFrame, shape (N, p)¶

Full origin-destination design matrix with \(N = n^2\) rows. DataFrame columns are preserved as feature names.

col_names : list of str, optional

Column labels for X. Inferred from a DataFrame if omitted; otherwise defaults to ["x0", "x1", ...].

k : int, optional

Number of regional attribute columns (destination/origin variable pairs). Inferred from dest_*/orig_* column names when the standard LeSage layout is used.

logdet_method : str, default "traces"

Log-determinant method. Only "traces" is supported here.

restrict_positive : bool, default True

If True, use pm.Dirichlet("lam_simplex", a=ones(4)) to enforce \(\\lambda_d, \\lambda_o, \\lambda_w \\geq 0\) and \(\\lambda_d + \\lambda_o + \\lambda_w \\leq 1\). If False, three independent pm.Uniform(lam_lower, lam_upper) priors are used with a differentiable quadratic-wall stability potential.

miter : int, default 30

Trace polynomial order for the log-determinant.

titer : int, default 800

Geometric tail cutoff for the log-determinant series.

trace_riter : int, default 50

Number of Monte Carlo probes for trace estimation.

trace_seed : int, optional

Random seed for trace estimation reproducibility.

symmetric_xo_xd : bool, optional

If None (default), origin and destination design blocks are compared and symmetry is auto-detected.

priors : dict, optional

Override default priors. Supported keys:

• beta_mu : float, default 0.0 — Normal prior mean for beta.

• beta_sigma : float, default 1e6 — Normal prior std for beta.

• sigma_sigma : float, default 10.0 — HalfNormal prior std for sigma.

• lam_lower : float, default -1.0 — Lower bound of Uniform prior on each λ (only when restrict_positive=False).

• lam_upper : float, default 1.0 — Upper bound of Uniform prior on each λ (only when restrict_positive=False).

Notes

Implementation: PyMC body uses precomputed lags of both y and X (self._Wd, self._Wo, self._Ww applied to self._X_design) so that the residual \(B u = B y - B X \\beta\) is expressible as a linear combination of fixed quantities — no symbolic sparse mat-vec is required. The Jacobian \(\\log|B|\) reuses the same trace-based polynomial as SARFlow.

__init__(y, G, X, **kwargs)[source]¶

Methods

__init__(y, G, X, **kwargs)
fit([draws, tune, chains, target_accept, ...])	Draw samples from the posterior.
fit_approx([draws, n, method, random_seed, ...])	Fit a variational approximation and return posterior draws.
posterior_predictive([n_draws, random_seed, ...])	Draw posterior-predictive samples y_rep.
spatial_diagnostics()	Run Bayesian LM specification tests for flow models.
spatial_diagnostics_decision([alpha, ...])	Return a model-selection decision from Bayesian LM test results.
spatial_effects([draws, ...])	Summarise posterior origin/destination/intra/network/total effects.
summary([var_names])	Return posterior summary table via ArviZ.

Attributes

approximation	Return the most recent PyMC variational approximation, if any.
inference_data	Return ArviZ InferenceData from the most recent fit, or None.
pymc_model	Return the PyMC model used for the most recent fit, or None.

property approximation[source]¶

Return the most recent PyMC variational approximation, if any.

fit(draws=2000, tune=1000, chains=4, target_accept=0.9, random_seed=None, store_lambda=False, idata_kwargs=None, **sample_kwargs)[source]¶

Draw samples from the posterior.

Parameters:¶

draws : int, default 2000¶

Number of posterior samples per chain (after tuning).

tune : int, default 1000¶

Number of tuning (warm-up) steps per chain.

chains : int, default 4¶

Number of parallel chains.

target_accept : float, default 0.9¶

Target acceptance rate for NUTS.

random_seed : int, optional¶

Seed for reproducibility.

store_lambda : bool, default False¶

If True, include the high-dimensional fitted mean lambda in the stored posterior. Leaving this False reduces memory and conversion overhead for Poisson flow models.

idata_kwargs : dict, optional¶

Forwarded to pm.sample. Defaults to {"log_likelihood": True} so that az.loo / az.waic / az.compare work out of the box; for SAR flow variants the captured Gaussian log-likelihood is post-processed to add the Jacobian contribution from log|I_N - rho_d W_d - rho_o W_o - rho_w W_w|.

**sample_kwargs¶

Additional keyword arguments forwarded to pm.sample.

Return type:¶

arviz.InferenceData

fit_approx(draws=2000, n=10000, method='advi', random_seed=None, store_lambda=False, compute_log_likelihood=True, **fit_kwargs)[source]¶

Fit a variational approximation and return posterior draws.

Parameters:¶

draws : int, default 2000¶

Number of samples to draw from the fitted approximation.

n : int, default 10000¶

Number of optimisation iterations for pm.fit.

method : {"advi", "fullrank_advi"}, default "advi"¶

Variational inference family to fit.

random_seed : int, optional¶

Seed for optimisation and posterior sampling.

store_lambda : bool, default False¶

If True, keep the high-dimensional fitted mean lambda in the posterior draws.

compute_log_likelihood : bool, default True¶

If True, compute pointwise log-likelihood after sampling and attach to the InferenceData (with Jacobian correction for SAR flow variants), enabling az.loo / az.waic.

**fit_kwargs¶

Additional keyword arguments forwarded to pm.fit.

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

Return ArviZ InferenceData from the most recent fit, or None.

posterior_predictive(n_draws=None, random_seed=None, parallel=-1)[source]¶

Draw posterior-predictive samples y_rep.

For each (subsampled) posterior draw, simulates a new flow vector y_rep from the implied data-generating process by solving the sparse system A(rho) y_rep = X β + ε (Gaussian) or y_rep ~ Poisson(exp(A^{-1} X β)) (Poisson variants).

Parameters:¶

n_draws : int, optional¶

Number of posterior draws to use. Defaults to all available.

random_seed : int, optional¶

Seed for the noise/Poisson sampler.

parallel : int or None, default -1¶

Number of worker threads for the per-draw loop. -1 uses os.cpu_count(); None/0/1 forces sequential execution. Reproducibility under a fixed random_seed is preserved across worker counts via SeedSequence.spawn.

Returns:¶

Array of shape (n_draws, N) with posterior-predictive flows.

Return type:¶

np.ndarray

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

Return the PyMC model used for the most recent fit, or None.

spatial_diagnostics()[source]¶

Run Bayesian LM specification tests for flow models.

Iterates over the class-level _spatial_diagnostics_tests registry and returns a tidy DataFrame with one row per test. See bayespecon.models.base.SpatialModel.spatial_diagnostics() for the column schema.

Raises:¶

RuntimeError – If the model has not been fit yet.

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

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

Walks the flow decision tree using Bayesian p-values from spatial_diagnostics() and recommends either OLSFlow (no spatial dependence detected) or SARFlow (at least one direction is significant).

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 tree. "graphviz" returns a graphviz.Digraph (with ASCII fallback if graphviz is not installed).

Return type:¶

str or graphviz.Digraph

spatial_effects(draws=None, return_posterior_samples=False, ci=0.95, mode='auto', parallel=-1)[source]¶

Summarise posterior origin/destination/intra/network/total effects.

Wraps _compute_spatial_effects_posterior() to produce a tidy DataFrame indexed by predictor with posterior means, credible-interval bounds, and Bayesian p-values for each effect type (origin, destination, intra, network, total). Following Thomas-Agnan & LeSage (2014, §83.5.2), when destination and origin design blocks differ the decomposition is reported separately for shocks applied to each side.

Parameters:¶

draws : int, optional¶

Maximum number of posterior draws to use. Defaults to all.

return_posterior_samples : bool, default False¶

If True, also return the underlying posterior-draw arrays.

ci : float, default 0.95¶

Credible-interval coverage.

mode : {"auto", "combined", "separate"}, default "auto"¶

Controls whether destination- and origin-side effects are summed or reported separately. "auto" collapses to combined when the destination and origin design blocks are identical (self._symmetric_xo_xd) and reports both sides otherwise. "combined" always sums; "separate" always reports both.

parallel : int or None, default -1¶

Number of worker threads for the per-draw effects loop. -1 uses os.cpu_count(); None/0/1 forces sequential execution. Ignored by closed-form (OLSFlow, SEMFlow) variants.

Returns:¶

Long-format summary indexed by (predictor, side, effect) where side is one of "combined", "dest", "orig".

Return type:¶

pandas.DataFrame, or (DataFrame, dict)

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

Return posterior summary table via ArviZ.

Parameters:¶

var_names : list, optional¶

Variable names to include. Defaults to all parameters.

**kwargs¶

Additional keyword arguments forwarded to az.summary.

Return type:¶

pandas.DataFrame