Spatial Diagnostics

GearyCResult

Geary's c with its reference moments, returned by geary_c.

Fields

• C_stat: the statistic. Its null expectation is 1; below 1 indicates positive spatial autocorrelation and above 1 negative — the opposite direction to Moran's I.
• E_C: expectation under the null, 1.
• Var_C: variance under the null.
• z_stat: (C_stat - E_C) / sqrt(Var_C), so a negative z_stat means positive autocorrelation.
• p_value: two-sided.

LISAResult

Local indicators of spatial association, returned by moran_local. All four vectors are length N and aligned with the units of y.

Fields

• local_I: the local Moran statistic per unit. These sum to the global statistic up to a scale factor.
• pseudo_p: conditional-permutation p-value per unit.
• cluster_type: one of :HH, :LL, :HL, :LH or :ns — note these are Symbols, not strings. :HH and :LL mark clusters of high and of low values; :HL and :LH mark spatial outliers; :ns marks units with pseudo_p > 0.05.
• z_scores: standardized y, the basis of the H/L classification.

LMTestResult

One Lagrange-multiplier specification test, returned by lm_test_lag and its siblings.

Fields

• stat: the test statistic, asymptotically chi-squared with 1 degree of freedom under the null of no spatial dependence.
• p_value: upper-tail probability.
• test_name: which test produced it.

MoranResult

Global Moran's I with its reference moments, returned by moran_global.

Fields

• I_stat: the statistic. Positive means like values cluster together, negative means neighbours tend to differ.
• E_I: expectation under no spatial autocorrelation, -1/(N-1) — note that it is slightly negative, not zero.
• Var_I: variance under the same null.
• z_stat: (I_stat - E_I) / sqrt(Var_I).
• p_value: two-sided.
• method: :asymptotic or :permutation, recording how p_value was obtained.

geary_c(y, W)

Geary's c, a test for spatial autocorrelation built on differences between neighbours.

\[c = ((N-1) / (2 S_0)) * sum_ij w_ij (y_i - y_j)^2 / sum_i (y_i - ybar)^2\]

Where Moran's I is a cross-product statistic and so responds to the global pattern, Geary's c sums squared differences across linked pairs and is therefore more sensitive to local, short-range variation. Using both is informative when they disagree.

Returns

A GearyCResult.

Examples

g = geary_c(y, W)
g.C_stat < 1      # positive autocorrelation

See also moran_global.

References

Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer.

lm_test_error(y, X, W)

Anselin's LM test for spatially autocorrelated errors.

Tests H0: lambda = 0 in y = X*beta + u, u = lambda*W*u + e, from OLS residuals. Chi-squared with 1 degree of freedom.

Like lm_test_lag it is not robust to the other form of dependence; see lm_test_robust_error and lm_tests. X must include an intercept column; none is added for you.

Returns

An LMTestResult.

References

Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer.

lm_test_lag(y, X, W)

Anselin's LM test for an omitted spatial lag of the dependent variable.

Fitted by OLS under the null of no spatial dependence, this tests H0: rho = 0 in y = rho*W*y + X*beta + e against the alternative that a lag belongs in the model. The statistic is chi-squared with 1 degree of freedom.

It is not robust: a spatial error process also inflates it, so a rejection on its own does not distinguish lag from error dependence. Use lm_test_robust_lag, or lm_tests for the whole battery. X must include an intercept column; none is added for you.

Returns

An LMTestResult.

References

Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer.

lm_test_robust_error(y, X, W)

LM test for spatially autocorrelated errors, robust to the presence of an omitted spatial lag.

The counterpart to lm_test_robust_lag; together they are what separates an error process from a lag process when both plain tests reject. X must include an intercept column; none is added for you.

Returns

An LMTestResult.

References

Anselin, L., Bera, A., Florax, R. & Yoon, M. (1996). Regional Science and Urban Economics, 26(1).

lm_test_robust_lag(y, X, W)

LM test for an omitted spatial lag, robust to the presence of spatial error dependence.

This is the test to read when both plain statistics reject. It corrects lm_test_lag for local misspecification of the error process, so a rejection points at the lag specifically rather than at spatial dependence in general. X must include an intercept column; none is added for you.

Returns

An LMTestResult.

See also lm_tests, lm_test_robust_error.

References

Anselin, L., Bera, A., Florax, R. & Yoon, M. (1996). Simple diagnostic tests for spatial dependence. Regional Science and Urban Economics, 26(1).

lm_tests(y, X, W) -> NamedTuple

Run the full Anselin specification-search battery and return all four tests as (; lag, error, robust_lag, robust_error).

This is the usual way to decide which spatial model to fit. The standard reading:

Each element is an LMTestResult.

Examples

t = lm_tests(y, hcat(ones(N), X), W)      # X needs the intercept column
t.lag.p_value, t.robust_lag.p_value

# fit_spatial does add its own intercept, so pass the bare X there
m = t.robust_lag.p_value < t.robust_error.p_value ?
        fit_spatial(SAR(), W, X, y) : fit_spatial_error(W, X, y)

References

Anselin, L., Bera, A., Florax, R. & Yoon, M. (1996). Regional Science and Urban Economics, 26(1).

moran_global(y, W; inference = :asymptotic, n_perm = 999, rng = Random.default_rng())

Global Moran's I, the standard test for spatial autocorrelation in y.

\[I = (N/S_0) * (z' W z) / (z' z)\]

where z is y in deviations from its mean and S_0 is the sum of all weights. When W is row-standardized S_0 = N and I reduces to the slope of Wz on z, which is why it can be read off a Moran scatter plot (plot_moran_scatter).

Keywords

• inference: :asymptotic uses the normal approximation with the exact finite-sample variance, correcting for the kurtosis of y. :permutation instead randomly reassigns y across locations n_perm times and reports the rank of the observed statistic, which makes no distributional assumption and is the safer choice for small N or heavy-tailed y.
• n_perm, rng: used only when inference = :permutation.

Returns

A MoranResult.

Examples

W = make_rook_weights(10)
moran_global(y, W)
moran_global(y, W; inference = :permutation, n_perm = 9999)

m = fit_spatial(SAR(), W, X, y)
moran_global(residuals(m), W)      # did the model absorb the dependence?

See also moran_local, geary_c, lm_tests.

References

Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer.

moran_local(y, W; n_perm = 999, rng = Random.default_rng())

Local Moran's I per unit, with conditional-permutation inference and LISA cluster classification.

Where moran_global asks whether there is clustering anywhere, this asks where it is. Inference is conditional: unit i's own value is held fixed while the remaining N-1 values are permuted among the other locations, which is the appropriate null for a local statistic.

Each unit is then labelled by the sign of its own standardized value and of its spatial lag: :HH and :LL for clusters, :HL and :LH for spatial outliers, and :ns where pseudo_p > 0.05.

Returns

A LISAResult.

Examples

lisa = moran_local(y, W; n_perm = 9999)
count(==(:HH), lisa.cluster_type)
plot_local_moran(lisa, coords)          # needs a Makie backend

References

Anselin, L. (1995). Local indicators of spatial association — LISA. Geographical Analysis, 27(2).