netneurotools.spatial.morans_i

netneurotools.spatial.morans_i(annot, weight, use_numba=False, standardized=False)[source]

Calculate Moran’s I for spatial autocorrelation.

Parameters:

  • annot (array-like, shape (n,)) – Array of annotations to calculate Moran’s I for.

  • weight (array-like, shape (n, n)) – Spatial weight matrix. Note that we do not explicitly check for symmetry in the weight matrix, nor zero-diagonal elements.

  • use_numba (bool, optional) – Whether to use numba for calculation. Default: True (if numba is installed).

  • standardized (bool, optional) – Whether to standardize the Moran’s I statistic with respect to its expectation and variance under the null assumption that annotation values are sampled from a normal distribution.

Returns:

morans_i – Moran’s I value for the given annotations and weight matrix.

Return type:

float

See also

netneurotools.spatial.spatial_stats.local_morans_i

Notes

Moran’s I [1,2] is calculated as:

\[I = \frac{n}{\sum_{i=1}^{n} \sum_{j=1}^{n} w_{ij}} \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} w_{ij} (x_i - \bar{x})(x_j - \bar{x})}{\sum_{i=1}^{n} (x_i - \bar{x})^2}\]

where \(n\) is the number of observations, \(w_{ij}\) is the spatial weight between observations \(i\) and \(j\), \(x_i\) is the annotation for observation \(i\), and \(\bar{x}\) is the mean annotation value.

The value can be tested using the R pacakge spdep:

x <- rnorm(100)
m <- matrix(runif(100*100), nrow=100)
w <- mat2listw(m)
moran(v, w, 100, Szero(w))
# or
moran.test(x, w)

Standardized Moran’s I [2] is calculated as:

\[Z_I = \frac{I - \mathbb{E}[I]}{\sqrt{\mathrm{Var}(I)}}\]

where

\[\mathbb{E}[I] = -\frac{1}{n-1}\]

and

\[\mathrm{Var}(I) = \frac{n^2 S_1 - n S_2 + 3 S_0^2} {(n-1)(n-2)(n-3) S_0^2} - \left( \frac{1}{n-1} \right)^2,\]

with

\[S_0 = \sum_i\sum_j w_{ij},\]
\[S_1 = \frac{1}{2} \sum_{ij} (w_{ij} + w_{ji})^2,\]

and

\[S_2 = \sum_i \left( \sum_j w_{ij} + \sum_j w_{ji} \right)^2.\]

References