Spatially informed dimension reduction is actually not new, and dates
back to at least 1985, with Wartenberg’s crossover of Moran’s I and PCA
(Wartenberg 1985), which was generalized
and further developed as MULTISPATI PCA (Dray,
Saı̈d, and Débias 2008), implemented in the `adespatial`

package on CRAN. In short, while PCA tries to maximize the variance
explained by each PC, MULTISPATI maximizes the product of Moran’s I and
variance explained. Also, while all the eigenvalues from PCA are
non-negative, because the covariance matrix is positive semidefinite,
MULTISPATI can give negative eigenvalues, which represent negative
spatial autocorrelation, which can be present and interesting but is not
as common as positive spatial autocorrelation and is often masked by the
latter (Griffith 2019).

In single cell -omics conventions, let \(X\) denote a gene count matrix whose columns are cells or Visium spots and whose rows are genes, with \(n\) columns. Let \(W\) denote the row normalized \(n\times n\) adjacency matrix of the spatial neighborhood graph of the cells or Visium spots, which does not have to be symmetric. MULTISPATI diagonalizes a symmetric matrix

\[ H = \frac 1 {2n} X(W^t+W)X^t \]

However, the implementation in `adespatial`

is more
general and can be used for other multivariate analyses in the duality
diagram paradigm, such as correspondence analysis; the equation
above is simplified just for PCA, without having to introduce the
duality diagram here.

Voyager 1.2.0 (Bioconductor 3.17) has a much faster implementation of
MULTISPATI PCA based on `RSpectra`

.
See benchmark here.

Below is a list of vignettes that use MULTISPATI PCA. The links point to the sections that use MULTISPATI PCA. The corresponding Google Colab notebooks are also linked to. The list is sorted by technology.

Vignette | Colab Notebook | Description |
---|---|---|

MULTISPATI PCA and negative spatial autocorrelation | Colab Notebook | Run MULTISPATI PCA on MERFISH mouse liver dataset, and compare the results to those from non-spatial PCA |

## References

*J. Veg. Sci.*19 (1): 45–56.

*Stats*2 (3): 388–415.

*Geogr. Anal.*17 (4): 263–83.