**Change Detection in Spatial Data and Multivariate
Statistics**

**Allan
Aasbjerg Nielsen**

**Technical**** ****University**** of ****Denmark**

Informatics
and Mathematical Modelling

http://www.imm.dtu.dk/~aa

Abstract

Change
detection is an important subject in many computer vision and remote sensing
applications based on temporally dynamical data. Computer vision examples include industrial
inspection and process control. Remote
sensing application areas include for example agriculture, forestry,
oceanography and environmental monitoring.

Simple change detection methods are often inadequate since they are typically sensitive to differences in calibration such as offset and gain in the measuring device used to record the data. Also, simple change detection methods applied to spatial data are typically non-spatial. This indicates a need for methods based on multivariate and spatial statistics.

Change
detection methods typically include (in this notation **X** is a column vector of *p* variables recorded at each observation or pixel at one point in
time, and **Y** is the same vector at the same geographical location at
another point in time; **a** and **b** are column vectors of coefficients
for calculating the linear combinations involved)

•
Use of *ad hoc* transformations such as
the normalized difference vegetation index (NDVI).

•
Principal component analysis (PCA)
on concatenated data: maximize the variance Var{**a**^{T}[**X**^{T}**Y**^{T}^{T}}.

•
Simple differencing: calculate **X
**– **Y**.

•
PCA on simple differences: maximize
the variance Var{**a**^{T}(**X **– **Y**)}.

•
Canonical correlation analysis
(CCA) based multivariate alteration detection (MAD): maximize the variance Var{**a**^{T}**X
**– **b**^{T}**Y**}. CCA
maximizes linear combinations **a**^{T}**X **of **X **and **b**^{T}**Y
**of **Y**,** **Corr{**a**^{T}**X**,**
b**^{T}**Y**}.

•
MAF post-processing of MAD
variables: maximize the correlation between a linear combination **a**^{T}**X**
of **X** (after MAD transformation) and the same linear combination of the
same data spatially shifted, Corr{**a**^{T}**X**(**r**), **a**^{T}**X**(**r**+**D**)}.

PCA, CCA, MAD
and MAF transformations can all be found by eigenvalue decomposition. As opposed to simple differencing and
derivatives thereof, MAD and MAF/MAD variables are invariant to linear and affine
transformations, which means that they are insensitive to for example differences
in offset and gain in a measuring device at the two points in time. MAF post-processing introduces a desired spatial
element. Simultaneous inspection of spatial
patterns and correlations with the original variables facilitates interpretation
of MAD and MAF/MAD variables.