Nonstationarity

A random variable is a mathematical function that translates a construct into numbers that behave in some random way. A space-time stochastic process is a mathematical function that yields a time-series random variable for each location on a map. A geographic distribution is a map (e.g., a GIS layer) taken from some space-time stochastic process. Each of its location-specific variables possesses statistical quantities, such as measures of central tendency and dispersion. When a map for only a single point in time is observed, these statistical properties often are assumed to be constant across all locations. In contrast, spatial nonstationarity means statistical parameters of interest are dependent upon and unstable across location. Most spatial statistical techniques require data to be either stationary or modified in some way that mimics stationarity.

Spatial nonstationarity may materialize as systematic geographic trends in a variable across locations that may be described with mathematical functions of the relative arrangement of these locations (e.g., their coordinates). Regionalized trends may materialize as patches that can be denoted with regional indicator variables. Correlational trends can mirror important or influential covariates that may have complex map patterns. Meanwhile, variation in the variance of a variable can occur from location to location and may be detected by partitioning a landscape into a relatively small number of arbitrary subregions and then calculating a homoscedasticity test statistic (e.g., Levene or Bartlett) that compares the resulting set of regional variances.

Prevailing levels of positive spatial autocorrelation can trick a researcher into thinking that stationary data are nonstationary by interacting with a variable's frequency distribution to inflate its variance: Bell-shaped curves flatten; Poisson distributions acquire increasing numbers of zeroes and outliers; and binomial distributions become uniform, then bimodal, and ultimately dichotomous in shape. Positive spatial autocorrelation frequently translates nonconstant variance across the aspatial magnitude of values into nonstationary spatial variance; this variance instability often can be handled with a Box-Cox power transformation. Meanwhile, spatial dependence that varies across subregions of a landscape renders anomalies in a Moran scatterplot and a oneand two-dimensional semivariogram (e.g., distinct concomitant patterns) and may be measured with LISA and Getis-Ord G statistics.

Spatial nonstationarity can be modeled in five ways: specifying a nonconstant mean response, yielding stationary residuals; applying a mathematical transformation (e.g., Box-Cox) to modify a measurement scale so that the transformed variable behaves as though it were stationary; employing a suitable nonnormal probability model; weighting or stratifying [Page 320]observations in separate data subsets that are small enough to be considered stationary; and specifying relatively simple nonstationary spatial dependence. Employing conceptually relevant covariates coupled with a spatial autocorrelation term in a mathematical function describing a variable frequently captures much of any detected spatial nonstationarity; spatial autocorrelation often is the source of detected spatial nonstationarity. Statistical normal curve theory has motivated methodology that attempts to sculpt a variable to be more bell-shaped, with backtransformed calculations recapturing nonstationarity. But a mathematical transformation may not exist that can achieve this end (e.g., if a variable is binary). Generalized linear models allow Poisson and binomial rather than only normal probability models to be utilized, capturing nonstationarity with nonlinear relationships and in many (but not all) cases making mathematical transformation usage obsolete. Weighting, which is especially useful when nonconstant variance is present, effectively is division by standard deviations, much as is done when calculating z-scores. Stratification dramatically increases the number of parameters to be estimated, one set for each subregion, using pooling for their simultaneous estimation. Finally, anisotropic spatial autoregressive or semivariogram models attempt to account for directionality in spatial dependencies.

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