S Patio - Temporal H Ypothesis T Esting in M Odel R Esiduals

Spatio-temporal models add complexity, but not necessarily value, to some climate analyses. To confirm the presence of spatio-temporal dependence, a hypothesis test should be conducted. The Space-Time Index is one statistic to detect such dependence; this statistic is simple, easily interpretable, and used in several disciplines. In an application to Indian monsoon precipitation thresholds, residuals from logit-normal mixed models were tested for spatio-temporal dependence. No evidence of dependence was detected in high thresholds. I. SPACE-TIME INDEX (STI) METHODOLOGY Spatio-temporal dependence in climate related data should not be ignored in modeling efforts. Along with graphical diagnostics, it is advantageous to have formal hypothesis testing procedures in place to understand the exact nature of spatio-temporal dependence, and to evaluate whether a given statistical model is adequate in capturing such dependencies in the data. Elegant procedures exist for testing separability, symmetry [1], or stationarity [2] in the data. Another simple method still in current use ([3], [4], [5]) is the Space-Time Index (STI) [6] which combines Moran’s I [7] and the Durbin-Watson statistic [8]. The STI is interpretable and is useful for conveying information to stakeholders who may not be experts on spatiotemporal data patterns and related statistical models. Our goal is to evaluate the performance of STI for modeling dependencies in climate data. Based on that, we will propose and conduct future studies on computational methodology-based generalizations to overcome the limitations of STI as it applies to non-stationary, highdimensional data from climate applications. In its current form, the STI tests the null hypothesis of no spatio-temporal dependence in a vector autoregression process over the entire spatial field given by: yt = Ayt−1 + t where t ∈ {1, ..., T} represents discrete time. Let i, j ∈ {1, ..., n} represent stations, ȳ = 1 nT ∑T t=1 ∑n i=1 yi,t, and cij,t−1 = 1 if stations i and j are neighbors during time t − 1 and cij,t−1 = 0 otherwise. Then, Corresponding author: L.R. Dietz ([email protected]), School of Statistics, University of Minnesota, Minneapolis, MN STI = n(T − 1) ∑T t=2 ∑n i=1 ∑n j=1 cij,t−1 ∗ ∑T t=2 ∑n i=1 ∑n j=1 cij,t−1(yi,t − ȳ)(yj,t−1 − ȳ) ∑T t=1 ∑n i=1(yi,t − ȳ)2 Under an asymptotic normality assumption, the sampling distribution of STI can be used to conduct the test. II. STI SIMULATION RESULTS Structural assumptions are imposed to run simulations. First, assume the neighbors of a point remain constant, i.e. cij,t−1 = cij,t for all t. Neighbors of station i (Ni) are weighted by scaled distances (wi) where wi satisfies: 1) wij=0 if j 6∈ Ni, wij= ∑ j∈Ni distj ∑ j∈Ni distj for j ∈ Ni 2) ∑n j=1 wij = 1 for all i. Next, generate the ith vector with time parameter (ρtime) and space parameter (ρspace) as: t = 1 : yi,1 i.i.d ∼ N (0, σ) for all i t > 1 : yi,t = ρtime · yi,t−1 + ρspace · T ∑