J Geograph Syst (2002) 4:43–51
Modeling spatial dependence in high spatial resolution hyperspectral data sets Daniel A. Griffith Department of Geography, Syracuse University, Syracuse, NY 13244-1020, USA (e-mail: griffi
[email protected]) Received: 25 February 2001 / Accepted: 2 August 2001
Abstract. As either the spatial resolution or the spatial scale for a geographic landscape increases, both latent spatial dependence and spatial heterogeneity also will tend to increase. In addition, the amount of georeferenced data that results becomes massively large. These features of high spatial resolution hyperspectral data present several impediments to conducting a spatial statistical analysis of such data. Foremost is the requirement of popular spatial autoregressive models to compute eigenvalues for a row-standardized geographic weights matrix that depicts the geographic configuration of an image’s pixels. A second drawback arises from a need to account for increased spatial heterogeneity. And a third concern stems from the usefulness of marrying geostatistical and spatial autoregressive models in order to employ their combined power in a spatial analysis. Research reported in this paper addresses all three of these topics, proposing successful ways to prevent them from hindering a spatial statistical analysis. For illustrative purposes, the proposed techniques are employed in a spatial analysis of a high spatial resolution hyperspectral image collected during research on riparian habitats in the Yellowstone ecosystem. Key words: Eigenvalue, spatial autocorrelation, spatial autoregression, geostatistics, spatial heterogeneity, high spatial resolution hyperspectral JEL classification: C49, C13, R15 1 Introduction Cressie et al. (1996) note that ‘‘the sheer size of a massive data set may challenge and, ultimately, defeat a statistical methodology that was designed for smaller data sets.’’ In addition, Cressie et al. (1998) contrast geostatistical and spatial autoregressive model applications to environmental data. This latter work relates to that reported by Griffith and Csillag (1993), Griffith and Layne (1997, 1999), and Griffith et al. (1996). The purpose of this paper is to discuss these two features with respect to high spatial resolution,
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hyperspectral (HSHR) data. Accordingly, it contributes to the set of analytic tools for assessing the statistical significance of patterns. 2 Spatial dependence One of the drawbacks associated with popular spatial autoregressive models is the need to compute eigenvalues for a row-standardized geographic weights matrix that can be used to analyze an image with spatial statistics. Current computer technology supports this task for matrices associated with images involving thousands of pixels, but not hundreds-of-thousands, millions or billions of pixels. Griffith (2000) offers an excellent approximation solution for this problem. The necessary eigenvalues are approximately equal to k^pq;^c ¼ Isign ½ðI Icorrect ÞjðkQ 1P þ 1Q kP Þ=2jdiag c 1pq þ Icorrect ðkQ 1P þ 1Q kP Þ=2;
ð1Þ
where the initial values associated with the geographic weights matrix are pk kk ¼ cosðK1 Þ=2 ðk ¼ 0; 1; 2; . . . ; K 1Þ, where K ¼ P or Q and n ¼ PQ; Q P P P k^2 ¼ 18PQþ11P þ11Qþ12, 0 < c 1 (with c^ iteratively estimated so that p¼1 q¼1
pq;c
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when P 4 and Q 4), subscript ‘‘diag’’ denotes a diagonal matrix, j ? j denotes absolute value, the diagonal entries of diagonal matrix Icorrect ¼ 1 if an approximate eigenvalue is correct (note: a subset of the eigenvalues are known analytically) and 0 otherwise, and of diagonal matrix Isign ¼ )1 if equation (1) is negative and 1 otherwise. The exponent c must be contained in the interval (0, 1) to inflate the sum of squares, and ð1; 1Þ to deflate the sum of squares, since all of the eigenvalues computed by expression (1) are contained in the interval [)1, 1]. The sign indicator variable, Isign, is used here because virtually all eigenvalues of matrix W appear as ± square root pairs. These eigenvalues allow the spatial autoregressive normalizing factor (i.e., the Jacobian term) to be simplified. They can be used to calibrate the equation approximating this factor, namely (see Griffith 1990, p. 187) J^ðqÞ ¼ 2an LN ðdn Þ an LN ðdn þ qÞ an LN ðdn qÞ;
ð2Þ
or the slightly better but discontinuous approximation (see Griffith 1999) J^ðqÞ ¼ 0; if q ¼ 0; and LN ð1 þ qÞ ^ þ 1 dn LN ð1 þ qÞ J ðqÞ ¼ an q LN ð1 qÞ þ 1 dn LN ð1 qÞ ; þ an q
q 6¼ 0:
ð3Þ
Equations (2) or (3) need to be calibrated only once for a given remotely sensed image, in order to determine the given pair of coefficients. Calibrated versions of these equations allow the n-by-n ¼ PQ-by-PQ matrix appearing in a spatial autoregressive model to be avoided in its full form. To illustrate the utility of equations (1)–(3), consider a rectangular set of pixels, having an east–west dimension of 3,500 pixels and a north–south dimension of 3,000 pixels; in other words, the image contains 10,500,000
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pixels. For this situation, c^ ¼ 0:99991 is the calibrated value calculated for Q 2 Q P P P P P P the exponent appearing in equation (1) (j k^pq;c k2pq j ¼ 1:8, with
p¼1 q¼1 p¼1 q¼1 18PQþ11P þ11Qþ12 ¼ 2625993:22Þ. The calibrated coefficients for equation (2) 72 are an ¼ 1599521:464 and dn ¼ 1:147 (RESS ¼ 5.2 · 10)4); the calibrated
coefficients for equation (3) are an ¼ 0:374 and dn ¼ 0:996 (RESS ¼ 2.1 · 10)5). Meanwhile, as spatial resolution becomes increasingly finer, the detected level of spatial autocorrelation tends to increase. Remotely sensed data with a coarser resolution than that for the new hyperspectral imagery already yield values of q^ in the 0.85-to-0.99 range. The finer spatial resolution may well result in values of q^ that try to exceed the maximum value of 1, implying the need for higher order spatial autoregressive models. Equations (2) and (3) immediately extend to only a limited family of such higher order models. Other extensions need to be established. 3 Spatial heterogeneity As either the spatial resolution (related to infill asymptotics, where n increases by drawing more and more samples from a region) or the spatial scale (related to increasing domain asymptotics, where n increases by extending the borders of a region) for a geographic landscape increases, spatial heterogeneity also will tend to increase. Part of this heterogeneity relates to a more detailed inventory of the mixture of attributes, while part relates to diversity across regional settings. One effective way of handling this increased variability is to focus on strata or subsets of the data, rendering a between-strata component of variance (Cressie et al. 1996, p. 116). In order to do so, meaningful regionalization schemes are needed; once established, they can be translated into a set of indicator variables. Such indicator variables allow ANOVA and ANCOVA types of models to be specified in order to account for heterogeneity through the mean response. Heterogeneity also can arise from an attribute displaying non-constant variance. One way of accommodating this type of heterogeneity is by employing a Box-Cox power transformation, which tends to inflate excessively small values and shrink excessively large values. Often geographically distributed environmental attribute measures exhibit approximately a log-normal distribution. But spectral bands from previous satellite sensors often have been combined in both linear and multiplicative ways, resulting in considerable heterogeneity that is not necessarily accommodated by a logarithmic transformation. Therefore, experiments need to be conducted that subject the hyperspectral data to Box-Cox transformations. 4 Relationships between spatial autoregressive and geostatistical models Cressie et al. (1998) state the following observations: (1) in their air pollution monitoring example, the spatial autoregressive model appears to be less sensitive to particular data values; (2) the geostatistical approach easily accommodates ‘‘replicate’’ values (i.e., multiple measures associated with a single location);
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(3) the more automated nature of spatial autoregressive model estimating allows easier implementation of subset analysis, such as cross-validation; (4) in their example, the spatial autoregressive model predictions offer considerable improvement over their kriging counterparts; and, (5) geostatistical models seem to produce a greater degree of smoothing than do spatial autoregressive models. One of their conclusions is that ‘‘a great amount of statistical research has focused on the geostatistical approach to spatial analysis, while much less attention has been devoted to the [spatial autoregressive] approach.’’ Meanwhile, Griffith and his colleagues (1993, 1997, 1999) establish several links between spatial autoregressive and geostatistical models, most notably: the simultaneous autoregressive model links to the Bessel semivariogram model, and the conditional autoregressive model links to the exponential semivariogram model. In addition, a spatial EM algorithm specification – which iteratively Estimates missing values followed by Maximizing a likelihood function into which they have been substituted–links to the kriging equations. Griffith and Layne (1999, pp. 430–431) illustrate the similarity of interpolated values rendered by these different techniques. The hyperspectral satellite data may well benefit from an approach that combines both modeling strategies. Geostatistical models supply a feasible means of checking for anisotropy, determining the angle of axis rotation associated with anisotropy, and determining the range of spatial dependence. Spatial autoregressive models supply a feasible means of implementing the general linear model for descriptive and inferential purposes; they offer a way of helping to differentiate between pattern and noise variation in massively large georeferenced data sets. Given that both categories of spatial statistics focus on the concept of spatial autocorrelation, future research needs to seek a clearer articulation between these two classes of model. 5 An example from the Yellowstone ecosystem The high resolution hyperspectral dataset for the Lamar River is for a 350by-450 rectangular region of 5-meter square pixels, is atmospherically corrected, and includes 128 spectral bands (see Jacquez et al. 2002 for a full description). Twelve of these spectral bands include negative values; because these values appear to be artifacts of perhaps the atmospheric correction technique applied to the raw data, they have been treated as missing values in this analysis. Most of the Moran Coefficient (MC) and Geary Ratio (GR) values for these 128 spectral bands indicate the presence of very strong, positive spatial autocorrelation (see Fig. 1). Most of these pairs of values are consistent in their measurements; spectral bands #1 and #128 deviate from this pattern, though. Of note is that all of the bands indicating less than very strong, positive spatial autocorrelation are ones containing suppressed (i.e., negative) values. Conspicuous differences between these incomplete maps and the complete maps is apparent in Fig. 2, too; only the incomplete bands (#1–#3, #64–#67, #123–#128) deviate from essentially horizontal plots of MC and GR values.
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The set of approximate eigenvalues for a complete 450-by-350 pixels map Q P P P k^2pq;c ¼ 39497:51; with have an exponent of c^ ¼ 0:9992ð Q P p¼1 q¼1 PP 2 18PQþ11P þ11Qþ12 kpq ¼ ¼ 39497:39Þ. The calibrated coefficients for 72
p¼1 q¼1
equation (2) are an ¼ 23948:6 and dn ¼ 1:1448 (RESS ¼ 5.3 · 10)2); the calibrated coefficients for equation (3) are an ¼ 0:3744 and dn ¼ 0:9965 (RESS ¼ 2.1 · 10)5). The plot of the Jacobian term for this geographic landscape appears in Fig. 3; its symmetry is attributable to the underlying regular square tessellation of pixels. Employing the Jacobian approximation given by equation (2), a simultaneous autoregressive (SAR) model was estimated for spectral bands #4, #70 and #122. This SAR model furnishes a spatial statistical description of each pixel value for a given spectral band as a
Fig. 1. Scatterplot of spatial autocorrelation indices calculated for each of the 128 spectral bands
Fig. 2. MC and GR values for the 128 spectral bands
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Fig. 3. Jacobian plot for the 350-by-450 Yellowstone image
function of the average of nearby pixel values (as identified by the geographic weights matrix) for the same spectral band. Results of this estimation exercise appear in Table 1. These results indicate that (1) the frequency distributions of the selected spectral band values fail to conform to a bellshaped curve, both before and after spatial autocorrelation is accounted for, (2) as indicated by the MC and GR values reported previously, very strong positive spatial autocorrelation is present in these spectral data, and (3) a spatial statistical model can be estimated for this case of n ¼ 350 450 ¼ 157;500. Heterogeneity across the Yellowstone geographic landscape is apparent from an inspection of the image itself. The northern portion of the image displays far more variability, whereas the southeastern portion of the image displays relatively little variability. A large triangular region (5,382 pixels) constituting the northwest corner of the image has spectral values of 0 for sample spectral bands #4, #70 and #122. Creating an indicator variable to differentiate this subregion from the rest of the image respectively results in 13%, 20% and 14% of the variance being accounted for in these spectral measures. In addition, spectral band #4 exhibits a pronounced gradient across the image, one that accounts for 10.5% of the variance in its measures. Besides this heterogeneity captured by indicator variables in a mean response term, often variance heterogeneity can be addressed with a Box-Cox power transformation. Two spectral bands have been selected for analysis here, namely band #1 and band #128. These two bands were selected because their respective pairs of MC and GR values are the two most inconsistent ones. Both spectral bands benefit considerably from being subjected to a logarithmic transformation; the two specific transformations are LN(band#1 + 15) and LN(band#128 + 94). Boxplots illustrating the impact of these transformations appear in Figs. 4 and 5. Of note is that while the log-band values exhibit far more symmetric distributions, these frequency distributions still markedly differ from bell-shaped curves: for band #1, K-S decreases from 0.191 to 0.129, and for band #128, K-S decreases from 0.215 to 0.176. Regardless, the MC and GR values change very little when the transformations are employed.
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Finally, a sample semivariogram was constructed by randomly selecting 1,000 pixel values from spectral band #70, computing inter-pixel distances and semivariances, and then clustering the resulting semivariances into 50 groups (see Fig. 6). This plot also indicates the presence of very strong, positive spatial autocorrelation. Next, selected semivariogram models were fit to the plot through a distance of 203 lattice units; results of this exercise appear in Table 2. While the spherical and circular models exhibit the best relative error sum of squares (RESS), both the exponential and the Bessel function models have curves that came much closer to the first semivariogram point. 6 Implications Elements of a research agenda are presented here that will help answer the question How can we best combine cutting-edge methods to achieve HSRH descriptions useful for assessing exposure to contaminants in the environment? The focus is on complications arising from the enormous size of and spatial dependencies latent in HSRH data sets. The theme is one of analytic tools for assessing the statistical significance of patterns. The illustrations presented here show that HSRH data sets can be effectively and efficiently analyzed Table 1. SAR model estimation results for selected hyperspectral bands, the Yellowstone image Band
Kolmogorov–Smirnov ðK–SÞ normality evaluation: spectral data
SAR q^
Kolmogorov–Smirnov ðK–SÞ normality evaluation: SAR residuals
Pseudo-R2
4 70 122
0.115 0.061 0.050
0.9584 0.9460 0.9390
0.113 0.047 0.033
0.9037 0.8789 0.8622
Fig. 4. Box-Cox transformation for spectral band #1
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Fig. 5. Box-Cox transformation for spectral band #128
Fig. 6. Sample semivariogram
Table 2. Isotropic semivariogram model parameter estimates Model
Intercept (104)
Slope (104)
Range parameter
RESS
Spherical Circular Exponential Gaussian Bessel function
4.82 5.26 0.83 7.41 4.17
16.70 16.20 21.50 14.20 17.90
120 107 44 59 31
0.006 0.007 0.033 0.014 0.024
with spatial autoregressive and geostatistical techniques. This has been achieved by providing a very accurate approximation to the auto-Gaussian normalizing factor, by promoting the use of Box-Cox power transformations
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to convert spectral band values to ones whose frequency distribution more closely resembles a bell-shaped curve, and by linking the Bessel function semivariogram model with the SAR model to best describe spatial autocorrelation latent in HSRH images. References Cressie N, Olsen A, Cook D (1996) Massive data sets: problems and possibilities, with application to environmental monitoring. In: National Research Council (ed) Massive Data Sets, Proceedings of a Workshop, pp 115–119. National Academy Press, Washington, DC Cressie N, Kaiser M, Daniels M, Aldworth J, Lee J, Lahiri S, Cox L (1998) Spatial analysis of particulate matter in an urban environment. In: Gomez-Hernandez J, Soares A, Froidevaux R (eds) Geostatistics for Environmental Applications, Proceedings of the 2nd European Conference on Geostatistics for Environmental (Valencia, Spain, November 18–20, 1998). Kluwer, Dordrecht Griffith D (1990) A numerical simplification for estimating parameters of spatial autoregressive models. In: Griffith D (ed) Spatial Statistics: Past, Present, and Future, Institute of Mathematical Geography, Ann Arbor, MI, pp 185–195 Griffith D (1999) So what is the actual size of your geographical sample? Paper presented to the 95th annual meeting of the Association of American Geographers, Honolulu, March 24 Griffith D (2000) Eigenfunction properties and approximations of selected incidence matrices employed in spatial analyses. Linear Algebra & Its Applications 321:95–112 Griffith D, Csillag F (1993) Exploring relationships between semi-variogram and spatial autoregressive models. Papers in Regional Science 72:283–296 Griffith D, Layne L (1997) Uncovering relationships between geostatistical and spatial autoregressive models, in the 1996 Proceedings on the Section on Statistics and the Environment, American Statistical Association, pp 91–96 Griffith D, Layne L (1999) A Casebook for Spatial Statistical Data Analysis: A Compilation of Analyses of Different Thematic Datasets. Oxford University Press, NY Griffith D, Layne L, Doyle P (1996) Further explorations of relationships between semivariogram and spatial autoregressive models. In: Mowrer H, Czaplewski R, Hamre R (eds) Spatial Accuracy in Natural Resource and Environmental Sciences: Second International Symposium, Rocky Mountain Forest and Range Experiment Station, General Technical Report RM-GTR-277, Fort Collins, CO, pp 147–154 Jacquez GM, Marcus WA, Aspinall RJ, Greiling DA (2002) Exposure assessment using high spatial resolution hyperspectral (HSRH) imagery. (In this issue) Journal of Geographical Systems 4:1–14
