GLS - Spatial stats

GLS - Spatial stats - Spatial statistics and Generalized...

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Spatial statistics and Generalized Least Squares Regression ESM 206C May 20, 2008

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pH and NO3 in Norwegian lakes 0 50 100 150 200 250 300 350 4.5 5.0 5.5 6.0 6.5 NO3.1981 pH.1981
Call: lm(formula = pH.1981 ~ NO3.1981, data = lake) Residuals: Min 1Q Median 3Q Max -0.888624 -0.410622 -0.007402 0.386811 1.237405 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.7130797 0.1218255 46.896 < 2e-16 *** NO3.1981 -0.0032446 0.0009186 -3.532 0.00106 ** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.5103 on 40 degrees of freedom Multiple R-Squared: 0.2377, Adjusted R-squared: 0.2187 F-statistic: 12.47 on 1 and 40 DF, p-value: 0.001056

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pH of Norwegian Lakes 6 8 10 12 58 59 60 61 62 Longitude (degrees E) Latititude (degrees N)
Questions about Norwegian lake data Is there spatial autocorrelation in pH values? How can we interpolate and smooth those values? Is there a relationship between pH and NO 3 , taking into account spatial autocorrelations in both variables?

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Semivariogram Semivariance is half the average squared difference between pairs of lakes a certain distance apart Measures variance among sites as a function of distance Also called empirical variogram 0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 distance semivariance
Theoretical variogram 0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 distance semivariance

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8 Theoretical variogram 0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 distance semivariance Nugget Sill Range
Theoretical variogram Summary of the parameter estimation ----------------------------------- Estimation method: WLS (weighted least squares) Parameters of the spatial component: correlation function: gaussian (estimated) variance parameter sigmasq (partial sill) = 0.3175 (estimated) cor. fct. parameter phi (range parameter) = 0.7817 Parameter of the error component: (estimated) nugget = 0.0773 Minimised weighted sum of squares: 4.2704 Call: variofit(vario = pH.vg, ini.cov.pars = c(0.4, 1), cov.model = "gaussian")

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0 1 2 3 4 0.0 0.2 0.4 0.6 Gaussian: SS=4.27 distance semivariance 0 1
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