University of California, Los AngelesDepartment of StatisticsStatistics C173/C273 Instructor: Nicolas ChristouHomework 3Exercise 1:Use gstat to fit a variogram model to the omnidrectional sample variogram of the variable V of the databelow. The variable V is measured in parts per million (ppm) at the Walker Lake area in the state of Nevada(from An Introduction to Applied Geostatistics by Issaks, E., and Srivastava, R. M. (1989)). See figure below:You can access the data at:a1 <- read.table("http://www.stat.ucla.edu/~nchristo/statistics_c173_c273/walker_lake_v.txt", header=TRUE)Details: For the variogram calculations use maximum distance 100 m. Clearly explain what variogrammodel, weights, cutoff points, direction, etc. you are using. Print the initial values for the minimization andthe final estimates of the parameters. Please submit the R commands, the plot of the sample variogram andthe fitted model variogram.Exercise 2:Let the spatial data Z(s1), Z(s2), · · · , Z(sn) be random variables from a normal distribution with mean µand variance σ2. The variogram is defined as2γ(h) = var (Z(s + h) − Z(s))a. What is the distribution ofQ = Z(s + h) − Z(s)p2γ(h)!2.b. Find the expected value and variance of Q.c. Findvar (Z(s + h) − Z(s))2.Exercise 3:Access the elevation data:a3 <- read.table("http://www.stat.ucla.edu/~nchristo/statistics_c173_c273/elevation_data.txt", header=TRUE)From Statistics and Data Analysis in Geology (Second edition), Davis, J. C. (1972).a. Use geoR to fit a quadratic trend surface model to the variogram of these data. Use least squares (noweights) and weighted least squares (number of pairs and Cressie’s weights).b. Plot the sample variogram and graph all the model variograms on the same plot.Exercise 4:Access the coal ash data:a4 <- read.table("http://www.stat.ucla.edu/~nchristo/statistics_c173_c273/coal_ash.txt", header=TRUE)From Statistics for Spatial Data, Noel Cressie, Revised Edition (1993).a. Compute the variogram cloud on the east-west direction for both the classical and robust estimators.Construct the box plots for each bin for both estimators. Print the two plots. What is your conclusion?b. Compute the sample variogram using the classical and robust estimators on the east-west direction.Print the two plots.c. Fit a model variogram to the sample variograms you constructed in (b). Print the estimated param-eters and the two plots.Exercise 5:Access the soil calcium data:a5 <- read.table("http://www.stat.ucla.edu/~nchristo/statistics_c173_c273/soil_ca_data.txt", header=TRUE)From Model-based Geostatistics, Peter Diggle, Paul Ribeiro, (2007).a. Use the packgage gstat. Compute the omnidirectional variogram using no trend and first ordertrend. Plot these sample variograms and fit a model variogram to them. What are the estimates ofthe parameters of the model variogram you are using?b. Run the regression for the second-order trend with a formula using I to treat powers and products“as is”:reg <- lm(a5$ca20 ~ a5$x+a5$y+I(a5$x^2)+I(a5$y^2))Construct a data frame with x, y, res, where res are the residuals of the regression above (reg$res).Finally create a gstat object, compute the sample variogram and fit a model variogram to