EXERCISE A phenomenon =(x) defined on a linear segment has been sampled with a regular mesh. The following measurements are available: 3. 9 10 0: UT =(x) 10 10 12 15 10 =(r ) 5 This phenomenon is considered as a realisation of a random function. The problem consists in the statistical inference of this random function. Answer the following questions keeping in mind that this is a small data set that cannot lead to good statistical results, however, enables to obtain valuable results without a long calculation process. Calculate the variogram y(4) (plot the values) and show that it indicates the occurrence of trend. Assuming that the drift is of the form u,(x) = ax + b, express the relationship using the least squares method and plot the relevant trend line. Calculate the estimated residuals r (x)= =(x)u (x) and the variograms of the estimated residuals y, (h). Assuming that the estimated drift is linear, function u, (x)=1.2x was found to represent the data using geostatistical analysis tools. Plot the data and the function and calculate the estimated residuals s (x)= =(x)u; (x) and the variograms of the estimated residuals y, (1). The data used were generated from a head and tails game, where every time the coin was thrown x was incremented by 1 and : was derived adding to the previous result 0 for heads and 2 for tails. This means that =(x) can be written as a random function of the form: =(x) = Ex where there is equal probability to derive 0 or 2 values (by tossing the coin) and the variance of the x, values equals to 1. P(x, =0)= P(x, =2)=0.5 and E[x]= Var[x ]=1 Establish the 'true' drift u, (x) = E[=], derive the true residuals r,(x)= =(x)u, (x) and calculate the corresponding variogram. Compare the performance of the us(x), , (x) and u, (x) functions in describing the random function