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Werner G. Müller: Collecting Spatial Data (2007)

Springer Verlag, Heidelberg


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Every chapter (in its last section) contains a list of questions that relates to a real example. The Südliche Tullnerfeld is a part of the Danube river basin in central Lower Austria and due to its homogeneous aquifer well suited for a model-oriented geostatistical analysis. It contains 36 official water quality measurement stations, which are irregularly spread over the region. A graphical representation of the sites (after rescaling to a unit of approximately 31 kms) on a 95 x 95 grid is given in the figure below.

The water quality monitoring network in the Südliche Tullnerfeld; solid circles represent sites, grid points represent region.

The exercise data set contains measurements of chlorid (Cl) over the period 1992-1997 on all time points, for which at least one measurement was taken (missing values are indicated by blanks). The rows and columns indicate the date and station number respectively. The measurements are daily averages of Cl concentrations in units of mg/l.

Actual data from this monitoring network can be found at

Chapter 2

  1. Select a station and check graphically whether the temporal mean can be considered a meaningful estimate of the spatial trend. If not try to divide up the periods to make it so.
  2. Check whether there is skewness in the data and remove it by an appropriate transformation.
  3. The data is suspect to anisotropy since there is a dominant flow direction (from west to east) in the aquifer. Guttorp and Sampson (1994) suggest to fit the variogram after a linear coordinate transformation x → Qx. Select a particular day and estimate the appropriate 2 x 2 transformation matrix ^Q by introducing a separate step in the iterative estimation procedure. Compare results for various days for consistency. Note that there also exist spatial models that explicitly utilize flow information (cf. Verhoef et al., 2006).
  4. In the transformed coordinate system perform a local linear regression fit for the mean surface by selecting the smoothing parameter by cross validation. Can you take account of the differing number of nonmissing observations NT? If the result is not satisfactory try a different local model.
  5. Fit a Matérn model to the variogram cloud and perform according residual diagnostics.
  6. Construct a nonparametric kriging contour plot by adding simple kriging predictions to the smoothed surface. For comparison give a graphical representation of the transformed design region and site coordinates.

Chapter 3

  1. Fit an appropriate nonlinear global model to the data and use the resulting parameter estimates to find a locally optimum design. Vary the estimates a bit and compare the respective resulting designs.
  2. Compare the above design to one suitable for estimating a second order polynomial. Are there unexpected differences - if yes, to what could they be due to?
  3. Use the reparameterization technique based on model (3.12) to construct a design for estimating the location of maximum concentration in the field. What criterion would one use to find an optimum design for estimating the maximum concentration value?
  4. Use a random coefficient formulation of the quadratic model and estimate the corresponding covariance matrix of the parameters. With the resulting estimates construct an optimum design for the mean and the individual parameters with respect to a particular day. Interpret the difference, especially concerning the number of support points.
  5. Use the design technique by Der Megreditchian (1985) to reduce the number of sites accordingly. Which design does the result resemble?

Chapter 4

Perform all analyses for this chapter on both the original and the transformed design space and compare the respective results.

  1. Construct a `coffee-house' design with 36 points. Identify points a simple shift of which will improve the design criterion.
  2. Construct a couple of random and/or regular designs and compare their coverage properties. Some answers given by K. Danzer and A. Fussl in a seminar paper (in German, with R-code).
  3. For local linear regression find the respective optimum design by using sensitivity function (4.6). How good are its space-filling properties? Vary the set of points of interest and the locality.
  4. Assume a true global model (from an earlier chapter) and employ the technique from Fedorov et al. (1999) for finding an optimum local linear regression design. How does the result differ from the above? Some answers given by M. Hainy and G. Malsiner Walli in a seminar paper (in German, with R-code).
  5. Contrast the global model with a second order polynomial and calculate the optimum discriminating design. Can this design be also efficiently used for other purposes?

Chapter 5

  1. By the two methods given (and for both the nonlinear model selected and the second order polynomial) calculate 36 point replication-free designs. Are the results in accordance?
  2. Calculate 36 point replication-free designs for (a) localizing the peak, (b) estimating the mean parameter in a random coefficient setting, (c) discriminating the models.
  3. Assuming the correlation structure is according to your preestimated variogram. Construct an optimum design by the approximate information matrix algorithm and a Brimkulov et al. (1980) type procedure, and compare their efficiencies.
  4. For all points in X find a design that minimizes the average prediction variance (5.23) by assuming first uncorrelated errors and later relaxing this assumption.
  5. Let the 36 observations be distributed on 2 (later more) points in time. Find a design that maximizes a reasonable criterion. Are there any replications?

Chapter 6

  1. Construct a Fedorov-exchange procedure to optimize the power of the Moran's I test. Is there a number of points less than 36 that performs better?
  2. Using the augmentation procedure construct an `optimum' 36 point design for variogram estimation. Compare its efficiency (also with respect to trend estimation) to the other designs calculated.
  3. Construct an alternative design by the multidimensional scaling technique. Report both efficiencies and computing times.
  4. Employ both ML- and REML-based criteria for estimating the variogram parameters and compare to the above two approaches.

Chapter 7

  1. Use a varying portion of the 36 points for trend estimation versus variogram estimation. Give resulting designs and compare criterion values.
  2. Compute JEK and JEA 36 point `nearly optimum (by an exchange procedure) designs and compare.
  3. Employ the total information matrix to calculated D-optimum designs. Observe how their information content varies with varying covariance parameters.
  4. Use maximum entropy sampling to reduce the number of design points similarly to Chapter 3; compare with results therein.
  5. Employ criterion (7.5) for all your designs calculated so far. Which one performs best?