Homework

Homework 0

Visit The Software Web Page And Look At The Available R Packages For Spatial Analysis. Download GeoR; Fields And SpBayes And Become Familiar With Them.

Homework 1

(due 4/20/17)

  1. Prove the conditions for the existence of a Gaussian process
  2. Consider an isotropic correlation function. Consider a transformation that produces geometric anisotropy. Prove that the resulting correlation function is positive definite.
  3. Plot all the covariograms and variograms in the tables of the second set of slides. Take the variance to be 1, and take the range parameter to be such that the correlation is .05 at a distance of one unit
  4. Assume that the correlation functions in the previous point correspond to one dimensional Gaussian processes. Simulate one 100-points realization of the process corresponding to each of the plotted functions.
  5. Write explicitly the correlation function of a Matern with nu=1/2,3/2 and 5/2. 

 

Homework 2

(due 5/2/17)

  1. Prove the results about the smoothness of the members of the Matèrn family
  2. Use the spectral representation to show that the product of two valid correlation functions is a valid correlation function.
  3. The spectral density of a correlation in the Matèrn family has tails whose thickness depends on the smoothness parameter. Conjecture: the smoothness of the corresponding random field depends on the number of moments of the spectral density. What can you say about this conjecture?
  4. Use the K-L representation to approximate the exponential correlation for range parameter equal to 1. Plot the approximation for several orders and compare to the actual correlation.
  5. Repeat for the approximation given on Page 13 of the fifth set of slides.
  6. Generate 100 realizations of a univariate Gaussian process with exponential correlation with range parameter 1. Compare the empirically estimated eigenvalues and eigenfunctions to the ones given by the K-L and the approximation on Page 12.

     

     

Homework 3

(due 5/16/17) No extra days!!

Pollutans Data

To do this problem you need to: (a) coordinate your efforts with the other students; (b) search and process data; (c) perform the analysis. The goal is to study a set of data on air pollutants in the US. Data for this goal are available from the EPA web site.

  1. Start by choosing and area in the US of one of more contiguous states that has at least 100 monitoring stations. Each one of you needs to come up with a different non-overlapping area. The coordination is up to you.
  2. Download the annual summary data for the year 2015, and extract the measurements corresponding to ozone. There are many ozone data recorded. Make sure that you are all using the same variable.
  3. Based on graphical exploration, is there evidence of a first or second order trend function of location and altitude? Is there evidence that a transformation is needed in order to make the data closer to normality?
  4. Obtain the residuals after fitting the trend function resulting from the previous question, if any. Plot the variogram. Explore possible anisotropies using a directional variogram.
  5. Use least squares to fit the covariograms in the Matèrn family with smoothness equal to .5; 1; 1.5; 2.5. Plot the results. Use the plots and the values of the LSE to select the best fit.
  6. Plot the likelihood function for the sill and the range corresponding to each of the correlations in the previous point. If a nugget is needed, you can plug an estimated value.
  7. Plot the marginal likelihood for the range parameter for each of the examples above.

Homework 4

(due 5/25/17)

Perform a full Bayesian analysis of the Pollutans data. This is an open-ended problem. You can consider something as simple as fitting the average annual ozone with some sensibly chosen priors. You do get bonus points for using the reference prior for the range parameter. You do need to consider model assessment to convince me that your model fits the data reasonably well. You need to write a report, of no more than 8 pages, with an abstract, an introduction to the data and methods, an EDA, explanation of the models, discussion of results, conclusions, and references. The figures need to be included in the text, and count towards the 8 pages.

Homework 5

(due 6/8/17)

 This is the third part of the analysis of the pollutant data. To complete this part:

  1. Download the data for your selected area corresponding to PM2.5 and NO2
  2. Fit a model using predictive processes for Ozone, PM2.5 and NO2, separately. Compare the fits using the modified predictive process with the ones obtained with the non-modified one.
  3. Fit multivariate predictive process model to the variables simultaneously. Compare your results.

 

Homework 6

(due 6/15/17)

This is the fourth part of the analysis of the pollutant data. To complete this part:

  1. Fit a model using a discrete process convolution model for Ozone. Use the same grid points you used in the previous analysis of the data that used predictive processes. Use a spherical Bezier kernel. Use an i.i.d. set of normal priors for the convolution coefficients. 
  2. Fit the same model to the same data using a prior for the coefficients that uses a Markov random field that considers only the nearest neighbors. Compare.