Homework 5

Space/time covariance models

Date given: 9/30

Due: 10/14 noon

Part 1 (20 points)

Consider the space/time random field (S/TRF) X(p), with p=(s,t), where s=(s1,s2) is the 2D spatial location, and t is time.  Assume that X(p) (measured in ppm) has a zero mean (over space and time), and that it has the following covariance

cX(r,t) = c01 exp(-3r/ar1) exp(-3t/at1) + c02 exp(-3r/ar2) exp(-3t2/at22)

where r is the spatial lag (Km), t is the temporal lag (day), and the covariance parameters are c01=2.0 ppm2, ar1=1 Km and at1=3 days, c02=0.5 ppm2, ar2=30 Km and at2=700 days

1) Can this field be assumed to be homogeneous/stationary? Can you express this covariance model as a space/time separable model? What is the variance of X(p)? Describe physically the covariance model and each of its coefficients c01, ar1, etc.

2) Write the expression the covariance when the temporal lag t  is zero:

cX(r,t =0) = …

3) Write the expression the covariance when the spatial lag r is zero:

cX(r=0,t ) = …

Part2 (30 points)

1) Consider a homogenous space/time random field (S/TRF) X(p) where we know that it’s covariance model verifies the following two expressions

cX(r,t=0) = varX [0.80 exp(-3r/ar1) + 0.20 exp(-3r/ar2)]

cX(r=0,t) = varX [0.80 exp(-3t/at1) + 0.20 exp(-3t/at2) ]

where varX=2.5 ppm2, ar1=1 Km and at1=3 days, ar2=30 Km and at2=700 days.  Find an expression for the space/time covariance model cX(r,t ) such that the above two expressions are true.  Is that a space/time separable covariance model?

2) Now consider the following two expressions

cX(r,t=0) = varX [0.1 exp(-3r/ar1) + 0.9 exp(-3r/ar2)]

cX(r=0,t) = varX [0.4 exp(-3t/at1) + 0.6 exp(-3t/at2) ]

where varX=2.0 ppm2, ar1=1 Km and at1=3 days, ar2=30 Km and at2=700 days. Find one or more expressions for the space/time covariance model cX(r,t ) such that the above two expressions are true, or if this is not possible, explain why.

Part 3 (25 points)

1) Model the space/time covariance of the Arsenic dataset dataAS.txt.  Do an exploratory analysis of this data and model its covariance model.  Provide a write up describing your analysis.  This write up should describe the distribution of the data (is it normal?), it should have a figure showing the experimental and model covariance, it should provide the mathematical expression of the spatial and temporal components of your s/t covariance, and it should provide the expression for the full s/t covariance model.

2) Do the same analysis for the Flu dataset dataFLU.txt

Part 4 (25 points)

Model the space/time covariance for your project dataset and add this to your growing project report.  This write-up should include what you have produced in previous homework, i.e. it should lay out your research question with relevant citations, it should have an exploratory data analysis showing your project data, it should include your covariance analysis, your covariance model (including a figure of the covariance model on top of your experimental covariance values), and its interpretation. Provide citations that either relates your covariance modeling with previous works, or write something up claiming you are generating novel covariance results.  This write-up will serve as a part of your final project report.

Submit parts 1, 2 and 3 as a well-written word document named yourfirstname_yourlastname_hwk5_part123.docx that you send to the TA. Submit part 4 separately as a well-written word document named yourfirstname_yourlastname_hwk5_part4.docx that you also send to the TA.