Homework 7

Mapping using a dataset of hard data values

Date given: 11/6

Date Due for Part 1: 11/13 noon

Date Due for Part 2: 11/20 noon

Date Due for Part 3 and 4: 11/27 noon

You are asked to do problem 1 and 4 of this homework without help from classmates or students who have already taken the class. Doing otherwise on this homework will violate your honor code pledge.

Part 1

Due to a traffic accident, a truck carrying the toxic agent A contaminates a lake in Orange County. An infant resides in a house near this lake. Knowledge about the fate and transport of agent A indicates that if Xh is the concentration in the lake and Xk is the concentration at the house, then the joint PDF for Xh and Xk is There is no toxic effect on an n months old infant if xk<(n+27)/52, however an adverse health effect occurs as soon as xk>(n+27)/52.  Authorities stipulates that the infant needs to be evacuated if the risk of adverse health effect is greater than 0.5.

Consider the situation where you do not have any measurement for Xh, and that the age n (months) of the infant is the month of your birth date.  For example, if you are born on 11/30/1975, then n=11.

a.      Indicate what n is in your case.

b.      What is the expected value of Xk (i.e. calculate E[Xk])?

c.      What is the probability that the infant will suffer from an adverse health effect

(i.e. calculate P[ Xk > (n+27)/52 ]) ?

d.      Should you request evacuation of the infant?

Now consider that the concentration measured in the lake is Xh=0.1.

e.      What is the expected value of Xk given that Xh=0.1 (i.e. calculate E[Xk | Xh=0.1])?

f.       What is the probability that the infant will suffer from an adverse health effect given that Xh=0.1

(i.e. calculate P[ Xk > (n+27)/52 |  Xh=0.1 ]) ?

g.      Should you request evacuation of the infant given that Xh=0.1?

Part 2

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) has a zero mean (over space and time), and that it has the following covariance ,

where r is the spatial lag, t is the temporal lag, and the covariance parameters are c01=1.5, ar1=1 and at1=3, c02=0.5, ar2=30 and at2=700.

Do you think this field can be assumed to be homogeneous/stationary, and space/time separable?  Justify your answer.

Measurements of hard data values are available for this S/TRF, as follow

At p1=(-1,1,1),  X(p1)= 3.3,

At p2=(0,3,2),  X(p2)= 1.8,

Your task is to use the BME conceptual framework to provide the mathematical steps to obtain a stochastic characterization of the S/TFR at the location pk=(0,-1,3).  Then, if you can, provide the solution to the problem (though this might require advanced mathematical derivation). Finally, if you can, comment on the comparison between BME and kriging for this special mapping situation.

Hints:

• The general knowledge consists in the covariance equation, and there are 3 mapping points.  Using the prior step of the BME framework, derive the formulae of the prior PDF, and write down the set of equations used to solve for the Lagrange coefficient
• The site specific knowledge consists of two hard data values.  Using the posterior step of the BME analysis, write the formulae of the BME posterior PDF
• For the more advanced students: Using properties of the multivariate Gaussian PDF, provide a closed form formulae for the prior PDF, as well as the posterior PDF, and numerically calculate its moments (mean, variance, and skewness).

Part 3

Use what you have learned from the BMEGUI tutorials and previous homework to refine the space/time analysis of your project dataset. Prepare and submit a short preliminary draft (less than 8 pages of text and figures) of your final class project report. This report should be well written, it should have an introduction providing background about the environmental contaminant and the research question you plan to address, it should have a materials and method section describing the data you obtained (i.e. exploratory data analysis) and the method and tools you will use to analyze that data (i.e. the BME framework and the BMEGUI tool), and it should have some preliminary results (covariance and maps). Use a “future work” section to describe what you plan to do to complete or further improve the analysis and address your research question.

Part 4

Prepare a 5 minute PowerPoint presentation of your final project that you will present to the class on the last week of the semester.

Submit parts 1 to 3 as a well-written word documents (named yourfirstname_yourlastname_hwk7_part1.docx, etc.) and send them to the instructor. Submit part 4 in a PowerPoint document named yourfirstname_yourlastname_hwk7_part4.pptx and send it to the instructor.