Homework
7* *

Mapping using a dataset of hard data values

Date given: 11/5

Date Due for Part 1: 11/12 noon

Date Due for Part 2: 11/19 noon

Date Due for Part 3: 11/26 noon

Date Due for Part 4: 11/25 noon

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**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.**

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**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 *X** _{h}*
is the concentration in the lake and

There is no
toxic effect on an *n* months old infant if *x** _{k}*<(

Consider the
situation where you do not have any measurement for *X** _{h}*,
and that the age

a.
Indicate what *n* is in your case.

b.
What is the expected value of *X** _{k}* (i.e.
calculate E[

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

(i.e. calculate P[ *X** _{k
}*> (

d. Should you request evacuation of the infant?

Now consider
that the concentration measured in the lake is *X** _{h}*=0.1.

e.
What is the expected value of *X** _{k}* given that

f.
What is the probability that the infant will suffer from an adverse
health effect given that *X** _{h}*=0.1

(i.e. calculate P[ *X** _{k
}*> (

g.
Should you request evacuation of the infant given that *X** _{h}*=0.1?

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**Part 2**

Consider the space/time random
field (S/TRF) *X( p)*, with

*c _{X}*(

where *r* is the spatial
lag, *t* is the temporal lag, and
the covariance parameters are *c _{01}*=1.5,

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 *p _{1}*

At *p _{2}*

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 *p _{k}*

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 slides / 3 minute PowerPoint preliminary presentation of your project that you will present to the class on the last week of the semester. This preliminary presentation should cover your research question and exploratory data analysis (required), and any preliminary results available on covariance, maps, and mean trend scenarios (optional).

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.