Homework
7* *

Mapping using a dataset of hard data values

Date given: 11/4

Date Due for Part 1&2: 11/11 noon

Date Due for Part 3&4: 11/18 noon

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**You are asked to do parts 1 and
2 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**

Let the SRF *X*(** s**) represent the prevalence of
COVID-19 amongst students living at location

Recently published studies indicate that the spread of COVID-19 substantially increases once more than one third of students living in a dorm test positive for COVID-19, and therefore your directive is that you need to ask all students to move out of the UNC dorm if there is more than 70% chance that the prevalence at the UNC dorm exceeds 1/3.

a. State
your Month Of Birth (MOB), write the PDF using *d*=MOB, and show that it
is a valid PDF.

b. You
know that the students at the dorm located at ** s**’ are being tested
and you will receive their results soon. However, you do not have that
information yet. While you are waiting, what do you expect the prevalence to be
at the UNC dorm? What is the probability that the prevalence at the UNC dorm
exceeds 1/3=33.333%? Based on that probability should you already ask students
to move out?

c.
You learn that the prevalence at the other dorm is 0.9 (i.e. 90% of the
students tested positive for COVID-19 at the other dorm). Given that
information (i.e. given that *X*’=0.9), what do you expect the prevalence
to be at the UNC dorm? What is the probability that the UNC dorm prevalence *X*
exceeds 1/3 given that* X*’=0.9? Should you ask students to move out from
the UNC dorm given that* X*’=0.9?

d. Make a plot of the marginal PDF and the conditional PDF . Explain what these plots show and use these plots to explain what you found in questions a and b above.

e. Is
the SRF *X*(** s**) homogeneous?

f.
Is the correlation between *X* and *X*’ increasing with *d*,
or decreasing with *d*? Make a proof supporting your answer.

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**Part 2**

Consider the space/time random
field (S/TRF) *X( p)* representing log-PM2.5 (log-ppm) at space/time
location

*c _{X}*(

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

Samples of PM2.5 were collected at the following space/time locations:

At *p _{1}*

At *p _{2}*

* *

Your task is to use simple
kriging to estimate log-PM2.5 at *p _{k}*

a. Do
you think that the S/TRF *X( p)* can be assumed to be
homogeneous/stationary, and space/time separable? Justify your answer.

b. Derive
the value of mean value m_{k} and the column mean vector m_{h}.
Likewise derive the values of the covariance value c* _{kk}*, and
covariance matrix C

c. You
have not analyzed sample 1 nor sample 2. What is the expected value of *X _{k}*=

d. What
is the correlation between *X _{k}* and

e. You
are instructed to analyze sample 1 and your measurement indicates that *X_{1}*=

f.
You need to quarantine due to COVID-19. Your lab mate analyzes sample 2
and her measurement indicates that *X_{2}*=

g. Your
quarantine is over, you get back to the lab, and you now know that *X_{1}*=

**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 3 slides / 3 minutes PowerPoint preliminary final presentation that you will present to the class on the last week of the semester. Generally, this presentation should have one slide on intro and data, one slide on mean trend and covariance model, and one slide on mapping results

Create a file containing a scan of your hand written work for parts 1 and 2 (named yourfirstname_yourlastname_hwk7_part1and2.pdf) and email it to the TA at the deadline for parts 1 and 2. Create a well written report part 3 and 4 in a file named yourfirstname_yourlastname_hwk7_part3and4.docx and email it to the TA at the deadline for parts 3 and 4.