Homework 7 BMElib track:

Mapping using a dataset of hard data values

Date given: 11/10

Date Due: 12/01 noon

You are asked to do part 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.

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 X_k is the concentration at the house, then the joint PDF for X_h and X_k is

There is no toxic effect on an n months old infant if x_k<(n+27)/52, however an adverse health effect occurs as soon as x_k>(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 X_h, 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 X_k (i.e. calculate E[X_k])?

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

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

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 X_h=0.1 (i.e. calculate E[X_k | X_h=0.1])?

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 > (n+27)/52 | X_h=0.1 ]) ?

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

Part 2

Before you start, review the examplesBMEPROBALIB.m program providing a tutorial on using BMElib to obtain the posterior pdf at estimation points using measurements at data 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) 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, a_r1=1 and a_t1=3, c₀₂=0.5, a_r2=30 and a_t2=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 p_k=(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 (i.e. calculate its mean, variance, and skewness using BMEprobaMoments.m).

Part 3

Create a directory called hwk7, and from that directory create the following three sub-directories: hwk7\resAlongRiver, hwk7\resMonitoringStation, and hwk7\resValidation

In the hwk7 directory download the following files which perform a BME mapping analysis of surface water Phosphorus data in the Raritan river basin:

analysis.m, estPhosTutorial.m, plotStationTutorial.m, plotMapTutorial.m, raritanmask.m, and raritanbuffer_2.mat

You will also need to add to the following files from the homework with the phosphorus data and the exploratory analysis function:

Raritan_Phos.txt, raritan_river.e00, exploreDataTutorial.m,

as well as the following function from the homework on covariance modeling:

estCovTutorial.m

Review the estPhosTutorial.m function, which calculates the BME estimates of phosphorus over space and time.  Note that this function has the option to calculate the BME estimate as a function of time for a selected monitoring station and plot the resulting time series using plotStationTutorial.m, or the option to calculate the BME estimate across space for a selected estimation time and plot the resulting map using plotMapTutorial.m and raritanmask.m.  Review also the analysis.m function, which calls estPhosTutorial.m to estimate several time series of Phosphorus at a set of pre-selected monitoring stations, and calculates the map of phosphors for pre-selected time instances.

Use the analysis.m function to create time-series and maps of BME estimated phosphorus (i.e. use analysis(1) and analysis(3), respectively). Make sure that you run analysis.m from the hwk7 directory after you have created the sub-directories: hwk7\resAlongRiver, hwk7\resMonitoringStation, and hwk7\resValidation, otherwise you will get an error.  The first time you run analysis.m it will take a long moment, but after that the estimation results are stored in the subdirectories, and subsequent runs of analysis will be rapid.  Then using the plotStationTutorial.m and plotMapTutorial.m functions explore the time series and maps that you have calculated.

Inspiring yourself from the above analysis of Phosphorus, use a similar approach to improve the estimation mapping analysis of your project dataset.  Provide a write up describing your project dataset analysis and the results you obtain, including several time-series and maps of the BME estimates.  This write up should be an improvement over the previous homework for the mapping analysis section of your project report.