ST440/540 – Mid-term exam 2 - Due April 15
THIS IS AN EXAM - DO NOT DISCUSS THE PROBLEM WITH ANYONE (INCLUDING
OTHER STUDENTS OR THE TA)! If you have questions, please email me.
The enhanced vegetation index (EVI) is a satellite-derived measure of the greenness at a
spatial location. In this exam, you will use EVI measurements at one location1 to estimate
the “green-up time” (GUT) each year and test whether the timing of GUT has changed over
the study period. EVI is a unitless quantity that is between zero and one. At this site it is
measured 802 times (between 6 and 38 times per year) from 1984 to 2019. As shown in the
plot below for 2000-2005, EVI exhibits strong seasonality.
2000 2001 2002 2003 2004 2005 2006
0.
0
0.
8
Time
EV
I
Let Yi be the EVI measured at time ti. We assume the true EVI at time t is the curve
µ(t), and that Yi is a noisy measurement of µ(ti). Define the GUT for year i as the first time
in year i that µ(t) > 0.5 (this a simplification of the real definition). For example, the GUT
for the year 2000 (this not real data) is the time value for the dashed vertical line below.
2000.0 2000.4 2000.8
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
Time
EV
I
mu
Y
1Thanks to Xiaojie Gao for providing data!
Analyze these data in the following sections:
1. Model definition: Define a statistical model for the analysis, including the distribu-
tion of the data Yi given the true EVI curve µ(t), the model for the true EVI curve
and all prior distributions. Write the model description in a paragraph with at least
two equations and conceptual justification that the model is reasonable.
2. MCMC convergence: For one model fit, describe the MCMC algorithm and con-
vergence diagnostics.
3. Model comparisons: Compare the fit of 2-3 models.
4. Model fit: Summarize (with uncertainty) the estimate of µ(t).
5. GUT analysis: Summarize the posterior distribution of GUT for each year.
6. Time-trend analysis: Evaluate whether there is a change in the distribution of GUT
across the years.
Organize your report with sections labeled as above. With the exception of the first section,
each section should have the format:
1. One sentence (no more than three lines) describing the analysis
2. One clearly-labeled (with informative caption) table or plot with the main results
3. One sentence (no more than three lines) summarizing the results
Summarize your analysis in a PDF document that is no more than four pages long
(excluding code). Append your code to the end of this document and submit a single
document on moodle.
HAVE FUN!