Stat 406 – Midterm I answers
Spring 2016
1. Yellow perch abundance (3 pts each).
a) You can’t say anything about bias from one result per method.
b) Combination C is the least biased. In fact it appears to be unbiased. The mean of many estimates
equals the population value.
c) Combination E is the most precise. It has the smallest standard deviation (table from part b)
Note: Standard error is not appropriate. It tells you the precision of the estimated mean, not the
precision of the method. Standard error goes to zero as you look at more samples. Standard
deviation does not.
2. Singular model fit error (3 pts each).
a) You can’t estimate both the nugget and measurement error variances. Set one of them to a fixed
value and you can estimate the other.
b) You can’t estimate both the range and the nugget when there is only one semivariance value below
the sill. Various things you could do: evaluate the empirical semivariance at more distances between
0 and 3, fix the range based on outside information, assume the nugget is 0.
Note: assuming there is enough data, evaluating the semivariance at more distances is by far the best
option.
3. Kriging concepts (3 pts each).
a) P1 is closest to 2.0. Values within the practical range (0.5) are between 1.3 and 2.7.
b) P2 is closed to 10.0. There are no observed values within the practical range, so the predicted value
will be the GLS mean.
c) Var P1 will be closest to 10.0. It will be substantially less than 𝜎 2 because there are nearby
observed values.
d) Var P2 will be closest to 38.0. It will be slightly larger than 𝜎 2 because of the additional uncertainty
estimating the GLS mean.
4. Residuals from a kriged fit. 3 pts.
You can’t conclude anything from this plot (other than the fitted SV model did not include
measurement error). Without measurement error, a kriged predictions always equals the observed
values (“kriging honors the data”), so the non-cross validation residuals will be identically zero.
Note: To learn about model assumptions, you need to use cross-validated predictions and residuals, or
have a test data set not used to make the predictions.
5. Sea surface temperature in the Gulf of Mexico (45 pts).
There is no single right answer because there are legitimate alternatives to the choices I made. My
scoring rubric was:
a) description of methods. 10 pts. (See notes at end)
Was this detailed enough for someone else to recreate your analysis?
It should include the software you used (no points off this time if not done).
b) elaboration on choices. 20 pts.
which empirical variogram estimator? (I chose Matheron, but Cressie quite similar)
do you need to worry about anisotropy? (I chose to ignore anisotropy because different sills hard)
which distance bins and cutoff, if not the default?
which SV model?
what criterion did you use to choose the SV model?
Information that may inform your choice of SV model:
SV Model SV SSErr x 1000 MS Pred observations
rMSEP’s are square root of these
SS Prediction are 250 times these
Sph
7.88
0.152
Mat, k=1
2.06
0.155
Mat, k=2
0.89
0.154
Mat, k=4
0.99
0.165
Gau
1.86
0.190
Exp
13.97
0.155
FYI, estimated parameters for 3 reasonable SV models:
Model
Sph
Mat k=1
Mat k=2
Nugget Partial Sill Range (km)
0
2.89
502
0
3.20
148
0.051
2.87
84
c) Predictions and sd’s: 10 pts.
2 pts off if you gave me variance not sd.
2 pts each if your predictions and sd values were close to mine, which were, using the fitted Spherical,
Matern (k=1) and Matern (k=2) models:
Spherical model
Matern, k=1, model
Matern, k=2, model
location
prediction standard prediction standard prediction standard
deviation
deviation
deviation
(-91.1, 28.6)
18.73
0.69
18.33
0.51
18.24
0.42
(-86.2, 26.35)
25.91
0.45
25.90
0.24
25.96
0.27
(-87.6, 24.6)
26.30
0.61
26.40
0.42
26.43
0.35
(-85.1, 24.0)
26.55
0.74
26.52
0.59
26.62
0.48
d) Map: 5 pts. Full points if your map was close in appearance to mine.
Using the fitted Spherical CV model, (Dots are the prediction points, not expected on your map):
26
24
22
20
18
16
Using the fitted Matern, k=2 model
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24
22
20
18
16
Notes:
I got many different sorts for answers for part a. I was hoping for the statistical methods paragraph that
you would write if you wrote a scientific paper about this analysis. That is not a step-by-step ‘what I
did’, but it is complete enough (in the important details) that someone else could recreate your analysis.
I only deducted points if major pieces were missing. Everyone will have a different writing style, so
there is no single correct answer. Here’s one possibility:
The sample variogram was estimated using the Materon estimator with 15 equally-spaced distance bins.
Inspection of the data and the variogram cloud indicated no issues with outliers. We assumed isotropic
semivariance although inspection of the variogram map and direction variograms indicated some
evidence of different sills in different directions. We considered four models, Spherical, Exponential,
Matern (k=1) and Matern (k=2), for the semivariance and chose the model with the smallest weighted
sum-of-squared errors. Visual assessment of the fit appeared reasonable. Sea surface temperatures at
a fine grid of locations in the northern GOM and at four locations of concern were predicted by universal
kriging
In a real paper, this paragraph would be preceded by a paragraph describing all the preliminary analysis,
i.e. starting with “We have SST at 250 locations” and ending with “all subsequent analyses included a
linear model for the trend”. Sometimes, more “results” that drive the analysis are included in the
methods paragraph, e.g., we chose the Matern k=2 model.
For part b, some folks apparently thought I had dealt with anisotropy in the “preliminary evaluation”. I
did not mean to imply that. Here is a plot of predictions from the linear trend without any spatial
correlation model.
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25
24
23
22
21
20
19
18
This is quite different from the trend+spatial correlation maps (previous page). Although the trend picks
up the major features of variation in SST, there clearly is a lot of residual from the trend. The spatial
correlation model accounts for the variation in those residuals. There may well still be some anisotropy
in those residuals.
Depending on how you looked at anisotropy, you may have decided that it was minimal (variogram
map) or present (directional variograms, indicating different sills in different directions). Since it is really
hard to model directional sills, everyone who looked at anisotropy decided to ignore it. I would do the
same.
For part c, the predictions are quite similar for the three SV models; the standard deviations are
moderately different. That’s because the three models have different behaviour at small distances.