Mean trend and covariance modeling for Space/Time Random Fields (S/TRF) Z(s,t)
Space/Time Random Field (S/TRF) Z(s,t) takes values that change both as a function of the spatial location
s=[x,y] as well as time t.
We assume that a non-homogeneous/non-stationary S/TRF Z can be modeled as the sum of a space/time mean
trend and a homogeneous/stationary residual X as follow
Z(s,t) = mZ(s,t) + X(s,t)
Where X is a homogeneous/stationary.
Steps to model the mean trend of Z and covariance of the residual X
1) Model the mean trend mZ(s,t)of Z(s,t)
2) Calculate the residual data X(si,t) = Z(si,t)-mZ(si,t)
3) Model the Covariance cX(r,t) of the homogeneous/stationary residual X(s,t)
Modeling the space/time mean trend of Z
BMEGUI uses the following additive model to model the space/time mean trend
mZ(s,t) = ms(s) + mt(t)
where ms(s) is the spatial component and mt(t) is the temporal component
1
Space/Time Random Field (S/TRF) Z(s,t).
Temporal plot of Z versus time t for Monitoring Station 1 and 2, i.e. Z(si,t) for i=1, 2
There is a temporal trend of increasing values with time.
2
Spatial plot of Z versus Monitoring Event 1 and 2, i.e. Z(s,tj) for j=1, 2
There is also a spatial trend of increasing values from left to right
The mean trend model proposed is
mZ(s,t) = ms(s) + mt(t)
3
Temporal plot of X versus time t for Monitoring Station 1 and 2,
i.e. Z(si,t) and mZ(si,t) for i=1, 2
4
Spatial plot of residual X-m versus Monitoring Event 1 and 2,
i.e. X(s,tj)=Z(s,tj)-mZ(s,tj) for j=1, 2
5
The model selected is
cX(r,)=c0 exp(-3 r/ar) exp(-3 t/at)
where the sill (variance) c0=1.5, the spatial range ar =50 Km, and the temporal range at =11 days.
6
Mean Trend Modeling in BMEGUI:
i.
We can model and plot the mean trend in BMEGUI by clicking on the “Model mean trend and remove it
from data” button. BMEGUI then calculates the mean trend using default parameters.
ii.
The mean trend is smoothed using smoothing parameters (the search radius and the smoothing range)
specified for the spatial and temporal components of the mean trend. The mean trend model is equal
(within a constant) to the sum of these spatial and temporal mean trend components, i.e. it is a
space/time separable additive function.
iii.
To recalculate the mean trend using new parameters, the user needs to change the defaults with new
values, and then click on the “Recalculate Mean Trend” button. Following in an example of user defined
values
Search Radius
Spatial
Temporal
2.0
2000
Smoothing Range
0.5
300
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The raw spatial mean trend (calculated by averaging the data at each spatial location of interest) is shown in one
screen, while its smoothed counterpart is shown in a different screen. Examples of these two screens are shown
in the two figures below.
Spatial (raw) “Mean Trend Analysis” screen
Spatial (smoothened) “Mean Trend Analysis” screen
8
The raw temporal mean trend (calculated by averaging the data at each time of interest) is shown in dotted line,
while its smoothed counterpart is shown in solid line.
Temporal “Mean Trend Analysis” screen
9
The effect of increased smoothing is shown in the two examples below.
Search Radius
Smoothing Range
Temporal, example 1 3 days
1 days
Temporal, example 2 5days
3 days
Temporal “Mean Trend Analysis” for 3 days of search radius and 1 day of smoothening range
Temporal “Mean Trend Analysis” for 5 days of search radius and 3 days of smoothening range
10
Space/Time Covariance Analysis of mean-trend removed data in BMEGUI:
Once a mean trend model is used, BMEGUI automatically models the space/time covariance of the mean trend
removed data. The procedure to fit a covariance model for the mean trend removed data is similar to that
presented earlier for the case when a mean trend model was not used.
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Case study using PM2.5 in New Jersey
Let’s consider the case of atmospheric daily PM2.5 concentrations measured at monitoring station in New
Jersey and its neighboring states.
Exploratory Spatial Plots: Days 1, 2, and 4
12
Exploratory Time Series Plots: Stations 1, 2, and 4
Superimposed Time Series
60
Measured PM2.5 MS=1
Measured PM2.5 MS=2
Measured PM2.5 MS=30
50
Measured PM2.5 MS=1
Measured PM2.5 MS=2
Measured PM2.5 MS=30
35
30
log(PM25) ug/m3
log(PM25) ug/m3
40
30
20
25
20
15
10
10
5
0
0
100
200
300
400
Days
500
600
700
800
520
540
560
580
600
620
Days
640
660
680
700
720
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Mean trend and covariance analysis: Trade off in estimation efficiency
There PM2.5 S/TRF Z(s,t) is modeled as the sum of a mean trend mZ(s,t). and its residual X(s,t), i.e.
Z(s,t) = mZ(s,t) + X(s,t)
The estimation can be though as a two step procedure: Selecting a mean trend that explains as much of the
variability of PM2.5, and that results in a residual that is as auto correlated as possible.
The inefficiency of the first step, i.e. the degree to which the mean trend fails to explain the variability in
PM2.5, is measured by the standard deviation X of the residual X(s,t). The less the mean trend explain
variability in PM2.5, the higher X will be. In the extreme case that the mean trend is constant we get the
highest possible value for X.
The efficiency of the second step, i.e. the degree to which the residual X(s,t).is autocorrelated, is measured by
the range of the covariance
60
Std deviation weighted range
50
40
30
20
10
0
0.45
0.5
0.55
0.6
0.65
0.7
Std deviation
0.75
0.8
0.85
0.9
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Case1: very Flat mean Trend
15
16
Flat Mean trend
Sill
Model
Range
Structure 1
Spatial
0.2
exp
4
Temporal
exp
7
Structure 2
Spatial
0.19
exp
100
Temporal
exp
75
17
Time 1 to 20
Station ID: 360610062(43) and 360470122(40)
18
Case2: Raw mean trend
19
20
Very
smoothened
Mean trend
Sill
Model
Range
Structure 1
Spatial
0.05
exp
1.5
Structure 2
Temporal
exp
5
Spatial
0.0619
exp
3
Temporal
exp
25
21
Time 1 to 20
Station ID: 360610062(43) and 360470122(40)
22
Case 3: Moderate mean trend
23
24
Very
smoothened
Mean trend
Sill
Model
Range
Structure 1
Spatial
0.18
exp
3.9
Structure 2
Temporal
exp
2
Spatial
0.15318
exp
95
Temporal
exp
30
25
26
Case Another moderate Trend
27
28
29
Case 22:
30
Very
smoothened
Mean trend
Sill
Model
Range
Structure 1
Spatial
0.157
exp
3.7
Structure 2
Temporal
exp
2
Spatial
0.13
exp
85
Temporal
exp
20
31
Case222:
32
Very
smoothened
Mean trend
Sill
Model
Range
Structure 1
Spatial
0.11
exp
3
Structure 2
Temporal
exp
2
Spatial
0.1312
exp
30
Temporal
exp
15
33