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 trendmZ(s,t) = ms(s) + mt(t) where ms(s) is the spatial component and mt(t) is the temporal component

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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.

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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 ismZ(s,t) = ms(s) + mt(t)

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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

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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

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The model selected iscX(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.

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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 Smoothing RangeSpatial 2.0 0.5

Temporal 2000 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

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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

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The effect of increased smoothing is shown in the two examples below.Search Radius Smoothing Range

Temporal, example 1 3 days 1 daysTemporal, 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

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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

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Exploratory Time Series Plots: Stations 1, 2, and 4

Superimposed Time Series

0 100 200 300 400 500 600 700 8000

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log(

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Measured PM2.5 MS=1Measured PM2.5 MS=2Measured PM2.5 MS=30

520 540 560 580 600 620 640 660 680 700 720

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log(

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Measured PM2.5 MS=1Measured PM2.5 MS=2Measured PM2.5 MS=30

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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

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.90

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Std deviation

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Case1: very Flat mean Trend

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Flat Mean trend Structure 1 Structure 2Spatial Temporal Spatial Temporal

Sill 0.2 0.19Model exp exp exp expRange 4 7 100 75

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Time 1 to 20Station ID: 360610062(43) and 360470122(40)

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Case2: Raw mean trend

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Very smoothened Mean trend

Structure 1 Structure 2

Spatial Temporal Spatial TemporalSill 0.05 0.0619Model exp exp exp expRange 1.5 5 3 25

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Time 1 to 20Station ID: 360610062(43) and 360470122(40)

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Case 3: Moderate mean trend

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Very smoothened Mean trend

Structure 1 Structure 2

Spatial Temporal Spatial TemporalSill 0.18 0.15318Model exp exp exp expRange 3.9 2 95 30

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Case Another moderate Trend

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Case 22:

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Very smoothened Mean trend

Structure 1 Structure 2

Spatial Temporal Spatial TemporalSill 0.157 0.13Model exp exp exp expRange 3.7 2 85 20

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Case222:

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Very smoothened Mean trend

Structure 1 Structure 2

Spatial Temporal Spatial TemporalSill 0.11 0.1312Model exp exp exp expRange 3 2 30 15

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