The SIM2D Procedure

Example 91.3 Risk Analysis with Simulation

This example is in the field of environmental risk assessment. A square region of size 500 km $\times $ 500 km has been sampled for arsenic in drinking water. The logAsData data set consists of 138 simulated arsenic logarithm concentration observations represented by the logAs variable. Section Aspects of Semivariogram Model Fitting in Chapter 109: The VARIOGRAM Procedure, treats these observations as actual data in order to determine the spatial continuity structure for illustration in the examples.

A preliminary analysis indicates that the population exposure to the pollutant is currently at a relatively low level, as shown in the section Spatial Prediction of Pollutant Concentration in Chapter 55: The KRIGE2D Procedure, procedure. In particular, spatial prediction with kriging suggests that less than 1% of the region is affected by arsenic concentration in water that exceeds the World Health Organization (WHO) standard of 10 $\mu $g/lt.

You want to simulate the random field of the arsenic logarithm concentration so that you can gain insight into the characteristics of this field. Your objective is to assess whether the occurrence of above-threshold arsenic concentrations is a localized phenomenon, and whether you might expect more such occurrences across the region. For the simulation you need the outcome of the spatial continuity analysis in section Aspects of Semivariogram Model Fitting in Chapter 109: The VARIOGRAM Procedure. These results are the fitted semivariance models that are stored in the SemivAsStore item store by PROC VARIOGRAM.

Based on the discussion in the section Simulation and Economic Feasibility, you simulate the arsenic logarithm concentration on the premise of the ergodicity assumption. In brief, this assumption relates the mean and covariances of each individual realization to the corresponding values of the random field; see also the section Ergodicity in Chapter 109: The VARIOGRAM Procedure. In the present case you are interested in the average of a large size of realizations, rather than individual ones. A single realization could suggest possible locations where the WHO regulatory standard might be violated. The average of multiple realizations has a smoothing effect on individual excessive-concentration episodes in single realizations. The smoothing enables you to see general characteristics of the arsenic concentration field behavior on the basis of its spatial correlation description.

For illustration, assume a rectangular grid of nodes with an equal spacing of 10 km between neighboring nodes in the north and east directions. Then, simulated values are produced at a total of $51 \times 51 = 2601$ locations.

You begin by reading the logAsData data set with the following DATA step:

title 'Risk Assessment with Simulation';

data logAsData;
   input East North logAs @@;
   label logAs='log(As) Concentration';
   193.0 296.6 -0.68153   232.6 479.1  0.96279   268.7 312.5 -1.02908
    43.6   4.9  0.65010   152.6  54.9  1.87076   449.1 395.8  0.95932
   310.9 493.6 -1.66208   287.8 164.9 -0.01779   330.0   8.0  2.06837
   225.7 241.7  0.15899   452.3  83.4 -1.21217   156.5 462.5 -0.89031
    11.5  84.4 -0.24496   144.4 335.7  0.11950   149.0 431.8 -0.57251
   234.3 123.2 -1.33642    37.8 197.8 -0.27624   183.1 173.9 -2.14558
   149.3 426.7 -1.06506   434.4  67.5 -1.04657   439.6 237.0 -0.09074
    36.4 175.2 -1.21211   370.6 244.0  3.28091   452.0  96.5 -0.77081
   247.0  86.8  0.04720   413.6 373.2  1.78235   253.5 291.7  0.56132
   129.7 111.9  1.34000   352.7  42.1  0.23621   279.3  82.7  2.12350
   382.6 290.7  0.86756   188.2 222.8 -1.23308   382.8 154.5 -0.94094
   304.4 309.2 -1.95158   337.5 387.2 -1.31294   490.7 189.8  0.40206
   159.0 100.1 -0.22272   245.5 329.2 -0.26082   372.1 379.5 -1.89078
   417.8  84.1 -1.25176   173.9 407.6 -0.24240   121.5 107.7  1.54509
   453.5 313.6  0.65895   143.5 346.7 -0.87196   157.4 125.5 -1.96165
   371.8 353.2 -0.59464   358.9 338.2 -1.07133     8.6 437.8  1.44203
   395.9 394.2 -0.24144   149.5  58.9  1.17459   453.5 420.6 -0.63951
   182.3  85.0  1.00005    21.0 290.1  0.31016    11.1 352.2 -0.88418
   131.2 238.4 -0.57184   104.9   6.3  1.12054   247.3 256.0  0.14019
   428.4 383.7  0.92448   327.8 481.1 -2.72543   199.2  92.8 -0.05717
   453.9 230.1  0.16571   205.0 250.6  0.07581   459.5 271.6  0.93700
   229.5 262.8  1.83590   370.4 228.6  2.96611   330.2 281.9  1.79723
   354.8 388.3 -3.18262   406.2 222.7  2.41594   254.4 393.1  2.03221
    96.7  85.2 -0.47156   407.2 256.8  0.66747   498.5 273.8  1.03041
   417.2 471.4 -1.42766   368.8 424.3 -0.70506   303.0  59.1  1.43070
   403.1 264.1  1.64554    21.2 360.8  0.67094   148.2  78.1  2.15323
   305.5 310.7 -1.47985   228.5 180.3 -0.68386   161.1 143.3  1.07901
    70.5 155.1  0.54652   363.1 282.6 -0.43051    86.0 472.5 -1.18855
   175.9 105.3 -2.08112    96.8 426.3  1.56592   475.1 453.1 -1.53776
   125.7 485.4  1.40054   277.9 201.6 -0.54565   406.2 125.0 -1.38657
    60.0 275.5 -0.59966   431.3 494.6 -0.36860   399.9 399.0 -0.77265
    28.8 311.1  0.91693   166.1 348.2 -0.49056   266.6  83.5  0.67277
    54.7 356.3  0.49596   433.5 460.3 -1.61309   201.7 167.6 -1.40678
   158.1 203.6 -1.32499    67.6 230.4  1.14672    81.9 250.0  0.63378
   372.0  50.7  0.72445    26.4 264.6  1.00862   300.1  91.7 -0.74089
   303.0 447.4  1.74589   108.4 386.2  1.12847    55.6 191.7  0.95175
    36.3 273.2  1.78880    94.5 298.3 -2.43320   366.1 187.3 -0.80526
   130.7 389.2 -0.31513    37.2 324.2  0.24489   295.5 211.8  0.41899
    58.6 206.2  0.18495   346.3 142.8 -0.92038   484.2 215.9  0.08012
   451.4 415.7  0.02773    58.9  86.5  0.17652   212.6 363.9  0.17215
   378.7 407.6  0.51516   265.9 305.0 -0.30718   123.2 314.8 -0.90591
    26.9 471.7  1.70285    16.5   7.1  0.51736   255.1 472.6  2.02381
   111.5 148.4 -0.09658   440.4 375.0  1.23285   406.4  19.5  1.01181
   321.2  65.8 -0.02095   466.4 357.1 -0.49272     2.0 484.6  0.50994
   200.9 205.1  0.43543    30.3 337.0  1.60882   297.0  12.7  1.79824
   158.2 450.7  0.05295   122.8 105.3  1.53936   417.8 329.7 -2.08124

For this simulation you use the spatial correlation information that is saved in the SemivAsStore item store in the section Aspects of Semivariogram Model Fitting in Chapter 109: The VARIOGRAM Procedure, with the following statements:

ods graphics on;
proc variogram data=logAsData plots=none;
   store out=SemivAsStore / label='LogAs Concentration Models';
   compute lagd=5 maxlag=40;
   coord xc=East yc=North;
   model form=auto(mlist=(exp,gau,mat) nest=1 to 2);
   var logAs;

In PROC SIM2D you specify the container item store with the IN= option of the RESTORE statement. You request correlation input from an item store by specifying the STORESELECT option in the SIMULATE statement. You request 5,000 realizations for this simulation by specifying the number in the NUMREAL= option of the SIMULATE statement.

Assume that you first want to review the saved models in the item store. Use the INFO option of the RESTORE statement to produce a table with information about the top-ranking fitted model in the item store. Use the DET and ONLY suboptions of the INFO option to request details about all fitted models included in the item store. The ONLY suboption suppresses the simulation tasks and produces only the tables about the item store. You specify the following statements:

proc sim2d data=logAsData outsim=Outsim plots=none;
   restore in=SemivAsStore / info(det only);
   coordinates xc=East yc=North;
   simulate var=logAs numreal=5000 storeselect seed=39841;
   grid x=0 to 500 by 10 y=0 to 500 by 10;

PROC SIM2D produces a table with general information about the input item store identity, as shown in Output 91.3.1.

Output 91.3.1: PROC SIM2D and Input Item Store General Information

Risk Assessment with Simulation

The SIM2D Procedure

Correlation Model Item Store Information
Item Store Label LogAs Concentration Models
Data Set Created From WORK.LOGASDATA
By-group Information No By-groups Present
Date Created 15MAY14:14:38:46

Output 91.3.2 displays the item store variables, in addition to the mean and standard deviation of their data set of origin. In this case, the logAs values come from the logAsData data set. By default, the SIM2D procedure uses the variable mean in the item store for the simulation, unless you explicitly specify the MEAN statement.

Output 91.3.2: Variables in the Input Item Store

Item Store Variables
Variable Mean Std Deviation
logAs 0.084309 1.527707

The models in the SemivAsStore item store that have been fitted to the arsenic logarithm logAs empirical semivariance are shown in Output 91.3.3. The default item store model selection is the model on top of the list in Output 91.3.3.

Output 91.3.3: Angle and Models Information in the Input Item Store

Item Store Models
For logAs
Class Model
1 Gau-Gau
2 Exp-Gau
3 Exp-Mat
4 Mat
5 Gau
6 Exp

You run again the SIM2D procedure without the INFO option in the RESTORE statement. This action prompts PROC SIM2D to run the simulation tasks that you specify with the SIMULATE statement. You specify the STORESELECT option in the SIMULATE statement without any suboptions, so that the simulation uses the default model selection in the SemivAsStore item store. You save the simulation output in the Outsim output data set. You also specify the SIM and the SEMIVAR suboptions in the PLOTS option of the PROC SIM2D statement to obtain output plots. You use the following statements:

proc sim2d data=logAsData outsim=Outsim plots=(sim semivar);
   restore in=SemivAsStore;
   coordinates xc=East yc=North;
   simulate var=logAs numreal=5000 storeselect seed=89702;
   grid x=0 to 500 by 10 y=0 to 500 by 10;

Note: This step can take several minutes to run, and it produces a data set with over 10 million observations. When you run these statements, PROC SIM2D again produces a table about the input item store identity. This output is followed by a number of observations table and information about the simulation task, as shown in Output 91.3.4.

Output 91.3.4: Number of Observations and Simulation Information Tables

Risk Assessment with Simulation

The SIM2D Procedure
Simulation: SIM1, Dependent Variable: logAs

Number of Observations Read 138
Number of Observations Used 138

Simulation Information
Simulation Grid Points 2601
Type Conditional
Number of Realizations 5000

The SIM2D procedure uses the selected fitted Gaussian-Gaussian model in the SemivAsStore item store. Output 91.3.5 shows the saved parameter values of the model that are used in the simulation.

Output 91.3.5: Information about the Gaussian-Gaussian Model

Covariance Model Information
Nested Structure 1 Type Gaussian
Nested Structure 1 Sill 0.3276646
Nested Structure 1 Range 62.312728
Nested Structure 1 Effective Range 107.92881
Nested Structure 2 Type Gaussian
Nested Structure 2 Sill 1.261545
Nested Structure 2 Range 21.459563
Nested Structure 2 Effective Range 37.169053
Nugget Effect 0.0830758

Output 91.3.6 shows a plot of the correlation model used in the simulation. Its parameters come from the stored information in the SemivAsStore item store.

Output 91.3.6: Gaussian-Gaussian Semivariogram Model Used in Simulation

Gaussian-Gaussian Semivariogram Model Used in Simulation

The PLOTS=SIM option produces contours of the mean value of the 5,000 realizations for each one of the output grid locations, and a surface of the associated standard deviation. The resulting plot is shown in Output 91.3.7. The mean value contours exhibit reduced variation compared to a SIM plot of one single realization. In fact, Output 91.3.7 is very similar to the prediction plot of the same Gaussian model in the section Spatial Prediction of Pollutant Concentration in Chapter 55: The KRIGE2D Procedure.

Clearly, there appears to be a small area of increased arsenic concentration values, which is located in the central-eastern part of the domain. The WHO threshold of 10 $\mu $g/lt for the maximum allowed arsenic concentration in water translates into about 2.3 in the log scale. The indicated area is the only one within the region where the simulated means exceed the value 3. In addition, there appear to be two smaller areas of simulated means values at or above 2 in the southern part of the region.

Output 91.3.7: Simulated Arsenic Logarithm Values with Gaussian-Gaussian Covariance

Simulated Arsenic Logarithm Values with Gaussian-Gaussian Covariance

The simulation plot indicates that arsenic concentration values in excess of the WHO health standard is a rather localized phenomenon. If it were not so, then the plot would suggest that increased arsenic concentrations extend across larger areas. In turn, this means that individual realizations would tend to produce systematically higher arsenic concentrations at different neighboring locations, and their average would appear as higher logAs values in the SIM plot. Instead, you conclude that the Gaussian-Gaussian correlation that describes the spatial continuity for the arsenic logarithm observations leads to only localized occurrence of the WHO standard violation.

Now you want to find out whether additional localized occurrences might take place on the basis of the Gaussian-Gaussian spatial continuity behavior. The distribution of the simulation standard deviation values in Output 91.3.7 offers important feedback about this issue. Observe the areas in the plot where the means gradient indicates increasing values. The means contours create several pools where the values increase to simulated means higher than 1.

With the clear exceptions of the isolated area in the central-eastern part of the domain and the pool in the south-west that exhibit logAs values above 2, the simulation standard deviation seems to be around 1 or less in almost all other pools with logAs values above 1. Due to those relatively low standard deviations, this visual inspection suggests that even in individual realizations you might rarely expect the arsenic logarithm to exceed the threshold value of around 2.3.

However, in the few pools of logAs values above 1, there can be patches of increased standard deviation. In these cases, you suspect that arsenic concentration could exceed occasionally the WHO regulatory standard, even if the plot of the simulated means does not explicitly portray so.

Based on these remarks, you want to quantify the percentage of the region that could be affected by excessive arsenic concentration. You begin with a DATA step that takes as input the simulation Outsim output data set. The DATA step marks the simulated arsenic logarithm means values that are in excess of the WHO concentration threshold of 10 $\mu $g/lt, and saves the outcome into an indicator variable OverLimit. At the same time, the arsenic logarithm values are transformed back into arsenic concentration values so that they can be compared to the threshold value. The simulated means in the Outsim data set are stored in the svalue variable. You use the following statements:

data AsOverLimit;
   set Outsim;
   OverLimit = (exp(svalue) > 10) * 100;

Then, you use the MEANS procedure to express the selected nodes population (where the WHO standard violation occurs) as a percentage of the entire domain area. You invoke PROC MEANS twice. The first time, you average over each realization in the PROC SIM2D output. For that purpose, you use the variable _ITER_ in the simulation sim1 output data set as a BY variable in PROC MEANS. You save the iteration-averaged indicator values in the PctOverLimit variable. The second time, PROC MEANS averages the PctOverLimit variable to obtain the percentage you want. You also specify the P5 and P95 options in the PROC MEANS statement to request the lower and upper confidence limits, respectively, in this computation. The interval between those limits expresses the 90% interval for the true area percentage based on the stochastic simulation. You use the following statements:

proc means data=AsOverLimit noprint;
   by _ITER_;
   var OverLimit;
   output out=OverLimitData mean=PctOverLimit;
proc means data=OverLimitData mean p5 p95;
   var PctOverLimit;
   label PctOverLimit="Percent above threshold";

The result is shown in Output 91.3.8. PROC MEANS accounts for all individual occurrences throughout the simulation where the WHO arsenic concentration threshold is exceeded. This happens at about 3.9% of the total area in 5,000 realizations. Compare this value to the less than 1% percentage in the PROC KRIGE2D prediction, as reported in the beginning of this section. On one hand, the prediction outcome tells you what goes on currently across the region within a degree of certainty given by the prediction error. On the other hand, the simulation provides you with an indicator that under the given spatial continuity estimate a relatively larger area is in potential danger of being affected by above-threshold arsenic concentration.

In fact, the 5% and 95% confidence limits in Output 91.3.8 show the area percentage limits within which you can expect excessive arsenic concentration. Based on the stochastic simulation, at the 90% confidence level the arsenic concentration in drinking water is expected to violate the WHO regulatory standard in anywhere from about 2.6% to 5.4% of the study region.

Output 91.3.8: Violation of Arsenic Concentration Threshold Using Gaussian-Gaussian Model

Risk Assessment with Simulation

The MEANS Procedure

Analysis Variable : PctOverLimit Percent
above threshold
Mean 5th Pctl 95th Pctl
3.9308727 2.6143791 5.4209919

You also want to compute the probability that individual areas in the region are expected to violate the WHO arsenic concentration standard. Start by identifying again the simulated arsenic logarithm means values in excess of the WHO concentration threshold of 10 $\mu $g/lt. The following DATA step saves the outcome into the variable LocalOverLimit:

data LocalAsOverLimit;
   set Outsim;
   LocalOverLimit = (exp(svalue) > 10);

Then you use the SORT procedure to sort the information based on location coordinates. This intermediate step is necessary so that you can use the MEANS procedure with the LocalAsOverLimit data set. In the following statements, PROC MEANS computes the expected violations of the WHO standard for each location in the region across all realizations of the simulation:

proc sort data=LocalAsOverLimit;
   by gxc gyc;
proc means data=LocalAsOverLimit noprint;
   by gxc gyc;
   var LocalOverLimit;
   output out=OverLimLoci mean=ProbOverLimit;

The output is the probability that the WHO standard is violated at each one of the simulation locations across the region. You create a plot of this probability with the help of the TEMPLATE and the SGRENDER procedures. You use the following statements:

proc template;
   define statgraph surfacePlot;
      entrytitle _TITLE;
      layout overlay /
         xaxisopts = (offsetmax=0)
         yaxisopts = (offsetmax=0);
         contourplotparm x=_VARX y=_VARY z=_VAR /
                         nhint=10 name='probplot';
         continuouslegend 'probplot' / title=_LEGENDLABEL;

proc sgrender data=OverLimLoci template=surfacePlot;
   dynamic _VARX ='gxc'
           _VARY ='gyc'
           _VAR  ='ProbOverLimit'
           _TITLE='Arsenic WHO Standard Violation'
           _LEGENDLABEL='Violation Probability';
   label gyc='North' gxc='East';

ods graphics off;

Output 91.3.9 shows the map of probability that the arsenic concentration WHO standard is violated in the region for the selected spatial correlation model of the pollutant. Based on the preceding analysis, the probability is very close to 1 in the area where the simulation mean values are above the standard limit. A few more areas that were indicated earlier in the analysis suggest a violation probability between 20% and 40%. The remaining areas in the region exhibit very low probability, which is notably nonzero at a few scattered locations. These findings suggest that throughout the simulation there have been realizations where the arsenic WHO standard was exceeded at locations whose simulated mean is well below that standard.

From the environmental risk assessment perspective, the preceding analysis can trigger a more detailed investigation into areas in the region where the health standard might be violated. Although the original observations in the logAsData data set indicate no existing problem in some of these areas, the present example illustrates that spatial analysis and simulation can raise flags of caution. This type of analysis can help to focus scientific, management, and policy efforts on these particular areas of the region and to monitor closely the pollutant concentration for potential health risks.

Output 91.3.9: Map of Violation of the Arsenic WHO Standard Probability

Map of Violation of the Arsenic WHO Standard Probability