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Page Subgrade Shear Failures JOINT CSHRP/SASKATCHEWAN BAYESIAN APPLICATION 1. Introduction 1.1 Problem Statement/Need for a Model In the spring of 1994, Saskatchewan Highways and Transportation (SHT), Technical Standards and Policies Branch, Materials Section, was given the task to investigate the issue of subgrade design and characterisation, as it relates to performance of unpaved roads. The intent was to take a 'quick and dirty' look at the factors that impact on subgrade performance and to attempt to collect the required field data, needed to develop a first generation performance prediction model. SHT wanted to develop a method of subgrade performance prediction under different loading conditions. The issue of subgrade performance had not been looked at by SHT since the 1960's and early 1 970's. SHT anticipated newer technology could be applied to the old problem of subgrade design, analysis and performance modelling. The impetus for the subgrade study was a direct result of concentrated grain hauls from elevators to inland terminals, causing significant damage on many Gravel and Thin Membrane Surfaces (TMS). SHT felt it may be possible to develop a method of predicting the performance of these roads, and that in the future, hauling could be managed to reduce or eliminate, the damage. Changes in the Crow grain transportation subsidy, the move to abandon rail lines and the rationalization of the elevator systems, are other factors leading Saskatchewan Highways and Transportation to place a high priority on developing methods to predict the performance of low volume roads. The performance of unpaved roads, or roads with only surface treatments, is a direct function of the road's subgrade materials. The factors that control the subgrade performance and the interaction between factors, have never totally been understood. There is no reliable method available to evaluate or predict the load carrying capability of a subgrade throughout the year. The influence of truck loading can be incorporated in pavement design, but its impact on unpaved roads are not understood. Truck loading factors that could be considered in low volume road design include: type of truck (number and configuration of axles), gross vehicle weight, axle weight, number of repetitions and the frequency of repetitions. Tire inflation pressure is also an important loading variable, but was not quantified in this subgrade study. There have been other developments related to material characterization, lab and field testing equipment, and instrumentation, that now allow the measurement of insitu strength, soil matrix suction, moisture content and temperature, over time and changing environmental conditions. There is no clear definition of what is meant by subgrade performance and how subgrade failure can be defined. However, to develop a model of subgrade performance, it is necessary to explicitly define the dependent variable. This variable must be intuitive and interpretable by experts and it must be possible to measure it in the field. For our model, subgrade performance was defined in terms of subgrade deflection. Part way through the data collection process, SHT had the opportunity to take part in a TAC sponsored Bayesian Methods and BSTAT Software training session. A proposal was put forward and accepted by TAC for a joint CSHRP / SHT Bayesian Application Project. The project's purpose was to learn the Bayesian methodology and to use the newly developed Bayesian software as a tool in developing a subgrade performance model. The problem was defined as being able to model subgrade deflection. 1.2 Project Objectives The Bayesian project had a number of objectives:
2. Team Members
Experts:
3. Schedule and Methodology Saskatchewan Highways and Transportation (SHT) was introduced to Bayesian regression during a CSHRP/TAC sponsored training session in Saskatoon in June 1 994. Prior to the development of the Bayesian project, SHT had started to collect field data as part of a Subgrade Study. This data was subsequently processed to meet the needs of the Bayesian project. Part way through the data collection process, SHT had the opportunity to take part in a CSHRP/TAC sponsored Bayesian Methods and BSTAT Software training session. A proposal was put forward and accepted by CSHRP for a joint CSHRP Saskatchewan Highways and Transportation Bayesian Application Project. The project's purpose was to learn the Bayesian methodology and to use the newly developed Bayesian software as a tool in developing a subgrade performance model. Expert judgement was requested from a number of internal and external experts. Judgement was elicited using a structured survey technique. Each expert was provided with an encoding package that explained the problem and the model, and led them through the encoding process. A total of three external and 10 internal experts provided input. The Bayesian Project provided an opportunity to become familiar with the model development process and software. Saskatchewan Highways and Transportation used this project to formally evaluate the new technique and determine if it had potential in other areas of the department. 4. Model Selection The first step was to select the variables and define the form of the model to be developed. A linear relationship of the following form was selected.
The Variables, their Coefficients, the Constant as well as the Random Error for this linear relationship, were to be determined. 4.1 Definition of Model Variables The first step in designing a Bayesian Modelling Project is to obtain expert input, to determine what variables contribute to the problem and to prioritize these variables. A list of possible variables was developed for the project by the analysis team. In order to test the full utilization of the Bayesian methodology, the experts were requested to provide their input into variable identification. A list of potential variables was provided in the Encoding Package (Appendix C) and the experts were asked to add any variables they felt were missing, and then to rate all the variables on a scale of one to 100 (100 having the most significance to subgrade performance). The response for independent variables, not surprisingly, had CBR ranked first. Subgrade material type, shear strength, truck axle weight, moisture content, standing water, tire pressure, and number of axle repetitions, were identified as being the top eight variables which were casually linked to subgrade deflection (or proxy for subgrade performance). The Expert's were asked for their input into subgrade performance or failure, in terms of a dependent variable. What did they feel could be predicted and measured? A list of possible performance measures was provided and they were asked to add any additional factors they felt could be defined or measured, and to rate them on a scale of one to 100 (100 having the most significance to subgrade performance). The experts' response for dependent variables, had deflection ranked as number eight, out of 15. However, some of the higher ranked dependent variables would be difficult to measure; specifically all weather capability, frost boils and heaves, soft spots and spring strength and reduction. A summary of the variables identified by the experts and the aggregate ranking for each, is provided in Figure 1 (filename = EXP_VAR.XLS). Typically, the field data collection program would have been designed before any data was collected. In this study, existing data was to be used. This data was not explicitly collected to support regression analysis. As such, for some sites, there were incomplete records (i.e.: data was missed for one or more of the independent variables). This constraint resulted in many of the observed records in the database being deleted from the regression analysis. The data available for this project had been collected over a short period of time, in order to develop data collection and handling techniques. The two construction haul roads were selected to obtain data during concentrated haul. A large quantity of aggregate was being hauled over gravel road, in a short period of time. A decision had been made to collect a variety of data that could be readily obtained, before, during and after the haul. Visual inspections were made and additional testing was completed at specific locations where changes in the road condition occurred. Performance was being measured in terms of deflections and visual condition, while the following data was collected: daily loading, nuclear density, insitu density, moisture content, unified classification, pocket penetration on spoon samples, and dynamic cone penetrations (DCP). The data collection was well underway before the Bayesian project was started, so it became necessary to select variables that had been measured, and to build a model based on available data. Deflection was selected as the dependent variable to be modelled. Loading data was very good, since the weight tickets could define any loading variable desired. The subgrade data was compiled in order to determine how best to include it in the analysis. It was found that the basic material properties gave a shotgun scatter and no trends could be defined over time. In order to define the subgrade in terms that an expert could relate to performance, the data was reassessed. It was determined pop gave very consistent results and showed a number of trends. The pop value is a measurement of penetration in units of mm per blow. Figure 2 shows a schematic of a pop with approximate dimensions. The output of the pop is a pop curve with blows as a function of depth. Two concerns were identified with respect to using pop in the Bayesian analysis:
To resolve these concerns a decision was made to convert the pop results to CBR. The following equation developed by Kleyn and Van Heerden in 1975 was used in the conversion.
If the penetration per blow becomes very low (less than three mm per blow), the CBR exceeds 100 and becomes meaningless. In this situation, the result was recorded as a maximum CBR(CBR=100). The data showed in almost all cases, there is a distinct change in strength from a strong surface layer to a weaker underlayer. This strong surface layer was defined as the 'crust'. As a result, a thickness and strength value could be determined for the crust. The underlying material tended to have a uniform strength throughout the 1.2m depth measured. Figure 3 shows a typical pop result with data, while Figure 4 shows the crust thickness and calculated CBR's. This CBR relationship was developed for all pop tests taken during the study. SHT's intent was to use the collected expert judgement in conjunction with the field data collected on the two haul roads. In order to ensure consistency between the two information sources, each variable and it's inference space was explicitly defined. It was important to have all experts view the problem in the same way and they make the same assumptions. 5. Expert Encoding Package In order to ensure consistent responses from the experts, a detailed expert encoding package was developed. The package introduced the problem, identified the selected variables and specifically indicated what was expected of the experts (Appendix C). Experts were asked to predict deflection, based on the variable matrix provided. The dependent variable selected to be modelled was rebound deflection in millimetres. It was measured by a Benkelman Beam (recorded to 0.lmm), under a standard axle of 80 KN. The road surface was generally very hard, so it was possible to brush away any loose material, in order to take a normal beam deflection. Measurements were taken in the outside wheel path, but at least 1m from the unsupported edge of the road. Readings were not corrected for temperature on the gravel roads, as they are for pavements. The deflections measured at the sites were distributed over a wide range. The inference space selected was a range from a low reading of 0. 1mm to a high reading of 9.5mm. The experts were asked to predict deflection for all variable combinations using this inference space. 5.2 Independent Variables The problem was defined to apply to a normal grid road standard, with an eight metre top width. Both test sections were old roads that have carried gravel hauls in the past, but would normally have ADTs of less than 100 vehicles per day and very few trucks. The roads were level, with a slight crown. Measurements were not taken on curves and there were no hills. The grade height ranged from 1.2m to 1.8m, with no standing water or groundwater discharge. There was no significant prolonged rainfall during the period of the haul and temperatures were in the normal summer range. Normal surface spot blading occurred during the haul, but there was no major blading or windrowing of gravel. The subgrade material varied slightly between holes and with depth. The material was a till with traffic gravel worked into the surface. The following properties would be typical of those measured over the sites.
After reviewing the data collected from the two sites, the independent variables were defined as:
5.2.1 Crust Thickness Crust thickness was defined as the depth or thickness, at which the slope of the pop penetration plot took a sharp break indicating a drop in strength. A thickness of less than 50mm was considered to be no thickness and was encoded as zero mm. In such cases, only the strength of the subgrade would be applicable. A crust thickness between 50mm to l50mm, was encoded as 100mm When the crust thickness was greater than 1 50mm, it was encoded as 1 75mm. 5.2.2 Crust Strength The crust strength was reported in terms of CBR, with values in the range of CBR 40 to CBR 70 being encoded as CBR 55, the range CBR 70 to CBR 100 being encoded as CBR 85, and CBR 100 was encoded as CBR 100. 5.2.3 Subgrade Material Strength The subgrade material strength was also reported in terms of CBR. Values less than CBR 15 were encoded as CBR 7, values in the range of CBR 15 to CBR 30 were encoded as CBR 22, and values greater than CBR 30 were encoded as CBR 30. 5.2.4 Total Loading Total loading was recorded in terms of Total Gross Tonnes which were hauled over the section throughout the term of monitoring. The trucks hauled at secondary weights and consisted of an almost even split of five axle semitrailer loaded to 34.5t (steering 5.5t, drive tandem 14.5t and rear tandem 14.5 t) and six axle semitrailers loaded to 40.0t (steering 5.5t, drive tandem 14.5t and rear tridem 20t). Both configurations have an ESAL equivalent of approximately four EASLs per load. The total loading was broken into three ranges. Less than 40,000t was encoded as 20kt (approximately 540 loads, 3000 axle passes or 2000 ESALs), 40,000t to 120,000t was encoded as 80kt (approximately 2000 loads, 12,000 axles or 8000 ESALs) and greater than 120,000t encoded as 150kt (approximately 4000 loads, 24,000 axles or 16,000 ESALs). 5.2.5 Repetitive Loading The repetitive loading was recorded in terms of axles per day. Less than 220 axles per day was encoded as 120 axles per day (20 loads, 800t or 80 ESALs), the range of 220 to 450 axles per day was encoded as 320 axles per day (60 loads, 2400 t, or 240 ESALs) and greater than 450 axles per day was encoded as 500 axles per day (90 loads, 3600 tonnes, or 360 ESALs). 5.3 Relationship Between Deflection and Independent Variables In order to understand the experts' reasoning in completing the Encoding package, they were asked to provide a short narrative describing the relationship of each independent variable to the predicted deflection. They were also asked to sketch the form of this relationship. These trends were important, as to whether the independent variable had a positive or negative impact on deflection. This information provided a valuable check on the expert's opinion encoded in the matrix and provided some insight into possible variable transformations. 5.4 Summary of Variables As a result of evaluating all of the Factual data, the following categorization, inferences and encoding values were adopted.
6. Modeling 6.1 Model Type and Functional Form A simple linear functional form was initially assumed and the subsequent Bayesian analysis was performed. The resulting regression equation was defined as: Deflection = Constant + A(Crust Thickness) + B(Crust Strength) + C(Subgrade Strength) + D(Total Loading) + E(Repetitive Loading) + E The regression analysis would determine the optimal values for the Constant and the five Coefficients based on both the field data (DATA) and the encoded expert judgement (PRIOR DATA). 6.2 Factual Data (Data) (DATA.XLS) The field data was obtained from two independent haul roads, the Regina Beach site (filename = DATA_REG.XLS) and the Denholm site (filename = DATA_DEN.XLS). The field test results provided only 61 data records, where all the independent variables were available. These 61 records corresponded to only having Factual data for 22 matrix boxes, out of a total of 243 boxes, i.e.: five variables with three options per variable (3*3*3*3*3 = 243). Figure 5 shows the 'coverage' of the Factual data, with respect to the encoding matrix. Figure 6 is a printout of the 61 records of Field data used in the Bayesian regression analysis. A regression analysis of the Factual data was performed to determine if the data was statistically valid. The Residual Variance was calculated to be 0.37, and the 95% Confidence Level to be 1.22mm. The variance of the Field data was thought to be in the range of + 1.22mm. This turned out to be misleading, since the TValues for a number of the independent variable were below 1.96 (for a 95% confidence level with five variables, a TValue of 1.96 or greater, was required) (Figure 7) (filename = 5DATA_RA.XLS). 6.3 Expert Data (Prior) (EXPERT_A.XLS) Thirteen experts were encoded. The experts completed the Encoding matrix, as shown in Figure 8. Figure 9 shows the subsequent conversion to a file format. The EXPERT _A.XLS file is provided as an example of a typical response. Performing a Sensitivity Analysis resulted in reducing the number of experts used for subsequent analysis, to 10. Three experts were removed from the analysis because the Sensitivity Analysis indicated their interpretation of the problem was significantly different than that of the remaining experts. This discrepancy was confirmed as a result of discussing it with each of these three experts. 6.4 Sensitivity Analysis The sensitivity analysis was performed on the Factual data, as well as for each expert's judgement. This procedure helped to confirm whether the Factual data indicated the same trends as what the experts were saying, and whether the experts agreed on the effects of the various independent variables. The sensitivity analysis was found to be an excellent tool in verifying the compatibility between the Factual data and the expert's judgement. The results of the sensitivity analysis are shown in Figure 10. The affects of the various independent variables, whether a variable has a positive or negative influence, indicated the Factual data was compatible with what the experts predicted. Increases in three of the independent variables, Crust Thickness, Crust Strength, Subgrade Material Strength, indicated reduced deflection, while an increase in the Total Loading independent variable increased the deflection. Also, of the 13 experts initially encoded, an assessment of which of the experts were 'weak', was determined using the Sensitivity analysis. Three experts were ultimately removed from the amalgamation of all the experts. These expert's judgement did not conform with the remaining experts. Their line slopes were significantly different, or their range of values did not match the inference space specified in the Encoding package. Note: The sensitivity analysis was performed on the initial five independent variables. A reevaluation of sensitivity was not performed when the independent variables were reduced to four. No significant change was expected on the remaining four independent variables. 6.5 Combining of Expert Judgments To permit Bayes to work with all of the various expert's judgement, all of the expert's matrixes had to be combined into one. A regression analysis on this combined file would then be input into XLBAYES. The first technique was to combine all the expert's files and run a regression analysis. Unfortunately, Excel and/or Windows did not allow 1890 records to be input. An unknown maximum number of input records, appears to have been exceeded. The second option was far more cumbersome and time consuming. Each expert's file had to be input as Factual data into XLBAYES. The resultant Posterior was then input as the Prior, and the next Expert was input as Factual data. The final Posterior became the combined Expert's regression file. Note: This exercise was performed a second time, when the independent variables were reduced from five to four. The end result of combining the experts resulted in the Degree of Freedom (DOF) = 1884 for the five variable analysis, and 1885 for the four variable analysis. In removing the fifth variable, we removed the 'column' associated with Repetitive Loading, from the Factual data and the expert judgements. This resulted in the experts having indicated three deflection values, for the fourth variable. The resultant model variance, can in part, be attributed to the way the data was encoded. As a result, the same number of records, 1890, were input for the four variable analysis (DOF = 1885). To ensure the Expert's data was statistically valid, a regression analysis was performed. The Residual Variance was found to be 1.93 and the 95% Confidence Level was calculated to be 2.78mm. These results are higher than those of the Factual data. Therefore, all the independent variables are statistically significant. This is shown in Figure 11 (filename = 5EXP_BAY.XLS). 7. Bayes Analysis 7.1 Five Variables (BAYES5.XLS) Factual Data (Data) The Factual data originated from 61 individual records obtained from pop Testing and Loading information, Figures 12 through 17 indicate the results of performing a classical regression on this data. Expert Data (Prior) The Bayesian classical regression for the 10 combined experts was used as input, Figure 18 23. Bayes Result (Posterior) The Factual Data and the Expert Data was used as input to Bayes and resulted in the following predictive model (PRIOR) (Figures 36 41). Deflection = 6.43 0.01(Crust Thickness) 0.02(Crust Strength) 0.09(Subgrade Material Strength) + 0.01(Total Loading) + 0.003(Repetitive Loading) Residual Variance = 2.06 95% Confidence Level = 2.87mm, i.e.: Deflection = n +/- 2.87mm All of the variable graphs, indicated the Posterior favoured the Expert data (Prior) and showed little confidence towards the Factual Data. 7.2 Four Variables (BAYES4.XLS) Factual Data (Data) The number of Factual data records remained the same, but the Repetitive Loading column of information was discarded (Figure 24). A regression analysis of the revised Factual data indicated minimal change to the Residual Variance, 0.38 versus 0.37, and the 95% Confidence Level was calculated to be 1.23mm versus 1.22mm. The TValues however, improved for all variables, except for Subgrade Material Strength, as shown in Figures 25 through 29 (filename = 4DATA_RA.XLS). Expert Data (Prior) The Expert's data was also modified to reflect four variables. The subsequent regression analysis for the 10 combined experts, only changed the Constant value from 6.94mm to 7.73mm. A regression analysis of the revised Expert data, obtained by having to input each expert's judgement individually into Bayes to create a Posterior, from which the next expert's judgement would go into Bayes as Factual data, indicated minimal change to the Residual Variance, 2.08 versus 1.93, and the 95% Confidence Level was calculated to be 2.88 versus 2.78mm. The TValues also remained statistically valid, Figures 30 through 34 (filename = 4EXP_BAY.XLS). Bayes Result (Posterior)
Residual Variance = 2.25 95% Confidence Level = 3.00mm, i.e.: Deflection = n +/ 3.00mm All of the variable graphs, indicated the Posterior once again favoured the Expert data (Prior) and showed little confidence towards the Factual data. An interesting observation is the Factual data for Total Loading, now graphs with a shift towards the Expert data! It would be expected that since the Factual data remained constant and did not change for the remaining four variables, why would dropping an independent variable, shift the Total Loading? The Expert's data went into Bayes with 1885 DOF, as a result of the large number of experts. The DOF is used by Bayes as a weighting factor; 1885 DOF from the Experts, influenced the significance of only 61 DOF for the Factual data. As a result, the next step was to reduce the significance of the Experts, by reducing the DOF. 7.3 Four Variables 61 DOF (BAYES4B.XLS) Factual Data (Data) The Factual data remained the same at 61 records. Expert Data (Prior) The Expert's data remained the same, except it was input with 61 DOF, rather than the original 1885, to match the Factual data's DOF. Bayes Result (Posterior)
Residual Variance = 34.56 95% Confidence Level = 11.76mm, i.e.: Deflection = n + / 11.76mm Note: The actual coefficients have not changed, only the Confidence Level. All of the variable graphs, indicated the Posterior once again favoured the Expert data (Prior) and showed little confidence towards the Factual data. Changing the Expert's DOF, did not provide us with a revised Model equation, as expected. All that occurred was the Confidence Level decreased substantially, from three mm with 1885 DOF, to 11.76mm with only 61 DOF. 7.4 Iterations Based On Degrees of Freedom The Expert's DOF were altered, so as to evaluate the changes in the 95% Confidence Level. The Model equation itself did not change. The Constant and the Coefficients for each independent variable remained the same. The results are summarized in Section 8.1 Model Runs. 7.5 Matrix Modification (4EXP_RAM.XLS) The purpose of performing this analysis was to determine if having reduced the Encoding Matrix for Crust Thickness = 0, had an effect on the final model equation. Expert Data (Prior) The Encoding Matrix has a number of columns crossed out. The reasoning for this, was if the Crust Thickness equalled zero, the overall strength of the subgrade was being provided by only the Subgrade Material Strength variable. However, if the Expert's data was duplicated, for the crossed out columns, would there be a change to the model equation? The Expert's data was modified, and a regression analysis was performed, Figure 35 (filename = 4EXP_RAM.XLS The Residual Variance equalled 2.39 and the 95% Confidence Level equalled 3.09mm. This compares to the original values of 2.08 and 2.88mm, Figure 30 (filename = 4EXP_RAO.XLS). The 95% Confidence Level dropped to 3.09mm, from 2.88mm. Therefore, no further analysis was performed on the premise that duplicating these repetitive values would reduce the overall confidence more! 7.6 Final Model
The 95% Confidence Level associated with the final result was as low as 3.00mm, if the weighting of the experts is based on 10 Experts versus 61 Factual data records. The 95% Confidence Level goes as high as 11.76mm, if the Experts are assigned the same 'weighting' as the 61 Factual data records (by changing the DOF to 61). 8. Interpretation and Discussion 8.1 Model Runs As indicated earlier, the first Bayesian analysis was with the five independent variables. The results are shown in Figures 36 through 41. The Bayesian analysis was performed after dropping the independent variable Repetitive Loading. The results are shown in Figures 42 through 46. The four variable analysis was then adjusted by changing the Expert's DOF, in an effort to minimize the Residual Variance. The Bayesian analysis was run with a number of different Expert DOF. The results are summarized in the following table.
As discussed earlier, changing the Expert's DOF, does not change the coefficients of the model. It only helped to indicate how much the Expert's data is favoured by Bayes. Saskatchewan Highways and Transportation will continue to 'fine tune' the model, with more field data, or Factual data, and possibly including other independent variables, in an effort to improve the 'balance' between the Expert Judgement and the Factual Data. 8.2 Transformations Two transformations were encountered; one minor and one rather major transformation. The Encoding Matrix was setup on an Excel spreadsheet, in order to key the Expert's judgement into a format similar to the work sheet. Transforming each matrix 'box' into an individual record, was a minor problem. The major transformation encountered was a result of having solicited a large number of experts. To be able to combine and consider the amalgamation as one Expert, was time consuming. To have to do it a second time, as a result of the decision to discard the Repetitive Loading variable, provided ample opportunity for human error. An expert who best 'fits' the average of all of the experts could have been selected, but the effort of combining them, was felt to be beneficial in the long run. 8.3 Factual Data Statistics The Factual data (filenames = DATA_DEN.XLS and DATA_REG.XLS) was reviewed to ensure it was statistically valid, by checking the maximums and minimums of each independent variable, to ensure they were within acceptable boundaries. A regression analysis was performed, (filenames = 5DATA_RA.XLS and 4DATA_RA.XLS) to evaluate the 'T' values and 95% Confidence Level' etc.. This was done prior to developing the Encoding package, because the inference space for each of the independent variables had to be determined. This inference space, or range of acceptable deflection values expected from the experts, had to be predetermined to ensure consistent expert responses. 8.4 Compatibility Problems No compatibility problems were encountered due to the fact that the Factual data was derived from pop readings and no previous field data was being used as Factual data. Also, since the Encoding package was tailored to match the Factual data, no compatibility problems were encountered. 8.5 Analysis The analysis started with the five independent variables as discussed earlier, but ultimately a four variable model was developed. There was a high standard error on the five variable model. This was because of the similarities between two of the variables: Total Loading and Repetitive loading. Repetitive Loading was excluded from the final four variable analysis because of 'poor' Factual data and its similarity to Total Loading. These two variables ran parallel with each other, and as a result, worked against each other when Bayes considered them as independent variables. When more Factual data is entered into the model, it will improve the confidence level. At the present time, the Factual data has a 95% confidence level of 1.23mrn. This is a result of only 61 Factual data records. The experts had a 95% confidence level of 2.88mm, based upon 1885 DOF. This is a moderate value for the amount of experts influencing the confidence level. The Posterior, or the final Bayes result, has a 95% confidence level of +/- 3.00mm. 8. 6 Interpretation of Results Interpreting the results was easy, with the direction and support provided by Vemax Management Inc. Gaining experience with Bayes, interacting with other jurisdiction's Bayesian successes and failures, and attending the MidCourse Workshop in Ottawa, will help to interpret future Bayesian applications. SHT feels comfortable with the Bayesian methodology. The final model however, has resulted in a large constant, 7.23, and a poor 95% Confidence Level. SHT intends to test the model with real data and confirm whether or not the predictions are valid. Additional Factual data will be collected to cover a broader range of the independent variable matrix. 8. 7 Inference from Analysis The model's Residual Variance could be attributed to a number of possible problems; the small sample size of Factual data records, a nonlinear functional form, other independent variable which were not encoded, and noisy or poor data. 9. Conclusions The initial objectives for this Bayesian project, as discussed in detail in the Project Objectives section, were to: 1. To learn Bayes and to incorporate the use of it within Saskatchewan Highways & Transportation. SHT has now applied the Bayesian methodology and learned how to use XLBAYES. The software appears to be an excellent tool. SHT needs to market this expertise within the department, in order to ensure others are aware of its potential. 2. Clearly define and prioritize all of the variables that characterise or contribute to the performance of subgrade materials. This was accomplished with the experts providing a priorized list of independent variables. Surveying the experts to rank candidate variables worked well. 3. To incorporate a reasonable number of variables to develop a manageable and supportable model. Our greatest success in performing this analysis is the 'simple' independent variables: Crust Thickness, Crust Strength, Subgrade Material Strength and Total Loading. They are all easily obtainable, since all that is needed is a pop and an indication of the Total Loading expected. 4. To develop data collection, handling and analysis techniques that are practical and economical. This objective can only be achieved after SHT gains confidence with the model, through 'fine tuning' it and applying it to real applications. 5. To develop a first generation performance model on which to build on. The following model was developed:
In general, SHT's initial objectives have been met. Efforts must continue, to ensure the Bayes expertise is not lost. 10. Satisfaction and Improvements 10.1 Use of Developed Model, Both Inside and/or Outside the Agency The final model derived for the purpose of completing the Joint CSHRP/Agency project, will be used as opportunities become available. Saskatchewan Highways and Transportation's objectives go beyond completing this report. The overall Subgrade Study continues and the resultant Bayesian model is only a small part of the Study's objectives. The whole principle behind Bayesian statistics is to prevent spending five years collecting data before starting the analysis. SHT has a 'first draft' model after collecting data for only a few months. Supplementing the Factual data with more results, will help to improve the accuracy of the model. There is also an opportunity to add another variable, such as truck tire pressure. SHT expects to continue evaluating the accuracy of the model by bench marking the model predictions to observations in the field. Saskatchewan Highways and Transportation welcomes any interest from outside parties, who may be interested in our findings, or who have had experience in similar modelling exercises. 10.2 Bayesian Methodology Saskatchewan Highways and Transportation's experience with the Bayesian methodology was positive. Incorporating expert judgement in a modelling exercise, early during the design of the experiment, will save time and money. The traditional method of collecting data for five years before evaluating it, is too costly and could result in collecting the wrong data. SHT cannot compare the Bayesian methodology to another similar process. However, the systematic stepbystep approach utilized by the Bayes methodology, was successful for this project. 10.3 XLBAYES Software The Bayesian methodology works, but improvements should be made to the XLBAYES software. The first priority for improving the interface between the user and the methodology, would be to streamline the amalgamation of experts. The process used was dictated because of the number of experts and the number of input records for each expert. Combining all the experts into one large database resulted in too large of a file for XLBAYES to analyze. This resulted in having to combine successive experts, which involved a lot of time and effort. This effort may reduce the number of experts incorporated in future modelling exercises. XLBAYES should work from the Expert's data, rather than having to perform a regression analysis first, and then having to input the resulting Variance/Covariance Matrix, the Residual Variance, and the DOF. The software would then become more user friendly. When evaluating a regression analysis, the critical values should be calculated by XLBAYES, based on the user having entered the level of significance required. As it is, reference to a 'stats book' is necessary . In determining if the data is statistically significant, XLBAYES should calculate the 'F' statistic, as well as the 't' statistic. There appears to be a preference towards the 'F' statistic. XLBAYES works, but is lacking in the area of 'user interfacing'. It is not a polished product, compared to other 'engineering' software packages that are available. Image is important, and when an Excel user can question why certain statistics are not provided, or why computer generated results have to be reentered, this image becomes tarnished quickly. 10.4 'Joint Application' Partnership Arrangement with CSHRP From SHT's perspective, the Joint Application technique, whereby CSHRP provided for technical assistance to the highway agencies, was an excellent means of ensuring all jurisdictions became involved in evaluating the Bayesian methodology and XLBAYES. For the jurisdictions who participated in the Joint Application Partnership, the support and guidance provided by Vemax Management Inc. was a key element to ensuring success. The opportunity for interaction in Ottawa was also helpful, SHT was able to learn what worked well and what the potential problem areas were. In hindsight, the overall success of the Joint Application Partnership can be attributed to the foresight and planning of the Steering Committee. 11. Future Modeling Needs and Direction Saskatchewan Highways and Transportation would ideally like to internally market the Bayesian modelling technique, as a result of our Bayesian experience. The increasing need for quick results, makes Bayes an excellent tool. Saskatchewan Highways and Transportation must promote Bayes within our own organization. The success of this Subgrade Shear Failure analysis will go a long way to ensuring acceptance. 12. Appendices 'A' Figures 'B' Filename Summary 'C' Expert Judgement Encoding Package |
