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Page Evaluation of Rutting in Nova Scotia's Special "B" Asphalt Concrete Overlays JOINT
CSHRP/NOVA SCOTIA BAYESIAN APPLICATION
4.0 MODEL RUNS Linear and the linearlog models were
developed based on combing the 'new data' with each
expert, separately. However, from a practical standpoint,
a single model was required which could be implemented
within NSDOT&C. Therefore, the individual responses
of the experts were aggregated to form a "group
prior" and a supplemental iteration of each
functional form of the model was performed (Models 1A and
2A). The regression coefficients of the posterior models
for all iterations were analysed in terms of their
mathematical sign and tstatistics. The reliability of
the models were evaluated in terms of their residual
variance. The "XLBAYES" computer
software generated all statistics for the posterior,
prior, and data models. These included the precision,
variance/covariance, and correlation matrices, a
statistical summary of the regression coefficients,
including the mean, standard deviation and tstatistics;
and, finally, the model's degrees of freedom and residual
variance. The program also generated a graphical
presentation of the regression coefficients for each of
the three models, thereby visually illustrating the
estimates of the posterior prior and data models. A copy
of all outputs from "XLBAYES" is included in
Appendices C, D, E, and, F. 4.1 Linear Model Model 1 was a linear model and had the following functional form This model was run separately for all five experts. Each predictive model is shown in Table 6. The Posterior model was developed through the Bayesian regression technique. To evaluate the Posterior model, the regression coefficients, bo, b1, b2, b3, b4, b5, were analysed in terms of their mathematical sign and tstatistics. The reliability of the model was evaluated in terms of its residual variance. A summary table of coefficient statistics for the Posterior, Prior and Classical (Data) models is presented in Table 7. Sign The mathematical sign of the posterior
coefficients are technically correct for all experts.
Pavement engineers generally agree that as % Passing 5000
um, overlay age and traffic/year increase, rutting
increases. Thus, these terms are positive in sign. The
sign for the air void and % fractured faces variables is
negative, indicating that rutting will decrease as these terms increase. This
is evident in the % fractured face term as most agencies
agree that a high percentage of fractured faced coarse
aggregate will produce a more rut resistant pavement.
With reference to instability rutting and insitu air
voids the negative sign of the posterior is correct;
however the sign on the data model is inconsistent. In comparing the posterior, prior and
classical models, the sign on the air voids term for the
data model is positive, however, the sign on the
posterior and prior models is negative. This indicates
that for this variable the posterior models reflect the
prior information. This implies it was the experts'
judgement that as the insitu air voids decrease rutting
increases. Pavement engineers generally agree that
instability rutting may occur if the inservice insitu
air voids of the AC pavement falls below 3%. The inservice insitu air voids are
generally calculated from core samples on the basis of
pavement compaction after it has been densified by
traffic The insitu air void data collected from the
CLTPP test sites for this model development represents
the air voids calculated on core samples taken
immediately after construction. Generally in Nova Scotia, the insitu
air voids of an AC pavement are calculated from a core
sample on the basis of compaction immediately after
construction. With reference to the Special "B"
mix, the density of a high friction AC mix will not
increase significantly with traffic if properly
compacted. Therefore, the air void content of the mix
should not change significantly after a few years of
traffic.. As a result, the insitu air void content
obtained from core samples may be used in this model,
provided the specified compaction has been achieved.
Otherwise, the field Marshall air void measurements may
be used as they represent the inservice air voids. The Marshall air voids for all CLTPP
test sections were not available for this project. Hence,
the insitu air voids immediately after construction were
utilized. Although the data does not support the effect
of air voids on instability rutting, Bayesian regression
corrects the sign on this term with the subjective data
from the experts. T Statistics A common statistical tool to evaluate the significance of the regression coefficients is the "student t distribution test". The "t test" indicates if a term can be removed from the model in a subsequent iteration. The significance of the coefficients is tested by the following hypotheses (6):
For Model 1 the posterior has 67
degrees of freedom (DOF) and, from statistical tables,the
null hypothesis will be accepted if the computed
tstatistic is less than 1.96. The results from the experts indicate
that for Expert 1, Expert 2, and Expert 5, the air void
term is not significant (t< l 1.96 l ) for their
models. Also, % Passing 5000 um is insignificant for the
Expert 2 and Expert 5 models, as is traffic for Expert
2's model. All other coefficients have t statistics
> l 1.96 l for all other experts. Based on these
results, the majority of the experts agree that all five
independent variables influence rutting. Therefore, the
regression coefficients of all independent variables were
considered significant. Consequently, a decision was made
to include all variables in any given iteration process.
All five independent variables were utilized in an
iteration of the model using a group prior (Model 1A). Residual Variance The residual variance is a measure of dispersion in the data set. In statistical analysis, the square root of the residual variance is also referred to as the standard deviation, and is used herein as a measure of this dispersion. The residual variance of the models (see Table 8 represents the degree of error associated with the model. The 95% confidence interval, commonly called the 95% highest (posterior) density region or 95% HDR in Bayesian regression (9), is also presented in Table 8. The 95 % HDR represents the assumption that 95% of the area of a normal distribution lies within +/ 1.96 standard deviations of the mean. For example, the 95% HDR for Expert 3's model can be calculated as follows: 95% HDR Calculations & Table 8 Therefore, Expert 3's model has a 95%
HDR of +/ 2. 1 mm of rutting and a confidence interval
between 3.6 mm and 7.8 mm for a base case prediction
model. The probability that the rutting will fall within
this range for the base case is 0.95. Hence, the residual
variance provides a means of evaluating the model in
terms of its predictive capabilities. 4.1.1 Sensitivity Analyses Model 1 Prediction sensitivity analysis was conducted for the models. This type of sensitivity analysis is useful to determine which variables make the greatest contribution (dominant variables) to the model, and to identify where the prior, posterior and classical models agree and disagree. Each variable was varied its low, high and base case setting to estimate its effect on the model's prediction of rutting (Table 9). The base case is considered the average value of each variable. The sensitivity analysis of predictions for the prior and posterior models are shown graphically in Figures 4 and 5, respectively. The slope of each line illustrates the
degree to which each variable is sensitive to rutting. A
steep slope indicates that the variable is extremely
sensitive to rutting. As the slope of the line decreases
the sensitivity of the variable to rutting decreases.
Hence, a horizontal line (no slope) indicates that a
change in the variable has no effect on rutting. The sign
of the regression coefficient is reflected in the
graphical positioning of the slope. A positive slope
indicates that the coefficient is positive, whereas a
negative slope indicates the opposite. Graphical
presentation of the slopes in Figures 4 and 5 affirms the
previously discussed mathematical sign of the model
regression coefficients by illustrating the sign
difference between the models. In comparing the sensitivity of
predictions for the posterior and the prior, it is clear
that the two graphs are very similar. This indicates that
the posterior model agrees with the prior model. This
occurrence is commonly described by saying "the
posterior buys into the prior" (6). That is, the
coefficients of the posterior models reflect those of the
prior models based on expert prediction. This illustrates
that the posterior model confirms the prior model and
corrects the sign on the air voids term. With one exception, the experts rutting
models generally agree. Expert 2's model provides the
only discrepancy. The predicted degree of rutting from
the posterior model, graphically illustrated in Figure 5
and tabulated in Table 9, verifies that Expert 2's model
disagrees with those of the other experts. Sensitivity analysis of the predictions
clearly demonstrates that the developed posterior models
influenced the subjective judgement of the prior data.
Therefore, it was necessary to test the assumptions of
the prior model inputs. These input parameters included
the prior residual variance and degrees of freedom (DOF).
The sensitivity of the posterior residual variance was
determined by varying the prior degrees of freedom and
the prior residual variance. The result sheets from
"XLBAYES" are dynamically linked to support
this type of analysis. As the prior input parameters are
varied, the posterior residual variance is automatically
recalculated. The prior DOF and residual variance were varied separately to determine their effect on each expert's posterior model. The results of these variations are presented in Figures 6 and 7. Increasing the DOF reduced the residual variance of each expert's posterior model, except in the case of Expert 2. This indicates that the posterior model is sensitive to any changes to the prior model's DOF. It is also evident that the prior assumption of DOF = 42 is acceptable as the posterior residual variance only changed marginally when the prior DOF = 50. As illustrated in Figure 7, the residual variance of the posterior is sensitive to changes in the prior residual variance. Increasing the residual variance of the prior model results in a direct increase to the residual variance of the posterior model. This illustrates the models reliability in the prior and indicates that the prior is a realistic representation of the expert's opinion.
4.2 Combined Linear Model A comparison of the residual variances
and the 95%HDRs (Table 8) indicates that all of the
models represent a realistic prediction interval (+ 2.1
mm to +/ 3.0 mm). It is therefore conceivable that any
one model could be utilized. However, from the practical
standpoint of NSDOT&C, a single model combining
opinions of the experts would be preferred. A single model (Model 1A) could be
generated by combining the models of the four experts who
are in agreement. As discussed earlier, four of the
experts created models which produce similar results.
Expert 2's model was the exception as illustrated by the
sensitivity analysis of Model 1. Yet among the four
similar models another exception was noted. Expert l's
model, like Expert 2's, had higher residual values than
did the models of the remaining three experts. As a
result of this observation, Expert 1 's model was not
used in the final combination. The single model was,
instead, based exclusively on the models generated by
Expert 3, Expert 4, and Expert 5. The functional form of the combined model (Model 1A) was the same as the model run (Model 1) for the individual experts and defined as: To develop Model 1 A, the expert judgement for all three chosen experts was combined to calculate the prior data and using the "XLBAYES" software the resultant model (posterior) was developed as shown in Table 10. The posterior model for the combined
experts is again analysed in terms of sign and
tstatistics for the regression coefficients and the
residual variance of the model. A summary of the
statistics of the regression coefficients is summarized
in Table 11 for the posterior, prior and classical data
model. Sign In terms of mathematical sign, the
resultant posterior model has the proper sign convention
in terms of the effect of each independent variable on
rutting. For example, as the percentage of coarse
aggregate fractured faces increases, pavement rutting
decreases, hence the negative sign. As discussed for
Model 1, the Bayesian methodology, incorporating the
subjective prior data, does not agree with the sign on
the air void term of the data. As indicated by the
negative sign on the regression coefficient the posterior
model in agreement with the subjective data confirms that
a decrease in insitu air voids (below 3%) will increase
rutting. The positive sign for the other terms (%>
5000 passing, overlay age and traffic/year indicates that
as these values increase rutting increases, a
relationship generally correct. T Statistics As previously discussed, the tstatistic value for the regression coefficients of the model is a valuable tool to evaluate the statistical significance of the independent variables. Again the significance of the coefficients is tested by the following hypothesis (6):
In this case the posterior model has
211 degrees of freedom (DOF). Hence, from statistical
tables, t = / 1.96 l . The results from the combined experts indicate that for the combined model, all five independent variables are statistically significant. All regression terms have tstatistics greater than the t = + 1.96. This is to be expected as the posterior models of the individual experts (Model 1) generally produced statistically significant regression coefficients. The tstatistics summary is also presented in Table 11.
Residual Variance The residual variance is a measure of
error associated with the model's predictive ability. The
combined model has a residual variance of 1.49 mm sq. In
turn, the model has a standard deviation of 1.22 mm and a
95% confidence interval (HDR) of +/2.4 mm. Hence, there
is a 0.95 probability that the predicted value generated
from this model will be within +/2.4 mm. For example,
for a set of given independent variable values, if the
model returns a predicted value of 10 mm of rutting, then
there is a 0.95 probability that the actual rutting is
within the 7.6 mm 12.4 mm (7.6, 12.4). 4.2.1 Sensitivity Analysis Model
1A The first sensitivity analysis of Model 1A was conducted for the predictions of the combined model. To estimate the predictive value of rutting, each variable was varied to include a low, high, and base case as shown in Table 12. The graphical presentation shown in Figure 8 illustrates the sensitivity of the predictions for the posterior, prior, and the classical regression model of the data. As in the analysis of Model 1, the graphical presentation of these predictions indicates that the posterior model agrees with the subjective prior model, and hence that both models yield rutting predictions in the same order of magnitude. The classical data model is the same as for Model 1. The graphs in Figure 8 show that the data generally agrees with that for the posterior on the %5000 passing, fractured faces, and age terms. However, the magnitude of difference increases on the traffic term and the disagreement on the air void term is again evident by the positive slope of the data model. The slope of the air void and fractured face terms is negative for the posterior and prior models which supports the negative sign of their regression coefficients. To attempt to identify which variables
in the prior have a significant impact on the posterior,
the sensitivity of the posterior's residual variance was
tested. As in Model 1, the DOF and residual variance were
varied for the prior of Model 1A. As the DOF of the prior
model increased the residual variance of the posterior
model decreased (Figure 9); however, the effect of
changing the prior DOF from 94 to 500 was very small.
This indicates that the sensitivity of the assumption
with respect to prior DOF is minimal and there is not a
great concern with the assumption of prior DOF. The sensitivity of the posterior's residual variance of Model 1A to that of the prior illustrates the models reliability in the prior. As the residual variance of the prior increased, so did the residual variance of the posterior model (Figure 10). Once again the implication here is that the residual variance of the prior is a realistic representation of the combined expert's opinion. 4.3 LinearLog Model In an effort to improve the linear model and account for the cumulative effect of traffic on the pavement a linearlog model of the data was utilized. The functional form of this model was derived from the CLTPP project and had the following functional form: Prior to combining the models, the linearlog model of each was run separately and analysed. A summary of these linearlog models is presented in Table 13. To analyse the posterior model, the regression coefficients were evaluated in terms of their sign convention and tstatistics, as well as the model's residual variance. A summary table of coefficient statistics for the posterior, prior, and classical (data) models is presented in Table 14.
Sign The posterior coefficients all have the
correct sign in terms of the effect of each independent
variable on rutting. The sign on the air void and
fractured face terms are negative, all other terms are
positive. The log(age*traffic) term is also positive
which indicates that cumulative loading over time results
in an increase in rutting. In comparing the posterior,
prior and classical models, the sign on the air voids
term for the data model is positive, however, the sign on
the posterior and prior models are negative. Again, the
Bayesian regression posterior model reflects the prior
model and corrects the sign generated by the data model.
All five experts agreed on the sign convention of the
model variables. T Statistics The common tstatistic value is utilized to evaluate the statistical significance of the variables to the model. The significance of the coefficients again tested by the following hypotheses (6):
For this model the posterior has 163
degrees of freedom (DOF). Hence, from statistical tables,
t= 1.96. The posterior results of the experts
indicate that for Expert 2's model the only statistically
significant variables (t> +/ 1.96) are the fractured
face and log(age*traffic) terms, which may explain why
his Model 1 varied from those of the other experts. In
Expert 2's judgement, then,% fractured faces, age, and
traffic have the most impact on rutting. For Expert 1,
the air void term is insignificant, as is the % 5000
passing term for Expert 4. For Expert 5's model both air
voids and % 5000 passing are less than t= |1.96| and
therefore are not statistically significant for his
model. However, in Expert 3's model all five variables
are statistically significant as tested by the above
hypotheses (t> |1.96|). Based on the lvalue analysis, it was
obvious that there was disagreement among the experts in
terms of the significance of the variables to their
models. Consequently, all variables were included in an
iteration process of developing a combined linearlog
model. The residual variance of the models and the
sensitivity analysis would be used as tools to select
experts for the combined linearlog Model 2A. Residual Variance The difference in the individual experts' models is evident in the resultant residual variance of each (Table 15). The residual variances range from as low as 1.50 mm sq for Expert 3 to as high as 5.77 mm sq for Expert 2. This results in an increase in the range of the 95% HDR as compared to Model 1. The interval for Expert 3's model is +/ 2.4 mm of rutting, while Expert 2's model has a 0.95 probability of +/ 4.7 mm rutting.
4.3.1 Sensitivity Analysis Model 2 The first sensitivity analysis of Model 2 was conducted for the predictions of the individual models. As previously described and as outlined in Table 16, the models were evaluated based on three predictions for each independent variable. The sensitivity of the predictions for both the prior and posterior models are very similar as illustrated graphically in Figures 11 and 12. The magnitude of difference between Expert 2 and the other experts is once again evident. Expert 2's model produces a higher value of rutting based on the varied prediction cases. The other experts are generally in agreement as their predictions differ only marginally for the different cases. The positive and negative slopes in Figure 12 verify the sign conventions of the regression coefficients of the posterior model. The sensitivity of the posterior
residual variance to the prior model input parameters was
also tested. As conducted for Models 1 and 1A, the DOF
and the residual variance of the prior models were
varied. Varying the prior DOF produced insignificant
changes to the posterior residual variance for the
experts. This once again illustrates that the prior
assumption of DOF is reasonable for the modelling
process. As expected, an increase in the prior residual variance results in a direct increase in the posterior residual variance. This once again supports the fact that the posterior model is exhibits the prior model and that the prior is a realistic representation of the expert's opinion. The graphical presentation of these analyses is presented in Figures 13 and 14. 4.4 Combined
LinearLog Model Analysis of the individual models indicated that Expert 3's model had the lowest residual variance. Expert 5's model was the second lowest, however the sensitivity of predictions clearly indicated that Expert 4's model was in closer agreement to that of Expert 3 on the predictive cases. Although, Expert 3's model was statistically the best of the groups, it was decided that a combined model would be less biased than one based on a single expert's model. Therefore, the expert judgement of Expert 3 and Expert 4 were combined to form Model 2A. The resultant posterior model is presented in Table 17. Sign In terms of sign convention, the regression coefficients of the posterior of Model 2A are correct and are the same as Models 1, 1A and 2. The posterior model again corrects the sign of the air void term of the classical regression data model from the prior model. This again provides evidence that the posterior model reflects the prior model. A summary of the regression coefficients and their statistics is presented in Table 18.
T Statistics The posterior model has strong tstatistics for the regression coefficients of the independent variables. Therefore, testing the hypotheses (6):
indicates that all tvalues are
greater than t = |1.96| (DOF = 91), and that the
regression coefficients of the independent variables are
all statistically significant to the model. A summary of
the model's t values is presented in Table 18. Residual Variance The posterior of Model 2A has a
residual variance of 2.07 mm sq and a standard deviation
of 1.44 mm. Assuming that 95 % HDR indicates that 95%
area of a normal distribution lies within +/ 1.96 standard
deviations of the mean, the 95% HDR for Model 2A is +/
2.8 mm. Hence, there is a 0.95 probability that the
actual rutting value will be within +/ 2.8 mm of the
Model 2A's predicted value. 4.4.1 Sensitivity Analysis Model 2A The first sensitivity analysis of Model 2A was conducted for the predictions of the combined model. As in the previous analyses, the model was evaluated on three prediction cases for each independent variable as seen in Table 19. Graphical presentation (Figure 15) illustrates the sensitivity of the predictions for the posterior, prior and the classical regression models of the data. Once again the predictions of the posterior model are marginally different than those for the prior model in terms of the four independent variables. However, the magnitude of difference increases with the classical regression model of the data. To illustrate the sensitivity of the
posterior model to the prior model inputs, the DOF and
the residual variance were altered. As seen in Figure 16,
an increase in the prior DOF results in a decrease in the
posterior residual variance; however, this decrease was
very small. For example, doubling the prior DOF from 91
to 182 resulted in the posterior residual variance
decreasing from 2.07 mm sq to 1.98 mm sq respectively.
This again illustrates that the sensitivity of the DOF
assumption is minimal and therefore the assumption of DOF
has little affect on the model's predictive capability. The residual variance of the posterior model increase with an increase in the prior model (Figure 17). This demonstrates that the posterior model is dependent on the prior model. As the error of the subjective model increases, the error of the resultant posterior model increases. This again implies that the prior's residual variance reasonably represents the opinions of the experts. 4.5 Comparison of Models Two single models (Table 20) were produced by combining expert models which were in general agreement in their individual models. As described in the analysis, both combined models have strong tvalues for the regression coefficients. In comparing the two models (Table 21), the linear model has a 95 % HDR of +/ 2.4 mm, while the linearlog model has a 95% HDR of +/ 2.8 mm. This indicates that, statistically, the linear model is the better of the two; however, the basis of prediction differs. That is, the linear model will predict the amount of rutting for any given overlay age and traffic/year, whereas the linearlog model accounts for the cumulative effect of traffic on the pavement. However, as seen in Tables 12 and 19, both models appear to predicting similar results. This implies that the experts accounted for the cumulative effect of traffic on rutting by increasing the rutting accordingly for their predictions at overlay ages of 2, 10 and 12 years. Therefore, the linear model is also representative of the cumulative effect of traffic on pavement rutting. Mathematically, however, the model does not account for a nonlinear relationship between rutting and overlay age in which there is less rutting in the later years of the overlay life. Nevertheless, the estimates of rutting in the later years of the overlay life are insignificantly different for Models 1 A and 2A with rutting predictions of 7.7 mm and 7.3 mm respectively.
5.0 SUMMARY Bayesian methodology was used to develop a predictive model to evaluate instability rutting in Nova Scotia's High Friction Special "B" AC mix. The methodology allows for the incorporation of objective and subjective data from experienced engineers. Hence, data from CSHRP test sections in Nova Scotia collected in CLTPP project, and subjective data from experienced pavement engineers using the "full orthogonal" matrix methods were collected to formulate the model. Two functional model forms were evaluated during the course of this project. These included linear and linearlog models. The "XLBAYES" software program was utilized to analyse the individual models of the experts. Based on this analysis the expert's predictions were combined to provide the following unbiased models: The analysis of the linear and
linearlog models indicated that both would provide
reasonable estimates of rutting. Nevertheless,
statistically the linear model was the better of the two
for evaluating rutting for the Special "B" AC
mix on Nova Scotia's 100 series highways. 5.1 Discussion and Conclusions Overall Conclusions From Analyses The Bayesian methodology produced
individual and combined models for the two functional
forms investigated in this project. Two single models
(Table 20) were produced by the combining experts which
were in general agreement in their individual models. As
described in the analysis, both combined models have
strong tvalues for the regression coefficients.. In
comparing the residual variances of these two models, the
linear model has a lower residual error than does the
linearlog model. This indicates that the experts, in
encoding their judgement, considered the model's linear
functional form which was provided for the initial
iteration of the project. The linear model presents an
equation that will predict rutting for a given set of
independent variables, including the overlay age and the
traffic/year (in kESALs). The linearlog model provides
an equation that evaluates rutting in terms of the
cumulative effect of traffic on the pavement. The
multiplication of the age and traffic terms accounts for
cumulative traffic. The logarithm of the age and traffic
terms accounts for less rutting in the later years of
pavement life. The linear model provides a similar
estimate of rutting when compared to the linear log
model. However, the linear model has a lower 95% HDR than
the linearlog model and combines the judgement of three
experts. Therefore, if a model is implemented by
NSDOT&C, it is recommended that Model 1A be utilized
as a tool to evaluate rutting in Nova Scotia's Special
"B" type asphalt. The analysis indicates that Special "B" performs well as a rut resistant AC mix. For the base case of independent variables over a 15 year period, the model shows that the AC pavement will exhibit 7.7 mm of rutting with a 95% confidence interval of (5.3,10.1). The average life span of an AC overlay is generally 15 years. Based on the CSHRP classification of wheel path rutting severity (Table 22), the model predicts that the severity of the ruts due to instability of the mix is "slight" (10). Therefore, in consideration of the model's independent variables, which influence rutting, as well as the fact that this model deals with rutting due to instability of the mix, it can be concluded that Special "B" is expected to perform well as a rut resistant AC mix.
Expected Uses of Model The developed model provides
NSDOT&C with a means of evaluating the Special
"B" mix for rutting. Hence, 100 Series Highways
can be evaluated as to the expected amount of instability
rutting to occur. It is also plausible that the
Department can use the model to investigate cases of
severe rutting of Special "B" overlays and
determine if the rutting is an instability problem with
the AC mix or one of the other types of rutting described
in Section 1.1. In addition, the model provides for the
design of a data collection program for the variables
that effect rutting. The involvement of NSDOT&C, and
consultants from private industry, and TUNS, has
facilitated the transfer of technology using the Bayesian
methodology. Bayesian Methodology Based on the findings of this project,
Bayesian methodology appears to have merit in modelling
as it provides a means to incorporate subjective and
objective data in the development of a regression
predictive model. In this project the objective data
sample size was very small, however, the Bayesian
methodology recognizes this and provides a usable model
with the inclusion of the prior. It became evident from
the sensitivity analyses that good quality data (prior
and new) are essential to the development of a good
model. Therefore, the Bayesian methodology has to include
a quality assurance check on both new and prior data.
From the prior data analyses it also became evident that
understanding the contributions of variables to the model
is essential. For example, in this project the posterior
model corrects the sign on the air void term with the
prior data. In this case, the expert's understanding of
the effect of low air voids on instability rutting is
evident in the posterior model. "XLBAYES" Software The "XLBAYES" software, used
in this project, provided an excellent user friendly
program to develop the model. It provides both graphical
and tabular statistical outputs for all three models
(posterior, prior and data) in the analysis. However, the
program could be improved to provide a more complete
analysis. Some suggested improvements discovered from
using "XLBAYES" may include: 1. An alternate method to compute the prior. 2. The incorporation of sensitivity
analysis into the program CSHRP Joint Application The "joint application"
partnership arrangement with CSHRP and NSDOT&C,
which provided the context for this project has been
successful. While training workshops and the availability
of a consultant from Vemax Management Inc. has aided the
transfer of Bayesian technology, further support is
needed. Such support could usefully take the form of an
explanatory manual to assist in the interpretation of the
model analysis. Future Modelling Needs /Direction Currently, NSDOT&C and TUNS through
graduate student work, are developing a model to evaluate
road roughness on Nova Scotia's 100 Series Highways.
Based on reviewing the two projects, NSDOT&C may opt
to initialize other modelling projects to assist in the
evaluation of pavement distress. A series of models
should provide the means for analysing the cost
effectiveness of various rehabilitation strategies in the
future. REFERENCES 1. Jacques Whitford Material Limited,
"Rutting Investigation, 100 Series Highways, Nova
Scotia", Project No. 7587 Report for Nova Scotia
Department of Transportation and Communications, 1992. 2. Smith, W., Finn, F., Kulkarni, R.,
Saraf, C., and Nair, K., "Bayesian Methodology for
Verifying Recommendations to Minimize Asphalt Pavement
Distress, National Research Program Report 213,
Transportation Research Board, Washington, D.C., June
1979. 3. Abowd, J., Moulton, B., Zellner, A.,
"The Bayesian Regression Analysis Package, BRAP
User's Manual", Version 2.0 of 12/03/85, H.G.B.
Alexander Research Foundation, Graduate School of
Business, University of Chicago, May 1987. 4. Canadian Strategic Highway Research
Program (CSHRP), "CLTPP Five Year Data Analysis
Briefing Tour", Transportation Association of
Canada, Ottawa, 1994. 5. Strategic Highway Research Program
(SHRP), "Distress Identification Manual For The
LongTerm Pavement Performance Project, 1993. 6. Canadian Strategic Highway Research
Program (CSHRP), "Training Sessions in Bayesian
Methods and Software", Transportation Association of
Canada, Ottawa, 1994. 7. Canadian Strategic Highway Research
Program (CSHRP), "Summary of Canadian LongTerm
Pavement Performance Project (CLTPP) Monitoring
Activities" Transportation Association of Canada,
Ottawa, 1993. 8. Nova Scotia Department of
Transportation and Communications," Section 4,
Asphalt Specifications", Nova Scotia 1995. 9. Lee, P.M., "Bayesian
Statistics: An Introduction", Provost of Wentworth
College, University of York, England, 1994. 10. Canadian Strategic Highway Research Program (CSHRP)," Severity Guide for Canadian LongTerm Pavement Performance Project (CLTPP)" Transportation Association of Canada, Ottawa, 1993. |
