Neural Networks and AASHO Road Test

INTRODUCTION

P =F(DI> D 2 , D 3 , L, W) (1)

where D lo D 2 , and D 3 = thicknesses of the surface, base, andsubbase layers, respectively; L = axle load; and W = numberof accumulated axle-load applications. Databased modeling as-

. Predicting the dur~bility of materials and structural systemsIS one of the most difficult problems in engineering. Modelsof durability based on physical principles have been slow toemerge and have not always proven useful at the level of design. On the other hand, it is often relatively easy to collectdata on the performance of a particular system. Models basedon measured data, often referred to as databased models, canbe very useful in situations where data are plentiful but thephysical principles underlying the observations have not yetyielded a concise mathematical description. Durability ofpavements is a case in point.

During the period of 1958 through 1960, the American Association of State Highway Officials (AASHO) conducted alarge-scale experiment for assessing the durability of pavementsystems. This experiment is known as the "AASHO roadtest" and has served as the basis for pavement-design practicefor more than 30 years. The principal goal of the test was toestablish relationships among pavement performance, struc~ural design characteristics, and loading. The primary variables10 the test were the axle load; number of applications of thatload; and t~e depths of the surface, base, and subbase layersthat compnse the pavement structure. Durability of the pavement structure was indexed through a single measure of performance called the present serviceability index. Many of thephysical and environmental factors were the same for all sections throughout the test. The factors that were not varied during the test have been the subject of considerable research anddebate since. In the present paper, we are concerned simplywith the processing of the AASHO road-test data.

The data collected at the AASHO road test support a databased mathematical model for a pavement durability that expresses the present serviceability index P as a function of thefive variables of the test as

NEURAL NETWORKS AND AASHO ROAD TEST

By M. R. Banan l and K. D. Hjelmstad/ Member, ASCE

(Reviewed by the Highway Division)

AB~TRACT: The American Association ~f State Highway Officials (AASHO) road test, conducted during thepenod of 1958 through 1.96~, was a facto?al test o~ pavement durability that considered layer depths, axle load,and n~mber of load apphcations as the pnmary vanables. These data were processed using traditional statisticaltechmques.. The AASHO fonnula is the resulting databased model of the road-test data. In the present paper,we reexamme the AASHO road-test data, using the Monte Carlo Hierarchical Adaptive Random Partitioning(MC-HARP) neural-network model developed by Banan and Hjelmstad (1995), and show that an MC-HARPmodel can represent the da~ far better than the AASHO fonnula can. We conclude that the MC-HARP neuralnetwork may be an appropnate tool for the development of databased models of pavement perfonnance in thefuture.

sumes that function F used in (1) is implicit in the data andcan be established from those data. In establishing a databasedmodel by traditional means, one typically selects the form offunction F and determines the parameters associated with thefunctional form through a process of minimizing the error between the predictions made by the function and the observations available. Finding the best straight line that passesthrough a collection of data pairs is an example of this typeof databased modeling. When the dimension of the data domain is large, the selection of an appropriate functional formcan be very difficult. When the behavior of the data is fundame~tal.ly different in different regions of the input domain,establ.lshl~g a .function~ form. that holds for the entire inputdomam IS virtually Impossible. These difficulties havespawned research into other methods of databased modelingthat have given rise to approaches like nearest-neighbor techniques, spline and radial-basis functions, and statistical learning n~tworks, to name a few. Recently, we have proposed analgonthm called Monte Carlo Hierarchical Adaptive RandomPartitioning (MC-HARP) that produces a model that can beclassified as a neural network (Banan and Hjelmstad 1995).

A neural network is a parallel distributed processing systemcomprising many simple, interconnected processors. The net:-vork parameters are adjusted to represent the mapping implicit10 the data through a process often called learning. The selfo:ganization, noise and fault tolerance, adaptivity, and masSIVely parallel structure are among the features that makeneural networks attractive for databased modeling. The MCHARP algorithm is a particularly effective means of generatinga neural network, combining some of the best features of neural networks with some of the most useful tools of traditionalstatistical methods. The aim of the present paper is to showthat an MC-HARP neural network can be constructed forAASHO road-test data and to compare the performance of theneural-network model with the AASHO formula.

This new look at the AASHO road test is relevant becausedatabased modeling of pavement durability continues to be animportant engineering tool. While we are not looking to replace the AASHO formula or any of the engineering resultsthat have emanated from it, these data provide an excellent

'Engr., Carter & Burgess. 2500 Michelson. Suite 100 Irvine CA forum for examining a new databased modeling technique that'Assoc. Prof.• Univ. of Illinois at Urbana-Champaign, 205 N. Math~ws, might be used in future tests. The AASHO road test allows us

Urbana, IL 61801. to compare the new model with a well-established, practice-Note. Discussion open until March I, 1997. To extend the closing date tested result. The Long-Term Pavement Performance project

one month, a written request must be filed with the ASCE Manager of f th S .Journals. The manuscript for this paper was submitted for review and 0 e trateglc Highway Research Program and the Highwaypossible publication on August 17,1995. This paper is part ofthelourllDl Performance Monitoring System provide examples of new andof Transportation Engineering, Vol. 122, No.5, September/October, emerging databases on pavement performance that could ben-1996. ©ASCE, ISSN 0733-947X19610005-0358-03661$4.00 + $.50 per efit from new databased modeling tools. These research pro-page. Paper No. 11426. grams involve monitoring the performance of a broad range

358/ JOURNAL OF TRANSPORTATION ENGINEERING / SEPTEMBER/OCTOBER 1996

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(3)

of existing in-service pavement sections that have varied design factors under a range of climate and traffic conditions.The gathered database is large, complex, and heterogeneous,including traffic and environmental data, pavement surface images, pavement profiles, and structural measurements. Traditional statistical techniques for building empirical models arenot well suited to the processing of these data.

In the following sections, we study the performance of themodel that is developed in AASHO (1962) and has been thebasis for pavement-design practices during the last 30 years.Then we use the AASHO road-test data to build two differentMC-HARP neural networks for the model defined in (1) andcompare its performance with the AASHO model.

TABLE 1 Value. of Coefficients of AASHO Formula

Log Ao A, A. 8 0 8, s.. 8, 8. ~ a. 130(1 ) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11 )

5.93 9.36 4.79 0.081 5.19 3.23 0.44 0.14 0.11 1.0 0.4

ability loss rate, controlled by ~, and the traffic capacity, represented by p, are assumed to be functions of the pavementdesign and traffic. The functional relationships for ~ and pwere expressed in the form

Ao(D + a4t'

p = (L + It'

AASHO MODEL FOR PAVEMENT SERVICEABILITY

A parametric approach was used in AASHO (1962) to builda mathematical model for the multivariate mapping defined in(1). The functional form of multivariate mapping F was presumed, and its parameters were estimated to acceptably fit theroad-test data. The basis of the AASHO model is a decaycurve for which serviceability deteriorates with accumulatedtraffic in the form

where Po = initial serviceability level; and P, = terminal serviceability. The initial serviceability Po represents the freshlyconstructed, untrafficked pavement. New flexible pavement atthe road test reflected an average serviceability index value ofPo = 4.2. Terminal serviceability P, is indicative of a pavementthat is in need of resurfacing or reconstruction. The terminalserviceability averaged P, = 1.5 in the road test. The service-

(5)

(4)= + Bo(L + Il'~ - ~o (D + a4)B,

where pavement-design parameter D = linear combination ofthe three layer depths as follows:

PREPROCESSING PERFORMANCE DATA OFROAD TEST

In general, the AASHO formula implies that the condition ofa pavement deteriorates with weighted accumulated traffic Wand axle load L. Furthermore, one can observe that the pavement serviceability improves by increasing thickness index D.The axle load L and design thickness D h D2 , and D3 areknown for each test section of the AASHO road test. Theunknown coefficients Ao_2 , Bo_2 , ~o, and the layer coefficientsal-4 were estimated through a regression analysis using theperformance data of the AASHO road test and are given inTable 1. From here on, we refer to (2)-(5) with the valuesgiven in Table 1 as the AASHO formula.

The AASHO road test has been the basis for pavementdesign practices for the last 30 years. In 1986, the AmericanAssociation of State Highway and Transportation Officials(AASHTO) published the AASHTO Guide for Design ofPavement Structures (1986, 1981). This document incorporates theoriginal development of the road-test data with more recentadditions relating to subsurface drainage, materials, reliability,and others. The AASHTO guide accepts the original AASHOformulations as a starting point and adds new factors for effects neglected in the AASHO model.

The road-test data set for single-axle traffic contains 4,788data points for 164 pavement sections. We refer to this dataset as the measured serviceability data set. Each data point hasfive input values (D h D2 , D3 , L and W) and one output value,the PSI P. There were 20 pairs of replicate sections for flexiblepavements under single-axle traffic in the road test.

We smoothed the measured serviceability trend for eachpavement section using a locally weighted regression scatterplot smoothing (LOWESS) procedure (Chambers 1983). Foreach pavement section, we computed the smoothed PSI for 43points equally spaced on the W-axis from zero to 1.26 X 106

•

We refer to the data set containing smoothed serviceabilitytrends as the smoothed serviceability data set. This set contains7,052 (164 X 43) data points corresponding to serviceabilitytrends for 164 pavement sections. Fig. 1 presents the smoothPSI plotted against the measured PSI for the 4,788 data pointsin the measured serviceability data set. The mean average andstandard deviation for the difference between the smooth PSIand measured PSI are -0.008 and 0.206, respectively. Thus,the smoothing process filters out disturbances whose rootmean-square (RMS) amplitudes are about 0.206 from the measured serviceability trends.

The training set contains smoothed serviceability trends for

(2)P =Po - (Po - P,)(W/p)~

AASHO ROAD TEST

The AASHO road test on flexible pavements comprised 164test sections. The test variables were surface thickness Dl> basethickness D 2 , subbase thickness D 3 , and axle load L. The variables took discrete values D 1 E {I, 2, 3,4, 5, 6} in., D2 E{O, 3, 6, 9} in., D 3 E {O, 4, 8, 12, 16} in. (1 in. = 2.54 cm),and L E {2, 6, 12, 18, 22.4, 30} kip (1 kip = 4.448 kN) forsingle-axle test traffic. AASHO (1962) gives the values ofpavement-design variables D h D2 , D3 , and axle load L for eachtest section in the main factorial experiment for flexible pavements.

A regression equation was developed for the AASHO roadtest to relate objective measurements of longitudinal and transverse profile variations and the amount of cracking and patching to subjective panel ratings of present serviceability. Themeasure computed from the developed regression equation using the distress measurements, was referred to as the presentserviceability index (PSI). A scale of 0-5 was established forPSI, with a value of 5 for the ideal pavement. PSI representsthe level of serviceability at any time during the life of thepavement. The present serviceability and number of accumulated load applications were recorded for each test section. Thedecay of PSI with W, called the serviceability trend, was takenas indicative of pavement performance. The serviceabilitytrend of each section was smoothed using a moving averagethat typically included five (and always had at least three)successive data points. A weight was applied to accumulatedaxle-load applications W to account for seasonal environmental effects (AASHO 1962). The life of a section started atcompletion of its construction and ended when its serviceability reached terminal serviceability (typically at a PSI of 1.5).The serviceability of some test sections with thick structuraldesign or light axle load were not exhausted during the testingperiod, and some sections reached their terminal serviceabilitywell before the end of the road test.

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Thickness Axle(In.) load AASHO

Section L test-boundingnumber D1 ~ Os (kip) sections

(1 ) (2) (3) (4) (5) (6)

1 1.5 0.0 0.0 2.0 721 7712 1.5 0.0 4.0 2.0 727 7294 1.5 3.0 4.0 2.0 717 741

10 2.5 3.0 4.0 2.0 741 745150 4.5 3.0 8.0 18.0 589 631153 4.5 6.0 8.0 18.0 577 591159 3.0 4.5 8.0 18.0 573 623160 3.0 4.5 12.0 18.0 617 601262 4.5 3.0 12.0 30.0 299 261265 4.5 6.0 12.0 30.0 323 307268 4.5 9.0 12.0 30.0 267 331271 5.5 3.0 12.0 30.0 261 335

TABLE 2. Test Sections and Their Bounding 1)'alnlng Sections

Note: 1 in. 2.54 em, 1 kip = 4.448 kN.

542 3Measured PSI

1

2

3

FIG. 1. Variation of Smooth Data

4

all nonreplicate sections and averages of smooth serviceabilitytrends for replicate section pairs. The serviceability data comprises 144 sections with 43 equally spaced points along theW-axis for each section, giving 6,192 data points for PSI inthe input domain (Db D2 , D 3 , L and W). In the sequel, theterm "section" shall refer to a 4-tuple (Db D 2 , D 3 , and L).We refer to sections that belong to the training set as trainingsections. Similarly, we refer to sections that belong to the testset as test sections.

To evaluate the performance of the databased models, weselected a number of points between the training data. Themeasured PSI values at neighboring points were used to evaluate predicted PSI values in the test set. The test set comprises6,762 points in the input domain corresponding to 322 testsections. The test sections were subjected to the same axleloads used in the road test and have layer thicknesses (D.,D 2 , and D 3) bounded by layer thicknesses of training sections.Table 2 shows the specifications of some ex~ple test sectionsand their corresponding bounding sections from the trainingset. For example, test section 1 had the same base thickness,subbase thickness, and axle load as sections 721 and 771 inthe AASHO road test, but its surface thickness was the averageof surface thicknesses for training sections 721 and 771. Thus,sections 721 and 771 represent bounds for test section 1.

Ultimately, we want to characterize how well the databasedmathematical models capture the main features of the data andhence the physical behavior implicit in those data. In the testing phase, we looked for two trends: the predicted P shoulddecrease as W increases, and the predicted P should increaseif anyone of the design variables (Db D 2 , and D 3) increaseswhile the others remain constant. We shall refer to these trendsas the testing trends. A mathematical model, such as an MCHARP neural network or AASHO formula, is a reasonablerepresentation of the road test if it adequately fits the data (in

the sense of high correlation coefficient) and if it follows thetesting trends for the majority of test sections.

PERFORMANCE OF AASHO FORMULA

In this section, we study the performance of the AASHOformula using the performance data measured during the roadtest to observe how well the AASHO formula represents thedata used to build it. Fig. 2 shows the scatter plot of theAASHO predicted PSI versus the measured PSI and smoothPSI with and without considering the replicate section pairs.The points in the scatter plots do not cluster on the diagonaland do not funnel in toward the origin, indicating that theAASHO formula does not accurately represent its trainingdata. The scatter measures for the plots in Fig. 2 are shownin the first row of Table 3, which uses three indices to measurethe scatter of plotted data points: the average of the residualse, root-mean-square error RMS, and the coefficient of determination R 2

• The coefficient of determination is negative forthe scatter plot of the AASHO predicted PSI versus the measured PSI, indicating that the variance of residuals betweenpredicted PSI values and the measured ones is larger than thevariance of measured PSI values. With respect to smooth PSIvalues, the AASHO formula has R2 equal to 0.788, consideringthe serviceability data of all 164 sections, and has R 2 equal to0.784, considering the serviceability data of 124 nonreplicatesections. The low R 2 values indicate poor fit of the AASHOformula to the data. In Figs. 4 and 5 (and again in Fig. 8), wehave plotted the serviceability trends predicted by the AASHOformula for a number of pavement sections of the road test.The AASHO formula fails to acceptably represent the measured serviceability trends of a significant number of pavementsections.

,,

,..~ ,.•..~'

.. '::::;;::i~~~.i1i. / .... :"'!::'~;"'~:~~:'i\:l

:~;i~;.)~;m1.lr

5,....--..----......--..,..--...,.---71

0-' ......

o Measured PSI 5 o Smooth PSIwithout replicate section pairs

5 o Smooth PSIwith replicate section pairs

5

FIG. 2. Performance of AASHO Formula for 1)'alnlng Set

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TABLE 3 Performance of Constructed Approximations for Training Set

Smooth PSI Smooth PSIMeasured PSI (without replicate pairs) (with replicate pairs)

Approximation N= 4,788 N= 5,332 N= 7,052(1 ) (2) (3) (4)

- i RMS R' i RMS R' i RMS R'AASHO -0.086 0.671 -0.134 0.136 0.763 0.784 0.129 0.756 0.788MC-HARP (linear) -0.185 0.458 0.470 0.027 0.338 0.958 0.022 0.388 0.944MC-HARP (AASHO) -0.053 0.385 0.626 -0.007 0.192 0.986 -0.010 0.260 0.975

For each replicate section pair, an adequately fit empiricalformula should predict a serviceability trend that passes between the replicate serviceability trends. Consequently, theRMS error for the plot based on the nonreplicate sectionsshould be smaller or at most equal to the RMS for the plotconsidering the replicate section pairs. It is evident that theAASHO formula does not show this characteristic as its RMSvalue increases by omitting replicate section pairs. This observation provides evidence that the AASHO predicted serviceability trends for some replicate section pairs do not passbetween the associated replicate serviceability trends.

An examination of the shapes of the measured serviceabilitytrends, such as those shown in Figs. 4 and 5, reveals someshortcomings of the AASHO formula. Instead of being smoothdecreasing curves, like the AASHO formula, the measuredserviceability trends decay sharply at either one or two reasonably well-defined events. The locations of these two criticalevents closely correspond with the periods of spring thaw during the two years of the road test (Coree and White 1990). Alarge number of sections failed rapidly during the first freeze/thaw event. A number of sections distinctly deteriorated duringthe first spring-thaw period and then failed at the beginningof the next winter, if they had cracked at the end of the firstyear, or failed during the second freeze/thaw event, if they hadsurvived the first year uncracked. Only pavement sections withthick layers or with light axle loads exhibited no significantloss of serviceability during the two-year period of the roadtest. It appears that time, as measured by the number of freeze/thaw events, is a significant factor missed in the AASHO formula (Coree and White 1990). Furthermore, the seasonalweight factor used for the accumulated load-axle applicationsdoes not adequately account for the effect of freeze/thawevents.

During the road test, environmental conditions inflicteddamage to the test pavements but were not measured or controlled. By definition, a databased model cannot characterizethese effects. In essence, the measured data represent a projection of the actual system behavior onto the five-dimensionalspace spanned by the measurements. The errors associatedwith this projectional manifest as noise in the data. The performance of a simplified model that attempts to follow the dataexactly will be dominated by noise, and its ability to generalizewill be poor. One can argue that, despite the fact that significant factors like the freeze/thaw event are not considered inthe AASHO simplified model, a databased approximation witha reasonable performance should still follow the main behaviorof the measured serviceability trends. In other words, a reasonable model for the pavement performance should predictserviceability trends that are close to the measured trends andfollow the global behavior of the measured trends, but thepredicted trends should not perfectly mimic the measuredtrends. The AASHO predicted serviceability trends follow theglobal behavior of the measured trends for a number of testsections like 129, 131, 327, 313, and 265 shown in Fig. 4.However, for a large number of pavement sections, theAASHO predicted trends are not close to the measured trendsand do not follow their global behavior, as shown in Fig. 4for sections 727, 743, 771, and 153.

It is evident that the AASHO model does not represent theobserved serviceability trends for pavement sections traffickedat the road test. One reason for this shortcoming is that environmental effects like the spring-thaw event are not considered in the AASHO formula. Another contributing factor isthe constraint imposed by the functional form of the AASHOparametric model. The kinks in the measured serviceabilitytrends indicate that the actual mapping is significantly morecomplex than the AASHO approximation, which is verysmooth. Furthermore, the significant discrepancy between theAASHO predicted serviceability trends and measured trendsfor a number of pavement sections exposes the poor performance of the AASHO formula in certain regions of the inputdomain. A local approximation might model the pavementperformance data over the entire input domain better than aglobal approximation like the AASHO formula. An MCHARP neural network is such a locally based approximator.In the following sections, we describe two MC-HARP neuralnetworks constructed from the road-test data and comparethem with the AASHO formula. We show that the MC-HARPapproach results in better models of pavement serviceabilitytrends.

NEURAL NETWORKS AND Me-HARP ALGORITHM

The MC-HARP strategy for building a databased information processing system is a novel combination of parallel distributed processing, fuzzy logic, and the Monte Carlo strategy(Banan and Hjelmstad 1994, 1995). The MC-HARP algorithmis self-organizing, adaptive, and is able to generalize from itstraining data. The algorithm can process heterogeneous dataand operate with minimal external adjustment. It can interactively accept knowledge and provide guidance for efficientlyimproving the database.

The MC-HARP strategy determines the functional structureand parameters of a mathematical model simultaneously. AMonte Carlo (MC) strategy combined with the concept of Hierarchical Adaptive Random Partitioning (HARP) and fuzzysubdomains determines a multivariate parallel distributed mapping. The constructed mapping has the features of a neuralnetwork. The HARP algorithm is based on a divide-and-conquer strategy that partitions the input space into measurableconnected subdomains and fits the data in each subdomainwith a local parametric approximation 6. The algorithm is hierarchical because each subdomain is a subset of its parentsubdomain; it is adaptive because the partitioning process isdriven by approximation error in each subdomain; it is randombecause the subdomain splits are generated randomly. The notions of squashing functions and fuzzy subdomains were introduced to improve the generalization of the approximatingfunction. Fuzziness promotes continuity of the mapping constructed by HARP and smooths the mismatching of the localapproximations in the neighboring subdomains. The MonteCarlo strategy improves the generalization of the HARP approximation for a fixed amount of data, combining the flexibility of a local approximation to adapt to complex, nonhomogeneous behavior and the smoothness of a globalapproximation to capture the coarse features inherent in the

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Smooth PSI 5with replicate section pairs

oo Smooth PSI 5without replicate section pairs

o Measured PSI 5

FIG. 3. Performance of MC-HARP Neural Network with Linear Subclomaln Approximations

5.-----r--~--.,....----.----,

- - AASHO Formula

',-- Me-HARP neural network

--- ---

-- ---17~r

5r--...,....---,.---r---r---,

0'----'----'----'---'----'

G~ 1~

Weighted Axle Load Applications. 1000'so 1~

Weighted Axle Load Applications. 1000',o 1~

Weighted Axle Load Applications. 1000's

FIG. 4. Performance of MC·HARP Neural Network with Linear Subclomaln Approximations for 1)'alnlng Set

data. In addition, the standard deviation of the Monte Carlosample gives a measure of approximation confidence at allpoints in the input domain, a feature not available in othermethods at this time.

The tree structure of the HARP modules and the independence of both the subdomain approximations and random partitions enable the MC-HARP environment to quickly convergeto a series of equally plausible solutions without user interaction. The MC-HARP environment enjoys a large-scale granularity produced by the Monte Carlo parallelism and the geometric parallelism achieved by partitioning the input space.Subdomain training processes are independent of one another,so these computations can be done in parallel.

Because the resulting approximation can be implemented asa parallel distributed processing system, it can be classified asa mapping neural network. An MC-HARP approximation canbe modeled as a modular neural network whose basic module

is a HARP neural network. As such, the MC-HARP algorithmprovides a method for simultaneously determining the architecture of the network and training the network on the inputdata on that architecture. The MC-HARP method is faster, byorders of magnitude, than traditional (e.g., backpropagation)neural networks, and yields considerably more informationabout the knowledge implicit in the data.

Me·HARP NEURAL NETWORK OF AASHOROAD-TEST DATA

In this section, we use the MC-HARP method to build adatabased mathematical model for pavement performance using the data of the road test. The training set contains smoothserviceability trends for all nonreplicate sections and averagesof smooth serviceability trends for replicate section pairs. Thetraining set contains N = 6,192 data points and the number of


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5,....---r-----r------,,...--..,...--,

0'-----'-----'----'-------'-----'

G0

0 1500Weighted Axle Load Applications, 1000's

MCHARP Predicted 'trend

Bounding 'trend

o 1500Weighted Axle Load Applications, 1000's

o 1500Weighted Axle Load Applications, 1000's

FIG. 5. Performance of Me-HARP Neural Network with Linear Subdomaln Approximations for Test Set

test data points is Nt = 6,762, as discussed earlier. The inputvariables are: the surface, base, and subbase thicknesses (D1,

D2 , and D3), the axle load L, and the logarithm of accumulatedsingle-axle load applications log W. The output variable isPSI P.

As mentioned earlier, an empirical model that perfectly fitsthe training serviceability trends tends to be overparameterizedand, as a result, does not generalize well. The MC-HARPmodel selection technique and framework for classifying datasets discussed by Banan and Hjelmstad (1994) limits the complexity of the o'cerall model through the tolerance on the fitof the local approximation in a subdomain. Banan and Hjelmstad have shown that there exists an optimal tolerance belowwhich the model tries to follow the noise and above which themodel is inaccurate. For the first MC-HARP model consideredhere, we choose the subdomain approximation function 9 tobe linear. The subdomain training tolerance, which controlsthe complexity of an MC-HARP neural network, was set at1.75.

The constructed MC-HARP neural network F'L (the subscript L indicates that the local functions were linear) has 693parameters and 116 subdomains on average for 15 HARP partitions. The scatter plots of predicted PSI values against measured and smooth PSI values are shown in Fig. 3. It is evidentthat points in the scatter plots are close to the diagonal butthat there is a slight curvature in the scatter plots, particularlynear the origin. This curvature could be easily eliminated bycomposing a nonlinear univariate function on the output of theconstructed approximation. The second row of Table 3 showsthe scatter measures for scatter plots. The small average andRMS errors and R 2 values close to one indicate good fit of thetraining data. For the training data, the performance of theMC-HARP neural network is superior to the AASHO formula.

Figs. 4 and 5 show serviceability trends and their confidencestrips predicted by the constructed MC-HARP neural networkFL for a number of training and test sections, respectively. TheMC-HARP predicted trends well but did not perfectly followthe smooth measured trends. The MC-HARP neural networkcomputes a deviation measure (1' for each predicted PSI. The(1' measure is the standard deviation of the sample of outputspredicted by a sample of p = 15 HARP neural networks for apoint in the input domain. A one-standard-deviation confidence interval for a predicted PSI value P is (P - (1', P + (1').

In Figs. 4 and 5, the shaded strip along each predicted serviceability trend represents its confidence strip for one standard deviation. The average of deviation measures for trainingdata points is 0.349, equal to the RMS error for the trainingset. Therefore, the average width of the confidence strip is0.698. The constructed MC-HARP empirical model representsthe training data with good precision and accuracy. For thetest data, the predicted serviceability trends are mostly nonincreasing and pass between the bounding trends for the majority of test sections. The MC-HARP deviation measure forthe test set is 0.639, and consequently, the average width ofconfidence strips is 1.278. The serviceability trends predictedby the MC-HARP neural network FL pass between the bounding trends for more test sections than the AASHO formula.Therefore, the constructed Me-HARP neural network has superior performance to the AASHO empirical model for thetraining and test data.

Me-HARP NEURAL NETWORK WITH AASHOSUBDOMAIN APPROXIMATION

We can improve the performance of an MC-HARP neuralnetwork by using a parametric subdomain approximation 9


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5.---.,.....--.--........-~--."

lC

I"", .:::;;0 .

o " .o Measured PSI 5 o Smooth PSI 5

without replicate section pairso Smooth PSI 5

with replicate section pairs

FIG. 6. Performance of MC-HARP Neural Network with AASHO Subdomaln Approximation

5r---.--~----,--"""'-",

- - AASHO Formula

... _-- MCHARP neural network

--- --- ---

... ,b............~I 7~ r

5r----,-----,---.,.....-~----,

~-------------•..

0'----'----'----'----'----'

5,-----r---.,.----.---.,.---,

00 1500

Weighted Axle Load Applications, 1000'so 1500


Weighted Axle Load Applications, 1000's

FIG. 7. Performance of MC-HARP Neural Network with AASHO Subdomain Approximations for Training Set

(8)

(7)

(6)(

x )~(X;W)

6A(x; w) =4.2 - 2.7 __5_

p(x; w)

where parametric functions 13 and p have the AASHO form

where D = WIXI - W2X2 - W3X3; the input variables XI throughX5 are Db D2 , D3 , L, and W, respectively, and the parametersWI through W ll are respectively the coefficients al-4, Ao _ 2,

Bo _ 2 , and 130 used in (2) through (5). We refer to 6A (x; w) asthe AASHO subdomain approximation. The 6A (x; w) approximation has 11 unknown parameters that are estimated during

that satisfies the testing trends (decay of P with increasing Wand decreasing D) a priori. In doing so, one can build an MCHARP neural network that inherently follows the testing trendswithout the need to learn them from the data. A good candidatefor such a subdomain approximation is the AASHO parametricmodel. The AASHO formula intrinsically satisfies the testingtrends but is not able to represent the performance data of theroad test as well as a global model. By using the AASHOparametric form for the subdomain approximation of an MCHARP neural network, one can expect a mutual improvement.MC-HARP helps the AASHO function represent data betterby lending it adaptivity, while the AASHO function helps theMC-HARP neural network to reliably capture the testingtrends.

We choose the subdomain approximation function that hasthe AASHO parametric functional form defined in (2) through(5). The selected subdomain approximation takes the form


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5,...---r----r-----r------r--

----~.-- ----

5.---,---,-----,----r----,

Me-HARP Predicted 'Itend

Bounding 'nend

5,...--,-----,----r----,-----,

~--

G00 1500



Weighted Axle Load Applications, 1000's

FIG. 8. Performance of Me-HARP Neural Network with AASHO Subdomaln Approximations for Test Set

the subdomain training process, using the least squares estimator. The subdomain parameter estimation problem for a subdomain concerning T data points can be expressed as

T

minimize 2: [9A (Xi; w) - PI]2 subject to 0 :S Wj :S hj ,Wi_I

j = 1, ... , 11 (9)

where PI = measured PSI for the ith data point; XI and theupper bounds {h,}J~l are set to be {I, 1, 1, 2, 107

, 30, 10, 1,6, 10, I}, respectively. The parameters of 9A are constrainedto be nonnegative. Such a constraint is explained on the basisthat 9A should satisfy the testing trends. For example, the layercoefficients Wl_3 are positive because an increasing layer thickness must lead to an increase in the overall serviceability. TheAASHO subdomain approximation and consequently, the objective function of the estimation problem (9) are highly nonlinear with respect to optimization variables w. We use recursive quadratic programming (Hanan et al. 1994) to solve theconstrained nonlinear minimization problem in (9). The initialvalues for the parameters W are set to be the values recommended by the AASHO formula, which are {0.44, 0.14, 0.11,1.0,10,·93,9.36,4.79,0.081,5.19,3.23, 0.4}, respectively. Weuse the MC-HARP model-selection technique and frameworkfor classifying data sets to build an MC-HARP neural networkwith suitable complexity. Again, the tolerance for subdomaintraining was set to 1.75 to prevent building an approximationwith unreasonable complexity.

The constructed MC-HARP neural network with theAASHO subdomain approximation FA (the subscript A indicates that the local functions were the AASHO function) has602 parameters and 55 subdomains on average for 15 HARP

partitions. The MC-HARP neural network FA is slightly simpler and has fewer subdomains than the MC-HARP neuralnetwork with linear subdomains FL'

The scatter plots of predicted PSI values against measuredand smooth PSI values for FA are shown in Fig. 6. Points inthe scatter plots are clustered along the diagonal. This observation suggests a good fit of the training data. The fourth rowof Table 3 shows the scatter measures for scatter plots. Thesmall average and RMS errors and R2 values close to oneexhibit good fit of training data. According to Table 3, fortraining data, the MC-HARP neural network with the AASHOsubdomain approximation fits the data better than MC-HARPneural network FL , and its performance is superior to theAASHO formula.

Figs. 7 and 8 show the serviceability trends and their confidence strips predicted by the constructed MC-HARP neuralnetwork with the AASHO subdomain approximation for anumber of training and test sections, respectively. The MCHARP predicted trends closely follow the smooth measuredtrends. The MC-HARP deviation measure for training datapoints is 0.106. The average width of the confidence strip is0.212. Hence, the constructed MC-HARP empirical model represents the training data with good precision and accuracy.Furthermore, the approximation FA has better approximationaccuracy (lower RMS) and precision (lower deviation) for thetraining set than the MC-HARP neural network with linearsubdomain approximations PL' For test sections, the predictedserviceability trends are mostly nonincreasing and pass between their bounding trends for the majority of test sectionswith good confidence. The MC-HARP deviation measure forthe test set is 0.430, and consequently, the average width ofconfidence strips is 0.860. In comparison to the PL approximation, the FA approximation has better precision (lower de-


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viation) for test-data points, and its predicted trends aresmoother and satisfy the second testing trend better. Furthermore, the serviceability trends predicted by the MC-HARPneural network with the AASHO subdomain approximationpass between their bounding trends for more test sections thanthe AASHO formula.

By selecting a subdomain approximation that inherently exhibits a known global feature of the data, we have built anMC-HARP neural network FA that is better than MC-HARPneural network FL that does not use the prior knowledge. Bothneural-network models are superior to the AASHO empiricalmodel.

CONCLUSION

In this paper, we have examined the question of developingdatabased mathematical models of long-term pavement performance by taking another look at the AASHO road-test data.We have developed two neural networks based on the MCHARP algorithm and compared the performance of these twomodels with the venerable AASHO formula. Throughout thispaper, we have insisted that an empirical formula for the roadtest is reasonable only if the predicted serviceability trend fora pavement is a nonincreasing curve and if the predicted PSIincreases by increasing the overall thickness of a pavementstructure and by decreasing the axle load.

We have studied the performance of the databased modelpresented in AASHO (1962) that has served as the basis forpavement design practice during the last 30 years. TheAASHO model does not represent the observed serviceabilitytrends for pavement sections trafficked at the road test well.Significant discrepancies between the AASHO predicted serviceability trends and measured trends for a number of pavement sections indicate poor performance of the AASHO formula for some regions of the input domain. We have shownthat a local approximation like an MC-HARP neural networkcan model the pavement performance data for the entire inputdomain better than a global approximation like the AASHOformula.

The MC-HARP method was used to build a nonparametricmodel for pavement performance using the data of the roadtest and compared its performance with the AASHO model.The MC-HARP model-selection technique and framework forclassifying data sets has enabled us to build a neural networkwith suitable complexity to represent the pavement-performance data. The constructed MC-HARP empirical model rep-

resents the trammg data with good preClSlon and accuracy.Furthermore, its performance for the road-test is superior tothe AASHO formula and satisfies the testing trends for moretest sections that the AASHO formula.

We built a better MC-HARP neural network by selecting asubdomain approximation that inherently exhibits the trendsof decrease of P with W and increase of P with an increasein pavement thickness or decrease in axle weight. We used theMC-HARP method to construct a neural network whose parametric subdomain approximation was taken to be theAASHO model. For training and test sets, the constructed empirical model is better than the MC-HARP neural network withlinear subdomain approximation and is superior to theAASHO empirical model. MC-HARP helps the AASHO formula to better represent data, and on the other hand, theAASHO model helps an MC-HARP neural network to reliablycapture the testing trends, improve its generalization, andcurve unrealistic performance outside the training set.

Databased mathematical modeling is likely to play a keyrole in durability assessment of pavements in the future. Wehope that an algorithm like MC-HARP will prove useful as ameans of processing data and building useful mathematicalmodels for engineering design.

APPENDIX. REFERENCES

American Association of State Highway Officials (AASHO). (1962)."The AASHO road test: report 5, pavement research." Publication No.954, National Academy of Science-National Research Council,Washington, D.C.

American Association of State Highway and Transportation Officials(AASHTO). (1981). AASHTO interim guide for design of pavementstructures. Washington, D.C.

American Association of State Highway and Transportation Officials(AASHTO). (1986). AASHTO guide for design ofpavement structures.Washington, D.C.

Banan, M. R., Banan, M. R., and Hjelmstad, K. D. (1994). "Parameterestimation of structures from static response. Part I: Computationalaspects." J. Struct. Engrg., ASCE, 120(11),3243-3258.

Banan, M. R., and Hjelmstad, K. D. (1994). "Data-based mathematicalmodeling or How to build a mapping neural network." Struct. Res.Ser. No. 590, UILU-ENG-94-200B, Univ. of 111inois, Urbana, Ill.

Banan, M. R., and Hjelmstad, K. D. (1995). "A Monte Carlo strategyfor data-based mathematical modeling." J. Math. and Compo Modelling, 22(8), 73-90.

Chambers, J. M. (1983). Graphical methods for data analysis. WadsworthInternational Group, Belmont, Calif.

Coree, B. J., and White, T. D. (1990). "AASHTO flexible pavementdesign method: fact or fiction?" Transp. Res. Rec., 1286, 206-216.


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Documents

Neural Networks and AASHO Road Test