Technical Note 141
Using Statistical Cluster Method to Group Deflection
Data for the Purpose of Pavement Performance
Assessment and Structural Overlay Design
January 2015
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© State of Queensland (Department of Transport and Main Roads) 2015
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Technical Note, Transport and Main Roads, January 2015
TN141 Using Statistical Clutter Method to Group Deflection Data for the Purpose of Pavement Performance
Assessment and Structural Overlay Design
1
Statement
This Technical Note provides guidance on the application of statistical cluster analysis to deflection
data used for pavement structural design.
2
Background
Statistical cluster analysis has been applied to classification of multivariate data since the 1930s [1]
and the hierarchical agglomerative clustering referred to in this document has since been expanded to
include a variety of metrics and linkage methods [2].
Traditionally, either individual deflection bowls or the mean bowl of a group of ‘similarly shaped’
deflection bowls have been used as input to back analysis procedures. The first method relies on the
assumption of very small measurement errors associated with each deflection. Whatever the case, it
can be demonstrated that subsequent back analysis will magnify such errors, typically by a factor of
approximately 3. The second method attempts to reduce the effect of measurement errors by taking
the mean [3].
With the mean bowl method judgement as to what constitutes a ‘similarly shaped’ bowl, in the past,
has been somewhat subjective. Cluster analysis provides a scientific (repeatable) method by which
groups of bowls can be identified.
3
Methodology
The hierarchical clustering procedures of R Statistical Computing [4], available under a GNU General
Public License (free) [5], have been used to implement the clustering.
AutoIt [6] performs the minor task of ensuring that R has completed the cluster analysis before
passing control back to MS Access.
MS Access has been used to generate the bowl combinations required.
Bowls are compared by means of a simple linear regression, excluding the intercept. The remaining
regression parameters of Slope and the Coefficient of Determination (R²) serve as measures of the
relative amplitude and shape, respectively, of the bowls being compared. This can be seen as a
dimensional reduction of n deflection measurements per bowl to just two per bowl pair.
Since these two parameters are defined over different ranges, Slope from 0 to +∞, and R² from 0 to 1,
they need to be transformed and scaled in order to be used as the components of a metric. An
absolute logarithmic transformation is adopted for both parameters. The absolute value of the
logarithm of the slope is employed to ensure that the reciprocal of the slope is treated as an
equivalent. In the rare event that R² is negative, it should be replaced with a small positive value,
(1E- 9), prior to applying the logarithm and the absolute value of the logarithm ensures a positive
contribution to the metric.
If two bowls are identical the slope and R² will have a value of 1 and the absolute logarithm will be 0,
the appropriate contribution to the metric.
However, since the clustering procedure requires a metric ‘cut off’ value to determine group
membership, the relative weighting of the slope and R² need to be indicated by some scaling factor. A
metric cut off of 1 is adopted. Let 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 be the slope discriminant, the maximum regression slope that
will be considered for two bowls to belong to the same group. The transformed slope must be divided
2
by the logarithm of 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 to result in a metric cut off of 1. Let 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚
be the R² discriminant, the minimum
Technical Note, Transport and Main Roads, January 2015
1
TN141 Using Statistical Clutter Method to Group Deflection Data for the Purpose of Pavement Performance
Assessment and Structural Overlay Design
regression R² that will be considered for two bowls to belong to the same group. The transformed R²
2
to result in a metric cut off of 1.
must be divided by the logarithm of 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚
The resulting metric r is indicated in Equation 1.
Equation 1
r = ���
2
2
𝑎𝑎𝑎𝑎𝑎𝑎�𝑙𝑙𝑙𝑙(𝑚𝑚)�
𝑎𝑎𝑎𝑎𝑎𝑎�𝑙𝑙𝑙𝑙(𝑅𝑅2 )�
� +�
2 ) � �
𝑙𝑙𝑙𝑙(𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 )
𝑙𝑙𝑙𝑙(𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚
Typical discriminant values are 1.4 for slope and 0.95 for R²; however, values up to 2 for slope and as
low as 0.9 for R² may be considered depending on the age and extent of the pavement.
A ‘Complete’ linkage method is adopted. Alternative linkage methods produce similar results.
4
Procedure
1. Capture deflection data to a format suitable for processing.
2. Separate bowls into sites where a structural difference is known or likely. Sites are typically
identified based on a gap in chainage (longitudinal), but can also be defined based on
carriageway, lane or even wheelpath (transverse).
3. Identify wheelpaths which require a Moisture Adjustment Factor (MAF) to be applied prior to
back analysis.
4. Separate each site into longitudinal sections based on relative deflection levels, historical
records and Ground Penetrating Radar (GPR) results. To assist with sectioning, deflection
data may be further processed to produce a Cumulative Difference [7] plot. It is acceptable to
use several wheelpaths in the sectioning process.
5. Commence statistical clustering of bowl shapes in each section with the ‘Complete’ linkage
method with a slope discriminant not greater than 2 and an R² discriminant not less than 0.9.
Typical values are 1.4 and 0.95 respectively.
6. Increase the R² discriminant until bowl groups contain bowls of predominantly the same
shape. One or two incongruent bowls can be tolerated so long as they do not extend
significantly beyond the general envelope of the group because the overall bowl shape of the
group is derived from the mean of all bowls in the group.
7. Review bowl charts and, if necessary, increase R² to reduce the occurrence of incongruent
bowls.
8. Select bowl groups with a higher representative deflection for back analysis. Groups
containing a single bowl may be the result of geophone measurement errors, therefore groups
containing at least two bowls (preferably more) are recommended. Generally, only one or two
bowl groups with the largest representative deflection are selected for back analysis. Both
inner and outer wheelpaths should be sampled by the selected bowl groups to allow the
effects of moisture adjustment to be represented.
9. Compute the mean bowl for each group by calculating the arithmetic mean of deflections at
each geophone. The mean bowl for each group will include the mean maximum deflection
D0g.
Technical Note, Transport and Main Roads, January 2015
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TN141 Using Statistical Clutter Method to Group Deflection Data for the Purpose of Pavement Performance
Assessment and Structural Overlay Design
10. For each section, calculate the representative maximum deflection and representative
subgrade deflection for each wheelpath. For each wheelpath, estimate the subgrade California
Bearing Ratio (CBR) using device specific equations. For the FWD device, representative
maximum deflection D0r, representative subgrade deflection D900r and a 40 kN load
Equation 2 should be used.
Equation 2
𝐶𝐶𝐶𝐶𝐶𝐶 =
0.5911
𝐷𝐷9001.4759
𝑟𝑟
Note that most pavements exhibit approximate linear elastic behaviour allowing deflections for
loads other than 40 kN can be ‘normalised’ to 40 kN by multiplying by 40 and dividing by the
measured load.
11. The moisture adjustment method of the Transport and Main Roads Pavement Rehabilitation
Manual should be applied to wheelpaths, except that the method is confined to bowl groups
rather than sections. Therefore, the bowl group or groups selected for back analysis should
include both inner and outer wheelpaths. Normally, the bowl group with the largest
representative maximum deflection will be selected for each section and if this bowl group
contains both wheelpaths no further bowl groups need be selected. Each selected bowl group
can thus be represented by a moisture adjusted representative maximum deflection D0rma.
12. Multiply the mean bowl of each group by a factor D0rma/D0g to raise the maximum deflection
to the value of the moisture corrected representative maximum deflection of the group.
13. Submit the moisture adjusted mean bowl of each selected group to a back analysis process.
14. Bowl groups can also be used in a geospatial context to assist in the identification of locations
for test pits, cores and trenches.
5
References
1. Cluster analysis http://en.wikipedia.org/wiki/Cluster_analysis
2. Hierarchical clustering http://en.wikipedia.org/wiki/Hierarchical_clustering
3. Standard error http://en.wikipedia.org/wiki/Standard_error_(statistics)
4. R Statistical Computing http://www.r-project.org/
5. GNU General Public License http://en.wikipedia.org/wiki/GNU_Public_License
6. AutoIt automation and scripting language https://www.autoitscript.com/site/autoit/
7. AASHTO Design of Pavement Structures, Ch. 3, Guides for Field Data Collection, 1993
Technical Note, Transport and Main Roads, January 2015
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