## 3 Cluster Analysis for Data Classification  
It is well known in traffic theory that flow and density of traffic are linearly related until congestion occurs and the relationship breaks down. At this point, mean traffic speed slows, resulting in reduced flow. From a bridge loading perspective, the density of traffic determines the number of vehicles on the bridge. Truck percentage is also key as trucks are much heavier than cars. Figure 4 illustrates the data extracted from images in terms of traffic flow (vehicles per hour) and the average percentage of trucks (per hour). In the hours when flow is highest, the % trucks tends to be less, and the really high truck percentages are when flow is less than about 500 veh/h--clearly many truck drivers choose to  
Figure 3: Weight inference process illustrated with one month of slow lane vehicle weights: (**a**) Contours of relative frequency of weight/length combinations; (**b**) Gross weight (GVW) histogram for segment 34 (9.90–10.20 m); (**c**) Fitted probability density function; (**d**) Cumulative distribution function. make the journey when traffic flow is low. Cluster analysis [44; 45] is used here to separate the data into a number of clusters whose implications for bridge loading can be considered separately. In cluster analysis, similarities between objects are measured as distances between points on the graph [46]. Thus, a cluster is a group of points that are relatively close to each other and are distinctly less close to points in other clusters. While graphical methods can provide evidence of cluster structure in data, they tend to be subjective [47]. A hierarchical approach, consisting of a series of data segmentations, provides an objective approach. This method involves progressive data subdivision or synthesis at each stage of segmentation [48]. The hierarchical classification is often represented by a tree structure referred to as a dendrogram [49] which shows the clustering structure at each stage-Figure 5. In a 'bottom up' approach, it starts by assuming that each point is an individual cluster. At each stage, the two most similar clusters generate a new cluster. Each object is attributed to one cluster only. The dendrogram of Figure 5 shows the objects (vehicle flow/truck density) on the X-axis and the distances (or differences) between clusters on the Y-axis. The horizontal lines in the dendrogram represent the clusters while the vertical lines are the distances that clusters are connected  
Figure 4: Correlation between traffic flow and percentage of trucks. Figure 5: Dendrogram: the traffic flow–percentage of trucks pairs attributed to correct/natural/optimal clusters. at. The correct/natural/optimum number of clusters is indicated by horizontal lines positioned at approximate the same level on the Y-axis. The black dashed line in Figure 5 shows there are four correct/natural/optimal clusters in this dataset. Each cluster is highlighted in this figure by a different colour. Figure 6 shows the four clusters on the scatter plot and Table 1 provides some details. The LTMF (low percentage of trucks and medium flow) cluster (red circles) has a low % of trucks (less than about 7%) and up to a medium flow (less than about 1400 veh/hour). The LTHF (low percentage of trucks and high flow) cluster (liac stars) has a low % trucks but a high flow. Due to the low % trucks, neither the LTMF nor the LTHF clusters are critical for bridge traffic loading. The HTLF (high percentage of trucks and low flow) cluster (green squares) has a high % trucks (from about 7% to 25%) but low flow (below 700 veh/hour). Because of the low flow, the inter-vehicle gaps will be large so, despite the high % of trucks, this cluster will only be important for bridge traffic loading in the event of non-recurrent congestion (e.g., accident) that causes the gaps to be reduced. Finally, the MTHF (medium percentage of trucks and high flow) cluster (navy crosses) has a medium % of trucks (from about 3% to 16%) and a high flow (about 700 to 3500 veh/h).