# Cluster Analysis for Evaluating Trading Strategies 1

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3 3 METHODOLOGY Our methodology uses the intuition of a progress chart when characterizing a trading strategy, but applies a common clustering technique called k-means to divide the aggregate strategy into its component strategies in the same way a prism divides light into its component colors (as shown in Figure 2). The process begins by creating a progress chart for each order. Specifically, for each 15-minute period in the trading day (26 in total), it computes the cumulative fraction of the order that was completed by the end of that period, i.e., the progress of the order at that point. The trading strategy itself is represented by the collection of these 26 progress points, an example of which is given in Figure 1. These charts will always begin at 0% and end at 100%, and will increase as we move from left to right along the x-axis to represent the order s cumulative fill progress over the day. We then apply k-means to group them into k distinct trading strategies. Figure 2. The methodology takes an aggregate progress chart and splits it into its underlying component strategies. To understand how k-means works intuitively, assume that we break the trading day into 3 bins instead of 26 bins. For each order, we determine the percent of the order that was complete at the end of each bin. For example, suppose the trader executed a 10,000-share order by executing 2000 shares in bin 1, 1000 shares in bin 2, and 7000 shares in bin 3. Our methodology would characterize this order as a progress chart with the values 20%, 30%, and 100%, to represent the percent complete at the end of each bin. Since all orders are completed by the end of the last bin, all orders will have a value of 100% in bin 3. For this reason, we only need to look at the progress at the end of the first two bins when attempting to distinguish between strategies. 2 In Figure 3, we plot a sample of orders, where each black dot on the graph represents an order. The x-axis represents the percent of the order completed by the end of bin 1, and the y-axis represents the percent completed by the end of bin 2. In the 2 Adding the third bin where all orders take on a value of 100% to the k-means methodology does not provide any useful information in helping us differentiate between how the different orders were traded. So one can exclude the third bin from the k-means methodology without influencing the results.

4 4 example of the 10,000-share order above, the order can be represented graphically as the dot labeled X in Figure 3A. Since this order was 20% complete at the end of bin 1 and 30% complete by the end of bin 2, the point is represented with an x-axis value of 20% and a y-axis value of 30%. Figure 3. Illustration of k-means algorithm. In Figure 3A, the black dots are the existing, classified observations. The triangle in Figure 3B represents a new order that must be classified, and the squares represent the centers of the two existing clusters. The grey arrows show the distance between the new point and the existing clusters centers. The algorithm classifies the new point with the cluster whose center is the shortest distance from it. The black squares in Figure 3C represent the original cluster centers. The grey square is the updated center of the cluster with the additional order. Looking at Figure 3A, there are clearly two distinct groups of dots one cluster in the lower left quadrant and another in the upper right quadrant. Intuitively, these clusters represent the two distinct strategies that the trader used. The former represents orders that are executing slowly, i.e., those that have made relatively little progress after both bin 1 (x axis) and bin 2 (y axis). The latter represents orders that are being executed more quickly, where progress in both bin 1 and bin2 is significantly higher. In two-dimensions with a small amount of data, one could do cluster analysis visually, as in Figure 3A. When the data set is large or the number of dimensions is higher, as is the case here where we could have thousands of orders each split into 26 distinct bins, one must rely on statistical techniques to manage the clustering. This is where k-means methodology comes into play. The k-means algorithm begins by assigning k initial cluster centers, which can be specified by the user or selected randomly by the algorithm. Iteratively, the algorithm works through the sample, using a distance metric to assign each observation to the nearest cluster. Figure 3B provides an example of an iteration of k-means. Suppose we were to add a new observation, represented by the triangle in Figure 3B. K-means computes the distance between that point and the two existing cluster centers, represented by the squares in Figure 3B, to determine the nearest cluster. Since the triangle is closer to the left cluster, k-means assigns it to the left cluster. With the addition of a new data point, however, k-means must now compute a new cluster center. Figure 3C shows the new cluster center, represented by the grey square, which has shifted in the direction of the new observation. When cluster centers and assignments of observations stop changing dramatically, the algorithm stops. At this point, the output contains information on the k cluster centers, which can be used to characterize the group itself, as well as the assignment of each observation into a cluster. 3 In our specific application, the center point of a group characterizes the average progress chart of that strategy and the assignments indicate the strategy that each order most closely resembles. 3 See Johnson & Wichern (2007) and MacQueen (1967) for a detailed discussion of k-means.

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