## 6 Evaluation
### Results  
In order to compare and evaluate the rankings computed, we looked at three tickers from the technology sector: AAPL, INTC and GOOG. Figure 10 shows the relative price change of these stocks over the entire time period, normalized to start from the same relative value and overlaid on the same figure. This helps visualize the performance of the three stocks relative to each other. Figure 11 shows the computed ranks using snapshot aggregation for our three stocks. The data are noisy enough that we chose not to connect the points; otherwise, we would have been faced with a solid block of color. Much like every graph seen so far, snapshotting gives us noisy data: the rank for a given stock will vary significantly from one day to the next. This may or may not be a disadvantage; there may be a fixed cost to switching services (e.g., a transaction fee), which could overwhelm the benefits of frequent change. However, there may be an advantage to changing the selected service frequently, since switching the provider load balances the services nicely. Therefore, the high frequency of change in ranks in the snapshotting technique cannot, in and of itself, be counted against the technique. However, in Figure 12, by computing the histogram of the data over a period of time, it becomes more readily visible. The snapshot skyline ranks are similar to Figure 11 in that they are scattered with no obvious trend. Figure 13 instead shows the ranks computed using the Dynamis skyline techniques. Figure 9: INTC Dynamis skyline rank as its score changes over time in a pool of 500 tickers. Figure 11: Snapshot aggregation rank of our three stocks over the time period considered. Figure 10: Relative price change of three stock tickers over the time period considered, normalized to start from the same relative value. On a specific day \(d\) we select the top \(k\) stocks using our four methods. Specifically, we obtain the following top \(k\) rankings:  
* L-SAgg using SAgg
* L-SSkyline using SSkyline
* L-DAgg using DAgg
* L-DSkyline using DSkyline  
Figure 12: Dynamis aggregation rank of our three stocks over the time period considered. Figure 13: Dynamis skyline rank of our three stocks over the time period considered. \[ \begin{cases}L-\text{{SAgg}}=(s_{1},s_{2},\cdots,s_{k})\\ L-\text{{SSkyline}}=(s^{\prime}_{1},s^{\prime}_{2},\cdots,s^{\prime}_{k})\\ L-\text{{DAGg}}=(s^{\prime\prime}_{1},s^{\prime\prime}_{2},\cdots,s^{\prime \prime}_{k})\\ L-\text{{DSkyline}}=(s^{\prime\prime\prime}_{1},s^{\prime\prime\prime}_{2}, \cdots,s^{\prime\prime\prime}_{k})\end{cases} \]  
In order to evaluate the success of these four methods, we now evaluate the cumulative gain obtained from purchasing the top \(k\) ranked stocks. Using the historical data gathered for stock prices, we are able to follow the evolution of a hypothetical investment on day \(d\) for each of our top \(k\) picks; the period over which we follow that investment we call window \(w\). We can compute the gain for stock \(s\) purchased on day \(d\) over the investment period \(w\) as:  
\[\text{{Gain}}^{s}_{d}=\frac{\text{P}^{s}_{d+w}-\text{P}^{s}_{d}}{\text{P}^{s}_ {d}}\]  
We divide the gain by the initial price in order for our computations to remain independent of the magnitude of the price. This quantity is known as the arithmetic return or yield. We assume a sale on the last day of the window, when the gains or losses are realized. At that time, we add the gains for each one of the stocks selected as part of the top \(k\) pick. In order to keep our results independent of the number of stocks selected, we divide the total gain at the end of the investment period by \(k\). The cumulative gain (\(CG\)) is therefore:  
\[CG_{d}=\frac{1}{k}\sum_{s=s_{1}\ldots s_{k}}\text{Gain}^{s}_{d}\]  
This quantity is technically known as the arithmetic average rate of return. Thus, a result greater than zero represents a net gain, and a result less than zero represents a net loss over the period.