Data Visualization

Our initial motivation for developing dependency networks concerned the visualization of predictive relationships. In this section, we examine this application in more detail and describe a tool developed at Microsoft Research, called DNetViewer, that employs dependency networks for data visualization. For illustration, we use a real data set, provided by Media Metrix, that contains demographic and internet-use data for about 5,000 individuals during the month of January 1997. Figure 2 shows DNetViewer's display of a dependency-network structure learned from this data. After only a short inspection, an interesting relationship becomes apparent: there are many dependencies among demographics, and many dependencies among frequency-of-use, but there are few dependencies between demographics and frequency-of-use. We have found numerous interesting dependency relationships such as this one across a wide variety of datasets using dependency networks for visualization. In fact, we have given dependency networks this name because they have been so useful in this regard.

Figure 2: A dependency network for Media Metrix data. The dataset contains demographic and internet-use data for about 5,000 individuals during the month of January 1997. The node labeled Overall.Freq represents the overall frequency-of-use of the internet during this period. The nodes Search.Freq, Edu.Freq, and so on represent frequency-of-use for various subsets of the internet.

DNetViewer allows a user to display both the dependency-network structure and the probabilistic decision tree associated with each variable. Navigation between the views is straightforward. To view a decision tree for a variable, a user double clicks on the corresponding node in the dependency network. Figure 3 shows the tree for Shopping.Freq. Note that there is an interesting relationship between the dependency-network structure and the individual decision-tree structures. Namely, there will be a split on variable $X$ in the decision tree for $Y$ if and only if there is an arc from $X$ to $Y$ in the dependency network. We have found that this correspondence facilitates the process of data visualization.

Figure 3: The probabilistic decision tree for Shopping.Freq obtained by double-clicking the corresponding node in the dependency-network graph. The histograms at the leaves correspond to probabilities of Shopping.Freq use being zero, one, and greater than one visit per month, respectively.

Besides avoiding the sometimes confusing semantics of Bayesian networks, a dependency network--in particular, an inconsistent dependency network--learned from data offers an additional advantage for visualization over Bayesian networks. If there is an arc from $X$ to $Y$ in such a network, we know that $X$ is a significant predictor of $Y$--significant in whatever sense was used to learn the network with finite data. Under this interpretation, a uni-directional link from $X$ to $Y$ is not confusing, but rather informative. For example, in Figure 2, we see that Socioeconomic status is a significant predictor of Sex, but not vice versa--an interesting observation. Of course, when making such interpretations, one must always be careful to recognize that statements of the form ``$X$ helps to predict $Y$'' are made in the context of the other variables in the network. In DNetViewer, we enhance the ability of dependency networks to reflect strength of dependency by including a slider (on the left). As a user moves the slider from bottom to top, arcs of decreasing strength are added to the graph. When the slider is in its upper-most position, all arcs (i.e., all significant dependencies) are shown. There are several reasonable methods for ranking arc strength. The one we use determines the order in which arcs would be added during a (greedy) structure search that grows all decision trees in parallel. (In practice, we construct the trees one after the other, but we can imagine a parallel procedure.) At each step of this imagined construction process, we compute the increase in score (log posterior probability) of each tree for every possible new split. We accept the split with the largest increase in score and iterate. As the slider is moved up, we add arcs in the order in which this procedure accepts corresponding splits. Figure 4 shows the dependency network for the Media Metrix data with the slider at half position. At this setting, we find the interesting observation that the dependence between Sex and XXX.Freq (frequency of hits to pornographic pages) is the strongest among all dependencies between demographics and internet use.

Figure 4: The dependency network in Figure 2 with the slider set at half position.