Analysts and data scientists are constantly seeking new ways to parse increasingly intricate datasets, many of which are deemed “high dimensional”, i.e., contain many (sometimes hundreds or more) individual variables. Machine learning has recently emerged as one such technique due to its exceptional ability to process massive quantities of data. A particularly useful machine learning method is *t-distributed stochastic neighbor embedding (t-SNE)*, used to summarize very high-dimensional data using comparatively few variables. T-SNE visualizations allow analysts to identify hidden structures that may have otherwise been missed.

## Traditional Data Visualization

The first step in tackling any analytical problem is to develop a solid understanding of the dataset in question. This process often begins with calculating descriptive statistics that summarize useful characteristics of each variable, such as the mean and variance. Also critical to this pursuit is the use of data visualizations that can illustrate the relationships between observations and variables and can identify issues that must be corrected. For example, the chart below shows a series of pairwise plots between a set of variables taken from a loan-level dataset. Along the diagonal axis the distribution of each individual variable is plotted.

The plot above is useful for identifying pairs of variables that are highly correlated as well as variables that lack variance, such as original loan term. When dealing with a larger number of variables, heatmaps like the one below can summarize the relationships between the data in a compact way that is also visually intuitive.

The statistics and visualizations described so far are helpful for summarizing and identifying issues, but they often fall short in telling the entire narrative of the data. One issue that remains is a lack of understanding of the underlying structure of the data. Gaining this understanding is often key to selecting the best approach for problem solving.

## Enhanced Data Visualization with Machine Learning

Humans can visualize observations plotted with up to three variables (dimensions), but with the exponential rise in data collection it is now abnormal to only be dealing with a handful of variables. Thankfully, there are new machine learning methods that can help overcome our limited capacity and deliver new insights never seen before.

T-SNE is a type of non-linear dimensionality reduction algorithm. While this is a mouthful, the idea behind it is straightforward: t-SNE takes data that exists in very high dimensions and produces a plot in two or three dimensions that can be observed. The plot in low dimensions is created in such a way that observations close to each other in high dimensions remain close together in low dimensions. Additionally, t-SNE has proven to be good at preserving both the global and local structures present within the data^{1}, which is of critical importance.

The full technical details of t-SNE are beyond the scope of this blog, but a simplified version of the steps for t-SNE are as follows:

- Compute the Euclidean distance between each pair of observations in high-dimensional space.
- Using a Gaussian distribution, convert the distance between each pair of observations into a probability that represents similarity between the points.
- Randomly place the observations into low-dimensional space (usually 2 or 3).
- Compute the distance and similarity (as in steps 1 and 2) for each pair of observations in the low-dimensional space. Crucially, in this step a Student t-distribution is used instead of a normal Gaussian.
- Using gradient based optimization, iteratively nudge the observations in the low-dimensional space in such a way that the probabilities between pairs of observations are as close as possible to the probabilities in high dimensions.

Two key consideration are the use of the Student t-distribution in step four as opposed to the Gaussian in step two, and the random initialization of the data points in low dimensional space. The t-distribution is critical to the success of the algorithm for multiple reasons, but perhaps most importantly in that it allows clusters that initially start far apart to re-converge^{2}. Given the random initialization of the points in low dimensional space, it is common practice to run the algorithm multiple times with the same parameters to observe the best mapping and ensure that the gradient descent optimization does not get stuck in a local minima.

We applied t-SNE to a loan-level dataset comprised of approximately 40 variables. The loans are a random sample of originations from every quarter dating back to 1999. T-SNE was used to map the data into just three dimensions and the resulting plot was color-coded based on the year of origination.

In the interactive visualization below many clusters emerge. Rotating the figure reveals that some clusters are comprised predominantly of loans within similar origination years (groups of same-colored data points). Other clusters are less well-defined or contain a mix of origination years. Using this same method, we could choose to color loans with other information that we may wish to explore. For example, a mapping showing clusters related to delinquencies, foreclosure, or other credit loss events could prove tremendously insightful. For a given problem, using information from a plot such as this can enhance the understanding of the problem separability and enhance the analytical approach.

Crucial to the t-SNE mapping is a parameter set by the analyst called perplexity, which should be roughly equal to the number of expected nearby neighbors for each data point. Therefore, as the value of perplexity increases, the number of resulting clusters should generally decrease and vice versa. When implementing t-SNE, various perplexity parameters should be tried as the appropriate value is generally not known beforehand. The plot below was produced using the same dataset as before but with a larger value of perplexity. In this plot four distinct clusters emerge, and within each cluster loans of similar origination years group closely together.