Wednesday, January 15, 2014

VisuMap 4.2.896 Released

I have just released our software package VisuMap version 4.2.896.  In this release we have upgraded the graphics library DirectX from version 9 to version 11.  Due to this upgrade some of 3D animation data views have improved performance by more than 10 folds. VisuMap is now capable to interactively explore more than one million data points in 3D view on a normal desktop computer.

Also with this upgrade, VisuMap will require OS system Windows 7 or higher. Windows XP will be continually supported, but without new features which take advantage of DirectX 11 library.  In the future, we'll likly implement more performance enhancements by tackling the computation power of GPU.

Monday, November 4, 2013

On Multidimensional Sorting

In a previous blog post I have talked about a method to convert photos to high dimensional datasets for analysis with MDS methods. This post will address the opposite problem: given a high dimensional dataset, can we convert it to a photo alike 2D image for easy feature recognition by human?

Let's consider our sample dataset S&P 500. A direct approach to create a 2D map is to simply convert the matrix of real values to a matrix of gray-scale pixels. This can be easily done in VisuMap with the heatmap view and the gray-scale spectrum. The following picture shows the S&P 500 dataset in normal line diagram view on the left side and the corresponding heatmap on the right side:

Each curve in the left picture corresponds to the price history of a S&P-500 component stock in a year. We notice heavy overlaps among the curves. On the right side, the heatmap represents the stock prices with row strips with different brightness. In the heatmap, we can spot roughly the two systematical downturns at the two slightly darkened vertical strips. There is however no discernible pattern between the rows, as the rows are just ordered by their stock ticks, so that neighboring relationship don't indicate any relationships between their price history.

One simple way to improve the heatmap is to group stocks with similar price history together. This is the task of most clustering algorithms. We can do this pretty easily in VisuMap. The following heatmap shows the same dataset after we have applied k-Mean clustering algorithm on the rows and re-shoveled the rows according to cluster assignments:

The k-Mean algorithm has grouped the rows into about 8 clusters, we can see that rows within each group represent rows with, more or less, similar values. However, the clusters are randomly ordered, and as well as the rows within a cluster. The clustering algorithm does not provide ordering information for individual data points.

Can we do better job than the clustering algorithm? To answer this question, let's recall what we actually tried to do above: we want to reorder the rows of the heatmap, so that neighboring rows are similar to each other. This kind task is basically also the task of MDS (Multdimensional Scaling) algorithms, which aim to map high dimensional data to low dimensional spaces while preserving similarity. Particularly in our case, the low dimensional space is just the one-dimensional space, whereas MDS in general have been used to create 2 or 3 dimensional maps.

Thus, to improve our heatmap,  we apply a MDS algorithm to our dataset to map it to an one-dimensional space, then use their coordinates in that one-dimensional space to re-order the rows in the heatmap. For this test, I have adapted the RPM mapping (Relational Perspective Map) algorithm to reduce the dimensionality of the dataset from 50 to 1, then used the one-dimensional RPM map to order the rows in the heatmap. The following picture shows a result heatmap:

We notice this heatmap is much smoother than the previous two.  In fact, we can see that the data rows gradually change from bright to dark color, as they go from the top to the bottom. More importantly, we don't see clear cluster structure among the rows, this is in clear contrary to the k-Mean algorithm that indicates 8 or 10 clusters. In this case, it is clear that k-Mean algorithm delivered the wrong artificial cluster information.

Now, a few words about the implementation. The multidimensional sorting has been implemented with RPM algorithm, since RPM allows gradually shrinking the size of the map by starting with 2D or 3D map. This special feature enables RPM to search for optimal 1D map configuration among a large solution space. We could easily adapt other MDS algorithms to perform multidimensional sorting, but we probably will have to restrict solely on one-dimensional space with those algorithms. A few years ago, I have already blogged about this kind of dimensionality reduction by shrinking dimension with some simple illustrations.

Since high dimensional datasets are generally not sequentially ordered, there are in general not an unique way to order the data. What we want to do is just find a solution that preserves as much as possible similarity information with the order information. Thus, an important question arises here: How do we validate the sorting results? How do we compare two such sorting algorithms?  A simple way to validate the sorting result is illustrated in the following picture:
So, in order to test the sorting algorithm we first take simple photo and convert it to high dimensional dataset that represents the gray-scale levels of the photo. We then randomize the order of the rows in the data table. Then, we apply the sorting algorithm on the randomized data table, and check how good the original photo can be recovered. I have done this test for some normal photos, the multidimensonal sorting was able to recover pretty well the original photo (see also the short attached video), as long as enough time is given to the learning process.

We have described an application of MDS algorithm for heatmap by using it to reduce data dimension to a single dimension. We might go a step further to use MDS to reduce data dimension to 0, so that data points will be mapped to a finite discrete points set. In this way, we would have archived the service of clustering algorithms. If this works, clustering algorithms can be considered a special case of MDS algorithms; and MDS algorithms might lead us to group of new clustering algorithms.

The multidimsional sorting service has been implemented in VisuMap version 4.0.895 as integrated service. The following short video shows how to use this service for the sample datasets mentioned above.

A more practical application of MDS sorting is for microarray data analysis where heatmaps are often used to visualize the expression levels of a collection of gens for a collection of samples. Normally, neither the gens nor  the samples are ordered according to their expression similarity, so that those heatmaps often appear rather random (even after applying additional grouping with hierarchical clustering algorithm.) The following picture shows, for instance, a heatmap for expressions of 12000 genes for about 200 samples:

After applying MDS sorting both on the rows and columns of above heatmap, the heatmap looks  as following:

The above heatmap contains theoretically the same expression information. But, just by re-ordering the rows and columns, it shows more discernible structure and information for human perception.

Friday, November 1, 2013

VisuMap version 4.0.895 released

We have just released VisuMap version 4.0.895. The following is a list of major changes:
  • Implemented the multidimensional sorting and on-fly mapping & clustering services.
  • Extended the RPM algorithms with dark-pressure mechanism.
  • Reduced memory usage of the application significantly, so that it can load much large dataset.
  • Various defect fixing and enchantments

Saturday, October 19, 2013

On the shape of pictures

This blog post is going to talk about a simple way to convert normal 2D pictures or photos to high dimensional datasets; and then use MDS tool to analyze those data. At the end, I'll mention some potential uses of this type of analysis for data from practical cases.

Assuming that we have a picture that is represented by a MxN matrix of pixels, The pixel matrix can be simply converted to a matrix of real numbers by replacing each pixel with its intensity (or brightness) using the formula: a = 0.114*R + 0.587*G + 0.299*B; where R, G, B are the red, green and blue intensities of the pixel. We consider this MxN matrix as a N-dimensional dataset with M data points, where each data point is just a row vector in the matrix.

Obtained such a high dimensional dataset for our picture, we can then visualize this dataset with various MDS methods. The following picture shows visualizations of a sample photo with three different MDS methods, namely, PCA, tSNE and RPM:
We see that all three methods reveal, more or less, a serial structure among the data points (i.e. rows in the pictures.) When using MDS methods to visualize high dimensional data, the first question we ask is often: Are the maps reliable? Asked differently, Do those geometrical shapes in maps show some nature of the initial dataset, or are those shapes just random result of those algorithms? To answer this question, I have run RPM and tSNE multiple times, all runs produced, more or less, similar maps (the PCA algorithm is deterministic, will therefore always produce the same 2D map.) The following picture shows two more maps produced by RPM and tSNE algorithms:
Comparing above two pictures we see that both RPM and tSNE algorithms produced repeatable results by different runs despite their non-deterministic nature. Going one step further, we could ask what kind maps we'll get when we alter the sample dataset slightly. To answer this question, I have created a RPM map and a tSNE map with half of the rows and half of the columns by chosen alternately each other row and column of the matrix (i.e. only with 1/4 of the data matrix). The following picture shows the result maps:

Above maps show apparent similarity with the maps in previous picture. Thus, we have here strong evidence that the geometrical characteristics in these maps correlate with characteristics in the initial sample picture. We can see such correlation more clearly when exam how regions of the map represent sections in the sample picture. The following picture shows how some sections of the tSNE map correspond to sections in the sample picture:

We notice that MDS maps topologically have simple sequential structure, so no information are embedded into the topological structure. MDS maps rather carry information through geometrical characteristics. For instance, a section of smooth curves; a large arch; a section of points with less density; etc.;  Also, two distant sections winding together in particular way may indicate special feature of the underlying data. Furthermore, when we use 3D MDS maps, those secondary structure over the sequential structure may reveal much more rich complex information about the high dimensional data. Thus, MDS maps may offer new ways to describe and analyse the initial picture.

We might ask here: why are the MDS maps of the picture sequential?  To answer this question, let's image that we scan the picture from top to bottom, line by line, and store the line in a vector, then this vector will gradually change from one line to the next line. Thus, this line vector manifest a kind random-walk in the high dimensional space mostly with relatively small steps. Those random-walk alike datasets exist in many areas of data analysis. For instance, a while ago I have blogged about one such case where the state of the whole stock market has been considered as an random-walk process. We notice also that the MDS maps shown above resemble, more or less, those map in the blog self-similarity of high dimensional random-walk.

The following picture shows the price history of  S&P 500 components for a year, and the corresponding RPM map where each dot represents the state of the stock market at a day:

We notice the apparent similarity of the above RPM map with that produced for our sample picture previously. Thus, notions and methods developed for analyzing pictures may apply to a broader scope of data.

One interesting question arises in view of above picture is: Can we produce 2D picture (maybe not as nice as the photo of Lena but with easier recognizable features) out from a high dimensional dataset like the S&P 500 dataset? The answer is pretty much positive, and I'll blog about this in another post where I will talk about about multidimensional sorting.

At last, the sample picture and maps in this blog is produced by VisuMap version 4.0.895 and the sample dataset can be downloaded from the link here. The following short video shows some basic steps to apply mentioned services: