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Octave clustering demo part 5: hierarchical clustering

[This post is part of the Octave clustering demo series]

So, if you read this you probably have already unpacked the tgz file containing the demo and read the previous article about k-medoids clustering. If you want to check out the older Octave clustering articles, here they are: part 0 - part 1 - part 2 - part 3.

Run the hierarchicalDemo script (if launched without input parameters, it will open the "crescents.mat" data file by default). The script will plot the dataset, then it will try to perform clustering using first k-means and then an agglomerative hierarchical clustering algorithm. Run the algorithm with all the datasets available (note that you will need to modify the code to make it work with different numbers of clusters).

Questions:

1. how does k-means perform on the non-globular datasets (crescents, circles01, and circles02)? What is the reason of this performance? how does hierarchical algorithm perform instead?
2. now let us play with the "agglomerative" dataset. Information is missing so you know neither the number of clusters in advance nor the ground truth about points (i.e. which class they belong to). First of all, run the hclust function with different numbers of clusters and plot the results. Are you satisfied with them? What do you think is the reason of this "chaining" phenomenon?
3. complete the code in hclust.m so it also calculates complete and average linkage, and run the algorithm again. How do clustering results change? Plot and comment them.

Octave clustering demo part 4: k-Medoids

[This post is part of the Octave clustering demo series]

Here follows the description of the first of two new Octave clustering demos. If you have attended the PAMI class this year you probably know what I am talking about. If instead you are new to these demos please (1) check the page linked above and (2) set up your Octave environment (feel free to download the old package and play with the old demos too ;-)). Then download this new package.

Run the kMedoidsDemo script (if launched without input parameters, it will open the "noisyBlobs01.mat" data file by default). The script will plot the dataset, then it will try to perform clustering using first k-means and then k-medoids. Run the experiment many times (at least 4-5 times) for each of the three noisyBlobs datasets (you can pass the dataset path and name as a parameter to the kMedoidsDemo function).

Questions:

1. comment the "worst case" results you get from k-means: what happens and how does noise influence this behavior?
2. compare the results you get with k-means with the ones you get with k-medoids: how does k-medoids deal with the presence of noise?
3. what would happen if noise points were much farther from the original clusters? Try to get few (1-2) of them and bring them very far from the rest of the data… what are the results you get from k-means and k-medoids?

Hints:

• the demo stops at each step waiting for you to press a key. You can disable this feature by setting the "interactive" variable to zero at the beginning of the script;
• there are two different k-means implementations you can run: feel free to try both just by uncommenting their respective code.

Octave clustering demo part 3: spectral clustering

[This post is part of the Octave clustering demo series]

This is (currently) the last of my posts devoted to clustering demos. If you haven't downloaded the file containing Octave code yet, you can get it here. If you want to read the previous articles, here they are: part 0 - part 1 - part 2.

The last demo in the package shows you how spectral clustering works. Spectral Clustering is a relatively recent approach to clustering that I found very interesting, especially because it addresses some of the limitations of k-means but, at the same time, it relies on k-means itself! What it does in practice is finding a new space where localities are preserved, emphasizing both the cohesion and the separation of clusters, thus allowing k-means to work at its best. If you are curious and want to delve deeper into this topic, you will find more material in the following page: Paper Review: A Tutorial on Spectral Clustering.

I was quite surprised not to find ready code for spectral clustering in Octave, but probably I just had not searched (well) enough. By the way, if you came here searching for it here is an(other?) implementation, together with some examples and data to play with: feel free to try it and to leave your feedback!

The spectralDemo function can be called as follows:

    spectralDemo(dataset,nn,t)

where dataset is the usual dataset path+name, while nn and t are parameters used to build the (weighted) adjacency matrix describing similarity relationships between data points. Respectively, nn specifies the number of nearest neighbors for a point (a point is connected to another if it appears as one of its top nn nearest neighbors) and t is the parameter for the Gaussian similarity function s(xi,xj)=exp(-(xi-xj)^2 / t). My suggestion is to play with both parameters, but once you understood how they work you can just keep t=0 for the automatic tuning of the kernel and focus only on nn. Good values of nn range between 10 and 40, but of course you are free to test others (but keep in mind that the bigger the value, the slower the whole algorithm is! Can you say why?). The script performs the following tasks:

• it calculates the weighted adjacency matrix and plots both the distances between points and the weights between nodes in the adjacency graph (this allows one to inspect whether parameters need to be trimmed to have a more/less connected graph, and also to understand when the graph approach might work better than a classical euclidean distance approach);
• it builds the graph Laplacian and calculates its eigenvalues and eigenvectors. Then, it plots eigenvalues for inspection (this allows one to understand --if the dataset is clean enough-- how many connected components/natural clusters are present in the graph);
• to compare results between k-means and spectral clustering, it runs k-means both in the euclidean space and in the eigenspace built out of the top k eigenvectors, then shows both clusterings with related SSEs and accuracies.

And now, here are the usual maieutical questions:

1. what are the datasets in which spectral clustering performs definitely better than k-means?
2. how much do the results of spectral clustering depend on the input parameters?
3. how would you evaluate the quality of a labeling done with spectral clustering? How would you apply that to find which is the best number of nearest neighbors to take into account?
4. how often (according to your tests with different datasets) does the rule "the number of 0-valued eigenvalues matches the number of clusters" holds? How ofted does its "relaxed version" hold? (if the question is not clear check spectral clustering notes/papers)
5. how worth would be to use algorithms other than k-means to do clustering in the eigenspace?

That's all... have fun with the demos and with spectral clustering :-)

Octave clustering demo part 2: cluster evaluation

[This post is part of the Octave clustering demo series]

So, if you read this you probably have already unpacked the tgz file containing the demo, read the previous article, and ran demos 1 and 2. This post describes in detail the third demo, devoted to cluster evaluation.

In the previous article you saw how you can use an internal evaluation index to get better results from k-means. The main difference between internal and external indices is that the former (e.g. SSE) do not require additional information (such as the original labeling of data points), while the latter (e.g. accuracy) typically rely on that. SSE was used to evaluate the quality of different k-means executions, then the result providing the lowest SSE was considered the best one.

All of this was done supposing we already knew the correct number of clusters in the dataset (the variable clusters we used was actually provided, together with the data points and their classes). In the real world, however, this is not always true: you will typically lack not only the original classification (which is obvious, otherwise you would not need to run a clustering algorithm!), but also the best number of groups you want to divide your data into. So, you will have to find it... But what does best mean, in this case?

In kMeansDemo3 we employ the same SSE function to compare different executions of k-means. These are the main steps performed by the script:

• for k varying between 2 and 10
• run k-means 10 times (to address the random initialization problem, by choosing the lowest-SSE clustering)
• save the best (lowest) SSE and the matching clustering result
• plot the best SSE values for k=2, 3, ..., 10
• allow the user to decide which k is best, then plot that result and show both its SSE and its accuracy.

… Yes, there is nothing automatic in this :-). The reason is that I wanted you to get a grasp of how SSE changes with k and how to interpret it. This is strongly connected to what I wrote early about what best means in this case.

If you look at the SSE plot, you will see that it always decreases with increasing values of k. Which is reasonable, if you think that by increasing the number of clusters you will have points which are closer and closer to their centroids, up to the case when every point is the centroid of its own 1-point cluster and SSE becomes zero. This is totally different from the case in which the number of clusters was given and we just had to take the lowest-SSE results. How can we choose, then, the best k?

By looking again at the SSE plot, you will notice that the value of SSE always decreases, but for some values of k the change is bigger than for others. To find the best k, what we should look for is where the decrease changes most, that is the "elbow" inside the plot where the difference in change is maximum. Of course, this is quite easy to spot visually but not always trivial in terms of calculations. Some code that roughly does the trick follows:

    diffs = [sse(2:end) 0]-sse;    % calculates differences in SSE between each (k,k-1) pair
    ratios = [diffs(2:end) 0]./diffs;    % calculate ratios between differences
    k = find(ratios == min(ratios(ratios>0))) + 1;    % get lowest ratio (constrained to be >0 to ignore case k=1 and k=max)

... however, there are still cases (even in the provided datasets) where it does not perform well, especially when k-means is not the best algorithm for a given dataset. Run this demo both with blobs and circles/crescents datasets and try to answer the following questions:

1. were you always able to calculate the correct number of clusters using the code provided above?
2. were you always able to visually detect the correct number of clusters by looking at the SSE plot?
3. was the value of k you estimated confirmed by the final plot/the accuracy calculated against the ground truth?
4. how would you fix the code provided above to automatically estimate k even in the "blobs03" example? (note: if the code already gives you the correct k do not worry, just skip this question (see comments below ;)).
5. how would you fix the code provided above to automatically estimate k in the circles/crescents examples?

This time I really wrote a lot of questions :) Don't worry, some of them will be (partially) answered in the next tutorial!