KMeans as a classifier for the WIFI and MNIST datasets – V – cluster based classification of the MNIST dataset

In this series about KMeans

KMeans as a classifier for the WIFI and MNIST datasets – I – Cluster analysis of the WIFI example
KMeans as a classifier for the WIFI and MNIST datasets – II – PCA in combination with KMeans for the WIFI-example
KMeans as a classifier for the WIFI and MNIST datasets – III – KMeans as a classifier for the WIFI-example
KMeans as a classifier for the WIFI and MNIST datasets – IV – KMeans on PCA transformed data

we have so far studied the application of KMeans to the WIFI dataset of the UCI Irvine. We now apply the Kmeans clustering algorithm to the MNIST dataset – also in an extended form, namely as a classifier. The MNIST dataset – a collection of 28x28px images of handwritten numbers – has already been discussed in other sections of this blog and is well documented on the Internet. I, therefore, do not describe its basic properties in this post. A typical image of the collection is

Load MNIST – dimensionality of the feature space and scaling of the data

Due to the ease of use, I loaded the MNIST data samples via TF2 and the included Keras interface. Otherwise TF2 was not used for the following experiments. Instead the clustering algorithms were taken from „sklearn“.

Each MNIST image can be transformed into a one-dimensional array with dimension 784 (= 28 * 28). This means the MNIST feature space has a dimension of 784 – which is much more than the seven dimensions we dealt with when analyzing the WIFI data in the last post. All MNIST samples were shuffled for individual runs.

Scaling of MNIST data for clustering?

A good question is whether we should scale or normalize the sample data for clustering – and if so by what formula. I could not answer this question directly; instead I tested multiple methods. Previous experience with PCA and MNIST indicated that Sklearn’s „Normalizer“ would be helpful, but I did not take this as granted.

A simple scaling method is to just divide the pixel values by 255. This brings all 784 data array elements of each image into the value range [0,1]. Note that this scaling does not change relative length differences of the sample vectors in the feature space. Neither does it shift or change the width of the data distribution around its mean value. Other methods would be to standardize the data or to normalize them, e.g. by using respective algorithms from Scikit-Learn. Using either method in combination with a cluster analysis corresponds to a theory about the cluster distribution in the feature space. Normalization would mean that we assume that the clusters do not so much depend on the vector length in the feature space but mainly on the angle of the sample vectors. We shall later see what kind of scaling helps when we classify the MNIST data based on clusters.

In a first approach we leave the data as they are, i.e. unscaled.

Parameters for clustering

All the following cluster calculations were done on 3 out of 8 available (hyperthreaded) CPU cores. For Kmeans and MiniBatchKMeans we used

n_init       = 100       # number of initial cluster configurations to test 
max_iter     = 100       # maximum number of iterations  
tol          = 1.e-4     # final deviation of subsequent results (= stop condition)  
random_state = 2         # a random state nmber for repeatable runs
mb_size      = 200       # size of minibatches (for MiniBatchKMeans) 

The number of clusters „num_clus“ was defined individually for each run.

Analysis by KMeans? Too expensive …

A naive approach to perform an elbow analysis, as we did for the WIFI-data, would be to apply KMeans of Sklearn directly to the MNIST data. But a test run on the CPU shows that such an endeavor would cost too much time. With 3 CPU cores and only a very limited number of clusters and iterations

n_init   = 10      # only a few initial configurations
max_iter = 50 
tol      = 1.e-3  
num_clus = 25      # only a few clusters

a KMeans fit() run applied to 60,000 training samples [len(X_train) => 60,000]

kmeans.fit(X_train)

requires around 42 secs. For 200 clusters the cluster analysis requires around 214 secs. Doing an elbow analysis would therefore require many hours of computational time.
To overcome this problem I had to use MiniBatchKMeans. It is by factors > 80 faster.

Elbow analysis with the help of MiniBatchKmeans

When we use the following setting for MiniBatchKMeans

n_init = 50 # only a few initial configurations
max_iter = 100
tol = 1.e-4
mb_size = 200 

I could perform an elbow analysis for all cluster-numbers 1 < k <= 250 in less than 20 minutes. The following graphics shows the resulting intertia curve vs. cluster number:

The „elbow“ is not very pronounced. But I would say that by using a cluster number around 200 we are on the safe side. By the way: The shape of the curve does not change very much when we apply Sklearn’s Normalizer to the MNIST data ahead of the cluster analysis.

Classifying unscaled data with the help of clusters

We now perform a prediction of our adapted cluster algorithm regarding the cluster membership for the training data and for k=225 clusters:

n_clu    = 225
mb_size  = 200
max_iter = 120
n_init   = 100
tol      = 1.e-4

Based on the resulting data we afterward apply the same type of algorithm which we used for the WIFI data to construct a „classifier“ based on clusters and a respective predictor function (see the last post of this series).

The data distribution for the 10 different digits of the training set was:

class 0 :  5905
class 1 :  6721
class 2 :  6031
class 3 :  6082
class 4 :  5845
class 5 :  5412
class 6 :  5917
class 7 :  6266
class 8 :  5860
class 9 :  5961

How good is the cluster membership of a sample for a digit class defined?
Well, out of 225 clusters there were only around 15 for which I got an „error“ above 40%, i.e. for which the relative fraction of data samples deviating from the dominant class of the cluster was above 40%. For the vast majority of clusters, however, samples of one specific digit class dominated the clusters members by more than 90%.

The resulting confusion matrix of our new „cluster classifier“ for the (unscaled) MNIST data looks like

[[5695    4   37   21    7   57   51   15   15    3]
 [   0 6609   33   21   11    2   15   17    2   11]
 [  62   45 5523  120   14   10   27  107  116    7]
 [  11   43  114 5362   15  153    8   60  267   49]
 [   5   60   62    2 4752    3   59   63    5  834]
 [  54   18  103  777   25 4158  126    9  110   32]
 [  49   20   56    4    6   38 5736    0    8    0]
 [   5   57   96    2   86    1    0 5774    7  238]
 [  30   76  109  416   51  152   39   35 4864   88]
 [  25   20   37   84  706   14    6  381   46 4642]]

This confusion matrix comes at no surprise: The digits „4“, „5“, „8“, „9“ are somewhat error prone. Actually, everybody familiar with MNIST images knows that sometimes „4“s and „9“s can be mixed up even by the human eye. The same is true for handwritten „5“s, „8“s and „3“s.

Another representation of the confusion matrix is:

The calculation for the matrix elements was done in a standard way – the sum over percentages in a row gives 100% (the slight deviation in the matrix is due to rounding). I.e. we look at erors of the type TN (True Negatives).

The confusion matrix for the remaining 10,000 test data samples is:

The relative errors we get for our classifier when applied to the train and test data is

rel_err_train = 0.115 ,
rel_err_test = 0.112

All for unscaled MNIST data. Taking into account the crudeness of the whole approach this is a rather convincing result. It proves that it is worth the effort to perform a cluster analysis on high dimensional data:

  • It provides a first impression whether the data are structured in the feature space such that we can find relatively good separable clusters with dominant members belonging to just one class.
  • It also shows that a cluster based classification for many datasets cannot reach accuracy levels of CNNs, but that it may still deliver good results. Without any supervised training …

The second point also proves that the distance of the data points to the various cluster centers contains valuable information about the class membership. So, a MLP or CNN based classification could be performed on transformed MNIST data, namely distance vectors of sample datapoints to the different cluster centers. This corresponds to a dimension reduction of the classification problem. Actually, in a different part of this blog, I have already shown that such an approach delivers accuracy values beyond 98%.

For MNIST we can say that the samples define a relatively well separable cluster structure in the feature space. The granularity required to resolve classes sufficiently well means a clsuter number of around 200 < k < 250. Then we get an accuracy close to 90% for cluster based classification.

t-SNE representation of the MNIST data

Can we somehow confirm this finding about a good cluster-class-relation independently? Well, in a limited way. The t-SNE algorithm, which can be used to „project“ multidimensional data onto a 2-dimensional plane, respects the vicinity of vectors in the original feature space whilst deriving a 2-dim representation. So, a rather well structured t-SNE diagram is an indication of clustering in the feature space. And indeed for 10,000 randomly selected samples of the (shufffled) training data we get:

The colorization was done by classes, i.e. digits. We see a relatively good separation of major „clusters“ with data points belonging to a specific class. But we also can identify multiple problem zones, where data points belonging to different classes are intermixed. This explains the confusion matrix. It also explains why we need so many fine-grained clusters to get a reasonable resolution regarding a reliable class-cluster-relation.

Classifying scaled and normalized MNIST data with the help of clusters

Can we improve the accuracy of our cluster based classification a bit? This would, e.g., require some transformation which leads to a better cluster separation. To see the effect of two different scalers I tried the „Normalizer“ and then also the „StandardScaler“ of Sklearn. Actually, they work in opposite direction regarding accuracy:

The „Normalizer“ improves accuracy by more than 1.5%, while the „Standardizer“ reduces it by almost the same amount.

I only discuss results for „Normalization“ below. The confusion matrix for the training data becomes:

and for the test data:

The relative error for the test data is

Error for trainings data:
avg_err_train = 0.085 :: num_err_train = 5113
 
Error for test data:
avg_err_test = 0.083 :: num_err_test = 832

So, the relative accuracy is now around 91.5%.
The result depends a bit on the composition of the training and the test dataset after an initial shuffling. But the value remains consistently above 90%.

Data compression by Autoencoders and clustering

Just for interest I also had a look at a very different approach to invoke clustering:

I first applied a simple CNN-based AutoEncoder [AE] to compress the MNIST data into a 25-dimensional space and applied our clustering methods afterwards.

I shall not discuss the technology of autoenconders in this post. The only relevant point in our context is that an autoencoder provides an efficient non-linear way of data compression and dimensionality reduction. Among many other useful properties and abilities … . Note: I did not use a „Variational Autoencoder“ which would have allowed for even better results. The loss function for the AE was a simple quadratic loss. The autoencoder was trained on 50,000 training samples and for 40 epochs.

A t-SNE based plot of the „clusters“ for test data in the 25-dimensional space looks like:

We see that the separation of the data belonging to different classes is somewhat better than before. Therefore, we expect a slightly better classification based on clusters, too. Without any scaling we get the following confusion data:

[[5817    7   10    3    1   14   15    2   27    1]
 [   3 6726   29    2    0    1   10    5   12   10]
 [  49   35 5704   35   14    4   10   61   87    7]
 [   8   78   48 5580   22  148    2   40  111   29]
 [  47   27   18    0 4967    0   44   38    3  673]
 [  32   20   10  150    8 5039   73    4   43   28]
 [  31   11   23    2    2   47 5746    0   15    1]
 [   6   35   35    6   32    0    1 5977    7  163]
 [  17   67   22   86   16  217   24   22 5365   52]
 [  35   32   11   92  184   15    1  172   33 5406]]

Error averaged over (all) clusters :  6.74

The resulting relative error for the test data was:

avg_err_test = 0.0574 :: num_err_test = 574

With Normalization:

Error for test data:
avg_err_test = 0.054 :: num_err_test = 832

So, after performing the autoencoder training on normalized data we consistently get

an accuracy of around 94%.

This is not too much of a gain. But remember:
We performed a cluster analysis on a feature space with only 25 dimensions – which of course is much cheaper. However, we paid a prize, namely the Autoencoder training which lasted about 150 secs on my old Nvidia 960 GTX.

And note: Even with only 100 clusters we get above 92% on the AE-compressed data.

Conclusion

We have shown that using a non-supervised cluster analysis of the MNIST data with around 225 clusters allows for classifying images with an accuracy around 90.5%. In combination with an Autoencoder compression we even reaches values around 94%. This is comparable with other non-optimized standard algorithms aside of neural networks.

This means that the MNIST data samples are organized in a well separable cluster structure in their feature space. A test run with normalized data showed that the clusters (and their centers) differ mostly by their direction relative to the origin of the feature space and not so much by their distance from the origin. With a relatively fine grained resolution we could establish a simple cluster-class-relation which allowed for cluster based classification.

The accuracy is, of course, below the values reachable with optimized MLPS (98%) and CNNs (above 99%). But, clustering is a fast, reliable and non-supervised method. In addition in combination with t-SNE we can create plots which can easily be understood by the customers. So, even for more complex data I would always recommend to try a cluster based classification approach if you need to provide plots and quick results. Sometimes the accuracy may even be sufficient for your customer’s purposes.

KMeans as a classifier for the WIFI and MNIST datasets – IV – KMeans on PCA transformed data

In the last posts of this series

KMeans as a classifier for the WIFI and MNIST datasets – I – Cluster analysis of the WIFI example
KMeans as a classifier for the WIFI and MNIST datasets – II – PCA in combination with KMeans for the WIFI-example
KMeans as a classifier for the WIFI and MNIST datasets – III – KMeans as a classifier for the WIFI-example

we applied the KMeans algorithm to perform a cluster analysis of the WIFI dataset of the UCI Irvine. The results gave us insights into the spatial grouping and the separability of the data samples in their 7-dimensional feature space. An additional PCA analysis helped to understand why projections of the data into some selected 2-dimensional sub-spaces of the feature space revealed the four or five dominant clusters very well. In the third post I discussed a simple method to transform KMeans into a classifier. In the WIFI case a set of 9 to 11 clusters provided a good resolution of the data distribution and we reached a convincing classifier accuracy.

What we have not done, yet, is to transform and project the WIFI data into the coordinate system of the most important main components and afterward apply clustering by the help of KMeans. We know already that three primary components fit the data very well and give us around 90% of the „explained variance„. See the second post for these basic PCA results. We, therefore, expect comparably accurate prediction results of a cluster classifier for the PCA transformed data as the accuracy values given in the last post. For 500 test samples after a KMeans fit of 1500 training samples in the original feature space we found a prediction accuracy of around 98%.

In this post we first perform a PCA analysis for three primary components of the WIFI data distribution and then transform the vectors of 1500 randomly selected training samples to the 3-dimensional main component space. Then we apply KMeans onto the data in the reduced vector space and establish a classifier predictor based on the methods described in the last article. Eventually, we check the accuracy and display the resulting confusion matrix for the 500 test samples.

As a side-step for readers who look for real world use cases regarding signals I want to mention an article in „Nature“, which I found today via a newspaper podcast. There neural firing rates of a brain region, i.e. some very special signals, were used to enable an ALS patient to select letters from presented sequences and form statements – by his „thoughts“. This looks like an environment where Machine Learning really could contribute more in the future.

KMeans as a classifier on the PCA transformed WIFI data

Below I give you the results for the WIFI data transformed and projected to the most important three primary components and 11 clusters:

Results for 1500 training samples

  
Confusion matrix for training data - 11 clusters, 3 PCA components 
A confusion matrix for the classes according to the clustering
[[374   0   1   0]
 [  0 362  13   0]
 [  4   7 359   5]
 [  1   0   0 374]]

Number of wrongly predicted train samples:  31  :: avg_err =  0.020

So, just from counting wrongly classified examples the average error is measured to be around 2% and the relative accuracy is something like 98%.

And for the test data I got:

Number of wrongly predicted test samples:  6  :: avg_err =  0.012

This gives us the following confusion matrix:

This actually proves that our assumption about combining a PCA transformation with a KMeans classifier was correct. The reduction of the dimensionality of the problem did not affect the prediction accuracy very much.

Just for completeness the data for only 2 primary components:

The accuracy of around 97% is still convincing. The reason is that the two most important primary components already deliver around 85% of the „explained variance“.

Why is the Wifi-example not so boring as one may think?

A reader wrote me that he finds the WIFI example too simple and boring. OK, but … The principles and methods remain the same when more complex data are analyzed for clusters. Especially in the case of binary classification. But are there interesting real world use cases for other types of signals? Oh, yes. I just want to refer to an interesting example which I read about this morning.

The WIFI example works with samples which describe 7 signals. Now, imagine that such signals come from a sensor implant measuring electric potentials of a human brain and that we do not analyze for the location of rooms but for the selection of letters by „Yes/No“ decision-„imaginations“ – made by a human who was trained via frequency based audio-feedback for the brain regions covered by the implants. Science fiction? No, reality. And of huge help for ALS patients. See

Chaudhary, U., Vlachos, I., Zimmermann, J.B. et al. Spelling interface using intracortical signals in a completely locked-in patient enabled via auditory neurofeedback training. Nat Commun 13, 1236 (2022). https://doi.org/10.1038/s41467-022-28859-8

and
https://www.nature.com/articles/s41467-022-28859-8

There, signals were measured from two implant arrays with 64 electrodes. OK, these are somewhat more signals than just 7. But if I understood the text correctly not all channels were used or useful. Just a few. Reminds us of PCA? In addition the time structure of the signal (firing rates) are important – but these are just different signal characteristics. And we have different labels. But, at least in principle, we speak of nothing else than pattern detection based on signal values.

I only had a brief look into the supplementary data of the experiment (an Excel file) and I am not at all familiar with the the experimental setup – but from reading my impression was that just threshold values for the firing rate of some channels were used to distinguish „Yes“ from „No“. Maybe we could do a bit better with AI (PCA and classifying according to multidimensional pattern analysis)? Does this look like an interesting use case?

Conclusion

In the case of the WIFI example KMeans can be used as an efficient classifier for samples in a feature space which describes characteristics of multiple signal sources. We have seen that the basic concept also works when we apply KMeans after a PCA based transformation to the most important primary components.

The question is: Does this work equally well for other data sets? The answer depends upon the accuracy by which clusters reside completely within regions of the feature space filled by samples of a specific label.
A data set whose samples show grouping in a multidimensional feature space and appear relatively well separable by their labels is the MNIST data set. In the next post of this series we shall therefore try and apply a clustering algorithm to the MNIST data ensemble.

Stay tuned …

Ceterum censeo: The worst fascist today who must be isolated and denazified is the Putler.