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

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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.

KMeans as a classifier for the WIFI and MNIST datasets – III – KMeans as a classifier for the WIFI-example

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In the previous articles 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

I applied KMeans to the “WIFI” dataset – a small and rather simple training set of the UCI Irvine. The 7 dimensional feature space for this example is defined by the strength of seven different WLAN-signals. The data samples are labeled by numbers specifying four different rooms. We just have 2000 samples. But “simple” does not mean that one cannot learn something of it.

An elbow and a silhouette analysis indicated that the data can well be described by 4 to 5 clusters.
A detailed PCA analysis helped us to understand that the data could basically be described by two primary components whose axes were defined by a diagonal in a two-dimensional sub-space of the original feature space (defined by the features “WLAN-0” and “WLAN-3”) plus an orthogonal axis (defined by “WLAN-4”). As the data points segregated into well separated clusters in the coordinate system of the primary components we could expect that they also segregate well when projected onto two 2-dim sub-spaces of the feature space defined by the signal combinations {WLAN-0/WLAN-4} and {WLAN-3/WLAN-4}.

So we projected the results of a KMeans analysis into a 2-dim sub-space of the 7-dim feature space spanned by two selected “features” (WLAN signals WLAN-0 and WLAN-4). And indeed: For 2 special signal- or feature-combinations we saw a field with 4 to 5 well separated clusters.

As we have well separated clusters – either in sub-planes of the feature space or the space defined by the dominant two PCA components – a legitimate question is: Can we use the cluster information for classifying?

Clusters and classification

In the special example of the WIFI data the data points can obviously be divided into different “groups” filled by data points which belong to a certain “label” or class (in the WIFI case: a room):

In the plots above the colorization of the data points is given by their label in the plot above. However:
Does this mean that these spatially separated “groups” coincide with the clusters detected by KMeans?

In the next plots I superimposed the data points – first colorized by class and then by cluster with different colors and a slight spatial deviation.

We see that the border of the class related points on average overlaps well with the border of the clusters. Only a few points show a major deviation.

Is this always the case? Of course not!

We see this already when we we just use 2 clusters and colorize the data points according to cluster membership in one of the two clusters:

Oops, all the beauty is gone! Members of the previously visual dark-red, green and pink “clusters” would now certainly be members of the first defined “green” cluster. And members of the visual pink and orange clusters would now be members of the “defined” blue cluster. So: Labels may but do strong>not always define clusters.

An important thing we learned by this consideration is that we should at least use a number of cluster greater or equal the number of labels.

But even then the separation may not work. Let us imagine three rooms: In room “A” we only have male persons, in room “B” only female persons and in the third room “C” female persons on one side of the room and male persons on the other – like in some old fashioned dancing courses. If we used the persons’ coordinates to define features and the “male”/”female” attributes as labels and tried a cluster analysis in the feature space we would clearly identify three clusters. However, in the cluster for room “C” we would find a mixture of samples with different labels. Spatial vicinity in the feature space does not mean class identity and a label does not necessarily mean a big distance in the feature space.

Hyperplanes and clusters

The only thing you can hope for with respect to a solvable ML problem is that the data points for different labels may be distributed such that a complicated and curved hyperplane can be found in the multidimensional feature space which separates groups of data points with the same label sufficiently well. But this hyperplane does not necessarily coincide with fictitious borders of some clusters identified by KMeans. We are lucky if the topologically closed surface of a cluster lies completely in a region of the hyperspace separated by complex and topologically open hyperplanes separating data points belonging to one class from other feature space regions which contain data points with different labels.

Not all data ensembles in ML will fall apart in extended clusters each of which dominated by some specific label. Ring like distributions of data samples with identical labels may pose a problem to cluster algorithms – mot only in 2 dimensions.

On the other side: If the labeled data – after some useful transformation – are not well separable in the feature space, then the posed ML task is problematic anyway and the chosen feature definitions may not be appropriate.

Granularity: Hope for label dominance in sufficiently fine grained clusters

What does a label-cluster correlation depend on? Well, if there is an important factor regarding the position of the centroids of KMeans then it is the number of clusters. In the extreme case of as many clusters as data points a super-well defined cluster/label-association exists – but it is not of much use for ML-task. It just represents the most extreme case of overfitting you can think of. It will be useless for new unknown samples.

But the example with the distribution of male/female samples gives you an indication of what we should try: With a growing number of cluster the chances for a fine grained separation into clusters residing on one side of a hyperplane separating samples of different labels rises and with it the chance for a clearer dominance of one label per cluster. This is even true for ring like data distributions: With more then 4 clusters we may even describe a ring like data distribution quite well.

Therefore: If the sample distribution has some reasonable separation hyperplane at all there is a chance that you may find a number of clusters for which each cluster is dominated by a specific label.

In the case of the WIFI example this is pretty obvious. The following first plot shows the data points colorized according to their labels:

The next plot shows four cluster – with data points colorized according to their cluster membership:

The colors are different (we have no control of it) – but the different data point “clouds” in the pictures coincide rather well. Only some data points do not behave well. We can again use the trick we iused in 2 dimensions and superimpose data points with colorization according to cluster and class:

How do we define a classifier by the help of KMeans?

We need a predict()-function which predicts a label from a predicted cluster membership. The recipe is rather simple:

Define the number of cluster you want to use.

  • Use KMeans for a cluster analysis of a training set of samples.
  • Predict the cluster-membership of a sample with the help of the fitted kmeans-object and its predict()-function.
  • Get the labels of the samples belonging to a specific cluster.
  • Find out what the amount of samples with a certain label for a cluster is and compared the data.
  • Find the label which contributes most samples to a cluster. Check the relative amount of deviating data points with other labels. Should be sufficiently small…
  • Build a dictionary which associates the cluster number with its dominant label.
  • Build a prediction function for yet unknown data points. It first predicts the cluster and then the associated label.

You should then test the accuracy of your new cluster-based classifier model on test data.

More clusters than labels?

What will happen to such an algorithm if you use more clusters than labels? Short answer: Nothing – as long as each cluster is really dominated by a specific label. More precisely: A vast majority of your clusters should exhibit contributions from data points with one specific label to an amount significantly far beyond the statistical average. Having more clusters means that under reasonable conditions we just have more than one cluster predicting a certain label. For the case of the WIFI example this means that we can work with five clusters without getting nervous.

Application to the WIFI example

As the clusters represent the labels quite well we expect very good values for the recall values. The experiment is pretty simple. I just present the resulting confusion matrices for the four labels (identifying rooms) and different numbers of clusters “nclus”. I first show you a presentation with seaborn, which explains, how to interpret rows and columns:

nclus = 5

Here I used five (5) clusters and included all samples in the calculation. I.e. I did not separate a training set from a test set of data points.

Confusion matrices for different numbers of clusters

nclus = 4

[[496   0   4   0]
 [  0 425  75   0]
 [  2   0 492   6]
 [  2   0   2 496]]
num of wrongly predicted samples:  91  :: avg_err =  0.0455

The “avg_err” gives you the number of wrongly predicted samples divided by the total number of samples. We see that 4 clusters have a problem with the differentiation of the data points for the room “Diele”.

nclus = 5

[[495   0   5   0]
 [  0 455  45   0]
 [  2   0 493   5]
 [  2   0   2 496]]
num of wrongly predicted samples:  61  :: avg_err =  0.0305

nclus = 6

[[499   0   1   0]
 [  0 452  48   0]
 [  5   0 490   5]
 [  2   0   2 496]]
num of wrongly predicted samples:  63  :: avg_err =  0.0315

nclus = 7

[[499   0   1   0]
 [  0 492   8   0]
 [  4  38 455   3]
 [  2   0   3 495]]
 num of wrongly predicted samples:  59  :: avg_err =  0.0295

nclus = 

num wrongly predicted samples:  58  :: avg_err =  0.029

nclus = 9

[[499   0   1   0]
 [  0 484  16   0]
 [  4  16 476   4]
 [  2   0   2 496]]
num of wrongly predicted samples:  45  :: avg_err =  0.0225

All data above were derived for the same random_state-variable to KMeans.

However, this result depends on the initial shuffling of the samples, the random_state parameter and the initial statistical distribution of the centroids by KMeans. And other hyperparameters … The next plot shows a slightly different result for nclus = 9:

nclus = 9

This means: Nine cluster give us a reasonably fine grained resolution in the case of the WIFI example.

But: Note that the problem areas in the confusion matrix have changed. “Diele” is handled a bit better, but “Wohnzimmer” is not so sharply separated as before. This is not astonishing as we saw a significant mix of data of different labels for different clusters above already:

After nclus = 9 we do not get much better – and dance around an average error of 0.025 => 2.5%:

nclus = 10

num of wrongly predicted samples:  55  :: avg_err =  0.0275

nclus = 11

num of wrongly predicted samples:  52  :: avg_err =  0.026

nclus = 25

num of wrongly predicted samples:  40  :: avg_err =  0.02

Being conservative we can say that our simple cluster based classifier approaches an accuracy around 97.4%. Not so bad regarding the crudeness of our approach! A random forest algorithm reaches something above 98.2%. This is not so much better.

Results after a separation of test data samples

I divided the data set than into 1500 train and 500 test samples.

For 9 clusters I got:

nclus = 9

[[374   0   1   0]
 [  0 362  13   0]
 [  3  17 352   3]
 [  2   0   2 371]]
num wrongly predicted train samples:  41  :: avg_err =  0.027333333333333334
num wrongly predicted train samples:  8  :: avg_err =  0.016

But these numbers depend strongly on the splitting – even if we split stratified. Another run gives:

nclus = 9

[[374   0   1   0]
 [  0 352  23   0]
 [  3   0 369   3]
 [  1   0   2 372]]
num wrongly predicted train samples:  33  :: avg_err =  0.022 
num wrongly predicted test samples:  11  :: avg_err =  0.022

Other runs may even give higher average error values. The variation depend upon of how many critical data points in the intermixing zone were omitted in the train samples. At least the results are not too different from the ones named above.

Conclusion

In this article I presented a very simple method by which a cluster algorithm can be used as a classifier. When we applied the approach to the rather simple WIFI example we saw that this worked pretty well.

What we in addition should try is to combine the classifier with a dimension reduction based on a PCA-analysis. From the results of a previous post we would expect that a cluster classifier should work well on the WIFI data after a transformation and projection of the samples’ data points into a 3-dimensional space spanned by the most important orthogonal main component axes. This is the topic of the next post in this series:

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

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

 

KMeans as a classifier for the WIFI and MNIST datasets – II – PCA in combination with KMeans for the WIFI-example

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I continue with my series of posts about using KMeans as a classifier for some simple ML datasets. In the last post

KMeans as a classifier for the WIFI and MNIST datasets – I – Cluster analysis of the WIFI example

I applied KMeans to the “WIFI” dataset which was discussed in the German “Linux Magazin”; see the Nov and Dec editions of 2021. We found 4 to 5 well separated clusters in a projection of the data points onto a 2-dimensional space defined by just two out of seven features.

I briefly discuss how this result is related to a PCA analysis. In my opinion the discussion in the “Linux Magazin” was a bit misleading regarding this point.

PCA analysis

A PCA analysis helps to identify the main orthogonal axes of the distribution of the samples’ data-points in their multidimensional “feature space”. If the data points for the samples are distributed in all directions and over all regions of the feature space in a similar way we may indeed need all of the feature spaces’s dimensions to describe the data distribution. Still, we could find some complicated curved hyperplanes which separate groups of data points with identical label quite well.

But often the data points are positioned along certain preferred directions, i.e. the data points are located along specific lines or (multidimensional) flat planes in the feature space – not withstanding an additional clustering. In such cases the data distribution may exhibit some intrinsic major axes in the feature space AND/OR the distribution may be confined to a subspace of the original feature space. The sub-space is defined and spanned by fewer axes than the original space. We speak of the “primary components” of the data distribution when we refer to the (orthogonal) axes of such a sub-space.

Of course, we can span the full original feature space by the most important primary component axes plus some extra orthogonal axes. I.e., we just need the same number of main components as the dimension of the original feature space. Then we get a new coordinate system which describes the vector space of the original features in a different way: The differences are

  • another orientation of the main component axes in comparison with the original axes,
  • a difference in the position of the origins of the two coordinate systems.

Of course, there is a mathematically well defined transformation which maps coordinates of data points with respect to the original feature spaces axes to coordinates in a coordinate system defined by the main component axes.

Of course, we can project a vector describing a data point onto unit vectors along the axes of either coordinate system. By using these special components we can describe the data distribution in terms of a sub-space spanned by only the most important primary component axes. A projection of ML data points to primary component axes is equivalent to a reduction of the dimensionality of the ML problem. Whenever we find that we can use fewer main component axes than the feature space’s number of dimensions to describe the data distribution reasonably well we can reduce the dimensionality of the problem: We project the original data vectors in the feature space to the axes of the main components’ coordinate system with fewer dimensions than the feature space.

How do we measure the “importance” of a main or primary component?

A metric for the importance of a primary component axis is its contribution to the so called “explained variance”. This quantity measures correlations of data in the original feature space and thus also the amount of information residing in the data.

From a mathematical point of view determining the main component axes corresponds to the diagonalization of the so called covariance matrix and the identification of eigenvectors. (You can find a short explanation in the books of S. Rashka on “Python Machine Learning”, 2016, PACKT or the book of J. Frochte on “Maschinelles Lernen”, 2019, Carl Hanser.) The eigenvectors define the directions of the main components’ axes. The variation of the data along such a main component axis contributes to the total variance by its measure of specific correlations, i.d. weighted quadratic data point distances. We are interested in finding those main component axes for which the projected data distribution explains most of the data variation.

Note that the axes which a PCA-analysis determines are orthogonal axes. Thus PCA describes a transformation of vectors in the feature space from one coordinate system with orthogonal axes to another coordinate system with orthogonal axes. Geometrically, this can be described by a sequence of a translation, followed by a rotation – and in case of a dimensionality reduction in addition by a projection.

Let us visualize an example. The following picture shows the surface of an asymmetric “ellipsoidal” data distribution in a 3-dim feature space. (Actually, the image does not display a real ellipsoid – but we ignore the differences for reasons of simplicity.) The dark arrows in the picture indicate the orthogonal axes and unit-vectors per dimension of the 3-dim feature space. The colored arrows of the ellipsoid show the direction of the main component axes of the “ellipsoid”.

Regarding the dimensions of the ellipsoid the elongation along the red vector is biggest. The width of the data point distribution in the direction of the blue vector is significantly smaller, but still bigger than in the direction of the green vector. So, we would expect that the two main component axes in the directions of the red and the blue arrows explain most of the data variation in the feature space. The projections of the data point vectors in a coordinate system defined by the main component axes onto the “green” axis would only give us rather small values. Therefore, we can reduce the dimensionality of the problem by describing the data distribution in only two dimensions: We project the vectors of the original data points in the feature space onto the two most important main component axes – in the directions of the red and the blue unit vectors.

Note: The axis in the direction of the dominant elongation of the ellipsoid has a diagonal orientation in the feature space. This means that all of the three original features contribute to the data points’ distribution in this direction. Therefore, the following point must be underlined:

A PCA analysis is not a selection process with respect to the original features. A vector describing a main component axis of the data distribution is a
linear combination of all unit vectors along various original axes of the feature-space. Normally many – if not all – original features contribute to a main component axis.. Without a detailed analysis you can not assume that only one or two original features determine the primary component axis.

In particular: You cannot assume that only one special feature dominates a main component axis. Unfortunately, the text in the Linux Magazin on the WIFI example could be read and interpreted in this way. So, let us have a closer look at the reason, why the projections of the WIFI-data from the original 7-dimensional feature-space onto a reduced 2-dim space of the signal-combinations [WLAN-4, WLAN-0] or [WLAN-3, WLAN-0] worked so well.

How many main components dominate the variance of the WIFI data distribution?

How many main components explain most of the variation of the samples’ data of the WIFI example? How do we get the required information about the contribution to the explained variance from PCA applications? Sklearn’s PCA implementation provides the usual “fit()”-function to perform the PCA calculation for a given data set. But it also provides an array named “explained_variance_” which contains the individual contributions of the main components to the “explained variance“.

So, as a first step, we simply try and apply Sklearn’s PCA()-function directly to the original WIFI data given in their 7-dimensional feature space. I.e. without any scaling. Just as it was done in the Linux Magazin. The bar plot below displays the “importance” of a maximum of 7 main components. The importance of each component is measured by its normalized contribution to the “explained variance”:

An accumulation gives the following percentages:

We see that only two of the main PCA components already explain 85% of the data variance in the feature space. We thus could choose these two main components as our “primary” components. But this does NOT automatically mean that only two features dominate the data distribution in the feature space.

Which features determine the primary components’ orientation?

As we have discussed above the projection of a unit vector along the main component axis onto the original axes of the feature-space may give similarly big values for each of the features. It is rather seldom that only a few features determine the direction of a main component’s axis.

But: The WIFI data are (on purpose?) indeed distributed in a very special way. Actually, only two original features determine the direction of the most important primary component’s axis already quite well. And just one feature dominates the second most important PCA component. So, in the WIFI example the first and the second main components are oriented more or less within a 2-dim feature plane and along a special feature axis, respectively. I.e. the data are more or less confined to a 3-dimensional sub-space of the feature space. How and where from did I get this information?

Sklearn’s function “PCA()” returns a reference to an object, which after a call to its method “fit()” has a filled property “components_“. This array gives us the 7 vector-components of each unit vector oriented along a PCA main component axis with respect to the various original axes of the feature space. I.e. we get the components of unit vectors along main component axes in terms of the original coordinate system spanning our feature space. The related coefficients of the vector tell us whether the main component axes are confined to a subspace spanned by just a few original features.

Below, the elements (rows) of the array were reversely sorted by the the contribution of the main component to the “explained variance”. So the first two rows correspond to the two most important PCA main components.

[6.22e-01 7.47e-04 1.03e-02 6.29e-01 2.02e-01 3.03e-01 2.91e-01]
 [1.71e-01 9.05e-02 4.31e-01 1.95e-01 8.50e-01 1.02e-01 7.62e-02]
 [2.41e-01 2.78e-01 2.18e-01 2.90e-01 9.80e-02 5.28e-01 6.67e-01]
 [1.67e-01 3.44e-01 7.05e-01 1.69e-01 4.22e-01 9.65e-02 3.75e-01]
 [1.13e-01 1.77e-01 3.15e-01 1.33e-01 1.82e-01 6.97e-01 5.66e-01]
 [6.37e-01 2.31e-01 3.28e-01 6.45e-01 1.26e-01 1.32e-02 3.22e-02]
 [2.80e-01 8.43e-01 2.50e-01 1.42e-01 2.43e-02 3.52e-01 5.12e-02]]

The first primary component has vector-components along the original axes of the feature space given by

6.22e-01, 7.47e-04, 1.03e-02, 6.29e-01, 2.02e-01, 3.03e-01, 2.91e-01

The features, i.e. the WLAN signals, are numbered in the given order from left to right. Obviously, the features “WLAN-0” and “WLAN-3” dominate the direction of the first primary component.

The second main component

1.71e-01, 9.05e-02, 4.31e-01, 1.95e-01, 8.50e-01, 1.02e-01, 7.62e-02

is instead dominated by he feature “WLAN-4”.

Actually, a closer look shows that the signal “WLAN-6” dominates the third main component by a relatively big value. This might be an indication that we had better used three primary components instead of two … I come back to this point below.

So, what do we learn from the results of the projections of the PCA unit vectors onto the axes of the feature space?

  1. In the very special case of the WIFI example around 3 original features dominate the overall data distribution.
  2. In the WLAN-0/WLAN-3 plane we should see an approximate diagonal distribution of data. Reason: the vector components are of almost equal size.
  3. As we already know from an elbow-analysis we have 4 to 5 clusters. So, we should see them clearly in a 3D-plot for the axes WLAN-0, WLAN-3 und WLAN-4.
  4. If the data are well separated into clusters along the diagonal in the WLAN-0/WLAN-3 plane AND the WLAN-4 direction then they will also be well separated in the 2-dim space of the 2 main PCA components.

Ok, let us visualize it. The next plot shows the data distribution from a view almost perpendicular to the WLAN-3/WLAN-4 plane. The colors indicate the labels of the data (i.e. the rooms where the strength of each of the 7 WLAN signals has been measured).

3D-plot of the WIFI data distribution in the space of the dominant 3 original features – the WLAN-3/WLAN-4 plane

We already see the clustering. The next plot shows the data distribution from a direction almost perpendicular to the WLAN-0/WLAN-4 plane.

3D-plot of the WIFI data distribution in the space of the dominant 3 original features – the WLAN-0/WLAN-4 plane

And now a view from above showing the diagonal distribution of very many data points. We clearly see that there is something strange going on in the “orange” room.

3D-plot of the WIFI data distribution in the space of the dominant 3 original features – the WLAN-3/WLAN-0 plane

So far, so good! We have again identified the clusters which we already got familiar with in my last post.

Data distribution in the vector space of the three most important PCA components

In full consistency with the results derived above we expect a good cluster separation in the plane of the first two main PCA components. These components are called “PCA-1” and “PCA-2” in the following plot:

3D-plot of the transformed WIFI data distribution in the space of the dominant 3 PCA components – the PCA-1/PCA-2 plane

But looking from a different perspective, we see that there still is a significant distribution along the axis of the third main component – at least with the scaling used along the PCA-3 axis.

3D-plot of the transformed WIFI data distribution in the space of the dominant 3 PCA components – the PCA-1/PCA-3 plane

Even when we take into account the different scales of the axes: The spread in z-direction (PCA-3) is relatively big compared with the data spread in the PCA-2 direction. Again, we see that the data indicate three main components. Why did we not get this information already in our bar plot for the “explained variance”?

Working on scaled data

Well, part of the answer to the last question is that we did not really treat the various WLAN signals equally well. Actually, for very simple reasons, a PCA analysis of really independent features with different measurement units should be applied to scaled data. So, just for curiosity’s sake, let us apply Sklearn’s StandardScaler() to our WIFI data ahead of a PCA-analysis. Then, we indeed get a different bar plot:

The elbow is now centered at a point corresponding to 3 main components! Below I show respective 3D-plots for the standardized and PCA-transformed data:

3D-plot of the scaled WIFI data distribution in the space of the dominant 3 PCA components – the PCA-1/PCA-3 plane

3D-plot of the scaled WIFI data distribution in the space of the dominant 3 PCA components – the PCA-1/PCA-2 plane

However, the 5th cluster – a subcluster of the orange one – is no longer so clearly visible as before. This is due to the fact that the standard deviation of the data around the mean value of each feature is adjusted to a value of 1.0 with StandardScaler(). A MinMaxScaler() does a better job:

In the case of the WIFI example there is also a strong counter-argument against scaling:
The individual features and their scales are NOT really independent of each other. A weaker signal or a specific spread around the mean value of a specific signal do actually mean something! When we have multiple maxima in a signal distribution (see the previous post) this carries some important information – and then the adjustment of the standard deviation to a standard value is not a really good idea. This means that it depends on the data and their meaning which kind of scaler one should use ahead of a PCA analysis.

Conclusions

The existence of a few primary components does not automatically mean that only a few features contribute to the data distribution’s variance in the features space. However, in the case of the WIFI data example we have a special situation for which only three out of seven features do determine the primary components and the direction of the respective preferred axes of the data distribution. We also saw that we may have to scale feature data properly before applying a PCA analysis.

In the next post of this series

KMeans as a classifier for the WIFI and MNIST datasets – III – KMeans as a classifier for the WIFI-example

we shall answer the question whether and how we can use the cluster algorithm KMeans also as a classifier for the WIFI data.

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