What is difference between PCA , TruncatedSVD and ICA in details? - machine-learning

Can anybody tell me the differences between PCA(Principal component analysis ) , TruncatedSVD(Truncated singular value decomposition) and ICA(Independent component analysis) in detail?

Doing it in detail will require long pages PDF document :-).
But the idea is simple:
Principal Component Analysis (PCA) - analyze the data native coordinates. Namely the coordinated which along the data has its most energy (Variance). For n Samples of dimension d there will be $ d $ orthogonal directions. Namely the data projected on them has no correlation. If we look on the data as Random Variables, it means we found a coordinate system where the Cross Correlation (First Moment) of any pair from the projected data is vanished.
This is a very efficient way to approximate the data in lower dimensionality by keeping most of its energy.
Truncated SVD - One could show that one of the ways of calculating those system of coordinate is using the SVD. Hence this is method to apply the ideas behind PCA.
Independent Component Analysis (ICA) - This is one step farther from PCA. While in PCA we dealt with only First Order Moments of the data (Correlation) in ICA we're looking into higher moments and try to find a projection of data which vanishes higher moments (Think of Lack of Correlation vs. Probability Independence).

Related

Gaussian Progress Regression usecase

while reading the paper :" Tactile-based active object discrimination and target object search in an unknown workspace", there is something that I just can not understand:
The paper is about finding object's position and other properties using only tactile information. In the section 4.1.2, the author says that he uses GPR to guide the exploratory process and in section 4.1.4 he describes how he trained his GPR:
Using the example from the section 4.1.2, the input is (x,z) and the ouput y.
Whenever there is a contact, the coresponding y-value is stored.
This procedure is repeated several times.
This trained GPR is used to estimate the next exploring point, which is the point where the variance is maximum at.
In the following link, you also can see the demonstration: https://www.youtube.com/watch?v=ZiLq3i-BJcA&t=177s . In the first part of video (0:24-0:29), the first initalization takes place where the robot samples 4 times. Then in the next 25 seconds, the robot explores explores from the corresponding direction. I do not understand how this tiny initialization of GPR can guide the exploratory process. Could someone please explain how the input points (x,z) from the first exploring part could be estimated?
Any regression algorithm simply maps the input (x,z) to an output y in some way unique to the specific algorithm. For a new input (x0,z0) the algorithm will likely predict something very close to the true output y0 if many data points similar to this was included in the training. If only training data was available in a vastly different region, the predictions will likely be very bad.
GPR includes a measure of confidence of the predictions, namely the variance. The variance will naturally be very high in regions where no training data has been seen before and low very close to already seen data points. If the 'experiment' takes much longer than evaluating the Gaussian Process, you can use the Gaussian Process fit to make sure you sample regions where you are very uncertain of your answer.
If the goal is to fully explore the entire input space, you could draw a lot of random values of (x,z) and evaluate the variance at these values. Then you could perform the costly experiment at the input point where you are most uncertain in y. Then you can retrain the GPR with all the explored data so far and repeat the process.
For optimization problems (Not the OP's question)
If you wish to find the lowest value of y across the input space, you are not interested in doing the experiment in regions that you know give high values of y, but you are just uncertain of how high these values will be. So instead of choosing the (x,z) points with the highest variance, you might choose the predicted value of y plus one standard deviation. Minimizing values this way is named Bayesian Optimization and this specific scheme is named Upper Confidence Bound (UCB). Expected Improvement (EI) - the probability of improving the previously best score - is also commonly used.

an algorithm for clustering visually separable clusters

I have visualized a dataset in 2D after employing PCA. 1 dimension is time and the Y dimension is First PCA component. As figure shows, there is relatively good separation between points (A, B). But unfortunately clustering methods (DBSCAN, SMO, KMEANS, Hierarchical) are not able to cluster these points in 2 clusters. As you see in section A there is a relative continuity and this continuous process is finished and Section B starts and there is rather big gap in comparison to past data between A and B.
I will be so grateful if you can introduce me any method and algorithm (or devising any metric from data considering its distribution) to be able to do separation between A and B without visualization. Thank you so much.
This is plot of 2 PCA components for the above plot(the first one). The other one is also the plot of components of other dataset which I get bad result,too.
This is a time series, and apparently you are looking for change points or want to segment this time series.
Do not treat this data set as a two dimensional x-y data set, and don't use clustering here; rather choose an algorithm that is actually designed for time series.
As a starter, plot series[x] - series[x-1], i.e. the first derivative. You may need to remove seasonality to improve results. No clustering algorithm will do this, they do not have a notion of seasonality or time.
If PCA gives you a good separation, you can just try to cluster after projecting your data through your PCA eigenvectors. If you don't want to use PCA, then you will need anyway an alternative data projection method, because failing clustering methods imply that your data is not separable in the original dimensions. You can take a look at non linear clustering methods such as the kernel based ones or spectral clustering for example. Or to define your own non-euclidian metric, which is in fact just another data projection method.
But using PCA clearly seems to be the best fit in your case (Occam razor : use the simplest model that fits your data).
I don't know that you'll have an easy time devising an algorithm to handle this case, which is dangerously (by present capabilities) close to "read my mind" clustering. You have a significant alley where you've marked the division. You have one nearly as good around (1700, +1/3), and an isolate near (1850, 0.45). These will make it hard to convince a general-use algorithm to make exactly one division at the spot you want, although that one is (I think) still the most computationally obvious.
Spectral clustering works well at finding gaps; I'd try that first. You might have to ask it for 3 or 4 clusters to separate the one you want in general. You could also try playing with SVM (good at finding alleys in data), but doing that in an unsupervised context is the tricky part.
No, KMeans is not going to work; it isn't sensitive to density or connectivity.

What methods are best for clustering multidimensional data that has irregular shape?

I am new to machine learning and data analysis and I'm struggling to cluster my data. I'm working with about 40,000 observations with 6 features.
I have tried various clustering methods including K-Means, DBSCAN, and also attempted scipy hierarchical clustering with linkage. During preprocessing missing data is imputed and all of the data is normalized. Once I complete PCA to reduce the dimensions from 4 to 6 my data looks like a crescent moon shape that can be seen below as the blue dots.
I determined that using 10 clusters for K-means would be best based on silhouette coefficient analysis and this is the result:
The result does not change much when performing PCA after the data has been clustered.
DBSCAN itself decides on 4 clusters and gives 4 clusters but with most of the data excluded from these clusters and depicted as noise.
For the hierarchical method the data usage was too much when trying to perform linkage() and kept providing a memory error message.
Is there any way I can cluster my data? Is the shape of my data (a crescent moon) lend itself to other modelling methods?
Don't run clustering without thinking first
Clustering algorithms must not be used as black boxes. They need to be carefully used or you get out only garbage. And to use them right, you need to understand the objective of each algorithm. K-means is a least squares approach. if you use it on badly normalized data, it fails.
Judging from your plot, there is a bad record in your database, largely causing that "moon" shape: everything needs tp be as far away as possible from that bad record.
Apart from that: 1. did you scale the data correctly for your problem? 2. did you choose the appropriate distance measure?

How do I use principal component analysis in supervised machine learning classification problems?

I have been working through the concepts of principal component analysis in R.
I am comfortable with applying PCA to a (say, labeled) dataset and ultimately extracting out the most interesting first few principal components as numeric variables from my matrix.
The ultimate question is, in a sense, now what? Most of the reading I've come across on PCA immediately halts after the computations are done, especially with regards to machine learning. Pardon my hyperbole, but I feel as if everyone agrees that the technique is useful, but nobody wants to actually use it after they do it.
More specifically, here's my real question:
I respect that principle components are linear combinations of the variables you started with. So, how does this transformed data play a role in supervised machine learning? How could someone ever use PCA as a way to reduce dimensionality of a dataset, and THEN, use these components with a supervised learner, say, SVM?
I'm absolutely confused about what happens to our labels. Once we are in eigenspace, great. But I don't see any way to continue to move forward with machine learning if this transformation blows apart our concept of classification (unless there's some linear combination of "Yes" or "No" I haven't come across!)
Please step in and set me straight if you have the time and wherewithal. Thanks in advance.
Old question, but I don't think it's been satisfactorily answered (and I just landed here myself through Google). I found myself in your same shoes and had to hunt down the answer myself.
The goal of PCA is to represent your data X in an orthonormal basis W; the coordinates of your data in this new basis is Z, as expressed below:
Because of orthonormality, we can invert W simply by transposing it and write:
Now to reduce dimensionality, let's pick some number of components k < p. Assuming our basis vectors in W are ordered from largest to smallest (i.e., eigenvector corresponding to the largest eigenvalue is first, etc.), this amounts to simply keeping the first k columns of W.
Now we have a k dimensional representation of our training data X. Now you run some supervised classifier using the new features in Z.
The key is to realize that W is in some sense a canonical transformation from our space of p features down to a space of k features (or at least the best transformation we could find using our training data). Thus, we can hit our test data with the same W transformation, resulting in a k-dimensional set of test features:
We can now use the same classifier trained on the k-dimensional representation of our training data to make predictions on the k-dimensional representation of our test data:
The point of going through this whole procedure is because you may have thousands of features, but (1) not all of them are going to have a meaningful signal and (2) your supervised learning method may be far too complex to train on the full feature set (either it would take too long or your computer wouldn't have a enough memory to process the calculations). PCA allows you to dramatically reduce the number of features it takes to represent your data without eliminating features of your data that truly add value.
After you have used PCA on a portion of your data to compute the transformation matrix, you apply that matrix to each of your data points before submitting them to your classifier.
This is useful when the intrinsic dimensionality of your data is much smaller than the number of components and the gain in performance you get during classification is worth the loss in accuracy and the cost of PCA. Also, keep in mind the limitations of PCA:
In performing a linear transformation, you implicitly assume that all components are expressed in equivalent units.
Beyond variance, PCA is blind to the structure of your data. It may very well happen that the data splits along low-variance dimensions. In that case, the classifier won't learn from transformed data.

Clustering Method Selection in High-Dimension?

If the data to cluster are literally points (either 2D (x, y) or 3D (x, y,z)), it would be quite intuitive to choose a clustering method. Because we can draw them and visualize them, we somewhat know better which clustering method is more suitable.
e.g.1 If my 2D data set is of the formation shown in the right top corner, I would know that K-means may not be a wise choice here, whereas DBSCAN seems like a better idea.
However, just as the scikit-learn website states:
While these examples give some intuition about the algorithms, this
intuition might not apply to very high dimensional data.
AFAIK, in most of the piratical problems we don't have such simple data. Most probably, we have high-dimensional tuples, which cannot be visualized like such, as data.
e.g.2 I wish to cluster a data set where each data is represented as a 4-D tuple <characteristic1, characteristic2, characteristic3, characteristic4>. I CANNOT visualize it in a coordinate system and observes its distribution like before. So I will NOT be able to say DBSCAN is superior to K-means in this case.
So my question:
How does one choose the suitable clustering method for such an "invisualizable" high-dimensional case?
"High-dimensional" in clustering probably starts at some 10-20 dimensions in dense data, and 1000+ dimensions in sparse data (e.g. text).
4 dimensions are not much of a problem, and can still be visualized; for example by using multiple 2d projections (or even 3d, using rotation); or using parallel coordinates. Here's a visualization of the 4-dimensional "iris" data set using a scatter plot matrix.
However, the first thing you still should do is spend a lot of time on preprocessing, and finding an appropriate distance function.
If you really need methods for high-dimensional data, have a look at subspace clustering and correlation clustering, e.g.
Kriegel, Hans-Peter, Peer Kröger, and Arthur Zimek. Clustering high-dimensional data: A survey on subspace clustering, pattern-based clustering, and correlation clustering. ACM Transactions on Knowledge Discovery from Data (TKDD) 3.1 (2009): 1.
The authors of that survey also publish a software framework which has a lot of these advanced clustering methods (not just k-means, but e.h. CASH, FourC, ERiC): ELKI
There are at least two common, generic approaches:
One can use some dimensionality reduction technique in order to actually visualize the high dimensional data, there are dozens of popular solutions including (but not limited to):
PCA - principal component analysis
SOM - self-organizing maps
Sammon's mapping
Autoencoder Neural Networks
KPCA - kernel principal component analysis
Isomap
After this one goes back to the original space and use some techniques that seems resonable based on observations in the reduced space, or performs clustering in the reduced space itself.First approach uses all avaliable information, but can be invalid due to differences induced by the reduction process. While the second one ensures that your observations and choice is valid (as you reduce your problem to the nice, 2d/3d one) but it loses lots of information due to transformation used.
One tries many different algorithms and choose the one with the best metrics (there have been many clustering evaluation metrics proposed). This is computationally expensive approach, but has a lower bias (as reducting the dimensionality introduces the information change following from the used transformation)
It is true that high dimensional data cannot be easily visualized in an euclidean high dimensional data but it is not true that there are no visualization techniques for them.
In addition to this claim I will add that with just 4 features (your dimensions) you can easily try the parallel coordinates visualization method. Or simply try a multivariate data analysis taking two features at a time (so 6 times in total) to try to figure out which relations intercour between the two (correlation and dependency generally). Or you can even use a 3d space for three at a time.
Then, how to get some info from these visualizations? Well, it is not as easy as in an euclidean space but the point is to spot visually if the data clusters in some groups (eg near some values on an axis for a parallel coordinate diagram) and think if the data is somehow separable (eg if it forms regions like circles or line separable in the scatter plots).
A little digression: the diagram you posted is not indicative of the power or capabilities of each algorithm given some particular data distributions, it simply highlights the nature of some algorithms: for instance k-means is able to separate only convex and ellipsoidail areas (and keep in mind that convexity and ellipsoids exist even in N-th dimensions). What I mean is that there is not a rule that says: given the distributiuons depicted in this diagram, you have to choose the correct clustering algorithm consequently.
I suggest to use a data mining toolbox that lets you explore and visualize the data (and easily transform them since you can change their topology with transformations, projections and reductions, check the other answer by lejlot for that) like Weka (plus you do not have to implement all the algorithms by yourself.
In the end I will point you to this resource for different cluster goodness and fitness measures so you can compare the results rfom different algorithms.
I would also suggest soft subspace clustering, a pretty common approach nowadays, where feature weights are added to find the most relevant features. You can use these weights to increase performance and improve the BMU calculation with euclidean distance, for example.

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