I am studying Support Vector Machines (SVM) by reading a lot of material. However, it seems that most of it focuses on how to classify the input 2D data by mapping it using several kernels such as linear, polynomial, RBF / Gaussian, etc.
My first question is, can SVM handle high-dimensional (n-D) input data?
According to what I found, the answer is YES!
If my understanding is correct, n-D input data will be
constructed in Hilbert hyperspace, then those data will be
simplified by using some approaches (such as PCA ?) to combine it together / project it back to 2D plane, so that
the kernel methods can map it into an appropriate shape such a line or curve can separate it into distinguish groups.
It means most of the guides / tutorials focus on step (3). But some toolboxes I've checked cannot plot if the input data greater than 2D. How can the data after be projected to 2D?
If there is no projection of data, how can they classify it?
My second question is: is my understanding correct?
My first question is, does SVM can handle high-dimensional (n-D) input data?
Yes. I have dealt with data where n > 2500 when using LIBSVM software: http://www.csie.ntu.edu.tw/~cjlin/libsvm/. I used linear and RBF kernels.
My second question is, does it correct my understanding?
I'm not entirely sure on what you mean here, so I'll try to comment on what you said most recently. I believe your intuition is generally correct. Data is "constructed" in some n-dimensional space, and a hyperplane of dimension n-1 is used to classify the data into two groups. However, by using kernel methods, it's possible to generate this information using linear methods and not consume all the memory of your computer.
I'm not sure if you've seen this already, but if you haven't, you may be interested in some of the information in this paper: http://pyml.sourceforge.net/doc/howto.pdf. I've copied and pasted a part of the text that may appeal to your thoughts:
A kernel method is an algorithm that depends on the data only through dot-products. When this is the case, the dot product can be replaced by a kernel function which computes a dot product in some possibly high dimensional feature space. This has two advantages: First, the ability to generate non-linear decision boundaries using methods designed for linear classifiers. Second, the use of kernel functions allows the user to apply a classifier to data that have no obvious fixed-dimensional vector space representation. The prime example of such data in bioinformatics are sequence, either DNA or protein, and protein structure.
It would also help if you could explain what "guides" you are referring to. I don't think I've ever had to project data on a 2-D plane before, and it doesn't make sense to do so anyway for data with a ridiculous amount of dimensions (or "features" as it is called in LIBSVM). Using selected kernel methods should be enough to classify such data.
Related
I have a 115*8000 data where 115 is the number of features. When I use pca function of matlab like this
[coeff,score,latent,tsquared,explained,mu] = pca(data);
on my data. I get some values. I read on here that how can I reduce my data but one thing confuses me. The explained data shows how much a feature weighs on calculation but do features get reorganized in this proces or features are exactly in same order as I give it to function?
Also I give 115 features but explained shows 114. Why does it happen?
The data is not "reorganized" in PCA, is transformed to a new space. When you crop the PCA space, that is your data, but you are not going to be able to visualize/understand it there, you need to convert it back to "normal" space, using eigenvectors and such.
explained gives you 114 because you now what is the answer with 115! 100% of the data can be explained with the whole data!
Read about it further in this answer: Significance of 99% of variance covered by the first component in PCA
PCA does not "choose" some of your features and remove the rest.
So you should not still be thinking about the original features after running PCA.
It is well-explained here on Wikipedia. You are converting your samples from the space defined by your original features to a space where features are linearly uncorrelated and called "principal components". Note: these components are no longer the original features.
An example of this in 2D could be: you have a vector z=(2,3) defined in your Euclidean space. It needs 2 features (the x and the y). If we change the space and define it using the coordinate vectors v=(2,3) and w an orthogonal vector to v, then z=(1,0) i.e. z=1.v+0.w and can now be represented with only 1 feature (the first coordinate!).
The link that you shared explains exactly (in the selected answer) how you can go about using the outputs of the pca function to reduce your dimensionality.
(As noted by Ander you do not care about the last components since these are the weakest anyway and you want to drop them)
I was reading this particular paper http://www.robots.ox.ac.uk/~vgg/publications/2011/Chatfield11/chatfield11.pdf and I find the Fisher Vector with GMM vocabulary approach very interesting and I would like to test it myself.
However, it is totally unclear (to me) how do they apply PCA dimensionality reduction on the data. I mean, do they calculate Feature Space and once it is calculated they perform PCA on it? Or do they just perform PCA on every image after SIFT is calculated and then they create feature space?
Is this supposed to be done for both training test sets? To me it's an 'obviously yes' answer, however it is not clear.
I was thinking of creating the feature space from training set and then run PCA on it. Then, I could use that PCA coefficient from training set to reduce each image's sift descriptor that is going to be encoded into Fisher Vector for later classification, whether it is a test or a train image.
EDIT 1;
Simplistic example:
[coef , reduced_feat_space]= pca(Feat_Space','NumComponents', 80);
and then (for both test and train images)
reduced_test_img = test_img * coef; (And then choose the first 80 dimensions of the reduced_test_img)
What do you think? Cheers
It looks to me like they do SIFT first and then do PCA. the article states in section 2.1 "The local descriptors are fixed in all experiments to be SIFT descriptors..."
also in the introduction section "the following three steps:(i) extraction
of local image features (e.g., SIFT descriptors), (ii) encoding of the local features in an image descriptor (e.g., a histogram of the quantized local features), and (iii) classification ... Recently several authors have focused on improving the second component" so it looks to me that the dimensionality reduction occurs after SIFT and the paper is simply talking about a few different methods of doing this, and the performance of each
I would also guess (as you did) that you would have to run it on both sets of images. Otherwise your would be using two different metrics to classify the images it really is like comparing apples to oranges. Comparing a reduced dimensional representation to the full one (even for the same exact image) will show some variation. In fact that is the whole premise of PCA, you are giving up some smaller features (usually) for computational efficiency. The real question with PCA or any dimensionality reduction algorithm is how much information can I give up and still reliably classify/segment different data sets
And as a last point, you would have to treat both images the same way, because your end goal is to use the Fisher Feature Vector for classification as either test or training. Now imagine you decided training images dont get PCA and test images do. Now I give you some image X, what would you do with it? How could you treat one set of images differently from another BEFORE you've classified them? Using the same technique on both sets means you'd process my image X then decide where to put it.
Anyway, I hope that helped and wasn't to rant-like. Good Luck :-)
I have a 40X3249 noisy dataset and 40X1 resultset. I want to perform simple sequential feature selection on it, in Matlab. Matlab example is complicated and I can't follow it. Even a few examples on SoF didn't help. I want to use decision tree as classifier to perform feature selection. Can someone please explain in simple terms.
Also is it a problem that my dataset has very low number of observations compared to the number of features?
I am following this example: Sequential feature selection Matlab and I am getting error like this:
The pooled covariance matrix of TRAINING must be positive definite.
I've explained the error message you're getting in answers to your previous questions.
In general, it is a problem that you have many more variables than samples. This will prevent you using some techniques, such as the discriminant analysis you were attempting, but it's a problem anyway. The fact is that if you have that high a ratio of variables to samples, it is very likely that some combination of variables would perfectly classify your dataset even if they were all random numbers. That's true if you build a single decision tree model, and even more true if you are using a feature selection method to explicitly search through combinations of variables.
I would suggest you try some sort of dimensionality reduction method. If all of your variables are continuous, you could try PCA as suggested by #user1207217. Alternatively you could use a latent variable method for model-building, such as PLS (plsregress in MATLAB).
If you're still intent on using sequential feature selection with a decision tree on this dataset, then you should be able to modify the example in the question you linked to, replacing the call to classify with one to classregtree.
This error comes from the use of the classify function in that question, which is performing LDA. This error occurs when the data is rank deficient (or in other words, some features are almost exactly correlated). In order to overcome this, you should project the data down to a lower dimensional subspace. Principal component analysis can do this for you. See here for more details on how to use pca function within statistics toolbox of Matlab.
[basis, scores, ~] = pca(X); % Find the basis functions and their weighting, X is row vectors
indices = find(scores > eps(2*max(scores))); % This is to find irrelevant components up to machine precision of the biggest component .. with a litte extra tolerance (2x)
new_basis = basis(:, indices); % This gets us the relevant components, which are stored in variable "basis" as column vectors
X_new = X*new_basis; % inner products between the new basis functions spanning some subspace of the original, and the original feature vectors
This should get you automatic projections down into a relevant subspace. Note that your features won't have the same meaning as before, because they will be weighted combinations of the old features.
Extra note: If you don't want to change your feature representation, then instead of classify, you need to use something which works with rank deficient data. You could roll your own version of penalised discriminant analysis (which is quite simple), use support vector machines, or other classification functions which don't break with correlated features as LDA does (by virtue of requiring matrix inversion of the covariance estimate).
EDIT: P.S I haven't tested this, because I have rolled my own version of PCA in Matlab.
I am trying to differentiate two populations. Each population is an NxM matrix in which N is fixed between the two and M is variable in length (N=column specific attributes of each run, M=run number). I have looked at PCA and K-means for differentiating the two, but I was curious of the best practice.
To my knowledge, in K-means, there is no initial 'calibration' in which the clusters are chosen such that known bimodal populations can be differentiated. It simply minimizes the distance and assigns the data to an arbitrary number of populations. I would like to tell the clustering algorithm that I want the best fit in which the two populations are separated. I can then use the fit I get from the initial clustering on future datasets. Any help, example code, or reading material would be appreciated.
-R
K-means and PCA are typically used in unsupervised learning problems, i.e. problems where you have a single batch of data and want to find some easier way to describe it. In principle, you could run K-means (with K=2) on your data, and then evaluate the degree to which your two classes of data match up with the data clusters found by this algorithm (note: you may want multiple starts).
It sounds to like you have a supervised learning problem: you have a training data set which has already been partitioned into two classes. In this case k-nearest neighbors (as mentioned by #amas) is probably the approach most like k-means; however Support Vector Machines can also be an attractive approach.
I frequently refer to The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition (Springer Series in Statistics) by Trevor Hastie (Author), Robert Tibshirani (Author), Jerome Friedman (Author).
It really depends on the data. But just to let you know K-means does get stuck at local minima so if you wanna use it try running it from different random starting points. PCA's might also be useful how ever like any other spectral clustering method you have much less control over the clustering procedure. I recommend that you cluster the data using k-means with multiple random starting points and c how it works then you can predict and learn for each the new samples with K-NN (I don't know if it is useful for your case).
Check Lazy learners and K-NN for prediction.
I have a dataset of n data, where each data is represented by a set of extracted features. Generally, the clustering algorithms need that all input data have the same dimensions (the same number of features), that is, the input data X is a n*d matrix of n data points each of which has d features.
In my case, I've previously extracted some features from my data but the number of extracted features for each data is most likely to be different (I mean, I have a dataset X where data points have not the same number of features).
Is there any way to adapt them, in order to cluster them using some common clustering algorithms requiring data to be of the same dimensions.
Thanks
Sounds like the problem you have is that it's a 'sparse' data set. There are generally two options.
Reduce the dimensionality of the input data set using multi-dimensional scaling techniques. For example Sparse SVD (e.g. Lanczos algorithm) or sparse PCA. Then apply traditional clustering on the dense lower dimensional outputs.
Directly apply a sparse clustering algorithm, such as sparse k-mean. Note you can probably find a PDF of this paper if you look hard enough online (try scholar.google.com).
[Updated after problem clarification]
In the problem, a handwritten word is analyzed visually for connected components (lines). For each component, a fixed number of multi-dimensional features is extracted. We need to cluster the words, each of which may have one or more connected components.
Suggested solution:
Classify the connected components first, into 1000(*) unique component classifications. Then classify the words against the classified components they contain (a sparse problem described above).
*Note, the exact number of component classifications you choose doesn't really matter as long as it's high enough as the MDS analysis will reduce them to the essential 'orthogonal' classifications.
There are also clustering algorithms such as DBSCAN that in fact do not care about your data. All this algorithm needs is a distance function. So if you can specify a distance function for your features, then you can use DBSCAN (or OPTICS, which is an extension of DBSCAN, that doesn't need the epsilon parameter).
So the key question here is how you want to compare your features. This doesn't have much to do with clustering, and is highly domain dependant. If your features are e.g. word occurrences, Cosine distance is a good choice (using 0s for non-present features). But if you e.g. have a set of SIFT keypoints extracted from a picture, there is no obvious way to relate the different features with each other efficiently, as there is no order to the features (so one could compare the first keypoint with the first keypoint etc.) A possible approach here is to derive another - uniform - set of features. Typically, bag of words features are used for such a situation. For images, this is also known as visual words. Essentially, you first cluster the sub-features to obtain a limited vocabulary. Then you can assign each of the original objects a "text" composed of these "words" and use a distance function such as cosine distance on them.
I see two options here:
Restrict yourself to those features for which all your data-points have a value.
See if you can generate sensible default values for missing features.
However, if possible, you should probably resample all your data-points, so that they all have values for all features.