I have a concern about support vector machines, namely their classification scores:
Do these classification scores have an upper bound?
I think no, since an SVM is just a hyperplane, and the score basically a point's distance to that hyperplane. Without restrictions, a point could lie anywhere in the space and thus the distance does not have any bound, does it?
I am asking, because I have read the following line:
"When decision scores are bounded — and SVM scores are bounded by the margin — ..."
Could you explain what is meant by that? I don't see how the margin is a bound on the detection score...
Thanks for your help, I appreciate it!
Your intuition is correct. Whatever you read is misleading at best or plain wrong (some context is required in any case). SVM decision values do not have an upper bound. It depends entirely on the test instances.
SVM decision values are a linear combination of inner products in feature space of the test instance and the support vectors. If the test instance has infinite norm, these inner products will be infinite as well.
Related
What are some of the reasons that a Neural Network (Feed Forward) might have a general tendency to overshoot the real output rather than undershoot, and vice versa?
As an example, the below figure (right) shows an almost perfect gaussian distribution of the prediction errors, however it seems there is a tendency for the neural network to predict values higher than the actual value:
Only overshoot/undershoot can happen if there is an offset in the data but you don't have a constant term (in NNs called "bias") in your model to compensate for that. This is not a problem if you center your data first.
(What also happens in NNs, especially in deep ones, is that the variance increases layer by layer and the distribution widens, since the variance of a variable Y = aX is V(Y) = a^2*V(X).)
Outliers on either side of the mean can also lead to a rotation (in your left plot), which means that small values are over-/undererstimated and larger values are under-/overestimated, depending on the exact position of the outlier(s).
However, in your right plot, the prediction error is so much smaller than the values of Y that it could be just a numerical/machine precision issue.
I hope that I could help you.
I'm going through CS231n to understand the basics of neural networks.
Attached is the slide in which Justin (the tutor) gives the reasoning for why data preprocessing is required and I don't completely understand. The explanation given is similar to the one given on the slide and I don't get it. The slide is below.
The second question I have is: is it actually normalisation or standardisation? This link implies that it is standardisation, whereas the course material says it is normalisation.
Any help will be appreciated.
A) The meaning of "less sensitive to small changes in weights" can easily be visualized. Imagine to operate a little change in the weights of the drawn hyperplane, i.e. rotate it a bit. If the samples are located around the origin, you'll notice that they can still be correctly classified. If they're far away from the origin, the same little change in weights will lead to bigger misclassifications.
B) Sometimes standardization and normalization are used interchangeably.
Standardization: I quote from Machine Learning and Pattern Recognition by Bishop : "For the purposes of this example, we have made a linear re-scaling of the data, known as standardizing, such that each of the variables has zero mean and unit standard deviation."
Normalization could be e.g. min-max normalization when you scale all feature values to the [0,1] range, or feature vector normalization when you divide the feature vector by its modulus.
I am trying to implement a binary classifier using logistic regression for data drawn from 2 point sets (classes y (-1, 1)). As seen below, we can use the parameter a to prevent overfitting.
Now I am not sure, how to choose the "good" value for a.
Another thing I am not sure about is how to choose a "good" convergence criterion for this sort of problem.
Value of 'a'
Choosing "good" things is a sort of meta-regression: pick any value for a that seems reasonable. Run the regression. Try again with a values larger and smaller by a factor of 3. If either works better than the original, try another factor of 3 in that direction -- but round it from 9x to 10x for readability.
You get the idea ... play with it until you get in the right range. Unless you're really trying to optimize the result, you probably won't need to narrow it down much closer than that factor of 3.
Data Set Partition
ML folks have spent a lot of words analysing the best split. The optimal split depends very much on your data space. As a global heuristic, use half or a bit more for training; of the rest, no more than half should be used for testing, the rest for validation. For instance, 50:20:30 is a viable approximation for train:test:validate.
Again, you get to play with this somewhat ... except that any true test of the error rate would be entirely new data.
Convergence
This depends very much on the characteristics of your empirical error space near the best solution, as well as near local regions of low gradient.
The first consideration is to choose an error function that is likely to be convex and have no flattish regions. The second is to get some feeling for the magnitude of the gradient in the region of a desired solution (normalizing your data will help with this); use this to help choose the convergence radius; you might want to play with that 3x scaling here, too. The final one is to play with the learning rate, so that it's scaled to the normalized data.
Does any of this help?
How exactly is an U-matrix constructed in order to visualise a self-organizing-map? More specifically, suppose that I have an output grid of 3x3 nodes (that have already been trained), how do I construct a U-matrix from this? You can e.g. assume that the neurons (and inputs) have dimension 4.
I have found several resources on the web, but they are not clear or they are contradictory. For example, the original paper is full of typos.
A U-matrix is a visual representation of the distances between neurons in the input data dimension space. Namely you calculate the distance between adjacent neurons, using their trained vector. If your input dimension was 4, then each neuron in the trained map also corresponds to a 4-dimensional vector. Let's say you have a 3x3 hexagonal map.
The U-matrix will be a 5x5 matrix with interpolated elements for each connection between two neurons like this
The {x,y} elements are the distance between neuron x and y, and the values in {x} elements are the mean of the surrounding values. For example, {4,5} = distance(4,5) and {4} = mean({1,4}, {2,4}, {4,5}, {4,7}). For the calculation of the distance you use the trained 4-dimensional vector of each neuron and the distance formula that you used for the training of the map (usually Euclidian distance). So, the values of the U-matrix are only numbers (not vectors). Then you can assign a light gray colour to the largest of these values and a dark gray to the smallest and the other values to corresponding shades of gray. You can use these colours to paint the cells of the U-matrix and have a visualized representation of the distances between neurons.
Have also a look at this web article.
The original paper cited in the question states:
A naive application of Kohonen's algorithm, although preserving the topology of the input data is not able to show clusters inherent in the input data.
Firstly, that's true, secondly, it is a deep mis-understanding of the SOM, thirdly it is also a mis-understanding of the purpose of calculating the SOM.
Just take the RGB color space as an example: are there 3 colors (RGB), or 6 (RGBCMY), or 8 (+BW), or more? How would you define that independent of the purpose, ie inherent in the data itself?
My recommendation would be not to use maximum likelihood estimators of cluster boundaries at all - not even such primitive ones as the U-Matrix -, because the underlying argument is already flawed. No matter which method you then use to determine the cluster, you would inherit that flaw. More precisely, the determination of cluster boundaries is not interesting at all, and it is loosing information regarding the true intention of building a SOM. So, why do we build SOM's from data?
Let us start with some basics:
Any SOM is a representative model of a data space, for it reduces the dimensionality of the latter. For it is a model it can be used as a diagnostic as well as a predictive tool. Yet, both cases are not justified by some universal objectivity. Instead, models are deeply dependent on the purpose and the accepted associated risk for errors.
Let us assume for a moment the U-Matrix (or similar) would be reasonable. So we determine some clusters on the map. It is not only an issue how to justify the criterion for it (outside of the purpose itself), it is also problematic because any further calculation destroys some information (it is a model about a model).
The only interesting thing on a SOM is the accuracy itself viz the classification error, not some estimation of it. Thus, the estimation of the model in terms of validation and robustness is the only thing that is interesting.
Any prediction has a purpose and the acceptance of the prediction is a function of the accuracy, which in turn can be expressed by the classification error. Note that the classification error can be determined for 2-class models as well as for multi-class models. If you don't have a purpose, you should not do anything with your data.
Inversely, the concept of "number of clusters" is completely dependent on the criterion "allowed divergence within clusters", so it is masking the most important thing of the structure of the data. It is also dependent on the risk and the risk structure (in terms of type I/II errors) you are willing to take.
So, how could we determine the number classes on a SOM? If there is no exterior apriori reasoning available, the only feasible way would be an a-posteriori check of the goodness-of-fit. On a given SOM, impose different numbers of classes and measure the deviations in terms of mis-classification cost, then choose (subjectively) the most pleasing one (using some fancy heuristics, like Occam's razor)
Taken together, the U-matrix is pretending objectivity where no objectivity can be. It is a serious misunderstanding of modeling altogether.
IMHO it is one of the greatest advantages of the SOM that all the parameters implied by it are accessible and open for being parameterized. Approaches like the U-matrix destroy just that, by disregarding this transparency and closing it again with opaque statistical reasoning.
I'm trying to read through PCA and saw that the objective was to maximize the variance. I don't quite understand why. Any explanation of other related topics would be helpful
Variance is a measure of the "variability" of the data you have. Potentially the number of components is infinite (actually, after numerization it is at most equal to the rank of the matrix, as #jazibjamil pointed out), so you want to "squeeze" the most information in each component of the finite set you build.
If, to exaggerate, you were to select a single principal component, you would want it to account for the most variability possible: hence the search for maximum variance, so that the one component collects the most "uniqueness" from the data set.
Note that PCA does not actually increase the variance of your data. Rather, it rotates the data set in such a way as to align the directions in which it is spread out the most with the principal axes. This enables you to remove those dimensions along which the data is almost flat. This decreases the dimensionality of the data while keeping the variance (or spread) among the points as close to the original as possible.
Maximizing the component vector variances is the same as maximizing the 'uniqueness' of those vectors. Thus you're vectors are as distant from each other as possible. That way if you only use the first N component vectors you're going to capture more space with highly varying vectors than with like vectors. Think about what Principal Component actually means.
Take for example a situation where you have 2 lines that are orthogonal in a 3D space. You can capture the environment much more completely with those orthogonal lines than 2 lines that are parallel (or nearly parallel). When applied to very high dimensional states using very few vectors, this becomes a much more important relationship among the vectors to maintain. In a linear algebra sense you want independent rows to be produced by PCA, otherwise some of those rows will be redundant.
See this PDF from Princeton's CS Department for a basic explanation.
max variance is basically setting these axis that occupy the maximum spread of the datapoints, why? because the direction of this axis is what really matters as it kinda explains correlations and later on we will compress/project the points along those axis to get rid of some dimensions