## KL divergence estimate for dataset, help calculating nearest neighbour (KDTree) - scipy

I am implementing the algorithm suggested in Perez-Cruz, 2008 (https://www.tsc.uc3m.es/~fernando/bare_conf3.pdf) to estimate the KL divergence of high-D data. (I understand this is not preferable but we are using it as a benchmark for another solution).
An implementation of this algorithm exists in https://gist.github.com/atabakd/ed0f7581f8510c8587bc2f41a094b518. I have used this code as a base. However this implementation misses some details of the algorithm that I believe is giving me erroneous results even when I try to recreate the results of the paper on MNIST data. Specifically, the paper author states:
where we employ the nearest-neighbour less than xi from X to estimate
p(xi) and the two nearest-neighbours, one less than and the other
larger than xi, from X to estimate q(xi).
However based on my reading of the code, the nearest neighbor is taken for p(xi) and q(xi) irrespective if it is greater than x.
Do people know how I can select neighbours less than x in the code. I have looked at the documentation and this is not something you can specify.
For now, the only change to the code linked on github i have made is averaging 2 nearest neighbour for q(xi) i.e:
s = ytree.query(x, k=2, eps=.01, p=2)[0].mean(axis = 1)

## Related

### Estimate geo-coordinates based on tweets using sparse coding

I'm trying to estimate Geo-coordinates of tweets on Twitter based only on characteristics of tweets' content. I used an algorithm from this paper. Basically, the tweets from users are collected and pre-processed to create a sequence/word-count vectors. Sub-vectors (patches) are extracted and an unsupervised learning method is used for learning Dictionary (KSVD). Using the learned Dictionary, sparse codes can be found. After that, a max pooling scheme is introduced. Finally, a look-up table is created with entries of key (sparse code)/value (Geo-coordinates). For estimating Geo-coordinates of tweets (same user), we calculate corresponding sparse codes, then use kNN to find the neighbors. The geo-coordinates can be estimated by the average value of these neighbor vectors. Following is how I implement the algorithm: I extracted data from GeoText dataset, separating data by user (say,80% of users for the training set, and 20% for the validation set) Patches/sub-vectors are put together to create a big matrices of the training set and the validation set For learning dictionary, I used KSVD-BOX from: http://www.cs.technion.ac.il/~ronrubin/software.html For sparse coding, I used OMP-BOX from the same aforementioned website Some necessary parameters: N = 64 (dimension of patch or sub-vector) K = 600 (number of atoms) T = 10 (sparsity) kNN = 30 (number of the nearest neighbours) epsilon = 0.1 (whitening constant) It can be seen that the algorithm runs pretty fast, taking 10 minutes for training, and 5 minutes for testing. However, I've never obtained a high accuracy. In fact, the mean value distance errors is always around 1000 km, which is not as good as in the paper (500km). I've followed every points in the paper, including enhancement options. Here is my Matlab source code. Well, I know the description is pretty long, but I tried to explain what I understand in an easy way. I hope you can help me improve the accuracy. Thank for your patient.

### What's the best way to calculate a numerical derivative in MATLAB?

(Note: This is intended to be a community Wiki.) Suppose I have a set of points xi = {x0,x1,x2,...xn} and corresponding function values fi = f(xi) = {f0,f1,f2,...,fn}, where f(x) is, in general, an unknown function. (In some situations, we might know f(x) ahead of time, but we want to do this generally, since we often don't know f(x) in advance.) What's a good way to approximate the derivative of f(x) at each point xi? That is, how can I estimate values of dfi == d/dx fi == df(xi)/dx at each of the points xi? Unfortunately, MATLAB doesn't have a very good general-purpose, numerical differentiation routine. Part of the reason for this is probably because choosing a good routine can be difficult! So what kinds of methods are there? What routines exist? How can we choose a good routine for a particular problem? There are several considerations when choosing how to differentiate in MATLAB: Do you have a symbolic function or a set of points? Is your grid evenly or unevenly spaced? Is your domain periodic? Can you assume periodic boundary conditions? What level of accuracy are you looking for? Do you need to compute the derivatives within a given tolerance? Does it matter to you that your derivative is evaluated on the same points as your function is defined? Do you need to calculate multiple orders of derivatives? What's the best way to proceed?

These are just some quick-and-dirty suggestions. Hopefully somebody will find them helpful! 1. Do you have a symbolic function or a set of points? If you have a symbolic function, you may be able to calculate the derivative analytically. (Chances are, you would have done this if it were that easy, and you would not be here looking for alternatives.) If you have a symbolic function and cannot calculate the derivative analytically, you can always evaluate the function on a set of points, and use some other method listed on this page to evaluate the derivative. In most cases, you have a set of points (xi,fi), and will have to use one of the following methods.... 2. Is your grid evenly or unevenly spaced? If your grid is evenly spaced, you probably will want to use a finite difference scheme (see either of the Wikipedia articles here or here), unless you are using periodic boundary conditions (see below). Here is a decent introduction to finite difference methods in the context of solving ordinary differential equations on a grid (see especially slides 9-14). These methods are generally computationally efficient, simple to implement, and the error of the method can be simply estimated as the truncation error of the Taylor expansions used to derive it. If your grid is unevenly spaced, you can still use a finite difference scheme, but the expressions are more difficult and the accuracy varies very strongly with how uniform your grid is. If your grid is very non-uniform, you will probably need to use large stencil sizes (more neighboring points) to calculate the derivative at a given point. People often construct an interpolating polynomial (often the Lagrange polynomial) and differentiate that polynomial to compute the derivative. See for instance, this StackExchange question. It is often difficult to estimate the error using these methods (although some have attempted to do so: here and here). Fornberg's method is often very useful in these cases.... Care must be taken at the boundaries of your domain because the stencil often involves points that are outside the domain. Some people introduce "ghost points" or combine boundary conditions with derivatives of different orders to eliminate these "ghost points" and simplify the stencil. Another approach is to use right- or left-sided finite difference methods. Here's an excellent "cheat sheet" of finite difference methods, including centered, right- and left-sided schemes of low orders. I keep a printout of this near my workstation because I find it so useful. 3. Is your domain periodic? Can you assume periodic boundary conditions? If your domain is periodic, you can compute derivatives to a very high order accuracy using Fourier spectral methods. This technique sacrifices performance somewhat to gain high accuracy. In fact, if you are using N points, your estimate of the derivative is approximately N^th order accurate. For more information, see (for example) this WikiBook. Fourier methods often use the Fast Fourier Transform (FFT) algorithm to achieve roughly O(N log(N)) performance, rather than the O(N^2) algorithm that a naively-implemented discrete Fourier transform (DFT) might employ. If your function and domain are not periodic, you should not use the Fourier spectral method. If you attempt to use it with a function that is not periodic, you will get large errors and undesirable "ringing" phenomena. Computing derivatives of any order requires 1) a transform from grid-space to spectral space (O(N log(N))), 2) multiplication of the Fourier coefficients by their spectral wavenumbers (O(N)), and 2) an inverse transform from spectral space to grid space (again O(N log(N))). Care must be taken when multiplying the Fourier coefficients by their spectral wavenumbers. Every implementation of the FFT algorithm seems to have its own ordering of the spectral modes and normalization parameters. See, for instance, the answer to this question on the Math StackExchange, for notes about doing this in MATLAB. 4. What level of accuracy are you looking for? Do you need to compute the derivatives within a given tolerance? For many purposes, a 1st or 2nd order finite difference scheme may be sufficient. For higher precision, you can use higher order Taylor expansions, dropping higher-order terms. If you need to compute the derivatives within a given tolerance, you may want to look around for a high-order scheme that has the error you need. Often, the best way to reduce error is reducing the grid spacing in a finite difference scheme, but this is not always possible. Be aware that higher-order finite difference schemes almost always require larger stencil sizes (more neighboring points). This can cause issues at the boundaries. (See the discussion above about ghost points.) 5. Does it matter to you that your derivative is evaluated on the same points as your function is defined? MATLAB provides the diff function to compute differences between adjacent array elements. This can be used to calculate approximate derivatives via a first-order forward-differencing (or forward finite difference) scheme, but the estimates are low-order estimates. As described in MATLAB's documentation of diff (link), if you input an array of length N, it will return an array of length N-1. When you estimate derivatives using this method on N points, you will only have estimates of the derivative at N-1 points. (Note that this can be used on uneven grids, if they are sorted in ascending order.) In most cases, we want the derivative evaluated at all points, which means we want to use something besides the diff method. 6. Do you need to calculate multiple orders of derivatives? One can set up a system of equations in which the grid point function values and the 1st and 2nd order derivatives at these points all depend on each other. This can be found by combining Taylor expansions at neighboring points as usual, but keeping the derivative terms rather than cancelling them out, and linking them together with those of neighboring points. These equations can be solved via linear algebra to give not just the first derivative, but the second as well (or higher orders, if set up properly). I believe these are called combined finite difference schemes, and they are often used in conjunction with compact finite difference schemes, which will be discussed next. Compact finite difference schemes (link). In these schemes, one sets up a design matrix and calculates the derivatives at all points simultaneously via a matrix solve. They are called "compact" because they are usually designed to require fewer stencil points than ordinary finite difference schemes of comparable accuracy. Because they involve a matrix equation that links all points together, certain compact finite difference schemes are said to have "spectral-like resolution" (e.g. Lele's 1992 paper--excellent!), meaning that they mimic spectral schemes by depending on all nodal values and, because of this, they maintain accuracy at all length scales. In contrast, typical finite difference methods are only locally accurate (the derivative at point #13, for example, ordinarily doesn't depend on the function value at point #200). A current area of research is how best to solve for multiple derivatives in a compact stencil. The results of such research, combined, compact finite difference methods, are powerful and widely applicable, though many researchers tend to tune them for particular needs (performance, accuracy, stability, or a particular field of research such as fluid dynamics). Ready-to-Go Routines As described above, one can use the diff function (link to documentation) to compute rough derivatives between adjacent array elements. MATLAB's gradient routine (link to documentation) is a great option for many purposes. It implements a second-order, central difference scheme. It has the advantages of computing derivatives in multiple dimensions and supporting arbitrary grid spacing. (Thanks to #thewaywewalk for pointing out this glaring omission!) I used Fornberg's method (see above) to develop a small routine (nderiv_fornberg) to calculate finite differences in one dimension for arbitrary grid spacings. I find it easy to use. It uses sided stencils of 6 points at the boundaries and a centered, 5-point stencil in the interior. It is available at the MATLAB File Exchange here. Conclusion The field of numerical differentiation is very diverse. For each method listed above, there are many variants with their own set of advantages and disadvantages. This post is hardly a complete treatment of numerical differentiation. Every application is different. Hopefully this post gives the interested reader an organized list of considerations and resources for choosing a method that suits their own needs. This community wiki could be improved with code snippets and examples particular to MATLAB.

I believe there is more in to these particular questions. So I have elaborated on the subject further as follows: (4) Q: What level of accuracy are you looking for? Do you need to compute the derivatives within a given tolerance? A: The accuracy of numerical differentiation is subjective to the application of interest. Usually the way it works is, if you are using the ND in forward problem to approximate the derivatives to estimate features from signal of interest, then you should be aware of noise perturbations. Usually such artifacts contain high frequency components and by the definition of the differentiator, the noise effect will be amplified in the magnitude order of $i\omega^n$. So, increasing the accuracy of differentiator (increasing the polynomial accuracy) will no help at all. In this case you should be able to cancelt the effect of noise for differentiation. This can be done in casecade order: first smooth the signal, and then differentiate. But a better way of doing this is to use "Lowpass Differentiator". A good example of MATLAB library can be found here. However, if this is not the case and you're using ND in inverse problems, such as solvign PDEs, then the global accuracy of differentiator is very important. Depending on what kind of bounady condition (BC) suits your problem, the design will be adapted accordingly. The rule of thump is to increase the numerical accuracy known is the fullband differentiator. You need to design a derivative matrix that takes care of suitable BC. You can find comprehensive solutions to such designs using the above link. (5) Does it matter to you that your derivative is evaluated on the same points as your function is defined? A: Yes absolutely. The evaluation of the ND on the same grid points is called "centralized" and off the points "staggered" schemes. Note that using odd order of derivatives, centralized ND will deviate the accuracy of frequency response of the differentiator. Therefore, if you're using such design in inverse problems, this will perturb your approximation. Also, the opposite applies to the case of even order of differentiation utilized by staggered schemes. You can find comprehensive explanation on this subject using the link above. (6) Do you need to calculate multiple orders of derivatives? This totally depends on your application at hand. You can refer to the same link I have provided and take care of multiple derivative designs.

### Using cross-validation to find the right value of k for the k-nearest-neighbor classifier

I am working on a UCI data set about wine quality. I have applied multiple classifiers and k-nearest neighbor is one of them. I was wondering if there is a way to find the exact value of k for nearest neighbor using 5-fold cross validation. And if yes, how do I apply that? And how can I get the depth of a decision tree using 5-fold CV? Thanks!

I assume here that you mean the value of k that returns the lowest error in your wine quality model. I find that a good k can depend on your data. Sparse data might prefer a lower k whereas larger datasets might work well with a larger k. In most of my work, a k between 5 and 10 have been quite good for problems with a large number of cases. Trial and Error can at times be the best tool here, but it shouldn't take too long to see a trend in the modelling error. Hope this Helps!

### How to define the maximum k of the kNN classifier?

I am trying to use kNN classifier to perform some supervised learning. In order to find the best number of 'k' of kNN, I used cross validation. For example, the following codes load some Matlab standard data and run the cross validation to plot various k values with respect to the cross validation error load ionosphere; [N,D] = size(X) resp = unique(Y) rng(8000,'twister') % for reproducibility K = round(logspace(0,log10(N),10)); % number of neighbors cvloss = zeros(numel(K),1); for k=1:numel(K) knn = ClassificationKNN.fit(X,Y,... 'NumNeighbors',K(k),'CrossVal','On'); cvloss(k) = kfoldLoss(knn); end figure; % Plot the accuracy versus k plot(K,cvloss); xlabel('Number of nearest neighbors'); ylabel('10 fold classification error'); title('k-NN classification'); The result looks like The best k in this case is k=2 (it is not an exhaustive search). From the figure, we can see that the cross validation error goes up dramatically after k>50. It gets to a large error and become stable after k>100. My question is what is the maximum k we should test in this kind of cross validation framework? For example, there are two classes in the 'ionosphere' data. One class labeled as 'g' and one labeled as 'b'. There are 351 instances in total. For 'g' there are 225 cases and for 'b' there are 126 cases. In the codes above, it chooses the largest k=351 to be tested. But should we only test from 1 to 126 or up to 225? Is there a relation between the test cases and the maximum number of k? Thanks. A.

The best way to choose a parameter in a classification problem, is to choose it by expertness. What you are doing certainly is not this. If your data is small enough to do a lot of classification with different values of parameters, you will do that, but to be reasonable, you need to show that the parameter you chose is not randomly chosen, you need to explain the behavior of plot you drawn. In this case, the function is ascending, so you can tell 2 is the best choice. In most cases you will not choose K more than 20, but there is no proof and you need to do the classification until you can proof your choice.

You don't want k to be too large (i.e. too close to the number of examples), because then the k neighborhood of each query example contains a large fraction of the space, so the prediction depends less and less on the actual location of the query and more on the overall statistics. This explains why the performance is not good for large k. Your classifier essentially chooses always 'g', and gets it wrong 126/351=35% as you see in the plot. Theory suggests that k needs to grow as the number of labeled examples grow, but sub-linearly. When you have lots of training data, you want k to be large because you want to have a good estimate of the likelihood of a point near the query point to get each label. This allows to imitate the maximum aposteriori decision rule (which is optimal, assuming you know the actual distribution). So here are some practical tips: Get more data if you can. Then run the experiment again. Focus on small values of k. My bet is that k=3 is better than k=2. Usually for binary classification k is at least 3, and usually an odd number (to avoid ties). The fact that you see that k=2 is better does not make sense. Therefore the only case in which k=1 is different than k=2 is when the 2 nearest neighbors have different labels. However, in this case the decision is made either randomly or arbitrarily (e.g. always choose 'g'). It depends on the implementation of the knn algorithm. My guess is that in the algorithm you are using the decision is fixed, and that in cases of a tie it chooses 'g' which just happens to be more likely overall. If you switch the roles of the labels you will probably see that k=1 is better than k=2. Would be interesting to see the the plot for small values of k (e.g. 1 - 20). References: nearest neighbor classification

Increasing the number of neighbors to be taken into account during the classification makes your classifier a mean value choice. You only need to check the ratio of your classes to see that it is equal to the error rate. Since you are using cross validation the k that corresponds to the minimum of your error rate is what you should select as value. In this case it is 3 if not mistaken. Keep in mind that the cross validation parameter introduces bias in your selection of k. A more elaborate analysis is needed there, but your 10 should be fine for this case.

### Measuring density for three dimensional data (in Matlab)

I have a dataset consisting of a large collection of points in three dimensional euclidian space. In this collection of points, i am trying to find the point that is nearest to the area with the highest density of points. So my problem consists of two steps: 1: Determine where density of the distribution of points is at its highest 2: Determine which point is nearest to the point found in 1 Point 2 i can manage, but i'm not sure how to solve point 1. I know there are a lot of functions for density estimation in Matlab, but i'm not sure which one would be the most suitable, or straightforward to use. Does anyone know? My command of statistics is a little bit rusty, but as far as i can tell, this type of problem calls for multivariate analysis. Someone suggested i use multivariate kernel density estimation, but i'm not really sure if that's the best solution.

Density is a measure of mass per unit volume. On the assumption that your points all have the same mass then you are, I suppose, trying to measure the number of points per unit volume. So one approach is to divide your subset of Euclidean space into lots of little unit volumes (let's call them voxels like everyone does) and count how many points there are in each one. The voxel with the most points is where the density of points is at its highest. This is, of course, numerical integration of a sort. If your points were distributed according to some analytic function (and I guess they are not) you could solve the problem with pencil and paper. You might make this approach as sophisticated as you like, perhaps initially dividing your space into 2 x 2 x 2 voxels, then choosing the voxel with most points and sub-dividing that in turn until your criteria are satisfied. I hope this will get you started on your point 1; you seem to be OK with point 2 so I'll stop now. EDIT It looks as if triplequad might be what you are looking for.