Category: Open Source

OxyPlot.Axes.LinearAxis is obsolete

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Warning 461 ‘OxyPlot.Axes.LinearAxis.LinearAxis(OxyPlot.Axes.AxisPosition, double, double, string)’ is obsolete

If you try to use OxyPlot.Axes.LinearAxis‘ constructor with parameters, the compiler will complain telling you the method has been deprecated and shouldn’t be used. I couldn’t find on the web which was the alternative solution to resolve this issue, but then it occurred to me that, what really is being deprecated, is the passage of arguments through the constructor’s parameters.

As such, the solution is simply to rewrite your code and call the axis’ default constructor instead, using C# object initialization syntax to configure your object:

Hope it can be of help if you were getting those warnings like me.

Liblinear algorithms in C#

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The Accord.NET Framework is not only an image processing and computer vision framework, but also a machine learning framework for .NET. One of its features is to encompass the exact same algorithms that can be found in other libraries, such as LIBLINEAR, but offer them in .NET withing a common interface ready to be incorporated in your application.

 

What is LIBLINEAR?

As its authors put, LIBLINEAR is a library for large linear classification. It is intended to be used to tackle classification and regression problems with millions of instances and features, although it can only produce linear classifiers, i.e. linear support vector machines.

The framework now offers almost all liblinear algorithms in C#, except for one. Those include:

Continue reading →

Screencast Capture

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I’ve recently started to record videos to demonstrate some capabilities of the Accord.NET Framework. Surprisingly, there were only a few, free, open source applications to achieve this goal – and none of them had all the features I needed.

It is, until I decided to roll my own.

 Screencast Capture Lite is a tool for recording the desktop screen and saving it to a video file, preserving quality as much as possible. However, this does not mean it produces gigantic files which take a long time to be uploaded to the web. The application encodes everything using solely H624 in an almost lossless setting.

As a demonstration, please take a look on the Youtube video sample shown below. However, note that Youtube actually reduced the quality of the video, even if you watch it in HD. The local copy produced by Screencast Capture has an even higher quality than what is being shown, while the generated video file occupied less than 2 megabytes on disk.

And by the way what would be a better approach to demonstrate the capabilities of the Accord.NET frameworks other than writing this application using them?

Well, actually this application has been created specifically for two things:

  • to aid in the recording of instructional videos for the Accord.NET Framework, and;
  • to serve itself as a demonstration of the use and capabilities of the Accord.NET Framework.

This means the application is written entirely in C# making extensive use of both aforementioned frameworks. The application is completely open source and free, distributed under the terms of the GPL, and a suitable project page is already being served on GitHub.

Hope you will find it interesting!

Deep Neural Networks and Restricted Boltzmann Machines

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diagram

The new version of the Accord.NET brings a nice addition for those working with machine learning and pattern recognition: Deep Neural Networks and Restricted Boltzmann Machines.

Class diagram for Deep Neural Networks in the Accord.Neuro namespace.

Deep neural networks have been listed as a recent breakthrough in signal and image processing applications, such as in speech recognition and visual object detection. However, is not the neural networks which are the new things here; but rather, the learning algorithms. Neural Networks have existed for decades, but previous learning algorithms were unsuitable to learn networks with more than one or two hidden layers.

But why more layers?

The Universal Approximation Theorem (Cybenko 1989; Hornik 1991) states that a standard multi-layer activation neural network with a single hidden layer is already capable of approximating any arbitrary real function with arbitrary precision. Why then create networks with more than one layer?
To reduce complexity. Networks with a single hidden layer may arbitrarily approximate any function, but they may require an exponential number of neurons to do so. We can borrow a more tactile example from the electronics field. Any boolean function can be expressed using only a single layer of AND, OR and NOT gates (or even only NAND gates). However, one would hardly use only this to fully design, let’s say, a computer processor. Rather, specific behaviors would be modeled in logic blocks, and those blocks would then be combined to form more complex blocks until we create a all-compassing block implementing the entire CPU.
The use of several hidden layers is no different. By allowing more layers we allow the network to model more complex behavior with less activation neurons; futhermore the first layers of the network may specialize on detecting more specific structures to help in the later classification. Dimensionality reduction and feature extraction could have been performed directly inside the network on its first layers rather than using specific separate algorithms. 

Do computers dream of electric sheep?

The key insight in learning deep networks was to apply a pre-training algorithm which could be used to tune individual hidden layers separately. Each layer is learned separately without supervision. This means the layers are able to learn features without knowing their corresponding output label. This is known as a pre-training algorithm because, after all layers have been learned unsupervised, a final supervised algorithm is used to fine-tune the network to perform the specific classification task at hand.

As shown in the class diagram on top of this post, Deep Networks are simply cascades of Restricted Boltzmann Machines (RBMs). Each layer of the final network is created by connecting the hidden layers of each RBM as if they were hidden layers of a single activation neural network.

Now, the most interesting part about this approach will given now. It is about one specific detail on how the RBMs are learned, which in turn allows a very interesting use of the final networks. As each layer is a RBM learned using an unsupervised algorithm, they can be seen as standard generative models. And if they are generative, they can be used to reconstruct what they have learned. And by sequentially alternating computation and reconstruction steps initialized with a random observation vector, the networks may produce patterns which have been created using solely they inner knowledge about the concepts it has learned. This may be seen fantastically close to the concept of a dream.

—

At this point I would also like to invite you to watch the video linked above. And if you like what you see, I also invite you to download the latest version of the Accord.NET Framework and experiment with those newly added features.

The new release also includes k-dimensional trees, also known as kd-trees, which can be use to speed up nearest neighbor lookups in algorithms which need it. They are particularly useful in algorithms such as the mean shift algorithm for data clustering, which has been included as well; and in instance classification algorithms such as the k-nearest neighbors.

Quadratic Programming in C#

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I have manually translated and adapted the QuadProg solver for quadratic programming problems made by Berwin A. Turlach. His code was originally published under the GNU Library License, which has now been superseded by the GNU Lesser License. This adapted version honors the original work and is thus distributed under the same license.

The code in this article is part of the Accord.NET Framework. Download the full framework through NuGet and use this code in your own applications.

Introduction

Despite the name, the terms linear or quadratic programming have little resemblance to the set of activities most people now know as programming. Those terms usually usually refers to a specific set of function optimization methods, i.e. methods which can be used to determine the maximum or minimum points of special kinds of functions under a given number of solution constraints. For example, suppose we would like to determine the minimum value of the function:

f(x, y) = 2x + y + 4

Under the constraints that x and y must be non-negative (i.e. either positive or zero). This may seem fairly simple and trivial, but remember that practical linear programming problems may have hundreds or even thousands of variables and possibly million constraints.

When the problem to be solved involves a quadratic function instead of a linear function, but still presents linear constraints, this problem can be cast as a quadratic programming problem. Quadratic functions are polynomial functions in each each term may have at most a total degree of 2. For example, consider the function

f(x, y, z) = 2x² + 5xy + y² – z² + x – 5.

Now let’s check the sum of the degrees for each variable on the polynomial terms. We start by writing the missing terms of the polynomial

f(x, y, z) = 2 x2y0z0 + 5 x1y1z0 + 2 x0y2z0 – x0y0z2 + x1y0z0 – 5 x0y0z0

and then proceed to check the sum of the degrees at each term. In the first term, 2+0+0 = 2. For the second, 1+1+0 = 2, and so on. Those functions have a nice property that they can be expressed in a matrix form

f(x) = 1/2 xT Q x + cTx.

Here, x and c are vectors. The matrix Q is a symmetric matrix specifying how the variables combine in the quadratic terms of the function. If this matrix is positive definite, then the function is convex, and the optimization has a single, unique optimum (we say it has a global optimum). The coefficients c specify the linear terms of our function.

Source code

The available source code is based on a translation of the Fortran code written by Berwin A. Turlach. However, some modifications have been made. Fortran uses column-major ordering for matrices, meaning that matrices are stored in memory in sequential order of column elements. Almost all other languages use row-major ordering, including C, C++, Java and C#. In order to improve data locality, I have modified the code to use the transpose of the original matrices D and A. I have also modified the QP formulation adopted in the Goldfarb and Idnani paper to reflect the form presented in the introduction.

In this article, we will be using the GoldfarbIdnani class from the Accord.NET Framework, as well as the QuadraticObjectiveFunction. Their source code are available at GoldfarbIdnani.cs and QuadraticObjectiveFunction.cs, respectively. Please note that the sample application version available in this blog post will most likely not be the most recently, updated, fixed and enhanced version of the code. For the latest version, be sure to download the latest version of the framework on the project site or through a NuGet package.

Using the code

The first step in solving a quadratic programming problem is, well, specifying the problem. To specify a quadratic programming problem, one would need two components: a matrix D describing the relationship between the quadratic terms, and a vector d describing the linear terms. Perhaps this would work better with an example.

Suppose we are trying to solve a minimization problem. Given a function, the goal in such problems is to find the correct set of function arguments which would result in the minimum possible value for the function. An example of a quadratic minimization problem is given below:

min f(x, y) = 2x² – xy + 4y² – 5x – 6y

subject to the constraints:

x – y = 5
x = 10
wolfram(generated with Wolfram Alpha)

However, note that this problem involves a set of constraints. The required solution for this minimization problem is required to lie in the interval specified by the constraints. More specifically, any x and y pair candidate for being a minimal of the function must respect the relations x – y = 5 and x >= 10. Thus, instead of lying in the unconstrained minimum of the function surface shown above, the solution lies slightly off the center of the surface. This is an obvious easy problem to solve manually, but it will fit for this demonstration.


wolfram2

As it can be seen (and also live demonstrated by asking Wolfram Alpha) the solution lies on the point (10,5), and the constrained minimum of the function is given by 170. So, now that we know what a quadratic programming problem looks like, how can we actually solve it?

Specifying the objective function

The first step in solving a quadratic programming problem is to specify the objective function. Using this code, there are three ways to specify it. Each of them has their own advantages and disadvantages.

1. Manually specifying the QP matrix.

This is the most common approach for numerical software, and probably the most cumbersome for the user. The problem matrix has to be specified manually. This matrix is sometimes denoted Q, D or H as it actually denotes the Hessian matrix for the problem.

The matrix Q is used to describe the quadratic terms of our problem. It is a n x n matrix, in which n corresponds to the number of variables in our problem, covering all possible combinations of variables. Recall our example given on the start of this section. We have 2 variables, x and y. Thus, our matrix Q is 2 x 2. The possible combinations for x and y are expressed in the table below.

x y
x x*x x*y
y y*x y*y

To form our matrix Q, we can take all coefficients associated with each pair mentioned on the table above. The diagonal elements should also be multiplied by two (this is actually because the matrix is the Hessian matrix of the problem: it is the matrix of all second-order derivatives for the function. Since we have only at most quadratic terms, the elementary power rule of derivation “drops” the ² from the x² and y² terms – I think a mathematician would hit me with a stick for explaining it like this, but it serves well for a quick, non-technical explanation).

Remember our quadratic terms were 2x²  – 1xy + 4y². Writing the terms on their proper position and differentiating, we have:


As it can be seen, the matrix is also symmetric (and often, but not always, positive definite). The next step, more trivial, is to write a vector d containing the linear terms. The linear terms are –5x –6y, and thus our vector d can be given by:


Therefore our C# code can be created like this:

2. Using lambda expressions

This approach is a bit more intuitive and less error prone. However, it involves lambdas functions and some people find it hard to follow them. Another disadvantage is that we will lose the edit & continue debugging ability of visual studio. The advantage is that the compiler may catch some obvious syntax errors automatically. Below we are using the QuadraticObjectiveFunction from Accord.NET to specify our target objective function.

Note that the x and y variables could have been initialized to any value. They are only used as symbols, and not used in any computations.

3. Using text strings

This approach is more intuitive but a bit more error prone. The function can be specified using strings, as in a standard mathematical formula. However, since all we have are strings, there is no way to enforce static, compile time checking.

Couldn’t be easier.

Specifying the constraints

The next step in specifying a quadratic programming problem is to specify the constraints. The constraints can be specified in almost the same way as the objective function.

1. Manually specifying the constraints matrix

The first option is to manually specify the constraints matrix A and vector b. The constraint matrix expresses the way the variables should be combined when compared to corresponding value on vector b. It is possible to specify either equality constraints or inequality constraints. The formulation used in this code is slightly different from the one used in Turlach’s original implementation. The constraints are supposed to be in the form:

A1 x = b1
A2 x = b2

This means that each line of matrix A expresses a possible combination of variables x which should be compared to the corresponding line of b. An integer variable m can be specified to denote how many of the first rows of matrix A should be treated as equalities rather than inequalities. Recall that in our example the constraints are given by 1x –1y = 5 and 1x = 10. Lets write this down in a tabular form:

# x y ? b
q1 1 -1 = 5
q2 1 0 = 10

Thus our matrix A and vector b can be specified as:

And not forgetting that m = 1, because the first constraint is actually an equality.

2. Using classes and objects

A more natural way to specify constraints is using the classes and objects of the Accord.NET Framework. The LinearConstraint class allows one to specify a single constraint using an object-oriented approach. It doesn’t have the most intuitive usage on earth, but has much more expressiveness. It can also be read aloud, it that adds anything! 🙂

The specification is centered around the notion that variables are numbered and have an associated index. For example, x is the zero-th variable of the problem. Thus x has an index of 0 and y has an index of 1. So for example, reading aloud the last constraint, it is possible to express how the variables at indices 0 and 1, when combined as 1x and –1y, should be equal to value 5.

2. Using lambda expressions

A more intuitive way to express constraints is again using lambda expressions. And again the problems are the same: some people find it hard to follow and we lose edit & continue.

3. Using text strings

Same as above, but with strings.

Finally, creating and solving the problem

Once we have specified what do we want, we can now ask the code for a solution. In case we have opted for manually specifying the objective function Q and d, and the constraint matrix A, vector b and integer m, we can use:

In case we have opted for creating a QuadraticObjectiveFunction object and a list of constraints instead, we can use:

After the solver object has been created, we can call Minimize() to solve the problem. In case we have opted for manually specifying Q and d, we can use:

The solution will be available in the Solution property of the solver object, and will be given by:

Sample application

The Accord.NET Framework now includes a sample application demonstrating the use of the Goldfarb-Idnani Quadratic Programming Solver. It can be downloaded at the Accord.NET Framework site, and also comes together with recent versions of the framework (> 2.6).

Solver

Solver sample application included in the Accord.NET Framework.

Remarks

Because the code has been translated by hand (in contrast of using automatic translators such as f2c) there could be potential bugs in the code. I have tested the code behavior against R’s quadprog package and still didn’t find errors. But this does not mean the code is bug-free. As always, as is the case of everything else in this blog, this code is published in good faith, but I can not guarantee the correctness of everything. Please read the disclaimer for more information.

References

Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) for Unconstrained Optimization in C#

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MSP120691a0731732i9i4h220000462e1a4c173g8b95-255B6-255D

The Limited-memory Broyden-Fletcher-Goldfarb-Shanno method is an optimization method belonging to the family of quasi-Newton methods for unconstrained non-linear optimization. In short terms, it is an off-the-shelf optimizer for seeking either minimum or maximum points of a any differentiable and possibly non-linear function, requiring only an expression of the function and its gradient. The goal of this article is to provide and demonstrate a C# port of the L-BFGS method originally written by Jorge Nocedal in Fortran.

Introduction

Function optimization is a common problem found in many numerical applications. Suppose we have a differentiable function f : Rn → R and we would like to obtain its minimal or maximum value while traversing its space of possible input values. Those kind of problems arise often in parameter optimization of machine learning models and other statistic related applications (and in a myriad of other applications too – however, we will pay special attention to cases related to machine learning).

The problem of maximizing or minimizing a function can be simply stated as max f(x) or min f(x). When there aren’t any constraints limiting possible values for x, it is widely known from calculus that the maximum or a minimum of such a function would occur when the first derivatives f’(x) of the function f(x) in respect to x are zero. Newton’s method is a common method for finding the roots of a differentiable function and is indeed a suitable method for finding those values such that first derivatives of a function are zero.

 

Surface of the example function. Contour lines of the example function.

Example of a convex, but non-linear function f(x,y) = exp{-(x-1)²} + exp{-(y-2)²/2}. Images have been created using Wolfram Alpha.

 

However, while Newton’s method may work very well with functions of a single variable, the generalization to higher dimensions will require the replacement of the first derivative f’(x) with the function’s gradient vector g of partial derivatives, and the second derivative f’’(x) with the inverse Hessian matrix H-1. The problem is that the Hessian is a square matrix with as many rows and columns as parameters of our function f, and, besides being very costly and challenging to compute, even its size may be infeasible to accommodate in main memory when dealing with very high dimensional problems.

To overcome those limitations, several methods attempt to replace the Hessian matrix with an approximation extracted from the first partial derivatives alone. Those methods are called quasi-Newton methods, as they replace H in Newton’s method by an approximation instead of using the full Hessian. In case the function being optimized has any particular form or special characteristic which could be exploited, it is also possible to use more specialized methods such as the Gauss-Newton, Levenberg-Marquardt or even lower-bound approximation methods to avoid computing the full Hessian.

The quasi-Newton BFGS method for non-linear optimization builds an approximation for the Hessian matrix based on estimates extracted solely from first order information. However, even being cheaper to compute, the memory requirements are still high as the method still needs to accommodate a dense N x N matrix for the inverse Hessian approximation (where N is the number of variables for the function). To overcome this problem, the L-BFGS method, or the limited-memory version of the BFGS method, never forms or stores the approximation matrix, but only a few vectors which can be used to represent it implicitly.

Source code

The code presented here is based upon a direct translation of the original code by Jorge Nocedal. His original code was released under the permissive BSD license, and honoring the original license and the author’s considerations, this code is released under the BSD as well.

L-BFGS Class diagrams

The L-BFGS method is implemented within the BroydenFletcherGoldfarbShanno class. Upon object creation, it is possible to specify a function and its gradient through either the use of Lambda expressions or by specifying handles for named functions. After the object has been initialized, a simple call to Minimize runs the standard L-BFGS method until convergence. The details for the Minimize method is given below.

The version of the code detailed here only supports Minimization problems. However, it is possible to note that any minimization problem can be converted into a maximization problem and vice-versa by taking the opposite of the function and its gradient.

Using the code

Code usage is rather simple. Suppose we would like to maximize the function g(x,y) = exp{-(x-1)²} + exp{-(y-2)²/2}:

Since we would like to perform a maximization, we first have to convert it to a minimization problem. The minimization version of the function is simply given by taking f(x,y) = –g(x,y):

Which is a convex function, as can be seen by plotting its surface. As it can be seen, the minimum of the function lies on the point (x,y) = (1,2). As expected, this point coincides with the roots of the partial derivative functions, as shown in the line plots below:

 

Example optimization cast as an minimization problem.

The minimization problem min f(x,y) = -(exp(-(x-1)²) + exp(-(y-2)²/2)), computed by Wolfram Alpha.

Partial derivative for x. Partial derivative for y.

The roots of the partial derivatives in respective to x (left) and y (right). It is possible to see that the roots occur at 1 and 2, respectively. Images computed using Wolfram Alpha. 

 

The first step in solving this optimization problem automatically is first to specify the target function. The target function f, in this case, can be specified through a lambda function in the form:

The next step is to specify the gradient g for the function f. In this particular case, we can manually compute the partial derivatives to be df/dx = -2 e^(-(x-1)^2) (x-1) and df/dy = -e^(-1/2 (y-2)^2) (y-2), in respect to x and y, respectively. Writing a lambda function to compute the gradient vector g, we have:

We can also note that this example was rather simple, so the gradient vector was easy to calculate. However, the gradient could also have been computed automatically using Accord.NET‘s FiniteDifferences class. In either case, all we have to do now is to create our L-BFGS solver and call its Minimize() method to begin optimization:

After the optimization is complete, the solution parameter will be available in the Solution property of the lbfgs object, and will be equal to { 1.0, 2.0 }. And also in case one is interested in progress reports (such as in the case of optimizing very large functions), it is also possible to register a listener to the Progress event handler. The complete version of the sample application accompanying the source code is given below:

Conclusion

In this post, we showed how to use a reduced set of the Accord.NET Framework‘s to perform non-linear optimization. This routine is also used by the Conditional Random Fields and Hidden Conditional Random Fields trainers to optimize parameters for such models. The solver routines have been adapted from the original Fortran’s source code from Nocedal, which, tracing back to a 2001 message from the author, also have been reported to be available under the public domain. Nevertheless, the reduced set of the framework available for download within this post is available under a BSD license, as an alternative to the version available within the Accord.NET Framework which is available only under a LGPL license.

As always, I hope someone will find it useful 🙂

References

Gaussian Mixture Models and Expectation-Maximization

/ cesarsouza / 48 Comments

Like K-Means, Gaussian Mixture Models (GMM) can be regarded as a type of unsupervised learning or clustering methods. They are among the most statistically mature methods for clustering. But unlike K-Means, GMMs are able to build soft clustering boundaries, i.e., points in space can belong to any class with a given probability.

The code presented here is part of the Accord.NET Framework. The Accord.NET Framework is a framework for developing machine learning, computer vision, computer audition, statistics and math applications in .NET. To use the framework in your projects, install it by typing Install-Package Accord.MachineLearning in your IDE’s NuGet package manager.

Contents

gaussians

  1. Introduction
  2. Source code
    1. Normal Distributions
    2. Mixture Distributions
    3. Expectation-Maximization
    4. Gaussian Mixture Models
  3. Using the code
  4. Sample application
  5. Conclusion
  6. References
  7. See also

Introduction

In statistics, a mixture model is a probabilistic model which assumes the underlying data to belong to a mixture distribution. In a mixture distribution, its density function is just a convex combination (a linear combination in which all coefficients or weights sum to one) of other probability density functions:

The individual pi(x) density functions that are combined to make the mixture density p(x) are called the mixture components, and the weights w1, w2, …, wn associated with each component are called the mixture weights or mixture coefficients.

The most common mixture distribution is the Gaussian (Normal) density function, in which each of the mixture components are Gaussian distributions, each with their own mean and variance parameters.

To give a more solid and concrete understanding of the Gaussian mixture models, we will be jumping directly on how to represent those abstractions in source code through the use of class diagrams. Hopefully class diagrams may give a better and direct overview on the subject than a lengthy theoretical discussion.

Source code

First we will discuss about the characteristics of any probability distribution. The IDistribution interface (depicted below) says any probability distribution may have a probability function and a distribution function. It also says that probability distributions can also be fitted to sets of observations through a parameter estimation method.

IDistribution

Since distribution functions can be either univariate or multivariate, the methods above accept any number of scalar values as input through the use of the params keyword. The Fit method for parameter estimation also accepts a general System.Array because we could be given either an array of univariate observations in the form of a double[] or an array of multivariate observations in the form of a jagged double[][] array. Since each observation may have a different weight in the estimation process, the weights parameter can be used to inform those weights to the fitting method.

Probability distributions may also have some associated measures, mainly the Expectation and the Variance of the modeled random variable. Depending if the distribution is univariate or multivariate, the expected values and variances can be either a single scalar or a vector of scalars. For multivariate distributions the variance can be also represented by a symmetric, positive-definite matrix known as the variance-covariance matrix.

IDistributions

Moreover, probability distributions can also be continuous or discrete. In continuous distributions the probability function is known as the Probability Density Function (pdf), and in discrete distributions the probability function is known as the Probability Mass Function (pmf). However, since our distribution of interest, the Gaussian, is a continuous distribution, be focusing only on continuous multivariate distributions:

All

The Normal (Gaussian) distribution and the mixture distributions fall under the multivariate continuous distributions category and are implemented as such.

Normal distributions

The Normal (or Gaussian) multivariate distribution is a multivariate distribution whose parameters are the mean vector μ and a variance-covariance matrix Σ. Its probability density function is given by:

The more detailed class diagram for the normal distribution shows the multivariate characteristics of most of its methods, in particular of the Fit method for estimating Gaussian parameters from a multivariate set of data. Each sample, given in the form of a double[], may also have different associated weights. In this case, the weight associated with each of the samples can be passed through the weights parameter. The source code for the Gaussian distribution can be found in the NormalDistribution.cs class.

NormalDistribution

To estimate the parameters of a Gaussian distribution all we have to do is compute the mean and the variance-covariance matrix out of the given data. This is a very straightforward procedure and can be done by using the standard methods of the Tools class in the Accord.Statistics namespace. The Fit method updates the distribution parameters, overwriting the current distribution.

Mixture distributions

Mixture distributions are just convex combinations of other probability distributions. Thus said, mixture distributions have an array of component distributions and a coefficient array which contains the weights of the each of the component probability distributions.

Mixture

In the generic implementation above, T is the type of the distributions used in the mixture. The most common mixture distribution, the Gaussian Mixture Distribution, could then be created by instantiating a Mixture object passing the initial Normal distributions as constructor parameters.

Alternatively, the parameters of the mixture distributions could be estimated from a set of observations by calling the Fit method. To estimate the parameters of a mixture distribution we will be using a common technique known as the Expectation-Maximization algorithm.

Expectation Maximization

Expectation-Maximization (EM) is a well established maximum likelihood algorithm for fitting a mixture model to a set of training data. It should be noted that EM requires an a priori selection of model order, namely, the number of M components to be incorporated into the model. In the Accord.NET Framework, it can be called implicitly by using the Mixture.Fit method or explicitly using the ExpectationMaximization class. The source code for Expectation Maximization can be found in the ExpectationMaximization.cs file.

The general E-M algorithm is comprised of the following simple steps:

  1. Initialization

    Initialize the distribution parameters, such as the means, covariances and mixing coefficients and evaluate the initial value of the log-likelihood (the goodness of fit of the current distribution against the observation dataset)’;

  2. Expectation

    Evaluate the responsibilities (i.e. weight factors of each sample) using the current parameter values;

  3. Maximization

    Re-estimate the parameters using the responsibilities found in the previous step;

  4. Repeat

    Re-evaluate the log-likelihood and check if it has changed; if it has changed less than a given threshold, the algorithm has converged.

 

Below is the actual realization of the algorithm as implemented in the Fit method of the Mixture<T> class.

Gaussian Mixture Models

Gaussian mixture models are among the most commonly used examples of mixture distributions. The GaussianMixtureModel class encompasses a Mixture<NormalDistribution> object and provides methods to learn from data and to perform actual classification through a simplified interface.

Class diagram for Gaussian Mixture Model

Moreover, a common problem which rises in mixture model fitting through E-M is the proper initialization of the mixture parameters. One popular approach is to initialize the mixture parameters using the centroids detected by the K-Means algorithm. The code below shows how the mixture parameters are computed by first creating a KMeans object and then passing the clusters to the mixture model for further processing.

Using the code

Code usage is rather simple. First, instantiate a new GaussianMixtureModel object passing the desired number of components in the Gaussian mixture. Then, call the Compute(double[][], double) method passing the learning samples and a desired convergence threshold.

After training has been completed, a new sample can be classified by calling the Classify(double[]) method.

Sample application

The accompanying sample application demonstrates how Gaussian Mixture Models can be used to cluster normally-distributed samples of data. By clicking the “Create random data” button, a given number of Normal distributions will be created in the two-dimensional space using random mean and covariance matrices.

After the data has been created, you can use the “Fit a Gaussian Mixture Model” button to fit a mixture of Gaussians to the data. Each of the Gaussians will receive a random color and the samples which have the greatest probability of belonging to any of the Gaussians will be colored accordingly.

Sample7 Sample8

Left: a random dataset containing three Gaussian clusters. Right: the clusters as identified by the Gaussian Mixture Model.

Sample9 Sample10

Left: a random dataset containing 10 Gaussian clusters. Right: the clusters as identified by the Gaussian Mixture Model.

Conclusion

In this article we found how Gaussian Mixture Models can be successfully used to create soft clustering boundaries around data. Those soft boundaries are possible because in a mixture model each sample is said to belong to a cluster only within certain probability.

A main drawback of GMMs is that the number of Gaussian mixture components, as in K-Means, is assumed known as prior, so it cannot be considered as totally unsupervised clustering method. To aid in this situation, one could use additional algorithms such as the Minimum Message Length (MML) criteria.

Another problem arises with the correct initialization of the mixture components. A poor choice of initial parameters will invariably lead to poor results, as the E-M algorithm converges only to a local optimization point. A commonly used solution is initialization by randomly sampling in the mixture data. Other common solution, covered by the implementation presented here, is to use K-Means to perform initialization. However, K-Means itself may also converge to a poor solution and impact negatively in the mixture model fitting.

References

  • Raja, Y., Shaogang, G. Gaussian Mixture Models. Department of Computer Science, Queen Mary and Westfield College, England.
  • Bishop, C. M. (2007), Pattern Recognition and Machine Learning (Information Science and Statistics), Springer.
  • Wikipedia contributors, “Normal distribution,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Normal_distribution (accessed October 18, 2010).
  • Wikipedia contributors, “Probability density function,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Probability_density_function (accessed October 18, 2010).
  • Wikipedia contributors, “Mixture density,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Mixture_density (accessed October 18, 2010).

See also

K-Means clustering in C#

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K-Means is one of the most popular, classic and simple approaches to clustering. Clustering is a method of unsupervised learning, and a common technique for statistical data analysis used in many fields, including machine learning, data mining, pattern recognition, image analysis and bioinformatics [3].

The code presented here is also part of the Accord.NET Framework. The Accord.NET Framework is a C# framework for developing machine learning, computer vision, computer audition, statistics and math applications in .NET. It is based on the AForge.NET Framework. Please see the starting guide for mode details. The latest version of the framework includes the latest version of this code plus many other statistics and machine learning tools.

Contents

  1. Introduction
    1. Lloyd’s K-Mean algorithm
  2. Source code
  3. Using the code
  4. Sample application
  5. Conclusion
  6. Acknowledgements
  7. References
  8. See also

Introduction

In statistics and machine learning, k-means clustering is a method of cluster analysis which aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean [4]. In its most common form, the algorithm is an iterative greedy algorithm which converges to a local optimum after a certain number of iterations.

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Illustration of the K-Means algorithm.

The algorithm works by first selecting k locations at random to be the initial centroids for the clusters. Each observation is then assigned to the cluster which has the nearest centroid, and the centroids are recalculated using the mean value of assigned values. The algorithm then repeats this process until the cluster centroids do not change anymore, or until the change less than a given threshold.

There are other refinements and extensions of the algorithm. The version depicted above is its most common form, also referred as Lloyd’s algorithm.

Lloyd’s K-Means algorithm

  1. Place K points into the space represented by the objects that are being clustered. These points represent initial group centroids.
  2. Assign each object to the group that has the closest centroid.
  3. When all objects have been assigned, recalculate the positions of the K centroids.
  4. Repeat Steps 2 and 3 until the centroids no longer move. This produces a separation of the objects into groups from which the metric to be minimized can be calculated.

 

Source code

The source code has been implemented using Accord.NET and is now part of the framework. In the current version (2.1.2), the following classes related to K-Means are contained inside the Accord.MachineLearning namespace. The source code available in this page contains only the parts of the framework actually needed to support the algorithm.

Class diagram for K-MeansClass diagram for the K-Means algorithm.

The KMeans class is the main class representing the K-Means algorithm. The algorithm itself is implemented in the Compute(double[][] data, double threshold) method, which accepts a set of observations and a convergence threshold to determine when the method should stop.

 

The implementation is quite straightforward and does not use additional techniques to avoid convergence problems. More refined techniques may be added to the implementation in the future, so please make sure to download the latest version of Accord.NET Framework for the most up-to-date revision of the code.

Using the code

To use the code, create a new instance of the KMeans class passing the desired number of clusters to its constructor. Additionally, you may also pass a distance function to be used as a distance metric during clustering. The default is to use the square Euclidean distance.

 

Sample application

The k-means clustering algorithm is commonly used in computer vision as a form of image segmentation. The results of the segmentation are often used to aid border detection and object recognition. The sample application performs image segmentation using the standard squared Euclidean distance over RGB pixel color space. There are, however, better distance metrics to be used for image segmentation, such as weighted distances and other color spaces, which will not be addressed in this example.

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Original image (from Ossi Petruska Flickr page*).

To perform image segmentation, we will first translate our image into an array of pixel values. The single image will be read, pixel by pixel, into a jagged array where each element is a double array of length 3. Each element of those double array will contain one of the three RGB values scaled to the interval [–1,1].

After we perform clustering on this array of pixel values, each pixel will have an associated cluster label. Each of these values will then be swapped by its corresponding cluster centroid. The source code below is called when one clicks the Run button in the application.

After segmentation, the following resulting images can be obtained:

kmeans5

Same image after K-Means clustering with k = 5.

kmeans10

Image after K-Means clustering with k = 10.

 

* The sample image used above has been licensed by Ossi Petruska in a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Generic license.

Conclusion

K-Means is a very simple and popular approach to clustering. The implementation presented here is the same implementation used in Accord.NET. As it can be seem, the method can be easily extended with custom distance functions through delegates or lambda expressions, and can be used in different contexts, such as image segmentation, without further modifications. As a suggestion for improvement, the method can be further speeded up by using the triangle inequality as suggested on the paper "Using the triangle inequality to accelerate k-means", by Charles Elkan.

In the next article, we will see how we can use K-Means to initialize the Expectation-Maximization algorithm for estimating Gaussian Mixture densities in Gaussian Mixture Models. Those articles will form the basis for Continuous density Hidden Markov Models.

Acknowledgements

To Antonino Porcino, who provided the first version of the code and for the valuable information about many other methods and algorithms.

References

See also

Handwritten Digits Recognition with C# SVMs in Accord.NET

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handwritten-recognition

I’ve posted a new article on CodeProject, entitled “Handwriting Recognition Revisited: Kernel Support Vector Machines”. It is a continuation of a previous article on handwritten digits recognition but this time using SVMs instead of KDA.

handwritten-recognition

The code uses the SVM library in Accord.NET Framework. The framework, which runs on .NET and is written mostly in C#, supports standard or multiclass support vector machines for either classification or regression, having more than 20 kernel functions available to choose from.

In the article, the same set and the same amount of training and testing samples have been used as in the previous Kernel Discriminant Analysis article for comparison purposes. The experimental results shows that  SVMs can outperform KDA both in terms of efficiency and accuracy, requiring much less processing time and memory available while still producing robust results.

Automatic Image Stitching with Accord.NET

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I have just posted a new article on CodeProject, entitled Automatic Image Stitching with Accord.NET. It is a demonstration of automatic image stitching by interest point matching using the Accord and AForge.NET Frameworks.

 

Automatic Image Stitching in C# using Accord.NET Framework

 

The method used is pretty classic in the computer vision literature. There are many other variations of the method with their own advantages and disadvantages, but most of them are built around the same ideas – feature detection, matching and blending. This is one of the most straightforward and free implementations available, since some of those most sophisticated methods are patented.