Implementation of Dynamic Logic Algorithm for Detection of EM Fields Scattered by Langmuir Soliton

In plasma diagnostics, scattering of electromagnetic waves is widely used for identification of density and wave field perturbations. Very often difficulties with implementations of this type of diagnostics are connected with the fact that scattered signals have small amplitudes comparable with the noise level. Under such conditions it is very important to implement the algorithm which allows us to identify and separate the scattered wave spectrum from the noise pattern. In the present work we use a powerful mathematical approach, dynamic logic (DL) [1], to identify the spectra of electromagnetic (EM) waves scattered by a Langmuir soliton – a one-dimensional localized wave structure. Such wave structures are understood to be basic elements of strong plasma turbulence.


Introduction
In plasma diagnostics, scattering of electromagnetic waves is widely used for identification of density and wave field perturbations. Very often difficulties with implementations of this type of diagnostics are connected with the fact that scattered signals have small amplitudes comparable with the noise level. Under such conditions it is very important to implement the algorithm which allows us to identify and separate the scattered wave spectrum from the noise pattern. In the present work we use a powerful mathematical approach, dynamic logic (DL) [1], to identify the spectra of electromagnetic (EM) waves scattered by a Langmuir soliton -a one-dimensional localized wave structure. Such wave structures are understood to be basic elements of strong plasma turbulence.
We are interested in the implementation of the Dynamic Logic algorithm for diagnostics of electromagnetic waves scattered by Langmuir soliton in the presence of random noise with amplitudes comparable with the amplitudes of the scattered waves. Scattering of EM waves on Langmuir soliton was previously analyzed by many authors (see for example Sinitzin, 1978Sinitzin, , 1979Mendoca, 1983).
For our purposes we will use the results for electromagnetic wave (EM) scattering on Langmuir soliton obtained in Mendonca, 1983, where scattering on low frequency part of density perturbations (density well) as well as scattering on high frequency part of density perturbations was examined and corresponding expressions for the amplitudes of the scattered waves were obtained.

EM scattering by Langmuir soliton
Langmuir soliton is a solution to the well known Zakharov equations (Zakharov, 1972 Fundamental object of the theory of strong Langmuir turbulence, Langmuir soliton (Rudakov, 1972), can be obtained from the equations (1)-(2) when we are looking for traveling wave solutions in the form: with imposed boundary conditions () 0 Ex  and with LS pe the unit vector along the x-axis.
Below we will use a well known solution to (1)-(2) which is given by: where v Te and De r are electron thermal speed and Debye radius.
Electric field of an EM wave interacting with a Langmuir soliton can be written in the following form: where p e is the polarization unit vector, 0  and 0 k are the frequency and the wave vector of the EM wave. These are connected through the dispersion relation: In the process of scattering the EM wave will interact with electron density perturbations associated with a Langmuir soliton. These density oscillations consist of high and low frequency parts: It is well known that the ratio 12 / nn   is of the order of unity and we need to take into account contributions to the scattered EM fields from both high and low frequency parts of density perturbations. As a result far away from the soliton it is possible to obtain the scattered wave field in the following asymptotic form (Mendonca, 1983): As it follows from (13)-(16) at large distances from the soliton it is possible to detect four scattered waves. Below for illustrative purposes we are presenting these four scattered waves which appear due to interaction of the incident EM wave, propagating at the 45 0 angle in x-y plane with the moving along the x -axis soliton.
The pattern of the total normalized electric field amplitudes of scattered EM waves given in (13) is presented in Fig. 2.
The Fourier spectrum of scattered EM waves is presented in Figure 3. Below we will use this spectrum in implementation of the Dynamic Logic algorithm.
We are interested in implementation of Dynamic Logic algorithm for diagnostics of electromagnetic waves scattered by Langmuir soliton in the presence of random noise with amplitudes comparable with the amplitudes of the scattered waves. Imposing the random   www.intechopen.com noise pattern on top of the Fourier spectrum in Figure 3 with amplitudes comparable with that of the scattered EM waves we will obtain the following 2D pattern in the Fourier space: Fig. 4. Combined Fourier spectrum, consisting from the spectrum of scattered EM waves and random noise pattern with comparable amplitudes.
As it clearly seen in Figure 4, scattered EM wave spectrum is hidden in the noise pattern.
Our aim is to apply Dynamic Logic algorithm in order to extract information on scattered EM waves from the pattern in Figure 4.

Implementation of the dynamic logic algorithm
In order to demonstrate the basics of the DL algorithm we will present below one illustrative example. Suppose we have two classes 1  and 2  . We also have samples . In Figure 5 two groups of samples are presented. Red dots define samples from class k=1 and blue dots -from class k=2. Typically, due to random variations there is good reason to expect the features in every class to be distributed normally (that is, according to the well-known Gaussian function, or "bell curve"). A normal distribution is completely specified by its set of parameters , where M is the mean vector and  is the covariance matrix (related to the width, or "spread", of the distribution). The equation for a normally-distributed probability density function (pdf) for the class k=1 is: In (17) is the probability that sample n with the feature vector and components of the mean vector are defined as: Components of the covariance matrix 1  for the class 1  are defined as: When we have N training samples 1,n X which belong only to class 1  the procedure to find the parameters of the Gaussian distribution 11 is as follows [6]. We need to introduce the likelihood function Logarithm of the likelihood function : 11 1 , 1 1 can be maximized by setting to zero its partial derivatives with respect to the parameters 11 (23) gives the probability (|) Pk n that sample n belongs to class k. To classify the unlabelled sample n, we simply choose the class k', for which probability is the highest.
Equations (23) is just an expression of Baye's rule. Note that in equation (23) it is assumed we know the a priori probabilities P(k) for each class. Obviously, for each sample the probabilities must add up to one, i.e. After we described how to estimate the mean and covariance of a single normal (Gaussian) distribution from a collection of data samples, below we will describe how to estimate parameters when the pdf is a weighted mixture, or summation, of Gaussian functions. Gaussian mixtures can be used to model complicated data sets. In such a case, each source corresponds to a class k which is parameterized by its own mean and covariance which we need to estimate. In addition to the mean and covariance, we also need to estimate the a priori probabilities for each class, i.e. what fraction of the samples belong to each class. The estimation of parameters for mixture models is complicated by the fact that the data samples are unlabelled, i.e. we don't know which data samples correspond to which class. Because of this inconvenience, it will be impossible to compute direct analytical expressions for the maximum-likelihood parameter estimates. However, we will be able to derive a convergent, iterative process to estimate the parameters. To do so we will perform the following basic procedure. First, a generic model for the data is defined in which the parameters for each class k are assumed to be the only unknowns. Next, a metric is selected which will be used to evaluate the "goodness of fit" between the model and the data for certain set of parameter values. Finally, an algorithm is developed to maximize the "goodness of fit" with respect to the parameters. Below we will describe maximum likelihood (ML) parameter estimation, in www.intechopen.com which we seek to maximize the log-likelihood of the parameters with respect to a collection of data samples. This approach is appropriate for standard data in which all samples are weighted equally in terms determining model parameters.
Next, we introduce an alternative known as maximum entropy (ME) parameter estimation.
Here the parameters are chosen to maximize the Einsteinian log-likelyhood. This metric is appropriate for processing digitized image data, in which samples are weighted according to their pixel values.

Maximum likelihood (ML) estimation
We now describe maximum-likelihood (ML) parameter estimation for a Gaussian Mixture [1,6]. Mathematically, the mixture model for two classes k=1 and k=2 is represented by the equation: where k r is the mixture weight for class k and 11 22 {, , , } rr    is the total set of parameters for the mixture. Also (, ) kk k   M is the subset of mean and covariance parameters for class k. Each class k has the normal (Gaussian) distribution:

XX M X M
Since k r denotes the fraction of the total samples that belong to class k, it is, by definition, equivalent to the a priori probability P(k) for class k. It is a well-known fact that these probabilities should sum to one, i.e.: Our goal is to compute the maximum likelihood (ML) parameter estimates 11 22 {, , , } rr        given the additional constraint in equation (25). As in the previous Section, this is accomplished by a process that incorporates the derivatives of the loglikelihood with respect to each parameter: 11 1 () l n( |) l n ( | ) The maximum-likelihood parameter estimates exactly satisfy the equations: Equation (31) is actually a form of Baye's and can be interpreted as the association probability, i.e. the probability that sample n belongs to class k. Equations (28) and (29) are intuitively pleasing in the sense that they are identical to their counterparts Eqs. (21) and (22) for the normal distribution, except that here each sample is weighted by the association probability P(k/n).
Although Eqs. (28)-(31) look simple enough, they are actually a set of coupled nonlinear equations, because the quantity P(k/n) on the right-hand sides is a function of all the parameters   , as can be seen from equations (31)   To initialize the procedure, parameter values are assigned at random, or by other means. Then the above set of equations is repeated until convergence, typically on the order of 100 iterations.
Below in Figures 6 -7 results of implementation of the DL algorithm for the problem of identification of samples which belong to two different classes 1  and 2  described above and illustrated in Figure 5, are presented. Fig. 6. With increasing iterations, the Dynamic Logic (DL) model adapts to segment data samples drawn from multiple random processes. In this example, the data samples are drawn from two Gaussian random processes, and the DL model consists of two components. The dots and circles represent the data samples, while the black ellipses represent the 2-sigma extent of the DL covariance, centered at the mean for each DL component. (a) Initially, the mean parameters for the components are assigned random initial values, and the covariance parameters are initialized in order to roughly span the sample data. (b) After several iterations the DL components begin to partition the data in a fuzzy manner as their evolving mean and covariance parameters cause them to "shrink" to fit each sample cluster. (c) Upon convergence, the DL model correctly segments the samples, and the mean and covariance parameters of the DL components provide good estimates for the mean and covariance of the underlying random processes.
In practice, one can decide whether convergence has been obtained by plotting the loglikelihood vs. iteration number and looking for a leveling of in the curve. The procedure is guaranteed to converge, in the sense that LL will never increase between successive iterations. However, convergence to an extraneous local minimum is sometimes possible. If this error occurs, and can be detected, the user must repeat the procedure after choosing a new set of initial conditions. The iterative procedure described above was pioneered by Perlovsky [1] and, independently, by others. In fact, the procedure has been shown to be an application of the well-known expectation-maximization (or EM) algorithm which is described, for example, in [20] and the references therein.

Maximum entropy (ME) estimation
In ML estimation, discussed above, we utilized the log-likelihood to determine the "goodness of fit" between the model and the data. This well-known metric is appropriate for standard data in which all samples are weighted equally in terms determining model parameters. However for some data sets, for example those involving digitized images, we require a different metric that will incorporate the natural weighting of samples corresponding to their pixel values [1,6]. For these problems we now introduce the metric known as the Einsteinian log-likelihood: For convenience we will assume that the data 0 () n p X is normalized, i.e.: so that the data and model have the same total energy, the k r must sum to one, i.e.: Our goal is to compute the maximum entropy (ME) parameters   which maximize LL E given by equation (39) subject to the constraint in equation (42). The mathematical derivation for the ME parameter estimates is almost identical to the mathematics governing the ML estimates. The result is that the ME parameters exactly satisfy the equations: As described in Section IV, the parameter estimation equations, above, are actually a set of coupled nonlinear equations, due to the implicit dependence within the bracket notation on the unknown parameters. However, we can compute the ME parameter estimates using a convergent, iterative process, identical to the process described in Section IV for the ML parameter estimates. In this process we alternate, in each iteration, between updating the probabilities (|) Pk n and the parameters { k M , k  , k r }.

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Applying maximum entropy estimation method to the problem of detection of scattered by a Langmuir soliton EM waves hidden in a noise pattern (see Fig. 5) we will obtain the following result, presented in Fig. 7. (lower left) After a few iterations the DL model begins to learn the structure of the data, and model components begin to adapt to fit the soliton spectra. (lower right) Upon convergence, the DL model is able to adaptively segment the data into background noise vs. multiple soliton spectra, despite the low signal-to-clutter ratio.