Statistical Simulation: Power Method Polynomials and Other Transformations

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In the scientific literature, two different approaches are typically applied toward the solution of this problem. One approach is based on the use of a PDF for the description of random processes from which the signal detection methods are developed [ 4 , 5 , 6 , 21 ]. Despite the fact that PDFs provide a complete description of stochastic processes, these methods have some limitations, and the computational complexity associated with non-Gaussian processes is notable. Another approach toward describing random processes is based on the use of the moment and cumulant functions.

In this case, the properties of decision functions can be described using other characteristics, such as the mean and variance of decision rules DRs. For example, a deflection criterion was developed for a class of linear-quadratic L-Q systems [ 3 , 18 , 19 ]. Further development of this criterion is shown in [ 2 ].

It is worth noting that the deflection criterion and its modifications are weakly connected with the classical criteria that are based on the use of PDFs. This second approach can be represented in the form of higher-order statistics HOS , see [ 12 ], such as moments and cumulant semivariants functions [ 14 , 20 ]. Such functions allow the description of the statistical properties of non-Gaussian processes with reasonable accuracy [ 10 , 14 ]. The HOS techniques are used for the development of the signal detection methods [ 15 , 22 ].

However, these methods have some restrictions, for instance the detection of deterministic signals, and the imposition that only third-order statistics can be used. A new method was proposed for signal detection based on the use of the moment quality criterion for decision making [ 13 ].

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This approach has led to improvements in the accuracy of non-Gaussian signal processing relative to traditional methods, along with a reduction in the complexity of signal detection algorithms [ 16 , 17 ]. In those papers, the signal detection methods and algorithms were proposed only for uncorrelated non-Gaussian noise. However, in practical cases, the signal often propagates through turbulent media or along multiple paths. With that in mind, one should model a signal more rigorously in the form of a correlated non-Gaussian process. The main objective of this paper is the synthesis and analysis of the signal detection methods in correlated non-Gaussian noise based on the moment-cumulant description of random variables and polynomial DRs.

This approach seems to optimally satisfy the adapted moment quality criterion of statistical hypotheses testing.

Such an approach provides an opportunity to create effective algorithms for the operation of data receiving and processing systems. A different approach is based on the moment and cumulant characteristics [ 10 , 12 , 14 , 20 ]. Suppose there are sample values of a stationary random process that can be considered to be different random variables. Then the relationship between two random variables is a simple and widely used example of statistically dependent variables.

This is equivalent to using a 2D PDF. The moment-cumulant description of the correlated non-Gaussian processes requires additional research and development. For this purpose it will be convenient to work with the additional new concept of punched random variables developed in [ 10 ]. The cumulants of the series expansion of any characteristic function can be separated into various classes in which the characteristic function has similar properties. Non-Gaussian random variables classified as described above are called punched random variables. The mathematical models of random variables were proposed and approved in classes of uncorrelated non-Gaussian random processes using punched random variables when the moment-cumulant models are represented only by a part of the cumulants from all possible sets that can match the real existing process.

Using this punched classification, there are also various skewness, kurtosis, and skewness-kurtosis statistically independent random variables [ 10 ]. A variable is said to be a punched random variable if in its cumulant description one part of cumulant coefficients of the 3rd order is distinct from zero and another part is strictly equal to zero.

The remaining higher order cumulant coefficients can assume arbitrary values [ 10 ]. In this paper we propose development of new moment and cumulant models of the non-Gaussian statistically dependent random variables. On the basis of our approach, it is then possible to create signal detection DRs using the adapted new moment quality criterion for statistical hypothesis testing. The other cumulants and joint cumulants of higher orders must equal zero.

The other cumulants and joint cumulants of the higher orders must equal zero. The proposed models are different from well-known models as they account for the properties of non-Gaussian correlation random processes using higher order cumulant coefficients. These models will be used for the development and adaptation of the new moment quality criterion for statistical hypothesis testing and signal detection methods. Such a Ku criterion provides the minimum of the sum probability of the first and second kind of errors. The decision rule DR will be optimal if the sum of variances for the hypothesis and the alternative is minimal, and the distance between the mean functions is as large as possible.

The new moment quality criterion 28 is different from the well-known probabilistic quality criterion, but it has a definite correlation with them. The Ku criterion is used to create effective methods and algorithms for signal detection in uncorrelated non-Gaussian noise [ 16 , 17 ]. In order to solve the problem of signal processing in correlated non-Gaussian noise, the moment quality criterion 34 , defined below, needs to be adapted.

This can be done using the new moment and cumulant models that were obtained previously in Sect. The quality criterion in parameter estimation theory is a variance of parameter estimation of random variables. In [ 11 ] it was shown that the minimum variance is inversely proportional to the Fisher information function. This information function is defined as a PDF form of the sample values. It is easy to show that the mean and variance of the polynomial stochastic DR 20 can be represented as Kullback—Leibler information number using a PDF for the hypothesis and the alternative.

Our new method of signal detection in correlated non-Gaussian noise is developed based on new moment-cumulant models and an adapted moment quality criterion for statistical hypotheses testing. This method will be used for synthesis and analysis of the non-linear polynomial stochastic DR. In this paper we have developed a linear DR of signal detection in correlated non-Gaussian noise. We have used the exponential correlation function for the simulation.

Smaller values of the criterion correspond to smaller values for the probability of the errors of the polynomial DR. Research also focused on other types of correlated noise: asymmetrical, excess and asymmetrical-excess non-Gaussian. For these cases, the efficiency of signal processing improved in comparison to the well-known results under the assumption of Gaussian noise. The complexity associated with description of non-Gaussian processes in the theory of signal processing requires a new approach toward solving the problems of signal detection.

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The approach described herein is based on the application of the moment-cumulant function of random processes and moment criterion quality for the statistical hypotheses testing. New mathematical models of correlated non-Gaussian processes have been developed. An adaptation of the moment quality criterion of upper bounds of error probability was also proposed.

Furthermore, power polynomial algorithms were developed for the correlated non-Gaussian processes based on the new method. This approach enables description of the characteristics of correlated non-Gaussian stochastic processes while taking into account the cumulant coefficients of the third and higher orders as well as joint cumulants. Skip to main content Skip to sections. Advertisement Hide.

Lesson 3.4 - Transformations of Polynomial Functions

Download PDF. Open Access. First Online: 22 August One way to describe statistically dependent random variables involves the use of MD moments and cumulants. Definition 1 A variable is said to be a punched random variable if in its cumulant description one part of cumulant coefficients of the 3rd order is distinct from zero and another part is strictly equal to zero.

The classification of new mathematical models is obtained from 2D moments and cumulants of non-Gaussian correlated processes. In the classical approach, the optimal Bayesian algorithm of signal detection is determined as the minimum average risk [ 23 ]. In this case, the adapted criterion should take into account the correlation of sample values. Let us consider the efficiency of the method presented in this paper by using an example of signal detection. Note that then the Eq. It is easy to show that if the sample values are independent, i.

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