Random Variables

The increasing wind power generation requires the updating of the computational tools that support decision making in the operation and planning of electric power systems. This work describes a methodology to model the spatial correlations among wind speeds probability distributions in probabilistic power flow. The proposed methodology is based on Nataf transformation and Monte Carlo Simulation. The proposed methodology is illustrated in case studies with the IEEE 24 bus test system, highlighting the importance of adequately considering such correlations.

We use empowerment, a recently introduced biologically inspired measure, to allow an AI player to assign utility values to potential future states within a previously un-encountered game without requiring explicit specification of goal states. We further introduce strategic affinity, a method of grouping action sequences together to form 'strategies', by examining the overlap in the sets of potential future states following each such action sequence. Secondly, we demonstrate an information-theoretic method of predicting future utility. Combining these methods, we extend empowerment to soft-horizon empowerment which enables the player to select a repertoire of action sequences that aim to maintain anticipated utility.

In this paper, we consider testing for additivity in a class of nonparametric stochastic regression models. Two test statistics are constructed and their asymptotic distributions are established. We also conduct a small sample study for one of the test statistics through a simulated example. © 2002 Elsevier Science (USA)

We study the two-way communication problem for the relay channel. Hereby, two terminals communicate simultaneously in both directions with the help of one relay. We consider the restricted two-way problem, i.e., the encoders at both terminals do not cooperate. We provide achievable rate regions for different cooperation strategies, such as decode-and-forward based on block Markov superposition coding and compress-andforward based on Wyner-Ziv source coding. We also evaluate the regions for the special case of additive white Gaussian noise channels. We show that a combined strategy of block Markov superposition coding and Wyner-Ziv coding achieves the cut-set upper bound on the sum-rate of the two-way relay channel when the relay is in the proximity of one of the terminals.

The sum of lognormal distributions is a well-known problem that no closed-form expression exists and it is difficult to evaluate numerically. In this paper, Log Shifted Gamma (LSG) approximation method is proposed to represent the sum of lognormal distributions and to derive a closed-form expression of the typical value of the sum. Illustrative results show that the LSG model provides much more accurate approximation than other previous methods for a wide range of lognormal variances.

This paper focuses on the implementation of various diagnostics for optimizing chemical oxygen iodine laser. This paper also dwells on the measurement methodologies and instrumentation employed in these diagnostic systems. The prime diagnostics are for the measurement of vital species in form of iodine, singlet oxygen, and chlorine. Iodine concentration measurement (absorption at λ = 499 nm), chlorine utilization (absorption at λ = 330 nm), and singlet oxygen yield (emission at λ = 1.27 µm) have been implemented based on optical absorption/emission principle. Furthermore, online Mach number determination, which is critical for supersonic gas flows, laser pulse detection (at λ = 1315 nm), and flow rate measurement and control over wide range, have been carried out. A dedicated diagnostics and data acquisition system (DDAS) customized for parameter monitoring have been developed. The developed DDAS also serves the purpose of precise operation sequencing and parameter control. It is a 168-channel personal-computer-based system with customized interface electronics employing Visual C++ programming language with user-friendly graphical user interfaces. A detailed uncertainty analysis of various critical parameters has been also presented.

This study investigates the ability of Monte Carlo simulation (MCs) to predict stock market returns in Amman Stock Exchange (ASE). Specifically, we compare the in-sample forecasting ability of MCs with the Simple and Exponential Moving average techniques. The data of the study consists of the daily general float index of ASE over the period (2003-2012). Forecasting accuracy is measured by four proxies: Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Theil Inequality Coefficient (U). The results indicate that MCs is the most accurate forecasting technique among the others investigated. Moreover, ASE seems to be inefficient at the weak level, given that technical analysis approaches enable investors to predict stock market returns.

In many engineering applications ranging from engineering seismology to petroleum engineering and civil engineering, it is important to process seismic data. In processing seismic data, it turns out to be very efficient to describe the signal's spectrum as a linear combination of Ricker wavelet spectra. In this paper, we provide a possible theoretical explanation for this empirical efficiency. Specifically, signal propagation through several layers is discussed, and it is shown that the Ricker wavelet is the simplest non-trivial solution for the corresponding data processing problem, under the condition that the described properties of the approximation family are satisfied.

The recently introduced random walker segmentation algorithm of has been shown to have desirable theoretical properties and to perform well on a wide variety of images in practice. However, this algorithm requires user-specified labels and produces a segmentation where each segment is connected to a labeled pixel. We show that incorporation of a nonparametric probability density model allows for an extended random walkers algorithm that can locate disconnected objects and does not require user-specified labels. Finally, we show that this formulation leads to a deep connection with the popular graph cuts method of .

"Purpose– Effect size is an important determinant of statistical power. However, very few experimental studies in international marketing (IM) report effect sizes and no meta-analysis work in
this regard has been done. The main objective of this paper, therefore, is to quantitatively document effect sizes of experiments in IM and to provide directions for further methodological improvement. Design/methodology/approach– All articles published in the top three marketing journals and the top six IM-related journals during the period 1992-2005 were screened; this yielded 35 experiment-based papers within the domain of IM. For each study, ten methodological characteristics
relevant to IM experimental designs were coded.
Findings– The 35 studies reported 68 experiments, which produced a total of 1, 074 observations. Results reveal that, on average, for experiments in international business marketing, about 2.89 percent of the variance in a dependent variable (DV) is accounted for by experimental treatments, and a variance of 3.61 percent is shared by the independent and DV for experiments in international consumer marketing. Sampling method, type of subjects, type of design and number of countries are
found to have significant influences on effect sizes. Originality/value– This paper provides a quantitative, state-of-the-art review of effect sizes in IM experiments, points out problems such as inappropriate reliance on an overall effect size index, and
further offers useful suggestions on how to report and improve effect sizes."

For trace minimization, the optimal one in this family is better than the one obtained by CI except in some cases where they are equal. It is further proved that the best pair of combination gains that minimizes the above optimal-trace-in-the-family coincides with the one associated with the best value of omega in CI. Thus the CI Algorithm provides a convenient one-dimensional parameterization for the optimal solution in the n-square dimensional space. The results are also extended to multiple RV's and partial estimates.

Abstract-This paper demonstrates how the problem of frequency line tracking can be formulated in terms of hidden Markov models (HMM's). Frequency cells comprising a subset, or gate, of the spectral bins from FFT processing are identified with the states of the hidden Markov ...

Two classes of metrics are introduced for spaces of fuzzy sets. Their equivalence is discussed and basic properties established. A characterisation of compact and locally compact subsets is given in terms of boundedness and p-mean equileft-continuity, and the spaces shown to be locally compact, complete and separable metric spaces.

This book is an introductory textbook in undergraduate probability. It has a mission: to spell out the motivation, intuition, and implication of the probabilistic tools we use in science and engineering. From over half a decade of teaching the course, I have distilled what I believe to be the core of probabilistic methods. I put the book in the context of data science to emphasize the inseparability between data (computing) and probability (theory) in our time.

Competitive Electricity Markets are based on a set of hypotheses which sustains a vertical and horizontal segmentation of the generation, transmission and distribution sectors in different functions. This theoretical model was implemented in several countries and its operation for more than a decade produced a great amount of data which are analyzed by the Public Authorities to take further decisions. Proposals regarding reforms for this type of markets must be based on a deep knowledge of different factors influence on the market parameters. Consequently, it is important to have methodologies and statistical techniques to help Public Authorities take decisions. In this paper, the application of the methodology of analysis of variance of multivariate data (MANOVA) to detect the impact of the fuel consumption on the market price is presented. A two factors fixed effect model is used; two-level factors for fuel-oil and two-level factors for natural gas. The random variable observed is a vector of twentyfour components which correspond to the 2005 market price in the Argentinean Electricity Market.

Variabel acak adalah keluaran dari suatu percobaan yang dinyatakan secara numerik. Notasi variabel acak dinyatakan dalam huruf besar (misal X, Y, W) dan suatu nilai dari variabel acak tersebut ditulis dengan huruf kecil (misal x, y, w). Contoh: Var.acak dinyatakan dengan fungsi X = X (s) = s 2 ; s adalah bilangan real, elemen-elemen pada sample-space S, yaitu himpunan {0 < s < 12}; maka titik-titik pada S akan dipetakan pada himpunan real {0 < x < 144} Syarat suatu fungsi X sebagai variable acak ialah: 1. Himpunan {X < x} merupakan suatu event untuk semua nilai real x-∞< x < ∞ 2. P{X =-∞} = 0 dan P{X = ∞}= 0 Variabel acak diskrit hanya memiliki nilai-nilai diskrit (countable), sedangkan variabel acak kontinyu mempunyai nilai-nilai (manapun) pada suatu range yang kontinyu.

ABSTRACT The paper deals with state estimation of nonlinear non-Gaussian systems with a special focus on the Gaussian sum filters. To achieve a higher estimate quality, state and measurement predictive moments appearing in the filters are computed by the randomized unscented transform, which provides asymptotically exact estimates of the moments. The use of the Gaussian sum filter employing the randomized unscented transform is introduced and the proposed algorithm is illustrated in a numerical example. The analysis of the numerical example involves a comparison of several filters using a number of performance metrics both absolute and relative, assessing the point estimate quality, the estimate error quality, and the density estimate quality.

The paper deals with state estimation of nonlinear non-Gaussian systems with a special focus on the Gaussian sum filters. To achieve a higher estimate quality, state and measurement predictive moments appearing in the filters are computed by the randomized unscented transform, which provides asymptotically exact estimates of the moments. The use of the Gaussian sum filter employing the randomized unscented transform is introduced and the proposed algorithm is illustrated in a numerical example. The analysis of the numerical example involves a comparison of several filters using a number of performance metrics both absolute and relative, assessing the point estimate quality, the estimate error quality, and the density estimate quality.

This correspondence discusses the application of random autoregressive (AR) models to signal processing problems: specifically, to adaptive line enhancement (ALE). The advantage of this approach is that random AR models may reflect more accurately the uncertainty in the stochastic process that generates the received signal. It is shown, through Monte Carlo simulations, that by using random AR models one obtains better results than by using the conventional deterministic AR models under the same conditions.

The views expressed in this Working Paper are those of the author(s) and do not necessarily represent those of the IMF or IMF policy. Working Papers describe research in progress by the author(s) and are published to elicit comments and to further debate.

In this paper, we investigate the classical problem of finding the probability density function (pdf) of the sum of Nakagami-m random variables. Exact infinite series representations are derived for the sum of three and four identically and independently distributed (i.i.d.) Nakagami-m random variables, and subsequently, it is extended to the sum of general number of random variables. A useful pattern emerged as a result of these extensions, and a new Fourier transform pair that involves parabolic cylinder function and Gauss hypergeometric function is obtained. Bounds on the error resulting from truncation of the infinite series of pdfs are also presented. These pdfs are used to analyze the performance of dual, triple, and quadruple branch predetection equal gain combining (EGC) receivers over Nakagami-m fading environment. Furthermore, a hypergeometric relation is derived as a result of those derivations. Subsequently, the analysis is extended to the case of L branch EGC receiver performance as well. Selected simulation plots are provided to compare the results with the approximate results available in the literature and to illustrate the validity of the formulation.

The Extended Kalman Filter (EKF) has become a standard technique used in a number of nonlinear estimation and machine learning applications. These include estimating the state of a nonlinear dynamic system, estimating parameters for nonlinear system identification (e.g., learning the weights of a neural network), and dual estimation (e.g., the Expectation Maximization (EM) algorithm) where both states and parameters are estimated simultaneously.

A more general class of broadcast channels with conditionally independent message sets is defined and its capacity region is determined using Marton's inner bound and the outer bound considered by Nair and El Gamal. The capacity region for this class of channels includes the capacity region for broadcast channels with degraded message sets, semi-deterministic broadcast channels, deterministic broadcast channels with and without common information, less noisy and (special) more capable broadcast channels.

In this contribution, we employ the Mellin transform to derive the expressions for probability density function (PDF) of the product of Rayleigh and Nakagami-m distributed random variables. We exploit the fact that the Mellin transform of a product of independent and identically distributed random variables is the product of the Mellin transforms of the individual random variables. Using this approach, the PDF of the product of random variables is expressed in the form of an easily computable infinite series.

This book is a practical reference guide accompanied with an Excel Workbook. This book gives an elementary introduction of the weighted standard deviational ellipse. This book also presents the computational aspects of the weighted exponential distributions as well. For the examples given, calculations are performed using VBA for Excel. This book makes comparisons (and shows the computations via VBA for Excel) using the likelihood functions with spatial data of the weighted ellipses. Lastly, the book covers spherical statistics. Throughout the text, the reader can see how to perform these difficult calculations and learn to adapt the code for his research.
https://rogerlgoodwin.academia.edu/

This book is a work in progress.

This 10 hours class is intended to give students the basis to empirically solve statistical problems. Talk 1 serves as an introduction to the statistical software R, and presents how to calculate basic measures such as mean, variance, correlation and gini index. Talk 2 shows how  the central limit theorem and the law of the large numbers work empirically. Talk 3 presents the point estimate, the confidence interval and the hypothesis test for the most important parameters. Talk 4 introduces to the linear regression model and Talk 5 to the bootstrap world. Talk 5 also presents an easy example of a markov chains.
All the talks are supported by script codes, in R language.

The "covariance" of complex random variables and processes, when defined consistently with the corresponding notion for real random variables, is shown to be detc!rmioed by the usual (complex) covariance together with a quantity called the pseudo-covariance. A characterization of uncorrelatedness and wide-sense stationarity in terms of covariance and pseudocovariance is given. Complex random variables and processes with a vanishing pseudo-covariaqce are called proper. It is shown that properness is preserved under affine transformations and that the complex-multivariate Gaussian density assumes a natural form only for proper random variables. The maximumsotropy theorem is generalized to the complex-multivariate case. The differential entropy of a complex random vector with a tixed correlation matrix is shown to be maximum, if and only if the random vector is proper, Gaussian and zero-mean. The notion of chulur si'utio-is introduced. For the class of proper complex random processes, a discrete Fourier transform correspondence is derived dating circular stationarity in the time domain to uncomlatedness in the fmquency domain. As an application of the theory, the capacity of a discrete-time channel with complex inputs, proper complex additive white Gaussian noise, and a finite complex unit-sample response is determined. This derivation is considerably simpler than an earlier derivation for the red discrete-time Gaussian channel with intersymbol interference, whose capacity is obtained as a by-product of the results for the complex channel.

A strategy for adaptive control and energetic optimization of aerobic fermentors was implemented, with both air flow and agitation speed as manipulated variables. This strategy is separable in its components: control, optimization, estimation. We optimized parameter’s estimation (from the usual KLa correlation) using sinusoidal excitation of air flow and agitation speed. We have implemented parameter’s estimation trough recursive least squares algorithm with forgetting factor. We carried separate essays on control, optimization and estimation algorithms. We carried our essays using an original computational simulation environment, with noise and delay generating facilities for data sampling and filtering.

Our results show the convergence and robustness of the estimation algorithm used, improved with use of both forgetting factor and KLa dead-band facilities. Control algorithm used in our work compares favorably with PID using the integrated area criteria for deviation between oxygen molarity and critical molarity (set point). Optimization algorithm clearly reduces energetic consumption, respecting critical molarity. Integration of control, optimization and adaptive algorithms was implemented, but future work is needed for stability. Methods were defined and implemented for stability improvement. We have implemented data acquisition and computer manipulation of air flow and agitation speed for actual fermentors.

In this paper, we first provide a very accurate estimation of the capacity of a single-input single-output system operating in a log-normal environment. Then, hinging on the fact that the sum of log-normal Random Variables (RV) is well approximated by another log-normal RV, we apply the obtained results to find the capacity of Maximum Ratio Combining and Equal Gain Combining in a log-normal environment. The capacity in an interference-limited environment is also investigated in this paper. The analytical expressions obtained match perfectly the capacity given by simulations.

Frequency-hopping communications over channels with partial band noise jamming, both with and without side information, is considered. The capacity and the cutoff rate are calculated both for the case of ideal interleaving and the case of non-ideal interleaving. It is shown that both the capacity and the cutoff rate decrease with an increase of the interleaving span. When side information is not available, the capacity in the case of non-ideally interleaved channel approaches monotonically the capacity of the ideally interleaved channel as the interleaving span tends to infinity, whereas the cutoff rate behaves in an opposite manner.

When (X, 0), --. , (Xn, On) are independent identically distributed random vectors from IRd X {0, 1 } distributed as (X, 0), and when 0 is estimated by its nearest neighbor estimate 0(1), then Cover and Hart have shown that P{0o() * 0} n -'-2E {1(X) (1 -r(X))} < 2R*(1 -R*) where R* is the Bayes probability of error and n(x) = P {O = II X = x}. They have conditions on the distribution of (X, 0). We give two proofs, one due to Stone and a short original one, of the same result for all distributions of (X, 8).

In this paper we discuss verification and validation of simulation models. Four different approaches to deciding model validity are described; two different paradigms that relate verification and validation to the model development process are presented; various validation techniques are defined; conceptual model validity, model verification, operational validity, and data validity are discussed; a way to document results is given; a recommended procedure for model validation is presented; and model accreditation is briefly discussed.

In this letter, a modified weighted bit-flipping decoding algorithm for low-density parity-check codes is proposed. Improvement in performance is observed by considering both the check constraint messages and the intrinsic message for each bit.

In this paper, we analyze Bayes fusion rule in details from a fusion standpoint, as well as the emblematic Dempster’s rule of combination introduced by Shafer in his Mathematical Theory of evidence based on belief functions. We propose a new interesting formulation of Bayes rule and point out some of its properties. A deep analysis of the compatibility of Dempster’s fusion rule with Bayes fusion rule is done. We show that Dempster’s rule is compatible with Bayes fusion rule only in the very particular case where the basic belief assignments (bba’s) to combine are Bayesian, and when the prior information is modeled either by a uniform probability measure, or by a vacuous bba. We show clearly that Dempster’s rule becomes incompatible with Bayes rule in the more general case where the prior is truly informative (not uniform, nor vacuous). Consequently, this paper proves that Dempster’s rule is not a generalization of Bayes fusion rule.

This paper sets simulation model verification and validation (V&V) in the context of the process of performing a simulation study. Various different forms of V&V need to take place depending on the stage that has been reached. Since the phases of a study are performed in an iterative manner, so too are the various forms of V&V. A number of difficulties with verifying and validating models are discussed, after which a series of V&V methods are described. V&V is seen as a process of increasing confidence in a model, and not one of demonstrating absolute accuracy.

Accept/reject sampling is a well-known method to generate random samples from arbitrary target probability distributions. It demands the design of a suitable proposal probability density function (pdf) from which candidate samples can be drawn. The main limitation to the use of RS is the need to find an adequate upper bound for the ratio of the target pdf over the proposal pdf from which the samples are generated. There are no general methods to analytically find this bound, except when the target pdf is log-concave. In this paper we introduce a novel procedure using the ratio of uniforms method to efficiently perform rejection sampling for a large class of target densities. The candidate samples are generated using only two independent uniform random variables. In order to illustrate the application of the proposed technique, we provide a numerical example.

A method for estimating the parameters of hidden Markov models of speech is described. Parameter values are chosen to maximize the mutual information between an acoustic observation sequence and the corresponding word sequence. Recognition results are presented comparing this method with maximum likelihood estimation.

: The method of successive substitution is shown applicable to solve for the classical minimum distortion quantizer parameters. The method is self converging and parameters can be calculated to any desired accuracy. The minimum distortion quantizer parameters, as well as parameters based on other criterion such as quantizers for signal detection, minimum risk quantizers and quantizers for LMMS estimation can be solved by Max&#39;s trial and error technique. There are also many other approximation methods to calculate the quantizer parameters. This report applies the method of successive substitution and its modifications to solve for the Lloyd-Max quantizer parameters. It is more accurate and computationally more efficient than the previously reported methods. It is shown to easily generate 7 bit (128 level) optimum quantization. The Lloyd-Max minimum mean square distortion quantizer problem deals with transforming a random variable X of differentiable probability density function f...