Yes / Probability density function (PDF) control strategy investigates the controller design approaches where the random variables for the stochastic processes were adjusted to follow the desirable distributions. In other words, the shape of the system PDF can be regulated by controller design.Different from the existing stochastic optimization and control methods, the most important problem of PDF control is to establish the evolution of the PDF expressions of the system variables. Once the relationship between the control input and the output PDF is formulated, the control objective can be described as obtaining the control input signals which would adjust the system output PDFs to follow the pre-specified target PDFs. Motivated by the development of data-driven control and the state of the art PDF-based applications, this paper summarizes the recent research results of the PDF control while the controller design approaches can be categorized into three groups: (1) system model-based direct evolution PDF control; (2) model-based distribution-transformation PDF control methods and (3) data-based PDF control. In addition, minimum entropy control, PDF-based filter design, fault diagnosis and probabilistic decoupling design are also introduced briefly as extended applications in theory sense. / De Montfort University - DMU HEIF’18 project, Natural Science Foundation of Shanxi Province [grant number 201701D221112], National Natural Science Foundation of China [grant numbers 61503271 and 61603136]

Yes / This note presents a novel data-based approach to investigate the non-Gaussian stochastic distribution control problem. As the motivation of this note, the existing methods have been summarised regarding to the drawbacks, for example, neural network weights training for unknown stochastic distribution and so on. To overcome these disadvantages, a new transformation for dynamic probability density function is given by kernel density estimation using interpolation. Based upon this transformation, a representative model has been developed while the stochastic distribution control problem has been transformed into an optimisation problem. Then, data-based direct optimisation and identification-based indirect optimisation have been proposed. In addition, the convergences of the presented algorithms are analysed and the effectiveness of these algorithms has been evaluated by numerical examples. In summary, the contributions of this note are as follows: 1) a new data-based probability density function transformation is given; 2) the optimisation algorithms are given based on the presented model; and 3) a new research framework is demonstrated as the potential extensions to the existing st

Statistical distributions also known as probability distributions are used to model a random experiment. Probability distributions consist of probability density functions (pdf) and cumulative density functions (cdf). Probability distributions are widely used in the area of engineering, actuarial science, computer science, biological science, physics, and other applicable areas of study. Statistics are used to draw conclusions about the population through probability models. Sample statistics such as the minimum, first quartile, median, third quartile, and maximum, referred to as the five-number summary, are examples of order statistics. The minimum and maximum observations are important in extreme value theory. This paper will focus on the probability distribution of the minimum observation, also known as the first order statistic, when the sample size is random.

Now a days the conditions at which the image taken may lead to near zero visibility for the human eye. They may usually due to lack of clarity, just like effects enclosed on earth’s atmosphere which have effects upon the images due to haze, fog and other day light effects. The effects on such images may exists, so useful information taken under those scenarios should be enhanced and made clear to recognize the objects and other useful information. To deal with such issues caused by low light or through the imaging devices experience haze effect many image processing algorithms were implemented. These algorithms also provide nonlinear contrast enhancement to some extent. We took pre-existed algorithms like SMQT (Successive mean Quantization Transform), V Transform, histogram equalization algorithms to improve the visual quality of digital picture with large range scenes and with irregular lighting conditions. These algorithms were performed in two different method and tested using different image facing low light and color change and succeeded in obtaining the enhanced image. These algorithms helps in various enhancements like color, contrast and very accurate results of images with low light. Histogram equalization technique is implemented by interpreting histogram of image as probability density function. To an image cumulative distribution function is applied so that accumulated histogram values are obtained. Then the values of the pixels are changed based on their probability and spread over the histogram. From these algorithms we choose histogram equalization, MATLAB code is taken as reference and made changes to implement in API (Application Program Interface) using JAVA and confirms that the application works properly with reduction of execution time.

Este trabalho apresenta um estudo acerca dos principais parâmetros que caracterizam o fenômeno da biestabilidade em dois tubos dispostos lado a lado submetidos a escoamento cruzado turbulento. A técnica experimental da anemometria de fio quente em canal aerodinâmico é aplicada na medição das flutuações de velocidade do escoamento após os tubos. As séries temporais obtidas são utilizadas como dados de entrada para determinação das funções densidade de probabilidade (PDF) usando um modelo de mistura finita, de acordo com uma função t de Student assimétrica e com o auxílio do método de Monte Carlo. Transformadas de ondaletas discretas e contínuas são aplicadas na filtragem das séries temporais para determinadas bandas de frequências e na análise do conteúdo de energia destes sinais. Através de conceitos de sistemas caóticos, é realizada a reconstrução do atrator do problema pelo método dos atrasos temporais, a partir das séries experimentais de velocidade, permitindo a determinação da dimensão de imersão e o cálculo do maior expoente de Lyapunov. Os resultados mostram a existência de dois patamares distintos de velocidade média nas séries temporais, correspondentes aos dois modos do escoamento, cada qual com números de Strouhal e funções densidade de probabilidade distintas. Uma análise conjunta das componentes axial e transversal do escoamento e suas PDF indicam as regiões no plano de medições onde o fenômeno se manifesta, sendo que reconstruções da trajetória filtrada das séries temporais para determinadas bandas de frequências apresentam características caóticodeterminísticas. O maior expoente de Lyapunov das séries experimentais é positivo, o que é um indício de comportamento caótico. / This work presents a study of the main parameters that characterize the phenomenon of bistability in two tubes placed side by side submitted to turbulent crossflow. The experimental technique of hot wire anemometry in aerodynamic channel is applied in the measurement of velocity fluctuations of the flow after the tubes. The time series obtained are used as input data for determining the probability density functions (PDF) using a finite mixture model, according to an asymmetric Student t function and with the aid of a Monte Carlo method. Wavelet transforms are applied in discrete and continuous filtering of time series for certain frequency bands and in the analysis of the energy content of these signals. By means of chaotic systems concepts, the attractor reconstruction of the problem is performed using the method of time delays from the experimental series of velocity, allowing the determination of the embedding dimension and calculating the largest Lyapunov exponent. The results show the existence of two different levels of mean velocity in time series, corresponding to two flow modes, each one with different Strouhal numbers and probability density functions. A joint analysis of axial and transverse components of flow and its PDF indicate the regions in the measurement plan where the phenomenon is manifested, and reconstructions of the trajectory of the filtered time series for certain frequency bands have chaotic-deterministic characteristics. The largest Lyapunov exponent of experimental series is positive, which is an indication of chaotic behavior.

Auditory physiology is nearly unique in the human body because of its small-diameter neurons. When considering a single node on one neuron, the number of channels is very small, so ion fluxes exhibit randomness. Hodgkin and Huxley, in 1952, set forth a system of Ordinary Differential Equations (ODEs) to track the flow of ions in a squid motor neuron, based on a circuit analogy for electric current. This formalism for modeling is still in use today and is useful because coefficients can be directly measured. To measure auditory properties of Firing Efficiency (FE) and Post Stimulus Time (PST), we can simply measure the depolarization, or "upstroke," of a node. Hence, we reduce the four-dimensional squid neuron model to a two-dimensional system of ODEs. The stochastic variable m for sodium activation is allowed a random walk in addition to its normal evolution, and the results are drastic. The diffusion coefficient, for spreading, is inversely proportional to the number of channels; for 130 ion channels, D is closer to 1/3 than 0 and cannot be called negligible. A system of Partial Differential Equations (PDEs) is derived in these pages to model the distribution of states of the node with respect to the (nondimensionalized) voltage v and the sodium activation gate m. Initial conditions describe a distribution of (v,m) states; in most experiments, this would be a curve with mode at the resting state. Boundary conditions are Robin (Natural) boundary conditions, which gives conservation of the population. Evolution of the PDE has a drift term for the mean change of state and a diffusion term, the random change of state. The phase plane is broken into fired and resting regions, which form basins of attraction for fired and resting-state fixed points. If a stimulus causes ions to flow from the resting region into the fired region, this rate of flux is approximately the firing rate, analogous to clinically measuring when the voltage crosses a threshold. This gives a PST histogram. The FE is an integral of the population over the fired region at a measured stop time after the stimulus (since, in the reduced model, when neurons fire they do not repolarize). This dissertation also includes useful generalizations and methodology for turning other ODEs into PDEs. Within the HH modeling, parameters can be switched for other systems of the body, and may present a similar firing and non-firing separatrix (as in Chapter 3). For any system of ODEs, an advection model can show a distribution of initial conditions or the evolution of a given initial probability density over a state space (Chapter 4); a system of Stochastic Differential Equations can be modeled with an advection-diffusion equation (Chapter 5). As computers increase in speed and as the ability of software to create adaptive meshes and step sizes improves, modeling with a PDE becomes more and more efficient over its ODE counterpart.

<p>The numerical path integration method for solving stochastic differential equations is extended to solve systems up to six spatial dimensions, angular variables, and highly nonlinear systems - including systems that results in discontinuities in the response probability density function of the system. Novel methods to stabilize the numerical method and increase computation speed are presented and discussed. This includes the use of the fast Fourier transform (FFT) and some new spline interpolation methods. Some sufficient criteria for the path integration theory to be applicable is also presented. The development of complex numerical code is made possible through automatic code generation by scripting. The resulting code is applied to chaotic dynamical systems by adding a Gaussian noise term to the deterministic equation. Various methods and approximations to compute the largest Lyapunov exponent of these systems are presented and illustrated, and the results are compared. Finally, it is shown that the location and size of the additive noise term affects the results, and it is shown that additive noise for specific systems could make a non-chaotic system chaotic, and a chaotic system non-chaotic.</p>

The numerical path integration method for solving stochastic differential equations is extended to solve systems up to six spatial dimensions, angular variables, and highly nonlinear systems - including systems that results in discontinuities in the response probability density function of the system. Novel methods to stabilize the numerical method and increase computation speed are presented and discussed. This includes the use of the fast Fourier transform (FFT) and some new spline interpolation methods. Some sufficient criteria for the path integration theory to be applicable is also presented. The development of complex numerical code is made possible through automatic code generation by scripting. The resulting code is applied to chaotic dynamical systems by adding a Gaussian noise term to the deterministic equation. Various methods and approximations to compute the largest Lyapunov exponent of these systems are presented and illustrated, and the results are compared. Finally, it is shown that the location and size of the additive noise term affects the results, and it is shown that additive noise for specific systems could make a non-chaotic system chaotic, and a chaotic system non-chaotic.

A new statistical algorithm, the "modal distribution method", is proposed to statistically quantify the significance of changes in mean frequencies of individual modal vibrations of measured structural response data. In this new method, a power spectrum of measured structural response is interpreted as being a series of independent modal responses, each of which is isolated over a frequency range and treated as a statistical distribution. Pairs of corresponding individual modal distributions from different segments are compared statistically. The first version is the parametric MDM. This method is applicable to well- separated modes having Gaussian shape. For application to situations in which the signal is corrupted by noise, a new noise reduction methodology is developed and implemented. Finally, a non-parametric version of the MDM based on the Central Limit Theorem is proposed for application of MDM to general cases including closely spaced peaks and high noise. Results from all three MDMs are compared through application to simulated clean signals and the two extended MDMs are compared through application to simulated noisy signals. As expected, the original parametric MDM is found to have the best performance if underlying requirements are met: signals that are clean and have well-separated Gaussian mode shapes. In application of nonparametric methods to Gaussian modes with high noise corruption, the noise reduction MDM is found to have lower probability of false alarms than the nonparametric MDM, though the nonparametric is more efficient at detecting changes. In closely related work, the Hermite moment model is extended to highly skewed data. The aim is to enable transformation from non-Gaussian modes to Gaussian modes, which would provide the possibility of applying parametric MDM to well- separated non-Gaussian modes. A new methodology to combine statistical moments using a histogram is also developed for reliable continuous monitoring by means of MDM. The MDM is a general statistical method. Because of its general nature, it may find a broad variety of applications, but it seems particularly well suited to structural health monitoring applications because only very limited knowledge of the excitation is required, and significant changes in computed power spectra may indicate changes, such as structural damage.

This paper presents a simple recursive formula to compute the Taylor polynomial of quantile for a continuous random variable. It is very easy to implement the formula in standard symbolic programming system, for example Mathematica (Wolfram, 2003). Applications of the formula to standard normal distribution and to the generation of random variables for continuous distribution with bounded support are illustrated.