Gaussian processes are certainly not a new tool in the field of science. However, alongside the quick increasing of computer power during the last decades, Gaussian processes have proved to be a successful and flexible statistical tool for data analysis. Its practical interpretation as a nonparametric procedure to represent prior beliefs about the underlying data generating mechanism has gained attention among a variety of research fields ranging from ecology, inverse problems and deep learning in artificial intelligence. The core of this thesis deals with multivariate Gaussian process model as an alternative method to classical methods of regression analysis in Statistics. I develop hierarchical models, where the vector of predictor functions (in the sense of generalized linear models) is assumed to follow a multivariate Gaussian process. Statistical inference over the vector of predictor functions is approached by means of the Bayesian paradigm with analytical approximations. I developed also ...

Many model-based clustering methods are based on a finite Gaussian mixture model. The Gaussian mixture model implies that the data scatter within each group is elliptically shaped. Hence non-elliptical groups are often modeled by more than one component, resulting in model over-fitting. An alternative is to use a mean-variance mixture of multivariate normal distributions with an inverse Gaussian mixing distribution (MNIG) in place of the Gaussian distribution, to yield a more flexible family of distributions. Under this model the component distributions may be skewed and have fatter tails than the Gaussian distribution. The MNIG based approach is extended to include a broad range of eigendecomposed covariance structures. Furthermore, MNIG models where the other distributional parameters are constrained is considered. The Bayesian Information Criterion is used to identify the optimal model and number of mixture components. The method is demonstrated on three sample data sets and a novel variation ...

Gaussian processes provide a powerful Bayesian approach to many machine learning tasks. Unfortunately, their application has been limited by the cubic computational complexity of inference. Mixtures of Gaussian processes have been used to lower the computational costs and to enable inference on more complex data sets. In this thesis, we investigate a certain finite Gaussian process mixture model and its applicability to clustering and prediction tasks. We apply the mixture model on a multidimensional data set that contains multiple groups. We perform Bayesian inference on the model using Markov chain Monte Carlo. We find the predictive performance of the model satisfactory. Both the variances and the trends of the groups are estimated well, bar the issues caused by poor clustering. The model is unable to cluster some of the groups properly and we suggest improving the prior of the mixing proportions or incorporating more prior information as remedies for the issues in clustering ...

Extended from superpixel segmentation by adding an additional constraint on temporal consistency, supervoxel segmentation is to partition video frames into atomic segments. In this work, we propose a novel scheme for supervoxel segmentation to address the problem of new and moving objects, where the segmentation is performed on every two consecutive frames and thus each internal frame has two valid superpixel segmentations. This scheme provides coarse-grained parallel ability, and subsequent algorithms can validate their result using two segmentations that will further improve robustness. To implement this scheme, a voxel-related Gaussian mixture model (GMM) is proposed, in which each supervoxel is assumed to be distributed in a local region and represented by two Gaussian distributions that share the same color parameters to capture temporal consistency. Our algorithm has a lower complexity with respect to frame size than the traditional GMM. According to our experiments, it also outperforms the state

Stationarity is often an unrealistic prior assumption for Gaussian process regression. One solution is to predefine an explicit nonstationary covariance function, but such covariance functions can be difficult to specify and require detailed prior knowledge of the nonstationarity. We propose the Gaussian process product model (GPPM) which models data as the pointwise product of two latent Gaussian processes to nonparametrically infer nonstationary variations of amplitude. This approach differs from other nonparametric approaches to covariance function inference in that it operates on the outputs rather than the inputs, resulting in a significant reduction in computational cost and required data for inference, while improving scalability to high-dimensional input spaces. We present an approximate inference scheme using Expectation Propagation. This variational approximation yields convenient GP hyperparameter selection and compact approximate predictive distributions.

Khansari Zadeh, S. M. and Billard, A. (2010) BM: An Iterative Method to Learn Stable Non-Linear Dynamical Systems with Gaussian Mixture Models. Proceeding of the International Conference on Robotics and Automation (ICRA 2010), 2010, p. 2381-2388. unknonw date. ...

Multi-task learning remains a difficult yet important problem in machine learning. In Gaussian processes the main challenge is the definition of valid kernels (covariance functions) able to capture the relationships between different tasks. This paper presents a novel methodology to construct valid multi-task covariance functions (Mercer kernels) for Gaussian processes allowing for a combination of kernels with different forms. The method is based on Fourier analysis and is general for arbitrary stationary covariance functions. Analytical solutions for cross covariance terms between popular forms are provided including Matern, squared exponential and sparse covariance functions. Experiments are conducted with both artificial and real datasets demonstrating the benefits of the approach.

We propose an active set selection framework for Gaussian process classification for cases when the dataset is large enough to render its inference prohibitive. Our scheme consists of a two step alternating procedure of active set update rules and hyperparameter optimization based upon marginal likelihood maximization. The active set update rules rely on the ability of the predictive distributions of a Gaussian process classifier to estimate the relative contribution of a datapoint when being either included or removed from the model. This means that we can use it to include points with potentially high impact to the classifier decision process while removing those that are less relevant. We introduce two active set rules based on different criteria, the first one prefers a model with interpretable active set parameters whereas the second puts computational complexity first, thus a model with active set parameters that directly control its complexity. We also provide both theoretical and ...

The single-parameter Gamma matrix of force constants proposed by the Gaussian Network Model (GNM) is iteratively modified to yield native state fluctuations that agree exactly with experimentally observed values. The resulting optimized Gamma matrix contains residue-specific force constants that may be used for an accurate analysis of ligand binding to single or multiple sites on proteins. Bovine Pancreatic Trypsin Inhibitor (BPTI) is used as an example. The calculated off-diagonal elements of the Gamma matrix, i.e., the optimized spring constants, obey a Lorentzian distribution. The mean value of the spring constants is approximately -0.1, a value much weaker than -1 of the GNM. Few of the spring constants are positive, indicating repulsion between residues. Residue pairs with large number of neighbors have spring constants around the mean, -0.1. Large negative spring constants are between highly correlated pairs of residues. The fluctuations of the distance between anticorrelated pairs of residues are

This program is part of Netpbm(1) pamgauss generates a one-plane PAM image whose samples are a gaussian function of their distance from the center of the image.

A value, x, from a normal distribution specified by a mean of m and a standard deviation of s can be converted to a corresponding value, z, in a standard normal distribution with the transformation z=(x-m)/s. And, of course, in reverse, any value from a standard normal graph, say z, can be converted to a corresponding value on a normal distribution with a mean of m and a standard deviation of s by the formula x=m+z*s. Remember that the standard normal distribution has a mean of 0 and a standard deviation of 1, i.e., m=0, s=1.. The ability to carry out this transformation is very important since we can do all our analysis with the standard normal distribution and then apply the results to every other normal distribution, including the one of interest. For example, to draw a normal curve with a mean of 10 and a standard deviation of 2 (m=10, s=2), draw the standard normal distribution and just re-label the axis. The first figure below is the standard normal curve and the next figure is the curve ...

The second part of the program is used to make the gaussian distribution. I defined l to be the width of a single interval, that is I wrote l=0.1/h where h is the number of intervals I want between 0 and 0.1. After that I computed p=int(pron(i)/l). In this way I should be able to compute the integer part of pron(i)/l where pron(i) are my shifted random numbers of the second attempt. Doing so I know that the pth interval contains one of my random numbers, in this case pron(i) (rigorously speaking it should be the (p+1)th but I dont think it should change much), and to keep track of this I add 1 to the (p+n/2+1)th component of the array a that I defined to be a=0 before the do cycle. I added n/2+1 to the index of a so that, when p=0 I obtain that the random number is assigned to the (n/2+1)th component. Finally, to do the plot of the gaussian distributions, I defined an index ltot(j) to be ltot(j)=-(n/2.0)+j so that, when I plot the points (a(i), ltot(i)), I should obtain a gaussian ...

Definition of Gaussian distribution with photos and pictures, translations, sample usage, and additional links for more information.

Using the result above, let us evaluate \[ \int_{-\infty}^{\infty}e^{-ax^2+bx+c}dx \] This is easily done by completing the square: \(-ax^2+bx+c=-a\left ( x-\tfrac{b}{2a} \right )^2+\tfrac{b^2}{4a}+c\). This immediately gives \[ \int_{-\infty}^{\infty}e^{-ax^2+bx+c}dx=\int_{-\infty}^{\infty}e^{-a\left ( x-\tfrac{b}{2a} \right )^2+\tfrac{b^2}{4a}+c}dx=e^{\tfrac{b^2}{4a}+c}\int_{-\infty}^{\infty}e^{-au^2}du \] \[ \bbox[5px,border:2px solid red] { \int_{-\infty}^{\infty}e^{-ax^2+bx+c}dx=e^{\tfrac{b^2}{4a}+c}\sqrt{\pi/a} } \] Based on this, we can easily find: \[ \bbox[5px,border:2px solid red] { \int_{-\infty}^{\infty}g_{\mu,\sigma^2}(x)e^{t x}dx=e^{\mu t+\tfrac{1}{2}\sigma^2t^2} } \] This is the same as the moment generating function for a Gaussian distribution. Several results can be deduced from this. For instance, the Fourier transform of a Gaussian function: \[ \bbox[5px,border:2px solid red] { \int_{-\infty}^{\infty}g_{\mu,\sigma^2}(x)e^{-i\omega ...

TY - GEN. T1 - Clustering patient length of stay using mixtures of Gaussian models and phase type distributions. AU - Garg, Lalit. AU - McClean, Sally. AU - Meenan, BJ. AU - El-Darzi, Elia. AU - Millard, Peter. PY - 2009. Y1 - 2009. N2 - Gaussian mixture distributions and Coxian phase type distributions have been popular choices model based clustering of patients length of stay data. This paper compares these models and presents an idea for a mixture distribution comprising of components of both of the above distributions. Also a mixed distribution survival tree is presented. A stroke dataset available from the English Hospital Episode Statistics database is used as a running example.. AB - Gaussian mixture distributions and Coxian phase type distributions have been popular choices model based clustering of patients length of stay data. This paper compares these models and presents an idea for a mixture distribution comprising of components of both of the above distributions. Also a mixed ...

In parametric estimation of covariance function of Gaussian processes, it is often the case that the true covariance function does not belong to the parametric set used for estimation. This situation is called the misspecified case. In this case, it has been observed that, for irregular spatial sampling of observation points, Cross Validation can yield smaller prediction errors than Maximum Likelihood. Motivated by this comparison, we provide a general asymptotic analysis of the misspecified case, for independent observation points with uniform distribution. We prove that the Maximum Likelihood estimator asymptotically minimizes a Kullback-Leibler divergence, within the misspecified parametric set, while Cross Validation asymptotically minimizes the integrated square prediction error. In a Monte Carlo simulation, we show that the covariance parameters estimated by Maximum Likelihood and Cross Validation, and the corresponding Kullback-Leibler divergences and integrated square prediction errors, can be

Lazaro-Gredilla et al. (2010) suggested an alternative approximation to the GP model. In their paper they suggest the decomposition of the GPs stationary covariance function into its Fourier series. The infinite series is then approximated with a finite one. They optimise over the frequencies of the series to minimise some divergence from the full Gaussian process. This approach was named a sparse spectrum approximation. This approach is closely related to the one suggested by Rahimi & Recht (2007) in the randomised methods community (random projections). In Rahimi & Recht (2007)s approach, the frequencies are randomised (sampled from some distribution rather than optimised) and the Fourier coefficients are computed analytically. Both approaches capture globally complex behaviour, but the direct optimisation of the different quantities often leads to some form of over-fitting (Wilson et al., 2014). Similar over-fitting problems that were observed with the sparse pseudo-input approximation ...

Recognition of motions and activities of objects in videos requires effective representations for analysis and matching of motion trajectories. In this paper, we introduce a new representation speciï¬ cally aimed at matching motion trajectories. We model a trajectory as a continuous dense flow ï¬ eld from a sparse set of vector sequences using Gaussian Process Regression. Furthermore, we introduce a random sampling strategy for learning stable classes of motions from limited data. Our representation allows for incrementally predicting possible paths and detecting anomalous events from online trajectories. This representation also supports matching of complex motions with acceleration changes and pauses or stops within a trajectory. We use the proposed approach for classifying and predicting motion trajectories in trafï¬ c monitoring domains and test on several data sets. We show that our approach works well on various types of complete and incomplete trajectories from a variety of ...

Gaussian process (GP) models are a flexible means of performing nonparametric Bayesian regression. However, GP models in healthcare are often only used to model a single univariate output time series, denoted as single-task GPs (STGP). Due to an increasing prevalence of sensors in healthcare settings, there is an urgent need for robust multivariate time-series tools. Here, we propose a method using multitask GPs (MTGPs) which can model multiple correlated multivariate physiological time series simultaneously. The flexible MTGP framework can learn the correlation between multiple signals even though they might be sampled at different frequencies and have training sets available for different intervals. Furthermore, prior knowledge of any relationship between the time series such as delays and temporal behavior can be easily integrated. A novel normalization is proposed to allow interpretation of the various hyperparameters used in the MTGP. We investigate MTGPs for physiological monitoring with synthetic

In this paper, we introduce four different combinations of EWMA schemes, each based on a single plotting statistic for simultaneous monitoring of the mean and variance of a Gaussian process. We compare the four schemes and address the problem of adopting the best combining mechanism. We consider that the actual process parameters are unknown and estimated from a reference sample. We take into account the effects of estimation of unknown parameters in designing the proposed schemes. We consider the maximum likelihood estimators based pivot statistics for monitoring both the parameters and combine them into a single statistic through the max and the distance type combining functions. Also, we examine two different adaptive approaches to introduce pivot statistics into the EWMA-structure. Results show that the distance-type schemes outperform the max-type schemes. Generally, the proposed schemes are useful in detecting small-to-moderate shifts in either or both of the process parameters. ...

If you have a question about this talk, please contact [email protected].. UNQW02 - Surrogate models for UQ in complex systems. Routine diagnostic checking of stationary Gaussian processes fitted to the output of complex computer codes often reveals nonstationary behaviour. There have been a number of approaches, both traditional and more recent, to modelling or accounting for this nonstationarity via the fitted process. These have included the fitting of complex mean functions to attempt to leave a stationary residual process (an idea that is often very difficult to get right in practice), using regression trees or other techniques to partition the input space into regions where different stationary processes are fitted (leading to arbitrary discontinuities in the modelling of the overall process), and other approaches which can be considered to live in one of these camps. In this work we allow the fitted process to be continuous by modelling the covariance kernel as a finite mixture of ...

RECOMMENDED: If you have Windows errors then we strongly recommend that you download and run this (Windows) Repair Tool.. hence known as cumulative errors - Tend to change. It is taken as a measure of the accuracy of measurement. of readings 2 12 2 2. of readings. 1 2. Normal or Gaussian distribution • Random effects in.. Uncertainty, Measurements and Error Analysis. 1. A normal distribution is described by the mean. What are some sources of measurement errors?. Scale mixtures of the skew-normal (SMSN) distribution is a class of asymmetric thick-tailed distributions that includes the skew-normal (SN) distribution as a.. Provides detailed reference material for using SAS/STAT software to perform statistical analyses, including analysis of variance, regression, categorical data.. The red curve is the standard normal distribution: Cumulative distribution function. (such as measurement errors) often have distributions that are nearly normal.. The measurement error with normal distribution is ...

In recent years, several methods have been proposed for the discovery of causal structure from non-experimental data. Such methods make various assumptions on the data generating process to facilitate its identification from purely observational data. Continuing this line of research, we show how to discover the complete causal structure of continuous-valued data, under the assumptions that (a) the data generating process is linear, (b) there are no unobserved confounders, and (c) disturbance variables have non-Gaussian distributions of non-zero variances. The solution relies on the use of the statistical method known as independent component analysis, and does not require any pre-specified time-ordering of the variables. We provide a complete Matlab package for performing this LiNGAM analysis (short for Linear Non-Gaussian Acyclic Model), and demonstrate the effectiveness of the method using artificially generated data and real-world data. [abs][pdf][bib ...

View Notes - normal distribution material from STAT 225 at Purdue. Section 8.5 Normal Random Variables The Normal Distribution is sometimes referred to as the Gaussian Distribution after Carl

In this paper we propose a feasible way to price American options in a model with time-varying volatility and conditional skewness and leptokurtosis, using GARCH processes and the Normal Inverse Gaussian distribution. We show how the risk-neutral dynamics can be obtained in this model, we interpret the effect of the risk-neutralization, and we derive approximation procedures which allow for a computationally efficient implementation of the model. When the model is estimated on financial returns data the results indicate that compared to the Gaussian case the extension is important. A study of the model properties shows that there are important option pricing differences compared to the Gaussian case as well as to the symmetric special case. A large scale empirical examination shows that our model out-performs the Gaussian case for pricing options on the three large US stocks as well as a major index. In particular, improvements are found when it comes to explaining the smile in implied standard ...

Downloadable! In this paper we propose a feasible way to price American options in a model with time varying volatility and conditional skewness and leptokurtosis using GARCH processes and the Normal Inverse Gaussian distribution. We show how the risk neutral dynamics can be obtained in this model, we interpret the effect of the riskneutralization, and we derive approximation procedures which allow for a computationally efficient implementation of the model. When the model is estimated on financial returns data the results indicate that compared to the Gaussian case the extension is important. A study of the model properties shows that there are important option pricing differences compared to the Gaussian case as well as to the symmetric special case. A large scale empirical examination shows that our model outperforms the Gaussian case for pricing options on three large US stocks as well as a major index. In particular, improvements are found when considering the smile in implied standard deviations.

If you have a question about this talk, please contact Rachel Fogg.. Although Monte Carlo based particle filters and smoothers can be used for approximate inference in almost any kind of probabilistic state space models, the required number of samples for a sufficient accuracy can be high. The efficiency of sampling can be improved by Rao-Blackwellization, where part of the state is marginalized out in closed form, and only the remaining part is sampled. Because the sampled space has a lower dimension, fewer particles are required. In this talk I will discuss on Rao-Blackwellization in the context of conditionally linear Gaussian models, and present efficient Rao-Blackwellized versions of previously proposed particle smoothers.. This talk is part of the Signal Processing and Communications Lab Seminars series.. ...

This thread is about the add-in BayesLinear. This add-in estimates a linear Gaussian model using Gibbs sampling. The add-in requests the user input the dependent variable and regressors. The user may select default or custom options for the priors, the number of MCMC draws, and the burn-in sample. The output is a table that reports posterior means, standard deviations, and 95% credibility intervals. In addition, histograms of the draws are displayed ...

end{code} \section{Algebra} We are following the homomorphic learning framework. In this section, we will concern ourselves with algebraic manipulations of fully trained Gaussian models. In particular, we will see how to convert them into other fully trained Gaussian models. %We will do this by working backwards from a known batch trainer for the Gaussian distribution. Knuth presents the %following recurrence relations in pg 232 of Vol2 AoCP: %\begin{align*} %m1_k &= m1_{k-1}+(x_k-m1_{k-1})/k\\ %m2_k &= m2_{k-1}+(x_k-m1_{k-1})/(x_k-m1_k) %\end{align*} \subsection{Semigroup} We want to construct the semigroup operation for our Gaussian distribution so that our batch trainer will be a semigroup homomorphism. That is, we want the property: \begin{spec} (train xs)(train ys) = train (xs++ys) \end{spec} To do this, we must construct appropriate definitions of ,n,, ,m1,, and ,m2, below: \begin{spec} (Gaussian na m1a m2a) (Gaussian nb m1b m2b) = Gaussian n m1 m2 \end{spec} This is a somewhat ...

comparison of a bayesian som with the em algorithm for gaussian mixtures a bayesian som (bsom) [8], is proposed and applied to the unsupervised learni

Yes, there are many techniques which produce probabilities of membership.. One class of techniques is generative techniques. Instead of estimating membership given data, these estimate probability densities for each class, as well as a probability distribution on classes. For example, a Gaussian mixture model may assume that each class is a Gaussian distribution with some mean and covariance. From such a generative model, you can determine the membership probabilities in each class by a proportion $ p(i) = w_i d_i / \sum_j w_j d_j $ where $w_i$ represents the weight of class $i$ and $d_i$ represents the density of the modeled distribution for class $i$ at the input.. Logistic regression and neural networks with a logistic or softmax output also estimate probabilities of membership.. ...

Researchr is a web site for finding, collecting, sharing, and reviewing scientific publications, for researchers by researchers.. Sign up for an account to create a profile with publication list, tag and review your related work, and share bibliographies with your co-authors. ...

The methods of evaluating the singular multivariate normal distribution have been commonly applied even though the complete analytical proofs are not found. Recently, those evaluation methods are shown to have some errors. In this paper we present a new approach with a complete proof for evaluating the exact two-sided percentage points of a standardized m-variate normal distribution with a singular negative product correlation structure for m = 3 and with a singular negative equi-correlated structure for m ≥ 3. The results are then applied to modify the existing procedures for estimating joint confidence intervals for multinomial proportions and for determining sample sizes. By extending the results from the multivariate normal distribution to the multivariate t-distribution with the corresponding singular correlation structure, we obtain the corrected two-sided exact critical values for the Analysis of Means for m = 4, 5 ...

I describe the standard normal distribution and its properties with respect to the percentage of observations within each standard deviation. I also make reference to two key statistical demarcation points (i.e., 1.96 and 2.58) and their relationship to the normal distribution. Finally, I mention two tests that can be used to test normal distributions for statistical significance ...

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one of the goals is to find the matrix A and thus the projections of the data giving the independent components. If the non-Gaussian signals are extremely strong, then one may find the latent dimension k and approximate the subspace spanned by the k columns of A by projecting the data into its k top principal components. If the non-Gaussian components are weak, then the top PCA directions, which maximize the empirical variance, will fail to detect them (17, 18).. Our results are significantly stronger and imply that in the presence of weak signals, any ICA procedure would in general fail. To see this, note that the projection of x onto the subspace orthogonal to the column space of A is only due to noise. Hence, by our results, there exist, for example, 2D projections whose empirical bivariate distribution is highly non-Gaussian with seemingly independent univariate marginals. In other words, we could fit a valid non-Gaussian component model even to the purely Gaussian part of the data.. On the ...

Signal detection in non-Gaussian noise is fundamental to design signal processing systems like decision making or information extraction....

View more ,Until now, marginalization-based Missing Feature Theory (MFT) for speech classification has been limited to the use of Log Spectral Subband Energies (LSSEs) as features. These features are highly correlated, thus suboptimal for classification with diagonal-covariance Gaussian Mixture Models (GMMs), a common classifier in marginalization-based MFT. In this paper, we propose that Spectral Subband Centroids (SSCs) are more apt for marginalization-based MFT, as they are both decorrelated and spectrally local. Our results show that SSCs as features produce a more robust marginalization-based MFT, diagonal-covariance GMM-based, Automatic Speaker Identification (ASI) system than LSSEs as features, for at all tested SNR values (with Additive White Gaussian Noise (AWGN)). It is also shown that a fully-connected Deep Neural Network (DNN) can accurately estimate the Ideal Binary Mask (IBM) used for MFT ...

Gaussian mixture distributions and Coxian phase type distributions have been popular choices model based clustering of patients length of stay data. This paper compares these models and presents an idea for a mixture distribution comprising of components of both of the above distributions. Also a mixed distribution survival tree is presented. A stroke dataset available from the English Hospital Episode Statistics database is used as a running example.. ...

A continuous probability distribution whose probability density function has a bell shape. A normal distribution is symmetric, and has zero skewness. A normal distribution is fully described with two parameters: its mean and standard deviation.

In this work our aim is to estimate the distribution of the maximum between variables representing the natural logarithm of the PM10 emission in two stations of the town of Cagliari in Italy in 2004. It turns out that such order statistic has a skew normal distribution with skew parameter which can be expressed as a function of the correlation coefficient between the two initial variables. The skew-normal distribution belongs to a family of distributions which includes the normal one along with an extra parameter to regulate skewness. Azzalini (1985) was the first to introduce the skew-normal distribution and studied some of its properties. Loperfido (2002) showed that the distribution of the maximum between two standardized correlated normal variables, with correlation coefficient rho, is Skew-Normal with parameter lambda which depend of the correlation coefficient rho. In this specific case we show how is possible, using some theoretical results involving the correlation coefficient, to find ...

Assume that $X=X_1 + X_2 +...+X_n$, where $X_i \sim CN(0,\sigma^2)$ and independent. Here $CN$ means circular complex Gaussian.. The question is, what is the distribution for. $Z = \frac{\left,X\right,^2}{\left,X_1\right,^2 + \left,X_2\right,^2+...+\left,X_n\right,^2}$. How can we benefit from the results obtained here: Distribution of the ratio of dependent chi-square random variables. ...

We then present a natural application to learning mixture models in the PAC framework. For learning a mixture of k axis-aligned Gaussians in d dimensions, we give an algorithm that outputs a mixture of O(k/ϵ3) Gaussians that is ϵ-close in statistical distance to the true distribution, without any separation assumptions. The time and sample complexity is roughly O(kd/ϵ3)d. This is polynomial when d is constant -- precisely the regime in which known methods fail to identify the components efficiently ...

Using the result above, let us evaluate \[ \int_{-\infty}^{\infty}e^{-ax^2+bx+c}dx \] This is easily done by completing the square: \(-ax^2+bx+c=-a\left ( x-\tfrac{b}{2a} \right )^2+\tfrac{b^2}{4a}+c\). This immediately gives \[ \int_{-\infty}^{\infty}e^{-ax^2+bx+c}dx=\int_{-\infty}^{\infty}e^{-a\left ( x-\tfrac{b}{2a} \right )^2+\tfrac{b^2}{4a}+c}dx=e^{\tfrac{b^2}{4a}+c}\int_{-\infty}^{\infty}e^{-au^2}du \] \[ \bbox[5px,border:2px solid red] { \int_{-\infty}^{\infty}e^{-ax^2+bx+c}dx=e^{\tfrac{b^2}{4a}+c}\sqrt{\pi/a} } \] Based on this, we can easily find: \[ \bbox[5px,border:2px solid red] { \int_{-\infty}^{\infty}g_{\mu,\sigma^2}(x)e^{t x}dx=e^{\mu t+\tfrac{1}{2}\sigma^2t^2} } \] This is the same as the moment generating function for a Gaussian distribution. Several results can be deduced from this. For instance, the Fourier transform of a Gaussian function: \[ \bbox[5px,border:2px solid red] { \int_{-\infty}^{\infty}g_{\mu,\sigma^2}(x)e^{-i\omega ...

Normal distribution (Lillie.test()). Hi all, I have a dataset of 2000 numbers ( its noise measured with a scoop ) Now i want to know of my data is normal distributed (Gaussian distribution). I...

The use of a reference population to derive a reference interval is as old as clinical chemistry itself. The underlying concept is that the patient with disease will be distinguishable from individuals who are healthy, as the test results will fall outside the reference interval or normal range. This concept has a degree of validity when the analyte in question has a Gaussian distribution and there is a clear association between an abnormal result and a symptomatic disease state. For many analytes, the situation is more complicated. The distribution of the reference population may be non-Gaussian and the distinction between health and disease more nuanced. An example of this is cholesterol where the overlap between those with and without cardiovascular disease is marked, even in individuals with the extreme phenotype resulting from familial hypercholesterolemia (1). Use of patient self-reference (using the patient as their own normal) overcomes the problem of broad non-Gaussian reference ...

Discriminant analysis and data clustering methods for high dimensional data, based on the assumption that high-dimensional data live in different subspaces with low dimensionality proposing a new parametrization of the Gaussian mixture model which combines the ideas of dimension reduction and constraints on the model.. ...

This paper develops a logistic approximation to the cumulative normal distribution. Although the literature contains a vast collection of approximate functions for the normal distribution, they are very complicated, not very accurate, or valid for only a limited range. This paper proposes an enhanced approximate function. When comparing the proposed function to other approximations studied in the literature, it can be observed that the proposed logistic approximation has a simpler functional form and that it gives higher accuracy, with the maximum error of less than 0.00014 for the entire range. This is, to the best of the authors knowledge, the lowest level of error reported in the literature. The proposed logistic approximate function may be appealing to researchers, practitioners and educators given its functional simplicity and mathematical accuracy.

The NORMAL option requests the fitted curve. The VAXIS= option specifies the AXIS statement controlling the vertical axis. The AXIS1 statement is used to rotate the vertical axis label Cumulative Percent. The INSET statement requests an inset containing the mean, the standard deviation, and the percent of observations below the lower specification limit. For more information about the INSET statement, see Chapter 5, INSET Statement . The SPEC statement requests a lower specification limit at 6.8 with a line type of 2 (a dashed line). For more information about the SPEC statement, see Syntax for the SPEC Statement . The agreement between the empirical and the normal distribution functions in Output 2.1.1 is evidence that the normal distribution is an appropriate model for the distribution of breaking strengths. The CAPABILITY procedure provides a variety of other tools for assessing goodness of fit. Goodness-of-fit tests (see Printed Output ) provide a quantitative assessment of a proposed ...

1. Foundations of probability theory and limit theorems; 2. Systems of Gaussian random variables; 3. Stationary Gaussian processes and their representations; 4. Canonical representation of Gaussian processes{rm: }general theory and multiplicity; 5. Multiple Markov Gaussian processes; 6. Equivalence of Gaussian processes; 7.

Evidence suggests that magnetoencephalogram (MEG) data have characteristics with non-Gaussian distribution, however, standard methods for source localisation assume Gaussian behaviour. We present a new general method for non-Gaussian source estimation of stationary signals for localising brain activity in the MEG data. By providing a Bayesian formulation for linearly constraint minimum variance (LCMV) beamformer, we extend this approach and show that how the source probability density function (pdf), which is not necessarily Gaussian, can be estimated. The proposed non-Gaussian beamformer is shown to give better spatial estimates than the LCMV beamformer, in both simulations incorporating non-Gaussian signal and in real MEG measurements. © 2013 IEEE.

Projection of a high-dimensional dataset onto a two-dimensional space is a useful tool to visualise structures and relationships in the dataset. However, a single two-dimensional visualisation may not display all the intrinsic structure. Therefore, hierarchical/multi-level visualisation methods have been used to extract more detailed understanding of the data. Here we propose a multi-level Gaussian process latent variable model (MLGPLVM). MLGPLVM works by segmenting data (with e.g. K-means, Gaussian mixture model or interactive clustering) in the visualisation space and then fitting a visualisation model to each subset. To measure the quality of multi-level visualisation (with respect to parent and child models), metrics such as trustworthiness, continuity, mean relative rank errors, visualisation distance distortion and the negative log-likelihood per point are used. We evaluate the MLGPLVM approach on the Oil Flow dataset and a dataset of protein electrostatic potentials for the Major ...

0041] Process 400 can begin at block 410 by receiving a video frame from a video source, such as an imaging device. At block 420, process 400 applies a Gaussian mixture model for excluding static background images and images with semantically insignificant motion (e.g., a flag waving in the wind). In this Gaussian mixture model, N Gaussian models are selected for each pixel and the current pixel is classified to be a foreground pixel or background pixel based on the probability of the model that the current pixel fits best. If a model appears more frequently than other models, the pixel will be classified as a background pixel; otherwise, the pixel will be classified as a foreground pixel. The foreground pixels are grouped into objects and tracked through frames to filter out various noise. At block 430, the foreground motion pixels are grouped into blobs by utilizing a connected component analysis method. At block 440, the labeled blobs are tracked in a plurality of consecutive frames. At block ...

Calibration of metal oxide (MOX) gas sensor for continuous monitoring is a complex problem due to the highly dynamic characteristics of the gas sensor signal when exposed to natural environment (Open Sampling System - OSS). This work presents a probabilistic approach to the calibration of a MOX gas sensor based on Gaussian Processes (GP). The proposed approach estimates for every sensor measurement a probability distribution of the gas concentration. This enables the calculation of confidence intervals for the predicted concentrations. This is particularly important since exact calibration is hard to obtain due to the chaotic nature that dominates gas dispersal. The proposed approach has been tested with an experimental setup where an array of MOX sensors and a Photo Ionization Detector (PID) are placed downwind w.r.t. the gas source. The PID is used to obtain ground truth concentration. Comparison with standard calibration methods demonstrates the advantage of the proposed approach.. ...

Global sensitivity analysis is now established as a powerful approach for determining the key random input parameters that drive the uncertainty of model output predictions. Yet the classical computation of the so-called Sobol indices is based on Monte Carlo simulation, which is not af- fordable when computationally expensive models are used, as it is the case in most applications in engineering and applied sciences. In this respect metamodels such as polynomial chaos expansions (PCE) and Gaussian processes (GP) have received tremendous attention in the last few years, as they allow one to replace the original, taxing model by a surrogate which is built from an experimental design of limited size. Then the surrogate can be used to compute the sensitivity indices in negligible time. In this chapter an introduction to each technique is given, with an emphasis on their strengths and limitations in the context of global sensitivity analysis. In particular, Sobol (resp. total Sobol) indices can be

Downloadable! In finite mixture model clustering, each component of the fitted mixture is usually associated with a cluster. In other words, each component of the mixture is interpreted as the probability distribution of the variables of interest conditionally on the membership to a given cluster. The Gaussian mixture model (GMM) is very popular in this context for its simplicity and flexibility. It may happen, however, that the components of the fitted model are not well separated. In such a circumstance, the number of clusters is often overestimated and a better clustering could be obtained by joining some subsets of the partition based on the fitted GMM. Some methods for the aggregation of mixture components have been recently proposed in the literature. In this work, we propose a hierarchical aggregation algorithm based on a generalisation of the definition of silhouette-width taking into account the Mahalanobis distances induced by the precison matrices of the components of the fitted GMM. The

The measurement changes the wave function, in both standard QM and dBB. If you measure the position at time t, then the wide Gaussian at t splits into a large number of narrow non-overlaping Gaussians at t+delta t, where delta t is time during which the measurement-causing interaction takes place. During the time delta t, the particle in dBB ends up in one and only one of these narrow Gaussians. Once the particle ends up in one of these Gaussians, the other narrow Gaussians do not longer influence the motion of the particle. From the point of view of the particle, it is effectively the same as if the wave function collapsed to the narrow Gaussian. Thats how dBB explans the illusion of wave function collapse, without the actual collapse ...

If you have a question about this talk, please contact Mustapha Amrani.. Advanced Monte Carlo Methods for Complex Inference Problems. Stochastic filtering is defined as the estimation of a partially observed dynamical system. A massive scientific and computational effort has been dedicated to the development of numerical methods for approximating the solution of the filtering problem. Approximating with Gaussian mixtures has been very popular since the 1970s, however the existing work is only based on the success of the numerical implementation and is not theoretically justified.. We fill this gap and conduct a rigorous analysis of a new Gaussian mixture approximation to the solution of the filtering problem. In particular, we construct the corresponding approximating algorithm, deduce the L2-convergence rate and prove a central limit type theorem for the approximating system. In addition, we show a numerical example to illustrate some features of this algorithm. This is joint work with Dan ...

Access Information: © Copyright 2019 American Meteorological Society (AMS). Permission to use figures, tables, and brief excerpts from this work in scientific and educational works is hereby granted provided that the source is acknowledged. Any use of material in this work that is determined to be fair use under Section 107 of the U.S. Copyright Act or that satisfies the conditions specified in Section 108 of the U.S. Copyright Act (17 USC §108) does not require the AMSs permission. Republication, systematic reproduction, posting in electronic form, such as on a website or in a searchable database, or other uses of this material, except as exempted by the above statement, requires written permission or a license from the AMS. All AMS journals and monograph publications are registered with the Copyright Clearance Center (http://www.copyright.com). Questions about permission to use materials for which AMS holds the copyright can also be directed to [email protected]. Additional details ...

1] R.J. Adler, An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. Lecture Notes of Inst. Math. Stat., Vol. 12, IMS, Hayword, 1990. , MR 1088478 , Zbl 0747.60039 [2] V.A. Dmitrowskii, On the Integrability of the Maximum and Conditions of Continuity and Local Properties of Gaussian Fields. In Grigelionis B. (Ed.), Probability Theory and Mathematical Statistics, Proceedings 5-th Vilnius Conference, VSP/Mokslas, Vilnius, Vol. 1, 1990, pp. 271-284. , Zbl 0726.60052 [3] V. Dobrič, M.B. Marcus and M. Weber, The Distribution of Large Values of the Supremum of a Gaussian Process, Astérisque, Vol. 157-158, 1988, pp. 95-127. , MR 976215 , Zbl 0659.60061 [4] R.M. Dudley, Sample Functions of Gaussian Processes, Ann. Probab., Vol. 1, 1973, pp. 66-103. , MR 346884 , Zbl 0261.60033 [5] A. Ehrhard, Symetrisation dans lespace de Gauss, Math. Scand., Vol. 53, 1983, pp. 281-301. , MR 745081 , Zbl 0542.60003 [6] X. Fernique, Regularité des trajectoires des fonctions ...

Editors Note: In this section, Ill break down some of the key aspects in probability theory that shape the basis for this website. First, I look at the basic concept behind normal distribution. Please note these explanations wont be 100% up to mathematical textbook standards, simply because these explanations need to be shaped into a sports context. If there are any concerns or criticisms about the process of applying probability theory into a sports context, please contact me at [email protected].. Pivotal to statistical analysis, normal distribution gets relatively overlooked by the stats freaks. Thats quite a surprise, considering that normal distribution can may be one of the best assets to any analyst looking to make a projection based on years of statistical research.. Fan graphs uses normal distribution to help build a range for their player projections, which is absolutely helpful in their process of making player projections for each MLB season. Meanwhile, there is some normal ...

This paper introduces a new unsupervised approach for dialogue act induction. Given the sequence of dialogue utterances, the task is to assign them the labels representing their function in the dialogue. Utterances are represented as real-valued vectors encoding their meaning. We model the dialogue as Hidden Markov model with emission probabilities estimated by Gaussian mixtures. We use Gibbs sampling for posterior inference. We present the results on the standard Switchboard-DAMSL corpus. Our algorithm achieves promising results compared with strong supervised baselines and outperforms other unsupervised algorithms ...

It has been reported that Gaussian functions could accurately and reliably model both carotid and radial artery pressure waveforms (CAPW and RAPW). However, the physiological relevance of the characteristic features from the modeled Gaussian function

View Notes - Chapter 6 from STT 200 at Michigan State University. CHAPTER 6: STANDARD DEVIATION & THE NORMAL MODEL Chapter 6. What is a normal distribution? The normal distribution is pattern for

The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes ...

The large amount of data collected by smart meters is a valuable resource that can be used to better understand consumer behavior and optimize electricity consumption in cities. This paper presents an unsupervised classification approach for extracting typical consumption patterns from data generated by smart electric meters. The proposed approach is based on a constrained Gaussian mixture model whose parameters vary according to the day type (weekday, Saturday or Sunday). The proposed methodology is applied to a real dataset of Irish households collected by smart meters over one year. For each cluster, the model provides three consumption profiles that depend on the day type. In the first instance, the model is applied on the electricity consumption of users during one month to extract groups of consumers who exhibit similar consumption behaviors. The clustering results are then crossed with contextual variables available for the households to show the close links between electricity consumption and

Autori: Adriana Birlutiu, Perry Groot and Tom Heskes. Editorial: Elsevier, Neurocomputing, 73, p.1177-1185, 2010.. Rezumat:. We present an EM-algorithm for the problem of learning preferences with semiparametric models derived from Gaussian processes in the context of multi-task learning. We validate our approach on an audiological data set and show that predictive results for sound quality perception of hearing-impaired subjects, in the context of pairwise comparison experiments, can be improved using a hierarchical model.. Cuvinte cheie: preference learning, multi-task learning, hierarchical modeling, Gaussian processes. URL: http://www.sciencedirect.com/science/article/pii/S0925231210000251. ...

Our goal is to understand the principles of Perception, Action and Learning in autonomous systems that successfully interact with complex environments and to use this understanding to design future systems.

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Human-object interaction (HOI) detection requires a large amount of annotated data. Current algorithms suffer from insufficient training samples and category imbalance within datasets. To increase data efficiency, in this paper, we propose an efficient and effective data augmentation method called DecAug for HOI detection. Based on our proposed object state similarity metric, object patterns across different HOIs are shared to augment local object appearance features without changing their states. Further, we shift spatial correlation between humans and objects to other feasible configurations with the aid of a pose-guided Gaussian Mixture Model while preserving their interactions. Experiments show that our method brings up to 3.3 mAP and 1.6 mAP improvements on V-COCO and HICO-DET dataset for two advanced models. Specifically, interactions with fewer samples enjoy more notable improvement. Our method can be easily integrated into various HOI detection models with negligible extra computational ...

Individual human carcinomas have distinct biological and clinical properties: gene-expression profiling is expected to unveil the underlying molecular features. Particular interest has been focused on potential diagnostic and therapeutic applications. Solid tumors, such as colorectal carcinoma, present additional obstacles for experimental and data analysis. We analyzed the expression levels of 1,536 genes in 100 colorectal cancer and 11 normal tissues using adaptor-tagged competitive PCR, a high-throughput reverse transcription-PCR technique. A parametric clustering method using the Gaussian mixture model and the Bayes inference revealed three groups of expressed genes. Two contained large numbers of genes. One of these groups correlated well with both the differences between tumor and normal tissues and the presence or absence of distant metastasis, whereas the other correlated only with the tumor/normal difference. The third group comprised a small number of genes. Approximately half showed an

A Goodness-of-Fit Test for Multivariate Normal Distribution Using Modified Squared Distance - Multivariate normal distribution;goodness-of-fit test;empirical distribution function;modified squared distance;

The model adaptation system of the present invention is a speaker verification system that embodies the capability to adapt models learned during the enrollment component to track aging of a users voice. The system has the advantage of only requiring a single enrollment for the user. The model adaptation system and methods can be applied to several types of speaker recognition models including neural tree networks (NTN), Gaussian Mixture Models (GMMs), and dynamic time warping (DTW) or to multiple models (i.e., combinations of NTNs, GMMs and DTW). Moreover, the present invention can be applied to text-dependent or text-independent systems.

This article presents advances in optimal experiment design, which are intended to improve the parameter identiï¬ cation of nonlinear state space models. Instead of using a sequence of samples from one or just a few coherent sequences, the idea of identifying nonlinear dynamic models at distinct points in the state space is considered. In this way, the placement of the experiment points is fully flexible with respect to the set of reachable points. Also, a method for model-based generation of prediction errors is proposed, which is used to compute an a-priori estimate of the sample covariance of the prediction error. This covariance matrix may be used to approximate the Fisher information matrix a-priori. The availability of the Fisher matrix a-priori is a prerequisite for experiment optimization with respect to covariance in the parameter estimates. This work is driven by the problem of parameter identiï¬ - cation of hydraulic models. There are methods for hydraulic systems regarding ...

Get an in-depth review and ask questions about Normal Distribution : Finding the mean and standard deviation : ExamSolutions. See what people are saying about Normal Distribution : Finding the mean and standard deviation : ExamSolutions.

1614: logcdf function of normal distribution (scipy.stats) can not handle a wide enough range of values -------------------------------------+-------------------------------------- Reporter: andrewschein , Owner: somebody Type: defect , Status: new Priority: normal , Milestone: Unscheduled Component: scipy.stats , Version: devel Keywords: normal distribution cdf , -------------------------------------+-------------------------------------- Comment(by andrewschein): A cursory glance of the R source code indicates that the log.p is implemented by taking the log() of the CDF (as opposed to some direct computation). The comments in the R code state: {{{ * The _both , lower, upper, and log_p variants were added by * Martin Maechler, Jan.2000; * as well as log1p() and similar improvements later on. }}} It is possible that more than mere interface changes went into the work of implementing log_p, and this could explain the results that follow. Since it appears that the cdflib directory is not fully ...

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During this past week I have been learning use graphics in Kyan Pascal on the Atari 800 emulator Altirra. A program to plot three overlapping normal distribution plots was written and it can illustrate the shift in mean or difference in spread of the plot of a normal distribution, given the mean and standard deviation…

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We provide forecasts for mortality rates by using two different approaches. First we employ dynamic non-linear logistic models based on the Heligman-Pollard formula. Second, we assume that the dynamics of the mortality rates can be modelled through a Gaussian Markov random field. We use efficient Bayesian methods to estimate the parameters and the latent states of the models proposed. Both methodologies are tested with past data and are used to forecast mortality rates both for large (UK and Wales) and small (New Zealand) populations up to 21 years ahead. We demonstrate that predictions for individual survivor functions and other posterior summaries of demographic and actuarial interest are readily obtained. Our results are compared with other competing forecasting methods. ...

Block distance, which is also known as Manhattan distance, computes the distance that would be traveled to get from one data point to the other if a grid-like path is followed. The Block distance between two items is the sum of the differences of their corresponding components [6]. Euclidean distance, or L2 distance, is the square root of the sum of squared differences between corresponding elements of the two vectors. Matching coefficient is a very simple vector based approach which simply counts the number of similar terms (dimensions), with which both vectors are non-zero. Overlap coefficient considers two strings as a full match if one is a subset of the other [7]. Gaussian model is a probabilistic model which can be used to characterize a group of feature vectors of any number of dimensions with two values, a mean vector, and a covariance matrix. The Gaussian model is one way of calculating the conditional probability [8]. Traditional spectral clustering algorithms typically use a Gaussian ...

Indeed, as J.C. said this has to do with the renormalization group (RG) which in the present context is a transformation $\mu\rightarrow \mu\ast\mu$ followed by rescaling by $\sqrt{2}$ to keep the variance the same. The orbits are the trajectories or sequences of iterates of a given probability measure by that RG transformation. The standard Gaussian is an attractive fixed point to which all these trajectories converge. This is one way to understand the central limit theorem. See this MO question for more info on this and in particular the paper by Anshelevich mentioned in the comment therein by Yemon Choi.. Also, one of the first references in this circle of ideas is the article The renormalization group: A probabilistic view by Jona-Lasinio. Finally you can find more explanations about the RG in my answer to this MO question.. ...

In the theoretical framework of the analysis of stochastic systems, statistical methodologies for random fluctuations have been proposed based on the inver

This section of the Engineering Statistics Handbook gives the normal probability density function as well as the standard normal distribution equations. Example graphs of the distributions are shown and a justification of the Central Limit Theorem is included ...

Short answer: to know a MVN distribution you need to know the mean vector and the covariance matrix. If you dont know a distribution you cannot simulate from it. So you need to know the marginal variances (the diagonal of the covariance matrix). If you have those, you can form the covariance matrix and use rmvnorm or mvrnorm. If you are willing to assume they are one, you have the covariance (= correlation matrix). If you dont know the marginal variances the problem is incompletely specified. On Fri, 25 Jun 2004, Matthew David Sylvester wrote: , Hello, , I would like to simulate randomly from a multivariate normal distribution using a correlation , matrix, rho. I do not have sigma. I have searched the help archive and the R documentation as , well as doing a standard google search. What I have seen is that one can either use rmvnorm in , the package: mvtnorm or mvrnorm in the package: MASS. I believe I read somewhere that the latter , was more robust. I have seen conflicting (or at least ...

Functions are like vectors. Actually you can define a vector space over functions. We can describe any vector in terms of 3 independent vector which may not be orthogonal to each other. Similarly, you can fit a function in terms of other linearly independent functions. The easiest case is fitting to a polynomial of order n. Depending on how well your data can be fit you can set n. Another example is fitting to a series of Sine and Cose functions(Discrete Fourier transform). One other example is fitting to Gaussian functions with different mean and standard deviation.. ...

The problem of calculation of electro and thermo static fields in an infinite homogeneous medium with a heterogeneous isolated inclusion (Kanaun et al) has shown to be reduced to the solution of integral equations for the fields inside the inclusion using Gaussian functions (V. Mazya) for the approximation of the unknown fields. Using this approach coefficients of the matrix of the discretized system will be obtained in closed analytical forms.

where s is the step index, t an index into the training sample, u is the index of the BMU for the input vector D(t), α(s) is a monotonically decreasing learning coefficient; Θ(u, v, s) is the neighborhood function which gives the distance between the neuron u and the neuron v in step s.[11] Depending on the implementations, t can scan the training data set systematically (t is 0, 1, 2...T-1, then repeat, T being the training samples size), be randomly drawn from the data set (bootstrap sampling), or implement some other sampling method (such as jackknifing). The neighborhood function Θ(u, v, s) depends on the grid-distance between the BMU (neuron u) and neuron v. In the simplest form, it is 1 for all neurons close enough to BMU and 0 for others, but a Gaussian function is a common choice, too. Regardless of the functional form, the neighborhood function shrinks with time.[9] At the beginning when the neighborhood is broad, the self-organizing takes place on the global scale. When the ...

In this note some well known asymptotic results for moments of order statistics from the normal distribution are treated. The results originates from the work of Cramér. A bias correction for finite sample sizes is proposed for the expected value of the largest observation ...

The graphs might be a little bit overkill, but its cool all the different ways you can visualize the this simple data. The number of pitches is distributed normally with a skew left. This skew occurs because there are instances when the pitcher has a bad day and gets pulled really early. To account for this, I excluded any outing that didnt have more than 50 pitches. We will consider these as rare events, which we shouldnt try to use in our prediction. The idea of the game is to hit the exact pitch count, and this would preclude a rare event from being factored in. I also used the median number of pitches instead of the average number of pitches for the same reason. We want to consistently pick numbers which are the most likely to get hit, not to try to predict every game.. The idea of using the median over the mean is important when there is a skew to the normal distribution of the data. This is important for something like income. There is a huge skew for incomes across the entire US ...

hey, now i using the Visual Studio 2010 C++ i would like to random generate a number from log normal distribution, but so far i only know to random ge

I handed out the following page today: Normal Distribution Practice Complete it for homework. Note that the table in your textbook only goes to z-scores up to +/-2.99, so there are a couple of probabilities you wont be able to find (sorry about that).