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Variance-stabilizing Transformation
In applied statistics, a variance-stabilizing transformation is a data transformation that is specifically chosen either to simplify considerations in graphical exploratory data analysis or to allow the application of simple regression-based or analysis of variance techniques. Overview The aim behind the choice of a variance-stabilizing transformation is to find a simple function ''ƒ'' to apply to values ''x'' in a data set to create new values such that the variability of the values ''y'' is not related to their mean value. For example, suppose that the values x are realizations from different Poisson distributions: i.e. the distributions each have different mean values ''μ''. Then, because for the Poisson distribution the variance is identical to the mean, the variance varies with the mean. However, if the simple variance-stabilizing transformation :y=\sqrt \, is applied, the sampling variance associated with observation will be nearly constant: see Anscombe transform for d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Data Transformation (statistics)
In statistics, data transformation is the application of a deterministic mathematical function to each point in a data set—that is, each data point ''zi'' is replaced with the transformed value ''yi'' = ''f''(''zi''), where ''f'' is a function. Transforms are usually applied so that the data appear to more closely meet the assumptions of a statistical inference procedure that is to be applied, or to improve the interpretability or appearance of graphs. Nearly always, the function that is used to transform the data is invertible, and generally is continuous. The transformation is usually applied to a collection of comparable measurements. For example, if we are working with data on peoples' incomes in some currency unit, it would be common to transform each person's income value by the logarithm function. Motivation Guidance for how data should be transformed, or whether a transformation should be applied at all, should come from the particular statistical analysis to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analysis Of Variance
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the ''t''-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means. History While the analysis of variance reached fruition in the 20th century, antecedents extend centuries into the past according to Stigler. These include hypothesis testing, the partitioning of sums of squares, experimental techniques and the additive model. Laplace was performing hypothesis test ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson (; ). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume. For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anscombe Transform
In statistics, the Anscombe transform, named after Francis Anscombe, is a variance-stabilizing transformation that transforms a random variable with a Poisson distribution into one with an approximately standard Gaussian distribution. The Anscombe transform is widely used in photon-limited imaging (astronomy, X-ray) where images naturally follow the Poisson law. The Anscombe transform is usually used to pre-process the data in order to make the standard deviation approximately constant. Then denoising algorithms designed for the framework of additive white Gaussian noise are used; the final estimate is then obtained by applying an inverse Anscombe transformation to the denoised data. Definition For the Poisson distribution the mean m and variance v are not independent: m = v. The Anscombe transform : A:x \mapsto 2 \sqrt \, aims at transforming the data so that the variance is set approximately 1 for large enough mean; for mean zero, the variance is still zero. It transf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: ''success'' (with probability ''p'') or ''failure'' (with probability q=1-p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., ''n'' = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size ''n'' drawn with replacement from a population of size ''N''. If the sampling is carried out without replacement, the draws are not independent and so the resulting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Transform
In statistics, a power transform is a family of functions applied to create a monotonic transformation of data using power functions. It is a data transformation technique used to stabilize variance, make the data more normal distribution-like, improve the validity of measures of association (such as the Pearson correlation between variables), and for other data stabilization procedures. Power transforms are used in multiple fields, including multi-resolution and wavelet analysis, statistical data analysis, medical research, modeling of physical processes, geochemical data analysis, epidemiology and many other clinical, environmental and social research areas. Definition The power transformation is defined as a continuously varying function, with respect to the power parameter ''λ'', in a piece-wise function form that makes it continuous at the point of singularity (''λ'' = 0). For data vectors (''y''1,..., ''y''''n'') in which each ''y''''i'' > 0, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Error
The approximation error in a data value is the discrepancy between an exact value and some '' approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value). An approximation error can occur because of computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.1 cm, so you measure it as 4.5 cm). In the mathematical field of numerical analysis, the numerical stability of an algorithm indicates how the error is propagated by the algorithm. Formal definition One commonly distinguishes between the relative error and the absolute error. Given some value ''v'' and its approximation ''v''approx, the absolute error is :\epsilon = , v-v_\text, \ , where the vertical bars denote the absolute value. If v \ne 0, the relative error is : \eta = \frac = \left, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Hyperbolic Sine
In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The size of the hyperbolic angle is equal to the area of the corresponding hyperbolic sector of the hyperbola , or twice the area of the corresponding sector of the unit hyperbola , just as a circular angle is twice the area of the circular sector of the unit circle. Some authors have called inverse hyperbolic functions "area functions" to realize the hyperbolic angles. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. It also occurs in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delta Method
In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator. History The delta method was derived from propagation of error, and the idea behind was known in the early 19th century. Its statistical application can be traced as far back as 1928 by T. L. Kelley. A formal description of the method was presented by J. L. Doob in 1935. Robert Dorfman also described a version of it in 1938. Univariate delta method While the delta method generalizes easily to a multivariate setting, careful motivation of the technique is more easily demonstrated in univariate terms. Roughly, if there is a sequence of random variables satisfying :, where ''θ'' and ''σ''2 are finite valued constants and \xrightarrow denotes convergence in distribution, then : for any function ''g'' satisfying the property that exists and is non-zero v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heteroscedasticity
In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The spellings ''homoskedasticity'' and ''heteroskedasticity'' are also frequently used. Assuming a variable is homoscedastic when in reality it is heteroscedastic () results in unbiased but inefficient point estimates and in biased estimates of standard errors, and may result in overestimating the goodness of fit as measured by the Pearson coefficient. The existence of heteroscedasticity is a major concern in regression analysis and the analysis of variance, as it invalidates statistical tests of significance that assume that the modelling errors all have the same variance. While the ordinary least squares estimator is still unbiased in the presence of heteroscedasticity, it is inefficient and generalized least squares should be use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
