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Nonparametric
Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as in parametric statistics. Nonparametric statistics can be used for descriptive statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated. Definitions The term "nonparametric statistics" has been defined imprecisely in the following two ways, among others: The first meaning of ''nonparametric'' involves techniques that do not rely on data belonging to any particular parametric family of probability distributions. These include, among others: * Methods which are ''distribution-free'', which do not rely on assumptions that the data are drawn from a given parametric family of probability distributions. * Statistics defined to be a function on a sample, without dependency on a pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Process
In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution. As an example, a bag of 100 real-world dice is a ''random probability mass function (random pmf)''—to sample this random pmf you put your hand in the bag and draw out a die, that is, you draw a pmf. A bag of dice manufactured using a crude process 100 years ago will likely have probabilities that deviate wildly from the uniform pmf, whereas a bag of state-of-the-art dice used by Las Vegas casinos may have barely perceptible imperfe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonparametric Regression
Nonparametric regression is a form of regression analysis where the predictor does not take a predetermined form but is completely constructed using information derived from the data. That is, no parametric equation is assumed for the relationship between predictors and dependent variable. A larger sample size is needed to build a nonparametric model having a level of uncertainty as a parametric model because the data must supply both the model structure and the parameter estimates. Definition Nonparametric regression assumes the following relationship, given the random variables X and Y: : \mathbb \mid X=x= m(x), where m(x) is some deterministic function. Linear regression is a restricted case of nonparametric regression where m(x) is assumed to be a linear function of the data. Sometimes a slightly stronger assumption of additive noise is used: : Y = m(X) + U, where the random variable U is the `noise term', with mean 0. Without the assumption that m belongs to a specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiparametric Regression
In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components. A statistical model is a parameterized family of distributions: \ indexed by a parameter \theta. * A parametric model is a model in which the indexing parameter \theta is a vector in k-dimensional Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ..., for some nonnegative integer k.. Thus, \theta is finite-dimensional, and \Theta \subseteq \mathbb^k. * With a nonparametric model, the set of possible values of the parameter \theta is a subset of some space V, which is not necessarily finite-dimensional. For example, we might consider the set of all distributions with mean 0. Such spaces are vector spaces with topological structure, but may not be finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Statistics
Parametric statistics is a branch of statistics which leverages models based on a fixed (finite) set of parameters. Conversely nonparametric statistics does not assume explicit (finite-parametric) mathematical forms for distributions when modeling data. However, it may make some assumptions about that distribution, such as continuity or symmetry, or even an explicit mathematical shape but have a model for a distributional parameter that is not itself finite-parametric. Most well-known statistical methods are parametric. Regarding nonparametric (and semiparametric) models, Sir David Cox has said, "These typically involve fewer assumptions of structure and distributional form but usually contain strong assumptions about independencies". Example The normal family of distributions all have the same general shape and are ''parameterized'' by mean and standard deviation. That means that if the mean and standard deviation are known and if the distribution is normal, the probability o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (statistics)
The term kernel is used in statistics, statistical analysis to refer to a window function. The term "kernel" has several distinct meanings in different branches of statistics. Bayesian statistics In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. Note that such factors may well be functions of the parameters of the pdf or pmf. These factors form part of the normalization factor of the probability distribution, and are unnecessary in many situations. For example, in pseudo-random number sampling, most sampling algorithms ignore the normalization factor. In addition, in Bayesian analysis of conjugate prior distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered. At the end, the form of the kernel is examined, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Histogram
A histogram is a visual representation of the frequency distribution, distribution of quantitative data. To construct a histogram, the first step is to Data binning, "bin" (or "bucket") the range of values— divide the entire range of values into a series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping interval (mathematics), intervals of a variable. The bins (intervals) are adjacent and are typically (but not required to be) of equal size. Histograms give a rough sense of the density of the underlying distribution of the data, and often for density estimation: estimating the probability density function of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the ''x''-axis are all 1, then a histogram is identical to a relative frequency plot. Histograms are sometimes confused with bar charts. In a his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Statistics
Robust statistics are statistics that maintain their properties even if the underlying distributional assumptions are incorrect. Robust Statistics, statistical methods have been developed for many common problems, such as estimating location parameter, location, scale parameter, scale, and regression coefficient, regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a Parametric statistics, parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly. Introduction Robust statistics seek to provide methods that emulate popular statistical methods, but are not unduly affected by outliers or other small departures from Statistical assumption, model assumptions. In statistics, classical e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Power
In frequentist statistics, power is the probability of detecting a given effect (if that effect actually exists) using a given test in a given context. In typical use, it is a function of the specific test that is used (including the choice of test statistic and significance level), the sample size (more data tends to provide more power), and the effect size (effects or correlations that are large relative to the variability of the data tend to provide more power). More formally, in the case of a simple hypothesis test with two hypotheses, the power of the test is the probability that the test correctly rejects the null hypothesis (H_0) when the alternative hypothesis (H_1) is true. It is commonly denoted by 1-\beta, where \beta is the probability of making a type II error (a false negative) conditional on there being a true effect or association. Background Statistical testing uses data from samples to assess, or make inferences about, a statistical population. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Data Envelopment Analysis
Data envelopment analysis (DEA) is a nonparametric method in operations research and economics for the estimation of production frontiers.Charnes et al (1978) DEA has been applied in a large range of fields including international banking, economic sustainability, police department operations, and logistical applicationsCharnes et al (1995)Emrouznejad et al (2016)Thanassoulis (1995) Additionally, DEA has been used to assess the performance of natural language processing models, and it has found other applications within machine learning.Zhou et al (2022)Guerrero et al (2022) Description DEA is used to empirically measure productive efficiency of decision-making units (DMUs). Although DEA has a strong link to production theory in economics, the method is also used for benchmarking in operations management, whereby a set of measures is selected to benchmark the performance of manufacturing and service operations. In benchmarking, the efficient DMUs, as defined by DEA, may not nece ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-nearest Neighbors Algorithm
In statistics, the ''k''-nearest neighbors algorithm (''k''-NN) is a Non-parametric statistics, non-parametric supervised learning method. It was first developed by Evelyn Fix and Joseph Lawson Hodges Jr., Joseph Hodges in 1951, and later expanded by Thomas M. Cover, Thomas Cover. Most often, it is used for statistical classification, classification, as a ''k''-NN classifier, the output of which is a class membership. An object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its ''k'' nearest neighbors (''k'' is a positive integer, typically small). If ''k'' = 1, then the object is simply assigned to the class of that single nearest neighbor. The ''k''-NN algorithm can also be generalized for regression analysis, regression. In ''-NN regression'', also known as ''nearest neighbor smoothing'', the output is the property value for the object. This value is the average of the values of ''k'' nearest neighbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Statistics
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and analysing those statistics. Descriptive statistics is distinguished from inferential statistics (or inductive statistics) by its aim to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent. This generally means that descriptive statistics, unlike inferential statistics, is not developed on the basis of probability theory, and are frequently nonparametric statistics. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example, in papers reporting on human subjects, typically a table is included giving the overall sample size, sample sizes in important subgroups (e.g., for each treatmen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
