|
Probability Plot Correlation Coefficient Plot
The probability plot correlation coefficient (PPCC) plot is a graphical technique for identifying the shape parameter for a distributional family that best describes the data set. This technique is appropriate for families, such as the Weibull, that are defined by a single shape parameter and location and scale parameters, and it is not appropriate or even possible for distributions, such as the normal, that are defined only by location and scale parameters. Many statistical analyses are based on distributional assumptions about the population from which the data have been obtained. However, distributional families can have radically different shapes depending on the value of the shape parameter. Therefore, finding a reasonable choice for the shape parameter is a necessary step in the analysis. In many analyses, finding a good distributional model for the data is the primary focus of the analysis. The technique is simply "plot the probability plot correlation coefficients for di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphical Technique
Statistical graphics, also known as statistical graphical techniques, are graphics used in the field of statistics for data visualization. Overview Whereas statistics and data analysis procedures generally yield their output in numeric or tabular form, graphical techniques allow such results to be displayed in some sort of pictorial form. They include plots such as scatter plots, histograms, probability plots, spaghetti plots, residual plots, box plots, block plots and biplots. Exploratory data analysis (EDA) relies heavily on such techniques. They can also provide insight into a data set to help with testing assumptions, model selection and regression model validation, estimator selection, relationship identification, factor effect determination, and outlier detection. In addition, the choice of appropriate statistical graphics can provide a convincing means of communicating the underlying message that is present in the data to others. Graphical statistical methods have fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lognormal Distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed, then has a normal distribution. Equivalently, if has a normal distribution, then the exponential function of , , has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics). The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution has also been associated with other names, such as Donald MacAlister#log-normal, McAlister, Gibrat's law, Gibrat and Cobb–Douglas. A l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as exponents of the variable and its complement to 1, respectively, and control the shape parameter, shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions. In Bayesian inference, the beta distribution is the conjugate prior distribution, conjugate prior probability distribution for the Bernoulli distribution, Bernoulli, binomial distribution, binomial, negative binomial distribution, negative binomial, and geometric distribution, geometric distributions. The formulation of the beta dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short-tailed Distribution
In probability theory and statistics, kurtosis (from , ''kyrtos'' or ''kurtos'', meaning "curved, arching") refers to the degree of “tailedness” in the probability distribution of a real-valued random variable. Similar to skewness, kurtosis provides insight into specific characteristics of a distribution. Various methods exist for quantifying kurtosis in theoretical distributions, and corresponding techniques allow estimation based on sample data from a population. It’s important to note that different measures of kurtosis can yield varying interpretations. The standard measure of a distribution's kurtosis, originating with Karl Pearson, is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak; hence, the sometimes-seen characterization of kurtosis as " peakedness" is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations (or outliers), and not the configuratio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Distribution
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together along the x-axis, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two Independent identically-distributed random variables, independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution. Definitions Probability density function A random variable has a \operatorname(\mu, b) distribution if its probability density function is : f(x \mid \mu, b) = \frac \exp\left( -\frac \rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long-tailed Distribution
In statistics and business, a long tail of some distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random numbers of occurrences of events with various probabilities, etc. The term is often used loosely, with no definition or an arbitrary definition, but precise definitions are possible. In statistics, the term ''long-tailed distribution'' has a narrow technical meaning, and is a subtype of heavy-tailed distribution. Intuitively, a distribution is (right) long-tailed if, for any fixed amount, when a quantity exceeds a high level, it almost certainly exceeds it by at least that amount: large quantities are probably even larger. Note that there is no sense of ''the'' "long tail" of a distribution, but only the ''property'' of a distribution being long-tailed. In business, the term ''long tail'' is applied to rank-size distributions or rank-fre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Uniform Distribution
In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (i.e. ,b/math>) or open (i.e. (a,b)). Therefore, the distribution is often abbreviated U(a,b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is f(x) = \begin \dfrac & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Distribution
In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis). The logistic distribution is a special case of the Tukey lambda distribution. Specification Cumulative distribution function The logistic distribution receives its name from its cumulative distribution function, which is an instance of the family of logistic functions. The cumulative distribution function of the logistic distribution is also a scaled version of the Hyperbolic function, hyperbolic tangent. :F(x; \mu, s) = \frac = \frac12 + \frac12 \operatorname \left(\frac\right). In this equation is the mean, and is a scale parameter proportional to the standard deviation. Probability density function The probability density function is the partia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Distribution
The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution f(x; x_0,\gamma) is the distribution of the -intercept of a ray issuing from (x_0,\gamma) with a uniformly distributed angle. It is also the distribution of the Ratio distribution, ratio of two independent Normal distribution, normally distributed random variables with mean zero. The Cauchy distribution is often used in statistics as the canonical example of a "pathological (mathematics), pathological" distribution since both its expected value and its variance are undefined (but see below). The Cauchy distribution does not have finite moment (mathematics), moments of order greater than or equal to one; only fractional absolute moments exist., Chapter 16. The Cauchy dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Gaussian Distribution
In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support (mathematics), support on (0,∞). Its probability density function is given by : f(x;\mu,\lambda) = \sqrt\frac \exp\biggl(-\frac\biggr) for ''x'' > 0, where \mu > 0 is the mean and \lambda > 0 is the shape parameter. The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an inverse only in that, while the Gaussian describes a Wiener process, Brownian motion's level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian motion with positive drift takes to reach a fixed positive level. Its cumulant generating function (logarithm of the characteristic function) is the inverse of the cumulant generating function of a Gaussian random variable. To indicate that a random variable ''X'' is inverse Gauss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Distribution
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: # With a shape parameter and a scale parameter # With a shape parameter \alpha and a rate parameter In each of these forms, both parameters are positive real numbers. The distribution has important applications in various fields, including econometrics, Bayesian statistics, and life testing. In econometrics, the (''α'', ''θ'') parameterization is common for modeling waiting times, such as the time until death, where it often takes the form of an Erlang distribution for integer ''α'' values. Bayesian statisticians prefer the (''α'',''λ'') parameterization, utilizing the gamma distribution as a conjugate prior for several inverse scale parameters, facilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reliability (statistics)
In statistics and psychometrics, reliability is the overall consistency of a measure. A measure is said to have a high reliability if it produces similar results under consistent conditions:It is the characteristic of a set of test scores that relates to the amount of random error from the measurement process that might be embedded in the scores. Scores that are highly reliable are precise, reproducible, and consistent from one testing occasion to another. That is, if the testing process were repeated with a group of test takers, essentially the same results would be obtained. Various kinds of reliability coefficients, with values ranging between 0.00 (much error) and 1.00 (no error), are usually used to indicate the amount of error in the scores. For example, measurements of people's height and weight are often extremely reliable.The Marketing Accountability Standards Board (MASB) endorses this definition as part of its ongoinCommon Language: Marketing Activities and Metrics Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
