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Smooth function in statistics
statistics, the variance function is a smooth function that depicts the variance of a random quantity as a function of its mean. The variance function is a measure
Variance_function
Statistical measure of how far values spread from their average
In probability theory and statistics, variance is a measure of dispersion, meaning it is a measure of how far a set of numbers are spread out from their
Variance
Probability distribution
described by this probability density function (or density): φ(z)=e−z2/22π. The variable z has a mean of 0 and a variance and standard deviation of 1. The
Normal_distribution
Class of statistical models
response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value. Generalized
Generalized_linear_model
Family of probability distributions
the function τ ( θ ) = κ ′ ( θ ) = μ . {\displaystyle \tau (\theta )=\kappa ^{\prime }(\theta )=\mu .} with cumulative function κ(θ). The variance function
Tweedie_distribution
Measure of frequency stability in clocks and oscillators
The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers. It is named after
Allan_variance
Property of a model
In statistics and machine learning, the bias–variance tradeoff describes the relationship between a model's complexity, the accuracy of its predictions
Bias–variance_tradeoff
Programming language concept
chosen variance determines the relationship between, for example, a list of Cats and a list of Animals, or a function returning Cat and a function returning
Type_variance
Class of probability distributions
NEF, called NEF with quadratic variance function (NEF-QVF) because the variance can be written as a quadratic function of the mean. NEF-QVF are discussed
Natural_exponential_family
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
Standard score Information ratio Omega ratio Sampling (statistics) Variance function Gini coefficient Everitt, Brian (1998). The Cambridge Dictionary of
Coefficient_of_variation
Mathematical model for stochastic processes
the conditional variance function, V a r ( Y ∣ X ) = σ 2 ( μ ) {\displaystyle {\rm {{Var}(Y\mid X)=\sigma ^{2}(\mu )}}} , as a function of the conditional
Generalized functional linear model
Generalized_functional_linear_model
Mathematical function
Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances: c
Gaussian_function
Variance of a random variable given value of other variables
econometrics, the conditional variance is also known as the scedastic function or skedastic function. Conditional variances are important parts of autoregressive
Conditional_variance
Set of probability distributions
{Var} [Y]=\sigma ^{2}A''(\theta )=\sigma ^{2}V(\mu )\,\!,} with unit variance function V ( μ ) = A ″ ( ( A ′ ) − 1 ( μ ) ) {\displaystyle V(\mu )=A''((A')^{-1}(\mu
Exponential_dispersion_model
Measure of variation in statistics
data set or probability distribution is the square root of its variance (the variance being the average of the squared deviations from the mean). A useful
Standard_deviation
Statistical property
(an example of a scedastic function), so the variance is proportional to the value of x. More generally, if the variance-covariance matrix of disturbance
Homoscedasticity and heteroscedasticity
Homoscedasticity_and_heteroscedasticity
Concept in applied statistics
regression-based or analysis of variance techniques. The aim behind the choice of a variance-stabilizing transformation is to find a simple function ƒ to apply to values
Variance-stabilizing transformation
Variance-stabilizing_transformation
Probability distribution
distribution when marginalizing over the variance parameter. Student's t distribution has the probability density function (PDF) given by f ( t ) = Γ ( ν + 1
Student's_t-distribution
Inexact statistical measure
mean and the variance is specified in the form of a variance function giving the variance as a function of the mean. Generally, this function is allowed
Quasi-likelihood
Fundamental theorem in probability theory and statistics
each with zero mean and unit variance (var(Y)=1). By independence of the random variables Y1,…,Yn, the characteristic function of Zn is given by
Central_limit_theorem
Statistical measure in mathematical model
In statistics, the variance inflation factor (VIF) is the ratio (quotient) of the variance of a parameter estimate when fitting a full model that includes
Variance_inflation_factor
Correlation of a signal with a time-shifted copy of itself, as a function of shift
{\displaystyle \mu } and the variance σ 2 {\displaystyle \sigma ^{2}} are time-independent, and further the autocovariance function depends only on the lag
Autocorrelation
Middle quantile of a data set or probability distribution
sample median from a population with a density function f(x) is asymptotically normal with mean m and variance 14nf(m)2 where m is the median of f(x) and
Median
Measure of the error of an estimator
the square root of the variance, known as the standard error. The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs
Mean_squared_error
Probability distribution
(1-\mu )} The accompanying plot of skewness as a function of variance and mean shows that maximum variance (1/4) is coupled with zero skewness and the symmetry
Beta_distribution
Fictional organization appearing in comic books published by Marvel Comics
The Time Variance Authority (TVA) is a fictional organization appearing in American comic books published by Marvel Comics. It is depicted as a group of
Time_Variance_Authority
Measure of covariance of components of a random vector
matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between
Covariance_matrix
Sales variance is the difference between actual sales and budgeted sales. It is used to measure the performance of a sales function, and/or analyze business
Sales_variance
Unbiased statistical estimator minimizing variance
minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than
Minimum-variance unbiased estimator
Minimum-variance_unbiased_estimator
Probability distribution
E[X]=E[X1+⋯+Xn]=E[X1]+⋯+E[Xn]=p+⋯+p=np. The variance is: Var(X)=npq=np(1−p). This similarly follows from the fact that the variance of a sum of independent random
Binomial_distribution
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Measure of linear correlation
ratio of two variances Mean cross-product of standardized variables Function of the angle between two standardized regression lines Function of the angle
Pearson correlation coefficient
Pearson_correlation_coefficient
Features that do not change if length or energy scales are multiplied by a common factor
model that asymptotically manifests a variance to mean power law will be required express a variance function that comes within the domain of attraction
Scale_invariance
Statistical hypothesis test
t-tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal; the form of the test
Student's_t-test
Statistical property
loss-functions. Any minimum-variance mean-unbiased estimator minimizes the risk (expected loss) with respect to the squared-error loss function (among
Bias_of_an_estimator
Family of statistical methods based on sampling of available data
package 'samplingVarEst': Sampling Variance Estimation. Implements functions for estimating the sampling variance of some point estimators. Paired
Resampling_(statistics)
Function related to statistics and probability theory
A likelihood function (often simply called the likelihood) gives the relative merit of various statistical models for describing a data set. Often the
Likelihood_function
Rule for calculating an estimate of a given quantity based on observed data
low variance means the arrows are clustered. Even if the variance is low, the cluster of arrows may still be far off-target, and even if the variance is
Estimator
Linear regression model with a single explanatory variable
finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent
Simple_linear_regression
Form of global sensitivity analysis
Variance-based sensitivity analysis (often referred to as the Sobol’ method or Sobol’ indices, after Ilya M. Sobol’) is a form of global sensitivity analysis
Variance-based sensitivity analysis
Variance-based_sensitivity_analysis
Collection of statistical models
Analysis of variance (ANOVA) is a family of statistical methods used to compare the means of two or more groups by analyzing variance. Specifically, ANOVA
Analysis_of_variance
Effect of variables' uncertainties on the uncertainty of a function based on them
{J} ^{\top }.} That is, the Jacobian of the function is used to transform the rows and columns of the variance-covariance matrix of the argument. Note this
Propagation_of_uncertainty
Method of data analysis
original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first
Principal_component_analysis
Probability distribution
diverge to infinity even faster than sample variance. If a probability distribution has a density function f ( x ) {\displaystyle f(x)} , then the mean
Cauchy_distribution
Approximation method in statistics
cases. Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable and the deviations
Least_squares
Continuous probability distribution
The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution is a continuous probability distribution that is defined
Variance-gamma_distribution
Set of statistical processes for estimating the relationships among variables
quartet Curve fitting Estimation theory Forecasting Fraction of variance unexplained Function approximation Generalized linear model Kriging (a linear least
Regression_analysis
Probability that random variable X is less than or equal to x
cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,
Cumulative distribution function
Cumulative_distribution_function
Important algorithms in numerical statistics
Algorithms for calculating variance play a major role in computational statistics. A key difficulty in the design of good algorithms for this problem is
Algorithms for calculating variance
Algorithms_for_calculating_variance
Discrete-variable probability distribution
a probability mass function (PMF), discrete probability density function, probability function, or frequency function is a function that gives the probability
Probability_mass_function
Method of estimating the parameters of a statistical model, given observations
the random errors are assumed to have normal distributions with the same variance. From the perspective of Bayesian inference, MLE is generally equivalent
Maximum_likelihood_estimation
Concept in probability theory and statistics
if Z and W are not independent. The variance is defined in terms of absolute squares as: Properties The variance is always a nonnegative real number.
Complex_random_variable
Statistical hypothesis test
statistical test that compares variances. It is used to determine if the variances of two samples, or if the ratios of variances among multiple samples, are
F-test
Statistical test of whether two populations have equal means
Welch's t-test, or unequal variances t-test in statistics is a two-sample location test which is used to test the (null) hypothesis that two populations
Welch's_t-test
Statistical method
estimated from the data. Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, etc.) to sample estimates. This
Bootstrapping_(statistics)
Measure of the joint variability
L2 inner product of real-valued functions on the sample space. As a result, for random variables with finite variance, the inequality | cov ( X , Y
Covariance
Statistical noise
In statistics, the fraction of variance unexplained (FVU) in the context of a regression task is the fraction of variance of the regressand (dependent variable)
Fraction of variance unexplained
Fraction_of_variance_unexplained
Description of continuous random distribution
probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function whose value at any given
Probability_density_function
Concept in mathematical modelling
given data set. Often, variation is quantified as variance; then, the more specific term explained variance can be used. The complementary part of the total
Explained_variation
Statistical property
has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided
Standard_error
Set of quantities in probability theory
μ and variance σ2, the cumulant generating function is K(t) = μt + σ2t2/2. The first and second derivatives of the cumulant generating function are K′(t)
Cumulant
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Statistical function that defines the quantiles of a probability distribution
the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution
Quantile_function
Statistical modeling method
response and explanatory variables with the best fit or least error, i.e. variance. After developing such a model, if additional values of the explanatory
Linear_regression
Estimate of an interval in which future observations will fall
the associated quantile function – for example, one could use the sample mean x‾ as an estimate for μ and the sample variance s2 as an estimate for σ2
Prediction_interval
Signal with equal energy per octave
"Asymptotic behaviour of the variance function". Scandinavian Journal of Statistics. 21: 223–243. Taylor LR (1961). "Aggregation, variance and the mean". Nature
Pink_noise
Method used in statistics, pattern recognition, and other fields
between groups and the function. Another popular measure of effect size is the percent of variance[clarification needed] for each function. This is calculated
Linear_discriminant_analysis
or equivalently as "systematic error variance shared among variables measured with and introduced as a function of the same method and/or source". For
Common-method_variance
Statistical property quantifying how much a collection of data is spread out
statistical dispersion are the variance, standard deviation, and interquartile range. For instance, when the variance of data in a set is large, the data
Statistical_dispersion
Measure of the shape of a function
statistics, the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized
Moment_(mathematics)
Mathematical function for the probability a given outcome occurs in an experiment
density function has a local peak. Quantile: the q-quantile is the value x {\displaystyle x} such that P ( X < x ) = q {\displaystyle P(X<x)=q} . Variance: the
Probability_distribution
Distribution function associated with the empirical measure of a sample
an empirical distribution function (a.k.a. an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical
Empirical distribution function
Empirical_distribution_function
Covariance and correlation
y)} Caution must be applied when using cross correlation function which assumes Gaussian variance for nonlinear systems. In certain circumstances, which
Cross-correlation
Generalization of the one-dimensional normal distribution to higher dimensions
X i {\displaystyle X_{i}} are independent and each is a zero-mean unit-variance normally distributed random variable, i.e. if X i ∼ N ( 0 , 1 ) {\displaystyle
Multivariate normal distribution
Multivariate_normal_distribution
Algorithm in computational quantum physics
energy-minimized wave functions on average yield more accurate values of other expectation values than variance minimized wave functions do. The optimization
Variational_Monte_Carlo
When variance is a random variable
\sigma } with a function ν t {\displaystyle \nu _{t}} that models the variance of S t {\displaystyle S_{t}} . This variance function is also modeled as
Stochastic_volatility
Range to estimate an unknown parameter
{\displaystyle \mu } and variance σ 2 . {\displaystyle \sigma ^{2}.} Define the sample mean X ¯ {\displaystyle {\bar {X}}} and unbiased sample variance S 2 {\displaystyle
Confidence_interval
Lower bound on variance of an estimator
distribution according to some probability density function f ( x ; θ ) {\displaystyle f(x;\theta )} . The variance of any unbiased estimator θ ^ {\displaystyle
Cramér–Rao_bound
Machine learning paradigm
function (classifier or regression function). If the true function is simple, then an "inflexible" learning algorithm with high bias and low variance
Supervised_learning
Variable representing a random phenomenon
quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution. However
Random_variable
Flaw in mathematical modelling
regression function can be split into random noise, approximation bias, and variance in the estimate of the regression function. The bias–variance tradeoff
Overfitting
Criterion for model selection
The BIC is an increasing function of the error variance σ e 2 {\displaystyle \sigma _{e}^{2}} and an increasing function of k. That is, unexplained
Bayesian information criterion
Bayesian_information_criterion
Artificial neural network node function
variances in variational autoencoders. The most common activation functions can be divided into three categories: ridge functions, radial functions and
Activation_function
Non-parametric method for testing whether samples originate from the same distribution
parametric equivalent of the Kruskal–Wallis test is the one-way analysis of variance (ANOVA). A significant Kruskal–Wallis test indicates that at least one
Kruskal–Wallis_test
Theorem in portfolio theory
then implications for the functioning of asset markets can be derived and tested. Portfolios can be analyzed in a mean-variance framework, with every investor
Mutual fund separation theorem
Mutual_fund_separation_theorem
Experimental design that is optimal with respect to some statistical criterion
which is related to the variance-matrix of the estimator. Specifying an appropriate model and specifying a suitable criterion function both require understanding
Optimal_experimental_design
Indicator function of positive numbers
The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function named after Oliver Heaviside
Heaviside_step_function
Inverse of the average of the inverses of a set of numbers
Assuming that the variance is not infinite and that the central limit theorem applies to the sample then using the delta method, the variance is Var ( H
Harmonic_mean
Measure of the asymmetry of random variables
distributions converge to a normal distribution with mean 0 and variance 6 (Fisher, 1930). The variance of the sample skewness is thus approximately 6 / n {\displaystyle
Skewness
Method for estimating variance of several different populations
In statistics, pooled variance (also known as combined variance, composite variance, or overall variance, and written σ 2 {\displaystyle \sigma ^{2}} )
Pooled_variance
Discrete probability distribution
positive real number λ is equal to the expected value of X and also to its variance. λ=E(X)=Var(X). The Poisson distribution can be applied to systems with
Poisson_distribution
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Statistical test comparing two probability distributions
standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is
Kolmogorov–Smirnov_test
Probability distribution
only exists for samples for which the sample variance is larger than the sample mean. The likelihood function for N iid observations (k1, ..., kN) is L (
Negative binomial distribution
Negative_binomial_distribution
Time series model
the variance of the current error term or innovation as a function of the actual sizes of the previous time periods' error terms; often the variance is
Autoregressive conditional heteroskedasticity
Autoregressive_conditional_heteroskedasticity
Probability distribution modeling a coin toss which need not be fair
1+\Pr(X{=}0)\cdot 0\\[1ex]&=p\cdot 1+q\cdot 0\\[1ex]&=p.\end{aligned}}} The variance of a Bernoulli distributed X {\displaystyle X} is Var [ X ] = p q = p
Bernoulli_distribution
Form of the utility function
economics and finance, exponential utility is a specific form of the utility function, used in some contexts because of its convenience when risk (sometimes
Exponential_utility
Average value of a random variable
Expected values can also be used to compute the variance, by means of the computational formula for the variance Var ( X ) = E [ X 2 ] − ( E [ X ] ) 2
Expected_value
Criterion applied in hierarchical cluster analysis
hierarchical cluster analysis. Ward's minimum variance method is a special case of the objective function approach originally presented by Joe H. Ward
Ward's_method
Uniform distribution on an interval
terms of mean μ {\displaystyle \mu } and variance σ 2 , {\displaystyle \sigma ^{2},} the probability density function of the continuous uniform distribution
Continuous uniform distribution
Continuous_uniform_distribution
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VARIANCE FUNCTION
VARIANCE FUNCTION
VARIANCE FUNCTION
VARIANCE FUNCTION
VARIANCE FUNCTION
VARIANCE FUNCTION
VARIANCE FUNCTION
VARIANCE FUNCTION
VARIANCE FUNCTION
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