NationMaster - Encyclopedia: Prediction interval

In statistics, a prediction interval bears the same relationship to a future observation that a confidence interval bears to an unobservable population parameter. Jump to: navigation, search Statistics is a type of data analysis which includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... In statistics, confidence intervals are the most prevalent form of interval estimation. ...

Example

WTF?

Suppose one has drawn a sample from a normally distributed population. The mean and standard deviation of the population are unknown except insofar as they can be estimated based on the sample. It is desired to predict the next observation. Let n be the sample size; let μ and σ be respectively the unobservable mean and standard deviation of the population. Let X₁, ..., X_n, be the sample; let X_n+1 be the future observation to be predicted. Let Jump to: navigation, search The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. ... Jump to: navigation, search In probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. ...

$overline{X}_n=(X_1+cdots+X_n)/n$

and

$S_n^2={1 over n-1}sum_{i=1}^n (X_i-overline{X}_n)^2.$

Then it is fairly routine to show that

${X_{n+1}-overline{X}_n over sqrt{S_n^2+S_n^2/n}} = {X_{n+1}-overline{X}_n over S_nsqrt{1+1/n}}$

has a Student's t-distribution with n − 1 degrees of freedom. Consequently we have Jump to: navigation, search In probability and statistics, the t-distribution or Students distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ...

$Pleft(overline{X}_n-A S_nsqrt{1+(1/n)}leq X_{n+1} leqoverline{X}_n+A S_nsqrt{1+(1/n)},right)=p$

where A is the 100(1 − (p/2))^th percentile of Student's t-distribution with n − 1 degrees of freedom. Therefore the numbers Jump to: navigation, search In probability and statistics, the t-distribution or Students distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ...

$overline{X}_npm A {S}_nsqrt{1+(1/n)}$

are the endpoints of a 100p% prediction interval for X_n+1.

4.1.3.2. Prediction (664 words)
	As mentioned earlier, the goal of prediction is to determine future value(s) of the response variable that are associated with a specific combination of predictor variable values.
	As in estimation, the predicted values are computed by plugging the value(s) of the predictor variable(s) into the regression equation, after estimating the unknown parameters from the data.
	Because the prediction interval is an interval for the value of a single new measurement from the process, the uncertainty includes the noise that is inherent in the estimates of the regression parameters and the uncertainty of the new measurement.

Linear prediction - Wikipedia, the free encyclopedia (413 words)
	Linear prediction is a mathematical operation where future values of a digital signal are estimated as a linear function of previous samples.
	In digital signal processing linear prediction is often called linear predictive coding (LPC) and can thus be viewed as a subset of filter theory.
	In system analysis (a subfield of mathematics), linear prediction can be viewed as a part of mathematical modelling or optimization.

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Encyclopedia > Prediction interval

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