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Prediction intervals are often used in regression analysis. Prediction intervals are used in both frequentist statistics and Bayesian statistics: Alternatively, in Bayesian terms , a prediction interval can be described as a credible interval for the variable itself, rather than for a parameter of the distribution thereof.

The concept of prediction intervals need not be restricted to inference about a single future sample value but can be extended to more complicated cases.

For example, in the context of river flooding where analyses are often based on annual values of the largest flow within the year, there may be interest in making inferences about the largest flood likely to be experienced within the next 50 years.

Since prediction intervals are only concerned with past and future observations, rather than unobservable population parameters, they are advocated as a better method than confidence intervals by some statisticians, such as Seymour Geisser , [ citation needed ] following the focus on observables by Bruno de Finetti.

Given a sample from a normal distribution , whose parameters are unknown, it is possible to give prediction intervals in the frequentist sense, i.

A general technique of frequentist prediction intervals is to find and compute a pivotal quantity of the observables X 1 , The most familiar pivotal quantity is the Student's t-statistic , which can be derived by this method and is used in the sequel.

This approach is usable, but the resulting interval will not have the repeated sampling interpretation [4] — it is not a predictive confidence interval.

This is necessary for the desired confidence interval property to hold. This simple combination is possible because the sample mean and sample variance of the normal distribution are independent statistics; this is only true for the normal distribution, and in fact characterizes the normal distribution.

One can compute prediction intervals without any assumptions on the population; formally, this is a non-parametric method. Suppose one randomly draws a sample of two observations X 1 and X 2 from a population in which values are assumed to have a continuous probability distribution.

Note that the assumption of a continuous distribution avoids the possibility that values might be exactly equal; this would complicate matters.

Notice that while this gives the probability that a future observation will fall in a range, it does not give any estimate as to where in a segment it will fall — notably, if it falls outside the range of observed values, it may be far outside the range.

See extreme value theory for further discussion. Formally, this applies not just to sampling from a population, but to any exchangeable sequence of random variables, not necessarily independent or identically distributed.

In parameter confidence intervals, one estimates population parameters; if one wishes to interpret this as prediction of the next sample, one models "the next sample" as a draw from this estimated population, using the estimated population distribution.

Prediction intervals are commonly used as definitions of reference ranges , such as reference ranges for blood tests to give an idea of whether a blood test is normal or not.

A common application of prediction intervals is to regression analysis. In regression, Faraway , p. Seymour Geisser , a proponent of predictive inference, gives predictive applications of Bayesian statistics.

In Bayesian statistics, one can compute Bayesian prediction intervals from the posterior probability of the random variable, as a credible interval.

In theoretical work, credible intervals are not often calculated for the prediction of future events, but for inference of parameters — i.

However, particularly where applications are concerned with possible extreme values of yet to be observed cases, credible intervals for such values can be of practical importance.

From Wikipedia, the free encyclopedia. A functional form , often linear, is hypothesized for the postulated causal relationship, and the parameters of the function are estimated from the data—that is, are chosen so as to optimize is some way the fit of the function, thus parameterized, to the data.

That is the estimation step. For the prediction step, explanatory variable values that are deemed relevant to future or current but not yet observed values of the dependent variable are input to the parameterized function to generate predictions for the dependent variable.

In science, a prediction is a rigorous, often quantitative, statement, forecasting what would happen under specific conditions; for example, if an apple fell from a tree it would be attracted towards the center of the earth by gravity with a specified and constant acceleration.

The scientific method is built on testing statements that are logical consequences of scientific theories. This is done through repeatable experiments or observational studies.

A scientific theory which is contradicted by observations and evidence will be rejected. New theories that generate many new predictions can more easily be supported or falsified see predictive power.

Notions that make no testable predictions are usually considered not to be part of science protoscience or nescience until testable predictions can be made.

Mathematical equations and models , and computer models , are frequently used to describe the past and future behaviour of a process within the boundaries of that model.

In some cases the probability of an outcome, rather than a specific outcome, can be predicted, for example in much of quantum physics. In microprocessors , branch prediction permits avoidance of pipeline emptying at branch instructions.

In engineering , possible failure modes are predicted and avoided by correcting the mechanism causing the failure. Accurate prediction and forecasting are very difficult in some areas, such as natural disasters , pandemics , demography , population dynamics and meteorology.

For example, it is possible to predict the occurrence of solar cycles , but their exact timing and magnitude is much more difficult see picture to right.

Established science makes useful predictions which are often extremely reliable and accurate; for example, eclipses are routinely predicted. New theories make predictions which allow them to be disproved by reality.

For example, predicting the structure of crystals at the atomic level is a current research challenge. The existence of this absolute frame was deemed necessary for consistency with the established idea that the speed of light is constant.

The famous Michelson-Morley experiment demonstrated that predictions deduced from this concept were not borne out in reality, thus disproving the theory of an absolute frame of reference.

The special theory of relativity was proposed by Einstein as an explanation for the seeming inconsistency between the constancy of the speed of light and the non-existence of a special, preferred or absolute frame of reference.

Albert Einstein 's theory of general relativity could not easily be tested as it did not produce any effects observable on a terrestrial scale.

However, the theory predicted that large masses such as stars would bend light, in contradiction to accepted theory; this was observed in a eclipse.

Mathematical models of stock market behaviour and economic behaviour in general are also unreliable in predicting future behaviour. Among other reasons, this is because economic events may span several years, and the world is changing over a similar time frame, thus invalidating the relevance of past observations to the present.

Thus there are an extremely small number of the order of 1 of relevant past data points from which to project the future.

In addition, it is generally believed that stock market prices already take into account all the information available to predict the future, and subsequent movements must therefore be the result of unforeseen events.

Consequently, it is extremely difficult for a stock investor to anticipate or predict a stock market boom , or a stock market crash.

In contrast to predicting the actual stock return, forecasting of broad economic trends tends to have better accuracy.

Such analysis is provided by both non-profit groups as well as by for-profit private institutions, including brokerage housesand consulting companies.

Some correlation has been seen between actual stock market movements and prediction data from large groups in surveys and prediction games.

An actuary uses actuarial science to assess and predict future business risk , such that the risk s can be mitigated. For example, in insurance an actuary would use a life table which incorporates the historical experience of mortality rates and sometimes an estimate of future trends to project life expectancy.

Predicting the outcome of sporting events is a business which has grown in popularity in recent years. Handicappers predict the outcome of games using a variety of mathematical formulas, simulation models or qualitative analysis.

Early, well known sports bettors, such as Jimmy the Greek , were believed to have access to information that gave them an edge. Information ranged from personal issues, such as gambling or drinking to undisclosed injuries; anything that may affect the performance of a player on the field.

Recent times have changed the way sports are predicted. Predictions now typically consist of two distinct approaches: Situational plays and statistical based models.

Situational plays are much more difficult to measure because they usually involve the motivation of a team. As situational plays become more widely known they become less useful because they will impact the way the line is set.

The widespread use of technology has brought with it more modern sports betting systems. These systems are typically algorithms and simulation models based on regression analysis.

Jeff Sagarin , a sports statistician, has brought attention to sports by having the results of his models published in USA Today.

He is currently paid as a consultant by the Dallas Mavericks for his advice on lineups and the use of his Winval system, which evaluates free agents.

Brian Burke , a former Navy fighter pilot turned sports statistician, has published his results of using regression analysis to predict the outcome of NFL games.

His website includes his College Basketball Ratings, a tempo based statistics system. Some statisticians have become very famous for having successful prediction systems.

In politics it is common to attempt to predict the outcome of elections via political forecasting techniques or assess the popularity of politicians through the use of opinion polls.

Prediction games have been used by many corporations and governments to learn about the most likely outcome of future events.

Predictions have often been made, from antiquity until the present, by using paranormal or supernatural means such as prophecy or by observing omens.

Methods including water divining , astrology , numerology , fortune telling , interpretation of dreams , and many other forms of divination , have been used for millennia to attempt to predict the future.

These means of prediction have not been proven by scientific experiments. In literature, vision and prophecy are literary devices used to present a possible timeline of future events.

They can be distinguished by vision referring to what an individual sees happen.

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