Prediction interval

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Standard score and prediction interval

Cumulative distribution function for normal distribution, mean 0 and sd 1

Prediction interval is a term used in statistics to describe a range within which a future observation or data point is expected to fall with a certain probability. Prediction intervals are used in both theoretical statistics and various applied fields such as econometrics, engineering, and environmental science to quantify the uncertainty of a single future observation given a model and past data. Unlike confidence intervals, which estimate the uncertainty of a population parameter, prediction intervals focus on the uncertainty of individual future observations.

Overview

A prediction interval provides an upper and lower bound on the expected value of a new observation based on a fitted statistical model. The interval is calculated to ensure that it covers the new observation with a specified probability, often referred to as the confidence level. For example, a 95% prediction interval aims to cover the future observation with a 95% probability, given the model and assumptions used.

Calculation

The calculation of a prediction interval generally involves the estimated mean or expected value of the future observation, the standard error of the prediction, and a critical value from a statistical distribution that corresponds to the desired confidence level. The specific formula and method for calculating a prediction interval can vary depending on the statistical model and assumptions made about the underlying data distribution.

For a simple linear regression model, the prediction interval for a new observation \(x_0\) can be expressed as:

\[ PI = \hat{y}_0 \pm t_{\alpha/2, n-2} \cdot S \cdot \sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{\sum(x_i - \bar{x})^2}} \]

where: - \( \hat{y}_0 \) is the predicted value of the new observation, - \( t_{\alpha/2, n-2} \) is the critical value from the t-distribution with \( n-2 \) degrees of freedom, - \( S \) is the standard error of the regression, - \( n \) is the number of observations in the data, - \( x_0 \) is the value of the new observation, - \( \bar{x} \) is the mean of the observed values, and - \( x_i \) are the individual observed values.

Applications

Prediction intervals are widely used in various fields to assess the uncertainty of predictions made by statistical models. In econometrics, they are used to forecast economic indicators such as GDP growth or inflation rates. In engineering, prediction intervals help in the design and testing of systems under uncertainty. Environmental scientists use prediction intervals to forecast weather conditions, pollution levels, or the effects of climate change.

Limitations

The accuracy and reliability of a prediction interval depend on the correctness of the statistical model and the assumptions about the data's distribution. If the model is poorly fitted or the data do not meet the assumptions (e.g., normality, independence), the prediction interval may not accurately reflect the uncertainty of future observations. Additionally, prediction intervals become wider as the prediction horizon extends, reflecting increased uncertainty about more distant future observations.

See Also

• Confidence interval
• Standard error
• Linear regression
• Statistical model