a minimum-variance Kalman filter and a minimum-variance smoother may be used to recover data of interest from noisy measurements. These techniques rely on one-step-ahead predictors (which minimise the variance of the prediction error). When the generating models are nonlinear then stepwise linearizations may be applied within Extended Kalman Filter and smoother recursions. However, in nonlinear cases, optimum minimum-variance performance guarantees no longer apply. To use regression analysis for prediction, data are collected on the variable that is to be predicted, called the dependent variable or response variable, and on one or more variables whose values are hypothesized to influence it, called independent variables or explanatory variables. 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. Science In science, a prediction is a rigorous, often quantitative, statement, forecasting what would be observed under specific conditions; for example, according to theories of gravity, if an apple fell from a tree it would be seen to move towards the center of the earth 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 whose predictions are 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 failure mechanism causing the failure. Accurate