Errors and residuals in the context of PID algorithm

In econometrics, endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term.

In simplest terms, endogeneity means that a factor or cause one uses to explain something as an outcome is also being influenced by that same thing. For example, education can affect income, but income can also affect how much education someone gets. When this happens, one's analysis might wrongly estimate cause and effect. The thing one thinks is causing change is also being influenced by the outcome, making the results unreliable.

In mathematics and statistics, deviation serves as a measure to quantify the disparity between an observed value of a variable and another designated value, frequently the mean of that variable. Deviations with respect to the sample mean and the population mean (or "true value") are called errors and residuals, respectively. The sign of the deviation reports the direction of that difference: the deviation is positive when the observed value exceeds the reference value. The absolute value of the deviation indicates the size or magnitude of the difference. In a given sample, there are as many deviations as sample points. Summary statistics can be derived from a set of deviations, such as the standard deviation and the mean absolute deviation, measures of dispersion, and the mean signed deviation, a measure of bias.

The deviation of each data point is calculated by subtracting the mean of the data set from the individual data point. Mathematically, the deviation d of a data point x in a data set with respect to the mean m is given by the difference:

In statistics, simple linear regression (SLR) is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and 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 variable.The adjective simple refers to the fact that the outcome variable is related to a single predictor.

It is common to make the additional stipulation that the ordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squared residual (vertical distance between the point of the data set and the fitted line), and the goal is to make the sum of these squared deviations as small as possible. In this case, the slope of the fitted line is equal to the correlation between y and x corrected by the ratio of standard deviations of these variables. The intercept of the fitted line is such that the line passes through the center of mass (x, y) of the data points.

In an economic model, an exogenous variable is one whose measure is determined outside the model and is imposed on the model, and an exogenous change is a change in an exogenous variable. In contrast, an endogenous variable is a variable whose measure is determined by the model. An endogenous change is a change in an endogenous variable in response to an exogenous change that is imposed upon the model.

The term 'endogeneity' in econometrics has a related but distinct meaning. An endogenous random variable is correlated with the error term in the econometric model, while an exogenous variable is not.

A proportional–integral–derivative controller (PID controller or three-term controller) is a feedback-based control loop mechanism commonly used to manage machines and processes that require continuous control and automatic adjustment. It is typically used in industrial control systems and various other applications where constant control through modulation is necessary without human intervention. The PID controller automatically compares the desired target value (setpoint or SP) with the actual value of the system (process variable or PV). The difference between these two values is called the error value, denoted as ${\displaystyle e(t)}$.

It then applies corrective actions automatically to bring the PV to the same value as the SP using three methods: The proportional (P) component responds to the current error value by producing an output that is directly proportional to the magnitude of the error. This provides immediate correction based on how far the system is from the desired setpoint. The integral (I) component, in turn, considers the cumulative sum of past errors to address any residual steady-state errors that persist over time, eliminating lingering discrepancies. Lastly, the derivative (D) component predicts future error by assessing the rate of change of the error, which helps to mitigate overshoot and enhance system stability, particularly when the system undergoes rapid changes. The PID output signal can directly control actuators through voltage, current, or other modulation methods, depending on the application. The PID controller reduces the likelihood of human error and improves automation.

In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Since the sample does not include all members of the population, statistics of the sample (often known as estimators), such as means and quartiles, generally differ from the statistics of the entire population (known as parameters). The difference between the sample statistic and population parameter is considered the sampling error. For example, if one measures the height of a thousand individuals from a population of one million, the average height of the thousand is typically not the same as the average height of all one million people in the country.

Since sampling is almost always done to estimate population parameters that are unknown, by definition exact measurement of the sampling errors will usually not be possible; however they can often be estimated, either by general methods such as bootstrapping, or by specific methods incorporating some assumptions (or guesses) regarding the true population distribution and parameters thereof.