# Residual Measure

(Redirected from Residual)

A Residual Measure is a deviation measure of a numerical approximation from the exact solution (observed value or theoretical value).

**AKA:**Regression Residual, Fitting Deviation, Numerical Residual.**Context:**- It can be defined as [math]r=b-f(x_0)[/math] for an numerical approximation [math]x_0[/math] of [math]x\lt math\gt where f(x)=b [/math].
- It [math]r(x)=\max_{x\in \mathcal X} |g(x)-T(f_{\rm A})(x)| [/math] where approximation [math]f_{A}~[/math] of the solution of [math]~f~[/math] of the equation [math] T(f)(x)=g(x) [/math].
- It can be defined as [math]r_i=X_i-\overline{X}[/math] where [math]X[/math] is a sample of random variables [math]X_i=\{X_1, \dots, X_n\}[/math] and [math]\overline{X}.[/math] is the sample mean.

**See:** Regression Algorithm, Least Squares Estimation Algorithm, Kernel Function, Studentized Residual, Numerical Approximation.

## References

### 2016

- (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Residual_(numerical_analysis) Retrieved 2016-08-07
- Loosely speaking, a
**residual**is the error in a result. To be precise, suppose we want to find*x*such that [math]f(x)=b.\,[/math]. Given an approximation*x*_{0}of*x*, the residual is [math]b - f(x_0)\,[/math] whereas the error is [math]x - x_0\,[/math]

- Loosely speaking, a

- If we do not know
*x*exactly, we cannot compute the error but we can compute the residual. (...) Similar terminology is used dealing with differential, integral and functional equations. For the approximation [math]~f_{\rm a}~[/math] of the solution [math]~f~[/math] of the equation [math] T(f)(x)=g(x) [/math], the residual can either be the function [math]~g(x)~ - ~T(f_{\rm a})(x)[/math] or can be said to be the maximum of the norm of this difference [math]\max_{x\in \mathcal X} |g(x)-T(f_{\rm a})(x)| [/math] over the domain [math]\mathcal X[/math], where the function [math]~f_{\rm a}~[/math] is expected to approximate the solution [math]~f~[/math], or some integral of a function of the difference, for example [math]~\int_{\mathcal X} |g(x)-T(f_{\rm a})(x)|^2~{\rm d} x.[/math] In many cases, the smallness of the residual means that the approximation is close to the solution, i.e., [math]~\left|\frac{f_{\rm a}(x) - f(x)}{f(x)}\right| \ll 1.~[/math] In these cases, the initial equation is considered as well-posed; and the residual can be considered as a measure of deviation of the approximation from the exact solution.

- If we do not know

- (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Errors_and_residuals_in_statistics Retrieved 2016-08-07
- In statistics and optimization,
**errors**and**residuals**are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "theoretical value". The**error**(or**disturbance**) of an observed value is the deviation of the observed value from the (unobservable)*true*value of a quantity of interest (for example, a population mean), and the**residual**of an observed value is the difference between the observed value and the*estimated*value of the quantity of interest (for example, a sample mean). The distinction is most important in regression analysis, where the concepts are sometimes called the**regression errors**and**regression residuals**and where they lead to the concept of studentized residuals (...) If we assume a normally distributed population with mean [math]\mu[/math] and standard deviation [math]\sigma[/math], and choose individuals independently, then we have [math]X_1, \dots, X_n\sim N(\mu,\sigma^2)\,[/math] and the sample mean

- In statistics and optimization,

- [math]\overline{X}={X_1 + \cdots + X_n \over n}[/math]
- is a random variable distributed thus:

- [math]\overline{X}\sim N(\mu, \sigma^2/n).[/math]
- The
*statistical errors*are then [math]e_i=X_i-\mu,\,[/math] whereas the*residuals*are [math]r_i=X_i-\overline{X}.[/math] - The sum of squares of the
**statistical errors**, divided by*σ*^{2}, has a chi-squared distribution with*n*degrees of freedom:

- The
- [math] \frac 1 {\sigma^2}\sum_{i=1}^n e_i^2\sim\chi^2_n.[/math]
- This quantity, however, is not observable. The sum of squares of the
**residuals**, on the other hand, is observable (...) In regression analysis, the distinction between*errors*and*residuals*is subtle and important, and leads to the concept of studentized residuals. Given an unobservable function that relates the independent variable to the dependent variable – say, a line – the deviations of the dependent variable observations from this function are the unobservable errors. If one runs a regression on some data, then the deviations of the dependent variable observations from the*fitted*function are the residuals.

- This quantity, however, is not observable. The sum of squares of the

- (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Residual Retrieved 2016-08-07
- A
**residual**is generally a quantity left over at the end of a process. It may refer to ...

- A

### 1997

- (Mitchell, 1997) ⇒ Tom M. Mitchell. (1997). “Machine Learning." McGraw-Hill.
- Much of the literature on nearest-neighbor methods and weighted local regression uses a terminology that has arisen from the field of statistical pattern recognition....
*Regression*means approximating a real-valued target function.*Residual*is the error*f*(^{^}*x*) - [math]f[/math](*x*) in approximating the target function.*Kernel function*is the function of distance that is used to determine the wight of each training example. In other words, the kernel function is the function [math]K[/math] such that w_{i}=*K*(*d*(*x*,_{i}*x*))._{q}

- Much of the literature on nearest-neighbor methods and weighted local regression uses a terminology that has arisen from the field of statistical pattern recognition....