Residual Measure

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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

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.
[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 σ2, has a chi-squared distribution with n degrees of freedom:
[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.

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 wi = K(d(xi, xq)).