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Mathematical method
least squares function approximation applies the principle of least squares to function approximation, by means of a weighted sum of other functions.
Least-squares function approximation
Least-squares_function_approximation
Approximating an arbitrary function with a well-behaved one
classification problem instead. Approximation theory Fitness approximation Kriging Least squares (function approximation) Radial basis function network Lakemeyer,
Function_approximation
Approximation method in statistics
In regression analysis, least squares is a method to determine the best-fit model by minimizing the sum of the squared residuals—the differences between
Least_squares
Least squares approximation of linear functions to data
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems
Linear_least_squares
Problem in recreational mathematics
mathematics. A 1975 attempt through least-squares function approximation required dozens of terms. An approximation using four parameters was found by
Von_Neumann's_elephant
Method for reconstructing continuous functions
Moving least squares is a method of reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares
Moving_least_squares
Approximation method in statistics
Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters
Non-linear_least_squares
Periodicity computation method
The relationship between the DFT and the approximation of trigonometric functions using the least-squares method is well explained in (Strutz, 2017)
Least-squares spectral analysis
Least-squares_spectral_analysis
Something roughly the same as something else
models assumption of facts Least squares – Approximation method in statistics Linear approximation – Approximation of a function by its tangent line at a
Approximation
Method for estimating the unknown parameters in a linear regression model
parameters in a linear regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable
Ordinary_least_squares
Technique in numerical linear algebra
Low-rank approximation is closely related to numerous other techniques, including principal component analysis, factor analysis, total least squares, latent
Low-rank_approximation
Type of mathematical function
curve approximation" (PDF). Computer Aided Geometric Design. 11 (3): 289. doi:10.1016/0167-8396(94)90004-3. Golovchenko, Nikolai. "Least-squares Fit of
Piecewise_linear_function
Optimization algorithm
objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient
Stochastic_gradient_descent
Measure of the error of an estimator
of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding
Mean_squared_error
Optimization algorithm
functions via an iterative recurrence formula much like the one for Newton's method, except using approximations of the derivatives of the functions in
Quasi-Newton_method
Algorithms for calculating square roots
may be used as the approximation, but a least-squares regression line intersecting the arc will be more accurate. A least-squares regression line minimizes
Square_root_algorithms
Statistical algorithm
Least mean squares (LMS) algorithms are a class of adaptive filter used to mimic a desired filter by finding the filter coefficients that relate to producing
Least_mean_squares_filter
Probability of shared birthdays
This is a result of the good approximation that an event with 1/k probability will have a 1/2 chance of occurring at least once if it is repeated k ln
Birthday_problem
Statistical technique
models. The total least squares approximation of the data is generically equivalent to the best, in the Frobenius norm, low-rank approximation of the data matrix
Total_least_squares
Approximations used in machine learning
Support vector machine Radial basis function kernel Regularized least squares Andreas Müller (2012). Kernel Approximations for Efficient SVMs (and other feature
Low-rank matrix approximations
Low-rank_matrix_approximations
Number whose square is a given number
at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier). A method for finding very good approximations to the square roots
Square_root
Sigmoid shape special function
this approximation is about 2×10−9. The parameters are obtained by fitting the extended approximation to the accurate values of the error function using
Error_function
Extension of the factorial function
gamma function Lemniscate constant Pseudogamma function Hadamard's gamma function Inverse gamma function Lanczos approximation Multiple gamma function Multivariate
Gamma_function
Assumption that motions of nuclei and electrons can be separated
and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the wave functions of atomic nuclei and electrons in a molecule can
Born–Oppenheimer approximation
Born–Oppenheimer_approximation
Probability distribution and special case of gamma distribution
distribution of a sum of the squares of k {\displaystyle k} independent standard normal random variables. The chi-squared distribution χ k 2 {\displaystyle
Chi-squared_distribution
Algorithm used to solve non-linear least squares problems
damped least-squares (DLS) method, is used to solve non-linear least squares problems. These minimization problems arise especially in least squares curve
Levenberg–Marquardt_algorithm
Linear map or polynomial function of degree one
Discontinuous linear map Linear least squares "The term linear function means a linear form in some textbooks and an affine function in others." Vaserstein 2006
Linear_function
Statistics concept
estimation, since the regression function is linear in terms of the unknown parameters β0, β1, .... Therefore, for least squares analysis, the computational
Polynomial_regression
Positive real number which when multiplied by itself gives 5
{\displaystyle x_{0}} , and at each step finds a new approximation by averaging the previous approximation and d {\displaystyle d} times its reciprocal
Square_root_of_5
Process of constructing a curve that has the best fit to a series of data points
compaction Discretization Estimation theory Function approximation Genetic programming Goodness of fit Least-squares adjustment Levenberg–Marquardt algorithm
Curve_fitting
moving least squares for the particular case of a global approximation (using all available data points). Using this function approximation method, partial
Diffuse_element_method
Method of data analysis
its singular value decomposition. Then the best rank‑k approximation to P in the least‑squares (Frobenius‑norm) sense is P k = U k Σ k V k T {\displaystyle
Principal_component_analysis
Type of non-sinusoidal waveform
effects similar to those of the σ-approximation. For a reasonable approximation to the square-wave shape, at least the fundamental and third harmonic
Square_wave_(waveform)
Class of statistical models
regression and Poisson regression. They proposed an iteratively reweighted least squares method for maximum likelihood estimation (MLE) of the model parameters
Generalized_linear_model
Numerical approximation algorithm
successive approximation include: Babylonian method, for finding square roots of numbers Fixed-point iteration Means of finding zeros of functions: Halley's
Iterative_method
Mathematical theorem in the study of analysis
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly
Stone–Weierstrass_theorem
can be plugged into a traditional function approximation framework. One such method is least-squares approximation. Let Φ G = { V 1 G , … , V k G } {\displaystyle
Proto-value_function
Technique to solve partial differential equations
of admissible solutions, increasing the generalizability of the function approximation. This way, embedding this prior information into a neural network
Physics-informed neural networks
Physics-informed_neural_networks
Function related to statistics and probability theory
likelihood function in order to proof asymptotic normality of the posterior probability, and therefore to justify a Laplace approximation of the posterior
Likelihood_function
Mathematical relation assigning a probability event to a cost
quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because
Loss_function
Technique in statistics
correlated with the error term (endogenous), in which case ordinary least squares and ANOVA give biased results. When used, a valid instrument changes
Instrumental_variables
Probability distribution
Error function#Approximation with elementary functions. In particular, small relative error on the whole domain for the cumulative distribution function
Normal_distribution
Regression analysis
}}^{-1}(\mathbf {d} -\mathbf {Y{\bar {m}})} } (see also linear least squares). The linear approximation introduces bias into the statistics. Therefore, more caution
Nonlinear_regression
Family of iterative methods
values of functions which cannot be computed directly, but only estimated via noisy observations. In a nutshell, stochastic approximation algorithms
Stochastic_approximation
measures smoothness of a function Least squares (function approximation) — minimizes the error in the L2-norm Minimax approximation algorithm — minimizes
List of numerical analysis topics
List_of_numerical_analysis_topics
relaxation are used in solving problems in differential equations, linear least-squares, and linear programming. However, iterative methods of relaxation have
Relaxation_(approximation)
Function in mathematical number theory
as Carmichael's λ function, the reduced totient function, and the least universal exponent function. The order of the multiplicative group of integers
Carmichael_function
Type of artificial neural network
radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series
Radial_basis_function_network
Method for estimating new data within known data points
leads to least squares approximation. Approximation theory studies how to find the best approximation to a given function by another function from some
Interpolation
Type of mathematical function
methods of linear least squares, because the approximating function is linear in the weights w i {\textstyle w_{i}} . Approximation schemes of this kind
Radial_basis_function
Statistical optimality criterion
analogous to the least squares technique, except that it is based on absolute values instead of squared values. It attempts to find a function which closely
Least_absolute_deviations
Set of statistical processes for estimating the relationships among variables
Forecasting Fraction of variance unexplained Function approximation Generalized linear model Kriging (a linear least squares estimation algorithm) Local regression
Regression_analysis
Unique positive real number which when multiplied by itself gives 2
square with side length a {\displaystyle a} will have an area equal to two squares of (lesser) side length b {\displaystyle b} . Call these squares A
Square_root_of_2
Statistical modeling technique
analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values
Quantile_regression
Computational geometry and optimization concept
settings, coresets often yield polynomial-time approximation schemes. In regression problems such as least-squares fitting, coresets provide smaller weighted
Coreset
Varying methods used to calculate pi
Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning
Approximations_of_pi
Computational quantum mechanical modelling method to investigate electronic structure
thermodynamic potential using known correlation functions of the uniform system. In the square gradient approximation a strong non-uniform density contributes
Density_functional_theory
Complex complementary error function
only the original values of the Faddeeva function, but also its derivative (e.g. in Non-linear least squares regression in spectroscopy). Its derivative
Faddeeva_function
Multi-dimensional version of a confidence interval
\mathbf {P} =\mathbf {V} } In effect, P is a square root of the covariance matrix V. The least-squares problem Y = X β + ε {\displaystyle \mathbf {Y}
Confidence_region
Matrix of partial derivatives of a vector-valued function
best linear approximation of the change of f along h in a neighborhood of x, if f(x) is differentiable at x. This means that the function that maps y
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
A fully polynomial-time approximation scheme (FPTAS) is an algorithm for finding approximate solutions to function problems, especially optimization problems
Fully polynomial-time approximation scheme
Fully_polynomial-time_approximation_scheme
Smooth function in statistics
parameter estimates. As in regular least squares, the goal is to estimate the unknown parameters in the regression function by finding values for parameter
Variance_function
Number, approximately 3.14
the accuracy of approximations. When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established
Pi
Mathematical problem
A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Failure of convergence in interpolation
phenomenon in Fourier series approximations. The Weierstrass approximation theorem states that for every continuous function f ( x ) {\displaystyle f(x)}
Runge's_phenomenon
Method in machine learning
{\mathcal {H}}} . That is, minimize the expected risk for a Least-squares loss function. Since E {\displaystyle {\mathcal {E}}} depends on the unknown
Early_stopping
Class of algorithms that find approximate solutions to optimization problems
program coming from the first level of the sum of squares hierarchy. A simple example of an approximation algorithm is one for the minimum vertex cover problem
Approximation_algorithm
Evaluates how likely it is that any difference between data sets arose by chance
chi-squared test is an approximation Lexis ratio, earlier statistic, replaced by chi-squared Mann–Whitney U test Median test Minimum chi-square estimation
Pearson's_chi-squared_test
Method of machine learning
regularization). The choice of loss function here gives rise to several well-known learning algorithms such as regularized least squares and support vector machines
Online_machine_learning
Fundamental theorem in probability theory and statistics
between the function and its approximation grows approximately as a2φ2(n). The idea is that dividing the function by appropriate normalizing functions, and looking
Central_limit_theorem
Regularization technique for ill-posed problems
variance and mean square estimator are often smaller than the least square estimators previously derived. In the ordinary least squares solution of Y =
Ridge_regression
Product of numbers from 1 to n
factorial function was developed beginning in the late 18th and early 19th centuries. Stirling's approximation provides an accurate approximation to the
Factorial
Type of artificial neural network
weight layer with linear activation functions. It was trained by the least squares method for minimising mean squared error, also known as linear regression
Feedforward_neural_network
Engineering model
surrogate models that are not available elsewhere: kriging by partial-least squares reduction and energy-minimizing spline interpolation. Python library
Surrogate_model
Statistics concept
mean square error (RMSE) is the square root of MSE. The sum of squares of errors (SSE) is the MSE multiplied by the sample size. Sum of squares of residuals
Errors_and_residuals
Probabilistic problem-solving algorithm
the approximation will be poor. The approximation improves as more points are randomly placed in the whole square. Uses of Monte Carlo methods require
Monte_Carlo_method
Number of values in the final calculation of a statistic that are free to vary
an ordinary least-squares fit (i.e. is not an orthogonal projection), these sums-of-squares no longer have (scaled, non-central) chi-squared distributions
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
Problem in combinatorial optimization
optimal for 2D knapsack (packing squares into a two-dimensional unit size square): when there are at most five squares in an optimal packing. Here, there
Knapsack_problem
Measure of variation in statistics
to obtain confidence intervals on the variance of residuals from a least squares fit under standard normal theory, where k is now the number of degrees
Standard_deviation
Technique to make a model more generalizable and transferable
advanced by gradient descent. The learning problem with the least squares loss function and Tikhonov regularization can be solved analytically. Written
Regularization_(mathematics)
Moving average and polynomial regression method for smoothing data
LOWESS thus build on "classical" methods, such as linear and nonlinear least squares regression. They address situations in which the classical procedures
Local_regression
Signal processing technique
Least-squares spectral analysis, based on least squares fitting to known frequencies Lomb–Scargle periodogram, an approximation of the Least-squares spectral
Spectral_density_estimation
Mathematics of real numbers and real functions
local rate of change of a function. In one variable, the derivative gives the slope of the best linear approximation to a function near a point. This point
Real_analysis
Methods for numerical approximations
measurement of the value of some function at these points (with an error), the unknown function can be found. The least squares-method is one way to achieve
Numerical_analysis
Branch of mathematics
f(a)) and (a + h, f(a + h)). The secant line is only an approximation to the behavior of the function at the point a because it does not account for what
Calculus
Simple rational approximation (SRA) is a subset of interpolating methods using rational functions. Especially, SRA interpolates a given function with a specific
Simple_rational_approximation
Unsolved problem about inscribing a square in a Jordan curve
the approximation are topologically separated from smaller inscribed squares that do not contain the center. The limit of a sequence of large squares must
Inscribed_square_problem
simple closed forms, and can be parameterized with data using linear least squares. The Marchenko–Pastur distribution is important in the theory of random
List of probability distributions
List_of_probability_distributions
Economic formula of productivity
Cobb–Douglas function. In some cases this simultaneous equation bias doesn't appear. However, it is apparent when least squares asymptotic approximations are used
Cobb–Douglas production function
Cobb–Douglas_production_function
Probability distribution
for N much larger than n, the binomial distribution remains a good approximation, and is widely used. If the random variable X follows the binomial distribution
Binomial_distribution
Empirical law on the variance of species in a habitat
{\displaystyle \log(s^{2})} and log m {\displaystyle \log m} . Ordinary least squares regression assumes that φ = ∞. This tends to underestimate the value
Taylor's_law
Statistical modeling method
version of the least squares cost function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Use of the Mean Squared Error (MSE) as
Linear_regression
Probability of survival beyond any specified time
non-parametric maximum likelihood and least squares estimates of survival functions, without lifetime data. Every survival function S ( t ) {\displaystyle S(t)}
Survival_function
Algorithm for finding zeros of functions
successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a real-valued function f, its derivative
Newton's_method
Family of statistical methods based on sampling of available data
populations), sample coefficient of variation, maximum likelihood estimators, least squares estimators, correlation coefficients and regression coefficients. It
Resampling_(statistics)
use of the discrete least squares method to discretize the governing differential equation. A Moving least squares (MLS) approximation method is used to
Discrete least squares meshless method
Discrete_least_squares_meshless_method
External links 1.96 2SLS (two-stage least squares) – redirects to instrumental variable 3SLS – see three-stage least squares 68–95–99.7 rule 100-year flood
List_of_statistics_articles
Study of mathematical algorithms for optimization problems
takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the Hessian matrix), the number of function evaluations
Mathematical_optimization
Statistical hypothesis test
continuous chi-squared distribution. This assumption is not quite correct and introduces some error. To reduce the error in approximation, Frank Yates suggested
Chi-squared_test
Method of estimating the parameters of a statistical model, given observations
first-order conditions of the likelihood function can be solved analytically; for instance, the ordinary least squares estimator for a linear regression model
Maximum_likelihood_estimation
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