For a derivation of this estimate see Linear least squares mathematics. There is, in some cases, a closed-form solution to a non-linear least squares problem — but in general there is not. Most algorithms involve choosing initial values for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation:. The Jacobian J is a function of constants, the independent variable and the parameters, so it changes from one iteration to the next. The residuals are given by. These are the defining equations of the Gauss—Newton algorithm.
These differences must be considered whenever the solution to a nonlinear least squares problem is being sought. The method of least squares is often used to generate estimators and other statistics in regression analysis.
Least Squares Method and Empirical Modeling: A Case Study in a Mexican Manufacturing Firm
Consider a simple example drawn from physics. A spring should obey Hooke's law which states that the extension of a spring y is proportional to the force, F , applied to it. Each experimental observation will contain some error. There are many methods we might use to estimate the unknown parameter k. Noting that the n equations in the m variables in our data comprise an overdetermined system with one unknown and n equations, we may choose to estimate k using least squares. The sum of squares to be minimized is. Here it is assumed that application of the force causes the spring to expand and, having derived the force constant by least squares fitting, the extension can be predicted from Hooke's law.
In regression analysis the researcher specifies an empirical model. For example, a very common model is the straight line model which is used to test if there is a linear relationship between dependent and independent variable. If a linear relationship is found to exist, the variables are said to be correlated. However, correlation does not prove causation , as both variables may be correlated with other, hidden, variables, or the dependent variable may "reverse" cause the independent variables, or the variables may be otherwise spuriously correlated.
For example, suppose there is a correlation between deaths by drowning and the volume of ice cream sales at a particular beach. Yet, both the number of people going swimming and the volume of ice cream sales increase as the weather gets hotter, and presumably the number of deaths by drowning is correlated with the number of people going swimming.
Perhaps an increase in swimmers causes both the other variables to increase. In order to make statistical tests on the results it is necessary to make assumptions about the nature of the experimental errors. A common but not necessary assumption is that the errors belong to a normal distribution. The central limit theorem supports the idea that this is a good approximation in many cases. However, if the errors are not normally distributed, a central limit theorem often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large.
For this reason, given the important property that the error mean is independent of the independent variables, the distribution of the error term is not an important issue in regression analysis. Specifically, it is not typically important whether the error term follows a normal distribution. Confidence limits can be found if the probability distribution of the parameters is known, or an asymptotic approximation is made, or assumed.
Likewise statistical tests on the residuals can be made if the probability distribution of the residuals is known or assumed. The probability distribution of any linear combination of the dependent variables can be derived if the probability distribution of experimental errors is known or assumed. Inference is particularly straightforward if the errors are assumed to follow a normal distribution, which implies that the parameter estimates and residuals will also be normally distributed conditional on the values of the independent variables.
The first principal component about the mean of a set of points can be represented by that line which most closely approaches the data points as measured by squared distance of closest approach, i. Thus, although the two use a similar error metric, linear least squares is a method that treats one dimension of the data preferentially, while PCA treats all dimensions equally. In some contexts a regularized version of the least squares solution may be preferable.
In a Bayesian context, this is equivalent to placing a zero-mean normally distributed prior on the parameter vector. In a Bayesian context, this is equivalent to placing a zero-mean Laplace prior distribution on the parameter vector.
One of the prime differences between Lasso and ridge regression is that in ridge regression, as the penalty is increased, all parameters are reduced while still remaining non-zero, while in Lasso, increasing the penalty will cause more and more of the parameters to be driven to zero. This is an advantage of Lasso over ridge regression, as driving parameters to zero deselects the features from the regression. Thus, Lasso automatically selects more relevant features and discards the others, whereas Ridge regression never fully discards any features.
The L 1 -regularized formulation is useful in some contexts due to its tendency to prefer solutions where more parameters are zero, which gives solutions that depend on fewer variables. An extension of this approach is elastic net regularization. From Wikipedia, the free encyclopedia. Method in statistics. It is not to be confused with Least-squares function approximation.
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Main article: Regularized least squares. Main article: Tikhonov regularization. Journal of the American Statistical Association. Linear Algebra With Applications 3rd ed.
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