least squares method: คุณกำลังดูกระทู้
Least Squares Fitting
A mathematical procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the offsets (“the residuals”) of
the points from the curve. The sum of the of the offsets is used instead
of the offset absolute values because this allows the residuals to be treated as
a continuous differentiable quantity. However, because squares of the offsets are
used, outlying points can have a disproportionate effect on the fit, a property which
may or may not be desirable depending on the problem at hand.
In practice, the offsets from a line (polynomial, surface, hyperplane, etc.) are almost always minimized instead of the perpendicular
offsets. This provides a fitting function for the independent variable that estimates
for a given (most often what
an experimenter wants), allows uncertainties of the data points along the – and -axes to be incorporated
simply, and also provides a much simpler analytic form for the fitting parameters
than would be obtained using a fit based on perpendicular
offsets. In addition, the fitting technique can be easily generalized from a
best-fit to a best-fit
when sums of vertical distances are used. In any case, for a reasonable number of
noisy data points, the difference between vertical and perpendicular fits is quite
The least squares fitting technique is the simplest and most commonly applied form of linear regression and provides
a solution to the problem of finding the best fitting line through
a set of points. In fact, if the functional relationship between the two quantities
being graphed is known to within additive or multiplicative constants, it is common
practice to transform the data in such a way that the resulting line a
straight line, say by plotting vs. instead
of vs. in the case of
analyzing the period of a pendulum as
a function of its length . For this reason,
standard forms for exponential,
logarithmic, and power
laws are often explicitly computed. The formulas for linear least squares fitting
were independently derived by Gauss and Legendre.
For nonlinear least squares fitting to a number of unknown parameters, linear least squares fitting may be applied iteratively
to a linearized form of the function until convergence is achieved. However, it is
often also possible to linearize a nonlinear function at the outset and still use
linear methods for determining fit parameters without resorting to iterative procedures.
This approach does commonly violate the implicit assumption that the distribution
of errors is normal, but often still gives
acceptable results using normal equations, a pseudoinverse,
etc. Depending on the type of fit and initial parameters chosen, the nonlinear fit
may have good or poor convergence properties. If uncertainties (in the most general
case, error ellipses) are given for the points, points can be weighted differently
in order to give the high-quality points more weight.
Vertical least squares fitting proceeds by finding the sum of the of the deviations of a set of
from a function . Note that this procedure does
minimize the actual deviations from the line (which would be measured perpendicular
to the given function). In addition, although the sum of distances
might seem a more appropriate quantity to minimize, use of the absolute value results
in discontinuous derivatives which cannot be treated analytically. The square deviations
from each point are therefore summed, and the resulting residual is then minimized
to find the best fit line. This procedure results in outlying points being given
disproportionately large weighting.
The condition for to be a minimum is that
for , …, . For a linear fit,
These lead to the equations
In matrix form,
(Kenney and Keeping 1962). These can be rewritten in a simpler form by defining the sums of squares
which are also written as
Here, is the covariance
and and are variances.
Note that the quantities
and can also be interpreted
as the dot products
In terms of the sums of squares, the regression coefficient is given by
and is given in terms of using (◇)
The overall quality of the fit is then parameterized in terms of a quantity known as the correlation coefficient, defined
which gives the proportion of which is
accounted for by the regression.
Let be the vertical coordinate of the
best-fit line with -coordinate , so
then the error between the actual vertical point and the fitted
point is given by
Now define as an estimator for the variance in
Then can be given by
(Acton 1966, pp. 32-35; Gonick and Smith 1993, pp. 202-204).
The standard errors for and are
Deriving the least squares estimators of the slope and intercept (simple linear regression)
I derive the least squares estimators of the slope and intercept in simple linear regression (Using summation notation, and no matrices.) I assume that the viewer has already been introduced to the linear regression model, but I do provide a brief review in the first few minutes. I assume that you have a basic knowledge of differential calculus, including the power rule and the chain rule.
If you are already familiar with the problem, and you are just looking for help with the mathematics of the derivation, the derivation starts at 3:26.
At the end of the video, I illustrate that sum(X_iX bar)(Y_i Y bar) = sum X_i(Y_i Y bar) =sum Y_i(X_i X bar) , and that sum(X_iX bar)^2 = sum X_i(X_i X bar).
There are, of course, a number of ways of expressing the formula for the slope estimator, and I make no attempt to list them all in this video.
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The Main Ideas of Fitting a Line to Data (The Main Ideas of Least Squares and Linear Regression.)
Fitting a line to data is actually pretty straightforward.
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What is LEAST SQUARES METHOD? What does LEAST SQUARES METHOD mean? LEAST SQUARES METHOD meaning
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Source: Wikipedia.org article, adapted under https://creativecommons.org/licenses/bysa/3.0/ license.
The method of least squares is a standard approach in regression analysis to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. \”Least squares\” means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation.
The most important application is in data fitting. The best fit in the leastsquares sense minimizes the sum of squared residuals, a residual being the difference between an observed value and the fitted value provided by a model. When the problem has substantial uncertainties in the independent variable (the x variable), then simple regression and least squares methods have problems; in such cases, the methodology required for fitting errorsinvariables models may be considered instead of that for least squares.
Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear leastsquares problem occurs in statistical regression analysis; it has a closedform solution. The nonlinear problem is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, and thus the core calculation is similar in both cases.
Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable and the deviations from the fitted curve.
When the observations come from an exponential family and mild conditions are satisfied, leastsquares estimates and maximumlikelihood estimates are identical. The method of least squares can also be derived as a method of moments estimator.
The following discussion is mostly presented in terms of linear functions but the use of leastsquares is valid and practical for more general families of functions. Also, by iteratively applying local quadratic approximation to the likelihood (through the Fisher information), the leastsquares method may be used to fit a generalized linear model.
For the topic of approximating a function by a sum of others using an objective function based on squared distances, see least squares (function approximation).
The leastsquares method is usually credited to Carl Friedrich Gauss (1795), but it was first published by AdrienMarie Legendre.
3.2: Linear Regression with Ordinary Least Squares Part 1 – Intelligence and Learning
This video is part of session 3 of my Spring 2017 ITP \”Intelligence and Learning\” course (https://github.com/shiffman/NOCS172IntelligenceLearning/tree/master/week3classificationregression)
Link to Part 2: https://youtu.be/_cXuvTQl090
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