|
Squares
Least squares or ordinary least squares (OLS) is a mathematical optimization technique which, when given a series of measured data, attempts to find a function which closely approximates the data (a "best fit"). more...
 Home
Building & Hardware
Dining & Bar
Electrical & Solar
Food & Wine
Heating, Cooling & Air
Home Security
Kitchen
Major Appliances
Outdoor Power Equipment
Plumbing & Fixtures
Tools
 Air Compressors
Air Tools
Flashlights
Generators, Home-Use
Hand Tools
 Clamps & Vises
 Bar Clamps
Bench, Table Vises
Other Clamps & Vises
Spring & C-Clamps
Files & Chisels
Hammers
Knives & Cutters
Other Knives & Cutters
Pocket Knives & Cutters
Utility Knives & Cutters
Leatherworking Tools
Levels & Plumb Bobs
Laser Levels
Other Levels & Plumb Bobs
Standard Levels
Measuring Tools
Calipers
Measuring Tapes & Rulers
Other Measuring Tools
Squares
Mixed Tool Sets
Other Hand Tools
Planes
Pliers
Other Pliers
Plier Sets
Single Pliers
Pry Bars
Pullers
Punches
Saws
Screwdrivers
Craftsman Sets
Other Screwdriver Sets
Other Screwdrivers
Screwdriver Sets
Single Screwdrivers
Snap-On Sets
Stanley Sets
Winches
Wrenches
Adjustable Wrenches
Allen & Hex Wrenches
Other Wrenches
Pipe Wrenches
Socket Wrenches
Torque Wrenches
Wrench Sets
Meters, Testers & Probes
Other Tools
Power Tools
Safety, Protective Gear
Tie Downs, Straps
Tool Boxes, Belts & Storage
Welding & Soldering Tools
Vacuum Cleaners &...
Wholesale Lots
Window Treatments
It attempts to minimize the sum of the squares of the ordinate differences (called residuals) between points generated by the function and corresponding points in the data. Specifically, it is called least mean squares (LMS) when the number of measured data is 1 and the gradient descent method is used to minimize the squared residual. LMS is known to minimize the expectation of the squared residual, with the smallest operations (per iteration). But it requires a large number of iterations to converge.
An implicit requirement for the least squares method to work is that errors in each measurement be randomly distributed. The Gauss-Markov theorem proves that least square estimators are unbiased and that the sample data do not have to comply with, for instance, a normal distribution. It is also important that the collected data be well chosen, so as to allow visibility into the variables to be solved for (for giving more weight to particular data, refer to weighted least squares).
The least squares technique is commonly used in curve fitting. Many other optimization problems can also be expressed in a least squares form, by either minimizing energy or maximizing entropy.
History
On New Year's Day in 1801, the Italian astronomer Giuseppe Piazzi discovered the asteroid Ceres. He was able to track its path for 40 days. In the course of the year, many scientists tried to estimate the trajectory on the basis of Piazzi's observations (solving Kepler's nonlinear equations of motion is very difficult). Most evaluations were useless; the only calculation precise enough to allow Zach, a German astronomer, to recover Ceres at the end of the year, was that of 24-year-old Carl Friedrich Gauss (the fundamentals of his approach had already been accomplished by him in 1795, when he was still 18 years old). But his method of least squares was not published until 1809, when it appeared in volume two of his work on celestial mechanics, Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium. Frenchman Adrien-Marie Legendre independently developed the same method in 1805, as did American Robert Adrain in 1808.
In 1829 Gauss was able to state the reason for this procedure's outstanding success: The method of least squares is simply optimal in many respects. The precise argument is known as the Gauss-Markov theorem.
Formulation of the problem
Suppose that the data set consists of the points (xi,yi) with  f such that  Read more at Wikipedia.org
|
|
