The method is based on minimizing the sum of squares of the regression residuals. one of the basic methods of regression analysis for estimating unknown parameters of regression models based on sample data.
Least square method ( OLS, OLS, Ordinary Least Squares) The results of measurements and calculations are recorded in Table 7. Then the equation takes the form y = a + bx. Λ is the wavelength of the incident light. Where d 0 is the thickness of the gap between the lens and the plane-parallel plate (or lens deformation), The radii of Newton's rings are related to the radius of curvature of the lens R and the number of the ring by the equation The radii of Newton's rings r m were measured and the numbers of these rings m were determined. It is required to determine the radius of curvature of the lens using Newton's rings.
Given the reliability P = 0.95, according to the table of Student's coefficients for n = 6, we find t = 2.57 and determine the absolute error Δα = 2.57 0.000132 = 0.000338 deg -1.Įxample 3. Let us find the error in the definition of α. The results of measurements and calculations are shown in the table ( see table 6).
#WEIGHTED LEAST SQUARES REGRESSION EXCEL FREE#
The free term defines the resistance R 0 at 0 ° C, and the slope is the product of the temperature coefficient α and the resistance R 0. Let's calculate the temperature coefficient of resistance of the metal using the least squares method. Given the reliability P = 0.95, according to the table of Student's coefficients for n = 5, we find t = 2.78 and determine the absolute error ΔJ = 2.78 0.05185 = 0.1441 ≈ 0.2 kg m 2.Įxample 2. To determine the mean square error, we use the formula (20) The results of measurements of the moment of force and angular acceleration are entered in the second and third columns. It is required to determine the moment of inertia of this body. For various values of the moment M, the angular acceleration ε of a certain body was measured. The basic equation of the dynamics of rotational motion ε = M / J (a straight line passing through the origin of coordinates) was investigated. The forms of these tables are shown in the examples discussed below.Įxample 1. When processing measurement results by this method, it is convenient to summarize all data in a table in which all the sums included in formulas (19) - (24) are preliminarily calculated. The root-mean-square errors in determining a and b are equal The joint solution of these equations gives The task is to find the best values of a and b from the available set of values x i, y i.Īgain, we compose the quadratic form φ, equal to the sum of the squares of the deviations of the points x i, y i from the straight lineĪnd find the values of a and b for which φ has a minimum Let us now consider a slightly more difficult case, when the points must satisfy the formula y = a + bx(straight line not passing through the origin). The calculation shows that the root-mean-square error in determining the value of k is equal to The least squares method states that for k one should choose such a value at which φ has a minimum The value of φ is always positive and turns out to be the smaller, the closer our points lie to the straight line. Let's compose the value φ - the sum of the squares of the deviations of our points from the straight line For example, if it is assumed that the refractive index of glass n is related to the length λ of the light wave by the ratio n = a + b / λ 2, then the dependence of n on λ -2 is plotted on the graph.Ĭonsider the dependency y = kx(straight line through the origin). And even when the relationship is non-linear, they usually try to plot the graph in such a way as to get a straight line. Linear dependence is very widespread in physics. In practice, this method is most often (and most simply) used in the case of a linear relationship, i.e.
The least squares method requires that the sum of the squares of the deviations of the experimental points from the curve, i.e. Experimental points, as a rule, do not fit exactly on the curve. The method of least squares allows you to determine them. However, the constant coefficients that are included in this function remain unknown. The resulting curve makes it possible to judge the form of the function ƒ (x). As a result of measurements, a number of values are obtained:īased on the data of such an experiment, it is possible to construct a graph of the dependence y = ƒ (x). If some physical quantity depends on another quantity, then this dependence can be investigated by measuring y at different values of x.