General linear least squares - UGAstro

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In the example in Figure 1, the parabolic fit gives g = 9.93 m s -2 , whereas the polynomial fitimplies that the initial position is -0.29 m, the initial velocity is 1.81 m s -1 and the accelerationis 6.18 m s -2 . Polynomial fitting fails to take account of our knowledge that the initial positionand velocity are zero, and as a consequence gives an inaccurate value for the acceleration.Example: Circular motionNow suppose our task is to determine the radius of a wheel by measuring the x-coordinate of apoint on the circumference as the wheel rotates at a known frequency ω. The position of thatpoint is given byx = x 0+ Rsin( ωt) .From measurements of (t, x) we want to find x 0 and R. For this example, the relevant fragmentof IDL code isb = transpose([[replicate(1.,npts)],[sin(omega*t)]])y = transpose(x)psi = invert( transpose(b) ## b)ans = psi ## transpose( b) ## yAn example of such a fit is shown in Figure 2.Figure 2: Time series of measurement of a point on the circumference of a wheel rotatingat known angular frequency ω. A linear least squares fit to x = x 0 + Rsin(ωt) yields theradius and the x-coordinate of the point of rotation.Note the limitation of this method—we cannot determine ω from the data; we have to know therotation rate. Problems where unknowns enter other than in linear combinations fall into thecategory of non-linear least squares. There are no closed-form solutions to non-linear
problems: they are solved using iterative methods that require an initial guess for the modelparameters.At first sight some problems appear non-linear, e.g., the case of the rotating wheel when thephase, φ, is unknownx = x 0+ Rsin( ωt + φ) .However, by use of trigonometric identities we can writewhich is a linear problem.x = x 0+ Rcos( φ )sin( ωt) + Rsin( φ )cos( ωt),
Page 1 and 2: General Linear Least SquaresJames R
Page 3: An simple example: uniform accelera

General linear least squares - UGAstro

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