059 - MIT Haystack Observatory

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0.03 0.025 Antenna temperature (K) 0.02 0.015 0.01 0.005 0 -0.005 8 10 12 14 16 18 20 22 24 26 Effective time (hr) Figure 4: Black: Best-fit model. Red: The same model with D replaced by D ± σ D , where σ D is the estimate of the error in fitting D. The value of σ D obtained from fitting is actually a very good estimate of the true random error in estimating D because D is only very weakly correlated with the other model parameters. See Section 5.5. t 0 not equal to 18 h exactly Hint: What is the elevation of the Sun at sunset at the surface How does this differ at the altitude of the mesosphere Why are just after sunset and just before sunrise the best times to see flashes from low-orbiting satellites (such as from the Iridium satellites) 6 Theory 6.1 Minimizing the sum of the square of the residuals In most contexts, random noise can be thought of as coming from a Gaussian (also called normal) distribution. A general Gaussian function with independent variable x has the form Ae −(x−µ)2 2σ 2 , (2) where A represents the amplitude, µ is the center of the distribution, and σ is the width of the distribution. A probability density function is required to have unit area. For a Gaussian pdf, A = 1/( √ 2πσ), since ∫ 1 √ e −(x−µ)2 2σ 2 = 1. (3) 2πσ Suppose we have a set of data points d i that can be fit by a model whose predicted value at each data point is m i . In general, d i = m i + ǫ i , where ǫ i is the contribution due to noise in the data point. Suppose that the noise is expected to follow a Gaussian distribution with errors σ i . 8
If we have correctly modelled our data, what is the probability that we would measure the data point d i The answer is P(d i ) = 1 √ 2πσi e − ǫ2 i 2σ 2 i = 1 √ 2πσi e −(d i −m i )2 2σ 2 i . (4) If we have n data points, the probablility that we would obtain all the data points d i is P(d 1 )P(d 2 )...(d n ) = = = n∏ P(d i ) i=0 n∏ 1 √ e −(d i −m i )2 2σ i 2 2πσi i=0 ( 1 √ 2π ) n 1 ∏ ni=0 σ i e −1 2 ∑ n (d i −m i ) 2 i=0 σ 2 i . (5) If the standard deviations σ i are all equal, the maximum of this probability occurs when the sum of the square of the residuals ∑ n i=0 (d i −m i ) 2 is minimized. If the models we are considering are parameterized by j different parameters p 1 through p j , the best fit model m(p 1 ,p 2 ,...p j ) is the one for which the sum of the square of the residuals ∑ n i=0 (d i − m i ) 2 is smallest. If the data points have different standard deviations, the probability is maximum if we first weight the square of the residuals by the inverse of the square of the standard deviation (σ 2 i ). 6.2 Why doesn’t the leasqr algorithm always find the best fit curve When the function you’re trying to fit is linear in the model parameters (which doesn’t necessarily have to be a linear function; y = ax 2 +bx+c is linear in model parameters a, b, and c even though the equation is for a parabola), linear least squares fitting can be achieved by solving a matrix equation (see VSRT Memo #042). If the model is not linear in all parameters, nonlinear least squares methods can be used. Nonlinear least squares methods are based on iterative application of the linear least squares technique. The algorithm is run once, and the sum of squares of the residuals is calculated. The algorithm takes the solutions from one iteration as the inputs to the next iteration. The algorithm stops when the sum of squares of the residuals converges to within a specified tolerance or the number of iterations exceeds a specified limit. The leasqr algorithm, like all nonlinear least squares algorithms, searches through a multidimensional parameter space by calculating the sum of the squares of the residuals and trying to find a path through parameter space that reduces this quantity. The problem when fitting nonlinear functions is that there’s no guarantee that the end result will be the global minimum of the sum of squares of residuals rather than just a local minimum. To put it another way, suppose you start at a random location in Asia and start walking in the steepest direction downward. After every step, you go in the new direction of steepest descent, which may change from step to step. Eventually you will end up at a valley (or the edge of the ocean), which is the lowest point locally. However, you won’t end up at the lowest point globally, the shore of the Dead Sea (at 418 m below sea level), unless you started near it in the first place. 9
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