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14.8 Savitzky-Golay Smoothing Filters. In this section we discuss a particular type of low-pass filter, well-adapted for data smoothing, and termed variously Savitzky-Golay [1. least-squares [2. Rather than having their properties defined in the Fourier domain, and then translated to the time domain, Savitzky-Golay filters derive directly from a particular formulation of the data smoothing problem in the time domain,...
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Sometimes the model is simply a convenient class of functions, such as polynomials or Gaussians, and the fit supplies the appropriate coefficients. Other times, the model’s parameters come from some underlying theory that the data are supposed to satisfy. The basic approach in all cases is usually the same: You choose or design a figure-of-merit function (“merit function,” for short)...
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When the third item suggests that the model is an unlikely match to the data, then items (i) and (ii) are probably worthless.. 15.1 Least Squares as a Maximum Likelihood Estimator. The model predicts a functional relationship between the measured independent and dependent variables,. What, exactly, do we want to minimize to get fitted values for the a j ’s?...
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If we take the derivative of equation (15.1.5) with respect to the parameters a k , we obtain equations that must hold at the chi-square minimum,. Equation (15.1.7) is, in general, a set of M nonlinear equations for the M unknown a k . Various of the procedures described subsequently in this chapter derive from (15.1.7) and its specializations.. 15.2...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. Equation (15.2.12).. 15.3 Straight-Line Data with Errors in Both Coordinates. If experimental data are subject to measurement error not only in the y i ’s, but also in the x i ’s, then the task...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. 15.4 General Linear Least Squares. An immediate generalization of § 15.2 is to fit a set of data points (x i , y i ) to a model that is not just a linear combination...
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Lawson, C.L., and Hanson, R. Forsythe, G.E., Malcolm, M.A., and Moler, C.B. 15.5 Nonlinear Models. On the other hand, (15.5.1) might be a poor local approximation to the shape of the function that we are trying to minimize at a cur . To use (15.5.2) or (15.5.3), we must be able to compute the gradient of the χ 2 function...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. 15.6 Confidence Limits on Estimated Model Parameters. We have given some formulas for computing standard deviations or variances of individual parameters (equations as well as some formulas for covariances between pairs of parameters (equation 15.2.10....
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w i 2 V ji V ki (15.6.10). 15.7 Robust Estimation. Various definitions of greater or lesser mathematical rigor are possible for the term, but in general, referring to a statistical estimator, it means “insensitive to small departures from the idealized assumptions for which the estimator is optimized.” [1,2] The word. M-estimates follow from maximum-likelihood arguments very much as equa-...
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Integration of Ordinary Differential Equations. 16.0 Introduction. Problems involving ordinary differential equations (ODEs) can always be reduced to the study of sets of first-order differential equations. Occasionally, it is useful to incorporate into their definition some other factors in the equation, or some powers of the independent variable, for the purpose of mitigating singular behavior that could result in overflows...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. 16.1 Runge-Kutta Method. There are several reasons that Euler’s method is not recommended for practical use, among them, (i) the method is not very accurate when compared to other, fancier, methods run at the equivalent...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. free_vector(dv,1,nvar);. free_vector(vout,1,nvar);. free_vector(v,1,nvar);. Abramowitz, M., and Stegun, I.A. Shampine, L.F., and Watts, H.A. 16.2 Adaptive Stepsize Control for Runge-Kutta. Usually the purpose of this adaptive stepsize control is to achieve some predetermined accuracy in the...
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16.3 Modified Midpoint Method. This section discusses the modified midpoint method, which advances a vector of dependent variables y(x) from a point x to a point x + H by a sequence of n substeps each of size h,. In principle, one could use the modified midpoint method in its own right as an ODE integrator. The number of right-hand...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. for (i=1;i<=nvar;i. for (n=2;n<=nstep;n. free_vector(yn,1,nvar);. free_vector(ym,1,nvar);. 16.4 Richardson Extrapolation and the Bulirsch-Stoer Method. Apart from those two caveats, we believe that the Bulirsch-Stoer method, discussed in this section, is the best known way to obtain...
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16.5 Second-Order Conservative Equations. 16.5 Second-Order Conservative Equations 733. Henrici showed how to rewrite equations (16.5.2) to reduce roundoff error by using the quantities ∆ k ≡ y k+1 − y k . (16.5.6) and set KMAXX = 12 in bsstep.. 16.6 Stiff Sets of Equations. 999u − 1999v (16.6.1) with boundary conditions. y + z (16.6.3) we find the...
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The right-hand side f is stored in the first n elements of the array d2y. (16.5.6) and set KMAXX = 12 in bsstep.. 16.6 Stiff Sets of Equations. Stiffness occurs in a problem where there are two or more very different scales of the independent variable on which the dependent variables are changing. 999u − 1999v (16.6.1) with boundary conditions....
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You could arrange for the stepsize to be reduced, say by a factor of the current nseq[k]. Kaps, P., and Rentrop, P. Enright, W.H., and Pryce, J.D. Bader, G., and Deuflhard, P. Enright, W.H., Hull, T.E., and Lindberg, B. Stoer, J., and Bulirsch, R. 16.7 Multistep, Multivalue, and Predictor-Corrector Methods. Predictor-corrector is a partic- ular subcategrory of these methods —...
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Two Point Boundary Value Problems. 17.0 Introduction. When ordinary differential equations are required to satisfy boundary conditions at more than one value of the independent variable, the resulting problem is called a two point boundary value problem. As the terminology indicates, the most common case by far is where boundary conditions are supposed to be satisfied at two points —...
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17.1 The Shooting Method 757. 17.1 The Shooting Method. In this section we discuss “pure” shooting, where the integration proceeds from x 1 to x 2 , and we try to match boundary conditions at the end of the integration. In the next section, we describe shooting to an intermediate fitting point, where the solution to the equations and boundary...
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17.2 Shooting to a Fitting Point. The shooting method described in § 17.1 tacitly assumed that the “shots” would be able to traverse the entire domain of integration, even at the early stages of convergence to a correct solution. (If you are confused by this, go back to § 17.1.) We can therefore define an n 2 -vector V (1)...