A. Ruhe. Fitting empirical data by positive sums of exponentials. SIAM Journal on Scientic and Statistical Computing, 1:481-498, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....appreciably di erent results. Some of the diculties and instabilities in convergence are inherent in the problem itself and are shared by other methods [25] Indeed, the same order of complexity has been documented in the companion problem of tting functions (or data) by sums of exponentials [20, 29, 36]. The diculties are mainly caused by the non linear dependence of the objective function on the exponential parameters, the number of parameters to be simultaneously optimized, the presence of local minima and the atness of the objective function. In order to reduce the variability of the ....

A. Ruhe. Fitting empirical data by positive sums of exponentials. SIAM Journal on Scientic and Statistical Computing, 1:481-498, 1980.

....data are generated in time steps of Deltat = 0:05 from 0 to 1:15 by the exponential sum f (t) 0:0951 exp ( Gammat) 0:8607 exp ( Gamma3t) 1:5576 exp ( Gamma5t) The data series is either used with full precision or rounded to two decimals or four decimals. For the two decimal case, Ruhe [17], finds the best model to be of order two, with the solution b = 1:75656; 4:54746) T . Using a maximum likelihood type of weighting, Ruhe finds the best solution to be b = 1:74990; 4:54551) T . We have also added one extra term, to get a fourth order Lanczos series, f (t) 0:0951 exp ....

Axel Ruhe. Fitting empirical data by positive sums of exponentials. SIAM Journal on Scientific and Statistical Computing, 1(4):481--498, 1980.

....test with inverse Laplace transform using Laguerre interpolation but found it totally unsatisfactory. In [30] the algorithm is extended to the case of non equidistant data points. Also a weighted norm is used in the least squares fitting. Two algorithms by Ruhe. The first algorithm by Ruhe [77] is a stepwise nonlinear regression algorithm. Define the inner product (x; y) W = x T Wy with W =diag Gamma 1=y 2 j Delta A Review of the Problem of Fitting Positive Exponential Sums to Empirical Data 16 and the exponential vector e (b) 0 exp ( Gammabt 1 ) exp ( Gammabt n ) ....

Axel Ruhe. Fitting empirical data by positive sums of exponentials. Compstat, pages 622--628, 1980.

....overcome by the regularization procedure. He uses complicated Morse theory to establish this. A Review of the Problem of Fitting Positive Exponential Sums to Empirical Data 13 7 Iteratively adding and removing terms Another type of method has been developed by Gustafson [35] Kaijser [45] Ruhe [76] and Evans et al. 30] Kaijser studies a fit of log normal functions and not an exponential sum, but the idea is the same and Kaijser has got inspiration and advice from Gustafson. The methods are stepwise nonlinear regression methods and the basic algorithmic steps are: 1. Make a grid with N ....

.... If there is no background information, the natural way to define a subgroup is to require that all components in a subgroup shall have indices which differ at most 5, say, or by some other small number (Small in comparison with the dimension N) This idea is expressed as a formula by Ruhe [76] and discussed later in this section. Kaijser omits the nonlinear optimization in step 3, while Gustafson suggests GaussNewton for this. Ruhe and Wedin [78] presents three algorithms for separable nonlinear least squares of Gauss Newton type. They separate the linear variables a from the ....

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Axel Ruhe. Fitting empirical data by positive sums of exponentials. SIAM Journal on Scientific and Statistical Computing, 1(4):481--498, 1980.

....might be unnecessary complex. The classical Lanczos exponential sum is a constructed example with three terms. When introducing errors by rounding data from this exponential sum, for example, Ruhe showed that a model of order two (four unknown parameters in the exponential sum) fits the data well [12]. Thus introduction of errors smooths the data giving best fit using a lower order model. This phenomenon has been illustrated by Van den Bos, who also gave an explanation to this in terms of catastrophe theory and structural stability [15] So, the problem is to determine ffl the model order p ....

....data are generated in time steps of Deltat = 0:05 from 0 to 1:15 by the exponential sum f (t) 0:0951 exp ( Gammat) 0:8607 exp ( Gamma3t) 1:5576 exp ( Gamma5t) The data series is either used with full precision or rounded to two decimals or four decimals. For the two decimal case, Ruhe [12], finds the best model to be of order two, with the solution b = 1:75656; 4:54746) T . Using a maximum likelihood type of weighting, Ruhe finds the best solution to be b = 1:74990; 4:54551) T . 6 The Boliden time series are all empirical data series. Boliden 1 t = 0 1 2:5 5 10 15 20 25 ....

Axel Ruhe. Fitting empirical data by positive sums of exponentials. SIAM Journal on Scientific and Statistical Computing, 1(4):481--498, 1980.

....is concerned, but may return a pair of damped sinusoids in place of two exponentials which are coalescing. In some applications the restriction to positive coefficients ff j is natural. A convex cone characterization is then possible, and special algorithms have been proposed in [6] 46] 15] 10] [35]. We prefer to treat the general problem with freely varying coefficients since this is appropriate for most compartment models. A common attempt to reduce the difficulty of the general problem has been to treat it as a separable regression, i.e. to estimate the coefficients by linear least ....

A. Ruhe, Fitting empirical data by positive sums of exponentials, SIAM J. Sci. Statist. Comp., 1 (1980), pp. 481--498.

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Ruhe A, Fitting empirical data by positive sums of exponentials, SIAM J. Sci. Stat. Comp. 1, 1980, 481 - 498.

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