Richard G. Cornell. A method for fitting linear combinations of exponentials. Biometrics, pages 104--113, 1962.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for different number of terms p in Section 3. For a single term, p = 1, we use a maximum likelihood (ML) type of initial value estimation. For the case of more terms, 2 p 4, we develop a new initial value estimate by dividing data equidistant in time into partial sums of equal length q. Cornell [2] has tried this analytically for p = 1 and numerically for p = 2. Here we derive an analytical expression for the case p = 2 and a numerical procedure for p = 3; 4. These sums can be rewritten as geometrical sums, which simplifies the expressions a lot. The initial value algorithm compares ....

....variables u 1 and u 2 . All together a system of order nine. Due to this high order system of equations, another method based upon geometrical sums was tried. This method assumes equally spaced data points and rewrites the exponential model in polynomial form as in 3. We used an idea from Cornell [2] which rewrite the geometrical sum formula for a partial sum from y m to y m l as l m X j=l e a i u j i = u l i Gamma u l m 1 i 1 Gamma u i = u l i 1 Gamma u i i 1 Gamma u m 1 i j : 5) Cornell investigated the idea for p = 1; 2 but found it too complicated for p = 2. The ....

Richard G. Cornell. A method for fitting linear combinations of exponentials. Biometrics, pages 104--113, 1962.

.... of finding initial values when fitting exponential sums f (t) P p i=1 a i exp ( Gammab i t) and f (t) P p i=1 a i (1 Gamma exp ( Gammab i t) to empirical data [7, 6] This idea originates in an article by Cornell who studied the fitting of f (t) P p i=1 a i exp ( Gammab i t) to data [2]. Generalized interpolation (GI) means solving the interpolation equations ( n P j2I k y j = P j2I k f (a; b; t j ) o I k 2I k = 1; k max (4) in partial sums S k = P j2I k y j of data points instead of solving them with only a single point for each equation. For example (2) have ....

Richard G. Cornell. A method for fitting linear combinations of exponentials. Biometrics, pages 104--113, 1962. 8

....parameters is to be found by an iterative method. Thus it is important to improve methods for parameter estimation by means of looking for good initial values. One way is to interpolate in 2p points f (t j ) Gamma y j = 0; j = 1; 2; 2p. In a literature study [15] we found that Cornell [3] used what we will call generalized interpolation. For the equation 8 : n P j2I k y j = P j2I k f (a; b; t j ) o I k 2I k = 1; 2p (3) he derived an analytical solution for p = 1 and a numerical method for solving the case p = 2. I = 1; 2; n is indices for the set of data ....

....If the number of estimated parameters is much less then the number of data points, we do not use much of the information stored in the data points. One possibility to overcome this is to instead use a sum of several data points for each equation. This is what Cornell did using partial sums [3]. He derived an analytical solution when p = 1 and a numerical procedure when p = 2. Inspired by Cornell, Agha did the same by using overlapping partial sums and he found it more effective [1] To make the equations for generalized interpolation solvable, data must be equidistant in time. First we ....

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Richard G. Cornell. A method for fitting linear combinations of exponentials. Biometrics, pages 104--113, 1962.

....This was not more successful then loglinear fit. In the context of transform methods, Smith et al. 82, page 477] also discusses cubic interpolation and splines for interpolation. Cubic interpolation can introduce false peaks (for transform methods) They have not tried splines. Cornell [21] has another approach of forming a geometrical sum. If the data is equally spaced in time, t j = t 0 d Delta (j Gamma 1) then f (t j ) p X i=1 a i Delta exp ( Gammab i t j ) p X i=1 ea i Delta u j i : 9) Here ea i = a i Delta exp ( Gammab i Delta (t 1 Gamma d) and u i = ....

Richard G. Cornell. A method for fitting linear combinations of exponentials. Biometrics, pages 104--113, 1962.

....If the number of estimated parameters is a lot fewer then the number of data points, we do not use much of the information stored in the data points. One possibility to overcome this is to instead use a sum of several data points for each equation. This is what Cornell did by partial sums [3]. Inspired by Cornell, Agha did the same by using overlapping partial sums [1] They both studied (3) We first formulate the generalized interpolation equations for 1 and solve them. Then we treat data not equidistant in time. 2.1 Rewriting the exponential sum for data equidistant in time For ....

....values and compute the roots of the denominator. d: 1:q: 1:t1: 1: b: seq(rand(50) 10, j=1. p) l: b( Yt: sum(exp( l[j] t) j=1. p) rotter: sort(evalf( seq(exp( l[j] d q) j=1. p) S: evalf( seq(sum(subs(t=t1 d (j 1) Yt) j=1 (k 1) q. k q) k=1. 2 p 1) Denr: solve(Den, z[3]) Make a log plot for eq31 in each subinterval defined by the roots of Den. if (p=3) then plot( log(abs(P) log(abs(sqrt(Q) log(abs(Den) z[3] 0. Denr[1] fi; A.1 Coefficients for exponential sum when p = 3 The denominator in (9) is D = 2( Gamma Gamma S 2 2 S 4 S 2 Gamma S ....

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Richard G. Cornell. A method for fitting linear combinations of exponentials. Biometrics, pages 104--113, 1962.

....by Smyth of the modified Prony algorithm in [29] An alternative to solve an overdetermined system of pointwise interpolation equations is to cluster data in partial sums and let each partial sum form an interpolation equation. This method we denote GI (Generalized Interpolation) In [1, 7, 34] the method of GI was developed for identifying initial values for parameters in exponential sum models (1.1) The results were algorithms for explicit solutions when p = 1, y Center for Mathematical Modeling, Department of Mathematics and Physics, Malardalen University, Vasteras, Sweden, ....

R. G. Cornell, A method for fitting linear combinations of exponentials, Biometrics, (1962), pp. 104--113.

....this is to instead use a sum of several data points for each equation. 8 : n P j2I k y j = P j2I k f (t j ; a; b) o I k 2I k = 1; k max : 2) I = 1; 2; n is indices for the set of data points and I k are subsets of indices from I. This is what Cornell did by partial sums [4]. Inspired by Cornell, Agha did the same by using overlapping partial sums [1] They both studied the exponential sum f (t) p X i=1 a i exp ( Gammab i t) 3) The point is that if data are equidistant in time, the exponentials in (2) can be reformulated as geometrical sums, which are ....

Richard G. Cornell. A method for fitting linear combinations of exponentials. Biometrics, pages 104--113, 1962.

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