Wikipedia:Reference desk/Archives/Mathematics/2016 November 9

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

November 9

Finding the coefficients of a sum of exponentials

The original problem stems from signal fitting. I have a function $f(t)$ which I expect to be a sum of decreasing exponentials $\sum _{k=1}^{n}A_{k}e^{-t/\tau _{k}}$ (but I ignore how many of them there are). How can one retrieve the coefficients $A_{k},\tau _{k}$ in a way that is somewhat robust to noise? Is there anything fundamentally better than plugging a least-square fit and trying different values of n?

It looks like a Fourier transform, but in the case of sines/cosines there is an orthonormal basis for the periodic functions, which means that one can recover the coefficients easily with the adequate scalar product. I do not think any such basis exists for exponential functions. Tigraan^{Click here to contact me} 10:28, 9 November 2016 (UTC)

Your $f(t)$ is essentially written as a Laplace transform (of a certain atomic measure consisting of a couple of Dirac deltas). There is an inversion formula, but I don't know how difficult it is to use in this case. —Kusma (t·c) 16:05, 9 November 2016 (UTC)

See linear prediction. Use singular value decomposition to get the recursion coefficients (try a lot of values of n to see which gives the best fit). From these you can solve for your wanted time-constants. --catslash (talk) 21:48, 9 November 2016 (UTC)

Regarding the selection of the value of n, in linear regression increasing n by 1 increases the number of parameters to be estimated, without forbidding the same regression results as occurred with the smaller n. So the "fit" in the sense of R-squared will go up. One needs to decide if it went up to a statistically significant extent. If I recall, an F-test is used for this. The article Stepwise regression#Selection criterion may be relevant here. Loraof (talk) 22:15, 9 November 2016 (UTC)

Any linear system, when it is not being driven by outside forces, has a response which is the sum of a number of decaying sinusoids (assuming it is stable). Identifying the frequencies and rates of decay (the poles of the system in the s-plane), is a fairly well-studied problem. In the special case of a linear system with all real poles, the response is composed of non-oscillating exponential decays - like the signal in your question.

If you are working with a linear system, then it is worth getting the signals from a number of different points in the system (different state variables). These signals will be different linear combinations (A_k) of the same set of exponential decays (τ_k) (show the same poles). A decay term which contributes little to the signal at one point, may be larger and more easily discerned at another. In any case, it can only increase your chances of having enough data to get a good fit. --catslash (talk) 01:01, 10 November 2016 (UTC)

That is not an option. For whoever cares, the real experiment is related to the decay signal of phosphor emission. Technically, the theory says it is a bunch of atomic orbitals with transition rates between them so it is a linear system, but there is no way to measure each level separately; what you get is (roughly) the population of one of the electronic states with time (because that is the starting point of a measurable transition with light emission). Tigraan^{Click here to contact me} 17:27, 10 November 2016 (UTC)

• Mellin's inverse formula will not cut it for me. The problem may mathematically be similar to a Laplace inverse transform, but I have only sampling of the transformed function over a finite real interval, and the reverse formulas (either Mellin's, or [1] which I found by hunting the links) seem to require evaluation on complex values.

Is there a way to perform an inverse Laplace transform evaluating only on real points of the transformed function? Is that even possible? Tigraan^{Click here to contact me} 17:27, 10 November 2016 (UTC)

You could try to compute the [n, n+1] Padé approximants of the Laplace transform of f(t). The Laplace transform has poles at $-{\frac {1}{\tau _{k}}}$ with residues of $A_{k}$, you can extract these from the Padé approximant fitting. Noise will affect the ability to recover the contribution of small $\tau _{k}$ terms. Count Iblis (talk) 21:24, 10 November 2016 (UTC)

Hypergeometric 2F1 algorithms

I'm looking for fast converging series of ${}_{2}F_{1}((1-n)/2,-n/2;1;4)$ and ${}_{2}F_{1}((1-n)/2,-n/2;2;4)$, where $n$ is a positive integer. 150.135.210.137 (talk) 21:34, 9 November 2016 (UTC)

What does it mean to ask for a fast-converging series of a finite sum? --JBL (talk) 23:08, 14 November 2016 (UTC)

P.S. Here are your sequences, with loads of references: c = 1, c = 2. --JBL (talk) 23:12, 14 November 2016 (UTC)