# Exponential fit for time series

I'm pretty new to time series, but I'm now practising with it in order to obtain an exponential envelope of my experimental data. I start by importing data, I create the time series, I calculate the upper and lower estimated background (which is a time series itself), then I extract local maxima using FindPeaks and finally I try to use FindFit or NonlinearModelFit, but both of them give the same error: time component is not a valid variable.

On the other hand I don't understand how to use the command TimeSeriesModelFit to obtain an exponential fit.

t = Take[Transpose[data], 1];
p1 = Take[Transpose[data], -1];
ts1 = TimeSeries[p1, {t}, TemporalRegularity -> True];
eb1ts1 = EstimatedBackground[ts1, 100, Method -> {"SNIP", 1}];
eb2ts1 = -EstimatedBackground[-ts1, 50, Method -> {"SNIP", 1}];
max1 = FindPeaks[eb1ts1, 500];
max2 = FindPeaks[eb2ts1, 500];
FindFit[max1, A Exp[-t/(120 τ)] + B, {A, B, τ}, t]; OR
funfit[t_] := A E^(-(t/(120 τ))) + B;
fitmax = NonlinearModelFit[max1, funfit[t], {τ, A, B}, t, MaxIterations -> 100000]

I try this way.

{{1/120, -0.0209557}, {1/60, -0.0209557}, {1/40, -0.0209557}, {1/
30, -0.019019}, {1/24, -0.0209557}, {1/20, -0.019019}, {7/
120, -0.019019}, {1/15, -0.019019}, {3/40, -0.019019}, {1/
12, -0.019019}, {11/120, -0.019019}, {1/10, -0.019019}, {13/
120, -0.019019}, {7/60, -0.019019}, {1/8, -0.019019}, {2/
15, -0.019019}, {17/120, -0.019019}, {3/20, -0.019019}, {19/
120, -0.019019}, {1/6, -0.019019}, {7/40, -0.019019}, {11/
60, -0.019019}, {23/120, -0.019019}, {1/5, -0.019019}, {5/
24, -0.019019}, {13/60, -0.019019}, {9/40, -0.019019}, {7/
30, -0.0170823}, {29/120, -0.0170823}, {1/4, -0.0170823}, {31/
120, -0.0170823}, {4/15, -0.0170823}, {11/40, -0.019019}, {17/
60, -0.0170823}, {7/24, -0.0170823}, {3/10, -0.0170823}, {37/
120, -0.0170823}, {19/60, -0.019019}, {13/40, -0.019019}, {1/
3, -0.019019}, {41/120, -0.019019}, {7/20, -0.019019}, {43/
120, -0.019019}, {11/30, -0.019019}, {3/8, -0.019019}, {23/
60, -0.019019}, {47/120, -0.019019}, {2/5, -0.019019}, {49/
120, -0.019019}, {5/12, -0.019019}, {17/40, -0.019019}, {13/
30, -0.019019}, {53/120, -0.019019}, {9/20, -0.019019}, {11/
24, -0.019019}, {29/60, -0.0170823}, {59/120, -0.0170823}, {1/
2, -0.019019}, {61/120, -0.019019}, {31/60, -0.019019}, {21/
40, -0.019019}, {11/20, -0.0170823}, {67/120, -0.0170823}, {17/
30, -0.019019}, {23/40, -0.019019}, {7/12, -0.019019}, {71/
120, -0.019019}, {37/60, -0.019019}, {5/8, -0.019019}, {19/
30, -0.019019}, {77/120, -0.019019}, {2/3, -0.019019}, {27/
40, -0.019019}, {41/60, -0.019019}, {83/120, -0.019019}, {7/
10, -0.019019}, {17/24, -0.019019}, {43/60, -0.019019}, {29/
40, -0.019019}, {11/15, -0.019019}, {89/120, -0.019019}, {3/
4, -0.0209557}, {91/120, -0.0209557}, {23/30, -0.0209557}, {31/
40, -0.0209557}, {47/60, -0.0209557}, {19/24, -0.0209557}, {4/
5, -0.019019}, {97/120, -0.019019}, {49/60, -0.019019}, {33/
40, -0.019019}, {5/6, -0.019019}, {101/120, -0.019019}, {17/
20, -0.0209557}, {103/120, -0.0209557}, {9/10, -0.0228924}, {109/
120, -0.0228924}, {11/12, -0.0228924}, {37/40, -0.0228924}, {14/
15, -0.0209557}}

This is a plot of the overall data (position as a function of time=

I'm interested in the damping phase, so in the second phase. The first phase is very short and cannot be even perceived in the overall plot. I would like to obtain an exponential envelop which show how fast the maximum oscillation amplitude goes to zero with respect to time. As you can see, the oscillation shows some beats; I would like to be able to select the local maxima of each beat and to "interpolate" those decreasing maxima with the best exponential.

Hello Hugh, I would like to follow your path in using NonlinearModelFit, but I don't know how to guess the initial parameter values. Is there a guideline to follow? Any practical hint you can provide in order to get closer to the correct ones? I appreciate your help.

• Thanks. That makes it much easier to read. But there are only 4 distinct (and equally spaced) y-values. Is this a time series with just 4 distinct states?
– JimB
Sep 7, 2017 at 19:59
• I don't understand how you are defining your phases. If there is just one interval you are interested in please extract that from the data and add a further plot. Then give us a clear description of what you wish to extract from this interval. Possibilities could be frequency, damping ratio, envelope etc
– Hugh
Sep 8, 2017 at 10:58
• Could you update the pastebin code so that I can be easily imported in Mathematica? For example using data = Import["http://pastebin.com/raw/M03RMCJk", "List"]. Sep 8, 2017 at 21:18
• sure, I do it now. Sep 8, 2017 at 21:33
• I must have been unclear. The first thing when trying to answer your question, is to manage to import the data. For now, when I go data = Import["http://pastebin.com/raw/M03RMCJk", "List"] from my notebook, I cannot import it properly (try it yourself and you'll see what I mean). I was asking if you could arrange that, either but changing the style of the uploaded data, or by providing a command to import it straightforwardly. Sep 8, 2017 at 21:44

I have imported your data and this is a plot

ListLinePlot[data]

The data is sampled over only a few quantisation levels of your analog to digital converter. You could do with some analogue gain before the converter to give you better resolution. However, that has nothing to do with Mathematica. The data is not an exponential but an oscillating function with possibly an exponential envelope. It has a mean offset. Let's fit an appropriate function.

nlm = NonlinearModelFit[data,
m + a E^(b t) Cos[ ω t + ϕ], {{m, -0.02}, {a,
1.5 10^-20}, {b, 0.0100}, {ω, 2.9}, ϕ}, t]

This is a nonlinear model so we have to make good initial guesses. I played around a bit to find good estimates.

Now we plot the data and the fitted curve

Show[ListPlot[data], Plot[nlm[t], {t, data[[1, 1]], data[[-1, 1]]}]]

This is a reasonable fit. More data would see if this model is any good.

The details of the fitted parameters can be found from

nlm["ParameterTable"]

Hope that helps.

• It is the output of a video tracking of a pendulum in a chain. And I've extracted a very short part of the motion. Sep 7, 2017 at 20:19
• I think it is too short to be any use. Can you see an exponential in this? You need to give us much more data. Consider using pastebin and giving us the link.
– Hugh
Sep 7, 2017 at 20:28
• pastebin.com/M03RMCJk Sep 7, 2017 at 20:47
• The error message would suggest that a variable was mistakenly assigned to the actual data. As others have noted, there are perhaps other issues with this. Sep 8, 2017 at 15:02
• One gets more stable results (i.e., less dependent on starting values) if Exp[a + b t] is used rather than a Exp[b t].
– JimB
Sep 8, 2017 at 22:49