Approximate Signal Derivative

Use MovingMap to approximate the derivative of a signal coming from irregularly sampled continuous time series.

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ts = TimeSeries[ Table[{t, EllipticTheta[1, t, 0.3]}, {t, Join[{0.}, RandomReal[{0, 2 Pi}, 254], {2. Pi}]}]]

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RegularlySampledQ[ts]

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ListPlot[ts, PlotTheme -> "Detailed"]

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Use the values and the times at the boundaries of each sliding window to compute the difference quotients.

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quotient[values_, times_] := First[Differences[values]/Differences[times]]

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mm = MovingMap[quotient[#BoundaryValues, #BoundaryTimes] &, ts, {.01, Right}]

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Compare to the theoretical derivative.

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prime = D[EllipticTheta[1, t, 0.3], t]

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Show[Plot[prime, {t, 0, 2 \[Pi]}, PlotStyle -> Thick, PlotTheme -> "Detailed", PlotLegends -> None], ListPlot[mm, PlotStyle -> Red]]

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Use MovingMap with Line in place of the quotient function to create a plot of secant lines that approximate the original time series.

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line[yvals_, xvals_] := Line[Transpose[{xvals, yvals}]];

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lines = MovingMap[ line[#BoundaryValues, #BoundaryTimes] &, ts, {1.3, Right, {0, 2. \[Pi], .1}}];

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Graphics[{Black, lines["Values"]}]

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