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Smoothing an irregular TimeSeries and calculate it's Derivative

Mathematica Asked by Revious on January 23, 2021

I’ve a data set consisting of two columns, the first column is a date and the second is a numeric cumulative amount. I tried to create a time series, resample it and smooth but I couldn’t manage to smooth it enough. I’m a noob with Mathematica. Can you help me to understand what I’m doing wrong?

enter image description here

data = Import[
    "https://1drv.ms/x/s!AiF0MVfYzFaAi3cO044kGlJW0Oam?e=Mt7qTv"][[1]];

data2 = data[[2 ;;, {1, 2}]] 

DateListPlot[data2]

ts = TimeSeries[data2, 
  ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}]

tr = TimeSeriesResample[ts, "Day", 
  ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}]

MovingMap[Median, tr, Quantity[3, "Months"]]

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

mm = 
 MovingMap[quotient[#BoundaryValues, #BoundaryTimes] &, 
  tr, {.1, Right}]


DateListPlot[mm]

One Answer

You might want to consider kernel regression. (It would be great if Mathematica would offer a function to do so. And it's likely that someone has already produced a package for this.)

Kernel regression is similar to using a "moving mean" but with weighting where each data point's influence decreases with distance from that data point.

Below is a very crude implementation of that for your data and your particular question. Note that I've converted the dates to the number of days since January 1, 1900 as I just don't want to deal with date objects.

data = {{42005, -107}, {42019, -429}, {42025, -459.5}, {42031, -496.1}, {42033, -573.14}, {42034, -625.14}, {42035, -650.53}, {42036, -655.53}, {42037, -675.53}, {42038, -680.53}, {42039, -685.53}, {42040, -790.53}, {42051, -798.33}, {42057, -805.93}, {42058, -815.69}, {42061, -820.56}, {42064, -900.56}, {42066, -924.56}, {42067, -954.96}, {42076, -960.16}, {42081, -969.56}, {42085, -993.08}, {42089, -997.16}, {42092, -1002.16}, {42093, -1007.16}, {42094, -1014.66}, {42095, -1027.79}, {42097, -1038.05}, {42099, -1043.05}, {42102, -1053.05}, {42103, -1058.14}, {42104, -1068.32}, {42106, -1078.45}, {42107, -1091.82}, {42108, -1097.05}, {42109, -1155.68}, {42110, -1171.28}, {42111, -1185.91}, {42114, -1195.6}, {42115, -1231.6}, {42116, -1247.16}, {42117, -1252.16}, {42118, -1262.16}, {42121, -1390.86}, {42122, -1405.86}, {42123, -1457.86}, {42124, -1478.93}, {42128, -1487.93}, {42130, -1497.9}, {42131, -1503.42}, {42132, -1508.42}, {42135, -1513.42}, {42136, -1570.41}, {42144, -1575.61}, {42146, -1596.96}, {42148, -1603.96}, {42149, -1661.72}, {42151, -1666.72}, {42153, -1679.48}, {42157, -1692.24}, {42159, -1697.24}, {42160, -1715}, {42163, -1745}, {42164, -1750.52}, {42165, -1763.28}, {42167, -1776.04}, {42170, -1781.04}, {42171, -1793.8}, {42172, -1873.8}, {42173, -1906.8}, {42179, -1918.8}, {42180, -1923.8}, {42181, -1953.8}, {42184, -1972.08}, {42185, -2034.15}, {42186, -2076.91}, {42187, -2086.91}, {42188, -2191.91}, {42190, -2209.67}, {42191, -2223.06}, {42193, -2228.1}, {42195, -2272.88}, {42198, -2282.76}, {42200, -2295.52}, {42201, -2303.98}, {42207, -2309.07}, {42209, -2324.11}, {42212, -2241.91}, {42213, -2270.91}, {42214, -2305.91}, {42215, -2310.91}, {42227, -2334.91}, {42228, -2351.41}, {42229, -2361.41}, {42234, -2411.86}, {42237, -2423.86}, {42240, -2458.86}, {42241, -2463.86}, {42243, -2398.17}, {42244, -2406.17}, {42247, -2447.17}, {42250, -2490.17}, {42251, -2495.17}, {42252, -2515.17}, {42255, -2550.17}, {42256, -2581.17}, {42258, -2592.17}, {42260, -2595.62}, {42261, -2614.07}, {42262, -2624.07}, {42263, -2657.52}, {42264, -2662.52}, {42265, -2667.01}, {42269, -2701.5}, {42274, -2623.75}, {42275, -2642.31}, {42276, -2656.31}, {42277, -2689.87}, {42279, -2704.87}, {42280, -2712.87}, {42281, -2716.87}, {42282, -2721.87}, {42283, -2731.87}, {42286, -2736.87}, {42290, -2767.56}, {42291, -2797.56}, {42292, -2812.05}, {42298, -2827.05}, {42300, -2831.54}, {42303, -2846.54}, {42304, -2760.11}, {42305, -2779.6}, {42306, -2830.6}, {42310, -2834.6}, {42311, -2839.6}, {42312, -2852.6}, {42314, -2916.6}, {42315, -3061.49}, {42319, -3072.98}, {42320, -3089.48}, {42325, -3104.48}, {42327, -3116.56}, {42328, -3146.56}, {42331, -3149.86}, {42332, -3164.86}, {42334, -3168.86}, {42335, -3131.68}, {42338, -3146.68}, {42339, -3155.68}, {42341, -3082.68}, {42345, -2967.45}, {42347, -2969.45}, {42348, -2999.45}, {42352, -3026.05}, {42353, -2996.05}, {42360, -2999.05}, {42361, -3009.05}, {42364, -3014.05}, {42365, -2877.05}, {42372, -2909.05}, {42374, -2896.75}, {42375, -2906.75}, {42376, -2929.25}, {42378, -2934.25}, {42381, -2929.25}, {42383, -2953.5}, {42388, -2956.85}, {42389, -2966.85}, {42390, -2981.85}, {42391, -3045.85}, {42394, -3052.85}, {42395, -3057.85}, {42396, -2963.85}, {42397, -2982.38}, {42401, -2988.38}, {42403, -3022.38}, {42404, -3042.38}, {42406, -3113.71}, {42408, -3143.71}, {42409, -3058.71}, {42410, -3066.71}, {42411, -3096.71}, {42412, -3042.71}, {42416, -3018.71}, {42417, -3008.91}, {42421, -3016.91}, {42423, -2978.67}, {42425, -2986.67}, {42426, -2891.55}, {42427, -2726.55}, {42429, -2730.55}, {42430, -2740.55}, {42431, -2748.55}, {42432, -2807.55}, {42435, -2827.55}, {42436, -2836.55}, {42438, -2874.55}, {42441, -2889.55}, {42444, -2944.55}, {42445, -2954.55}, {42446, -2992.55}, {42447, -2999.05}, {42449, -3014.05}, {42452, -3154.05}, {42456, -2900.05}, {42459, -2989.35}, {42462, -2998.15}, {42464, -3023.15}, {42465, -3031.15}, {42467, -3048.15}, {42471, -3156.15}, {42472, -3180.15}, {42475, -3164.15}, {42478, -3185.15}, {42479, -3188.15}, {42480, -3212.67}, {42481, -3283.17}, {42484, -3305.17}, {42486, -3321.17}, {42487, -3013.79}, {42489, -3021.79}, {42491, -3044.19}, {42492, -3063.19}, {42493, -3099.19}, {42494, -3089.99}, {42495, -3122.99}, {42496, -3139.99}, {42500, -3146.99}, {42501, -3149.99}, {42503, -3239.99}, {42505, -3200.99}, {42506, -3203.99}, {42507, -3223.99}, {42508, -3323.99}, {42509, -3349.99}, {42511, -3357.99}, {42513, -3388.99}, {42514, -3333.99}, {42515, -3371.99}, {42516, -3098.99}, {42517, -3137.79}, {42522, -3148.79}, {42524, -3172.79}, {42526, -3180.79}, {42528, -3188.79}, {42529, -3192.29}, {42536, -3209.54}, {42537, -3239.54}, {42538, -3217.54}, {42540, -3219.54}, {42541, -3249.54}, {42548, -2995.54}, {42549, -3035.54}, {42551, -3040.54}, {42552, -3086.54}, {42556, -3096.54}, {42558, -3129.54}, {42560, -3179.54}, {42562, -3356.54}, {42563, -3361.54}, {42566, -3429.04}, {42567, -3434.04}, {42570, -3449.04}, {42576, -3466.54}, {42577, -3501.54}, {42578, -3299.54}, {42579, -3179.68}, {42583, -3223.68}, {42584, -3232.68}, {42585, -3272.68}, {42586, -3208}, {42591, -3211.56}, {42592, -3229.06}, {42593, -3245.14}, {42597, -3253.14}, {42600, -3292.14}, {42601, -3312.14}, {42602, -3257.14}, {42605, -3307.14}, {42606, -3257.14}, {42608, -3274.14}, {42609, -3106}, {42611, -3149}, {42612, -3164}, {42613, -3222}, {42615, -3232}, {42618, -3240}, {42619, -3276.5}, {42620, -3166.5}, {42621, -3315.5}, {42622, -3326}, {42623, -3391}, {42626, -3494}, {42629, -3501}, {42634, -3373}, {42635, -3384}, {42636, -3399}, {42639, -3314}, {42640, -3202}, {42644, -3219}, {42645, -3234}, {42647, -3268}, {42650, -3279}, {42651, -3304}, {42653, -3327}, {42655, -3182.62}, {42657, -3200.62}, {42660, -3218.12}, {42662, -3227.12}, {42663, -3243.12}, {42664, -3279.12}, {42665, -3296.12}, {42667, -3339.12}, {42668, -3357.12}, {42669, -3252.12}, {42670, -3051.58}, {42675, -3057.58}, {42676, -3066.58}, {42677, -3081.18}, {42678, -3086.18}, {42682, -2976.67}, {42685, -2984.67}, {42691, -3001.67}, {42695, -3013.75}, {42696, -3022.75}, {42697, -2967.75}, {42698, -2982.75}, {42701, -2755.75}, {42702, -2783.21}, {42705, -2816.21}, {42706, -2834.71}, {42707, -2842.71}, {42710, -2757.71}, {42711, -2765.71}, {42719, -2773.71}, {42720, -2785.71}, {42726, -2793.71}, {42730, -2743.71}, {42731, -2455.71}, {42735, -2520.43}, {42736, -2528.43}, {42739, -2538.43}, {42740, -2548.43}, {42743, -2563.43}, {42751, -2508.43}, {42752, -2458.43}, {42760, -2355.67}, {42762, -2158.67}, {42767, -2198.67}, {42768, -2143.67}, {42769, -2153.67}, {42774, -2098.67}, {42775, -2113.67}, {42776, -2123.67}, {42780, -2138.67}, {42782, -2045.67}, {42790, -2060.67}, {42792, -2068.67}, {42793, -1709.67}, {42794, -1718.67}, {42795, -1750.67}, {42797, -1776.67}, {42800, -1786.67}, {42802, -1793.67}, {42807, -1733.67}, {42814, -1669.67}, {42815, -1599.67}, {42817, -1606.67}, {42818, -1506.67}, {42819, -1538.91}, {42821, -1412.91}, {42822, -1441.41}, {42825, -1413.41}, {42831, 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{43425, 9274.64}, {43426, 9268.64}, {43428, 9219.85}, {43431, 9202.46}, {43432, 9172.46}, {43433, 9163.28}, {43434, 9158.41}, {43438, 9116.41}, {43444, 9089.41}, {43445, 9015.41}, {43451, 8945.41}, {43453, 8931.91}, {43458, 9220.12}, {43461, 9160.12}, {43467, 9113.37}, {43472, 9053.37}, {43480, 8985.37}, {43483, 9042.19}, {43484, 9118.01}, {43487, 9714.51}, {43489, 9692.66}, {43493, 9632.66}, {43494, 9623.91}, {43498, 9577.91}, {43504, 9499.91}, {43507, 9499.91}, {43515, 9421.46}, {43517, 10287.46}, {43518, 10376.46}, {43519, 10354.46}, {43520, 10309.27}, {43523, 10239.27}, {43524, 10229.32}, {43526, 10183.82}, {43530, 10100.47}, {43537, 10090.77}, {43538, 10084.47}, {43539, 9993.47}, {43546, 11100.47}, {43549, 11140.41}, {43551, 11126.41}, {43553, 11111.41}, {43556, 11102.41}, {43557, 10966.41}, {43558, 10962.81}, {43564, 11062.81}, {43566, 11036.81}, {43567, 11030.81}, {43568, 11024.81}, {43570, 10947.81}, {43573, 10929.51}, {43574, 10924.01}, {43578, 12545.01}, {43579, 12518.21}, 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19903.95}, {43777, 19891.95}, {43780, 19867.73}, {43785, 19685.73}, {43787, 19670.73}, {43792, 20305.73}, {43794, 20260.73}, {43795, 20363.73}, {43796, 20172.73}, {43801, 20118.58}, {43802, 20023.6}, {43808, 20018.4}, {43809, 19959.9}, {43812, 20159.9}, {43814, 20052.7}, {43815, 20027.2}, {43816, 20024.4}, {43817, 20013.4}, {43818, 19518.42}, {43821, 19514.07}, {43822, 20365.07}, {43823, 20315.94}, {43833, 20235.94}, {43834, 20231.94}, {43835, 20189.94}, {43838, 20114.94}, {43839, 20099.94}, {43843, 20092.14}, {43845, 19842.14}, {43849, 19820.16}, {43850, 19773.22}, {43851, 19711.53}, {43852, 19671.53}, {43853, 19672.93}, {43854, 19437}, {43857, 20219}, {43859, 19950.6}, {43863, 19915.92}, {43864, 19975.92}, {43866, 19948.03}, {43877, 19888.23}, {43878, 19778.23}, {43880, 19763.25}, {43881, 19761.75}, {43883, 20217.75}, {43885, 20156.62}, {43888, 20595.4}, {43891, 20330.4}, {43894, 20253.86}, {43900, 20253.86}, {43901, 20205.16}, {43904, 20196.46}, {43905, 20018.6}, {43906, 20002.8}, {43908, 19985.2}, {43909, 19608.02}, {43911, 19567.72}, {43913, 20614.42}, {43914, 20569.42}, {43915, 20545.42}, {43916, 20509.12}, {43917, 20484.62}, {43919, 20452.02}, {43920, 20447.52}, {43921, 20207.52}, {43923, 20202.67}, {43924, 20418.67}, {43925, 20396.67}, {43927, 20360.07}, {43929, 20352.07}, {43934, 20326.57}, {43935, 20304.94}, {43936, 20296.94}, {43937, 20418.94}, {43941, 20364.88}, {43942, 20324.88}, {43943, 20904.88}, {43946, 20879.88}, {43949, 20911.88}, {43950, 20876.68}, {43951, 20848.48}, {43953, 20772.48}, {43955, 20493.48}, {43956, 20478.5}, {43959, 20455.5}, {43960, 20436.5}, {43963, 20418.5}, {43964, 20403.5}, {43967, 20382.5}, {43972, 20362.5}, {43973, 20753.5}, {43976, 20618.5}, {43979, 20636.5}, {43980, 20580.5}, {43983, 20558.5}, {43987, 20482.5}, {43990, 20340.5}, {43998, 20002.5}, {44004, 20544.9}, {44010, 20770.9}, {44017, 20750.2}, {44020, 20735.2}, {44022, 20711.2}, {44027, 20461.2}, {44029, 20431.2}, {44031, 20388}, {44032, 20372}, {44034, 20563.93}, {44035, 20548.95}, {44040, 20999.35}, {44041, 20939.35}, {44045, 20906.35}, {44046, 20729.35}, {44048, 20724.85}, {44052, 20678.85}, {44055, 20637.85}, {44060, 20600.85}, {44062, 20558.85}, {44063, 20421.85}, {44064, 20361.85}, {44067, 20269.85}, {44070, 20181.85}, {44071, 20942.39}, {44072, 20899.99}, {44074, 20649.99}, {44077, 20573.99}, {44081, 20535.99}, {44086, 20605.99}, {44090, 20715.99}, {44092, 20660.99}};

I also want to note that your data has some odd features (like jumps and serial correlation). Consider the following segment of data:

ListPlot[{data, data}, PlotRange -> {{43500, 43600}, {8000, 13000}}, 
 Joined -> {True, False}, PlotStyle -> {Blue, {Blue, PointSize[0.02]}}]

Oddities in data

(* Define a weighting function and the kernel estimator for predictions at date x *)
weights[x_, bandwidth_] := Exp[-(data[[All, 1]] - x)^2/bandwidth^2]
f[x_, bandwidth_] := Total[data[[All, 2]] weights[x, bandwidth]]/Total[weights[x, bandwidth]]

(* Try bandwidth = 100 *)
smoothed = Table[{x, f[x, 100]}, {x, Min[data[[All, 1]]], Max[data[[All, 1]]]}];
ListPlot[smoothed]

Smoothed data using kernel regression

Now find the derivative:

g = D[Total[data[[All, 2]] weights[x, h]]/Total[weights[x, h]], x];
Plot[g, {x, Min[data[[All, 1]]], Max[data[[All, 1]]]}]

Derivative of kernel regression fit

There are "automatic" choices for the bandwidth (try the R package np) but your mileage may vary. In short, choosing an optimal bandwidth is maybe more of an art as it will also depend on the subject matter and how the data was collected as opposed to depending on just the data values alone.

Correct answer by JimB on January 23, 2021

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