# How to find the starting point of an oscillation (with noise)?

I've got a series of coordinates which start to oscillate from some time (and with noise), like this:

data={{0, 105}, {8, 104}, {9, 105}, {28, 106}, {29, 105}, {31, 106}, {32,
105}, {56, 106}, {57, 105}, {59, 106}, {60, 105}, {66, 106}, {67,
105}, {69, 106}, {70, 105}, {81, 106}, {82, 105}, {106, 106}, {107,
105}, {119, 104}, {120, 105}, {127, 106}, {128, 105}, {129,
106}, {130, 105}, {131, 106}, {132, 105}, {139, 106}, {140,
105}, {152, 106}, {154, 105}, {156, 106}, {157, 105}, {178,
106}, {179, 105}, {186, 106}, {187, 105}, {206, 106}, {207,
105}, {231, 106}, {232, 105}, {241, 106}, {242, 105}, {253,
106}, {254, 105}, {256, 106}, {257, 105}, {281, 106}, {282,
105}, {306, 106}, {307, 105}, {310, 104}, {311, 105}, {317,
106}, {318, 105}, {337, 106}, {338, 105}, {356, 106}, {357,
105}, {383, 104}, {384, 105}, {400, 106}, {401, 105}, {404,
106}, {405, 105}, {406, 106}, {407, 105}, {426, 106}, {427,
105}, {429, 106}, {430, 105}, {431, 106}, {432, 105}, {456,
106}, {457, 105}, {460, 106}, {461, 105}, {464, 106}, {465,
105}, {466, 106}, {467, 105}, {476, 106}, {477, 105}, {482,
106}, {483, 105}, {506, 106}, {507, 105}, {526, 106}, {527,
105}, {528, 106}, {529, 105}, {531, 106}, {532, 105}, {540,
104}, {541, 105}, {556, 106}, {557, 105}, {574, 106}, {575,
105}, {606, 106}, {607, 105}, {612, 106}, {613, 105}, {616,
106}, {617, 105}, {618, 106}, {619, 105}, {631, 106}, {632,
105}, {637, 106}, {638, 105}, {656, 106}, {657, 105}, {668,
106}, {669, 105}, {679, 104}, {680, 105}, {681, 106}, {682,
105}, {706, 106}, {707, 105}, {756, 106}, {757, 105}, {766,
106}, {767, 105}, {777, 106}, {778, 105}, {779, 106}, {780,
105}, {785, 106}, {786, 105}, {806, 106}, {807, 105}, {809,
106}, {810, 105}, {837, 106}, {838, 105}, {856, 106}, {857,
105}, {858, 104}, {859, 105}, {877, 106}, {878, 105}, {883,
104}, {884, 105}, {901, 106}, {902, 105}, {912, 106}, {913,
105}, {921, 106}, {922, 105}, {931, 106}, {932, 105}, {941,
106}, {942, 105}, {947, 106}, {948, 105}, {962, 106}, {963,
105}, {964, 106}, {965, 105}, {976, 106}, {977, 105}, {981,
106}, {982, 105}, {985, 106}, {987, 105}, {1045, 106}, {1046,
105}, {1059, 106}, {1060, 105}, {1062, 106}, {1063, 105}, {1073,
106}, {1074, 105}, {1091, 106}, {1092, 105}, {1106, 106}, {1107,
105}, {1109, 106}, {1110, 105}, {1117, 106}, {1118, 105}, {1154,
106}, {1155, 105}, {1156, 106}, {1157, 105}, {1168, 106}, {1169,
105}, {1170, 106}, {1171, 105}, {1172, 106}, {1173, 105}, {1182,
106}, {1183, 105}, {1189, 106}, {1190, 105}, {1197, 106}, {1198,
105}, {1206, 106}, {1207, 105}, {1235, 106}, {1236, 105}, {1252,
106}, {1253, 105}, {1256, 106}, {1257, 105}, {1259, 106}, {1260,
105}, {1268, 106}, {1269, 105}, {1272, 106}, {1273, 105}, {1297,
106}, {1298, 105}, {1324, 106}, {1325, 105}, {1331, 106}, {1332,
105}, {1337, 106}, {1338, 105}, {1358, 104}, {1359, 105}, {1387,
106}, {1388, 105}, {1406, 106}, {1407, 105}, {1433, 104}, {1434,
105}, {1437, 106}, {1438, 105}, {1499, 104}, {1500, 105}, {1504,
104}, {1505, 105}, {1508, 104}, {1509, 105}, {1512, 104}, {1513,
105}, {1523, 104}, {1524, 105}, {1533, 104}, {1534, 105}, {1535,
106}, {1536, 105}, {1558, 104}, {1559, 105}, {1581, 106}, {1582,
104}, {1583, 105}, {1589, 104}, {1590, 105}, {1593, 106}, {1594,
105}, {1609, 106}, {1610, 105}, {1612, 104}, {1613, 105}, {1626,
106}, {1627, 105}, {1631, 106}, {1632, 105}, {1634, 106}, {1635,
105}, {1643, 106}, {1644, 105}, {1656, 106}, {1657, 105}, {1660,
106}, {1661, 105}, {1662, 106}, {1663, 105}, {1667, 106}, {1669,
105}, {1670, 106}, {1671, 105}, {1673, 106}, {1674, 105}, {1681,
106}, {1682, 105}, {1687, 106}, {1688, 105}, {1690, 106}, {1691,
105}, {1693, 106}, {1694, 105}, {1695, 106}, {1699, 105}, {1700,
106}, {1704, 105}, {1705, 106}, {1706, 105}, {1709, 106}, {1714,
105}, {1715, 106}, {1727, 105}, {1729, 106}, {1730, 105}, {1731,
106}, {1732, 105}, {1733, 106}, {1738, 105}, {1739, 106}, {1754,
105}, {1756, 106}, {1783, 105}, {1784, 106}, {1810, 107}, {1811,
106}, {1818, 107}, {1819, 106}, {1831, 107}, {1832, 106}, {1837,
107}, {1838, 106}, {1843, 107}, {1844, 106}, {1852, 107}, {1853,
106}, {1856, 107}, {1857, 106}, {1860, 107}, {1861, 106}, {1873,
107}, {1874, 106}, {1876, 107}, {1878, 106}, {1881, 107}, {1882,
106}, {1887, 107}, {1888, 106}, {1889, 107}, {1890, 106}, {1892,
107}, {1897, 106}, {1900, 107}, {1901, 106}, {1906, 107}, {1907,
106}, {1909, 107}, {1912, 106}, {1920, 107}, {1923, 106}, {1924,
107}, {1925, 106}, {1926, 107}, {1927, 106}, {1931, 107}, {1932,
106}, {1954, 105}, {1955, 106}, {1958, 105}, {1959, 106}, {1979,
105}, {1980, 106}, {1982, 105}, {1984, 106}, {1985, 105}, {1987,
106}, {1988, 105}, {1996, 106}, {1997, 105}, {2008, 104}, {2009,
105}, {2018, 104}, {2019, 105}, {2026, 104}, {2028, 105}, {2030,
104}, {2031, 105}, {2032, 104}, {2034, 105}, {2038, 104}, {2040,
105}, {2041, 104}, {2042, 105}, {2043, 104}, {2049, 105}, {2050,
104}, {2079, 103}, {2081, 104}, {2082, 103}, {2083, 104}, {2087,
103}, {2088, 104}, {2089, 103}, {2093, 104}, {2094, 103}, {2096,
104}, {2097, 103}, {2099, 104}, {2100, 103}, {2101, 104}, {2102,
103}, {2158, 102}, {2159, 103}, {2179, 102}, {2181, 103}, {2183,
102}, {2184, 103}, {2186, 102}, {2187, 103}, {2188, 102}, {2189,
103}, {2191, 102}, {2192, 103}, {2204, 102}, {2205, 103}, {2207,
102}, {2208, 103}, {2210, 102}, {2212, 103}, {2213, 102}, {2215,
103}, {2218, 102}, {2219, 103}, {2224, 102}, {2226, 103}, {2228,
102}, {2229, 103}, {2233, 102}, {2234, 103}, {2241, 102}, {2242,
103}, {2255, 102}, {2256, 103}, {2309, 104}, {2310, 103}, {2311,
104}, {2318, 103}, {2319, 104}, {2331, 105}, {2332, 104}, {2334,
105}, {2338, 104}, {2339, 105}, {2346, 106}, {2347, 105}, {2352,
106}, {2353, 105}, {2356, 106}, {2357, 105}, {2359, 106}, {2367,
107}, {2368, 106}, {2373, 107}, {2411, 108}, {2426, 109}, {2427,
108}, {2429, 109}, {2440, 110}, {2441, 109}, {2444, 110}, {2446,
109}, {2448, 110}, {2449, 109}, {2450, 110}, {2451, 109}, {2453,
110}, {2455, 109}, {2456, 110}, {2488, 111}, {2489, 110}, {2492,
111}, {2493, 110}, {2494, 111}, {2501, 110}, {2502, 111}, {2503,
110}, {2504, 111}, {2508, 110}, {2509, 111}, {2512, 110}, {2513,
111}, {2553, 110}, {2554, 111}, {2562, 110}, {2563, 111}, {2565,
110}, {2566, 111}, {2567, 110}, {2568, 111}, {2571, 110}, {2573,
111}, {2574, 110}, {2607, 109}, {2608, 110}, {2612, 109}, {2624,
108}, {2625, 109}, {2628, 108}, {2629, 109}, {2630, 108}, {2631,
109}, {2632, 108}, {2634, 109}, {2635, 108}, {2641, 107}, {2642,
108}, {2643, 107}, {2645, 108}, {2646, 107}, {2667, 106}, {2677,
105}, {2682, 104}, {2683, 105}, {2686, 104}, {2696, 103}, {2698,
104}, {2700, 103}, {2719, 102}, {2720, 103}, {2721, 102}, {2722,
103}, {2724, 102}, {2727, 101}, {2731, 102}, {2732, 101}, {2739,
100}, {2740, 101}, {2741, 100}, {2761, 99}, {2762, 100}, {2763,
99}, {2764, 100}, {2766, 99}, {2770, 100}, {2771, 99}, {2778,
98}, {2779, 99}, {2780, 98}, {2781, 99}, {2782, 98}, {2794,
97}, {2795, 98}, {2796, 97}, {2804, 96}, {2805, 97}, {2808,
96}, {2809, 97}, {2810, 96}, {2812, 97}, {2813, 96}, {2815,
97}, {2816, 96}, {2818, 97}, {2819, 96}, {2898, 97}, {2899,
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109}, {3055, 110}, {3058, 109}, {3059, 110}, {3067, 111}, {3073,
112}, {3084, 113}, {3093, 114}, {3094, 113}, {3095, 114}, {3109,
115}, {3120, 116}, {3131, 117}, {3132, 116}, {3133, 117}, {3153,
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119}, {3224, 118}, {3233, 117}, {3234, 118}, {3236, 117}, {3257,
116}, {3258, 117}, {3260, 116}, {3270, 115}, {3271, 116}, {3272,
115}, {3279, 114}, {3280, 115}, {3281, 114}, {3291, 113}, {3292,
114}, {3294, 113}, {3302, 112}, {3304, 111}, {3305, 112}, {3307,
111}, {3311, 110}, {3321, 109}, {3328, 108}, {3332, 107}, {3342,
106}, {3345, 105}, {3346, 106}, {3349, 105}, {3352, 104}, {3356,
103}, {3357, 104}, {3358, 103}, {3366, 102}, {3371, 101}, {3372,
102}, {3374, 101}, {3377, 100}, {3387, 99}, {3393, 98}, {3396,
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96}, {3413, 95}, {3418, 94}, {3419, 95}, {3420, 94}, {3424,
93}, {3431, 92}, {3444, 91}, {3445, 92}, {3446, 91}, {3450,
90}, {3452, 91}, {3453, 90}, {3457, 89}, {3458, 90}, {3459,
89}, {3460, 90}, {3461, 89}, {3478, 88}, {3499, 87}, {3500,
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150}, {5708, 151}, {5711, 152}, {5713, 153}, {5719, 154}, {5723,
155}, {5724, 156}, {5727, 157}, {5733, 158}, {5734, 157}, {5735,
158}, {5737, 159}, {5742, 160}, {5754, 161}, {5758, 162}, {5760,
161}, {5761, 162}, {5770, 163}, {5771, 162}, {5773, 163}, {5774,
162}, {5777, 163}, {5779, 162}, {5780, 163}, {5781, 162}, {5783,
163}, {5785, 162}, {5786, 163}, {5787, 162}, {5796, 161}, {5801,
160}, {5802, 161}, {5804, 160}, {5805, 161}, {5806, 160}, {5815,
159}, {5820, 158}, {5823, 157}, {5831, 156}, {5833, 157}, {5834,
156}, {5835, 155}, {5839, 154}, {5840, 153}, {5847, 152}, {5850,
151}, {5853, 150}, {5857, 149}, {5858, 150}, {5859, 149}, {5860,
148}, {5863, 147}, {5865, 146}, {5870, 145}, {5872, 144}, {5874,
143}, {5875, 142}, {5881, 141}, {5882, 140}, {5883, 139}, {5888,
138}, {5890, 136}, {5893, 135}, {5896, 134}, {5899, 133}, {5900,
132}, {5903, 131}, {5905, 130}, {5906, 129}, {5908, 128}, {5911,
127}, {5912, 126}, {5914, 125}, {5917, 124}, {5918, 123}, {5919,
122}, {5921, 121}, {5924, 120}, {5925, 119}, {5926, 118}, {5928,
117}, {5931, 116}, {5932, 115}, {5934, 114}, {5936, 113}, {5938,
112}, {5939, 111}, {5941, 110}, {5943, 109}, {5944, 108}, {5945,
107}, {5948, 106}, {5949, 105}, {5950, 104}, {5952, 103}, {5954,
102}, {5955, 101}, {5957, 100}, {5959, 99}, {5960, 98}, {5962,
97}, {5963, 96}, {5965, 95}, {5966, 94}, {5968, 93}, {5969,
92}, {5971, 91}, {5972, 90}, {5974, 89}, {5976, 88}, {5977,
87}, {5979, 86}, {5980, 85}, {5983, 84}, {5984, 83}, {5985,
82}, {5988, 81}, {5989, 80}, {5991, 78}, {5994, 77}, {5996,
76}, {5997, 75}};

It's graphic is shown below:

And now I want to find its starting time of oscillation. Here starting time can be defined as the place where lines rises (green arrow), or the n-th peak (red arrow).

Actually, I've tried many ways but the existence of noise failed them. So is there any neat ideas?

Attachment in Wolfram Cloud: Another 12 data examples is deployed on Wolfram Cloud, you can try on those too. (A list of 12 matrices).

Edit: Here we make it clear that the starting point is the first peak.

The problem becomes: how to find the first peak of data with noise, even the amplitude of peak might be close to the amplitude of noise.

• Implicit in your question is that there is a model of your function that includes a HeavisideTheta (step) function. I would simply fit your entire dataset with a function of the form HeavisideTheta[x0] f[x] and read off x0. Otherwise there is no unique answer to your question. After all, when does a growing exponential "start"? May 26 '15 at 16:31
• @DavidG.Stork Yes, it is. But I don't explicitly know the form of f[x]. May 26 '15 at 16:39
• Then there is no solution. After all, someone might claim that the function that "starts" is of the form 0 for some fixed interval $dx1$ followed by some variations. Another person might claim that the function that "starts" is of the form 0 for some fixed interval $dx2$ followed by the same variations. Both fit your data well but with different "starting points." Which person is "right"? Clearly your problem is under-specified. May 26 '15 at 16:55

As David commented, there is no unique answer. But here's one possibility: take the Hilbert transform (filter) of the signal and then threshold.

q1 = data[[All, 2]];
ListPlot[Abs[HilbertFilter[q + RandomReal[{-.1, .1}, Length[q]], 2.2]]]

To show the resiliency to noise, I've added randomness to the data. Even for reasonably large noise, you can still find a pretty good threshold (a little larger than 1 in this case) that gives an answer somewhere near the 400th sample.

If you want to keep the horizontal axes intact, use

q1 = data[[All, 1]];
q2 = data[[All, 2]];
ListPlot[Transpose[{q1,
Abs[HilbertFilter[q2 + RandomReal[{-.1, .1}, Length[q2]], 2.2]]}]]
• Thank you very much! I'll try your method. But I don't know what Hilbert Filter is...I'm going to do some research. May 26 '15 at 17:06