0
$\begingroup$

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, 
  96}, {2915, 97}, {2919, 96}, {2920, 97}, {2925, 98}, {2927, 
  97}, {2928, 98}, {2932, 97}, {2933, 98}, {2937, 99}, {2938, 
  98}, {2940, 99}, {2951, 100}, {2953, 99}, {2955, 100}, {2971, 
  101}, {2981, 102}, {2988, 103}, {3001, 104}, {3003, 103}, {3004, 
  104}, {3012, 105}, {3019, 106}, {3025, 107}, {3040, 108}, {3046, 
  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, 
  118}, {3154, 117}, {3155, 118}, {3157, 117}, {3159, 118}, {3160, 
  117}, {3162, 118}, {3169, 119}, {3170, 118}, {3171, 119}, {3172, 
  118}, {3174, 119}, {3176, 118}, {3178, 119}, {3179, 118}, {3180, 
  119}, {3182, 118}, {3183, 119}, {3204, 118}, {3205, 119}, {3207, 
  118}, {3208, 119}, {3216, 118}, {3217, 119}, {3220, 118}, {3222, 
  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, 
  97}, {3398, 98}, {3399, 97}, {3401, 96}, {3402, 97}, {3403, 
  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, 
  88}, {3502, 87}, {3506, 88}, {3507, 87}, {3509, 88}, {3512, 
  87}, {3515, 88}, {3516, 87}, {3519, 88}, {3521, 87}, {3522, 
  88}, {3543, 89}, {3544, 88}, {3545, 89}, {3549, 88}, {3550, 
  89}, {3566, 90}, {3567, 89}, {3569, 90}, {3575, 91}, {3576, 
  90}, {3578, 91}, {3582, 90}, {3583, 91}, {3584, 92}, {3597, 
  93}, {3605, 94}, {3608, 95}, {3610, 94}, {3611, 95}, {3616, 
  96}, {3625, 97}, {3630, 98}, {3634, 99}, {3638, 100}, {3640, 
  99}, {3641, 100}, {3647, 101}, {3648, 100}, {3650, 101}, {3654, 
  102}, {3656, 103}, {3665, 104}, {3669, 105}, {3672, 106}, {3676, 
  107}, {3684, 108}, {3689, 109}, {3693, 110}, {3700, 111}, {3701, 
  110}, {3702, 111}, {3706, 112}, {3709, 113}, {3714, 114}, {3720, 
  115}, {3721, 114}, {3722, 115}, {3729, 116}, {3734, 117}, {3743, 
  118}, {3748, 119}, {3749, 118}, {3750, 119}, {3755, 120}, {3761, 
  121}, {3762, 120}, {3764, 121}, {3773, 122}, {3776, 123}, {3777, 
  122}, {3781, 123}, {3788, 124}, {3789, 123}, {3790, 124}, {3797, 
  125}, {3820, 126}, {3854, 125}, {3855, 126}, {3857, 125}, {3858, 
  126}, {3860, 125}, {3862, 126}, {3864, 125}, {3885, 124}, {3893, 
  123}, {3895, 124}, {3896, 123}, {3899, 122}, {3901, 123}, {3902, 
  122}, {3910, 121}, {3920, 120}, {3929, 119}, {3932, 118}, {3933, 
  119}, {3935, 118}, {3936, 119}, {3937, 118}, {3941, 117}, {3950, 
  116}, {3954, 115}, {3955, 116}, {3956, 115}, {3958, 114}, {3959, 
  115}, {3960, 114}, {3971, 113}, {3974, 112}, {3978, 111}, {3982, 
  110}, {3990, 109}, {3994, 108}, {3998, 107}, {4004, 106}, {4007, 
  105}, {4010, 104}, {4015, 103}, {4020, 102}, {4024, 101}, {4025, 
  102}, {4026, 101}, {4028, 100}, {4035, 99}, {4038, 98}, {4041, 
  97}, {4045, 96}, {4052, 95}, {4057, 94}, {4060, 93}, {4064, 
  92}, {4075, 91}, {4077, 90}, {4082, 89}, {4090, 88}, {4094, 
  87}, {4095, 88}, {4096, 87}, {4101, 86}, {4102, 87}, {4103, 
  86}, {4105, 85}, {4118, 84}, {4125, 83}, {4131, 82}, {4147, 
  81}, {4148, 82}, {4150, 81}, {4151, 82}, {4153, 81}, {4155, 
  82}, {4156, 81}, {4162, 80}, {4170, 81}, {4171, 80}, {4181, 
  81}, {4182, 80}, {4183, 81}, {4185, 80}, {4187, 81}, {4188, 
  80}, {4189, 81}, {4202, 82}, {4204, 81}, {4205, 82}, {4221, 
  83}, {4226, 84}, {4230, 85}, {4243, 86}, {4246, 87}, {4250, 
  88}, {4255, 89}, {4262, 90}, {4266, 91}, {4270, 92}, {4277, 
  93}, {4281, 94}, {4284, 95}, {4285, 94}, {4286, 95}, {4288, 
  96}, {4297, 97}, {4301, 98}, {4303, 99}, {4305, 100}, {4313, 
  101}, {4316, 102}, {4319, 103}, {4324, 104}, {4327, 105}, {4329, 
  106}, {4332, 107}, {4338, 108}, {4341, 109}, {4343, 110}, {4348, 
  111}, {4351, 112}, {4353, 113}, {4355, 114}, {4362, 115}, {4365, 
  116}, {4368, 117}, {4373, 118}, {4375, 119}, {4378, 120}, {4381, 
  121}, {4387, 122}, {4389, 123}, {4393, 124}, {4395, 125}, {4402, 
  126}, {4405, 127}, {4408, 128}, {4416, 129}, {4421, 130}, {4424, 
  131}, {4428, 132}, {4437, 133}, {4441, 134}, {4442, 133}, {4444, 
  134}, {4448, 135}, {4459, 136}, {4460, 135}, {4461, 136}, {4470, 
  137}, {4471, 136}, {4472, 137}, {4473, 136}, {4474, 137}, {4491, 
  138}, {4493, 137}, {4498, 138}, {4499, 137}, {4513, 136}, {4521, 
  135}, {4533, 134}, {4538, 133}, {4542, 132}, {4543, 133}, {4544, 
  132}, {4550, 131}, {4554, 130}, {4558, 129}, {4560, 128}, {4567, 
  127}, {4569, 126}, {4572, 125}, {4573, 126}, {4574, 125}, {4578, 
  124}, {4581, 123}, {4583, 122}, {4587, 121}, {4591, 120}, {4594, 
  119}, {4597, 118}, {4598, 117}, {4603, 116}, {4606, 115}, {4609, 
  114}, {4614, 113}, {4617, 112}, {4619, 111}, {4622, 110}, {4626, 
  109}, {4629, 108}, {4630, 109}, {4631, 108}, {4632, 107}, {4637, 
  106}, {4639, 105}, {4643, 104}, {4645, 103}, {4649, 102}, {4650, 
  103}, {4651, 102}, {4653, 101}, {4656, 100}, {4661, 99}, {4664, 
  98}, {4666, 97}, {4669, 96}, {4673, 95}, {4678, 94}, {4679, 
  93}, {4682, 92}, {4687, 91}, {4689, 90}, {4692, 89}, {4697, 
  88}, {4701, 87}, {4702, 86}, {4705, 85}, {4711, 84}, {4713, 
  83}, {4716, 82}, {4723, 81}, {4726, 80}, {4727, 81}, {4728, 
  80}, {4731, 79}, {4735, 78}, {4742, 77}, {4747, 76}, {4753, 
  75}, {4764, 74}, {4767, 75}, {4768, 74}, {4776, 73}, {4785, 
  72}, {4786, 73}, {4787, 72}, {4791, 73}, {4792, 72}, {4819, 
  73}, {4820, 72}, {4821, 73}, {4824, 72}, {4825, 73}, {4831, 
  74}, {4832, 73}, {4833, 74}, {4835, 73}, {4836, 74}, {4842, 
  75}, {4844, 74}, {4845, 75}, {4855, 76}, {4860, 77}, {4865, 
  78}, {4866, 77}, {4867, 78}, {4875, 79}, {4877, 80}, {4880, 
  81}, {4883, 82}, {4884, 81}, {4885, 82}, {4892, 83}, {4894, 
  84}, {4895, 83}, {4896, 84}, {4897, 85}, {4902, 86}, {4905, 
  87}, {4908, 88}, {4911, 89}, {4914, 90}, {4917, 91}, {4920, 
  92}, {4924, 93}, {4927, 94}, {4929, 95}, {4930, 96}, {4936, 
  97}, {4938, 98}, {4940, 99}, {4942, 100}, {4947, 101}, {4949, 
  102}, {4952, 103}, {4955, 104}, {4958, 105}, {4959, 106}, {4962, 
  107}, {4967, 108}, {4969, 109}, {4972, 110}, {4975, 111}, {4978, 
  112}, {4980, 113}, {4983, 114}, {4988, 115}, {4991, 116}, {4993, 
  117}, {4998, 118}, {5000, 119}, {5002, 120}, {5005, 121}, {5009, 
  122}, {5013, 123}, {5014, 124}, {5017, 125}, {5023, 126}, {5026, 
  127}, {5027, 128}, {5033, 129}, {5034, 130}, {5037, 131}, {5040, 
  132}, {5045, 133}, {5048, 134}, {5051, 135}, {5056, 136}, {5058, 
  137}, {5060, 136}, {5061, 137}, {5062, 138}, {5065, 139}, {5072, 
  140}, {5077, 141}, {5079, 142}, {5088, 143}, {5094, 144}, {5100, 
  145}, {5108, 146}, {5110, 145}, {5111, 146}, {5160, 145}, {5166, 
  144}, {5174, 143}, {5181, 142}, {5188, 141}, {5193, 140}, {5196, 
  139}, {5203, 138}, {5207, 137}, {5210, 136}, {5213, 135}, {5218, 
  134}, {5221, 133}, {5223, 132}, {5228, 131}, {5230, 130}, {5233, 
  129}, {5235, 128}, {5239, 127}, {5241, 126}, {5243, 125}, {5247, 
  124}, {5249, 123}, {5251, 122}, {5253, 121}, {5256, 120}, {5257, 
  119}, {5260, 118}, {5262, 117}, {5264, 116}, {5266, 115}, {5268, 
  114}, {5272, 113}, {5274, 112}, {5275, 111}, {5277, 110}, {5281, 
  109}, {5282, 108}, {5284, 107}, {5287, 106}, {5288, 105}, {5291, 
  104}, {5292, 103}, {5296, 102}, {5297, 101}, {5299, 100}, {5303, 
  99}, {5304, 98}, {5307, 97}, {5309, 96}, {5313, 95}, {5315, 
  94}, {5316, 93}, {5318, 92}, {5321, 91}, {5323, 90}, {5325, 
  89}, {5329, 88}, {5331, 87}, {5333, 86}, {5335, 85}, {5339, 
  84}, {5341, 83}, {5343, 82}, {5347, 81}, {5350, 80}, {5352, 
  79}, {5355, 78}, {5359, 77}, {5360, 76}, {5363, 75}, {5368, 
  74}, {5370, 73}, {5372, 72}, {5374, 71}, {5379, 70}, {5382, 
  69}, {5385, 68}, {5386, 67}, {5387, 68}, {5388, 67}, {5391, 
  66}, {5395, 65}, {5397, 64}, {5403, 63}, {5406, 62}, {5407, 
  61}, {5408, 62}, {5410, 61}, {5411, 60}, {5412, 61}, {5413, 
  60}, {5420, 59}, {5426, 58}, {5432, 57}, {5433, 58}, {5434, 
  57}, {5445, 56}, {5446, 57}, {5447, 56}, {5448, 57}, {5449, 
  56}, {5458, 57}, {5459, 56}, {5462, 57}, {5463, 56}, {5472, 
  57}, {5486, 58}, {5492, 59}, {5494, 58}, {5495, 59}, {5498, 
  60}, {5505, 61}, {5508, 62}, {5511, 63}, {5515, 64}, {5520, 
  65}, {5523, 66}, {5525, 67}, {5530, 68}, {5533, 69}, {5534, 
  70}, {5536, 71}, {5541, 72}, {5543, 73}, {5545, 74}, {5548, 
  75}, {5552, 76}, {5553, 77}, {5555, 78}, {5558, 79}, {5561, 
  80}, {5562, 81}, {5564, 82}, {5568, 83}, {5569, 84}, {5571, 
  85}, {5575, 86}, {5576, 87}, {5578, 88}, {5579, 89}, {5581, 
  90}, {5584, 91}, {5585, 92}, {5588, 93}, {5589, 94}, {5591, 
  95}, {5592, 96}, {5595, 97}, {5597, 98}, {5598, 99}, {5600, 
  100}, {5603, 101}, {5604, 102}, {5605, 103}, {5608, 104}, {5609, 
  105}, {5611, 106}, {5612, 107}, {5615, 108}, {5616, 109}, {5618, 
  110}, {5620, 111}, {5622, 112}, {5623, 113}, {5625, 114}, {5628, 
  115}, {5629, 116}, {5630, 117}, {5633, 118}, {5634, 119}, {5636, 
  120}, {5638, 121}, {5640, 122}, {5642, 123}, {5643, 124}, {5645, 
  125}, {5648, 126}, {5649, 127}, {5651, 128}, {5654, 129}, {5655, 
  130}, {5657, 131}, {5658, 132}, {5661, 133}, {5664, 134}, {5665, 
  135}, {5669, 136}, {5670, 137}, {5672, 138}, {5674, 139}, {5679, 
  140}, {5680, 141}, {5682, 142}, {5685, 143}, {5688, 144}, {5690, 
  145}, {5692, 146}, {5696, 147}, {5698, 148}, {5701, 149}, {5703, 
  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: Plot of data

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).

Plot of data

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.

$\endgroup$
  • $\begingroup$ 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"? $\endgroup$ – David G. Stork May 26 '15 at 16:31
  • $\begingroup$ @DavidG.Stork Yes, it is. But I don't explicitly know the form of f[x]. $\endgroup$ – Louis Yu May 26 '15 at 16:39
  • 2
    $\begingroup$ 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. $\endgroup$ – David G. Stork May 26 '15 at 16:55
4
$\begingroup$

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]]]

enter image description here

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]]}]]
$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Louis Yu May 26 '15 at 17:06

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