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I am trying to fit a set of stochastic data to a sigmoidal function, but it's not working. Can you please help?

Here is the function I am trying to fit to my data.

1/(Exp[-4*(1 - X1^-1)*t]*(((1 - X1^-1) - z0)/z0) + 1)

and the normalized data is given below.

{{0.000405699, 0.0330812}, {0.000405699, 0.0161187}, {0.0126174, 
  0.0317448}, {0.0126174, 0.0161246}, {0.024829, 
  0.0310185}, {0.024829, 0.0161306}, {0.0370407, 
  0.029316}, {0.0370407, 0.0161366}, {0.0492524, 
  0.0285898}, {0.0492524, 0.0161425}, {0.061464, 
  0.0283517}, {0.061464, 0.0161485}, {0.0736757, 
  0.0286017}, {0.0736757, 0.0170087}, {0.0858874, 
  0.0300721}, {0.0980991, 0.0327627}, {0.110311, 0.0366737}, {0.12354,
   0.0386327}, {0.119469, 0.0296003}, {0.135752, 
  0.0433674}, {0.132699, 0.0344881}, {0.147408, 0.0454782}, {0.141858,
   0.0388857}, {0.160175, 0.050447}, {0.156104, 0.0430418}, {0.170758,
   0.0602799}, {0.171369, 0.0468322}, {0.1789, 0.0705345}, {0.182563, 
  0.0544647}, {0.182563, 0.045056}, {0.187448, 0.0759569}, {0.187651, 
  0.0659992}, {0.192739, 0.0904569}, {0.195792, 0.0520915}, {0.19681, 
  0.0752781}, {0.202153, 0.0984058}, {0.203934, 0.084312}, {0.206308, 
  0.0598579}, {0.209022, 0.105548}, {0.215128, 0.113605}, {0.213092, 
  0.0935421}, {0.218035, 0.0680014}, {0.22259, 0.121988}, {0.223269, 
  0.104823}, {0.227339, 0.133787}, {0.228357, 0.0758305}, {0.234317, 
  0.120936}, {0.229374, 0.111416}, {0.236611, 0.143961}, {0.238024, 
  0.0842737}, {0.240569, 0.13058}, {0.242604, 0.155232}, {0.249218, 
  0.164967}, {0.249218, 0.140988}, {0.248913, 0.0921538}, {0.247692, 
  0.131194}, {0.259349, 0.175277}, {0.258886, 0.15048}, {0.256851, 
  0.141815}, {0.260107, 0.101004}, {0.265331, 0.159026}, {0.267841, 
  0.189047}, {0.268045, 0.114272}, {0.271708, 0.203034}, {0.270589, 
  0.173855}, {0.273133, 0.126447}, {0.276186, 0.214037}, {0.272115, 
  0.165253}, {0.279849, 0.226541}, {0.279239, 0.139263}, {0.284618, 
  0.237066}, {0.284327, 0.20199}, {0.280256, 0.183683}, {0.284666, 
  0.152201}, {0.288107, 0.246377}, {0.28738, 0.211937}, {0.284327, 
  0.191251}, {0.291111, 0.260081}, {0.291111, 0.226644}, {0.290885, 
  0.164082}, {0.295521, 0.273995}, {0.296248, 0.239164}, {0.293486, 
  0.176612}, {0.300609, 0.253252}, {0.294503, 0.214687}, {0.301016, 
  0.287177}, {0.300609, 0.186552}, {0.304389, 0.297429}, {0.304389, 
  0.262389}, {0.300609, 0.229333}, {0.30468, 0.195508}, {0.30875, 
  0.306767}, {0.30875, 0.278801}, {0.30875, 0.267411}, {0.305901, 
  0.240172}, {0.309913, 0.205071}, {0.312239, 0.314901}, {0.311076, 
  0.25371}, {0.313839, 0.21494}, {0.315874, 0.329011}, {0.316688, 
  0.293847}, {0.316891, 0.223337}, {0.320962, 0.280139}, {0.318113, 
  0.270931}, {0.315874, 0.263754}, {0.320671, 0.344507}, {0.321301, 
  0.232638}, {0.323902, 0.308666}, {0.325033, 0.359218}, {0.326305, 
  0.24463}, {0.327831, 0.322701}, {0.327649, 0.293047}, {0.330121, 
  0.373224}, {0.331138, 0.335809}, {0.327068, 0.282796}, {0.33063, 
  0.255158}, {0.33487, 0.384941}, {0.334802, 0.343572}, {0.33487, 
  0.306719}, {0.335209, 0.263763}, {0.338407, 0.397233}, {0.338407, 
  0.354871}, {0.340443, 0.318408}, {0.340297, 0.274993}, {0.34335, 
  0.409265}, {0.344028, 0.368065}, {0.344804, 0.329456}, {0.344368, 
  0.284635}, {0.347014, 0.418994}, {0.349165, 0.382072}, {0.349049, 
  0.340504}, {0.349456, 0.295865}, {0.351491, 0.429857}, {0.356071, 
  0.443435}, {0.356071, 0.398863}, {0.351491, 0.388976}, {0.356172, 
  0.353321}, {0.356172, 0.309975}, {0.358411, 0.454877}, {0.358411, 
  0.41036}, {0.362798, 0.369538}, {0.360446, 0.325133}, {0.364155, 
  0.467571}, {0.362889, 0.420759}, {0.365331, 0.335387}, {0.368678, 
  0.487424}, {0.363703, 0.47831}, {0.367483, 0.439715}, {0.365738, 
  0.42962}, {0.368995, 0.388912}, {0.36913, 0.378246}, {0.368904, 
  0.34847}, {0.370623, 0.4994}, {0.371844, 0.453968}, {0.373653, 
  0.402931}, {0.370826, 0.359332}, {0.37566, 0.370683}, {0.376423, 
  0.511722}, {0.376932, 0.466235}, {0.379476, 0.410956}, {0.379694, 
  0.529885}, {0.375915, 0.519319}, {0.380276, 0.479155}, {0.381568, 
  0.425031}, {0.379985, 0.389722}, {0.375915, 0.380202}, {0.3858, 
  0.549448}, {0.384056, 0.538726}, {0.3858, 0.497881}, {0.38202, 
  0.488569}, {0.387109, 0.439954}, {0.386498, 0.403784}, {0.38958, 
  0.561792}, {0.389754, 0.510832}, {0.392197, 0.451965}, {0.392875, 
  0.57595}, {0.392197, 0.521523}, {0.393941, 0.463681}, {0.391906, 
  0.426652}, {0.390161, 0.414378}, {0.394232, 0.585225}, {0.398642, 
  0.596332}, {0.397963, 0.538244}, {0.394232, 0.528114}, {0.399783, 
  0.478993}, {0.400338, 0.466613}, {0.396267, 0.439031}, {0.402082, 
  0.611902}, {0.398303, 0.6028}, {0.402373, 0.556216}, {0.398303, 
  0.545688}, {0.403391, 0.495316}, {0.401355, 0.453068}, {0.406104, 
  0.623793}, {0.40814, 0.568879}, {0.409904, 0.509377}, {0.410967, 
  0.63787}, {0.406444, 0.630627}, {0.410514, 0.580254}, {0.410223, 
  0.481366}, {0.407122, 0.469301}, {0.414178, 0.523291}, {0.413364, 
  0.65245}, {0.413364, 0.591678}, {0.414924, 0.49908}, {0.411532, 
  0.49078}, {0.418655, 0.536546}, {0.418316, 0.663337}, {0.417638, 
  0.602321}, {0.420284, 0.674419}, {0.422726, 0.547165}, {0.419877, 
  0.512018}, {0.423404, 0.684939}, {0.422726, 0.622162}, {0.421369, 
  0.611353}, {0.42447, 0.557626}, {0.424422, 0.525078}, {0.424761, 
  0.696167}, {0.425779, 0.634297}, {0.428606, 0.569133}, {0.428323, 
  0.539663}, {0.429849, 0.706908}, {0.429849, 0.643269}, {0.433581, 
  0.722002}, {0.432563, 0.655778}, {0.433807, 0.586383}, {0.434938, 
  0.576825}, {0.435277, 0.55974}, {0.431885, 0.547535}, {0.436159, 
  0.733441}, {0.435955, 0.665054}, {0.43833, 0.598247}, {0.439687, 
  0.744987}, {0.439415, 0.675892}, {0.436973, 0.569138}, {0.442315, 
  0.609411}, {0.442061, 0.75292}, {0.442265, 0.684387}, {0.445566, 
  0.766834}, {0.445405, 0.697798}, {0.445792, 0.624056}, {0.4443, 
  0.589203}, {0.441043, 0.580122}, {0.450406, 0.782505}, {0.450202, 
  0.712776}, {0.452803, 0.641795}, {0.448506, 0.60136}, {0.453255, 
  0.793929}, {0.454476, 0.723346}, {0.456744, 0.651362}, {0.451898, 
  0.612832}, {0.458489, 0.805123}, {0.457326, 0.734037}, {0.459361, 
  0.661222}, {0.455697, 0.625379}, {0.463771, 0.744633}, {0.463431, 
  0.672024}, {0.460379, 0.638707}, {0.465467, 0.81429}, {0.467841, 
  0.755008}, {0.467095, 0.683667}, {0.465467, 0.650425}, {0.470555, 
  0.823713}, {0.46852, 0.660311}, {0.474626, 0.690883}, {0.476661, 
  0.832868}, {0.476254, 0.767727}, {0.47605, 0.704173}, {0.473608, 
  0.672395}, {0.479714, 0.712668}, {0.476661, 0.680816}, {0.481749, 
  0.842267}, {0.482767, 0.782716}, {0.487855, 0.71975}, {0.481749, 
  0.689971}, {0.487855, 0.851862}, {0.487855, 0.794556}, {0.487855, 
  0.734089}, {0.48582, 0.705349}, {0.492739, 0.742402}, {0.487855, 
  0.714137}, {0.493961, 0.861506}, {0.494251, 0.806082}, {0.498031, 
  0.754852}, {0.493961, 0.725122}, {0.501423, 0.874087}, {0.500575, 
  0.817931}, {0.502509, 0.767447}, {0.505155, 0.829465}, {0.500066, 
  0.743064}, {0.507868, 0.774796}, {0.509952, 0.885899}, {0.509225, 
  0.838254}, {0.511057, 0.787514}, {0.506172, 0.757345}, {0.515535, 
  0.89632}, {0.51567, 0.848142}, {0.517706, 0.798719}, {0.519402, 
  0.858394}, {0.522047, 0.809778}, {0.523133, 0.90851}, {0.522862, 
  0.819882}, {0.525762, 0.868648}, {0.529578, 0.830332}, {0.531468, 
  0.922519}, {0.530596, 0.878754}, {0.534259, 0.838486}, {0.535684, 
  0.889764}, {0.534666, 0.848151}, {0.537719, 0.932232}, {0.541111, 
  0.898431}, {0.541111, 0.861089}, {0.543825, 0.941632}, {0.543486, 
  0.906486}, {0.543825, 0.869755}, {0.550948, 0.951609}, {0.549931, 
  0.916252}, {0.549931, 0.879276}, {0.553798, 0.888138}, {0.558072, 
  0.923944}, {0.560107, 0.895146}, {0.561634, 0.959823}, {0.567231, 
  0.933638}, {0.566213, 0.905765}, {0.570792, 0.968413}, {0.570284, 
  0.900154}, {0.573744, 0.91685}, {0.577814, 0.941746}, {0.576389, 
  0.905648}, {0.581704, 0.975819}, {0.582834, 0.924322}, {0.582495, 
  0.914072}, {0.58894, 0.95005}, {0.589619, 0.934284}, {0.590491, 
  0.984994}, {0.592332, 0.943364}, {0.590636, 0.922496}, {0.598777, 
  0.959012}, {0.600958, 0.993262}, {0.600304, 0.934417}, {0.603575, 
  0.944416}, {0.606919, 0.96851}, {0.610424, 1.00115}, {0.611716, 
  0.95239}, {0.615399, 0.976161}, {0.612007, 0.944106}, {0.6206, 
  1.00954}, {0.622387, 0.957803}, {0.624727, 0.983925}, {0.63236, 
  1.01995}, {0.631487, 0.994114}, {0.631342, 0.967838}, {0.640246, 
  1.00407}, {0.640161, 0.9778}, {0.643554, 1.02869}, {0.642536, 
  0.997571}, {0.649223, 1.00981}, {0.651695, 0.985616}, {0.654748, 
  1.00618}, {0.655765, 1.03427}, {0.660853, 1.01344}, {0.663906, 
  0.989039}, {0.667977, 1.03818}, {0.672047, 1.01491}, {0.676118, 
  0.993438}, {0.680189, 1.04225}, {0.685277, 1.02208}, {0.681206, 
  1.01492}, {0.68833, 0.996983}, {0.6924, 1.04592}, {0.697488, 
  1.02606}, {0.694436, 1.01883}, {0.700541, 1.00041}, {0.704612, 
  1.04971}, {0.708682, 1.02323}, {0.711445, 1.0317}, {0.712753, 
  1.00444}, {0.716824, 1.05399}, {0.723075, 1.03307}, {0.719877, 
  1.02665}, {0.724965, 1.00896}, {0.729035, 1.0579}, {0.735955, 
  1.03489}, {0.737176, 1.01238}, {0.741247, 1.06095}, {0.74689, 
  1.03829}, {0.749388, 1.01739}, {0.753459, 1.06377}, {0.760027, 
  1.04149}, {0.7616, 1.02138}, {0.76567, 1.06536}, {0.771776, 
  1.04416}, {0.773811, 1.02666}, {0.777882, 1.06732}, {0.783988, 
  1.04621}, {0.786023, 1.03246}, {0.790094, 1.06891}, {0.795737, 
  1.05007}, {0.798235, 1.03784}, {0.802305, 1.07148}, {0.80719, 
  1.04473}, {0.809138, 1.05608}, {0.814517, 1.07283}, {0.818442, 
  1.04347}, {0.820962, 1.05819}, {0.822319, 1.04842}, {0.826729, 
  1.07466}, {0.832834, 1.06405}, {0.833598, 1.05077}, {0.83894, 
  1.07613}, {0.845046, 1.06222}, {0.845046, 1.05124}, {0.851152, 
  1.07602}, {0.857258, 1.06414}, {0.856444, 1.05111}, {0.863364, 
  1.07529}, {0.869469, 1.06407}, {0.869469, 1.04906}, {0.876593, 
  1.07191}, {0.880954, 1.06055}, {0.881681, 1.04687}, {0.887787, 
  1.07473}, {0.893893, 1.0592}, {0.893893, 1.04602}, {0.900461, 
  1.07728}, {0.906104, 1.0592}, {0.906104, 1.04578}, {0.91221, 
  1.0805}, {0.918316, 1.05836}, {0.918316, 1.04286}, {0.924422, 
  1.09033}, {0.924422, 1.07483}, {0.930528, 1.058}, {0.930528, 
  1.04213}, {0.936634, 1.09339}, {0.936634, 1.07228}, {0.943466, 
  1.06027}, {0.942739, 1.04165}, {0.948845, 1.09364}, {0.949049, 
  1.06728}, {0.95546, 1.06133}, {0.954951, 1.0408}, {0.961057, 
  1.09364}, {0.967977, 1.06106}, {0.967163, 1.03873}, {0.973269, 
  1.09328}, {0.978912, 1.061}, {0.979374, 1.03849}, {0.98548, 
  1.09329}, {0.989551, 1.05379}, {0.991586, 1.0673}, {0.991586, 
  1.03801}, {0.997692, 1.09354}, {1.00176, 1.04949}, {1.0038, 
  1.06865}, {1.0038, 1.03741}, {1.0099, 1.09367}, {1.01397, 
  1.04381}, {1.01601, 1.07122}, {1.01674, 1.03614}, {1.02212, 
  1.09233}, {1.02822, 1.07147}, {1.02822, 1.03711}, {1.03433, 
  1.08965}, {1.04043, 1.0683}, {1.04043, 1.03602}, {1.04654, 
  1.08697}, {1.05264, 1.06574}, {1.05264, 1.03621}, {1.05875, 
  1.08283}, {1.06486, 1.06319}, {1.06486, 1.03512}, {1.07096, 
  1.08064}, {1.07707, 1.05953}, {1.07707, 1.03473}, {1.08317, 
  1.0765}, {1.08928, 1.05722}, {1.08826, 1.03577}, {1.09437, 
  1.03264}, {1.09539, 1.07431}, {1.10149, 1.05421}, {1.10556, 
  1.03249}, {1.1076, 1.07151}, {1.1137, 1.05125}, {1.11777, 
  1.0335}, {1.11981, 1.06846}, {1.12591, 1.04784}, {1.131, 
  1.04156}, {1.12999, 1.02665}, {1.13202, 1.067}, {1.14118, 
  1.04352}, {1.1422, 1.022}, {1.14423, 1.06481}, {1.15543, 
  1.04454}, {1.15441, 1.01796}, {1.15644, 1.06372}, {1.16662, 
  1.04597}, {1.16662, 1.01284}, {1.16866, 1.06177}, {1.17849, 
  1.05519}, {1.17883, 1.0392}, {1.17883, 1.00869}, {1.19104, 
  1.05483}, {1.19104, 1.03689}, {1.19104, 1.00382}, {1.20398, 
  1.05795}, {1.20326, 1.0347}, {1.20326, 0.998943}, {1.215, 
  1.05836}, {1.21547, 1.03361}, {1.21547, 0.994922}, {1.22611, 
  1.0551}, {1.22768, 1.03142}, {1.22768, 0.989192}, {1.23785, 
  1.05924}, {1.23785, 1.04582}, {1.23989, 1.02861}, {1.23989, 
  0.983463}, {1.25007, 1.05742}, {1.25007, 1.04066}, {1.2521, 
  1.02728}, {1.2521, 0.978709}, {1.26228, 1.05523}, {1.26228, 
  1.03453}, {1.26504, 1.02731}, {1.26431, 0.972858}, {1.27449, 
  1.05304}, {1.27673, 1.02893}, {1.27551, 1.02259}, {1.27653, 
  0.968096}, {1.2867, 1.05097}, {1.28874, 1.0225}, {1.28874, 
  0.962485}, {1.29891, 1.04902}, {1.30138, 1.0185}, {1.30095, 
  0.956889}, {1.31112, 1.04659}, {1.31214, 1.01571}, {1.31316, 
  0.95299}, {1.32334, 1.04379}, {1.32639, 1.01392}, {1.32537, 
  0.951166}, {1.33555, 1.0416}, {1.33758, 1.01207}, {1.33758, 
  0.948975}, {1.34776, 1.03831}, {1.34903, 1.01476}, {1.3498, 
  0.947761}, {1.35244, 1.00636}, {1.35997, 1.0349}, {1.36201, 
  1.01903}, {1.36201, 0.945814}, {1.36404, 1.0028}, {1.37218, 
  1.03271}, {1.37422, 1.02026}, {1.37422, 0.943746}, {1.37625, 
  0.99919}, {1.38439, 1.02717}, {1.38716, 1.02147}, {1.38643, 
  0.941555}, {1.38847, 0.995494}, {1.39864, 1.02446}, {1.39864, 
  0.940097}, {1.40068, 0.992571}, {1.41158, 1.02355}, {1.41085, 
  0.93937}, {1.41289, 0.991723}, {1.42307, 1.02294}, {1.42307, 
  0.938522}, {1.4251, 0.99356}, {1.43528, 1.02883}, {1.43528, 
  1.01809}, {1.43528, 0.93682}, {1.43731, 0.99503}, {1.44749, 
  1.03213}, {1.44749, 1.01639}, {1.44749, 0.935239}, {1.44952, 
  0.995768}, {1.4597, 1.03397}, {1.4597, 1.01347}, {1.4597, 
  0.935001}, {1.46174, 0.997727}, {1.47191, 1.03568}, {1.47191, 
  1.01201}, {1.47191, 0.934397}, {1.47395, 0.998953}, {1.48412, 
  1.03874}, {1.48412, 1.00933}, {1.48412, 0.933549}, {1.48616, 
  1.00042}, {1.49634, 1.04045}, {1.4968, 1.00444}, {1.49634, 
  0.932212}, {1.50855, 1.03708}, {1.50855, 1.00462}, {1.50855, 
  0.930998}, {1.52076, 1.03034}, {1.52076, 1.00288}, {1.52076, 
  0.928197}, {1.53297, 1.024}, {1.53297, 1.00215}, {1.53297, 
  0.92564}, {1.54518, 1.01766}, {1.546, 0.999476}, {1.54518, 
  0.923572}, {1.55739, 1.01083}, {1.55638, 1.00038}, {1.55739, 
  0.920771}, {1.5619, 0.995405}, {1.56961, 1.00522}, {1.56961, 
  0.917482}, {1.57368, 0.994242}, {1.58182, 1.00335}, {1.58182, 
  0.91578}, {1.58661, 0.990605}, {1.59403, 1.00475}, {1.59403, 
  0.913345}, {1.5981, 0.990466}, {1.60353, 0.988277}, {1.60624, 
  1.00671}, {1.60624, 0.913717}, {1.61382, 0.993538}, {1.61866, 
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share|improve this question
    
This does what you want: FindFit[data, {1/((Exp[-4 (1 - 1/X1) t] ((1 - 1/X1) - z0))/z0 + 1), X1 < 0, z0 > 0}, {X1, z0}, t]. You have to add a couple of constraints to FindFit to point it towards the region where the solution lies. –  Stephen Luttrell Mar 29 at 12:12

1 Answer 1

Your model seems to be problematic. With a standard step-shaped model: ´model = a/(1 + b*Exp[-c (x - d)])´ you immediately find a simple enough solution. Assuming that ´lst´ is your list try this:

 model = a/(1 + b*Exp[-c (x - d)]);
ff = FindFit[lst, model, {a, b, c, d}, x]

(*  {a -> 1.00153, b -> 0.437, c -> 13.2399, d -> 0.454211}  *)

This shows how it looks like:

    Show[{
  ListPlot[lst],
  Plot[model /. ff, {x, 0, 3}, PlotStyle -> {Red, Thickness[0.007]}, 
   PlotRange -> All]
  }]

It yields the following:
enter image description here

All these things are standard. They require no tricks.

share|improve this answer
    
thank you. I will check my equations again. –  nitin Mar 28 at 19:07

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