0
$\begingroup$

I want to fit a data set:

data = {{1, 2.925*10^-6}, {2, 1.20625*10^-6}, {3, 6.78125*10^-7}, {4, 5.*10^-7}, {5,3.875*10^-7}, {6, 2.4375*10^-7}, {7, 2.*10^-7}, {8, 2.3125*10^-7}, {9, 2.*10^-7}, {10, 1.96875*10^-7}, {11, 1.625*10^-7}, {12, 1.25*10^-7}, {13, 1.40625*10^-7}, {14, 1.90625*10^-7}, {15, 1.71875*10^-7}, {16, 1.3125*10^-7}, {17, 1.5*10^-7}, {18, 1.84375*10^-7}, {19, 1.53125*10^-7}, {20, 1.28125*10^-7}, {21, 1.625*10^-7}, {22, 1.96875*10^-7}, {23, 1.8125*10^-7}, {24, 2.1875*10^-7}, {25, 1.8125*10^-7}, {26, 2.25*10^-7}, {27, 1.9375*10^-7}, {28, 1.8125*10^-7}, {29, 2.5*10^-7}, {30, 2.375*10^-7}, {31, 2.25*10^-7}, {32, 3.46875*10^-7}, {33, 3.21875*10^-7}, {34, 2.8125*10^-7}, {35, 3.4375*10^-7}, {36, 3.5*10^-7}, {37, 3.5*10^-7}, {38, 3.625*10^-7}, {39, 4.8125*10^-7}, {40, 4.125*10^-7}, {41, 3.90625*10^-7}, {42, 4.625*10^-7}, {43, 4.875*10^-7}, {44, 4.5*10^-7}, {45, 4.40625*10^-7}, {46, 5.46875*10^-7}, {47, 5.09375*10^-7}, {48, 4.90625*10^-7}, {49, 5.71875*10^-7}, {50, 5.34375*10^-7}, {51, 5.125*10^-7}, {52, 5.90625*10^-7}, {53, 5.34375*10^-7}, {54, 6.71875*10^-7}, {55, 6.0625*10^-7}, {56, 6.84375*10^-7}, {57, 6.9375*10^-7}, {58, 7.03125*10^-7}, {59, 7.78125*10^-7}, {60, 7.59375*10^-7}, {61, 7.3125*10^-7}, {62, 7.5625*10^-7}, {63, 7.3125*10^-7}, {64, 8.5625*10^-7}, {65, 8.59375*10^-7}, {66, 7.03125*10^-7}, {67, 7.40625*10^-7}, {68, 7.8125*10^-7}, {69, 7.65625*10^-7}, {70, 8.96875*10^-7}, {71, 8.875*10^-7}, {72, 8.875*10^-7}, {73, 7.65625*10^-7}, {74, 9.375*10^-7}, {75, 8.0625*10^-7}, {76, 8.96875*10^-7}, {77, 8.5625*10^-7}, {78, 8.1875*10^-7}, {79, 7.875*10^-7}, {80, 9.03125*10^-7}, {81, 8.84375*10^-7}, {82, 9.1875*10^-7}, {83, 9.09375*10^-7}, {84, 8.90625*10^-7}, {85, 8.71875*10^-7}, {86, 1.00625*10^-6}, {87, 9.28125*10^-7}, {88, 8.625*10^-7}, {89, 9.78125*10^-7}, {90, 8.53125*10^-7}, {91, 9.5*10^-7}, {92, 9.46875*10^-7}, {93, 9.875*10^-7}, {94, 9.375*10^-7}, {95, 9.46875*10^-7}, {96, 9.71875*10^-7}, {97, 9.15625*10^-7}, {98, 8.75*10^-7}, {99, 9.875*10^-7}, {100, 9.8125*10^-7}, {101, 8.5625*10^-7}, {102, 9.4375*10^-7}, {103, 8.78125*10^-7}, {104, 9.53125*10^-7}, {105, 9.71875*10^-7}, {106, 9.46875*10^-7}, {107, 9.34375*10^-7}, {108, 9.84375*10^-7}, {109, 8.75*10^-7}, {110, 1.01875*10^-6}, {111, 9.59375*10^-7}, {112, 9.03125*10^-7}, {113, 8.21875*10^-7}, {114, 8.3125*10^-7}, {115, 9.1875*10^-7}, {116, 1.02813*10^-6}, {117, 9.9375*10^-7}, {118, 9.625*10^-7}, {119, 8.9375*10^-7}, {120, 9.875*10^-7}, {121, 9.46875*10^-7}, {122, 8.5625*10^-7}, {123, 8.46875*10^-7}, {124, 9.09375*10^-7}, {125, 9.46875*10^-7}, {126, 1.00313*10^-6}, {127, 9.90625*10^-7}, {128, 8.1875*10^-7}, {129, 8.875*10^-7}, {130, 9.3125*10^-7}, {131, 9.125*10^-7}, {132, 9.3125*10^-7}, {133, 9.09375*10^-7}, {134, 9.5625*10^-7}, {135, 7.75*10^-7}, {136, 9.28125*10^-7}, {137, 9.3125*10^-7}, {138, 8.96875*10^-7}, {139, 8.96875*10^-7}, {140, 9.59375*10^-7}, {141, 8.75*10^-7}, {142, 9.5625*10^-7}, {143, 8.*10^-7}, {144, 8.75*10^-7}, {145, 8.5625*10^-7}, {146, 7.46875*10^-7}, {147, 8.1875*10^-7}, {148, 9.3125*10^-7}, {149, 8.*10^-7}, {150, 7.53125*10^-7}, {151, 8.21875*10^-7}, {152, 8.0625*10^-7}, {153, 8.40625*10^-7}, {154, 8.875*10^-7}, {155, 7.96875*10^-7}, {156, 8.46875*10^-7}, {157, 8.125*10^-7}, {158, 7.6875*10^-7}, {159, 9.*10^-7}, {160, 7.90625*10^-7}, {161, 8.28125*10^-7}, {162, 8.21875*10^-7}, {163, 8.15625*10^-7}, {164, 7.9375*10^-7}, {165, 8.625*10^-7}, {166, 7.5625*10^-7}, {167, 8.75*10^-7}, {168, 7.4375*10^-7}, {169, 7.875*10^-7}, {170, 7.6875*10^-7}, {171, 7.78125*10^-7}, {172, 7.78125*10^-7}, {173, 8.3125*10^-7}, {174, 8.09375*10^-7}, {175, 8.53125*10^-7}, {176, 7.8125*10^-7}, {177, 7.90625*10^-7}, {178, 7.6875*10^-7}, {179, 7.0625*10^-7}, {180, 7.03125*10^-7}, {181, 8.*10^-7}, {182, 7.53125*10^-7}, {183, 6.875*10^-7}, {184, 8.25*10^-7}, {185, 7.09375*10^-7}, {186, 7.75*10^-7}, {187, 7.3125*10^-7}, {188, 7.53125*10^-7}, {189, 6.8125*10^-7}, {190, 7.6875*10^-7}, {191, 6.8125*10^-7}, {192, 7.15625*10^-7}, {193, 5.8125*10^-7}, {194, 7.34375*10^-7}, {195, 7.34375*10^-7}, {196, 7.15625*10^-7}, {197, 7.4375*10^-7}, {198, 7.125*10^-7}, {199, 7.71875*10^-7}, {200, 5.9375*10^-7}, {201, 6.90625*10^-7}, {202, 7.6875*10^-7}, {203, 6.28125*10^-7}, {204, 6.09375*10^-7}, {205, 6.40625*10^-7}, {206, 6.96875*10^-7}, {207, 6.4375*10^-7}, {208, 7.75*10^-7}, {209, 6.5625*10^-7}, {210, 5.96875*10^-7}, {211, 6.46875*10^-7}, {212, 6.96875*10^-7}, {213, 7.*10^-7}, {214, 6.59375*10^-7}, {215, 5.4375*10^-7}, {216, 6.78125*10^-7}, {217, 5.65625*10^-7}, {218, 6.5625*10^-7}, {219, 5.96875*10^-7}, {220, 6.28125*10^-7}, {221, 6.34375*10^-7}, {222, 5.4375*10^-7}, {223, 5.96875*10^-7}, {224, 5.*10^-7}, {225, 5.75*10^-7}, {226, 6.1875*10^-7}, {227, 6.40625*10^-7}, {228, 5.53125*10^-7}, {229, 6.21875*10^-7}, {230, 5.34375*10^-7}, {231, 5.6875*10^-7}, {232, 4.71875*10^-7}, {233, 6.21875*10^-7}, {234, 6.0625*10^-7}, {235, 5.75*10^-7}, {236, 5.65625*10^-7}, {237, 5.40625*10^-7}, {238, 6.*10^-7}, {239, 6.75*10^-7}, {240, 6.*10^-7}, {241, 6.0625*10^-7}, {242, 6.03125*10^-7}, {243, 5.71875*10^-7}, {244, 5.1875*10^-7}, {245, 5.90625*10^-7}, {246, 5.21875*10^-7}, {247, 5.5*10^-7}, {248, 5.28125*10^-7}, {249, 5.34375*10^-7}, {250, 5.75*10^-7}, {251, 5.21875*10^-7}, {252, 5.40625*10^-7}, {253, 5.90625*10^-7}, {254, 5.34375*10^-7}, {255, 5.1875*10^-7}, {256, 5.71875*10^-7}, {257, 5.34375*10^-7}, {258, 5.375*10^-7}, {259, 4.40625*10^-7}, {260, 5.*10^-7}, {261, 4.875*10^-7}, {262, 5.1875*10^-7}, {263, 5.46875*10^-7}, {264, 5.21875*10^-7}, {265, 4.53125*10^-7}, {266, 4.53125*10^-7}, {267, 4.4375*10^-7}, {268, 5.40625*10^-7}, {269, 4.625*10^-7}, {270, 5.40625*10^-7}, {271, 4.6875*10^-7}, {272, 4.65625*10^-7}, {273, 4.46875*10^-7}, {274, 4.96875*10^-7}, {275, 4.53125*10^-7}, {279, 4.65625*10^-7}, {280, 4.3125*10^-7}, {281, 3.78125*10^-7}, {282, 4.40625*10^-7}, {283, 4.25*10^-7}, {284, 3.84375*10^-7}, {285, 4.125*10^-7}, {286, 3.84375*10^-7}, {287, 4.1875*10^-7}, {288, 4.15625*10^-7}, {289, 3.90625*10^-7}, {290, 4.21875*10^-7}, {291, 4.53125*10^-7}, {292, 4.15625*10^-7}, {293, 4.6875*10^-7}, {294, 4.34375*10^-7}, {295, 4.03125*10^-7}, {296, 4.3125*10^-7}, {297, 4.25*10^-7}, {298, 3.65625*10^-7}, {299, 4.28125*10^-7}, {303, 4.28125*10^-7}, {304, 4.15625*10^-7}, {305, 3.*10^-7}, {306, 3.8125*10^-7}, {307, 3.28125*10^-7}, {308, 4.21875*10^-7}, {309, 3.3125*10^-7}, {310, 3.71875*10^-7}, {311, 3.96875*10^-7}, {312, 3.875*10^-7}, {313, 3.34375*10^-7}, {314, 3.25*10^-7}, {318, 3.1875*10^-7}, {319, 3.53125*10^-7}, {320, 3.65625*10^-7}, {321, 3.15625*10^-7}, {322, 3.1875*10^-7}, {323, 3.78125*10^-7}, {324, 3.4375*10^-7}, {325, 3.375*10^-7}, {326, 2.90625*10^-7}, {327, 3.21875*10^-7}, {328, 2.8125*10^-7}, {329,3.4375*10^-7}, {330, 3.21875*10^-7}, {331, 2.6875*10^-7}, {332, 3.25*10^-7}, {333, 3.*10^-7}, {334, 3.0625*10^-7}, {335, 3.53125*10^-7}, {336, 2.75*10^-7}, {337, 3.375*10^-7}, {338, 3.5625*10^-7}, {339, 2.4375*10^-7}, {340, 3.*10^-7}, {341, 2.96875*10^-7}, {345, 2.375*10^-7}, {346, 2.59375*10^-7}, {347, 2.25*10^-7}, {348, 2.46875*10^-7}, {349, 2.4375*10^-7}, {350, 2.96875*10^-7}, {351, 2.875*10^-7}, {352, 2.59375*10^-7}, {353, 2.625*10^-7}, {354, 2.78125*10^-7}, {355, 2.84375*10^-7}, {356, 2.875*10^-7}, {357, 2.875*10^-7}, {358, 2.53125*10^-7}, {359, 3.1875*10^-7}, {360, 3.03125*10^-7}, {361, 2.78125*10^-7}, {362, 2.6875*10^-7}, {363, 2.21875*10^-7}, {364, 2.53125*10^-7}, {369, 2.46875*10^-7}, {370, 1.84375*10^-7}, {371, 2.125*10^-7}, {372, 2.1875*10^-7}, {373, 1.9375*10^-7}, {374, 2.46875*10^-7}, {375, 2.875*10^-7}, {376, 2.59375*10^-7}, {377, 2.21875*10^-7}, {378, 2.5625*10^-7}, {379, 2.375*10^-7}, {380, 1.84375*10^-7}, {381, 1.9375*10^-7}, {382, 2.875*10^-7}, {383, 1.75*10^-7}, {384, 2.0625*10^-7}, {385, 2.21875*10^-7}, {386, 2.0625*10^-7}, {387, 2.03125*10^-7}, {388, 1.90625*10^-7}, {389, 2.46875*10^-7}, {390, 2.25*10^-7}, {391, 1.5*10^-7}, {392, 2.5625*10^-7}, {393, 1.6875*10^-7}, {394, 2.15625*10^-7}, {395,1.90625*10^-7}, {396, 2.0625*10^-7}, {397, 2.03125*10^-7}, {398, 2.1875*10^-7}, {399, 1.78125*10^-7}, {400, 1.9375*10^-7}, {401, 2.125*10^-7}, {402, 1.9375*10^-7}, {403, 1.59375*10^-7}, {404, 2.09375*10^-7}, {405, 2.03125*10^-7}, {406, 2.0625*10^-7}, {407, 1.9375*10^-7}, {408, 1.96875*10^-7}, {409, 1.46875*10^-7}, {410, 1.875*10^-7}, {411, 1.875*10^-7}, {412, 1.75*10^-7}, {413, 1.5*10^-7}, {414, 1.4375*10^-7}, {415, 1.40625*10^-7}, {416, 1.84375*10^-7}, {417, 1.71875*10^-7}, {418, 1.28125*10^-7}, {419, 1.4375*10^-7}, {420, 1.71875*10^-7}, {421, 1.625*10^-7}, {422, 1.75*10^-7}, {423, 1.78125*10^-7}, {424, 1.8125*10^-7}, {425, 1.71875*10^-7}, {426, 1.3125*10^-7}, {427, 1.65625*10^-7}, {428, 1.53125*10^-7}, {429, 1.3125*10^-7}, {430, 1.03125*10^-7}, {431, 1.5625*10^-7}, {432, 2.0625*10^-7}, {433, 1.6875*10^-7}, {434, 1.59375*10^-7}, {435, 1.59375*10^-7}, {436, 1.6875*10^-7}, {437, 1.34375*10^-7}, {438, 1.5625*10^-7}, {439, 1.15625*10^-7}, {440, 1.1875*10^-7}, {441, 1.46875*10^-7}, {442, 1.46875*10^-7}, {443, 1.375*10^-7}, {444, 1.1875*10^-7}, {445, 1.3125*10^-7}, {446, 1.0625*10^-7}, {447, 1.5*10^-7}, {448, 1.28125*10^-7}, {449, 1.5*10^-7}, {450, 1.*10^-7}, {451, 1.40625*10^-7}, {452, 1.25*10^-7}, {453, 1.21875*10^-7}, {454, 1.0625*10^-7}, {455, 1.15625*10^-7}, {456, 1.0625*10^-7}, {457, 1.3125*10^-7}, {458, 1.5*10^-7}, {459, 1.125*10^-7}, {460, 5.625*10^-8}, {461, 1.625*10^-7}, {462, 1.125*10^-7}, {463, 1.09375*10^-7}, {464, 1.4375*10^-7}, {465, 1.34375*10^-7}, {466, 8.4375*10^-8}, {467, 1.21875*10^-7}, {468, 7.8125*10^-8}, {469, 1.*10^-7}, {470, 9.0625*10^-8}, {471, 9.0625*10^-8}, {472, 1.3125*10^-7}, {473, 6.875*10^-8}, {474, 8.125*10^-8}, {475, 9.0625*10^-8}, {476, 1.125*10^-7}, {477, 1.4375*10^-7}, {478, 1.03125*10^-7}, {479, 1.125*10^-7}, {480, 8.125*10^-8}, {481, 1.*10^-7}, {482, 1.21875*10^-7}, {483, 8.4375*10^-8}, {484, 8.75*10^-8}, {485, 9.0625*10^-8}, {486, 1.3125*10^-7}, {487, 9.375*10^-8}, {488, 9.375*10^-8}, {489, 1.125*10^-7}, {490, 8.4375*10^-8}, {491, 8.4375*10^-8}, {492, 6.5625*10^-8}, {493, 8.75*10^-8}, {494, 7.8125*10^-8}, {495, 9.0625*10^-8}, {496, 6.875*10^-8}, {497, 1.21875*10^-7}, {498, 7.5*10^-8}, {499, 6.875*10^-8}, {500, 7.8125*10^-8}, {501, 6.5625*10^-8}, {502, 1.125*10^-7}, {503, 1.0625*10^-7}, {504, 6.25*10^-8}, {505, 8.125*10^-8}, {506, 9.0625*10^-8}, {507, 8.125*10^-8}, {508, 7.8125*10^-8}, {509, 1.03125*10^-7}, {510, 5.3125*10^-8}, {511, 1.03125*10^-7}, {512, 6.875*10^-8}, {513, 5.625*10^-8}, {514, 8.4375*10^-8}, {515, 9.375*10^-8}, {516, 6.25*10^-8}, {517, 7.1875*10^-8}, {518, 5.*10^-8}, {519, 7.8125*10^-8}, {520, 9.375*10^-8}, {521, 7.1875*10^-8}, {522, 5.625*10^-8}, {523, 8.75*10^-8}, {524, 5.3125*10^-8}, {525, 6.875*10^-8}, {526, 7.5*10^-8}, {527, 7.8125*10^-8}, {528, 4.6875*10^-8}, {529, 7.5*10^-8}, {530, 6.5625*10^-8}, {531, 6.25*10^-8}, {532, 5.*10^-8}, {533, 5.625*10^-8}, {534, 5.9375*10^-8}, {535, 7.1875*10^-8}, {536, 7.8125*10^-8}, {537, 6.25*10^-8}, {538, 6.875*10^-8}, {539, 5.*10^-8}, {540, 5.625*10^-8}, {541, 6.5625*10^-8}, {542, 7.1875*10^-8}, {543, 6.5625*10^-8}, {544, 6.25*10^-8}, {545, 5.625*10^-8}, {546, 5.*10^-8}, {547, 4.375*10^-8}, {548, 5.9375*10^-8}, {549, 5.*10^-8}, {550, 5.9375*10^-8}, {551, 6.25*10^-8}, {552, 6.25*10^-8}, {553, 6.5625*10^-8}, {554, 5.625*10^-8}, {555, 5.9375*10^-8}, {556, 4.6875*10^-8}, {557, 3.4375*10^-8}, {558, 4.0625*10^-8}, {559, 6.25*10^-8}, {560, 5.625*10^-8}, {561, 5.9375*10^-8}, {562, 4.0625*10^-8}, {563, 4.375*10^-8}, {564, 4.6875*10^-8}, {565, 7.1875*10^-8}, {566, 4.375*10^-8}, {567, 4.6875*10^-8}, {568, 4.375*10^-8}, {569, 4.0625*10^-8}, {570, 5.9375*10^-8}, {571, 5.625*10^-8}, {572, 3.125*10^-8}, {573, 5.3125*10^-8}, {574, 5.3125*10^-8}, {575, 6.875*10^-8}, {576, 3.75*10^-8}, {577, 4.6875*10^-8}, {578, 4.6875*10^-8}, {579, 4.6875*10^-8}, {580, 1.875*10^-8}, {581, 2.5*10^-8}, {582, 3.125*10^-8}, {583, 2.8125*10^-8}, {584, 3.125*10^-8}, {585, 2.8125*10^-8}, {586, 4.375*10^-8}, {587, 4.0625*10^-8}, {588, 5.*10^-8}, {589, 5.3125*10^-8}, {590, 3.125*10^-8}, {591, 3.4375*10^-8}, {592, 3.75*10^-8}, {593, 3.4375*10^-8}, {594, 4.375*10^-8}, {595, 2.8125*10^-8}, {596, 3.4375*10^-8}, {597, 2.1875*10^-8}, {598, 4.375*10^-8}, {599, 3.4375*10^-8}, {600, 4.0625*10^-8}, {601, 3.125*10^-8}, {602, 5.3125*10^-8}, {603, 1.875*10^-8}, {604, 5.*10^-8}, {605, 2.5*10^-8}, {606, 5.625*10^-8}, {607, 5.625*10^-8}, {608, 3.75*10^-8}, {609, 4.6875*10^-8}, {610, 3.75*10^-8}, {611, 2.1875*10^-8}, {612, 1.875*10^-8}, {613, 4.0625*10^-8}, {614, 3.125*10^-8}, {615, 1.875*10^-8}, {616, 5.*10^-8}, {617, 2.1875*10^-8}, {618, 2.8125*10^-8}, {619, 2.1875*10^-8}, {620, 1.875*10^-8}, {621, 2.8125*10^-8}, {622, 4.0625*10^-8}, {623, 1.875*10^-8}, {624, 2.8125*10^-8}, {625, 2.8125*10^-8}, {626, 2.5*10^-8}, {627, 3.125*10^-8}, {628, 9.375*10^-9}, {629, 1.875*10^-8}, {630, 2.5*10^-8}, {631, 3.4375*10^-8}, {632, 3.125*10^-8}, {633, 1.875*10^-8}, {634, 4.0625*10^-8}, {635, 3.125*10^-9}, {636, 3.125*10^-8}, {637, 2.5*10^-8}, {638, 3.4375*10^-8}, {639, 1.5625*10^-8}, {640, 1.25*10^-8}, {641, 2.5*10^-8}, {642, 1.875*10^-8}, {643, 6.25*10^-9}, {644, 3.75*10^-8}, {645, 1.875*10^-8}, {646, 1.875*10^-8}, {647, 3.125*10^-8}, {648, 1.875*10^-8}, {649, 1.5625*10^-8}, {650, 6.25*10^-9}, {651, 2.1875*10^-8}, {652, 2.8125*10^-8}, {653, 2.1875*10^-8}, {654, 1.5625*10^-8}, {655, 2.5*10^-8}, {656, 6.25*10^-9}, {657, 1.875*10^-8}, {658, 1.875*10^-8}, {659, 2.1875*10^-8}, {660, 9.375*10^-9}, {661, 2.8125*10^-8}, {662, 2.1875*10^-8}, {663, 6.25*10^-9}, {664, 1.5625*10^-8}, {665, 1.5625*10^-8}, {666, 2.8125*10^-8}, {667, 2.5*10^-8}, {668, 1.5625*10^-8}, {669, 1.875*10^-8}, {670, 3.125*10^-8}, {671, 1.875*10^-8}, {672, 1.5625*10^-8}, {673, 9.375*10^-9}, {674, 1.25*10^-8}, {675, 1.5625*10^-8}, {676, 9.375*10^-9}, {677, 6.25*10^-9}, {678, 6.25*10^-9}, {679, 1.5625*10^-8}, {680, 6.25*10^-9}, {681, 1.25*10^-8}, {682, 9.375*10^-9}, {683, 2.1875*10^-8}, {684, 1.5625*10^-8}, {685, 1.25*10^-8}, {686, 9.375*10^-9}, {687, 9.375*10^-9}, {688, 6.25*10^-9}, {689, 1.875*10^-8}, {690, 2.1875*10^-8}, {691, 9.375*10^-9}, {692, 9.375*10^-9}, {693, 9.375*10^-9}, {694, 3.125*10^-9}, {695, 1.25*10^-8}, {697, 1.25*10^-8}, {698, 9.375*10^-9}, {699, 9.375*10^-9}, {700, 3.125*10^-9}, {701, 3.125*10^-9}, {702, 1.25*10^-8}, {703, 1.875*10^-8}, {704, 6.25*10^-9}, {705, 1.25*10^-8}, {706, 1.875*10^-8}, {707, 1.25*10^-8}, {708, 2.1875*10^-8}, {709, 2.8125*10^-8}, {710, 1.5625*10^-8}, {711, 1.875*10^-8}, {712, 9.375*10^-9}, {714, 2.5*10^-8}, {715, 1.25*10^-8}, {716, 1.5625*10^-8}, {717, 9.375*10^-9}, {718, 2.1875*10^-8}, {719, 1.25*10^-8}, {720, 1.5625*10^-8}, {722, 9.375*10^-9}, {723, 6.25*10^-9}, {724, 1.25*10^-8}, {725, 1.5625*10^-8}, {726, 9.375*10^-9}, {727, 9.375*10^-9}, {728, 1.25*10^-8}, {729, 9.375*10^-9}, {730, 9.375*10^-9}, {731, 1.5625*10^-8}, {732, 3.125*10^-9}, {733, 6.25*10^-9}, {734, 1.25*10^-8}, {735, 3.125*10^-9}, {736, 1.25*10^-8}, {737, 9.375*10^-9}, {738, 9.375*10^-9}, {739, 1.875*10^-8}, {741, 9.375*10^-9}, {742, 6.25*10^-9}, {743, 6.25*10^-9}, {744, 6.25*10^-9}, {745, 3.125*10^-9}, {746, 9.375*10^-9}, {747, 6.25*10^-9}, {748, 9.375*10^-9}, {749, 1.25*10^-8}, {750, 3.125*10^-9}, {751, 6.25*10^-9}, {752, 3.125*10^-9}, {753, 3.125*10^-9}, {754, 3.125*10^-9}, {755, 3.125*10^-9}, {756, 6.25*10^-9}, {757, 3.125*10^-9}, {758, 3.125*10^-9}, {759, 3.125*10^-9}, {760, 6.25*10^-9}, {761, 6.25*10^-9}, {762, 6.25*10^-9}, {763, 1.25*10^-8}, {764, 9.375*10^-9}, {766, 6.25*10^-9}, {767, 1.25*10^-8}, {768, 3.125*10^-9}, {769, 3.125*10^-9}, {770, 3.125*10^-9}, {772, 3.125*10^-9}, {773, 3.125*10^-9}, {774, 3.125*10^-9}, {775, 6.25*10^-9}, {776, 6.25*10^-9}, {777, 9.375*10^-9}, {779, 1.875*10^-8}, {780, 3.125*10^-9}, {782, 3.125*10^-9}, {783, 6.25*10^-9}, {784, 3.125*10^-9}, {785, 9.375*10^-9}, {786, 6.25*10^-9}, {788, 6.25*10^-9}, {790, 6.25*10^-9}, {791, 3.125*10^-9}, {792, 3.125*10^-9}, {793, 6.25*10^-9}, {794, 3.125*10^-9}, {795, 1.25*10^-8}, {796, 6.25*10^-9}, {798, 3.125*10^-9}, {799, 3.125*10^-9}, {800, 6.25*10^-9}, {801, 6.25*10^-9}, {803, 9.375*10^-9}, {804, 6.25*10^-9}, {805, 3.125*10^-9}, {806, 9.375*10^-9}, {807, 3.125*10^-9}, {808, 6.25*10^-9}, {809, 3.125*10^-9}, {811, 3.125*10^-9}, {812, 3.125*10^-9}, {813, 3.125*10^-9}, {814, 3.125*10^-9}, {815, 9.375*10^-9}, {817, 9.375*10^-9}, {819, 6.25*10^-9}, {821, 3.125*10^-9}, {822, 3.125*10^-9}, {823, 3.125*10^-9}, {825, 6.25*10^-9}, {828, 6.25*10^-9}, {832, 3.125*10^-9}, {833, 3.125*10^-9}, {835, 3.125*10^-9}, {836, 3.125*10^-9}, {837, 3.125*10^-9}, {842, 3.125*10^-9}, {843, 6.25*10^-9}, {844, 9.375*10^-9}, {852, 3.125*10^-9}, {853, 3.125*10^-9}, {856, 3.125*10^-9}, {868, 3.125*10^-9}, {872, 3.125*10^-9}, {874, 3.125*10^-9}, {876, 3.125*10^-9}, {877, 3.125*10^-9}, {883, 3.125*10^-9}, {885, 3.125*10^-9}, {887, 6.25*10^-9}, {890, 3.125*10^-9}, {891, 3.125*10^-9}, {892, 3.125*10^-9}, {893, 3.125*10^-9}, {898, 9.375*10^-9}, {901, 3.125*10^-9}, {903, 3.125*10^-9}, {905, 6.25*10^-9}, {912, 3.125*10^-9}, {914, 3.125*10^-9}, {915, 6.25*10^-9}, {919, 3.125*10^-9}, {921, 9.375*10^-9}, {922, 3.125*10^-9}, {925, 3.125*10^-9}, {930, 3.125*10^-9}, {931, 3.125*10^-9}, {932, 3.125*10^-9}, {933, 3.125*10^-9}, {940, 3.125*10^-9}, {941, 3.125*10^-9}, {942, 6.25*10^-9}, {945, 3.125*10^-9}, {947, 6.25*10^-9}, {967, 3.125*10^-9}, {971, 3.125*10^-9}, {978, 3.125*10^-9}, {981, 3.125*10^-9}, {989, 6.25*10^-9}, {993, 6.25*10^-9}, {994, 3.125*10^-9}, {1005, 3.125*10^-9}, {1008, 3.125*10^-9}, {1017, 3.125*10^-9}, {1024, 3.125*10^-9}, {1043, 3.125*10^-9}, {1048, 3.125*10^-9}, {1067, 3.125*10^-9}};

I tried to fit these data set using,

model = PDF[WeibullDistribution[\[Alpha], \[Beta], \[Gamma]], x];
fit = FindFit[data, model, {\[Alpha], \[Beta], \[Gamma]}, x];
modelf = Function[{x}, Evaluate[model /. fit]];
Plot[modelf[x], {x, 1, 1070}, Epilog -> Map[Point, data], 
 PlotRange -> All, 
 AxesLabel -> {x, Subscript[\[DoubleStruckCapitalP], x]}]

but doesn't work. Could someone help me to fit the data?

Thank you in advance.

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4
  • $\begingroup$ The main issue is that you have a function that might be a multiple of a Weibull density as opposed to random samples from a Weibull. That means you need to include a proportionality parameter. However, if the response might really consist of counts multiplied by 3.125 * 10^-9, then regression (i.e., using NonlinearModelFit) is not appropriate. Another issue is that the first 5 or 6 data points don't resemble samples from a Weibull distribution. Maybe giving more details about how the data was collected would help. $\endgroup$
    – JimB
    Jan 30, 2023 at 2:33
  • $\begingroup$ Hello, @JimB. Thank you for your comment. When I looked to shape exhibited by data, it seems similar to the Weibull distribution. So I tried that distribution. The data comes from a simulation and what really matters for our adjustment is the tail of the distribution. Therefore, the initial data can be removed. If you have other suggestion to fit that data, I would hear it. $\endgroup$
    – SAC
    Jan 30, 2023 at 19:39
  • $\begingroup$ I didn't see any connection to the FITS format. Sorry if I missed it. $\endgroup$
    – Michael E2
    Jan 30, 2023 at 22:07
  • $\begingroup$ It seems clear now that your response data (data[[All,2]]) is proportional to your real response data which is a count from 77,472 simulations. The first coordinate seems to be a binned version of what was actually measured (or categories of 1 unit width is what can be measured). No standard distribution (such as the Weibull) has two peaks as your data displays. For your subject matter does it make sense to have a mixture of two distributions? $\endgroup$
    – JimB
    Apr 9, 2023 at 1:50

1 Answer 1

1
$\begingroup$

To obtain a fit, I dropped the first few points. Then I made a hand fit just to get starting values for NonlinearModelFit.

(* Drop the first points *)

data = data[[6 ;;]];

dataPlot = ListPlot[data, PlotRange -> All];

model = a PDF[WeibullDistribution[\[Alpha], \[Beta], \[Gamma]], x];

(* Use a hand fit to estimate parameters *)

w[a_, \[Alpha]_, \[Beta]_, \[Gamma]_, x_] := Evaluate[model];

estimatePlot = 
 Show[Plot[w[.0002, 2, 200, 1, x], {x, 0, 1000}, PlotStyle -> Red], 
  dataPlot]

Hand fit

(* Fit with estimated starting point *)

fit = NonlinearModelFit[data, 
   model, {{a, .0002}, {\[Alpha], 2}, {\[Beta], 200}, {\[Gamma], 1}}, 
   x];

fit["BestFitParameters"];

fitPlot = 
 Show[Plot[fit[x], {x, 0, 1000}, 
   AxesLabel -> {"x", Subscript[\[DoubleStruckCapitalP], x]}, 
   PlotStyle -> Red], dataPlot]

Fit by NonlinearModelFit

$\endgroup$

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