我有matlab输出函数和输出曲线特征，我想从输出曲线特征中获取估计参数值，如何使用曲线拟合工具框获得95％置信度值
以下是输出功能
function c_t = output_function_conv(t, a1, a2, a3,b1,b2,b3,td, tmax,k1,k2,k3)
K_1 = (k1*k2)/(k2+k3);
K_2 = (k1*k3)/(k2+k3);
c_t = zeros(size(t));
ind = (t > td) & (t < tmax);
c_t(ind)= conv(
((t(ind) - td) ./ (tmax - td) * (a1 + a2 + a3)),
(K_1*exp(-(k2+k3)*t(ind)+K_2)),'same');
ind = (t >= tmax);
c_t(ind)= conv(
(a1 * exp(-b1 * (t(ind) - tmax))+ a2 * exp(-b2 * (t(ind) - tmax))) + a3 * exp(-b3 * (t(ind) - tmax)),
(K_1*exp(-(k2+k3)*t(ind)+K_2)),'same');
plot(t,c_t);
end
以下是我手动分配的输入值
output_function_conv(0:0.01:50,2501,18500,65000,0.9,0.2,0.5,0.7,0.3,0.036,0.02,0.12)
我想估计输出函数特征图的95％置信度值。
请告诉我如何进行估算
以下是曲线拟合的数据点
Output [ 0 0 21292.6393733205 20834.1763999840 20385.7118816737 19947.0305575470 19517.9214023497 19098.1775712551 18687.5963435309 18285.9790651943 17893.1310908080 17508.8617245528 17132.9841607124 16765.3154236868 16405.6763076516 16053.8913159642 15709.7886004177 15373.1999004294 15043.9604822502 14721.9090782703 14406.8878264928 14098.7422102414 13797.3209981621 13502.4761845740 13214.0629302211 12931.9395034707 12655.9672220004 12386.0103950142 12121.9362660198 11863.6149562013 11610.9194084135 11363.7253318255 11121.9111472338 10885.3579330675 10653.9493721007 10427.5716988901 10206.1136479486 9989.46640266844 9777.52354500148 9570.18100590553 9367.33701656270 9168.89206037408 8974.74882573439 8784.81215958863 8598.98902177138 8417.18844012873 8239.32146642148 8065.30113300742 7895.04241029970 7728.46216499746 7565.47911908413 7406.01380958819 7249.98854910055 7097.32738704227 6947.95607167558 6801.80201285096 6658.79424548257 6518.86339374380 6381.94163597454 6247.96267029150 6116.86168089246 5988.57530504527 5863.04160075221 5740.20001507995 5619.99135314564 5502.35774774896 5387.24262964033 5274.59069841522 5164.34789402439 5056.46136888997 4950.87946061717 4847.55166529148 4746.42861135114 4647.46203402467 4550.60475032357 4455.81063457978 4363.03459451807 4272.23254785336 4183.36139940305 4096.37901870450 4011.24421812800 3927.91673147547 3846.35719305551 3766.52711722518 3688.38887838929 3611.90569144786 3537.04159268280 3463.76142107467 3392.03080004070 3321.81611958540 3253.08451885496 3185.80386908720 3119.94275694843 3055.47046824929 2992.35697203117 2930.57290501549 2870.08955640789 2810.87885304966 2752.91334490887 2696.16619090368 2640.61114505071 2586.22254293112 2532.97528846746 2480.84484100443 2429.80720268671 2379.83890612722 2330.91700235940 2283.01904906694 2236.12309908497 2190.20768916629 2145.25182900688 2101.23499052463 2058.13709738563 2015.93851477229 1974.62003938789 1934.16288969200 1894.54869636156 1855.75949297240 1817.77770689617 1780.58615040766 1744.16801199767 1708.50684788667 1673.58657373467 1639.39145654262 1605.90610674094 1573.11547046102 1541.00482198505 1509.55975637042 1478.76618224438 1448.61031476500 1419.07866874474 1390.15805193257 1361.83555845114 1334.09856238523 1306.93471151803 1280.33192121178 1254.27836842933 1228.76248589333 1203.77295637992 1179.29870714364 1155.32890447051 1131.85294835633 1108.86046730722 1086.34131325951 1064.28555661621 1042.68348139733 1021.52558050131 1000.80255107504 980.505289989895 960.624889421189 941.152632528815 910.093552396909 880.074844716944 851.060744250693 823.016766773700 795.909660251508 769.707358052245 744.378934099714 719.894559876329 696.225463190064 673.343888624288 651.223059593511 629.837141932191 609.161208947531 589.171207870732 569.843927644531 551.156967988015 533.088709682665 515.618286026374 498.725555404831 482.391074932134 466.596075114855 451.322435495995 436.552661237327 422.269860600652 408.457723290306 395.100499621077 382.182980477300 369.690478030556 357.608807184818 345.924267719382 334.623627101217 323.694103939659 313.123352057611 302.899445154510 293.010862037469 283.446472397999 274.195523112734 265.247625047485 256.592740344874 248.221170176634 240.123542942462 232.290802898093 224.714199195994 217.385275322762 210.295858918002 203.438051960061 196.804221304642 190.386989562840 184.179226305776 178.174039583434 172.364767745896 166.744971555592 161.308426579671 156.049115852013 150.961222794856 146.039124390369 141.277384592926 136.670747973167 132.214133585317 127.902629049538 123.731484841434 119.696108781123 115.792060714591 112.015047380319 108.360917454464 104.825656768093 101.405383690273 98.0963446710064 94.8949099382621 91.7975693435645 88.8009283507997 85.9017041631206 83.0967219830068 80.3829114007349 77.7573029066858 75.2170245230854 72.7592985509469 70.3814384281314 68.0808456946014 65.8550070610841 63.7014915775029 61.6179478976678 59.6021016368462 57.6517528189581 55.7647734102607 53.9391049364990 52.1727561806128 50.4638009581927 48.8103759679823 47.2106787148207 45.6629655025121 44.1655494942032 42.7167988379332 41.3151348551059 39.9590302897151 38.6470076162314 37.3776374041319 36.1495367371296 34.9613676852239 33.8118358277656 32.6996888257886 31.6237150419265 30.5827422062891 29.5756361267334 28.6012994420155 27.6586704163679 26.7467217740917 25.8644595728099 25.0109221140670 24.1851788900145 23.3863295649593 22.6135029905981 21.8658562538011 21.1425737558466 20.4428663220491 19.7659703407560 19.1111469307271 18.4776811359438 17.8648811469268 17.2720775476747 16.6986225873647 16.1438894759878 15.6072717031179 15.0881823790414 14.5860535975035 14.1003358193474 13.6304972763527 13.1760233945998 12.7364162367108 12.3111939623395 11.8998903063055 11.5020540737853 11.1172486519962 10.7450515378264 10.3850538808823 10.0368600414446 9.70008716283935 9.37436475774730 9.05933430799276 8.75464887736619 8.45997273705098 8.17498100323961 7.89935928653744 7.63280335276621 7.37501879479276 7.12572071502023 6.88463341819175 6.65149011416815 6.42603263035222];
输入功能值：
input = [
10 -1 1
20 17956 1
30 61096 1
40 31098 1
50 18446 1
60 12969 1
95 7932 1
120 6213 1
188 4414 1
240 3310 1
300 3329 1
610 2623 1
1200 1953 1
1800 1617 1
2490 1559 1
3000 1561 1
3635 1574 1
4205 1438 1
4788 1448 1
];
calibrationfactor_wellcounter =1.841201569;
上述值的输入函数校准如下
function c = Input_function(t, a1, a2, a3, b1, b2, b3, td, tmax)
...some code to model the input function from the Above vector Input
Plot(t,c);
答案 0 :(得分：1)
我假设您想要对函数f(t; a,b,c,d,e,f)=y进行线性拟合（单输入 - >单输出）。在这里，t是您的输入，y是您的输出。现在，您有一组数据点，就像这样
data = [t1 ,y1;
t2, y2;
t3, y3;
......];
如果您scatter(data(:,1),data(:,2))，您应该能够看到您的数据点分布。通过数据拟合，您将估算最小化曲线与数据点之间距离的参数a,b,c,d,e,f。我认为MATLAB通常使用最小二乘法。现在定义辅助函数以提供给MATLAB的数据拟合例程
F = @(x,xdata) f(xdata, x(1),x(2),x(3),x(4),x(5),x(6) );
请注意，我们刚刚重新定义了a=x(1),b=x(2)，依此类推。然后定义您的初始猜测x0并进行非线性最小二乘拟合
x0 = [1 1 1 1 1 1];
[x,resnorm,~,exitflag,output] = lsqcurvefit(F,x0,data(:,1),data(:,2))
您可能需要调整初始猜测，收敛条件等，但如果成功，则会在向量a,b,c,d,e,f中获得x值。所以，你可以绘制
scatter(data(:,1),data(:,2))
hold on
plot(data(:,1),F(x,data(:,1));
hold off
你应该能够看到你的数据点和你的曲线。
您需要做的就是修改output_function_conv，以便每个t值获得一个c_t值，因为现在您的函数似乎返回了一个向量。此外，您的初始猜测和收敛标准对于使最小平方收敛非常重要。
参考：http://www.mathworks.com/help/optim/examples/nonlinear-data-fitting.html