OU plus noise automated EM multitau.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import Pkg\n",
    "Pkg.add(\"DifferentialEquations\")\n",
    "Pkg.add(\"Distributions\")\n",
    "Pkg.add(\"Polynomials\")\n",
    "Pkg.add(\"NPZ\")\n",
    "Pkg.add(\"DataFrames\")\n",
    "Pkg.add(\"StatsPlots\")\n",
    "Pkg.add(\"CSV\")\n",
    "print(VERSION)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1000×10001 Array{Float64,2}:\n",
       " 0.01  -1.87761    -2.18369   …   0.103991    0.446516   -0.797876\n",
       " 0.01   0.315095    0.391503      0.589589    1.07838     1.54724\n",
       " 0.01   1.26449    -0.11241       0.201999   -0.0330393   0.380695\n",
       " 0.01   0.270264   -0.166356     -1.12087    -0.147571    0.465649\n",
       " 0.01  -0.113972    0.828783     -0.273736   -0.498122   -1.10701\n",
       " 0.01  -1.46681     0.158524  …   0.614683   -0.314523    0.92442\n",
       " 0.01   1.9617      1.13969      -1.00993    -1.7245     -1.42317\n",
       " 0.01   0.0846742  -0.682009     -0.605891   -0.72488    -0.663328\n",
       " 0.01  -0.0267296   0.785891      0.875163    0.583334    0.671638\n",
       " 0.01   0.139125   -1.40436      -0.0740335  -1.12796    -1.30722\n",
       " 0.01  -3.00004    -2.06471   …   1.24806     0.621185    0.199754\n",
       " 0.01  -0.129275   -1.18372       0.132433   -0.21061    -0.153968\n",
       " 0.01   0.408069   -0.084261      1.72293     0.655718    0.804563\n",
       " ⋮                            ⋱                           ⋮\n",
       " 4.0   -0.436409    3.02066       1.54877    -0.784803   -0.709791\n",
       " 4.0   -0.0966333  -1.94439      -0.391366   -0.539594   -0.505375\n",
       " 4.0   -0.207699    0.585946  …   0.682742   -0.120337   -0.384004\n",
       " 4.0   -0.0414263   0.5743        2.58845     1.21164     0.784088\n",
       " 4.0   -0.192761    0.203791     -0.873768    2.53929    -0.85914\n",
       " 4.0    0.374108    1.7864        0.453001   -0.944999   -0.577324\n",
       " 4.0   -0.933886    0.371547     -2.08212     0.620644   -0.594266\n",
       " 4.0    0.879443    0.745473  …   0.260663    2.32986     0.702101\n",
       " 4.0   -0.961233   -1.16541       0.996468   -1.09454    -1.14596\n",
       " 4.0   -1.22069     0.230831     -0.120343   -0.481952    0.373715\n",
       " 4.0    0.277115   -0.817706     -0.86103     1.76765     0.783745\n",
       " 4.0    0.410251   -0.23056       0.346656   -1.6897      0.0433492"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using NPZ\n",
    "data = npzread(\"OUmt_sN05.npy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "ename": "UndefVarError",
     "evalue": "UndefVarError: data not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: data not defined",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[3]:2"
     ]
    }
   ],
   "source": [
    "using Plots\n",
    "plot(data[1,:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OUEMresult (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Polynomials\n",
    "# function to calculate A and B from the dataset\n",
    "function OUEM(data1sq,dataNsq,datasq,datacorr,N)\n",
    "    B_coef = Poly([N*datacorr,\n",
    "       -data1sq-(N+1)*datasq-dataNsq,\n",
    "       (2.0-N)*datacorr,     \n",
    "       (N-1)*datasq])\n",
    "    broots = roots(B_coef)\n",
    "#    println(broots)\n",
    "    B=broots[end-1]\n",
    "    Q=(data1sq+dataNsq)/(1-B^2)\n",
    "    Q=Q+datasq*(1+B^2)/(1-B^2)\n",
    "    Q=Q-datacorr*2*B/(1-B^2)\n",
    "    A = Q/N\n",
    "    P2A = -N/A^2/2\n",
    "    Btmp = B^2*(1+2*N)\n",
    "    tmp = (1+Btmp)*(data1sq+dataNsq) + (2*Btmp + N + 1 -B^4*(N-1))*datasq - 2*B*(1+B^2+2*N)*datacorr\n",
    "    P2B = -tmp/((1-B^2)^2*(data1sq+dataNsq + (1+B^2)*datasq - 2*B*datacorr))\n",
    "    PAB = (N-1)*B/A/(1-B^2)\n",
    "    dA = sqrt(-P2B/(P2A*P2B-PAB^2))\n",
    "    dB = sqrt(-P2A/(P2A*P2B-PAB^2))\n",
    "    return A,dA,B,dB\n",
    "end\n",
    "\n",
    "function OUEMresult(data1sq,dataNsq,datasq,datacorr,N,deltat)\n",
    "    A, dA, B ,dB = OUEM(data1sq,dataNsq,datasq,datacorr,N)\n",
    "    tau = -deltat/log(B)\n",
    "    dtau = deltat*dB/B/log(B)^2\n",
    "    return A,dA,tau,dtau\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "px2 (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# function to multiply two Gaussians\n",
    "function mul_gauss(mu1,mu2,sigma1,sigma2)\n",
    "    mu = (mu1*sigma2^2+mu2*sigma1^2)/(sigma1^2+sigma2^2)\n",
    "    sigma = sqrt(sigma1^2*sigma2^2/(sigma1^2+sigma2^2))\n",
    "    return mu,sigma\n",
    "end\n",
    "\n",
    "# function to convolute two Gaussians\n",
    "function con_gauss(mu1,mu2,sigma1,sigma2)\n",
    "    mu = mu1 + mu2\n",
    "    sigma = sqrt(sigma1^2+sigma2^2)\n",
    "    return mu,sigma\n",
    "end\n",
    "\n",
    "# function to calculate helper function alpha (forward)\n",
    "function alpha(y,sigmaN,A,B)\n",
    "    N = length(y)\n",
    "    sig_alpha = zeros(N)\n",
    "    mu_alpha = zeros(N)\n",
    "    mu_alpha[1],sig_alpha[1] = mul_gauss(0,y[1],sqrt(A),sigmaN)\n",
    "    for i=2:N\n",
    "        mu_temp,sig_temp = con_gauss(B*mu_alpha[i-1],0,sqrt(A*(1-B^2)),B*sig_alpha[i-1])\n",
    "        mu_alpha[i],sig_alpha[i] = mul_gauss(mu_temp,y[i],sig_temp,sigmaN)\n",
    "    end\n",
    "    return mu_alpha,sig_alpha\n",
    "end\n",
    "\n",
    "function beta(y,sigmaN,A,B)\n",
    "    N = length(y)\n",
    "    sig_beta = zeros(N-1)\n",
    "    mu_beta  = zeros(N-1)\n",
    "    mu_beta[N-1],sig_beta[N-1] = y[N]/B,sqrt(sigmaN^2+A*(1-B^2))/B\n",
    "    for i=(N-2):-1:1\n",
    "        mu_temp,sig_temp = mul_gauss(mu_beta[i+1],y[i+1],sig_beta[i+1],sigmaN)\n",
    "        mu_beta[i],sig_beta[i] = mu_temp/B,sqrt(sig_temp^2+A*(1-B^2))/B\n",
    "    end\n",
    "    return mu_beta,sig_beta\n",
    "end\n",
    "\n",
    "function px(y,sigmaN,A,B)\n",
    "    N = length(y)\n",
    "    mu_px = zeros(N)\n",
    "    sig_px = zeros(N)\n",
    "    exxpone = zeros(N-1)\n",
    "    rho = zeros(N-1)\n",
    "    mu_x = zeros(N-1)\n",
    "    mu_xpone = zeros(N-1)\n",
    "    mu_alpha,sig_alpha = alpha(y,sigmaN,A,B)\n",
    "    mu_beta,sig_beta = beta(y,sigmaN,A,B)\n",
    "    for i=1:N-1\n",
    "        mu_px[i],sig_px[i] = mul_gauss(mu_alpha[i],mu_beta[i],sig_alpha[i],sig_beta[i])\n",
    "        if i==N-1\n",
    "            mu_tmp,sig_tmp = y[N],sigmaN\n",
    "        else\n",
    "            mu_tmp,sig_tmp = mul_gauss(mu_beta[i+1],y[i+1],sig_beta[i+1],sigmaN)\n",
    "        end\n",
    "        # these are the polynomial coefficients of the distribution\n",
    "        a = 1.0/(2*sig_tmp)+1.0/(2*(A*(1-B^2)))\n",
    "        b = 1.0/(2*sig_alpha[i])+B^2/(2*(A*(1-B^2)))\n",
    "        c = -B/(A*(1-B^2))\n",
    "        d = -mu_tmp/sig_tmp^2\n",
    "        e = -mu_alpha[i]/sig_alpha[i]^2\n",
    "        rho[i] = -c/(2*sqrt(a*b))\n",
    "        exxpone[i] = -c/(4*a*b-c^2)\n",
    "        mu_xpone[i] = (2*b*d+c*e)/(4*a*b-c^2)\n",
    "        mu_x[i] = (2*a*e+d*c)/(4*a*b-c^2)\n",
    "    end\n",
    "    # for last one beta = 1\n",
    "    mu_px[N] = mu_alpha[N]\n",
    "    sig_px[N] = sig_alpha[N]\n",
    "    # it seems that the sign of mu_x and mu_xpone are not accuaratly determined by this method\n",
    "    # we need to correct the signs.  We chose the sign so that mu_x[i] is closest to mu_px[i]\n",
    "    # and mu_xpone[i] is closeset to mu_px[i+1]\n",
    "    nswitch=0\n",
    "    for i=1:N-1\n",
    "        dx1 = abs(mu_x[i]-mu_px[i])\n",
    "        dx2 = abs(-mu_x[i]-mu_px[i])\n",
    "        dxp1 = abs(mu_xpone[i]-mu_px[i+1])\n",
    "        dxp2 = abs(-mu_xpone[i]-mu_px[i+1])\n",
    "        if dx1>dx2\n",
    "            nswitch = nswitch+1\n",
    "            mu_x[i] = -mu_x[i]\n",
    "        end\n",
    "        if dxp1>dxp2\n",
    "            nswitch = nswitch+1\n",
    "            mu_xpone[i] = -mu_xpone[i]\n",
    "        end\n",
    "    end\n",
    "    return mu_px,sig_px,rho,exxpone,mu_x,mu_xpone,nswitch\n",
    "end\n",
    "\n",
    "function px2(y,sigmaN,A,B)\n",
    "    N = length(y)\n",
    "    mu_px = zeros(N)\n",
    "    sig_px = zeros(N)\n",
    "    exxpone = zeros(N-1)\n",
    "    mu_alpha,sig_alpha = alpha(y,sigmaN,A,B)\n",
    "    mu_beta,sig_beta = beta(y,sigmaN,A,B)\n",
    "    for i=1:N-1\n",
    "        mu_px[i],sig_px[i] = mul_gauss(mu_alpha[i],mu_beta[i],sig_alpha[i],sig_beta[i])\n",
    "        if i==N-1\n",
    "            mu_tmp,sig_tmp = y[N],sigmaN\n",
    "        else\n",
    "            mu_tmp,sig_tmp = mul_gauss(mu_beta[i+1],y[i+1],sig_beta[i+1],sigmaN)\n",
    "        end\n",
    "        term1 = A*(1-B^2)\n",
    "        term2 = term1^2 *mu_tmp*mu_alpha[i]\n",
    "        term3 = B^2*mu_tmp*mu_alpha[i]*sig_alpha[i]^2\n",
    "        term4 = B*mu_tmp^2*sig_alpha[i]^2\n",
    "        term5 = mu_tmp*mu_alpha[i]*sig_tmp^2\n",
    "        term6 = B*(mu_alpha[i]^2+sig_alpha[i]^2)*sig_tmp^2\n",
    "        term7 = 2*B*mu_tmp*mu_alpha[i]*sig_alpha[i]^2*sig_tmp^2\n",
    "        term8 = (mu_alpha[i]^2+sig_alpha[i]^2)*sig_tmp^4\n",
    "        term9 = B^2*sig_alpha[i]^4*(mu_tmp^2+sig_tmp^2)\n",
    "        term10 = (B^2*sig_alpha[i]^2+sig_tmp^2+term1)^2\n",
    "        exxpone[i] = (term2 +term1*(term3+term4+term5+term6)+B*(term7+term8+term9))/term10\n",
    "        \n",
    "    end\n",
    "    # for last one beta = 1\n",
    "    mu_px[N] = mu_alpha[N]\n",
    "    sig_px[N] = sig_alpha[N]\n",
    "    return mu_px,sig_px,exxpone\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OUWN (generic function with 2 methods)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function OUWN(data,A_s,B_s,sigN_s,convth,itermax=10000)\n",
    "    N=length(data)\n",
    "    A_p = A_s\n",
    "    B_p = B_s\n",
    "    sigN_p = sigN_s\n",
    "    for i=1:itermax\n",
    "        mu_px,sig_px,exxpone = px2(data,sigN_p,A_p,B_p)\n",
    "        # lets calculate all the values that we need for OU analysis\n",
    "        # data1sq,dataNsq,datasq,datacorr\n",
    "        data1sq = sig_px[1]^2 + mu_px[1]^2\n",
    "        dataNsq = sig_px[end]^2 + mu_px[end]^2\n",
    "        datasq = sum(sig_px[2:end-1].^2 .+ mu_px[2:end-1].^2)\n",
    "        datacorr = sum(exxpone)\n",
    "        global A,dA,B,dB,sigmaN\n",
    "        A,dA,B,dB=OUEM(data1sq,dataNsq,datasq,datacorr,N)\n",
    "        sigmaN = sqrt((sum(sig_px .^2) + sum(mu_px .^2) + sum(data .^2) -2*sum((mu_px .* data)))/length(data))\n",
    "#        println(\"A: \",A,\"  B: \",B,\"  SigmaN: \",sigmaN)\n",
    "        Arel = abs((A-A_p)/A_p)\n",
    "        Brel = abs((B-B_p)/B_p)\n",
    "        srel = abs((sigmaN-sigN_p)/sigN_p)\n",
    "        if (Arel<convth && Brel<convth && srel<convth)\n",
    "            break\n",
    "        end\n",
    "        A_p = A\n",
    "        B_p = B\n",
    "        sigN_p = sigmaN\n",
    "    end\n",
    "    return A,dA,B,dB,sigmaN\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>dt</th><th>A</th><th>dA</th><th>B</th><th>dB</th><th>sigmaN</th></tr><tr><th></th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th></tr></thead><tbody><p>1,000 rows × 6 columns</p><tr><th>1</th><td>0.01</td><td>1.08626</td><td>0.159642</td><td>0.990854</td><td>0.00134422</td><td>0.500037</td></tr><tr><th>2</th><td>0.01</td><td>1.07464</td><td>0.162053</td><td>0.99139</td><td>0.00129839</td><td>0.498834</td></tr><tr><th>3</th><td>0.01</td><td>0.925471</td><td>0.123341</td><td>0.988966</td><td>0.00147051</td><td>0.503385</td></tr><tr><th>4</th><td>0.01</td><td>1.07908</td><td>0.156615</td><td>0.990725</td><td>0.00134609</td><td>0.501257</td></tr><tr><th>5</th><td>0.01</td><td>1.04388</td><td>0.149837</td><td>0.990493</td><td>0.00136457</td><td>0.501161</td></tr><tr><th>6</th><td>0.01</td><td>0.961766</td><td>0.132662</td><td>0.989716</td><td>0.00141858</td><td>0.501622</td></tr><tr><th>7</th><td>0.01</td><td>1.13384</td><td>0.17078</td><td>0.991271</td><td>0.0013148</td><td>0.498584</td></tr><tr><th>8</th><td>0.01</td><td>0.97696</td><td>0.135632</td><td>0.989848</td><td>0.00140937</td><td>0.507725</td></tr><tr><th>9</th><td>0.01</td><td>1.02628</td><td>0.142659</td><td>0.989875</td><td>0.00140739</td><td>0.505748</td></tr><tr><th>10</th><td>0.01</td><td>0.784794</td><td>0.0965507</td><td>0.987004</td><td>0.00159884</td><td>0.499745</td></tr><tr><th>11</th><td>0.01</td><td>1.05709</td><td>0.154837</td><td>0.990596</td><td>0.00137768</td><td>0.500528</td></tr><tr><th>12</th><td>0.01</td><td>1.03362</td><td>0.146953</td><td>0.990317</td><td>0.0013767</td><td>0.501351</td></tr><tr><th>13</th><td>0.01</td><td>1.02386</td><td>0.144996</td><td>0.990217</td><td>0.00138545</td><td>0.504826</td></tr><tr><th>14</th><td>0.01</td><td>0.89694</td><td>0.116242</td><td>0.988344</td><td>0.00151053</td><td>0.499155</td></tr><tr><th>15</th><td>0.01</td><td>1.09595</td><td>0.160749</td><td>0.990855</td><td>0.00134138</td><td>0.495768</td></tr><tr><th>16</th><td>0.01</td><td>0.999807</td><td>0.140607</td><td>0.990116</td><td>0.00139007</td><td>0.492639</td></tr><tr><th>17</th><td>0.01</td><td>0.94064</td><td>0.122989</td><td>0.988553</td><td>0.00149666</td><td>0.500394</td></tr><tr><th>18</th><td>0.01</td><td>1.08409</td><td>0.158921</td><td>0.990757</td><td>0.00135514</td><td>0.503413</td></tr><tr><th>19</th><td>0.01</td><td>0.997493</td><td>0.139638</td><td>0.990025</td><td>0.00139637</td><td>0.499951</td></tr><tr><th>20</th><td>0.01</td><td>1.15075</td><td>0.161864</td><td>0.989939</td><td>0.00141536</td><td>0.49969</td></tr><tr><th>21</th><td>0.0914286</td><td>1.04314</td><td>0.0493899</td><td>0.914461</td><td>0.00404636</td><td>0.508947</td></tr><tr><th>22</th><td>0.0914286</td><td>1.06262</td><td>0.0499134</td><td>0.913273</td><td>0.00406958</td><td>0.494141</td></tr><tr><th>23</th><td>0.0914286</td><td>0.978303</td><td>0.0456756</td><td>0.912253</td><td>0.0040925</td><td>0.506486</td></tr><tr><th>24</th><td>0.0914286</td><td>1.05718</td><td>0.050938</td><td>0.91734</td><td>0.00397932</td><td>0.503604</td></tr><tr><th>25</th><td>0.0914286</td><td>1.05326</td><td>0.0492939</td><td>0.912566</td><td>0.00408801</td><td>0.502143</td></tr><tr><th>26</th><td>0.0914286</td><td>0.946711</td><td>0.0412176</td><td>0.899592</td><td>0.00436565</td><td>0.490893</td></tr><tr><th>27</th><td>0.0914286</td><td>1.03286</td><td>0.0494316</td><td>0.916295</td><td>0.00400232</td><td>0.506603</td></tr><tr><th>28</th><td>0.0914286</td><td>1.00308</td><td>0.0467404</td><td>0.911919</td><td>0.00409998</td><td>0.494778</td></tr><tr><th>29</th><td>0.0914286</td><td>1.01753</td><td>0.0492375</td><td>0.918014</td><td>0.00396385</td><td>0.499521</td></tr><tr><th>30</th><td>0.0914286</td><td>1.03965</td><td>0.0496168</td><td>0.915784</td><td>0.00401559</td><td>0.496663</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cccccc}\n",
       "\t& dt & A & dA & B & dB & sigmaN\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64 & Float64 & Float64 & Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 0.01 & 1.08626 & 0.159642 & 0.990854 & 0.00134422 & 0.500037 \\\\\n",
       "\t2 & 0.01 & 1.07464 & 0.162053 & 0.99139 & 0.00129839 & 0.498834 \\\\\n",
       "\t3 & 0.01 & 0.925471 & 0.123341 & 0.988966 & 0.00147051 & 0.503385 \\\\\n",
       "\t4 & 0.01 & 1.07908 & 0.156615 & 0.990725 & 0.00134609 & 0.501257 \\\\\n",
       "\t5 & 0.01 & 1.04388 & 0.149837 & 0.990493 & 0.00136457 & 0.501161 \\\\\n",
       "\t6 & 0.01 & 0.961766 & 0.132662 & 0.989716 & 0.00141858 & 0.501622 \\\\\n",
       "\t7 & 0.01 & 1.13384 & 0.17078 & 0.991271 & 0.0013148 & 0.498584 \\\\\n",
       "\t8 & 0.01 & 0.97696 & 0.135632 & 0.989848 & 0.00140937 & 0.507725 \\\\\n",
       "\t9 & 0.01 & 1.02628 & 0.142659 & 0.989875 & 0.00140739 & 0.505748 \\\\\n",
       "\t10 & 0.01 & 0.784794 & 0.0965507 & 0.987004 & 0.00159884 & 0.499745 \\\\\n",
       "\t11 & 0.01 & 1.05709 & 0.154837 & 0.990596 & 0.00137768 & 0.500528 \\\\\n",
       "\t12 & 0.01 & 1.03362 & 0.146953 & 0.990317 & 0.0013767 & 0.501351 \\\\\n",
       "\t13 & 0.01 & 1.02386 & 0.144996 & 0.990217 & 0.00138545 & 0.504826 \\\\\n",
       "\t14 & 0.01 & 0.89694 & 0.116242 & 0.988344 & 0.00151053 & 0.499155 \\\\\n",
       "\t15 & 0.01 & 1.09595 & 0.160749 & 0.990855 & 0.00134138 & 0.495768 \\\\\n",
       "\t16 & 0.01 & 0.999807 & 0.140607 & 0.990116 & 0.00139007 & 0.492639 \\\\\n",
       "\t17 & 0.01 & 0.94064 & 0.122989 & 0.988553 & 0.00149666 & 0.500394 \\\\\n",
       "\t18 & 0.01 & 1.08409 & 0.158921 & 0.990757 & 0.00135514 & 0.503413 \\\\\n",
       "\t19 & 0.01 & 0.997493 & 0.139638 & 0.990025 & 0.00139637 & 0.499951 \\\\\n",
       "\t20 & 0.01 & 1.15075 & 0.161864 & 0.989939 & 0.00141536 & 0.49969 \\\\\n",
       "\t21 & 0.0914286 & 1.04314 & 0.0493899 & 0.914461 & 0.00404636 & 0.508947 \\\\\n",
       "\t22 & 0.0914286 & 1.06262 & 0.0499134 & 0.913273 & 0.00406958 & 0.494141 \\\\\n",
       "\t23 & 0.0914286 & 0.978303 & 0.0456756 & 0.912253 & 0.0040925 & 0.506486 \\\\\n",
       "\t24 & 0.0914286 & 1.05718 & 0.050938 & 0.91734 & 0.00397932 & 0.503604 \\\\\n",
       "\t25 & 0.0914286 & 1.05326 & 0.0492939 & 0.912566 & 0.00408801 & 0.502143 \\\\\n",
       "\t26 & 0.0914286 & 0.946711 & 0.0412176 & 0.899592 & 0.00436565 & 0.490893 \\\\\n",
       "\t27 & 0.0914286 & 1.03286 & 0.0494316 & 0.916295 & 0.00400232 & 0.506603 \\\\\n",
       "\t28 & 0.0914286 & 1.00308 & 0.0467404 & 0.911919 & 0.00409998 & 0.494778 \\\\\n",
       "\t29 & 0.0914286 & 1.01753 & 0.0492375 & 0.918014 & 0.00396385 & 0.499521 \\\\\n",
       "\t30 & 0.0914286 & 1.03965 & 0.0496168 & 0.915784 & 0.00401559 & 0.496663 \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "1000×6 DataFrame\n",
       "│ Row  │ dt      │ A        │ dA        │ B          │ dB         │ sigmaN   │\n",
       "│      │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m  │ \u001b[90mFloat64\u001b[39m   │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mFloat64\u001b[39m  │\n",
       "├──────┼─────────┼──────────┼───────────┼────────────┼────────────┼──────────┤\n",
       "│ 1    │ 0.01    │ 1.08626  │ 0.159642  │ 0.990854   │ 0.00134422 │ 0.500037 │\n",
       "│ 2    │ 0.01    │ 1.07464  │ 0.162053  │ 0.99139    │ 0.00129839 │ 0.498834 │\n",
       "│ 3    │ 0.01    │ 0.925471 │ 0.123341  │ 0.988966   │ 0.00147051 │ 0.503385 │\n",
       "│ 4    │ 0.01    │ 1.07908  │ 0.156615  │ 0.990725   │ 0.00134609 │ 0.501257 │\n",
       "│ 5    │ 0.01    │ 1.04388  │ 0.149837  │ 0.990493   │ 0.00136457 │ 0.501161 │\n",
       "│ 6    │ 0.01    │ 0.961766 │ 0.132662  │ 0.989716   │ 0.00141858 │ 0.501622 │\n",
       "│ 7    │ 0.01    │ 1.13384  │ 0.17078   │ 0.991271   │ 0.0013148  │ 0.498584 │\n",
       "│ 8    │ 0.01    │ 0.97696  │ 0.135632  │ 0.989848   │ 0.00140937 │ 0.507725 │\n",
       "│ 9    │ 0.01    │ 1.02628  │ 0.142659  │ 0.989875   │ 0.00140739 │ 0.505748 │\n",
       "│ 10   │ 0.01    │ 0.784794 │ 0.0965507 │ 0.987004   │ 0.00159884 │ 0.499745 │\n",
       "⋮\n",
       "│ 990  │ 4.0     │ 0.994508 │ 0.0140751 │ 0.0274702  │ 0.00999601 │ 0.499602 │\n",
       "│ 991  │ 4.0     │ 0.996089 │ 0.014091  │ 0.0172606  │ 0.00999827 │ 0.49976  │\n",
       "│ 992  │ 4.0     │ 1.00031  │ 0.0141528 │ 0.0209964  │ 0.00999769 │ 0.500005 │\n",
       "│ 993  │ 4.0     │ 1.00413  │ 0.0142042 │ 0.0160857  │ 0.00999865 │ 0.500268 │\n",
       "│ 994  │ 4.0     │ 0.992651 │ 0.0140405 │ 0.0128595  │ 0.00999903 │ 0.499564 │\n",
       "│ 995  │ 4.0     │ 1.01048  │ 0.0143025 │ 0.0291645  │ 0.00999583 │ 0.500602 │\n",
       "│ 996  │ 4.0     │ 0.97496  │ 0.013788  │ 0.00143049 │ 0.0100001  │ 0.498474 │\n",
       "│ 997  │ 4.0     │ 1.0225   │ 0.0144626 │ 0.0126061  │ 0.00999961 │ 0.501418 │\n",
       "│ 998  │ 4.0     │ 0.989132 │ 0.0139949 │ 0.0215448  │ 0.0099979  │ 0.499307 │\n",
       "│ 999  │ 4.0     │ 0.981817 │ 0.0138932 │ 0.0243486  │ 0.00999696 │ 0.498823 │\n",
       "│ 1000 │ 4.0     │ 1.00402  │ 0.0142002 │ 0.00907142 │ 0.00999934 │ 0.500289 │"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using DataFrames\n",
    "# iterate over the dataset and estimate the best parameters\n",
    "paradf = DataFrame(dt=Float64[],A=Float64[],dA=Float64[],B=Float64[],dB=Float64[],sigmaN=Float64[])\n",
    "for i=1:size(data,1)\n",
    "    dt = data[i,1]\n",
    "    ts = data[i,2:end]\n",
    "    Bg = exp(-dt)\n",
    "    result = collect(OUWN(ts,1.0,Bg,0.5,0.0001))\n",
    "    push!(paradf,pushfirst!(result,dt))\n",
    "end\n",
    "paradf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>dt</th><th>A</th><th>dA</th><th>B</th><th>dB</th><th>sigmaN</th><th>tau</th><th>dtau</th></tr><tr><th></th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th></tr></thead><tbody><p>996 rows × 8 columns</p><tr><th>1</th><td>0.01</td><td>1.08626</td><td>0.159642</td><td>0.990854</td><td>0.00134422</td><td>0.500037</td><td>1.08836</td><td>0.160697</td></tr><tr><th>2</th><td>0.01</td><td>1.07464</td><td>0.162053</td><td>0.99139</td><td>0.00129839</td><td>0.498834</td><td>1.15643</td><td>0.175145</td></tr><tr><th>3</th><td>0.01</td><td>0.925471</td><td>0.123341</td><td>0.988966</td><td>0.00147051</td><td>0.503385</td><td>0.901303</td><td>0.120789</td></tr><tr><th>4</th><td>0.01</td><td>1.07908</td><td>0.156615</td><td>0.990725</td><td>0.00134609</td><td>0.501257</td><td>1.0732</td><td>0.156489</td></tr><tr><th>5</th><td>0.01</td><td>1.04388</td><td>0.149837</td><td>0.990493</td><td>0.00136457</td><td>0.501161</td><td>1.0469</td><td>0.150991</td></tr><tr><th>6</th><td>0.01</td><td>0.961766</td><td>0.132662</td><td>0.989716</td><td>0.00141858</td><td>0.501622</td><td>0.967341</td><td>0.134123</td></tr><tr><th>7</th><td>0.01</td><td>1.13384</td><td>0.17078</td><td>0.991271</td><td>0.0013148</td><td>0.498584</td><td>1.14065</td><td>0.172573</td></tr><tr><th>8</th><td>0.01</td><td>0.97696</td><td>0.135632</td><td>0.989848</td><td>0.00140937</td><td>0.507725</td><td>0.980044</td><td>0.136756</td></tr><tr><th>9</th><td>0.01</td><td>1.02628</td><td>0.142659</td><td>0.989875</td><td>0.00140739</td><td>0.505748</td><td>0.982679</td><td>0.137296</td></tr><tr><th>10</th><td>0.01</td><td>0.784794</td><td>0.0965507</td><td>0.987004</td><td>0.00159884</td><td>0.499745</td><td>0.764473</td><td>0.0946695</td></tr><tr><th>11</th><td>0.01</td><td>1.05709</td><td>0.154837</td><td>0.990596</td><td>0.00137768</td><td>0.500528</td><td>1.05835</td><td>0.15578</td></tr><tr><th>12</th><td>0.01</td><td>1.03362</td><td>0.146953</td><td>0.990317</td><td>0.0013767</td><td>0.501351</td><td>1.0277</td><td>0.146824</td></tr><tr><th>13</th><td>0.01</td><td>1.02386</td><td>0.144996</td><td>0.990217</td><td>0.00138545</td><td>0.504826</td><td>1.01718</td><td>0.144763</td></tr><tr><th>14</th><td>0.01</td><td>0.89694</td><td>0.116242</td><td>0.988344</td><td>0.00151053</td><td>0.499155</td><td>0.852948</td><td>0.11119</td></tr><tr><th>15</th><td>0.01</td><td>1.09595</td><td>0.160749</td><td>0.990855</td><td>0.00134138</td><td>0.495768</td><td>1.0885</td><td>0.160397</td></tr><tr><th>16</th><td>0.01</td><td>0.999807</td><td>0.140607</td><td>0.990116</td><td>0.00139007</td><td>0.492639</td><td>1.00669</td><td>0.142279</td></tr><tr><th>17</th><td>0.01</td><td>0.94064</td><td>0.122989</td><td>0.988553</td><td>0.00149666</td><td>0.500394</td><td>0.868597</td><td>0.114224</td></tr><tr><th>18</th><td>0.01</td><td>1.08409</td><td>0.158921</td><td>0.990757</td><td>0.00135514</td><td>0.503413</td><td>1.07684</td><td>0.158607</td></tr><tr><th>19</th><td>0.01</td><td>0.997493</td><td>0.139638</td><td>0.990025</td><td>0.00139637</td><td>0.499951</td><td>0.997509</td><td>0.140342</td></tr><tr><th>20</th><td>0.01</td><td>1.15075</td><td>0.161864</td><td>0.989939</td><td>0.00141536</td><td>0.49969</td><td>0.988882</td><td>0.139813</td></tr><tr><th>21</th><td>0.0914286</td><td>1.04314</td><td>0.0493899</td><td>0.914461</td><td>0.00404636</td><td>0.508947</td><td>1.02246</td><td>0.050595</td></tr><tr><th>22</th><td>0.0914286</td><td>1.06262</td><td>0.0499134</td><td>0.913273</td><td>0.00406958</td><td>0.494141</td><td>1.0078</td><td>0.0495017</td></tr><tr><th>23</th><td>0.0914286</td><td>0.978303</td><td>0.0456756</td><td>0.912253</td><td>0.0040925</td><td>0.506486</td><td>0.995543</td><td>0.0486309</td></tr><tr><th>24</th><td>0.0914286</td><td>1.05718</td><td>0.050938</td><td>0.91734</td><td>0.00397932</td><td>0.503604</td><td>1.05971</td><td>0.053281</td></tr><tr><th>25</th><td>0.0914286</td><td>1.05326</td><td>0.0492939</td><td>0.912566</td><td>0.00408801</td><td>0.502143</td><td>0.999275</td><td>0.0489256</td></tr><tr><th>26</th><td>0.0914286</td><td>0.946711</td><td>0.0412176</td><td>0.899592</td><td>0.00436565</td><td>0.490893</td><td>0.864052</td><td>0.0396279</td></tr><tr><th>27</th><td>0.0914286</td><td>1.03286</td><td>0.0494316</td><td>0.916295</td><td>0.00400232</td><td>0.506603</td><td>1.0459</td><td>0.0522604</td></tr><tr><th>28</th><td>0.0914286</td><td>1.00308</td><td>0.0467404</td><td>0.911919</td><td>0.00409998</td><td>0.494778</td><td>0.99159</td><td>0.0483513</td></tr><tr><th>29</th><td>0.0914286</td><td>1.01753</td><td>0.0492375</td><td>0.918014</td><td>0.00396385</td><td>0.499521</td><td>1.0688</td><td>0.0539488</td></tr><tr><th>30</th><td>0.0914286</td><td>1.03965</td><td>0.0496168</td><td>0.915784</td><td>0.00401559</td><td>0.496663</td><td>1.03925</td><td>0.0517986</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cccccccc}\n",
       "\t& dt & A & dA & B & dB & sigmaN & tau & dtau\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64 & Float64 & Float64 & Float64 & Float64 & Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 0.01 & 1.08626 & 0.159642 & 0.990854 & 0.00134422 & 0.500037 & 1.08836 & 0.160697 \\\\\n",
       "\t2 & 0.01 & 1.07464 & 0.162053 & 0.99139 & 0.00129839 & 0.498834 & 1.15643 & 0.175145 \\\\\n",
       "\t3 & 0.01 & 0.925471 & 0.123341 & 0.988966 & 0.00147051 & 0.503385 & 0.901303 & 0.120789 \\\\\n",
       "\t4 & 0.01 & 1.07908 & 0.156615 & 0.990725 & 0.00134609 & 0.501257 & 1.0732 & 0.156489 \\\\\n",
       "\t5 & 0.01 & 1.04388 & 0.149837 & 0.990493 & 0.00136457 & 0.501161 & 1.0469 & 0.150991 \\\\\n",
       "\t6 & 0.01 & 0.961766 & 0.132662 & 0.989716 & 0.00141858 & 0.501622 & 0.967341 & 0.134123 \\\\\n",
       "\t7 & 0.01 & 1.13384 & 0.17078 & 0.991271 & 0.0013148 & 0.498584 & 1.14065 & 0.172573 \\\\\n",
       "\t8 & 0.01 & 0.97696 & 0.135632 & 0.989848 & 0.00140937 & 0.507725 & 0.980044 & 0.136756 \\\\\n",
       "\t9 & 0.01 & 1.02628 & 0.142659 & 0.989875 & 0.00140739 & 0.505748 & 0.982679 & 0.137296 \\\\\n",
       "\t10 & 0.01 & 0.784794 & 0.0965507 & 0.987004 & 0.00159884 & 0.499745 & 0.764473 & 0.0946695 \\\\\n",
       "\t11 & 0.01 & 1.05709 & 0.154837 & 0.990596 & 0.00137768 & 0.500528 & 1.05835 & 0.15578 \\\\\n",
       "\t12 & 0.01 & 1.03362 & 0.146953 & 0.990317 & 0.0013767 & 0.501351 & 1.0277 & 0.146824 \\\\\n",
       "\t13 & 0.01 & 1.02386 & 0.144996 & 0.990217 & 0.00138545 & 0.504826 & 1.01718 & 0.144763 \\\\\n",
       "\t14 & 0.01 & 0.89694 & 0.116242 & 0.988344 & 0.00151053 & 0.499155 & 0.852948 & 0.11119 \\\\\n",
       "\t15 & 0.01 & 1.09595 & 0.160749 & 0.990855 & 0.00134138 & 0.495768 & 1.0885 & 0.160397 \\\\\n",
       "\t16 & 0.01 & 0.999807 & 0.140607 & 0.990116 & 0.00139007 & 0.492639 & 1.00669 & 0.142279 \\\\\n",
       "\t17 & 0.01 & 0.94064 & 0.122989 & 0.988553 & 0.00149666 & 0.500394 & 0.868597 & 0.114224 \\\\\n",
       "\t18 & 0.01 & 1.08409 & 0.158921 & 0.990757 & 0.00135514 & 0.503413 & 1.07684 & 0.158607 \\\\\n",
       "\t19 & 0.01 & 0.997493 & 0.139638 & 0.990025 & 0.00139637 & 0.499951 & 0.997509 & 0.140342 \\\\\n",
       "\t20 & 0.01 & 1.15075 & 0.161864 & 0.989939 & 0.00141536 & 0.49969 & 0.988882 & 0.139813 \\\\\n",
       "\t21 & 0.0914286 & 1.04314 & 0.0493899 & 0.914461 & 0.00404636 & 0.508947 & 1.02246 & 0.050595 \\\\\n",
       "\t22 & 0.0914286 & 1.06262 & 0.0499134 & 0.913273 & 0.00406958 & 0.494141 & 1.0078 & 0.0495017 \\\\\n",
       "\t23 & 0.0914286 & 0.978303 & 0.0456756 & 0.912253 & 0.0040925 & 0.506486 & 0.995543 & 0.0486309 \\\\\n",
       "\t24 & 0.0914286 & 1.05718 & 0.050938 & 0.91734 & 0.00397932 & 0.503604 & 1.05971 & 0.053281 \\\\\n",
       "\t25 & 0.0914286 & 1.05326 & 0.0492939 & 0.912566 & 0.00408801 & 0.502143 & 0.999275 & 0.0489256 \\\\\n",
       "\t26 & 0.0914286 & 0.946711 & 0.0412176 & 0.899592 & 0.00436565 & 0.490893 & 0.864052 & 0.0396279 \\\\\n",
       "\t27 & 0.0914286 & 1.03286 & 0.0494316 & 0.916295 & 0.00400232 & 0.506603 & 1.0459 & 0.0522604 \\\\\n",
       "\t28 & 0.0914286 & 1.00308 & 0.0467404 & 0.911919 & 0.00409998 & 0.494778 & 0.99159 & 0.0483513 \\\\\n",
       "\t29 & 0.0914286 & 1.01753 & 0.0492375 & 0.918014 & 0.00396385 & 0.499521 & 1.0688 & 0.0539488 \\\\\n",
       "\t30 & 0.0914286 & 1.03965 & 0.0496168 & 0.915784 & 0.00401559 & 0.496663 & 1.03925 & 0.0517986 \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "996×8 DataFrame. Omitted printing of 2 columns\n",
       "│ Row │ dt      │ A        │ dA        │ B          │ dB         │ sigmaN   │\n",
       "│     │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m  │ \u001b[90mFloat64\u001b[39m   │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mFloat64\u001b[39m  │\n",
       "├─────┼─────────┼──────────┼───────────┼────────────┼────────────┼──────────┤\n",
       "│ 1   │ 0.01    │ 1.08626  │ 0.159642  │ 0.990854   │ 0.00134422 │ 0.500037 │\n",
       "│ 2   │ 0.01    │ 1.07464  │ 0.162053  │ 0.99139    │ 0.00129839 │ 0.498834 │\n",
       "│ 3   │ 0.01    │ 0.925471 │ 0.123341  │ 0.988966   │ 0.00147051 │ 0.503385 │\n",
       "│ 4   │ 0.01    │ 1.07908  │ 0.156615  │ 0.990725   │ 0.00134609 │ 0.501257 │\n",
       "│ 5   │ 0.01    │ 1.04388  │ 0.149837  │ 0.990493   │ 0.00136457 │ 0.501161 │\n",
       "│ 6   │ 0.01    │ 0.961766 │ 0.132662  │ 0.989716   │ 0.00141858 │ 0.501622 │\n",
       "│ 7   │ 0.01    │ 1.13384  │ 0.17078   │ 0.991271   │ 0.0013148  │ 0.498584 │\n",
       "│ 8   │ 0.01    │ 0.97696  │ 0.135632  │ 0.989848   │ 0.00140937 │ 0.507725 │\n",
       "│ 9   │ 0.01    │ 1.02628  │ 0.142659  │ 0.989875   │ 0.00140739 │ 0.505748 │\n",
       "│ 10  │ 0.01    │ 0.784794 │ 0.0965507 │ 0.987004   │ 0.00159884 │ 0.499745 │\n",
       "⋮\n",
       "│ 986 │ 4.0     │ 0.994508 │ 0.0140751 │ 0.0274702  │ 0.00999601 │ 0.499602 │\n",
       "│ 987 │ 4.0     │ 0.996089 │ 0.014091  │ 0.0172606  │ 0.00999827 │ 0.49976  │\n",
       "│ 988 │ 4.0     │ 1.00031  │ 0.0141528 │ 0.0209964  │ 0.00999769 │ 0.500005 │\n",
       "│ 989 │ 4.0     │ 1.00413  │ 0.0142042 │ 0.0160857  │ 0.00999865 │ 0.500268 │\n",
       "│ 990 │ 4.0     │ 0.992651 │ 0.0140405 │ 0.0128595  │ 0.00999903 │ 0.499564 │\n",
       "│ 991 │ 4.0     │ 1.01048  │ 0.0143025 │ 0.0291645  │ 0.00999583 │ 0.500602 │\n",
       "│ 992 │ 4.0     │ 0.97496  │ 0.013788  │ 0.00143049 │ 0.0100001  │ 0.498474 │\n",
       "│ 993 │ 4.0     │ 1.0225   │ 0.0144626 │ 0.0126061  │ 0.00999961 │ 0.501418 │\n",
       "│ 994 │ 4.0     │ 0.989132 │ 0.0139949 │ 0.0215448  │ 0.0099979  │ 0.499307 │\n",
       "│ 995 │ 4.0     │ 0.981817 │ 0.0138932 │ 0.0243486  │ 0.00999696 │ 0.498823 │\n",
       "│ 996 │ 4.0     │ 1.00402  │ 0.0142002 │ 0.00907142 │ 0.00999934 │ 0.500289 │"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "paradf_clean = paradf[paradf[!,:B].>0,:]\n",
    "paradf_clean[!,:tau] = -paradf_clean[!,:dt]./log.(paradf_clean[!,:B])\n",
    "paradf_clean[!,:dtau] = paradf_clean[!,:dt].*paradf_clean[!,:dB]./log.(paradf_clean[!,:B]).^2 ./paradf_clean[!,:B]\n",
    "paradf_clean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>dt</th><th>A_mean</th><th>dA_mean</th><th>B_mean</th><th>dB_mean</th><th>sigmaN_mean</th><th>tau_mean</th><th>dtau_mean</th></tr><tr><th></th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th></tr></thead><tbody><p>50 rows × 8 columns</p><tr><th>1</th><td>0.01</td><td>1.01866</td><td>0.143878</td><td>0.989993</td><td>0.00140091</td><td>0.500791</td><td>1.00423</td><td>0.142688</td></tr><tr><th>2</th><td>0.0914286</td><td>1.00941</td><td>0.0473107</td><td>0.912545</td><td>0.00408678</td><td>0.500061</td><td>1.00185</td><td>0.0491564</td></tr><tr><th>3</th><td>0.172857</td><td>0.980956</td><td>0.0333261</td><td>0.839198</td><td>0.00543583</td><td>0.501871</td><td>0.987651</td><td>0.0365475</td></tr><tr><th>4</th><td>0.254286</td><td>1.00801</td><td>0.028505</td><td>0.774075</td><td>0.00632792</td><td>0.495993</td><td>0.995185</td><td>0.0318294</td></tr><tr><th>5</th><td>0.335714</td><td>0.992886</td><td>0.0245496</td><td>0.711732</td><td>0.00702217</td><td>0.498332</td><td>0.989206</td><td>0.0287481</td></tr><tr><th>6</th><td>0.417143</td><td>0.997061</td><td>0.022444</td><td>0.658406</td><td>0.00752353</td><td>0.497149</td><td>1.00045</td><td>0.0274041</td></tr><tr><th>7</th><td>0.498571</td><td>1.01301</td><td>0.0210865</td><td>0.606846</td><td>0.00794518</td><td>0.490659</td><td>1.00036</td><td>0.0262652</td></tr><tr><th>8</th><td>0.58</td><td>1.00014</td><td>0.0196236</td><td>0.562209</td><td>0.00826598</td><td>0.501508</td><td>1.01011</td><td>0.0258432</td></tr><tr><th>9</th><td>0.661429</td><td>1.00816</td><td>0.018836</td><td>0.520844</td><td>0.00853453</td><td>0.497203</td><td>1.01522</td><td>0.0255239</td></tr><tr><th>10</th><td>0.742857</td><td>0.997664</td><td>0.0178085</td><td>0.477934</td><td>0.00878104</td><td>0.501143</td><td>1.00831</td><td>0.0251282</td></tr><tr><th>11</th><td>0.824286</td><td>1.01325</td><td>0.0173375</td><td>0.433692</td><td>0.00900721</td><td>0.488809</td><td>0.989051</td><td>0.0246271</td></tr><tr><th>12</th><td>0.905714</td><td>0.987038</td><td>0.0165349</td><td>0.409845</td><td>0.00911753</td><td>0.508909</td><td>1.01817</td><td>0.0254385</td></tr><tr><th>13</th><td>0.987143</td><td>1.01737</td><td>0.0165293</td><td>0.37139</td><td>0.00928331</td><td>0.486525</td><td>0.997791</td><td>0.0252005</td></tr><tr><th>14</th><td>1.06857</td><td>0.980006</td><td>0.0156734</td><td>0.349921</td><td>0.00936438</td><td>0.521378</td><td>1.01978</td><td>0.026029</td></tr><tr><th>15</th><td>1.15</td><td>0.987138</td><td>0.0155209</td><td>0.324949</td><td>0.00945506</td><td>0.509722</td><td>1.02453</td><td>0.0265495</td></tr><tr><th>16</th><td>1.23143</td><td>0.973942</td><td>0.0150654</td><td>0.298893</td><td>0.0095402</td><td>0.518412</td><td>1.0213</td><td>0.0270276</td></tr><tr><th>17</th><td>1.31286</td><td>0.998898</td><td>0.0152207</td><td>0.272842</td><td>0.00961838</td><td>0.503121</td><td>1.0121</td><td>0.027505</td></tr><tr><th>18</th><td>1.39429</td><td>0.982421</td><td>0.0148292</td><td>0.255907</td><td>0.00965923</td><td>0.514913</td><td>1.02792</td><td>0.0285819</td></tr><tr><th>19</th><td>1.47571</td><td>1.01253</td><td>0.0150439</td><td>0.222018</td><td>0.00975021</td><td>0.479475</td><td>0.980859</td><td>0.0286399</td></tr><tr><th>20</th><td>1.55714</td><td>0.973752</td><td>0.014437</td><td>0.217453</td><td>0.00975741</td><td>0.521663</td><td>1.02219</td><td>0.0301379</td></tr><tr><th>21</th><td>1.63857</td><td>0.989689</td><td>0.0145152</td><td>0.190196</td><td>0.00981655</td><td>0.498601</td><td>0.987598</td><td>0.0307476</td></tr><tr><th>22</th><td>1.72</td><td>1.00215</td><td>0.0146206</td><td>0.176026</td><td>0.0098438</td><td>0.498819</td><td>0.990278</td><td>0.0319023</td></tr><tr><th>23</th><td>1.80143</td><td>1.00302</td><td>0.0145894</td><td>0.167166</td><td>0.00985924</td><td>0.500092</td><td>1.00725</td><td>0.0332537</td></tr><tr><th>24</th><td>1.88286</td><td>0.997016</td><td>0.0144324</td><td>0.152108</td><td>0.00988301</td><td>0.499803</td><td>0.99995</td><td>0.0345506</td></tr><tr><th>25</th><td>1.96429</td><td>1.00222</td><td>0.0144479</td><td>0.137574</td><td>0.00990428</td><td>0.500201</td><td>0.990316</td><td>0.0360382</td></tr><tr><th>26</th><td>2.04571</td><td>0.997269</td><td>0.0143401</td><td>0.128384</td><td>0.00991707</td><td>0.499839</td><td>0.996535</td><td>0.0375807</td></tr><tr><th>27</th><td>2.12714</td><td>1.00065</td><td>0.0143702</td><td>0.123679</td><td>0.00992333</td><td>0.499851</td><td>1.0177</td><td>0.0391021</td></tr><tr><th>28</th><td>2.20857</td><td>0.99794</td><td>0.0142743</td><td>0.106203</td><td>0.0099435</td><td>0.499989</td><td>0.984743</td><td>0.0411962</td></tr><tr><th>29</th><td>2.29</td><td>1.00051</td><td>0.0142896</td><td>0.0983403</td><td>0.00995095</td><td>0.500066</td><td>0.986751</td><td>0.0433091</td></tr><tr><th>30</th><td>2.37143</td><td>1.00403</td><td>0.0143236</td><td>0.0926649</td><td>0.00995656</td><td>0.500246</td><td>0.996393</td><td>0.0452235</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cccccccc}\n",
       "\t& dt & A\\_mean & dA\\_mean & B\\_mean & dB\\_mean & sigmaN\\_mean & tau\\_mean & dtau\\_mean\\\\\n",
       "\t\\hline\n",
       "\t& Float64 & Float64 & Float64 & Float64 & Float64 & Float64 & Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & 0.01 & 1.01866 & 0.143878 & 0.989993 & 0.00140091 & 0.500791 & 1.00423 & 0.142688 \\\\\n",
       "\t2 & 0.0914286 & 1.00941 & 0.0473107 & 0.912545 & 0.00408678 & 0.500061 & 1.00185 & 0.0491564 \\\\\n",
       "\t3 & 0.172857 & 0.980956 & 0.0333261 & 0.839198 & 0.00543583 & 0.501871 & 0.987651 & 0.0365475 \\\\\n",
       "\t4 & 0.254286 & 1.00801 & 0.028505 & 0.774075 & 0.00632792 & 0.495993 & 0.995185 & 0.0318294 \\\\\n",
       "\t5 & 0.335714 & 0.992886 & 0.0245496 & 0.711732 & 0.00702217 & 0.498332 & 0.989206 & 0.0287481 \\\\\n",
       "\t6 & 0.417143 & 0.997061 & 0.022444 & 0.658406 & 0.00752353 & 0.497149 & 1.00045 & 0.0274041 \\\\\n",
       "\t7 & 0.498571 & 1.01301 & 0.0210865 & 0.606846 & 0.00794518 & 0.490659 & 1.00036 & 0.0262652 \\\\\n",
       "\t8 & 0.58 & 1.00014 & 0.0196236 & 0.562209 & 0.00826598 & 0.501508 & 1.01011 & 0.0258432 \\\\\n",
       "\t9 & 0.661429 & 1.00816 & 0.018836 & 0.520844 & 0.00853453 & 0.497203 & 1.01522 & 0.0255239 \\\\\n",
       "\t10 & 0.742857 & 0.997664 & 0.0178085 & 0.477934 & 0.00878104 & 0.501143 & 1.00831 & 0.0251282 \\\\\n",
       "\t11 & 0.824286 & 1.01325 & 0.0173375 & 0.433692 & 0.00900721 & 0.488809 & 0.989051 & 0.0246271 \\\\\n",
       "\t12 & 0.905714 & 0.987038 & 0.0165349 & 0.409845 & 0.00911753 & 0.508909 & 1.01817 & 0.0254385 \\\\\n",
       "\t13 & 0.987143 & 1.01737 & 0.0165293 & 0.37139 & 0.00928331 & 0.486525 & 0.997791 & 0.0252005 \\\\\n",
       "\t14 & 1.06857 & 0.980006 & 0.0156734 & 0.349921 & 0.00936438 & 0.521378 & 1.01978 & 0.026029 \\\\\n",
       "\t15 & 1.15 & 0.987138 & 0.0155209 & 0.324949 & 0.00945506 & 0.509722 & 1.02453 & 0.0265495 \\\\\n",
       "\t16 & 1.23143 & 0.973942 & 0.0150654 & 0.298893 & 0.0095402 & 0.518412 & 1.0213 & 0.0270276 \\\\\n",
       "\t17 & 1.31286 & 0.998898 & 0.0152207 & 0.272842 & 0.00961838 & 0.503121 & 1.0121 & 0.027505 \\\\\n",
       "\t18 & 1.39429 & 0.982421 & 0.0148292 & 0.255907 & 0.00965923 & 0.514913 & 1.02792 & 0.0285819 \\\\\n",
       "\t19 & 1.47571 & 1.01253 & 0.0150439 & 0.222018 & 0.00975021 & 0.479475 & 0.980859 & 0.0286399 \\\\\n",
       "\t20 & 1.55714 & 0.973752 & 0.014437 & 0.217453 & 0.00975741 & 0.521663 & 1.02219 & 0.0301379 \\\\\n",
       "\t21 & 1.63857 & 0.989689 & 0.0145152 & 0.190196 & 0.00981655 & 0.498601 & 0.987598 & 0.0307476 \\\\\n",
       "\t22 & 1.72 & 1.00215 & 0.0146206 & 0.176026 & 0.0098438 & 0.498819 & 0.990278 & 0.0319023 \\\\\n",
       "\t23 & 1.80143 & 1.00302 & 0.0145894 & 0.167166 & 0.00985924 & 0.500092 & 1.00725 & 0.0332537 \\\\\n",
       "\t24 & 1.88286 & 0.997016 & 0.0144324 & 0.152108 & 0.00988301 & 0.499803 & 0.99995 & 0.0345506 \\\\\n",
       "\t25 & 1.96429 & 1.00222 & 0.0144479 & 0.137574 & 0.00990428 & 0.500201 & 0.990316 & 0.0360382 \\\\\n",
       "\t26 & 2.04571 & 0.997269 & 0.0143401 & 0.128384 & 0.00991707 & 0.499839 & 0.996535 & 0.0375807 \\\\\n",
       "\t27 & 2.12714 & 1.00065 & 0.0143702 & 0.123679 & 0.00992333 & 0.499851 & 1.0177 & 0.0391021 \\\\\n",
       "\t28 & 2.20857 & 0.99794 & 0.0142743 & 0.106203 & 0.0099435 & 0.499989 & 0.984743 & 0.0411962 \\\\\n",
       "\t29 & 2.29 & 1.00051 & 0.0142896 & 0.0983403 & 0.00995095 & 0.500066 & 0.986751 & 0.0433091 \\\\\n",
       "\t30 & 2.37143 & 1.00403 & 0.0143236 & 0.0926649 & 0.00995656 & 0.500246 & 0.996393 & 0.0452235 \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "50×8 DataFrame. Omitted printing of 3 columns\n",
       "│ Row │ dt        │ A_mean   │ dA_mean   │ B_mean    │ dB_mean    │\n",
       "│     │ \u001b[90mFloat64\u001b[39m   │ \u001b[90mFloat64\u001b[39m  │ \u001b[90mFloat64\u001b[39m   │ \u001b[90mFloat64\u001b[39m   │ \u001b[90mFloat64\u001b[39m    │\n",
       "├─────┼───────────┼──────────┼───────────┼───────────┼────────────┤\n",
       "│ 1   │ 0.01      │ 1.01866  │ 0.143878  │ 0.989993  │ 0.00140091 │\n",
       "│ 2   │ 0.0914286 │ 1.00941  │ 0.0473107 │ 0.912545  │ 0.00408678 │\n",
       "│ 3   │ 0.172857  │ 0.980956 │ 0.0333261 │ 0.839198  │ 0.00543583 │\n",
       "│ 4   │ 0.254286  │ 1.00801  │ 0.028505  │ 0.774075  │ 0.00632792 │\n",
       "│ 5   │ 0.335714  │ 0.992886 │ 0.0245496 │ 0.711732  │ 0.00702217 │\n",
       "│ 6   │ 0.417143  │ 0.997061 │ 0.022444  │ 0.658406  │ 0.00752353 │\n",
       "│ 7   │ 0.498571  │ 1.01301  │ 0.0210865 │ 0.606846  │ 0.00794518 │\n",
       "│ 8   │ 0.58      │ 1.00014  │ 0.0196236 │ 0.562209  │ 0.00826598 │\n",
       "│ 9   │ 0.661429  │ 1.00816  │ 0.018836  │ 0.520844  │ 0.00853453 │\n",
       "│ 10  │ 0.742857  │ 0.997664 │ 0.0178085 │ 0.477934  │ 0.00878104 │\n",
       "⋮\n",
       "│ 40  │ 3.18571   │ 1.00396  │ 0.0142293 │ 0.0460455 │ 0.0099895  │\n",
       "│ 41  │ 3.26714   │ 0.997665 │ 0.0141312 │ 0.0369846 │ 0.00999268 │\n",
       "│ 42  │ 3.34857   │ 0.997124 │ 0.0141248 │ 0.039466  │ 0.00999212 │\n",
       "│ 43  │ 3.43      │ 1.00575  │ 0.0142393 │ 0.031198  │ 0.00999489 │\n",
       "│ 44  │ 3.51143   │ 1.00243  │ 0.0141897 │ 0.0279945 │ 0.00999604 │\n",
       "│ 45  │ 3.59286   │ 0.996166 │ 0.0140982 │ 0.0260465 │ 0.00999654 │\n",
       "│ 46  │ 3.67429   │ 1.00723  │ 0.0142538 │ 0.0233079 │ 0.00999712 │\n",
       "│ 47  │ 3.75571   │ 0.996967 │ 0.0141108 │ 0.0238641 │ 0.00999596 │\n",
       "│ 48  │ 3.83714   │ 0.999336 │ 0.0141424 │ 0.0241162 │ 0.00999707 │\n",
       "│ 49  │ 3.91857   │ 1.00659  │ 0.0142406 │ 0.0168971 │ 0.00999886 │\n",
       "│ 50  │ 4.0       │ 0.997532 │ 0.0141139 │ 0.019533  │ 0.00999775 │"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Statistics\n",
    "df_mean = aggregate(paradf_clean, :dt, mean)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "plot(df_mean.dt, df_mean.dtau_mean,label=\"dtau\")\n",
    "plot!(df_mean.dt, df_mean.dA_mean,label=\"dA\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
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       "  660.328,378.33 702.147,765.423 743.967,317.485 785.786,869.26 827.605,763.944 869.425,958.805 911.244,590.28 953.063,833.6 994.883,389.019 1036.7,961.609 \n",
       "  1078.52,726.281 1120.34,542.254 1162.16,529.489 1203.98,618.073 1245.8,541.188 1287.62,614.343 1329.44,564.483 1371.26,604.427 1413.08,566.514 1454.9,514.529 \n",
       "  1496.71,562.426 1538.53,416.06 1580.35,535.518 1622.17,654.541 1663.99,593.061 1705.81,562.993 1747.63,445.849 1789.45,721.968 1831.27,690.779 1873.09,515.496 \n",
       "  1914.91,608.492 1956.73,616.486 1998.55,489.073 2040.37,538.153 2082.18,630.635 2124,467.179 2165.82,618.808 2207.64,583.825 2249.46,476.687 2291.28,610.454 \n",
       "  \n",
       "  \"/>\n",
       "<path clip-path=\"url(#clip1200)\" d=\"\n",
       "M1973.85 312.204 L2280.76 312.204 L2280.76 130.764 L1973.85 130.764  Z\n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<polyline clip-path=\"url(#clip1200)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  1973.85,312.204 2280.76,312.204 2280.76,130.764 1973.85,130.764 1973.85,312.204 \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip1200)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  1997.85,191.244 2141.85,191.244 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip1200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2165.85, 208.744)\" x=\"2165.85\" y=\"208.744\">tau</text>\n",
       "</g>\n",
       "<polyline clip-path=\"url(#clip1200)\" style=\"stroke:#e26f46; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  1997.85,251.724 2141.85,251.724 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip1200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2165.85, 269.224)\" x=\"2165.85\" y=\"269.224\">A</text>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "plot(df_mean.dt, df_mean.tau_mean,label=\"tau\")\n",
    "plot!(df_mean.dt, df_mean.A_mean,label=\"A\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Precompiling CSV [336ed68f-0bac-5ca0-87d4-7b16caf5d00b]\n",
      "└ @ Base loading.jl:1260\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\"OUEM_sN05.csv\""
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using CSV\n",
    "CSV.write(\"OUEM_sN05.csv\",df_mean)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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