noisy_correlation/correlation_demo.ipynb

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  {
   "cell_type": "markdown",
   "id": "9876bb8c",
   "metadata": {},
   "source": [
    "# Estimating the true correlations between activity patterns\n",
    "This jupyter notebook contains the simulations underlying the blog [How can we estimate the *true* correlation between two activity patterns?](http://diedrichsenlab.org/BrainDataScience/noisy_correlation/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "1f704544",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np \n",
    "import PcmPy as pcm\n",
    "from PcmPy import sim\n",
    "import pandas as pd\n",
    "from correlation_demo import plot_Figure2, plot_Figure3, plot_Figure4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "042cc2bf",
   "metadata": {},
   "source": [
    "## Simply correlating the mean activity patterns\n",
    "First look at the problem of simply using the correlation between two measured activty patterns. First, we write a little function to calculate the correlation between two mean activity patterns. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "4ce90db4",
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_corr(X,cond_vec):\n",
    "    \"\"\"\n",
    "        Get normal correlation\n",
    "    \"\"\"\n",
    "    p1 = X[cond_vec==0,:].mean(axis=0)\n",
    "    p2 = X[cond_vec==1,:].mean(axis=0)\n",
    "    return np.corrcoef(p1,p2)[0,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f81242a3",
   "metadata": {},
   "source": [
    "Now we use the pcm-toolbox to simulate data from a trye correlation model (M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "8e014409",
   "metadata": {},
   "outputs": [],
   "source": [
    "def do_sim(corr,signal=np.linspace(0,5,20),n_sim=50,randseed=None):\n",
    "    M = pcm.CorrelationModel('corr',num_items=1,corr=corr,cond_effect=False)\n",
    "    G,dG = M.predict([0,0])\n",
    "    cond_vec,part_vec = pcm.sim.make_design(2,2)\n",
    "    Lcorr = []\n",
    "    Lsign = []\n",
    "    rng = np.random.default_rng(randseed)\n",
    "\n",
    "    for s in signal:\n",
    "        D = pcm.sim.make_dataset(M, [0,0], cond_vec, n_sim=n_sim, signal=s,rng=rng)\n",
    "        for i in range(n_sim):\n",
    "            data = D[i].measurements\n",
    "            Lcorr.append(get_corr(data,cond_vec))\n",
    "            Lsign.append(s)\n",
    "\n",
    "    S = pd.DataFrame({'r_naive':Lcorr,'signal':Lsign})\n",
    "    S['true'] = np.ones((S.shape[0],))*corr\n",
    "    return S"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9f9cae9",
   "metadata": {},
   "source": [
    "Now we can run the simulation for Figure 2. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "4e1d492c",
   "metadata": {},
   "outputs": [
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    "# Make a spacing of different signals\n",
    "sig = np.linspace(0.1,5.1,21)# np.logspace(np.log(0.2),np.log(5),10)\n",
    "# Get the simulations\n",
    "D = do_sim(0.7,n_sim=200,signal=sig,randseed=10)\n",
    "# Summarize the mean and std deviation\n",
    "T = D.groupby(\"signal\").apply(np.mean)\n",
    "Tstd = D.groupby(\"signal\").apply(np.std)\n",
    "# Plot and show the Figure\n",
    "fig = plot_Figure2(D,T,Tstd)\n",
    "fig.update_layout(width=700,height=400)\n",
    "fig.show()\n"
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   "cell_type": "markdown",
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   "metadata": {},
   "source": [
    "This is a function that runs a specific simulation across a range of signal to noise levels. "
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   "id": "43cb4aa0",
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    "## Noise ceiling correction and cross-block estimates\n",
    "We can calculate the noise ceiling to correct for the bias in the correlation. Here a function that uses the cross-measurement correlations to obtain an estimate of the noise ceiling. "
   ]
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   "cell_type": "code",
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    "def get_noiseceil(X,cond_vec):\n",
    "    \"\"\"\n",
    "        Calculate noise ceiling over reliabilities\n",
    "    \"\"\"\n",
    "    rel = np.array([0.0,0.0])\n",
    "    for i in [0,1]:\n",
    "        N = np.sum(cond_vec==i) # Number of measurements\n",
    "        R = np.corrcoef(X[cond_vec==i,:]) # Correlation matrix\n",
    "        index_R = np.where(~np.eye(N,dtype=bool)) # Average cross-block correlations\n",
    "        r = np.mean(R[index_R]) # Average the non-diagnal elements.\n",
    "        rel[i] = r * N / (r*(N-1)+1) # Overall realibility of the mean\n",
    "\n",
    "    # Check if both are above zero\n",
    "    if rel[0]>0 and rel[1]>0:\n",
    "        noise_r = np.sqrt(rel[0]*rel[1])\n",
    "    else:\n",
    "        noise_r = np.nan\n",
    "    return noise_r\n"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Alternatively we can use the cross-block information direction to caluculate correct coefficients. Here we use `est_G_crossval` to calculate those covariances. "
   ]
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   "cell_type": "code",
   "execution_count": 31,
   "id": "217aa675",
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   "source": [
    "def get_crossblock(X,cond_vec,part_vec):\n",
    "    \"\"\"\n",
    "        calculate the cross-block correlation\n",
    "    \"\"\"\n",
    "    G = pcm.util.est_G_crossval(X,cond_vec,part_vec)\n",
    "    var = G[0][0,0]*G[0][1,1]\n",
    "    if var<=0:\n",
    "        crosscorr = np.nan\n",
    "    else:\n",
    "        crosscorr = G[0][0,1]/np.sqrt(var)\n",
    "    return crosscorr"
   ]
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   "cell_type": "markdown",
   "id": "96f1e818",
   "metadata": {},
   "source": [
    "Now we need to extend our do_sim function to get these addtional measures "
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   "cell_type": "code",
   "execution_count": 32,
   "id": "bd5d8df4",
   "metadata": {},
   "outputs": [],
   "source": [
    "def do_sim(corr,signal=np.linspace(0,5,20),n_sim=50,randseed=None):\n",
    "    M = pcm.CorrelationModel('corr',num_items=1,corr=corr,cond_effect=False)\n",
    "    G,dG = M.predict([0,0])\n",
    "    cond_vec,part_vec = pcm.sim.make_design(2,2)\n",
    "    Lcorr = []\n",
    "    LnoiseCeil = []\n",
    "    Lsign = []\n",
    "    LcrossBlock =[]\n",
    "    rng = np.random.default_rng(randseed)\n",
    "\n",
    "    for s in signal:\n",
    "        D = pcm.sim.make_dataset(M, [0,0], cond_vec, n_sim=n_sim, signal=s,rng=rng)\n",
    "        for i in range(n_sim):\n",
    "            data = D[i].measurements\n",
    "            Lcorr.append(get_corr(data,cond_vec))\n",
    "            LnoiseCeil.append(get_noiseceil(data,cond_vec))\n",
    "            LcrossBlock.append(get_crossblock(data,cond_vec,part_vec))\n",
    "            Lsign.append(s)\n",
    "    S = pd.DataFrame({'r_naive':Lcorr,'signal':Lsign,\n",
    "                        'noiseCeil':LnoiseCeil,'cross_block':LcrossBlock})\n",
    "    S['true'] = np.ones((S.shape[0],))*corr\n",
    "    return S"
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   "source": [
    "Now we can do the simulation - if you want more reliable results the number of simulations should be increase to 10000. "
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   "cell_type": "code",
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    "sig = np.array([0.1,0.15,0.2,0.25,0.3,0.35,0.4,0.5,0.6,0.7,0.8,0.9,1.5,2,2.5,3])# np.logspace(np.log(0.2),np.log(5),10)\n",
    "D = do_sim(0.7,n_sim=200,signal=sig,randseed=12)\n",
    "D['noiseceil_nan'] = np.isnan(D.noiseCeil)\n",
    "D['corr_corrected'] = D.r_naive / D.noiseCeil # Correction for noise ceiling \n",
    "D['corr_corrected_imp'] = D.corr_corrected\n",
    "D.loc[np.isnan(D.corr_corrected),'corr_corrected_imp']=0\n",
    "D['cross_block_imp'] = D.cross_block\n",
    "D.loc[np.isnan(D.cross_block),'cross_block_imp']=0\n",
    "T = D.groupby(\"signal\").apply(np.mean)\n",
    "Tstd = D.groupby(\"signal\").apply(np.std)\n",
    "fig = plot_Figure3(D,T,Tstd)\n",
    "fig.update_layout(width=700,height=400)\n",
    "fig.show()"
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   "source": [
    "## Solution 2: Using the likelihood\n",
    "First we define a function that simulates data and then fits it with a series of PCM models. "
   ]
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   "cell_type": "code",
   "execution_count": 3,
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    "def do_sim_corrpcm(corr=0.8,signal=0.5,n_sim=20,nsteps = 11,randseed=None):\n",
    "    # Make the design in this case it's 2 runs, 2 conditions!\n",
    "    cond_vec,part_vec = pcm.sim.make_design(2,2)\n",
    "    # Generate different models from 0 to 1\n",
    "    M=[]\n",
    "    for r in np.linspace(0,1,nsteps):\n",
    "        M.append(pcm.CorrelationModel(f\"{r:0.2f}\",num_items=1,corr=r,cond_effect=False))\n",
    "    Mflex = pcm.CorrelationModel(\"flex\",num_items=1,corr=None,cond_effect=False)\n",
    "    M.append(Mflex)\n",
    "    # Now do the simulations and fit with the models\n",
    "    rng = np.random.default_rng(randseed)\n",
    "    Mtrue = pcm.CorrelationModel('corr',num_items=1,corr=corr,cond_effect=False)\n",
    "    D = pcm.sim.make_dataset(Mtrue, [0,0], cond_vec,part_vec=part_vec,n_sim=n_sim, signal=signal, rng=rng)\n",
    "    T,theta = pcm.inference.fit_model_individ(D,M,fixed_effect=None,fit_scale=False,verbose = False)\n",
    "    return T,theta,M"
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   "source": [
    "T,theta,M = do_sim_corrpcm(corr=0.7,signal=0.5,n_sim=20,nsteps=21,randseed=11)\n",
    "fig=plot_Figure4(T,theta,M)\n",
    "fig.update_layout(width=700,height=400)\n",
    "fig.show()"
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  },
  {
   "cell_type": "markdown",
   "id": "e353f1a6",
   "metadata": {},
   "source": [
    "Note that the log-likelihood values are quite negative - and differ substantially across data sets. This is normal - the only thing we can interpret are differences between log-likelihoods across the same data set for different models. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d8009ad9",
   "metadata": {},
   "outputs": [
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       "      <th>0.00</th>\n",
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       "      <td>-73.188299</td>\n",
       "      <td>-73.082393</td>\n",
       "      <td>-72.984258</td>\n",
       "      <td>-72.894235</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>6.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-77.282291</td>\n",
       "      <td>-77.009676</td>\n",
       "      <td>-76.746336</td>\n",
       "      <td>-76.492943</td>\n",
       "      <td>-76.249959</td>\n",
       "      <td>-76.017729</td>\n",
       "      <td>-75.796557</td>\n",
       "      <td>-75.586744</td>\n",
       "      <td>-75.388627</td>\n",
       "      <td>-75.202605</td>\n",
       "      <td>...</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>8.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-83.491441</td>\n",
       "      <td>-83.240729</td>\n",
       "      <td>-82.997738</td>\n",
       "      <td>-82.763341</td>\n",
       "      <td>-82.538145</td>\n",
       "      <td>-82.322608</td>\n",
       "      <td>-82.117107</td>\n",
       "      <td>-81.921991</td>\n",
       "      <td>-81.737618</td>\n",
       "      <td>-81.564381</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>8.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-75.166672</td>\n",
       "      <td>-74.999432</td>\n",
       "      <td>-74.850917</td>\n",
       "      <td>-74.721235</td>\n",
       "      <td>-74.610569</td>\n",
       "      <td>-74.519197</td>\n",
       "      <td>-74.447497</td>\n",
       "      <td>-74.395945</td>\n",
       "      <td>-74.365090</td>\n",
       "      <td>-74.355498</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>4.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>-68.803865</td>\n",
       "      <td>-68.285783</td>\n",
       "      <td>-67.788955</td>\n",
       "      <td>-67.312309</td>\n",
       "      <td>-66.854954</td>\n",
       "      <td>-66.416196</td>\n",
       "      <td>-65.995559</td>\n",
       "      <td>-65.592813</td>\n",
       "      <td>-65.207999</td>\n",
       "      <td>-64.841480</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>7.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-84.549192</td>\n",
       "      <td>-84.274535</td>\n",
       "      <td>-84.006158</td>\n",
       "      <td>-83.745562</td>\n",
       "      <td>-83.493709</td>\n",
       "      <td>-83.251258</td>\n",
       "      <td>-83.018714</td>\n",
       "      <td>-82.796504</td>\n",
       "      <td>-82.585041</td>\n",
       "      <td>-82.384759</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>8.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>-64.024354</td>\n",
       "      <td>-63.773315</td>\n",
       "      <td>-63.537101</td>\n",
       "      <td>-63.316254</td>\n",
       "      <td>-63.111175</td>\n",
       "      <td>-62.922231</td>\n",
       "      <td>-62.749818</td>\n",
       "      <td>-62.594393</td>\n",
       "      <td>-62.456499</td>\n",
       "      <td>-62.336772</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>5.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-82.259471</td>\n",
       "      <td>-81.968971</td>\n",
       "      <td>-81.695224</td>\n",
       "      <td>-81.438186</td>\n",
       "      <td>-81.197853</td>\n",
       "      <td>-80.974303</td>\n",
       "      <td>-80.767729</td>\n",
       "      <td>-80.578462</td>\n",
       "      <td>-80.406992</td>\n",
       "      <td>-80.253985</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>5.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-69.506350</td>\n",
       "      <td>-69.254634</td>\n",
       "      <td>-68.993507</td>\n",
       "      <td>-68.727382</td>\n",
       "      <td>-68.459022</td>\n",
       "      <td>-68.190248</td>\n",
       "      <td>-67.922293</td>\n",
       "      <td>-67.656023</td>\n",
       "      <td>-67.392023</td>\n",
       "      <td>-67.130751</td>\n",
       "      <td>...</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>9.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>-76.422573</td>\n",
       "      <td>-76.265195</td>\n",
       "      <td>-76.116268</td>\n",
       "      <td>-75.976486</td>\n",
       "      <td>-75.846375</td>\n",
       "      <td>-75.726349</td>\n",
       "      <td>-75.616760</td>\n",
       "      <td>-75.517921</td>\n",
       "      <td>-75.430114</td>\n",
       "      <td>-75.353597</td>\n",
       "      <td>...</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>5.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>-82.964747</td>\n",
       "      <td>-82.891094</td>\n",
       "      <td>-82.820337</td>\n",
       "      <td>-82.753333</td>\n",
       "      <td>-82.690725</td>\n",
       "      <td>-82.632986</td>\n",
       "      <td>-82.580462</td>\n",
       "      <td>-82.533389</td>\n",
       "      <td>-82.491909</td>\n",
       "      <td>-82.456077</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>5.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>-71.393642</td>\n",
       "      <td>-71.100513</td>\n",
       "      <td>-70.822960</td>\n",
       "      <td>-70.561127</td>\n",
       "      <td>-70.315129</td>\n",
       "      <td>-70.085116</td>\n",
       "      <td>-69.871314</td>\n",
       "      <td>-69.674063</td>\n",
       "      <td>-69.493837</td>\n",
       "      <td>-69.331266</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>5.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>-76.469758</td>\n",
       "      <td>-76.166427</td>\n",
       "      <td>-75.888594</td>\n",
       "      <td>-75.635867</td>\n",
       "      <td>-75.408127</td>\n",
       "      <td>-75.205544</td>\n",
       "      <td>-75.028607</td>\n",
       "      <td>-74.878147</td>\n",
       "      <td>-74.755371</td>\n",
       "      <td>-74.661893</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>4.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>-81.981675</td>\n",
       "      <td>-81.750162</td>\n",
       "      <td>-81.546200</td>\n",
       "      <td>-81.369590</td>\n",
       "      <td>-81.220451</td>\n",
       "      <td>-81.099240</td>\n",
       "      <td>-81.006775</td>\n",
       "      <td>-80.944257</td>\n",
       "      <td>-80.913298</td>\n",
       "      <td>-80.915935</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>4.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>-88.695312</td>\n",
       "      <td>-88.212270</td>\n",
       "      <td>-87.744621</td>\n",
       "      <td>-87.291808</td>\n",
       "      <td>-86.853266</td>\n",
       "      <td>-86.428470</td>\n",
       "      <td>-86.016980</td>\n",
       "      <td>-85.618459</td>\n",
       "      <td>-85.232726</td>\n",
       "      <td>-84.859761</td>\n",
       "      <td>...</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>7.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>-71.802322</td>\n",
       "      <td>-71.625012</td>\n",
       "      <td>-71.443005</td>\n",
       "      <td>-71.259127</td>\n",
       "      <td>-71.075346</td>\n",
       "      <td>-70.893079</td>\n",
       "      <td>-70.713379</td>\n",
       "      <td>-70.537050</td>\n",
       "      <td>-70.364734</td>\n",
       "      <td>-70.196961</td>\n",
       "      <td>...</td>\n",
       "      <td>11.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>7.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>-62.851722</td>\n",
       "      <td>-62.766007</td>\n",
       "      <td>-62.687989</td>\n",
       "      <td>-62.619045</td>\n",
       "      <td>-62.559989</td>\n",
       "      <td>-62.511259</td>\n",
       "      <td>-62.472989</td>\n",
       "      <td>-62.445029</td>\n",
       "      <td>-62.426923</td>\n",
       "      <td>-62.417874</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>4.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>-59.295262</td>\n",
       "      <td>-58.983127</td>\n",
       "      <td>-58.696915</td>\n",
       "      <td>-58.436189</td>\n",
       "      <td>-58.200798</td>\n",
       "      <td>-57.990900</td>\n",
       "      <td>-57.806979</td>\n",
       "      <td>-57.649878</td>\n",
       "      <td>-57.520837</td>\n",
       "      <td>-57.421526</td>\n",
       "      <td>...</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>20 rows × 66 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "variable likelihood                                                         \\\n",
       "model          0.00       0.05       0.10       0.15       0.20       0.25   \n",
       "0        -80.213258 -79.975602 -79.743281 -79.518565 -79.302870 -79.097164   \n",
       "1        -80.890808 -80.451681 -80.034392 -79.638146 -79.262343 -78.906596   \n",
       "2        -73.952510 -73.814902 -73.679411 -73.547906 -73.421654 -73.301560   \n",
       "3        -77.282291 -77.009676 -76.746336 -76.492943 -76.249959 -76.017729   \n",
       "4        -83.491441 -83.240729 -82.997738 -82.763341 -82.538145 -82.322608   \n",
       "5        -75.166672 -74.999432 -74.850917 -74.721235 -74.610569 -74.519197   \n",
       "6        -68.803865 -68.285783 -67.788955 -67.312309 -66.854954 -66.416196   \n",
       "7        -84.549192 -84.274535 -84.006158 -83.745562 -83.493709 -83.251258   \n",
       "8        -64.024354 -63.773315 -63.537101 -63.316254 -63.111175 -62.922231   \n",
       "9        -82.259471 -81.968971 -81.695224 -81.438186 -81.197853 -80.974303   \n",
       "10       -69.506350 -69.254634 -68.993507 -68.727382 -68.459022 -68.190248   \n",
       "11       -76.422573 -76.265195 -76.116268 -75.976486 -75.846375 -75.726349   \n",
       "12       -82.964747 -82.891094 -82.820337 -82.753333 -82.690725 -82.632986   \n",
       "13       -71.393642 -71.100513 -70.822960 -70.561127 -70.315129 -70.085116   \n",
       "14       -76.469758 -76.166427 -75.888594 -75.635867 -75.408127 -75.205544   \n",
       "15       -81.981675 -81.750162 -81.546200 -81.369590 -81.220451 -81.099240   \n",
       "16       -88.695312 -88.212270 -87.744621 -87.291808 -86.853266 -86.428470   \n",
       "17       -71.802322 -71.625012 -71.443005 -71.259127 -71.075346 -70.893079   \n",
       "18       -62.851722 -62.766007 -62.687989 -62.619045 -62.559989 -62.511259   \n",
       "19       -59.295262 -58.983127 -58.696915 -58.436189 -58.200798 -57.990900   \n",
       "\n",
       "variable                                              ... iterations        \\\n",
       "model          0.30       0.35       0.40       0.45  ...       0.60  0.65   \n",
       "0        -78.902173 -78.718499 -78.546692 -78.387295  ...        4.0   4.0   \n",
       "1        -78.570757 -78.254942 -77.959566 -77.685386  ...        4.0   4.0   \n",
       "2        -73.188299 -73.082393 -72.984258 -72.894235  ...        4.0   4.0   \n",
       "3        -75.796557 -75.586744 -75.388627 -75.202605  ...        5.0   5.0   \n",
       "4        -82.117107 -81.921991 -81.737618 -81.564381  ...        4.0   4.0   \n",
       "5        -74.447497 -74.395945 -74.365090 -74.355498  ...        4.0   5.0   \n",
       "6        -65.995559 -65.592813 -65.207999 -64.841480  ...        4.0   4.0   \n",
       "7        -83.018714 -82.796504 -82.585041 -82.384759  ...        4.0   4.0   \n",
       "8        -62.749818 -62.594393 -62.456499 -62.336772  ...        4.0   4.0   \n",
       "9        -80.767729 -80.578462 -80.406992 -80.253985  ...        4.0   4.0   \n",
       "10       -67.922293 -67.656023 -67.392023 -67.130751  ...        5.0   5.0   \n",
       "11       -75.616760 -75.517921 -75.430114 -75.353597  ...        3.0   3.0   \n",
       "12       -82.580462 -82.533389 -82.491909 -82.456077  ...        4.0   4.0   \n",
       "13       -69.871314 -69.674063 -69.493837 -69.331266  ...        4.0   4.0   \n",
       "14       -75.028607 -74.878147 -74.755371 -74.661893  ...        4.0   4.0   \n",
       "15       -81.006775 -80.944257 -80.913298 -80.915935  ...        4.0   4.0   \n",
       "16       -86.016980 -85.618459 -85.232726 -84.859761  ...        5.0   5.0   \n",
       "17       -70.713379 -70.537050 -70.364734 -70.196961  ...       11.0  10.0   \n",
       "18       -62.472989 -62.445029 -62.426923 -62.417874  ...        4.0   4.0   \n",
       "19       -57.806979 -57.649878 -57.520837 -57.421526  ...        4.0   4.0   \n",
       "\n",
       "variable                                              \n",
       "model     0.70  0.75  0.80  0.85 0.90 0.95 1.00 flex  \n",
       "0          4.0   4.0   4.0   4.0  4.0  4.0  4.0  7.0  \n",
       "1          4.0   4.0   4.0   4.0  4.0  4.0  4.0  5.0  \n",
       "2          4.0   4.0   4.0   4.0  4.0  4.0  3.0  6.0  \n",
       "3          4.0   4.0   4.0   4.0  4.0  4.0  4.0  8.0  \n",
       "4          4.0   4.0   4.0   4.0  4.0  4.0  4.0  8.0  \n",
       "5          5.0   5.0   5.0   6.0  6.0  6.0  6.0  4.0  \n",
       "6          4.0   4.0   4.0   4.0  4.0  4.0  4.0  7.0  \n",
       "7          4.0   4.0   4.0   4.0  4.0  4.0  4.0  8.0  \n",
       "8          4.0   4.0   3.0   3.0  4.0  4.0  4.0  5.0  \n",
       "9          4.0   3.0   3.0   4.0  4.0  4.0  4.0  5.0  \n",
       "10         5.0   5.0   5.0   5.0  5.0  5.0  5.0  9.0  \n",
       "11         3.0   3.0   3.0   4.0  4.0  4.0  4.0  5.0  \n",
       "12         4.0   4.0   4.0   4.0  4.0  4.0  4.0  5.0  \n",
       "13         4.0   3.0   3.0   3.0  3.0  4.0  4.0  5.0  \n",
       "14         4.0   4.0   5.0   5.0  5.0  5.0  6.0  4.0  \n",
       "15         4.0   4.0   5.0   5.0  5.0  5.0  6.0  4.0  \n",
       "16         5.0   5.0   4.0   4.0  4.0  4.0  4.0  7.0  \n",
       "17        10.0  10.0  10.0  10.0  5.0  5.0  5.0  7.0  \n",
       "18         4.0   5.0   5.0   5.0  5.0  5.0  5.0  4.0  \n",
       "19         4.0   4.0   4.0   4.0  4.0  4.0  4.0  4.0  \n",
       "\n",
       "[20 rows x 66 columns]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T"
   ]
  }
 ],
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