juliaL19b.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear classifiers: `predict`\n",
    "\n",
    "CPSC 340: Machine Learning and Data Mining\n",
    "\n",
    "The University of British Columbia\n,
    "\n",
    "2018 Winter Term 1\n",
    "\n",
    "By Mike Gelbart (converted to Julia by Jason Chau)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: redefining constant LogisticRegression\n"
     ]
    }
   ],
   "source": [
    "using Distributions\n",
    "using ScikitLearn: fit!, predict, @sk_import\n",
    "@sk_import linear_model: LogisticRegression\n",
    "include(\"plot2Dclassifier.jl\");\n",
    "using Plots\n",
    "using Plots.PlotMeasures"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear decision boundaries and the coefficients\n",
    "\n",
    "- Key idea: compute $w^Tx$ (or $w^Tx+\\beta$) just like in linear regression, but then look at the **sign** of this function to make a prediction. \n",
    "\n",
    "$$\\hat{y}_i = \\textrm{sign}(w^Tx_i)$$\n",
    "\n",
    "with the convention that $y_i \\in \\{-1,+1\\}$.\n",
    "\n",
    "We're actually fitting a hyperplane through the space, that divides it in half. The $w$ represents the tilt of the hyperplane and $\\beta$ represents the intercept (offset from the origin)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 20\n",
    "X = rand(Normal(),N,2)\n",
    "y = rand([-1,+1],N)\n",
    "X[y.>0,1] .-= 1\n",
    "X[y.>0,2] .+= 1;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = LogisticRegression(solver=\"lbfgs\") # can change to LinearSVC and get the same plot\n",
    "fit!(model,X,y);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# # Let's play with the coefficients and see what they do.\n",
    "model.coef_ = [-2.0 2.0]  # this is w\n",
    "model.intercept_ = [0.0] # this is β\n",
    "\n",
    "# Plot the data and decision boundary\n",
    "plot2Dclassifier(X,y,model);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Demo: change the coefficients and see what happens.\n",
    "\n",
    "Question: why do we need 2 coefficients to encode a slope? Indeed, what if we double the coefficients? (Answer coming soon!)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Probabilities and `predict_proba`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {},
   "outputs": [],
   "source": [
    "lr = LogisticRegression(solver=\"lbfgs\") # can change to LinearSVC and get the same plot\n",
    "fit!(lr,X,y);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20-element Array{Int64,1}:\n",
       "  1\n",
       " -1\n",
       "  1\n",
       "  1\n",
       "  1\n",
       "  1\n",
       "  1\n",
       "  1\n",
       "  1\n",
       "  1\n",
       " -1\n",
       "  1\n",
       " -1\n",
       " -1\n",
       " -1\n",
       " -1\n",
       " -1\n",
       "  1\n",
       " -1\n",
       "  1"
      ]
     },
     "execution_count": 200,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predict(lr,X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20×2 Array{Float64,2}:\n",
       " 0.340397   0.659603 \n",
       " 0.933587   0.0664133\n",
       " 0.209852   0.790148 \n",
       " 0.488435   0.511565 \n",
       " 0.471468   0.528532 \n",
       " 0.235001   0.764999 \n",
       " 0.176042   0.823958 \n",
       " 0.0390351  0.960965 \n",
       " 0.488531   0.511469 \n",
       " 0.235268   0.764732 \n",
       " 0.65948    0.34052  \n",
       " 0.368628   0.631372 \n",
       " 0.723364   0.276636 \n",
       " 0.69795    0.30205  \n",
       " 0.739072   0.260928 \n",
       " 0.787553   0.212447 \n",
       " 0.682004   0.317996 \n",
       " 0.357656   0.642344 \n",
       " 0.90814    0.0918596\n",
       " 0.458527   0.541473 "
      ]
     },
     "execution_count": 201,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test = lr.predict_proba(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Logistic regression can output probabilities of each class. Note that the probabilities add up to 1 for each row. We can plot these probabilities (say the second column)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 2 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lr = LogisticRegression(C=1e6,solver=\"lbfgs\") # very little regularization, C=1/λ\n",
    "lr.fit(X,y)\n",
    "plot2Dclassifier(X,y,lr,proba=true);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Going back to the issue of the scale of the coefficents, let's try changing them now."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 2 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit!(lr,X,y)\n",
    "lr.coef_      *= 10\n",
    "lr.intercept_ *= 10\n",
    "\n",
    "plot2Dclassifier(X,y,lr, proba=true);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 2 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit!(lr,X,y)\n",
    "lr.coef_      /= 5\n",
    "lr.intercept_ /= 5\n",
    "\n",
    "plot2Dclassifier(X,y,lr, proba=true);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, we have an interpretation for regularization here: smaller coefficients means less confident predictions. \n",
    "\n",
    "This makes sense -- there's a relationship between overconfidence and overfitting."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Where do these probabilities come from?\n",
    "\n",
    "- The probabilities come from a softer version of taking $\\textrm{sign}(w^Tx)$. \n",
    "- Instead we take $h(w^Tx)$ where $$h(z)\\equiv \\frac{1}{1+\\exp(-z)}$$ \n",
    "- This is \"squishing\" the output from any number to $[0,1]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {},
   "outputs": [
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       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 994.583, 1385.83)\" x=\"994.583\" y=\"1385.83\">0.0</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 1232.79, 1385.83)\" x=\"1232.79\" y=\"1385.83\">2.5</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 1470.99, 1385.83)\" x=\"1470.99\" y=\"1385.83\">5.0</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:end;\" transform=\"rotate(0, 269.592, 1313.68)\" x=\"269.592\" y=\"1313.68\">0.00</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:end;\" transform=\"rotate(0, 269.592, 1038.01)\" x=\"269.592\" y=\"1038.01\">0.25</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:end;\" transform=\"rotate(0, 269.592, 762.351)\" x=\"269.592\" y=\"762.351\">0.50</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:end;\" transform=\"rotate(0, 269.592, 486.688)\" x=\"269.592\" y=\"486.688\">0.75</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:end;\" transform=\"rotate(0, 269.592, 211.025)\" x=\"269.592\" y=\"211.025\">1.00</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:74px; text-anchor:middle;\" transform=\"rotate(0, 994.583, 64.8)\" x=\"994.583\" y=\"64.8\">logistic (sigmoid) function, 1/(1+\u001bxp(-w^Tx))</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:81px; text-anchor:middle;\" transform=\"rotate(0, 994.583, 1488.2)\" x=\"994.583\" y=\"1488.2\">w^Tx (raw model output)</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip4000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:81px; text-anchor:middle;\" transform=\"rotate(-90, 70.8, 741.354)\" x=\"70.8\" y=\"741.354\">predicted probability</text>\n",
       "</g>\n",
       "<polyline clip-path=\"url(#clip4002)\" style=\"stroke:#009af9; stroke-width:5; stroke-opacity:1; fill:none\" points=\"\n",
       "  327.61,1291.17 328.945,1291.16 330.28,1291.14 331.616,1291.13 332.951,1291.12 334.286,1291.1 335.622,1291.08 336.957,1291.07 338.292,1291.05 339.627,1291.04 \n",
       "  340.963,1291.02 342.298,1291.01 343.633,1290.99 344.969,1290.97 346.304,1290.96 347.639,1290.94 348.974,1290.92 350.31,1290.9 351.645,1290.89 352.98,1290.87 \n",
       "  354.316,1290.85 355.651,1290.83 356.986,1290.81 358.321,1290.79 359.657,1290.77 360.992,1290.75 362.327,1290.73 363.663,1290.71 364.998,1290.69 366.333,1290.67 \n",
       "  367.668,1290.65 369.004,1290.63 370.339,1290.61 371.674,1290.58 373.01,1290.56 374.345,1290.54 375.68,1290.51 377.015,1290.49 378.351,1290.47 379.686,1290.44 \n",
       "  381.021,1290.42 382.356,1290.39 383.692,1290.37 385.027,1290.34 386.362,1290.32 387.698,1290.29 389.033,1290.27 390.368,1290.24 391.703,1290.21 393.039,1290.18 \n",
       "  394.374,1290.16 395.709,1290.13 397.045,1290.1 398.38,1290.07 399.715,1290.04 401.05,1290.01 402.386,1289.98 403.721,1289.95 405.056,1289.92 406.392,1289.88 \n",
       "  407.727,1289.85 409.062,1289.82 410.397,1289.79 411.733,1289.75 413.068,1289.72 414.403,1289.68 415.739,1289.65 417.074,1289.61 418.409,1289.58 419.744,1289.54 \n",
       "  421.08,1289.5 422.415,1289.46 423.75,1289.43 425.086,1289.39 426.421,1289.35 427.756,1289.31 429.091,1289.27 430.427,1289.23 431.762,1289.19 433.097,1289.14 \n",
       "  434.432,1289.1 435.768,1289.06 437.103,1289.01 438.438,1288.97 439.774,1288.92 441.109,1288.88 442.444,1288.83 443.779,1288.78 445.115,1288.74 446.45,1288.69 \n",
       "  447.785,1288.64 449.121,1288.59 450.456,1288.54 451.791,1288.49 453.126,1288.44 454.462,1288.38 455.797,1288.33 457.132,1288.28 458.468,1288.22 459.803,1288.17 \n",
       "  461.138,1288.11 462.473,1288.05 463.809,1287.99 465.144,1287.94 466.479,1287.88 467.815,1287.82 469.15,1287.75 470.485,1287.69 471.82,1287.63 473.156,1287.56 \n",
       "  474.491,1287.5 475.826,1287.43 477.161,1287.37 478.497,1287.3 479.832,1287.23 481.167,1287.16 482.503,1287.09 483.838,1287.02 485.173,1286.95 486.508,1286.87 \n",
       "  487.844,1286.8 489.179,1286.72 490.514,1286.65 491.85,1286.57 493.185,1286.49 494.52,1286.41 495.855,1286.33 497.191,1286.25 498.526,1286.17 499.861,1286.08 \n",
       "  501.197,1286 502.532,1285.91 503.867,1285.82 505.202,1285.73 506.538,1285.64 507.873,1285.55 509.208,1285.46 510.544,1285.36 511.879,1285.27 513.214,1285.17 \n",
       "  514.549,1285.07 515.885,1284.97 517.22,1284.87 518.555,1284.77 519.891,1284.66 521.226,1284.56 522.561,1284.45 523.896,1284.34 525.232,1284.23 526.567,1284.12 \n",
       "  527.902,1284.01 529.237,1283.9 530.573,1283.78 531.908,1283.66 533.243,1283.54 534.579,1283.42 535.914,1283.3 537.249,1283.18 538.584,1283.05 539.92,1282.92 \n",
       "  541.255,1282.79 542.59,1282.66 543.926,1282.53 545.261,1282.39 546.596,1282.26 547.931,1282.12 549.267,1281.98 550.602,1281.83 551.937,1281.69 553.273,1281.54 \n",
       "  554.608,1281.39 555.943,1281.24 557.278,1281.09 558.614,1280.94 559.949,1280.78 561.284,1280.62 562.62,1280.46 563.955,1280.29 565.29,1280.13 566.625,1279.96 \n",
       "  567.961,1279.79 569.296,1279.62 570.631,1279.44 571.966,1279.26 573.302,1279.08 574.637,1278.9 575.972,1278.72 577.308,1278.53 578.643,1278.34 579.978,1278.15 \n",
       "  581.313,1277.95 582.649,1277.75 583.984,1277.55 585.319,1277.35 586.655,1277.14 587.99,1276.93 589.325,1276.72 590.66,1276.5 591.996,1276.29 593.331,1276.07 \n",
       "  594.666,1275.84 596.002,1275.61 597.337,1275.38 598.672,1275.15 600.007,1274.91 601.343,1274.68 602.678,1274.43 604.013,1274.19 605.349,1273.94 606.684,1273.68 \n",
       "  608.019,1273.43 609.354,1273.17 610.69,1272.9 612.025,1272.64 613.36,1272.36 614.696,1272.09 616.031,1271.81 617.366,1271.53 618.701,1271.24 620.037,1270.95 \n",
       "  621.372,1270.66 622.707,1270.36 624.042,1270.06 625.378,1269.76 626.713,1269.45 628.048,1269.13 629.384,1268.81 630.719,1268.49 632.054,1268.16 633.389,1267.83 \n",
       "  634.725,1267.5 636.06,1267.16 637.395,1266.81 638.731,1266.46 640.066,1266.11 641.401,1265.75 642.736,1265.38 644.072,1265.01 645.407,1264.64 646.742,1264.26 \n",
       "  648.078,1263.88 649.413,1263.49 650.748,1263.1 652.083,1262.7 653.419,1262.29 654.754,1261.88 656.089,1261.46 657.425,1261.04 658.76,1260.62 660.095,1260.18 \n",
       "  661.43,1259.75 662.766,1259.3 664.101,1258.85 665.436,1258.4 666.771,1257.93 668.107,1257.47 669.442,1256.99 670.777,1256.51 672.113,1256.02 673.448,1255.53 \n",
       "  674.783,1255.03 676.118,1254.53 677.454,1254.01 678.789,1253.49 680.124,1252.97 681.46,1252.43 682.795,1251.89 684.13,1251.34 685.465,1250.79 686.801,1250.23 \n",
       "  688.136,1249.66 689.471,1249.08 690.807,1248.5 692.142,1247.91 693.477,1247.31 694.812,1246.7 696.148,1246.09 697.483,1245.46 698.818,1244.83 700.154,1244.19 \n",
       "  701.489,1243.55 702.824,1242.89 704.159,1242.23 705.495,1241.55 706.83,1240.87 708.165,1240.18 709.501,1239.48 710.836,1238.78 712.171,1238.06 713.506,1237.33 \n",
       "  714.842,1236.6 716.177,1235.85 717.512,1235.1 718.847,1234.34 720.183,1233.56 721.518,1232.78 722.853,1231.99 724.189,1231.19 725.524,1230.37 726.859,1229.55 \n",
       "  728.194,1228.72 729.53,1227.87 730.865,1227.02 732.2,1226.16 733.536,1225.28 734.871,1224.39 736.206,1223.5 737.541,1222.59 738.877,1221.67 740.212,1220.74 \n",
       "  741.547,1219.8 742.883,1218.84 744.218,1217.88 745.553,1216.9 746.888,1215.91 748.224,1214.91 749.559,1213.9 750.894,1212.87 752.23,1211.84 753.565,1210.79 \n",
       "  754.9,1209.72 756.235,1208.65 757.571,1207.56 758.906,1206.46 760.241,1205.34 761.577,1204.22 762.912,1203.08 764.247,1201.92 765.582,1200.75 766.918,1199.57 \n",
       "  768.253,1198.38 769.588,1197.17 770.923,1195.94 772.259,1194.7 773.594,1193.45 774.929,1192.18 776.265,1190.9 777.6,1189.61 778.935,1188.3 780.27,1186.97 \n",
       "  781.606,1185.63 782.941,1184.27 784.276,1182.9 785.612,1181.51 786.947,1180.11 788.282,1178.69 789.617,1177.26 790.953,1175.8 792.288,1174.34 793.623,1172.85 \n",
       "  794.959,1171.36 796.294,1169.84 797.629,1168.31 798.964,1166.76 800.3,1165.19 801.635,1163.61 802.97,1162.01 804.306,1160.39 805.641,1158.76 806.976,1157.1 \n",
       "  808.311,1155.43 809.647,1153.75 810.982,1152.04 812.317,1150.32 813.652,1148.58 814.988,1146.82 816.323,1145.04 817.658,1143.24 818.994,1141.43 820.329,1139.59 \n",
       "  821.664,1137.74 822.999,1135.87 824.335,1133.98 825.67,1132.07 827.005,1130.15 828.341,1128.2 829.676,1126.23 831.011,1124.25 832.346,1122.24 833.682,1120.22 \n",
       "  835.017,1118.18 836.352,1116.11 837.688,1114.03 839.023,1111.93 840.358,1109.8 841.693,1107.66 843.029,1105.5 844.364,1103.31 845.699,1101.11 847.035,1098.89 \n",
       "  848.37,1096.64 849.705,1094.38 851.04,1092.09 852.376,1089.79 853.711,1087.46 855.046,1085.12 856.382,1082.75 857.717,1080.36 859.052,1077.95 860.387,1075.52 \n",
       "  861.723,1073.07 863.058,1070.6 864.393,1068.11 865.728,1065.6 867.064,1063.06 868.399,1060.51 869.734,1057.94 871.07,1055.34 872.405,1052.72 873.74,1050.09 \n",
       "  875.075,1047.43 876.411,1044.75 877.746,1042.05 879.081,1039.33 880.417,1036.59 881.752,1033.83 883.087,1031.04 884.422,1028.24 885.758,1025.42 887.093,1022.57 \n",
       "  888.428,1019.71 889.764,1016.82 891.099,1013.92 892.434,1010.99 893.769,1008.05 895.105,1005.08 896.44,1002.09 897.775,999.088 899.111,996.062 900.446,993.017 \n",
       "  901.781,989.953 903.116,986.868 904.452,983.765 905.787,980.642 907.122,977.5 908.457,974.339 909.793,971.16 911.128,967.961 912.463,964.744 913.799,961.509 \n",
       "  915.134,958.256 916.469,954.984 917.804,951.695 919.14,948.388 920.475,945.064 921.81,941.722 923.146,938.363 924.481,934.987 925.816,931.595 927.151,928.186 \n",
       "  928.487,924.761 929.822,921.319 931.157,917.862 932.493,914.39 933.828,910.902 935.163,907.398 936.498,903.88 937.834,900.348 939.169,896.801 940.504,893.24 \n",
       "  941.84,889.665 943.175,886.077 944.51,882.475 945.845,878.86 947.181,875.233 948.516,871.593 949.851,867.941 951.187,864.277 952.522,860.602 953.857,856.916 \n",
       "  955.192,853.218 956.528,849.51 957.863,845.792 959.198,842.064 960.533,838.326 961.869,834.58 963.204,830.824 964.539,827.059 965.875,823.286 967.21,819.506 \n",
       "  968.545,815.717 969.88,811.922 971.216,808.12 972.551,804.311 973.886,800.496 975.222,796.675 976.557,792.849 977.892,789.018 979.227,785.182 980.563,781.341 \n",
       "  981.898,777.497 983.233,773.649 984.569,769.798 985.904,765.944 987.239,762.088 988.574,758.23 989.91,754.37 991.245,750.508 992.58,746.646 993.916,742.783 \n",
       "  995.251,738.92 996.586,735.057 997.921,731.194 999.257,727.333 1000.59,723.473 1001.93,719.615 1003.26,715.758 1004.6,711.904 1005.93,708.053 1007.27,704.206 \n",
       "  1008.6,700.361 1009.94,696.521 1011.27,692.685 1012.61,688.854 1013.94,685.028 1015.28,681.207 1016.62,677.392 1017.95,673.583 1019.29,669.781 1020.62,665.985 \n",
       "  1021.96,662.197 1023.29,658.416 1024.63,654.644 1025.96,650.879 1027.3,647.123 1028.63,643.376 1029.97,639.639 1031.3,635.91 1032.64,632.192 1033.97,628.484 \n",
       "  1035.31,624.787 1036.64,621.101 1037.98,617.425 1039.32,613.762 1040.65,610.11 1041.99,606.47 1043.32,602.843 1044.66,599.228 1045.99,595.626 1047.33,592.038 \n",
       "  1048.66,588.463 1050,584.902 1051.33,581.355 1052.67,577.822 1054,574.304 1055.34,570.801 1056.67,567.313 1058.01,563.84 1059.34,560.383 1060.68,556.942 \n",
       "  1062.01,553.517 1063.35,550.108 1064.69,546.715 1066.02,543.34 1067.36,539.981 1068.69,536.639 1070.03,533.315 1071.36,530.008 1072.7,526.718 1074.03,523.447 \n",
       "  1075.37,520.194 1076.7,516.958 1078.04,513.742 1079.37,510.543 1080.71,507.364 1082.04,504.203 1083.38,501.061 1084.71,497.938 1086.05,494.834 1087.39,491.75 \n",
       "  1088.72,488.685 1090.06,485.64 1091.39,482.615 1092.73,479.609 1094.06,476.623 1095.4,473.657 1096.73,470.712 1098.07,467.786 1099.4,464.881 1100.74,461.996 \n",
       "  1102.07,459.131 1103.41,456.287 1104.74,453.463 1106.08,450.66 1107.41,447.877 1108.75,445.115 1110.09,442.374 1111.42,439.654 1112.76,436.954 1114.09,434.275 \n",
       "  1115.43,431.617 1116.76,428.979 1118.1,426.363 1119.43,423.767 1120.77,421.192 1122.1,418.638 1123.44,416.105 1124.77,413.592 1126.11,411.101 1127.44,408.63 \n",
       "  1128.78,406.18 1130.11,403.751 1131.45,401.342 1132.78,398.954 1134.12,396.587 1135.46,394.241 1136.79,391.915 1138.13,389.609 1139.46,387.324 1140.8,385.06 \n",
       "  1142.13,382.816 1143.47,380.592 1144.8,378.388 1146.14,376.205 1147.47,374.042 1148.81,371.899 1150.14,369.776 1151.48,367.673 1152.81,365.589 1154.15,363.526 \n",
       "  1155.48,361.482 1156.82,359.458 1158.16,357.453 1159.49,355.468 1160.83,353.502 1162.16,351.555 1163.5,349.628 1164.83,347.72 1166.17,345.83 1167.5,343.96 \n",
       "  1168.84,342.108 1170.17,340.275 1171.51,338.46 1172.84,336.664 1174.18,334.887 1175.51,333.127 1176.85,331.386 1178.18,329.662 1179.52,327.957 1180.85,326.27 \n",
       "  1182.19,324.6 1183.53,322.947 1184.86,321.313 1186.2,319.695 1187.53,318.095 1188.87,316.512 1190.2,314.946 1191.54,313.396 1192.87,311.864 1194.21,310.348 \n",
       "  1195.54,308.848 1196.88,307.365 1198.21,305.898 1199.55,304.448 1200.88,303.013 1202.22,301.594 1203.55,300.191 1204.89,298.804 1206.23,297.432 1207.56,296.075 \n",
       "  1208.9,294.734 1210.23,293.408 1211.57,292.097 1212.9,290.8 1214.24,289.519 1215.57,288.252 1216.91,286.999 1218.24,285.761 1219.58,284.537 1220.91,283.328 \n",
       "  1222.25,282.132 1223.58,280.95 1224.92,279.782 1226.25,278.627 1227.59,277.486 1228.93,276.359 1230.26,275.244 1231.6,274.143 1232.93,273.054 1234.27,271.979 \n",
       "  1235.6,270.916 1236.94,269.866 1238.27,268.828 1239.61,267.803 1240.94,266.79 1242.28,265.789 1243.61,264.8 1244.95,263.823 1246.28,262.858 1247.62,261.905 \n",
       "  1248.95,260.963 1250.29,260.032 1251.62,259.113 1252.96,258.205 1254.3,257.308 1255.63,256.422 1256.97,255.546 1258.3,254.682 1259.64,253.828 1260.97,252.985 \n",
       "  1262.31,252.152 1263.64,251.329 1264.98,250.517 1266.31,249.714 1267.65,248.922 1268.98,248.139 1270.32,247.366 1271.65,246.603 1272.99,245.849 1274.32,245.105 \n",
       "  1275.66,244.37 1277,243.644 1278.33,242.927 1279.67,242.22 1281,241.521 1282.34,240.831 1283.67,240.15 1285.01,239.477 1286.34,238.813 1287.68,238.158 \n",
       "  1289.01,237.51 1290.35,236.871 1291.68,236.24 1293.02,235.617 1294.35,235.002 1295.69,234.395 1297.02,233.796 1298.36,233.204 1299.7,232.62 1301.03,232.043 \n",
       "  1302.37,231.474 1303.7,230.912 1305.04,230.358 1306.37,229.81 1307.71,229.27 1309.04,228.736 1310.38,228.21 1311.71,227.69 1313.05,227.177 1314.38,226.671 \n",
       "  1315.72,226.171 1317.05,225.678 1318.39,225.191 1319.72,224.711 1321.06,224.236 1322.39,223.768 1323.73,223.306 1325.07,222.85 1326.4,222.401 1327.74,221.956 \n",
       "  1329.07,221.518 1330.41,221.086 1331.74,220.659 1333.08,220.238 1334.41,219.822 1335.75,219.412 1337.08,219.007 1338.42,218.608 1339.75,218.213 1341.09,217.824 \n",
       "  1342.42,217.441 1343.76,217.062 1345.09,216.688 1346.43,216.319 1347.77,215.955 1349.1,215.596 1350.44,215.242 1351.77,214.892 1353.11,214.547 1354.44,214.207 \n",
       "  1355.78,213.871 1357.11,213.539 1358.45,213.212 1359.78,212.89 1361.12,212.571 1362.45,212.257 1363.79,211.947 1365.12,211.642 1366.46,211.34 1367.79,211.042 \n",
       "  1369.13,210.749 1370.46,210.459 1371.8,210.173 1373.14,209.891 1374.47,209.613 1375.81,209.338 1377.14,209.067 1378.48,208.8 1379.81,208.537 1381.15,208.277 \n",
       "  1382.48,208.02 1383.82,207.767 1385.15,207.517 1386.49,207.271 1387.82,207.028 1389.16,206.788 1390.49,206.551 1391.83,206.318 1393.16,206.088 1394.5,205.861 \n",
       "  1395.84,205.637 1397.17,205.416 1398.51,205.198 1399.84,204.983 1401.18,204.771 1402.51,204.561 1403.85,204.355 1405.18,204.151 1406.52,203.95 1407.85,203.752 \n",
       "  1409.19,203.557 1410.52,203.364 1411.86,203.174 1413.19,202.986 1414.53,202.801 1415.86,202.619 1417.2,202.438 1418.54,202.261 1419.87,202.086 1421.21,201.913 \n",
       "  1422.54,201.742 1423.88,201.574 1425.21,201.408 1426.55,201.245 1427.88,201.083 1429.22,200.924 1430.55,200.767 1431.89,200.612 1433.22,200.459 1434.56,200.309 \n",
       "  1435.89,200.16 1437.23,200.014 1438.56,199.869 1439.9,199.726 1441.23,199.586 1442.57,199.447 1443.91,199.31 1445.24,199.175 1446.58,199.042 1447.91,198.911 \n",
       "  1449.25,198.781 1450.58,198.653 1451.92,198.527 1453.25,198.403 1454.59,198.281 1455.92,198.16 1457.26,198.04 1458.59,197.923 1459.93,197.807 1461.26,197.693 \n",
       "  1462.6,197.58 1463.93,197.468 1465.27,197.359 1466.61,197.25 1467.94,197.144 1469.28,197.038 1470.61,196.934 1471.95,196.832 1473.28,196.731 1474.62,196.631 \n",
       "  1475.95,196.533 1477.29,196.436 1478.62,196.341 1479.96,196.246 1481.29,196.153 1482.63,196.062 1483.96,195.971 1485.3,195.882 1486.63,195.794 1487.97,195.707 \n",
       "  1489.31,195.622 1490.64,195.537 1491.98,195.454 1493.31,195.372 1494.65,195.291 1495.98,195.211 1497.32,195.133 1498.65,195.055 1499.99,194.978 1501.32,194.903 \n",
       "  1502.66,194.828 1503.99,194.755 1505.33,194.683 1506.66,194.611 1508,194.541 1509.33,194.471 1510.67,194.403 1512,194.335 1513.34,194.268 1514.68,194.203 \n",
       "  1516.01,194.138 1517.35,194.074 1518.68,194.011 1520.02,193.949 1521.35,193.887 1522.69,193.827 1524.02,193.767 1525.36,193.708 1526.69,193.65 1528.03,193.593 \n",
       "  1529.36,193.537 1530.7,193.481 1532.03,193.426 1533.37,193.372 1534.7,193.319 1536.04,193.266 1537.38,193.214 1538.71,193.163 1540.05,193.113 1541.38,193.063 \n",
       "  1542.72,193.014 1544.05,192.966 1545.39,192.918 1546.72,192.871 1548.06,192.824 1549.39,192.779 1550.73,192.733 1552.06,192.689 1553.4,192.645 1554.73,192.602 \n",
       "  1556.07,192.559 1557.4,192.517 1558.74,192.475 1560.07,192.434 1561.41,192.394 1562.75,192.354 1564.08,192.315 1565.42,192.276 1566.75,192.238 1568.09,192.2 \n",
       "  1569.42,192.163 1570.76,192.127 1572.09,192.09 1573.43,192.055 1574.76,192.02 1576.1,191.985 1577.43,191.951 1578.77,191.917 1580.1,191.884 1581.44,191.851 \n",
       "  1582.77,191.819 1584.11,191.787 1585.45,191.756 1586.78,191.725 1588.12,191.694 1589.45,191.664 1590.79,191.634 1592.12,191.605 1593.46,191.576 1594.79,191.548 \n",
       "  1596.13,191.52 1597.46,191.492 1598.8,191.465 1600.13,191.438 1601.47,191.411 1602.8,191.385 1604.14,191.359 1605.47,191.334 1606.81,191.308 1608.15,191.284 \n",
       "  1609.48,191.259 1610.82,191.235 1612.15,191.211 1613.49,191.188 1614.82,191.165 1616.16,191.142 1617.49,191.12 1618.83,191.097 1620.16,191.076 1621.5,191.054 \n",
       "  1622.83,191.033 1624.17,191.012 1625.5,190.991 1626.84,190.971 1628.17,190.951 1629.51,190.931 1630.84,190.911 1632.18,190.892 1633.52,190.873 1634.85,190.854 \n",
       "  1636.19,190.836 1637.52,190.818 1638.86,190.8 1640.19,190.782 1641.53,190.764 1642.86,190.747 1644.2,190.73 1645.53,190.714 1646.87,190.697 1648.2,190.681 \n",
       "  1649.54,190.665 1650.87,190.649 1652.21,190.633 1653.54,190.618 1654.88,190.603 1656.22,190.588 1657.55,190.573 1658.89,190.558 1660.22,190.544 1661.56,190.53 \n",
       "  \n",
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      ]
     },
     "metadata": {},
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    }
   ],
   "source": [
    "z = -7:0.014:7\n",
    "plt=Plots.plot(z, 1 ./(1 .+broadcast(exp,-z)),\n",
    "    xlabel=\"w^Tx (raw model output)\",\n",
    "    ylabel=\"predicted probability\",\n",
    "    title=\"logistic (sigmoid) function, 1/(1+\\exp(-w^Tx))\",\n",
    "    thickness_scaling=5,\n",
    "    size=(1800,1500),\n",
    "    legend=false,\n",
    "    titlefontsize=10,\n",
    "    left_margin=-95px,\n",
    "    bottom_margin=-45px\n",
    ");\n",
    "plot!([0,0],[0,0.5],\n",
    "    linestyle=:dash,\n",
    "    linecolor=\"black\"\n",
    ");\n",
    "plot!([-7,0],[0.5,0.5],\n",
    "    linestyle=:dash,\n",
    "    color=\"black\"\n",
    ");\n",
    "display(plt)\n",
    "# plt.yticks((0,0.25,0.5,0.75,1));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Thus, checking whether the raw model output is positive or negative corresponds to checking whether the probability is greater or less than 0.5. Makes sense!\n",
    "- Furthermore, making the coefficients larger (e.g. doubling) pushes $w^Tx$ away from zero and towards the edges of the sigmoid. Hence the behaviour we observed.\n",
    "- We can sort of see the sigmoid in the probability contours above: for small coefficients it appears linear; for large coefficients we see the saturation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Interpretation of the coefficients\n",
    "\n",
    "- Above we've seen a geometric interpretation\n",
    "- The interpretation from linear regression also holds:\n",
    "  - the $j$th coefficient tells us how feature $j$ affects the prediction\n",
    "  - if $w_j>0$ then increasing $x_{ij}$ moves us toward predicting $+1$\n",
    "  - if $w_j<0$ then increasing $x_{ij}$ moves us toward prediction $-1$\n",
    "  - if $w_j=0$ then the feature is not used in making a prediction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Preview of next lecture\n",
    "\n",
    "- We need a loss function for our linear classifiers.\n",
    "- Number of errors is not smooth/convex.\n",
    "- But, maybe, higher probabilities on the correct class?"
   ]
  }
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