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{
"cells": [
{
"cell_type": "markdown",
"id": "fce8933f-4594-4bb5-bffe-86fcb9ddd684",
"metadata": {},
"source": [
"# MLE of a Gaussian $p_{model}(x|w)$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "f6cd23f0-e755-48af-be5e-aaee83dda1e7",
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"import numpy as np\n",
"\n",
"data = [4, 5, 7, 8, 8, 9, 10, 5, 2, 3, 5, 4, 8, 9]\n",
"\n",
"\n",
"## imports\n",
"import numpy as np\n",
"import pandas as pd\n",
"from scipy.optimize import minimize\n",
"from scipy.stats import norm\n",
"import math\n",
"\n",
"\n",
"## Problem 1\n",
"data = [4, 5, 7, 8, 8, 9, 10, 5, 2, 3, 5, 4, 8, 9]\n",
"\n",
"data_mean = np.mean(data)\n",
"data_variance = np.var(data)\n",
"\n",
"\n",
"mu = 0.5\n",
"sigma = 0.5\n",
"w = np.array([mu, sigma])\n",
"\n",
"w_star = np.array([data_mean, data_variance])\n",
"mu_star = data_mean\n",
"sigma_star = np.sqrt(data_variance)\n",
"offset = 10 * np.random.random(2)\n",
"\n",
"w1p = w_star + 0.5 * offset\n",
"w1n = w_star - 0.5 * offset\n",
"w2p = w_star + 0.25 * offset\n",
"w2n = w_star - 0.25 * offset"
]
},
{
"cell_type": "markdown",
"id": "f3d8587b-3862-4e98-bbcc-99d57bb313c1",
"metadata": {},
"source": [
"Negative Log Likelihood is defined as follows: $-\\ln(\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp(-\\frac{1}{2}\\frac{(x-\\mu)}{\\sigma}^2))$. Ignoring the contribution of the constant, we find that $\\frac{\\delta}{\\delta \\mu} \\mathcal{N} = \\frac{\\mu-x}{\\sigma^2}$ and $\\frac{\\delta}{\\delta \\sigma} \\mathcal{N} = \\frac{\\sigma^2 + (\\mu-x)^2 - \\sigma^2}{\\sigma^3}$. We apply these as our step functions for our SGD. "
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "27bf27ad-031e-4b65-a44d-53c5c1a09d91",
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"loss = lambda mu, sigma, x: np.sum(\n",
" [-np.log(norm.pdf(xi, loc=mu, scale=sigma)) for xi in x]\n",
")\n",
"\n",
"loss_2_alternative = lambda mu, sigma, x: -len(x) / 2 * np.log(\n",
" 2 * np.pi * sigma**2\n",
") - 1 / (2 * sigma**2) * np.sum((x - mu) ** 2)\n",
"\n",
"\n",
"dmu = lambda mu, sigma, x: -np.sum([mu - xi for xi in x]) / (sigma**2)\n",
"dsigma = lambda mu, sigma, x: -len(x) / sigma + np.sum([(mu - xi) ** 2 for xi in x]) / (sigma**3)\n",
"\n",
"log = []\n",
"def SGD_problem1(mu, sigma, x, learning_rate=0.01, n_epochs=1000):\n",
" global log\n",
" log = []\n",
" for epoch in range(n_epochs):\n",
" mu += learning_rate * dmu(mu, sigma, x)\n",
" sigma += learning_rate * dsigma(mu, sigma, x)\n",
"\n",
" # print(f\"Epoch {epoch}, Loss: {loss(mu, sigma, x)}, New mu: {mu}, New sigma: {sigma}\")\n",
" log.append(\n",
" {\n",
" \"Epoch\": epoch,\n",
" \"Loss\": loss(mu, sigma, x),\n",
" \"Loss 2 Alternative\": loss_2_alternative(mu, sigma, x),\n",
" \"New mu\": mu,\n",
" \"New sigma\": sigma,\n",
" }\n",
" )\n",
" return np.array([mu, sigma])\n",
"\n",
"\n",
"def debug_SGD_1(wnn, data):\n",
" print(\"SGD Problem 1\")\n",
" print(\"wnn\", SGD_problem1(*wnn, data))\n",
" dflog = pd.DataFrame(log)\n",
" dflog[\"mu_star\"] = mu_star\n",
" dflog[\"mu_std\"] = sigma_star\n",
" print(f\"mu diff at start {dflog.iloc[0]['New mu'] - dflog.iloc[0]['mu_star']}\")\n",
" print(f\"mu diff at end {dflog.iloc[-1]['New mu'] - dflog.iloc[-1]['mu_star']}\")\n",
" if np.abs(dflog.iloc[-1][\"New mu\"] - dflog.iloc[-1][\"mu_star\"]) < np.abs(\n",
" dflog.iloc[0][\"New mu\"] - dflog.iloc[0][\"mu_star\"]\n",
" ):\n",
" print(\"mu is improving\")\n",
" else:\n",
" print(\"mu is not improving\")\n",
"\n",
" print(f\"sigma diff at start {dflog.iloc[0]['New sigma'] - dflog.iloc[0]['mu_std']}\")\n",
" print(f\"sigma diff at end {dflog.iloc[-1]['New sigma'] - dflog.iloc[-1]['mu_std']}\")\n",
" if np.abs(dflog.iloc[-1][\"New sigma\"] - dflog.iloc[-1][\"mu_std\"]) < np.abs(\n",
" dflog.iloc[0][\"New sigma\"] - dflog.iloc[0][\"mu_std\"]\n",
" ):\n",
" print(\"sigma is improving\")\n",
" else:\n",
" print(\"sigma is not improving\")\n",
"\n",
" return dflog"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "27dd3bc6-b96e-4f8b-9118-01ad344dfd6a",
"metadata": {
"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"SGD Problem 1\n",
"wnn [6.2142858 2.42541812]\n",
"mu diff at start 0.27610721776969527\n",
"mu diff at end 8.978893806244059e-08\n",
"mu is improving\n",
"sigma diff at start 8.134860851821205\n",
"sigma diff at end 1.7124079931818414e-12\n",
"sigma is improving\n",
"SGD Problem 1\n",
"wnn [6.21428571 2.42541812]\n",
"mu diff at start -0.24923650064862635\n",
"mu diff at end -6.602718372050731e-12\n",
"mu is improving\n",
"sigma diff at start -0.859536014291925\n",
"sigma diff at end -3.552713678800501e-15\n",
"sigma is improving\n",
"SGD Problem 1\n",
"wnn [6.21428572 2.42541812]\n",
"mu diff at start 0.13794086144778994\n",
"mu diff at end 1.0008935902305893e-09\n",
"mu is improving\n",
"sigma diff at start 5.786783512688555\n",
"sigma diff at end 4.440892098500626e-15\n",
"sigma is improving\n",
"SGD Problem 1\n",
"wnn [6.21428571 2.42541812]\n",
"mu diff at start -0.13668036978891251\n",
"mu diff at end -8.528289185960602e-12\n",
"mu is improving\n",
"sigma diff at start 1.091241177336173\n",
"sigma diff at end 4.440892098500626e-15\n",
"sigma is improving\n"
]
}
],
"source": [
"_ = debug_SGD_1(w1p, data)\n",
"_ = debug_SGD_1(w1n, data)\n",
"_ = debug_SGD_1(w2p, data)\n",
"_ = debug_SGD_1(w2n, data)"
]
},
{
"cell_type": "markdown",
"id": "30096401-0bd5-4cf6-b093-a688476e16f1",
"metadata": {
"tags": []
},
"source": [
"# MLE of Conditional Gaussian"
]
},
{
"cell_type": "markdown",
"id": "101a3c5e-1e02-41e6-9eab-aba65c39627a",
"metadata": {},
"source": [
"dsigma = $-\\frac{n}{\\sigma}+\\frac{1}{\\sigma^3}\\sum_{i=1}^n(y_i - (mx+c))^2$ \n",
"dc = $-\\frac{1}{\\sigma^2}\\sum_{i=1}^n(y_i - (mx+c))$ \n",
"dm = $-\\frac{1}{\\sigma^2}\\sum_{i=1}^n(x_i(y_i - (mx+c)))$ "
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "21969012-f81b-43d4-975d-13411e975f8f",
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
},
"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Epoch 0, Loss: 297.82677563555086\n",
"Epoch 1, Loss: 297.8267749215061\n",
"Epoch 2, Loss: 297.82677420752475\n",
"Epoch 3, Loss: 297.82677349360694\n",
"Epoch 4, Loss: 297.8267727797526\n",
"Epoch 5, Loss: 297.8267720659618\n",
"Epoch 6, Loss: 297.82677135223446\n",
"Epoch 7, Loss: 297.82677063857074\n",
"Epoch 8, Loss: 297.8267699249706\n",
"Epoch 9, Loss: 297.826769211434\n",
"Epoch 10, Loss: 297.8267684979611\n",
"Epoch 11, Loss: 297.82676778455175\n",
"Epoch 12, Loss: 297.82676707120623\n",
"Epoch 13, Loss: 297.82676635792427\n",
"Epoch 14, Loss: 297.8267656447061\n",
"Epoch 15, Loss: 297.8267649315517\n",
"Epoch 16, Loss: 297.82676421846105\n",
"Epoch 17, Loss: 297.82676350543414\n",
"Epoch 18, Loss: 297.82676279247113\n",
"Epoch 19, Loss: 297.82676207957184\n",
"Epoch 20, Loss: 297.82676136673643\n",
"Epoch 21, Loss: 297.826760653965\n",
"Epoch 22, Loss: 297.82675994125736\n",
"Epoch 23, Loss: 297.8267592286137\n",
"Epoch 24, Loss: 297.8267585160339\n",
"Epoch 25, Loss: 297.8267578035182\n",
"Epoch 26, Loss: 297.82675709106655\n",
"Epoch 27, Loss: 297.8267563786788\n",
"Epoch 28, Loss: 297.82675566635504\n",
"Epoch 29, Loss: 297.82675495409546\n",
"Epoch 30, Loss: 297.82675424189983\n",
"Epoch 31, Loss: 297.82675352976844\n",
"Epoch 32, Loss: 297.8267528177011\n",
"Epoch 33, Loss: 297.82675210569795\n",
"Epoch 34, Loss: 297.826751393759\n",
"Epoch 35, Loss: 297.8267506818843\n",
"Epoch 36, Loss: 297.8267499700737\n",
"Epoch 37, Loss: 297.8267492583274\n",
"Epoch 38, Loss: 297.82674854664543\n",
"Epoch 39, Loss: 297.8267478350277\n",
"Epoch 40, Loss: 297.8267471234743\n",
"Epoch 41, Loss: 297.82674641198514\n",
"Epoch 42, Loss: 297.8267457005605\n",
"Epoch 43, Loss: 297.82674498920017\n",
"Epoch 44, Loss: 297.82674427790425\n",
"Epoch 45, Loss: 297.82674356667275\n",
"Epoch 46, Loss: 297.82674285550564\n",
"Epoch 47, Loss: 297.8267421444032\n",
"Epoch 48, Loss: 297.82674143336516\n",
"Epoch 49, Loss: 297.8267407223916\n",
"Epoch 50, Loss: 297.8267400114827\n",
"Epoch 51, Loss: 297.82673930063817\n",
"Epoch 52, Loss: 297.8267385898584\n",
"Epoch 53, Loss: 297.8267378791432\n",
"Epoch 54, Loss: 297.8267371684925\n",
"Epoch 55, Loss: 297.8267364579067\n",
"Epoch 56, Loss: 297.8267357473855\n",
"Epoch 57, Loss: 297.82673503692894\n",
"Epoch 58, Loss: 297.82673432653723\n",
"Epoch 59, Loss: 297.8267336162103\n",
"Epoch 60, Loss: 297.826732905948\n",
"Epoch 61, Loss: 297.82673219575054\n",
"Epoch 62, Loss: 297.826731485618\n",
"Epoch 63, Loss: 297.82673077555023\n",
"Epoch 64, Loss: 297.82673006554734\n",
"Epoch 65, Loss: 297.8267293556093\n",
"Epoch 66, Loss: 297.82672864573624\n",
"Epoch 67, Loss: 297.82672793592815\n",
"Epoch 68, Loss: 297.82672722618497\n",
"Epoch 69, Loss: 297.8267265165068\n",
"Epoch 70, Loss: 297.8267258068936\n",
"Epoch 71, Loss: 297.8267250973455\n",
"Epoch 72, Loss: 297.8267243878624\n",
"Epoch 73, Loss: 297.8267236784444\n",
"Epoch 74, Loss: 297.82672296909146\n",
"Epoch 75, Loss: 297.8267222598038\n",
"Epoch 76, Loss: 297.8267215505811\n",
"Epoch 77, Loss: 297.8267208414237\n",
"Epoch 78, Loss: 297.82672013233156\n",
"Epoch 79, Loss: 297.8267194233045\n",
"Epoch 80, Loss: 297.8267187143427\n",
"Epoch 81, Loss: 297.8267180054462\n",
"Epoch 82, Loss: 297.82671729661496\n",
"Epoch 83, Loss: 297.82671658784903\n",
"Epoch 84, Loss: 297.8267158791485\n",
"Epoch 85, Loss: 297.82671517051335\n",
"Epoch 86, Loss: 297.8267144619435\n",
"Epoch 87, Loss: 297.8267137534391\n",
"Epoch 88, Loss: 297.82671304500013\n",
"Epoch 89, Loss: 297.82671233662654\n",
"Epoch 90, Loss: 297.82671162831855\n",
"Epoch 91, Loss: 297.8267109200759\n",
"Epoch 92, Loss: 297.82671021189896\n",
"Epoch 93, Loss: 297.8267095037876\n",
"Epoch 94, Loss: 297.8267087957417\n",
"Epoch 95, Loss: 297.82670808776135\n",
"Epoch 96, Loss: 297.82670737984665\n",
"Epoch 97, Loss: 297.8267066719976\n",
"Epoch 98, Loss: 297.82670596421417\n",
"Epoch 99, Loss: 297.8267052564966\n",
"Epoch 100, Loss: 297.82670454884465\n",
"Epoch 101, Loss: 297.82670384125834\n",
"Epoch 102, Loss: 297.8267031337379\n",
"Epoch 103, Loss: 297.8267024262833\n",
"Epoch 104, Loss: 297.82670171889436\n",
"Epoch 105, Loss: 297.8267010115713\n",
"Epoch 106, Loss: 297.82670030431416\n",
"Epoch 107, Loss: 297.82669959712285\n",
"Epoch 108, Loss: 297.82669888999743\n",
"Epoch 109, Loss: 297.8266981829379\n",
"Epoch 110, Loss: 297.8266974759444\n",
"Epoch 111, Loss: 297.8266967690169\n",
"Epoch 112, Loss: 297.8266960621552\n",
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"Epoch 902, Loss: 297.8261603534504\n",
"Epoch 903, Loss: 297.82615970692694\n",
"Epoch 904, Loss: 297.826159060491\n",
"Epoch 905, Loss: 297.82615841414264\n",
"Epoch 906, Loss: 297.82615776788197\n",
"Epoch 907, Loss: 297.826157121709\n",
"Epoch 908, Loss: 297.82615647562363\n",
"Epoch 909, Loss: 297.8261558296259\n",
"Epoch 910, Loss: 297.8261551837161\n",
"Epoch 911, Loss: 297.826154537894\n",
"Epoch 912, Loss: 297.82615389215977\n",
"Epoch 913, Loss: 297.82615324651323\n",
"Epoch 914, Loss: 297.8261526009547\n",
"Epoch 915, Loss: 297.826151955484\n",
"Epoch 916, Loss: 297.8261513101013\n",
"Epoch 917, Loss: 297.8261506648065\n",
"Epoch 918, Loss: 297.8261500195997\n",
"Epoch 919, Loss: 297.826149374481\n",
"Epoch 920, Loss: 297.82614872945044\n",
"Epoch 921, Loss: 297.8261480845078\n",
"Epoch 922, Loss: 297.8261474396533\n",
"Epoch 923, Loss: 297.8261467948871\n",
"Epoch 924, Loss: 297.826146150209\n",
"Epoch 925, Loss: 297.82614550561914\n",
"Epoch 926, Loss: 297.8261448611175\n",
"Epoch 927, Loss: 297.82614421670417\n",
"Epoch 928, Loss: 297.8261435723791\n",
"Epoch 929, Loss: 297.8261429281424\n",
"Epoch 930, Loss: 297.8261422839941\n",
"Epoch 931, Loss: 297.82614163993424\n",
"Epoch 932, Loss: 297.8261409959628\n",
"Epoch 933, Loss: 297.82614035207973\n",
"Epoch 934, Loss: 297.82613970828527\n",
"Epoch 935, Loss: 297.8261390645793\n",
"Epoch 936, Loss: 297.826138420962\n",
"Epoch 937, Loss: 297.8261377774331\n",
"Epoch 938, Loss: 297.82613713399303\n",
"Epoch 939, Loss: 297.82613649064143\n",
"Epoch 940, Loss: 297.8261358473787\n",
"Epoch 941, Loss: 297.8261352042046\n",
"Epoch 942, Loss: 297.8261345611193\n",
"Epoch 943, Loss: 297.8261339181227\n",
"Epoch 944, Loss: 297.826133275215\n",
"Epoch 945, Loss: 297.82613263239614\n",
"Epoch 946, Loss: 297.8261319896662\n",
"Epoch 947, Loss: 297.82613134702507\n",
"Epoch 948, Loss: 297.826130704473\n",
"Epoch 949, Loss: 297.8261300620098\n",
"Epoch 950, Loss: 297.8261294196356\n",
"Epoch 951, Loss: 297.8261287773505\n",
"Epoch 952, Loss: 297.8261281351545\n",
"Epoch 953, Loss: 297.8261274930475\n",
"Epoch 954, Loss: 297.82612685102976\n",
"Epoch 955, Loss: 297.8261262091011\n",
"Epoch 956, Loss: 297.82612556726167\n",
"Epoch 957, Loss: 297.8261249255115\n",
"Epoch 958, Loss: 297.8261242838506\n",
"Epoch 959, Loss: 297.8261236422789\n",
"Epoch 960, Loss: 297.8261230007966\n",
"Epoch 961, Loss: 297.8261223594037\n",
"Epoch 962, Loss: 297.82612171810007\n",
"Epoch 963, Loss: 297.82612107688584\n",
"Epoch 964, Loss: 297.8261204357612\n",
"Epoch 965, Loss: 297.82611979472597\n",
"Epoch 966, Loss: 297.8261191537804\n",
"Epoch 967, Loss: 297.82611851292415\n",
"Epoch 968, Loss: 297.8261178721576\n",
"Epoch 969, Loss: 297.8261172314807\n",
"Epoch 970, Loss: 297.8261165908933\n",
"Epoch 971, Loss: 297.82611595039566\n",
"Epoch 972, Loss: 297.8261153099878\n",
"Epoch 973, Loss: 297.82611466966955\n",
"Epoch 974, Loss: 297.8261140294412\n",
"Epoch 975, Loss: 297.8261133893026\n",
"Epoch 976, Loss: 297.82611274925375\n",
"Epoch 977, Loss: 297.82611210929485\n",
"Epoch 978, Loss: 297.82611146942594\n",
"Epoch 979, Loss: 297.8261108296469\n",
"Epoch 980, Loss: 297.8261101899578\n",
"Epoch 981, Loss: 297.8261095503587\n",
"Epoch 982, Loss: 297.8261089108496\n",
"Epoch 983, Loss: 297.82610827143054\n",
"Epoch 984, Loss: 297.8261076321016\n",
"Epoch 985, Loss: 297.8261069928628\n",
"Epoch 986, Loss: 297.8261063537141\n",
"Epoch 987, Loss: 297.8261057146557\n",
"Epoch 988, Loss: 297.8261050756874\n",
"Epoch 989, Loss: 297.82610443680943\n",
"Epoch 990, Loss: 297.82610379802173\n",
"Epoch 991, Loss: 297.82610315932436\n",
"Epoch 992, Loss: 297.8261025207173\n",
"Epoch 993, Loss: 297.8261018822007\n",
"Epoch 994, Loss: 297.8261012437744\n",
"Epoch 995, Loss: 297.8261006054387\n",
"Epoch 996, Loss: 297.82609996719333\n",
"Epoch 997, Loss: 297.8260993290385\n",
"Epoch 998, Loss: 297.8260986909742\n",
"Epoch 999, Loss: 297.8260980530006\n",
"final parameters: m=0.45136980910052144, c=0.49775672565271384, sigma=1562.2616856027405\n"
]
}
],
"source": [
"## Problem 2\n",
"x = np.array([8, 16, 22, 33, 50, 51])\n",
"y = np.array([5, 20, 14, 32, 42, 58])\n",
"\n",
"# $-\\frac{n}{\\sigma}+\\frac{1}{\\sigma^3}\\sum_{i=1}^n(y_i - (mx+c))^2$\n",
"dsigma = lambda sigma, c, m, x: -len(x) / sigma + np.sum(\n",
" [(xi - (m * x + c)) ** 2 for xi in x]\n",
") / (sigma**3)\n",
"# $-\\frac{1}{\\sigma^2}\\sum_{i=1}^n(y_i - (mx+c))$\n",
"dc = lambda sigma, c, m, x: -np.sum([xi - (m * x + c) for xi in x]) / (sigma**2)\n",
"# $-\\frac{1}{\\sigma^2}\\sum_{i=1}^n(x_i(y_i - (mx+c)))$\n",
"dm = lambda sigma, c, m, x: -np.sum([x * (xi - (m * x + c)) for xi in x]) / (sigma**2)\n",
"\n",
"\n",
"log2 = []\n",
"\n",
"\n",
"def SGD_problem2(\n",
" sigma: float,\n",
" c: float,\n",
" m: float,\n",
" x: np.array,\n",
" y: np.array,\n",
" learning_rate=0.01,\n",
" n_epochs=1000,\n",
"):\n",
" global log2\n",
" log2 = []\n",
" for epoch in range(n_epochs):\n",
" sigma += learning_rate * dsigma(sigma, c, m, x)\n",
" c += learning_rate * dc(sigma, c, m, x)\n",
" m += learning_rate * dm(sigma, c, m, x)\n",
"\n",
" log2.append(\n",
" {\n",
" \"Epoch\": epoch,\n",
" \"New sigma\": sigma,\n",
" \"New c\": c,\n",
" \"New m\": m,\n",
" \"dc\": dc(sigma, c, m, x),\n",
" \"dm\": dm(sigma, c, m, x),\n",
" \"dsigma\": dsigma(sigma, c, m, x),\n",
" \"Loss\": loss((m * x + c), sigma, y),\n",
" }\n",
" )\n",
" print(f\"Epoch {epoch}, Loss: {loss((m * x + c), sigma, y)}\")\n",
" return np.array([sigma, c, m])\n",
"\n",
"\n",
"result = SGD_problem2(0.5, 0.5, 0.5, x, y)\n",
"print(f\"final parameters: m={result[2]}, c={result[1]}, sigma={result[0]}\")"
]
},
{
"cell_type": "markdown",
"id": "0562b012-f4ca-47de-bc76-e0eb2bf1e509",
"metadata": {},
"source": [
"loss appears to be decreasing. Uncollapse cell for output"
]
},
{
"cell_type": "markdown",
"id": "bed9f3ce-c15c-4f30-8906-26f3e51acf30",
"metadata": {},
"source": [
"# Bike Rides and the Poisson Model"
]
},
{
"cell_type": "markdown",
"id": "975e2ef5-f5d5-45a3-b635-8faef035906f",
"metadata": {},
"source": [
"Knowing that the poisson pdf is $P(k) = \\frac{\\lambda^k e^{-\\lambda}}{k!}$, we can find the negative log likelihood of the data as $-\\log(\\Pi_{i=1}^n P(k_i)) = -\\sum_{i=1}^n \\log(\\frac{\\lambda^k_i e^{-\\lambda}}{k_i!}) = \\sum_{i=1}^n -\\ln(\\lambda) k_i + \\ln(k_i!) + \\lambda$. Which simplified, gives $n\\lambda + \\sum_{i=1}^n \\ln(k_i!) - \\sum_{i=1}^n k_i \\ln(\\lambda)$. Differentiating with respect to $\\lambda$ gives $n - \\sum_{i=1}^n \\frac{k_i}{\\lambda}$. Which is our desired $\\frac{\\partial L}{\\partial \\lambda}$!"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "3877723c-179e-4759-bed5-9eb70110ded2",
"metadata": {
"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"SGD Problem 3\n",
"l: [3215.17703224]\n",
"l diff at start 999.4184849065878\n",
"l diff at end 535.134976163929\n",
"l is improving\n",
"SGD Problem 3\n",
"l: [2326.70336987]\n",
"l diff at start -998.7262223631474\n",
"l diff at end -353.33868620734074\n",
"l is improving\n"
]
},
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>Epoch</th>\n",
" <th>New lambda</th>\n",
" <th>dlambda</th>\n",
" <th>Loss</th>\n",
" <th>l_star</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>0</td>\n",
" <td>1681.315834</td>\n",
" <td>-127.119133</td>\n",
" <td>-3.899989e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>1</td>\n",
" <td>1682.587025</td>\n",
" <td>-126.861418</td>\n",
" <td>-3.900150e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>2</td>\n",
" <td>1683.855639</td>\n",
" <td>-126.604614</td>\n",
" <td>-3.900311e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>3</td>\n",
" <td>1685.121685</td>\n",
" <td>-126.348715</td>\n",
" <td>-3.900471e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>4</td>\n",
" <td>1686.385173</td>\n",
" <td>-126.093716</td>\n",
" <td>-3.900631e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" <tr>\n",
" <th>...</th>\n",
" <td>...</td>\n",
" <td>...</td>\n",
" <td>...</td>\n",
" <td>...</td>\n",
" <td>...</td>\n",
" </tr>\n",
" <tr>\n",
" <th>995</th>\n",
" <td>995</td>\n",
" <td>2325.399976</td>\n",
" <td>-32.636710</td>\n",
" <td>-3.948159e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" <tr>\n",
" <th>996</th>\n",
" <td>996</td>\n",
" <td>2325.726343</td>\n",
" <td>-32.602100</td>\n",
" <td>-3.948170e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" <tr>\n",
" <th>997</th>\n",
" <td>997</td>\n",
" <td>2326.052364</td>\n",
" <td>-32.567536</td>\n",
" <td>-3.948180e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" <tr>\n",
" <th>998</th>\n",
" <td>998</td>\n",
" <td>2326.378040</td>\n",
" <td>-32.533018</td>\n",
" <td>-3.948191e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" <tr>\n",
" <th>999</th>\n",
" <td>999</td>\n",
" <td>2326.703370</td>\n",
" <td>-32.498547</td>\n",
" <td>-3.948201e+06</td>\n",
" <td>2680.042056</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"<p>1000 rows × 5 columns</p>\n",
"</div>"
],
"text/plain": [
" Epoch New lambda dlambda Loss l_star\n",
"0 0 1681.315834 -127.119133 -3.899989e+06 2680.042056\n",
"1 1 1682.587025 -126.861418 -3.900150e+06 2680.042056\n",
"2 2 1683.855639 -126.604614 -3.900311e+06 2680.042056\n",
"3 3 1685.121685 -126.348715 -3.900471e+06 2680.042056\n",
"4 4 1686.385173 -126.093716 -3.900631e+06 2680.042056\n",
".. ... ... ... ... ...\n",
"995 995 2325.399976 -32.636710 -3.948159e+06 2680.042056\n",
"996 996 2325.726343 -32.602100 -3.948170e+06 2680.042056\n",
"997 997 2326.052364 -32.567536 -3.948180e+06 2680.042056\n",
"998 998 2326.378040 -32.533018 -3.948191e+06 2680.042056\n",
"999 999 2326.703370 -32.498547 -3.948201e+06 2680.042056\n",
"\n",
"[1000 rows x 5 columns]"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import pandas as pd\n",
"\n",
"df = pd.read_csv(\"../data/01_raw/nyc_bb_bicyclist_counts.csv\")\n",
"\n",
"dlambda = lambda l, k: len(k) - np.sum([ki / l for ki in k])\n",
"\n",
"\n",
"def SGD_problem3(\n",
" l: float,\n",
" k: np.array,\n",
" learning_rate=0.01,\n",
" n_epochs=1000,\n",
"):\n",
" global log3\n",
" log3 = []\n",
" for epoch in range(n_epochs):\n",
" l -= learning_rate * dlambda(l, k)\n",
" # $n\\lambda + \\sum_{i=1}^n \\ln(k_i!) - \\sum_{i=1}^n k_i \\ln(\\lambda)$\n",
" # the rest of the loss function is commented out because it's a\n",
" # constant and was causing overflows. It is unnecessary, and a useless\n",
" # pain.\n",
" loss = len(k) * l - np.sum(\n",
" [ki * np.log(l) for ki in k]\n",
" ) # + np.sum([np.log(np.math.factorial(ki)) for ki in k])\n",
"\n",
" log3.append(\n",
" {\n",
" \"Epoch\": epoch,\n",
" \"New lambda\": l,\n",
" \"dlambda\": dlambda(l, k),\n",
" \"Loss\": loss,\n",
" }\n",
" )\n",
" # print(f\"Epoch {epoch}\", f\"Loss: {loss}\")\n",
" return np.array([l])\n",
"\n",
"\n",
"l_star = df[\"BB_COUNT\"].mean()\n",
"\n",
"\n",
"def debug_SGD_3(data, l=1000):\n",
" print(\"SGD Problem 3\")\n",
" print(f\"l: {SGD_problem3(l, data)}\")\n",
" dflog = pd.DataFrame(log3)\n",
" dflog[\"l_star\"] = l_star\n",
" print(f\"l diff at start {dflog.iloc[0]['New lambda'] - dflog.iloc[0]['l_star']}\")\n",
" print(f\"l diff at end {dflog.iloc[-1]['New lambda'] - dflog.iloc[-1]['l_star']}\")\n",
" if np.abs(dflog.iloc[-1][\"New lambda\"] - dflog.iloc[-1][\"l_star\"]) < np.abs(\n",
" dflog.iloc[0][\"New lambda\"] - dflog.iloc[0][\"l_star\"]\n",
" ):\n",
" print(\"l is improving\")\n",
" else:\n",
" print(\"l is not improving\")\n",
" return dflog\n",
"\n",
"\n",
"debug_SGD_3(data=df[\"BB_COUNT\"].values, l=l_star + 1000)\n",
"debug_SGD_3(data=df[\"BB_COUNT\"].values, l=l_star - 1000)"
]
},
{
"cell_type": "markdown",
"id": "c05192f9-78ae-4bdb-9df5-cac91006d79f",
"metadata": {},
"source": [
"l approaches the l_star and decreases the loss function."
]
},
{
"cell_type": "markdown",
"id": "4955b868-7f67-4760-bf86-39f6edd55871",
"metadata": {},
"source": [
"## Maximum Likelihood II"
]
},
{
"cell_type": "markdown",
"id": "cd7e6e62-3f64-43e5-bf2c-3cb514411446",
"metadata": {},
"source": [
"The partial of the poisson was found to be $nE^{w.x}*x - \\sum_{i=1}^{n}k x.x$"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "7c8b167d-c397-4155-93f3-d826c279fbb2",
"metadata": {
"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"SGD Problem 4\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/tmp/ipykernel_615396/2481416868.py:22: RuntimeWarning: divide by zero encountered in log\n",
" [ki * np.log(l) for ki in k]\n"
]
},
{
"ename": "ValueError",
"evalue": "operands could not be broadcast together with shapes (3,) (214,) ",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)",
"Cell \u001b[0;32mIn[18], line 44\u001b[0m\n\u001b[1;32m 40\u001b[0m dflog \u001b[38;5;241m=\u001b[39m pd\u001b[38;5;241m.\u001b[39mDataFrame(log4)\n\u001b[1;32m 41\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m dflog\n\u001b[0;32m---> 44\u001b[0m _ \u001b[38;5;241m=\u001b[39m \u001b[43mdebug_SGD_3\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m 45\u001b[0m \u001b[43m \u001b[49m\u001b[43mdata\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mdf\u001b[49m\u001b[43m[\u001b[49m\u001b[43m[\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mHIGH_T\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mLOW_T\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mPRECIP\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m]\u001b[49m\u001b[43m]\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mto_numpy\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 46\u001b[0m \u001b[43m \u001b[49m\u001b[43mw\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mnp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43marray\u001b[49m\u001b[43m(\u001b[49m\u001b[43m[\u001b[49m\u001b[38;5;241;43m1.0\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m1.0\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m1.0\u001b[39;49m\u001b[43m]\u001b[49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 47\u001b[0m \u001b[43m)\u001b[49m\n",
"Cell \u001b[0;32mIn[18], line 39\u001b[0m, in \u001b[0;36mdebug_SGD_3\u001b[0;34m(data, w)\u001b[0m\n\u001b[1;32m 37\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mdebug_SGD_3\u001b[39m(data, w\u001b[38;5;241m=\u001b[39mnp\u001b[38;5;241m.\u001b[39marray([\u001b[38;5;241m1\u001b[39m, \u001b[38;5;241m1\u001b[39m])):\n\u001b[1;32m 38\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mSGD Problem 4\u001b[39m\u001b[38;5;124m\"\u001b[39m)\n\u001b[0;32m---> 39\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mw: \u001b[39m\u001b[38;5;132;01m{\u001b[39;00m\u001b[43mSGD_problem4\u001b[49m\u001b[43m(\u001b[49m\u001b[43mw\u001b[49m\u001b[43m,\u001b[49m\u001b[38;5;250;43m \u001b[39;49m\u001b[43mdata\u001b[49m\u001b[43m)\u001b[49m\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m)\n\u001b[1;32m 40\u001b[0m dflog \u001b[38;5;241m=\u001b[39m pd\u001b[38;5;241m.\u001b[39mDataFrame(log4)\n\u001b[1;32m 41\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m dflog\n",
"Cell \u001b[0;32mIn[18], line 24\u001b[0m, in \u001b[0;36mSGD_problem4\u001b[0;34m(w, x, learning_rate, n_epochs)\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[38;5;66;03m# custom\u001b[39;00m\n\u001b[1;32m 20\u001b[0m \u001b[38;5;66;03m# loss = x.shape[0] * np.exp(np.dot(x, w))\u001b[39;00m\n\u001b[1;32m 21\u001b[0m loss_fn \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;01mlambda\u001b[39;00m k, l: \u001b[38;5;28mlen\u001b[39m(k) \u001b[38;5;241m*\u001b[39m l \u001b[38;5;241m-\u001b[39m np\u001b[38;5;241m.\u001b[39msum(\n\u001b[1;32m 22\u001b[0m [ki \u001b[38;5;241m*\u001b[39m np\u001b[38;5;241m.\u001b[39mlog(l) \u001b[38;5;28;01mfor\u001b[39;00m ki \u001b[38;5;129;01min\u001b[39;00m k]\n\u001b[1;32m 23\u001b[0m ) \u001b[38;5;66;03m# + np.sum([np.log(np.math.factorial(ki)) for ki in k])\u001b[39;00m\n\u001b[0;32m---> 24\u001b[0m loss \u001b[38;5;241m=\u001b[39m \u001b[43mloss_fn\u001b[49m\u001b[43m(\u001b[49m\u001b[43mx\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mnp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mexp\u001b[49m\u001b[43m(\u001b[49m\u001b[43mnp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mdot\u001b[49m\u001b[43m(\u001b[49m\u001b[43mx\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mw\u001b[49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 25\u001b[0m log4\u001b[38;5;241m.\u001b[39mappend(\n\u001b[1;32m 26\u001b[0m {\n\u001b[1;32m 27\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mEpoch\u001b[39m\u001b[38;5;124m\"\u001b[39m: epoch,\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 31\u001b[0m }\n\u001b[1;32m 32\u001b[0m )\n\u001b[1;32m 33\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mEpoch \u001b[39m\u001b[38;5;132;01m{\u001b[39;00mepoch\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mLoss: \u001b[39m\u001b[38;5;132;01m{\u001b[39;00mloss\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m)\n",
"Cell \u001b[0;32mIn[18], line 22\u001b[0m, in \u001b[0;36mSGD_problem4.<locals>.<lambda>\u001b[0;34m(k, l)\u001b[0m\n\u001b[1;32m 18\u001b[0m w \u001b[38;5;241m-\u001b[39m\u001b[38;5;241m=\u001b[39m learning_rate \u001b[38;5;241m*\u001b[39m dw(w, x)\n\u001b[1;32m 19\u001b[0m \u001b[38;5;66;03m# custom\u001b[39;00m\n\u001b[1;32m 20\u001b[0m \u001b[38;5;66;03m# loss = x.shape[0] * np.exp(np.dot(x, w))\u001b[39;00m\n\u001b[1;32m 21\u001b[0m loss_fn \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;01mlambda\u001b[39;00m k, l: \u001b[38;5;28mlen\u001b[39m(k) \u001b[38;5;241m*\u001b[39m l \u001b[38;5;241m-\u001b[39m np\u001b[38;5;241m.\u001b[39msum(\n\u001b[0;32m---> 22\u001b[0m [ki \u001b[38;5;241m*\u001b[39m np\u001b[38;5;241m.\u001b[39mlog(l) \u001b[38;5;28;01mfor\u001b[39;00m ki \u001b[38;5;129;01min\u001b[39;00m k]\n\u001b[1;32m 23\u001b[0m ) \u001b[38;5;66;03m# + np.sum([np.log(np.math.factorial(ki)) for ki in k])\u001b[39;00m\n\u001b[1;32m 24\u001b[0m loss \u001b[38;5;241m=\u001b[39m loss_fn(x, np\u001b[38;5;241m.\u001b[39mexp(np\u001b[38;5;241m.\u001b[39mdot(x, w)))\n\u001b[1;32m 25\u001b[0m log4\u001b[38;5;241m.\u001b[39mappend(\n\u001b[1;32m 26\u001b[0m {\n\u001b[1;32m 27\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mEpoch\u001b[39m\u001b[38;5;124m\"\u001b[39m: epoch,\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 31\u001b[0m }\n\u001b[1;32m 32\u001b[0m )\n",
"Cell \u001b[0;32mIn[18], line 22\u001b[0m, in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m 18\u001b[0m w \u001b[38;5;241m-\u001b[39m\u001b[38;5;241m=\u001b[39m learning_rate \u001b[38;5;241m*\u001b[39m dw(w, x)\n\u001b[1;32m 19\u001b[0m \u001b[38;5;66;03m# custom\u001b[39;00m\n\u001b[1;32m 20\u001b[0m \u001b[38;5;66;03m# loss = x.shape[0] * np.exp(np.dot(x, w))\u001b[39;00m\n\u001b[1;32m 21\u001b[0m loss_fn \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;01mlambda\u001b[39;00m k, l: \u001b[38;5;28mlen\u001b[39m(k) \u001b[38;5;241m*\u001b[39m l \u001b[38;5;241m-\u001b[39m np\u001b[38;5;241m.\u001b[39msum(\n\u001b[0;32m---> 22\u001b[0m [\u001b[43mki\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43m \u001b[49m\u001b[43mnp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mlog\u001b[49m\u001b[43m(\u001b[49m\u001b[43ml\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;28;01mfor\u001b[39;00m ki \u001b[38;5;129;01min\u001b[39;00m k]\n\u001b[1;32m 23\u001b[0m ) \u001b[38;5;66;03m# + np.sum([np.log(np.math.factorial(ki)) for ki in k])\u001b[39;00m\n\u001b[1;32m 24\u001b[0m loss \u001b[38;5;241m=\u001b[39m loss_fn(x, np\u001b[38;5;241m.\u001b[39mexp(np\u001b[38;5;241m.\u001b[39mdot(x, w)))\n\u001b[1;32m 25\u001b[0m log4\u001b[38;5;241m.\u001b[39mappend(\n\u001b[1;32m 26\u001b[0m {\n\u001b[1;32m 27\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mEpoch\u001b[39m\u001b[38;5;124m\"\u001b[39m: epoch,\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 31\u001b[0m }\n\u001b[1;32m 32\u001b[0m )\n",
"\u001b[0;31mValueError\u001b[0m: operands could not be broadcast together with shapes (3,) (214,) "
]
}
],
"source": [
"## pset 4\n",
"\n",
"dw = lambda w, x: np.sum([len(x) * np.exp(np.dot(xi, w)) * x - np.sum(np.dot(x.T,x)) for xi in x])\n",
"\n",
"#primitive = lambda xi, wi: (x.shape[0] * np.exp(wi * xi) * xi) - (xi**2)\n",
"#p_dw = lambda w, xi: np.array([primitive(xi, wi) for xi, wi in ])\n",
"\n",
"\n",
"def SGD_problem4(\n",
" w: np.array,\n",
" x: np.array,\n",
" learning_rate=0.01,\n",
" n_epochs=1000,\n",
"):\n",
" global log4\n",
" log4 = []\n",
" for epoch in range(n_epochs):\n",
" w -= learning_rate * dw(w, x)\n",
" # custom\n",
" # loss = x.shape[0] * np.exp(np.dot(x, w))\n",
" loss_fn = lambda k, l: len(k) * l - np.sum(\n",
" [ki * np.log(l) for ki in k]\n",
" ) # + np.sum([np.log(np.math.factorial(ki)) for ki in k])\n",
" loss = loss_fn(x, np.exp(np.dot(x, w)))\n",
" log4.append(\n",
" {\n",
" \"Epoch\": epoch,\n",
" \"New w\": w,\n",
" \"dw\": dw(w, x),\n",
" \"Loss\": loss,\n",
" }\n",
" )\n",
" print(f\"Epoch {epoch}\", f\"Loss: {loss}\")\n",
" return w\n",
"\n",
"\n",
"def debug_SGD_3(data, w=np.array([1, 1])):\n",
" print(\"SGD Problem 4\")\n",
" print(f\"w: {SGD_problem4(w, data)}\")\n",
" dflog = pd.DataFrame(log4)\n",
" return dflog\n",
"\n",
"\n",
"_ = debug_SGD_3(\n",
" data=df[[\"HIGH_T\", \"LOW_T\", \"PRECIP\"]].to_numpy(),\n",
" w=np.array([1.0, 1.0, 1.0]),\n",
")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7c00197d-873d-41b0-a458-dc8478b40f52",
"metadata": {},
"outputs": [],
"source": []
}
],
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|