assignment-2

assignment-2/assignment_2/GML.py

@@ -0,0 +1,305 @@
+## imports
+import numpy as np
+import pandas as pd
+from scipy.optimize import minimize
+from scipy.stats import norm
+import math
+
+
+## Problem 1
+data = [4, 5, 7, 8, 8, 9, 10, 5, 2, 3, 5, 4, 8, 9]
+
+data_mean = np.mean(data)
+data_variance = np.var(data)
+
+
+mu = 0.5
+sigma = 0.5
+w = np.array([mu, sigma])
+
+w_star = np.array([data_mean, data_variance])
+mu_star = data_mean
+sigma_star = np.sqrt(data_variance)
+offset = 10 * np.random.random(2)
+
+w1p = w_star + 0.5 * offset
+w1n = w_star - 0.5 * offset
+w2p = w_star + 0.25 * offset
+w2n = w_star - 0.25 * offset
+
+# 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.
+
+loss = lambda mu, sigma, x: np.sum(
+    [-np.log(norm.pdf(xi, loc=mu, scale=sigma)) for xi in x]
+)
+
+loss_2_electric_boogaloo = lambda mu, sigma, x: -len(x) / 2 * np.log(
+    2 * np.pi * sigma**2
+) - 1 / (2 * sigma**2) * np.sum((x - mu) ** 2)
+
+
+dmu = lambda mu, sigma, x: -np.sum([mu - xi for xi in x]) / (sigma**2)
+dsigma = lambda mu, sigma, x: -len(x) / sigma + np.sum([(mu - xi) ** 2 for xi in x]) / (
+    sigma**3
+)
+
+log = []
+
+
+def SGD_problem1(mu, sigma, x, learning_rate=0.01, n_epochs=1000):
+    global log
+    log = []
+    for epoch in range(n_epochs):
+        mu += learning_rate * dmu(mu, sigma, x)
+        sigma += learning_rate * dsigma(mu, sigma, x)
+
+        # print(f"Epoch {epoch}, Loss: {loss(mu, sigma, x)}, New mu: {mu}, New sigma: {sigma}")
+        log.append(
+            {
+                "Epoch": epoch,
+                "Loss": loss(mu, sigma, x),
+                "Loss 2 Alternative": loss_2_alternative(mu, sigma, x),
+                "New mu": mu,
+                "New sigma": sigma,
+            }
+        )
+    return np.array([mu, sigma])
+
+
+def debug_SGD_1(wnn, data):
+    print("SGD Problem 1")
+    print("wnn", SGD_problem1(*wnn, data))
+    dflog = pd.DataFrame(log)
+    dflog["mu_star"] = mu_star
+    dflog["mu_std"] = sigma_star
+    print(f"mu diff at start {dflog.iloc[0]['New mu'] - dflog.iloc[0]['mu_star']}")
+    print(f"mu diff at end {dflog.iloc[-1]['New mu'] - dflog.iloc[-1]['mu_star']}")
+    if np.abs(dflog.iloc[-1]["New mu"] - dflog.iloc[-1]["mu_star"]) < np.abs(
+        dflog.iloc[0]["New mu"] - dflog.iloc[0]["mu_star"]
+    ):
+        print("mu is improving")
+    else:
+        print("mu is not improving")
+
+    print(f"sigma diff at start {dflog.iloc[0]['New sigma'] - dflog.iloc[0]['mu_std']}")
+    print(f"sigma diff at end {dflog.iloc[-1]['New sigma'] - dflog.iloc[-1]['mu_std']}")
+    if np.abs(dflog.iloc[-1]["New sigma"] - dflog.iloc[-1]["mu_std"]) < np.abs(
+        dflog.iloc[0]["New sigma"] - dflog.iloc[0]["mu_std"]
+    ):
+        print("sigma is improving")
+    else:
+        print("sigma is not improving")
+
+    return dflog
+
+
+# _ = debug_SGD_1(w1p, data)
+# _ = debug_SGD_1(w1n, data)
+# _ = debug_SGD_1(w2p, data)
+# _ = debug_SGD_1(w2n, data)
+
+
+# TODO EXPLAIN WHY += WORKS HERE.
+
+
+## Problem 2
+x = np.array([8, 16, 22, 33, 50, 51])
+y = np.array([5, 20, 14, 32, 42, 58])
+
+# $-\frac{n}{\sigma}+\frac{1}{\sigma^3}\sum_{i=1}^n(y_i - (mx+c))^2$
+dsigma = lambda sigma, c, m, x: -len(x) / sigma + np.sum(
+    [(xi - (m * x + c)) ** 2 for xi in x]
+) / (sigma**3)
+# $-\frac{1}{\sigma^2}\sum_{i=1}^n(y_i - (mx+c))$
+dc = lambda sigma, c, m, x: -np.sum([xi - (m * x + c) for xi in x]) / (sigma**2)
+# $-\frac{1}{\sigma^2}\sum_{i=1}^n(x_i(y_i - (mx+c)))$
+dm = lambda sigma, c, m, x: -np.sum([x * (xi - (m * x + c)) for xi in x]) / (sigma**2)
+
+
+log2 = []
+
+
+def SGD_problem2(
+    sigma: float,
+    c: float,
+    m: float,
+    x: np.array,
+    y: np.array,
+    learning_rate=0.01,
+    n_epochs=1000,
+):
+    global log2
+    log2 = []
+    for epoch in range(n_epochs):
+        sigma += learning_rate * dsigma(sigma, c, m, x)
+        c += learning_rate * dc(sigma, c, m, x)
+        m += learning_rate * dm(sigma, c, m, x)
+
+        log2.append(
+            {
+                "Epoch": epoch,
+                "New sigma": sigma,
+                "New c": c,
+                "New m": m,
+                "dc": dc(sigma, c, m, x),
+                "dm": dm(sigma, c, m, x),
+                "dsigma": dsigma(sigma, c, m, x),
+                "Loss": loss((m * x + c), sigma, y),
+            }
+        )
+        print(f"Epoch {epoch}, Loss: {loss((m * x + c), sigma, y)}")
+    return np.array([sigma, c, m])
+
+
+# def debug_SGD_2(wnn, data):
+#     print("SGD Problem 2")
+#     print("wnn", SGD_problem2(*wnn, data))
+#     dflog = pd.DataFrame(log)
+#     dflog["m_star"] = m_star
+#     dflog["c_star"] = c_star
+#     dflog["sigma_star"] = sigma_star
+#     print(f"m diff at start {dflog.iloc[0]['New m'] - dflog.iloc[0]['m_star']}")
+#     print(f"m diff at end {dflog.iloc[-1]['New m'] - dflog.iloc[-1]['m_star']}")
+#     if np.abs(dflog.iloc[-1]["New m"] - dflog.iloc[-1]["m_star"]) < np.abs(
+#         dflog.iloc[0]["New m"] - dflog.iloc[0]["m_star"]
+#     ):
+#         print("m is improving")
+#     else:
+#         print("m is not improving")
+#     print(f"c diff at start {dflog.iloc[0]['New c'] - dflog.iloc[0]['c_star']}")
+#     print(f"c diff at end {dflog.iloc[-1]['New c'] - dflog.iloc[-1]['c_star']}")
+#     if np.abs(dflog.iloc[-1]["New c"] - dflog.iloc[-1]["c_star"]) < np.abs(
+#         dflog.iloc[0]["New c"] - dflog.iloc[0]["c_star"]
+#     ):
+#         print("c is improving")
+#     else:
+#         print("c is not improving")
+#     print(f"sigma diff at start {dflog.iloc[0]['New sigma'] - dflog.iloc[0]['sigma_star']}")
+#     print(f"sigma diff at end {dflog.iloc[-1]['New sigma'] - dflog.iloc[-1]['sigma_star']}")
+#     if np.abs(dflog.iloc[-1]["New sigma"] - dflog.iloc[-1]["sigma_star"]) < np.abs(
+#         dflog.iloc[0]["New sigma"] - dflog.iloc[0]["sigma_star"]
+#     ):
+#         print("sigma is improving")
+#     else:
+#         print("sigma is not improving")
+#     return dflog
+
+result = SGD_problem2(0.5, 0.5, 0.5, x, y)
+print(f"final parameters: m={result[2]}, c={result[1]}, sigma={result[0]}")
+
+
+## pset2
+# 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 $\frac{\partial L}{\partial \lambda}$!
+
+
+import pandas as pd
+
+df = pd.read_csv("../data/01_raw/nyc_bb_bicyclist_counts.csv")
+
+dlambda = lambda l, k: len(k) - np.sum([ki / l for ki in k])
+
+
+def SGD_problem3(
+    l: float,
+    k: np.array,
+    learning_rate=0.01,
+    n_epochs=1000,
+):
+    global log3
+    log3 = []
+    for epoch in range(n_epochs):
+        l -= learning_rate * dlambda(l, k)
+        # $n\lambda + \sum_{i=1}^n \ln(k_i!) - \sum_{i=1}^n k_i \ln(\lambda)$
+        # the rest of the loss function is commented out because it's a
+        # constant and was causing overflows. It is unnecessary, and a useless
+        # pain.
+        loss = len(k) * l - np.sum(
+            [ki * np.log(l) for ki in k]
+        )  # + np.sum([np.log(np.math.factorial(ki)) for ki in k])
+
+        log3.append(
+            {
+                "Epoch": epoch,
+                "New lambda": l,
+                "dlambda": dlambda(l, k),
+                "Loss": loss,
+            }
+        )
+        print(f"Epoch {epoch}", f"Loss: {loss}")
+    return np.array([l])
+
+
+l_star = df["BB_COUNT"].mean()
+
+
+def debug_SGD_3(data, l=1000):
+    print("SGD Problem 3")
+    print(f"l: {SGD_problem3(l, data)}")
+    dflog = pd.DataFrame(log3)
+    dflog["l_star"] = l_star
+    print(f"l diff at start {dflog.iloc[0]['New lambda'] - dflog.iloc[0]['l_star']}")
+    print(f"l diff at end {dflog.iloc[-1]['New lambda'] - dflog.iloc[-1]['l_star']}")
+    if np.abs(dflog.iloc[-1]["New lambda"] - dflog.iloc[-1]["l_star"]) < np.abs(
+        dflog.iloc[0]["New lambda"] - dflog.iloc[0]["l_star"]
+    ):
+        print("l is improving")
+    else:
+        print("l is not improving")
+    return dflog
+
+
+debug_SGD_3(data=df["BB_COUNT"].values, l=l_star + 1000)
+debug_SGD_3(data=df["BB_COUNT"].values, l=l_star - 1000)
+
+
+## pset 4
+
+# dw = lambda w, x: len(x) * np.exp(np.dot(x, w)) * x - np.sum()
+
+primitive = lambda xi, wi: (x.shape[0] * np.exp(wi * xi) * xi) - (xi**2)
+p_dw = lambda w, xi: np.array([primitive(xi, wi) for xi, wi in ])
+
+
+def SGD_problem4(
+    w: np.array,
+    x: np.array,
+    learning_rate=0.01,
+    n_epochs=1000,
+):
+    global log4
+    log4 = []
+    for epoch in range(n_epochs):
+        w -= learning_rate * p_dw(w, x)
+        # custom
+        # loss = x.shape[0] * np.exp(np.dot(x, w))
+        loss_fn = lambda k, l: len(k) * l - np.sum(
+            [ki * np.log(l) for ki in k]
+        )  # + np.sum([np.log(np.math.factorial(ki)) for ki in k])
+        loss = loss_fn(x, np.exp(np.dot(x, w)))
+        log4.append(
+            {
+                "Epoch": epoch,
+                "New w": w,
+                "dw": dw(w, x),
+                "Loss": loss,
+            }
+        )
+        print(f"Epoch {epoch}", f"Loss: {loss}")
+    return w
+
+
+def debug_SGD_3(data, w=np.array([1, 1])):
+    print("SGD Problem 4")
+    print(f"w: {SGD_problem4(w, data)}")
+    dflog = pd.DataFrame(log4)
+    return dflog
+
+
+_ = debug_SGD_3(
+    data=df[["HIGH_T", "LOW_T", "PRECIP"]].to_numpy(),
+    w=np.array([1.0, 1.0, 1.0]),
+)

assignment-2/assignment_2/Gaussian Maximum Likelihood.ipynb

@@ -0,0 +1,1682 @@
+{
+ "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",
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+      "Epoch 724, Loss: 297.82627680024444\n",
+      "Epoch 725, Loss: 297.8262761386215\n",
+      "Epoch 726, Loss: 297.8262754770806\n",
+      "Epoch 727, Loss: 297.826274815622\n",
+      "Epoch 728, Loss: 297.82627415424554\n",
+      "Epoch 729, Loss: 297.82627349295143\n",
+      "Epoch 730, Loss: 297.8262728317395\n",
+      "Epoch 731, Loss: 297.82627217061\n",
+      "Epoch 732, Loss: 297.82627150956284\n",
+      "Epoch 733, Loss: 297.8262708485981\n",
+      "Epoch 734, Loss: 297.82627018771575\n",
+      "Epoch 735, Loss: 297.8262695269158\n",
+      "Epoch 736, Loss: 297.8262688661983\n",
+      "Epoch 737, Loss: 297.82626820556334\n",
+      "Epoch 738, Loss: 297.826267545011\n",
+      "Epoch 739, Loss: 297.8262668845411\n",
+      "Epoch 740, Loss: 297.82626622415387\n",
+      "Epoch 741, Loss: 297.82626556384935\n",
+      "Epoch 742, Loss: 297.82626490362736\n",
+      "Epoch 743, Loss: 297.826264243488\n",
+      "Epoch 744, Loss: 297.82626358343146\n",
+      "Epoch 745, Loss: 297.82626292345765\n",
+      "Epoch 746, Loss: 297.82626226356655\n",
+      "Epoch 747, Loss: 297.8262616037582\n",
+      "Epoch 748, Loss: 297.8262609440329\n",
+      "Epoch 749, Loss: 297.8262602843903\n",
+      "Epoch 750, Loss: 297.82625962483047\n",
+      "Epoch 751, Loss: 297.82625896535376\n",
+      "Epoch 752, Loss: 297.82625830595987\n",
+      "Epoch 753, Loss: 297.8262576466491\n",
+      "Epoch 754, Loss: 297.82625698742123\n",
+      "Epoch 755, Loss: 297.82625632827643\n",
+      "Epoch 756, Loss: 297.8262556692147\n",
+      "Epoch 757, Loss: 297.82625501023597\n",
+      "Epoch 758, Loss: 297.8262543513405\n",
+      "Epoch 759, Loss: 297.8262536925281\n",
+      "Epoch 760, Loss: 297.82625303379893\n",
+      "Epoch 761, Loss: 297.8262523751529\n",
+      "Epoch 762, Loss: 297.82625171659015\n",
+      "Epoch 763, Loss: 297.82625105811076\n",
+      "Epoch 764, Loss: 297.8262503997146\n",
+      "Epoch 765, Loss: 297.82624974140174\n",
+      "Epoch 766, Loss: 297.82624908317234\n",
+      "Epoch 767, Loss: 297.8262484250262\n",
+      "Epoch 768, Loss: 297.82624776696355\n",
+      "Epoch 769, Loss: 297.8262471089844\n",
+      "Epoch 770, Loss: 297.82624645108865\n",
+      "Epoch 771, Loss: 297.8262457932765\n",
+      "Epoch 772, Loss: 297.82624513554777\n",
+      "Epoch 773, Loss: 297.8262444779027\n",
+      "Epoch 774, Loss: 297.8262438203412\n",
+      "Epoch 775, Loss: 297.8262431628633\n",
+      "Epoch 776, Loss: 297.8262425054691\n",
+      "Epoch 777, Loss: 297.82624184815865\n",
+      "Epoch 778, Loss: 297.82624119093174\n",
+      "Epoch 779, Loss: 297.82624053378873\n",
+      "Epoch 780, Loss: 297.82623987672946\n",
+      "Epoch 781, Loss: 297.826239219754\n",
+      "Epoch 782, Loss: 297.8262385628624\n",
+      "Epoch 783, Loss: 297.82623790605464\n",
+      "Epoch 784, Loss: 297.82623724933075\n",
+      "Epoch 785, Loss: 297.82623659269086\n",
+      "Epoch 786, Loss: 297.8262359361348\n",
+      "Epoch 787, Loss: 297.8262352796629\n",
+      "Epoch 788, Loss: 297.8262346232749\n",
+      "Epoch 789, Loss: 297.8262339669709\n",
+      "Epoch 790, Loss: 297.8262333107511\n",
+      "Epoch 791, Loss: 297.82623265461535\n",
+      "Epoch 792, Loss: 297.8262319985638\n",
+      "Epoch 793, Loss: 297.8262313425964\n",
+      "Epoch 794, Loss: 297.8262306867131\n",
+      "Epoch 795, Loss: 297.8262300309141\n",
+      "Epoch 796, Loss: 297.82622937519943\n",
+      "Epoch 797, Loss: 297.826228719569\n",
+      "Epoch 798, Loss: 297.82622806402287\n",
+      "Epoch 799, Loss: 297.82622740856107\n",
+      "Epoch 800, Loss: 297.8262267531837\n",
+      "Epoch 801, Loss: 297.82622609789064\n",
+      "Epoch 802, Loss: 297.826225442682\n",
+      "Epoch 803, Loss: 297.826224787558\n",
+      "Epoch 804, Loss: 297.8262241325183\n",
+      "Epoch 805, Loss: 297.8262234775633\n",
+      "Epoch 806, Loss: 297.8262228226928\n",
+      "Epoch 807, Loss: 297.82622216790685\n",
+      "Epoch 808, Loss: 297.8262215132056\n",
+      "Epoch 809, Loss: 297.8262208585888\n",
+      "Epoch 810, Loss: 297.8262202040569\n",
+      "Epoch 811, Loss: 297.8262195496096\n",
+      "Epoch 812, Loss: 297.82621889524705\n",
+      "Epoch 813, Loss: 297.82621824096935\n",
+      "Epoch 814, Loss: 297.8262175867764\n",
+      "Epoch 815, Loss: 297.8262169326682\n",
+      "Epoch 816, Loss: 297.82621627864495\n",
+      "Epoch 817, Loss: 297.82621562470666\n",
+      "Epoch 818, Loss: 297.8262149708532\n",
+      "Epoch 819, Loss: 297.82621431708463\n",
+      "Epoch 820, Loss: 297.8262136634011\n",
+      "Epoch 821, Loss: 297.82621300980264\n",
+      "Epoch 822, Loss: 297.82621235628915\n",
+      "Epoch 823, Loss: 297.82621170286075\n",
+      "Epoch 824, Loss: 297.8262110495175\n",
+      "Epoch 825, Loss: 297.8262103962594\n",
+      "Epoch 826, Loss: 297.8262097430863\n",
+      "Epoch 827, Loss: 297.8262090899985\n",
+      "Epoch 828, Loss: 297.82620843699596\n",
+      "Epoch 829, Loss: 297.8262077840786\n",
+      "Epoch 830, Loss: 297.8262071312466\n",
+      "Epoch 831, Loss: 297.82620647849984\n",
+      "Epoch 832, Loss: 297.8262058258384\n",
+      "Epoch 833, Loss: 297.82620517326245\n",
+      "Epoch 834, Loss: 297.8262045207719\n",
+      "Epoch 835, Loss: 297.82620386836675\n",
+      "Epoch 836, Loss: 297.82620321604696\n",
+      "Epoch 837, Loss: 297.8262025638128\n",
+      "Epoch 838, Loss: 297.82620191166416\n",
+      "Epoch 839, Loss: 297.8262012596011\n",
+      "Epoch 840, Loss: 297.82620060762355\n",
+      "Epoch 841, Loss: 297.8261999557316\n",
+      "Epoch 842, Loss: 297.8261993039254\n",
+      "Epoch 843, Loss: 297.8261986522048\n",
+      "Epoch 844, Loss: 297.82619800056995\n",
+      "Epoch 845, Loss: 297.8261973490208\n",
+      "Epoch 846, Loss: 297.82619669755746\n",
+      "Epoch 847, Loss: 297.8261960461799\n",
+      "Epoch 848, Loss: 297.82619539488826\n",
+      "Epoch 849, Loss: 297.82619474368244\n",
+      "Epoch 850, Loss: 297.82619409256245\n",
+      "Epoch 851, Loss: 297.8261934415284\n",
+      "Epoch 852, Loss: 297.82619279058036\n",
+      "Epoch 853, Loss: 297.82619213971833\n",
+      "Epoch 854, Loss: 297.8261914889422\n",
+      "Epoch 855, Loss: 297.82619083825216\n",
+      "Epoch 856, Loss: 297.82619018764836\n",
+      "Epoch 857, Loss: 297.8261895371304\n",
+      "Epoch 858, Loss: 297.82618888669873\n",
+      "Epoch 859, Loss: 297.8261882363531\n",
+      "Epoch 860, Loss: 297.8261875860939\n",
+      "Epoch 861, Loss: 297.8261869359208\n",
+      "Epoch 862, Loss: 297.826186285834\n",
+      "Epoch 863, Loss: 297.82618563583344\n",
+      "Epoch 864, Loss: 297.8261849859192\n",
+      "Epoch 865, Loss: 297.8261843360914\n",
+      "Epoch 866, Loss: 297.82618368634996\n",
+      "Epoch 867, Loss: 297.8261830366949\n",
+      "Epoch 868, Loss: 297.82618238712627\n",
+      "Epoch 869, Loss: 297.82618173764416\n",
+      "Epoch 870, Loss: 297.8261810882486\n",
+      "Epoch 871, Loss: 297.82618043893956\n",
+      "Epoch 872, Loss: 297.826179789717\n",
+      "Epoch 873, Loss: 297.8261791405811\n",
+      "Epoch 874, Loss: 297.82617849153183\n",
+      "Epoch 875, Loss: 297.82617784256917\n",
+      "Epoch 876, Loss: 297.8261771936933\n",
+      "Epoch 877, Loss: 297.8261765449042\n",
+      "Epoch 878, Loss: 297.8261758962016\n",
+      "Epoch 879, Loss: 297.826175247586\n",
+      "Epoch 880, Loss: 297.82617459905725\n",
+      "Epoch 881, Loss: 297.82617395061516\n",
+      "Epoch 882, Loss: 297.8261733022601\n",
+      "Epoch 883, Loss: 297.82617265399193\n",
+      "Epoch 884, Loss: 297.82617200581075\n",
+      "Epoch 885, Loss: 297.8261713577164\n",
+      "Epoch 886, Loss: 297.8261707097091\n",
+      "Epoch 887, Loss: 297.82617006178884\n",
+      "Epoch 888, Loss: 297.8261694139557\n",
+      "Epoch 889, Loss: 297.82616876620966\n",
+      "Epoch 890, Loss: 297.82616811855064\n",
+      "Epoch 891, Loss: 297.8261674709788\n",
+      "Epoch 892, Loss: 297.82616682349425\n",
+      "Epoch 893, Loss: 297.8261661760969\n",
+      "Epoch 894, Loss: 297.82616552878676\n",
+      "Epoch 895, Loss: 297.82616488156384\n",
+      "Epoch 896, Loss: 297.8261642344284\n",
+      "Epoch 897, Loss: 297.8261635873801\n",
+      "Epoch 898, Loss: 297.82616294041935\n",
+      "Epoch 899, Loss: 297.8261622935459\n",
+      "Epoch 900, Loss: 297.82616164676\n",
+      "Epoch 901, Loss: 297.8261610000614\n",
+      "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": "code",
+   "execution_count": 7,
+   "id": "7c8b167d-c397-4155-93f3-d826c279fbb2",
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [
+    {
+     "ename": "SyntaxError",
+     "evalue": "invalid syntax (3400372070.py, line 66)",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;36m  Cell \u001b[0;32mIn[7], line 66\u001b[0;36m\u001b[0m\n\u001b[0;31m    p_dw = lambda w, xi: np.array([primitive(xi, wi) for xi, wi in ])\u001b[0m\n\u001b[0m                                                                   ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n"
+     ]
+    }
+   ],
+   "source": [
+    "## pset 4\n",
+    "\n",
+    "# dw = lambda w, x: len(x) * np.exp(np.dot(x, w)) * x - np.sum()\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 * p_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": "markdown",
+   "id": "69e9148b-70fb-46e3-bc29-a08f471cccab",
+   "metadata": {},
+   "source": [
+    "Seek to maximize likelihood, which is equivalent to minimizing log likelihood, which is equivalent to maximizing MSE.\n",
+    "\n",
+    "-> Find MSE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "fb9d2e20-8a02-4f78-a3d3-6fc171f17af6",
+   "metadata": {},
+   "outputs": [
+    {
+     "ename": "SyntaxError",
+     "evalue": "incomplete input (3680490224.py, line 3)",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;36m  Cell \u001b[0;32mIn[2], line 3\u001b[0;36m\u001b[0m\n\u001b[0;31m    def MSE(x: List):\u001b[0m\n\u001b[0m                     ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m incomplete input\n"
+     ]
+    }
+   ],
+   "source": [
+    "from typing import List\n",
+    "from numpy import \n",
+    "\n",
+    "w = np.array(2)\n",
+    "\n",
+    "def partial_of_Loss(w, x, y):\n",
+    "    \n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7c00197d-873d-41b0-a458-dc8478b40f52",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.4"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

assignment-2/data/01_raw/nyc_bb_bicyclist_counts.csv

@@ -0,0 +1,215 @@
+Date,HIGH_T,LOW_T,PRECIP,BB_COUNT
+1-Apr-17,46.00,37.00,0.00,606
+2-Apr-17,62.10,41.00,0.00,2021
+3-Apr-17,63.00,50.00,0.03,2470
+4-Apr-17,51.10,46.00,1.18,723
+5-Apr-17,63.00,46.00,0.00,2807
+6-Apr-17,48.90,41.00,0.73,461
+7-Apr-17,48.00,43.00,0.01,1222
+8-Apr-17,55.90,39.90,0.00,1674
+9-Apr-17,66.00,45.00,0.00,2375
+10-Apr-17,73.90,55.00,0.00,3324
+11-Apr-17,80.10,62.10,0.00,3887
+12-Apr-17,73.90,57.90,0.02,2565
+13-Apr-17,64.00,48.90,0.00,3353
+14-Apr-17,64.90,48.90,0.00,2942
+15-Apr-17,64.90,52.00,0.00,2253
+16-Apr-17,84.90,62.10,0.01,2877
+17-Apr-17,73.90,64.00,0.01,3152
+18-Apr-17,66.00,50.00,0.00,3415
+19-Apr-17,52.00,45.00,0.01,1965
+20-Apr-17,64.90,50.00,0.17,1567
+21-Apr-17,53.10,48.00,0.29,1426
+22-Apr-17,55.90,52.00,0.11,1318
+23-Apr-17,64.90,46.90,0.00,2520
+24-Apr-17,60.10,50.00,0.01,2544
+25-Apr-17,54.00,50.00,0.91,611
+26-Apr-17,59.00,54.00,0.34,1247
+27-Apr-17,68.00,59.00,0.00,2959
+28-Apr-17,82.90,57.90,0.00,3679
+29-Apr-17,84.00,64.00,0.06,3315
+30-Apr-17,64.00,54.00,0.00,2225
+1-May-17,72.00,50.00,0.00,3084
+2-May-17,73.90,66.90,0.00,3423
+3-May-17,64.90,57.90,0.00,3342
+4-May-17,63.00,50.00,0.00,3019
+5-May-17,59.00,52.00,3.02,513
+6-May-17,64.90,57.00,0.18,1892
+7-May-17,54.00,48.90,0.01,3539
+8-May-17,57.00,45.00,0.00,2886
+9-May-17,61.00,48.00,0.00,2718
+10-May-17,70.00,51.10,0.00,2810
+11-May-17,61.00,51.80,0.00,2657
+12-May-17,62.10,51.10,0.00,2640
+13-May-17,51.10,45.00,1.31,151
+14-May-17,64.90,46.00,0.02,1452
+15-May-17,66.90,55.90,0.00,2685
+16-May-17,78.10,57.90,0.00,3666
+17-May-17,90.00,66.00,0.00,3535
+18-May-17,91.90,75.00,0.00,3190
+19-May-17,90.00,75.90,0.00,2952
+20-May-17,64.00,55.90,0.01,2161
+21-May-17,66.90,55.00,0.00,2612
+22-May-17,61.00,54.00,0.59,768
+23-May-17,68.00,57.90,0.00,3174
+24-May-17,66.90,57.00,0.04,2969
+25-May-17,57.90,55.90,0.58,488
+26-May-17,73.00,55.90,0.10,2590
+27-May-17,71.10,61.00,0.00,2609
+28-May-17,71.10,59.00,0.00,2640
+29-May-17,57.90,55.90,0.13,836
+30-May-17,59.00,55.90,0.06,2301
+31-May-17,75.00,57.90,0.03,2689
+1-Jun-17,78.10,62.10,0.00,3468
+2-Jun-17,73.90,60.10,0.01,3271
+3-Jun-17,72.00,55.00,0.01,2589
+4-Jun-17,68.00,60.10,0.09,1805
+5-Jun-17,66.90,60.10,0.02,2171
+6-Jun-17,55.90,53.10,0.06,1193
+7-Jun-17,66.90,54.00,0.00,3211
+8-Jun-17,68.00,59.00,0.00,3253
+9-Jun-17,80.10,59.00,0.00,3401
+10-Jun-17,84.00,68.00,0.00,3066
+11-Jun-17,90.00,73.00,0.00,2465
+12-Jun-17,91.90,77.00,0.00,2854
+13-Jun-17,93.90,78.10,0.01,2882
+14-Jun-17,84.00,69.10,0.29,2596
+15-Jun-17,75.00,66.00,0.00,3510
+16-Jun-17,68.00,66.00,0.00,2054
+17-Jun-17,73.00,66.90,1.39,1399
+18-Jun-17,84.00,72.00,0.01,2199
+19-Jun-17,87.10,70.00,1.35,1648
+20-Jun-17,82.00,72.00,0.03,3407
+21-Jun-17,82.00,72.00,0.00,3304
+22-Jun-17,82.00,70.00,0.00,3368
+23-Jun-17,82.90,75.90,0.04,2283
+24-Jun-17,82.90,71.10,1.29,2307
+25-Jun-17,82.00,69.10,0.00,2625
+26-Jun-17,78.10,66.00,0.00,3386
+27-Jun-17,75.90,61.00,0.18,3182
+28-Jun-17,78.10,62.10,0.00,3766
+29-Jun-17,81.00,68.00,0.00,3356
+30-Jun-17,88.00,73.90,0.01,2687
+1-Jul-17,84.90,72.00,0.23,1848
+2-Jul-17,87.10,73.00,0.00,2467
+3-Jul-17,87.10,71.10,0.45,2714
+4-Jul-17,82.90,70.00,0.00,2296
+5-Jul-17,84.90,71.10,0.00,3170
+6-Jul-17,75.00,71.10,0.01,3065
+7-Jul-17,79.00,68.00,1.78,1513
+8-Jul-17,82.90,70.00,0.00,2718
+9-Jul-17,81.00,69.10,0.00,3048
+10-Jul-17,82.90,71.10,0.00,3506
+11-Jul-17,84.00,75.00,0.00,2929
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assignment-2/data/nyc_bb_bicyclist_counts.csv

@@ -0,0 +1,215 @@
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assignment-2/pyproject.toml

@@ -0,0 +1,23 @@
+[tool.poetry]
+name = "assignment-2"
+version = "0.1.0"
+description = ""
+authors = ["Artur Janik <aj2614@nyu.edu>"]
+readme = "README.md"
+packages = [{include = "assignment_2"}]
+
+[tool.poetry.dependencies]
+python = "^3.10"
+black = "^23.1.0"
+jupyterlab = "^3.6.1"
+ipython = "^8.10.0"
+numpy = "^1.24.2"
+pandas = "^1.5.3"
+jax = "^0.4.4"
+seaborn = "^0.12.2"
+matplotlib = "^3.7.0"
+
+
+[build-system]
+requires = ["poetry-core"]
+build-backend = "poetry.core.masonry.api"