{ "cells": [ { "cell_type": "markdown", "id": "26826e61", "metadata": {}, "source": [ "# A1 · L3 — The line\n", "\n", "One channel, one line. What least squares actually minimises, and what the two\n", "numbers it returns mean to a marketer: **base sales** and **incremental sales per\n", "$1,000 of TV**.\n", "\n", "Every figure and number this notebook produces is what the lesson prints." ] }, { "cell_type": "code", "execution_count": 1, "id": "85667a5a", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:06.914936Z", "iopub.status.busy": "2026-08-19T09:12:06.914671Z", "iopub.status.idle": "2026-08-19T09:12:07.172569Z", "shell.execute_reply": "2026-08-19T09:12:07.171783Z" } }, "outputs": [], "source": [ "# hide — plumbing so this file runs both in the repo and in Colab.\n", "try:\n", " from _figkit import save_fig, record\n", "except ImportError: # Colab — no repo, no problem\n", " def save_fig(name, plot, **kw):\n", " import matplotlib.pyplot as plt\n", " fig, ax = plt.subplots(figsize=kw.get('figsize', (7, 4.2))); plot(ax); plt.show()\n", " def record(key, value):\n", " print(f'{key} = {value}'); return value\n", "\n", "from pathlib import Path\n", "REL = 'public/datasets/a1-regression/advertising-media-mix.csv'\n", "here = Path.cwd()\n", "LOCAL = next((p / REL for p in [here, *here.parents] if (p / REL).exists()), None)\n", "CSV = LOCAL or 'https://raw.githubusercontent.com/Dr-Shashank-S-Sharma/expedify-ai-courses/main/datasets/a1-regression/advertising-media-mix.csv'" ] }, { "cell_type": "code", "execution_count": 2, "id": "1339065b", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:07.175966Z", "iopub.status.busy": "2026-08-19T09:12:07.175585Z", "iopub.status.idle": "2026-08-19T09:12:07.386758Z", "shell.execute_reply": "2026-08-19T09:12:07.386542Z" } }, "outputs": [ { "data": { "text/html": [ "
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tv_spendsales
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" ], "text/plain": [ " tv_spend sales\n", "0 230.1 22.1\n", "1 44.5 10.4\n", "2 17.2 9.3\n", "3 151.5 18.5\n", "4 180.8 12.9" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#| caption: Load the data\n", "import numpy as np\n", "import pandas as pd\n", "\n", "df = pd.read_csv(CSV)\n", "x = df['tv_spend'] # TV budget, $ thousands\n", "y = df['sales'] # sales, thousands of units\n", "\n", "df[['tv_spend', 'sales']].head()" ] }, { "cell_type": "markdown", "id": "4acd5415", "metadata": {}, "source": [ "## 1. Any line has an error. Measure it.\n", "\n", "Before finding the best line, be able to score *a* line. Guess two, score both." ] }, { "cell_type": "code", "execution_count": 3, "id": "4f9d7d0a", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:07.387940Z", "iopub.status.busy": "2026-08-19T09:12:07.387856Z", "iopub.status.idle": "2026-08-19T09:12:07.390482Z", "shell.execute_reply": "2026-08-19T09:12:07.390280Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "intercept 14.02 slope 0.0000 -> SSE 5,417\n", "intercept 4.00 slope 0.0900 -> SSE 6,810\n", "intercept 7.00 slope 0.0475 -> SSE 2,103\n" ] } ], "source": [ "#| caption: Score a line by its total squared error\n", "def sse(intercept, slope):\n", " \"\"\"Sum of squared errors: how wrong this line is, over every market.\"\"\"\n", " predicted = intercept + slope * x\n", " return float(((y - predicted) ** 2).sum())\n", "\n", "GUESSES = [(y.mean(), 0.0), (4.0, 0.09), (7.0, 0.0475)]\n", "for intercept, slope in GUESSES:\n", " print(f'intercept {intercept:5.2f} slope {slope:.4f} -> SSE {sse(intercept, slope):,.0f}')" ] }, { "cell_type": "code", "execution_count": 4, "id": "73ee80ed", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:07.391385Z", "iopub.status.busy": "2026-08-19T09:12:07.391299Z", "iopub.status.idle": "2026-08-19T09:12:07.481438Z", "shell.execute_reply": "2026-08-19T09:12:07.481174Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "sse_flat = 5,417\n", "sse_steep = 6,810\n", "figure three-lines -> the-line.three-lines.{light,dark}.svg\n" ] } ], "source": [ "# hide — figure only\n", "record('sse_flat', f'{sse(y.mean(), 0.0):,.0f}')\n", "record('sse_steep', f'{sse(4.0, 0.09):,.0f}')\n", "\n", "def plot(ax):\n", " ax.scatter(x, y, s=18, alpha=0.45)\n", " grid = np.linspace(x.min(), x.max(), 50)\n", " styles = [(':', 'flat: ignore TV'), ('--', 'too steep'), ('-', 'least squares')]\n", " for (intercept, slope), (ls, label) in zip(GUESSES, styles):\n", " ax.plot(grid, intercept + slope * grid, ls, linewidth=1.9,\n", " label=f'{label} — SSE {sse(intercept, slope):,.0f}')\n", " ax.set_xlabel('TV budget ($ thousands)')\n", " ax.set_ylabel('Sales (thousands of units)')\n", " ax.legend(frameon=False, fontsize=9)\n", "\n", "save_fig('three-lines', plot, figsize=(7, 4.6))" ] }, { "cell_type": "markdown", "id": "a346c580", "metadata": {}, "source": [ "## 2. Every slope has an error. Plot the whole curve.\n", "\n", "Sweep the slope, hold the best intercept for each, and the error traces a bowl with\n", "exactly one bottom. That bowl is why there is a single right answer." ] }, { "cell_type": "code", "execution_count": 5, "id": "d4a36cc2", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:07.484011Z", "iopub.status.busy": "2026-08-19T09:12:07.483849Z", "iopub.status.idle": "2026-08-19T09:12:07.504716Z", "shell.execute_reply": "2026-08-19T09:12:07.504030Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "lowest error at slope ~ 0.0477\n" ] } ], "source": [ "#| caption: The error curve, and its one bottom\n", "slopes = np.linspace(0.0, 0.10, 200)\n", "errors = [sse(y.mean() - s * x.mean(), s) for s in slopes] # best intercept for each slope\n", "best = slopes[int(np.argmin(errors))]\n", "print(f'lowest error at slope ~ {best:.4f}')" ] }, { "cell_type": "code", "execution_count": 6, "id": "2cf7ae2a", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:07.506883Z", "iopub.status.busy": "2026-08-19T09:12:07.506657Z", "iopub.status.idle": "2026-08-19T09:12:07.579903Z", "shell.execute_reply": "2026-08-19T09:12:07.579655Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "figure error-curve -> the-line.error-curve.{light,dark}.svg\n" ] } ], "source": [ "# hide — figure only\n", "def plot(ax):\n", " ax.plot(slopes, errors)\n", " ax.axvline(best, linestyle='--', linewidth=1.2, color='#8a8a94')\n", " ax.annotate('the answer', xy=(best, min(errors)), xytext=(10, 26),\n", " textcoords='offset points', fontsize=9, color='#8a8a94')\n", " ax.set_xlabel('Slope — extra sales per $1,000 of TV')\n", " ax.set_ylabel('Total squared error')\n", "\n", "save_fig('error-curve', plot, figsize=(7, 3.8))" ] }, { "cell_type": "markdown", "id": "9a846e8d", "metadata": {}, "source": [ "## 3. Fit it properly\n", "\n", "`polyfit` finds the bottom of that bowl exactly, without searching." ] }, { "cell_type": "code", "execution_count": 7, "id": "613e045b", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:07.581070Z", "iopub.status.busy": "2026-08-19T09:12:07.581001Z", "iopub.status.idle": "2026-08-19T09:12:07.583609Z", "shell.execute_reply": "2026-08-19T09:12:07.583348Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "base sales (intercept) = 7.03 thousand units\n", "incremental (slope) = 0.0475 thousand units per $1,000 of TV\n", "R2 = 0.612\n" ] } ], "source": [ "#| caption: Fit the line\n", "slope, intercept = np.polyfit(x, y, 1)\n", "predicted = intercept + slope * x\n", "r2 = 1 - ((y - predicted) ** 2).sum() / ((y - y.mean()) ** 2).sum()\n", "\n", "print(f'base sales (intercept) = {intercept:.2f} thousand units')\n", "print(f'incremental (slope) = {slope:.4f} thousand units per $1,000 of TV')\n", "print(f'R2 = {r2:.3f}')" ] }, { "cell_type": "code", "execution_count": 8, "id": "dfdad9bc", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:07.584679Z", "iopub.status.busy": "2026-08-19T09:12:07.584593Z", "iopub.status.idle": "2026-08-19T09:12:07.660891Z", "shell.execute_reply": "2026-08-19T09:12:07.660290Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "slope = 0.0475\n", "intercept = 7.03\n", "r2 = 0.612\n", "sse_best = 2,103\n", "units_per_100k = 4.8\n", "units_per_1k = 47.5\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "figure fitted-line -> the-line.fitted-line.{light,dark}.svg\n" ] } ], "source": [ "# hide — figure + metrics\n", "record('slope', round(float(slope), 4))\n", "record('intercept', round(float(intercept), 2))\n", "record('r2', round(float(r2), 3))\n", "record('sse_best', f'{sse(intercept, slope):,.0f}')\n", "record('units_per_100k', round(float(slope) * 100, 1))\n", "record('units_per_1k', round(float(slope) * 1000, 1))\n", "\n", "def plot(ax):\n", " ax.scatter(x, y, s=20, alpha=0.55, label='markets')\n", " grid = np.linspace(0, x.max(), 50)\n", " ax.plot(grid, intercept + slope * grid, linewidth=2, label='fitted line')\n", " ax.axhline(intercept, linestyle=':', linewidth=1.2, color='#8a8a94')\n", " ax.annotate('base sales', xy=(2, intercept), xytext=(0, 8),\n", " textcoords='offset points', fontsize=9, color='#8a8a94')\n", " ax.set_xlabel('TV budget ($ thousands)')\n", " ax.set_ylabel('Sales (thousands of units)')\n", " ax.legend(frameon=False, fontsize=9)\n", "\n", "save_fig('fitted-line', plot, figsize=(7, 4.4))" ] }, { "cell_type": "markdown", "id": "de504760", "metadata": {}, "source": [ "## 4. The only chart a marketer needs\n", "\n", "Split predicted sales into what you'd have sold anyway and what the budget bought." ] }, { "cell_type": "code", "execution_count": 9, "id": "78b70296", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:07.662435Z", "iopub.status.busy": "2026-08-19T09:12:07.662339Z", "iopub.status.idle": "2026-08-19T09:12:07.664060Z", "shell.execute_reply": "2026-08-19T09:12:07.663867Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "$ 0k TV -> 7.03k units (7.03 base + 0.00 incremental)\n", "$ 50k TV -> 9.41k units (7.03 base + 2.38 incremental)\n", "$100k TV -> 11.79k units (7.03 base + 4.75 incremental)\n", "$200k TV -> 16.54k units (7.03 base + 9.51 incremental)\n" ] } ], "source": [ "#| caption: Base vs incremental at four budgets\n", "budgets = [0, 50, 100, 200]\n", "for budget in budgets:\n", " incremental = slope * budget\n", " print(f'${budget:3d}k TV -> {intercept + incremental:5.2f}k units '\n", " f'({intercept:.2f} base + {incremental:.2f} incremental)')" ] }, { "cell_type": "code", "execution_count": 10, "id": "bf7d1131", "metadata": { "execution": { "iopub.execute_input": "2026-08-19T09:12:07.664941Z", "iopub.status.busy": "2026-08-19T09:12:07.664871Z", "iopub.status.idle": "2026-08-19T09:12:07.729371Z", "shell.execute_reply": "2026-08-19T09:12:07.729128Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "pred_at_100 = 11.79\n", "incremental_at_100 = 4.75\n", "base_share_at_100 = 60%\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "figure base-vs-incremental -> the-line.base-vs-incremental.{light,dark}.svg\n" ] } ], "source": [ "# hide — figure + metrics\n", "record('pred_at_100', round(float(intercept + slope * 100), 2))\n", "record('incremental_at_100', round(float(slope * 100), 2))\n", "record('base_share_at_100', f'{100 * intercept / (intercept + slope * 100):.0f}%')\n", "\n", "def plot(ax):\n", " labels = [f'${b}k' for b in budgets]\n", " base = [intercept] * len(budgets)\n", " inc = [slope * b for b in budgets]\n", " ax.bar(labels, base, label='base sales')\n", " ax.bar(labels, inc, bottom=base, label='incremental — what TV bought')\n", " ax.set_xlabel('TV budget')\n", " ax.set_ylabel('Predicted sales (thousands of units)')\n", " ax.legend(frameon=False, fontsize=9)\n", "\n", "save_fig('base-vs-incremental', plot, figsize=(7, 4.0))" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.13.5" } }, "nbformat": 4, "nbformat_minor": 5 }