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495 lines
82 KiB
Plaintext
495 lines
82 KiB
Plaintext
1 year ago
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"cells": [
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{
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"cell_type": "markdown",
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"id": "eb335a6d",
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"metadata": {},
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"source": [
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"# Linear regression\n",
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"\n",
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"The linear regression is a training procedure based on a linear model. The model makes a prediction by simply computing a weighted sum of the input features, plus a constant term called the bias term (also called the intercept term, $\\theta_0$):\n",
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"\n",
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"$$ \\hat{y}=\\theta_0 x_0 + \\theta_1 x_1 + \\theta_2 x_2 + \\cdots + \\theta_n x_n$$\n",
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"\n",
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"\n",
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"This can be written more easy by the using vector notation form:\n",
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"\n",
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"$$\\hat{y}= h_\\theta(x) = \\theta^T x$$\n",
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"\n",
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"---\n",
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"**Now that we have our model, how do we train it?**\n",
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"\n",
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"*Please, consider that training the model means adjusting the parameters to reduce the error or minimizing the cost function.* \n",
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"\n",
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"The most common performance measure of a regression model is the **Mean Square Error (MSE)**. Therefore, to train a Linear Regression model, you need to find the value of $\\theta$ that minimizes the MSE:\n",
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"\n",
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"\n",
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"$$ MSE(X,h_\\theta) = \\frac{1}{m} \\sum_{i=1}^{m} \\left( \\theta^T x^{(i)}-y^{(i)} \\right)^2$$\n",
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"\n",
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"\n",
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"$$ MSE(X,h_\\theta) = \\frac{1}{m} \\sum_{i=1}^{m} \\left(\\hat{y}^{(i)}-y^{(i)} \\right)^2$$\n",
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"\n",
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"$$ MSE(X,h_\\theta) = \\frac{1}{m} \\left(\\hat{y}-y \\right)^2$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "12cf820d",
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"metadata": {},
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"source": [
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"# The normal equation\n",
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"\n",
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"To find the value of $\\theta$ that minimizes the cost function, there is a closed-form or direct solution that gives the result. This is called the **Normal Equation**; and can be found it by deriving the *MSE* equation as a function of $\\theta$ and making it equals to zero:\n",
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"\n",
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"\\begin{eqnarray*}\n",
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"\t\\frac{\\partial J(\\theta)}{\\partial \\theta} = 0\\\\\n",
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"\t\\frac{\\partial J(\\theta)}{\\partial \\theta}=\\frac{1}{m}\\left(\\theta x-y \\right)^2=\\frac{1}{m}\\left(\\theta x-y \\right)^T\\left(\\theta x-y \\right)\\\\\n",
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"\t\\frac{1}{m}\\left[(\\theta x)^T-y^T \\right] \\left[\\theta x-y \\right]\n",
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"\\end{eqnarray*}"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9139b1c9",
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"metadata": {},
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"source": [
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"just considers that:\n",
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"\\begin{align*}\n",
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"\t(A^T)^T = A\\\\\n",
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"\t(A+B)^T = A^T + B^T\\\\\n",
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"\t(kA)^T = kA^T\\\\\n",
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"\t(AB)^T =A^TB^T\n",
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"\\end{align*}"
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]
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},
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{
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"cell_type": "markdown",
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"id": "5b069092",
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"metadata": {},
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"source": [
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"just considers that $(\\theta x)^Ty=y^T(\\theta x)$\n",
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"\\begin{align*}\n",
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"\t0=\\frac{\\partial\\,}{\\partial \\theta}\\frac{1}{m}\\left[(\\theta x)^T\\theta x - (\\theta x)^Ty-y^T\\theta x+y^Ty \\right]\\\\\n",
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"\\end{align*}"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ae45a858",
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"metadata": {},
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"source": [
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"$$0=\\frac{\\partial\\,}{\\partial \\theta}\\frac{1}{m}\\left[(\\theta x)^T\\theta x - 2(\\theta x)^Ty+y^Ty \\right]$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ec27d3cd",
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"metadata": {},
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"source": [
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"$$0=\\frac{1}{m}\\left[2\\theta x^T x - 2(x)^Ty+0 \\right]$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7c8c8cfe",
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"metadata": {},
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"source": [
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"$$0=\\frac{2}{m}\\left[\\theta x^T x - (x)^Ty+0 \\right]$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c06b74fd",
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"metadata": {},
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"source": [
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"$$\\hat{\\theta} = (X^T X)^{-1} X^{T} y $$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1315356e",
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"metadata": {},
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"source": [
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"## A basic example\n",
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"\n",
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"First try to implement the linear regressor by only using two instances or points, $(x^0, y^0)$ and $(x^1,y^1)$, thus:\n",
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"\n",
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"$$\\hat{y}^{(i)}=\\theta_0 x_0+ \\theta_1 x_1^{(i)}$$\n",
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"\n",
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"but, considers that $x_0=1$, then:\n",
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"\n",
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"$$\\hat{y}^{(i)}=\\theta_0 + \\theta_1 x_1^{(i)}$$\n",
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"\n",
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"then, to write it in vector form\n",
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"\n",
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"$$\n",
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" \\hat{y}^0=\n",
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" \\begin{bmatrix}\n",
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" \\theta_0 & \\theta_1\n",
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" \\end{bmatrix} \n",
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" \\begin{bmatrix}\n",
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" 1 \\\\ \n",
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" x_0\n",
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" \\end{bmatrix}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "54dda83d",
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"metadata": {},
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"source": [
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"Now, we need a matrix form to manage all data:\n",
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"\n",
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"$$\n",
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" \\begin{bmatrix}\n",
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" \\hat{y}^0 \\\\ \n",
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" \\hat{y}^1\\\\\n",
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" \\end{bmatrix}\n",
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" =\n",
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" \\begin{bmatrix}\n",
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" \\theta_0 + \\theta_1 x^0_1 \\\\\n",
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" \\theta_0 + \\theta_1 x^0_1 \\\\\n",
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" \\end{bmatrix}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d86618e4",
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"metadata": {},
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"source": [
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"$$\n",
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" \\begin{bmatrix}\n",
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" \\hat{y}^0 \\\\ \n",
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" \\hat{y}^1\\\\\n",
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" \\end{bmatrix}\n",
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" =\n",
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" \\begin{bmatrix}\n",
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" 1 & x^0_1 \\\\\n",
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" 1 & x^1_1 \\\\\n",
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" \\end{bmatrix}\n",
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" \\begin{bmatrix}\n",
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" \\theta_0 \\\\\n",
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" \\theta_1 \\\\\n",
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" \\end{bmatrix}\n",
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"$$\n",
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"\n",
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"Now, let's compute $\\theta$ using the normal equation:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "d2670f5b",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[1.000e+00, 1.000e-03],\n",
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" [1.000e+00, 2.196e+00]])"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"import numpy as np\n",
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"X = np.array([[1,0.001],[1, 2.196]])\n",
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"Xo = np.array([[0.001],[2.196]])\n",
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"y = np.array([4.314825, 10.877373])\n",
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"X"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "d95f739b",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([4.31183523, 2.9897713 ])"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"theta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(y)\n",
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"theta"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "b241fb39",
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"metadata": {},
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"outputs": [],
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"source": [
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"t0 = theta[0]\n",
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"t1 = theta[1]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "152366f0",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"import matplotlib.pyplot as plt\n",
|
||
|
|
"plt.plot(Xo,y, '*k')\n",
|
||
|
|
"plt.show()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 8,
|
||
|
|
"id": "4f15df79",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"Xvec = np.linspace(0,2,2)\n",
|
||
|
|
"Xnew = np.c_[np.ones((2, 1)), Xvec]\n",
|
||
|
|
"ypre = Xnew.dot(theta)\n",
|
||
|
|
"plt.plot(Xo,y, '*k')\n",
|
||
|
|
"plt.plot(Xvec, ypre, '-r')\n",
|
||
|
|
"plt.show()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "a986cb4d",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"# My Linear Regressor\n",
|
||
|
|
"\n",
|
||
|
|
"**Instructions:** Create your own function train (estimate) the model's parameter for a Linear Regressor implementing the *Normal Equation*. Test your model with the next generated data."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 9,
|
||
|
|
"id": "5ca2686a",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"[<matplotlib.lines.Line2D at 0xffff82286040>]"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 9,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"np.random.seed(42)\n",
|
||
|
|
"Xo = 2 * np.random.rand(100, 1)\n",
|
||
|
|
"X = np.c_[np.ones((100, 1)), Xo] # add x0 = 1 to each instance\n",
|
||
|
|
"y = 4 + 3 * Xo + np.random.randn(100, 1)\n",
|
||
|
|
"plt.plot(Xo, y, '.k')"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "43a9426c",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Therefore, the model will become:\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
" \\begin{bmatrix}\n",
|
||
|
|
" \\hat{y}^0 \\\\ \n",
|
||
|
|
" \\hat{y}^1\\\\\n",
|
||
|
|
" \\vdots \\\\\n",
|
||
|
|
" \\hat{y}^n\n",
|
||
|
|
" \\end{bmatrix}\n",
|
||
|
|
" =\n",
|
||
|
|
" \\begin{bmatrix}\n",
|
||
|
|
" 1 & x^0_1 \\\\\n",
|
||
|
|
" 1 & x^1_1 \\\\\n",
|
||
|
|
" \\vdots & \\vdots \\\\\n",
|
||
|
|
" 1 & x^n_1\n",
|
||
|
|
" \\end{bmatrix}\n",
|
||
|
|
" \\begin{bmatrix}\n",
|
||
|
|
" \\theta_0 \\\\\n",
|
||
|
|
" \\theta_1 \\\\\n",
|
||
|
|
" \\vdots \\\\\n",
|
||
|
|
" \\theta_n\n",
|
||
|
|
" \\end{bmatrix}\n",
|
||
|
|
"$$\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "c07852f4",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"## Using Sklearn to train the model\n",
|
||
|
|
"Python already includes a linear regression function within Scikit-Learn."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 11,
|
||
|
|
"id": "17481af6",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"(array([4.21509616]), array([[0. , 2.77011339]]))"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 11,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"from sklearn.linear_model import LinearRegression\n",
|
||
|
|
"\n",
|
||
|
|
"lin_reg = LinearRegression()\n",
|
||
|
|
"lin_reg.fit(X, y)\n",
|
||
|
|
"lin_reg.intercept_, lin_reg.coef_"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 16,
|
||
|
|
"id": "c5e85264",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"array([[4.21509616],\n",
|
||
|
|
" [9.75532293]])"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 16,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"X_new = np.array([[0], [2]])\n",
|
||
|
|
"X_new_b = np.c_[np.ones((2, 1)), X_new] #Se agrega x0=1 para cada instancia\n",
|
||
|
|
"y_Pred=lin_reg.predict(X_new_b)\n",
|
||
|
|
"y_Pred"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 23,
|
||
|
|
"id": "20da996b",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"<matplotlib.legend.Legend at 0xffff72c2bfa0>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 23,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"plt.plot(Xo,y, '.k', label='data')\n",
|
||
|
|
"plt.plot(X_new, y_Pred, '-r', label='SK-model')\n",
|
||
|
|
"plt.legend()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "19ad47f6",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"# Gradient Descent"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": null,
|
||
|
|
"id": "1a8ec08d",
|
||
|
|
"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": "3.8.10"
|
||
|
|
}
|
||
|
|
},
|
||
|
|
"nbformat": 4,
|
||
|
|
"nbformat_minor": 5
|
||
|
|
}
|