" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Least Squares Linear Regression (Review)\n", "\n", "For the remainder of this lecture we will focus on fitting least squares linear regression models. Recall that **linear regression** models are functions of the form:\n", "\n", "\\begin{align}\\large\n", "f_\\theta(x) = x^T \\theta = \\sum_{j=1}^p \\theta_j x_j\n", "\\end{align}\n", "\n", "and **least squares** implies a loss function of the form:\n", "\n", "\\begin{align} \\large\n", "L_\\mathcal{D}(\\theta) = \\frac{1}{n} \\sum_{i=1}^n \\left(y_i - f_\\theta(x) \\right)^2\n", "\\end{align}\n", "\n", "In the previous lecture, we derived the **normal equations** which define the loss minimizing $\\hat{\\theta}$ for: \n", "\n", "\\begin{align}\\large\n", "\\hat{\\theta} \n", "& \\large = \\arg\\min L_\\mathcal{D}(\\theta) \\\\\n", "& \\large = \\arg\\min \\frac{1}{n} \\sum_{i=1}^n \\left(y_i - X \\theta \\right)^2 \\\\\n", "& \\large = \\left( X^T X \\right)^{-1} X^T Y\n", "\\end{align}\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "---\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear Regression with scikit-learn\n", "\n", "In this lecture we will use the [scikit-learn `linear_model` package](http://scikit-learn.org/stable/modules/linear_model.html) to compute the normal equations. This package supports a wide range of **generalized linear models**. For those who are interested in studying machine learning, I would encourage you to skim through the descriptions of the various models in the `linear_model` package. These are the foundation of most practical applications of supervised machine learning." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from sklearn import linear_model" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following block of code creates an instance of the Least Squares Linear Regression model and the fits that model to the training data. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "line_reg = linear_model.LinearRegression(fit_intercept=True)\n", "# Fit the model to the data\n", "line_reg.fit(train_data[['X']], train_data['Y'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Notice:** In the above code block we explicitly added a **bias** (**intercept**) term by setting `fit_intercept=True`. Therefore we will not need to add an additional `constant` feature.\n", "\n", "---\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Plotting the Model\n", "\n", "To plot the model we will predict the value of $Y$ for a range of $X$ values. I will call these **query points** `X_query`." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ "X_query = np.linspace(-10, 10, 500)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Use the regression model to render predictions at each `X_query` point." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Note that np.linspace returns a 1d vector therefore \n", "# we must transform it into a 2d column vector\n", "line_Yhat_query = line_reg.predict(\n", " np.reshape(X_query, (len(X_query),1)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To plot the residual we will also predict the $Y$ value for all the training points:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": true }, "outputs": [], "source": [ "line_Yhat = line_reg.predict(train_data[['X']])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following visualization code constructs a line as well as the residual plot" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Residual Analysis\n", "\n", "To help answer these questions it can often be helpful to plot the residuals in a **residual plot**. Residual plots plot the difference between the predicted value and the observed value in response to a particular covariate ($X$ dimension). The residual plot can help reveal patterns in the residual that might support additional modeling.\n" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": true }, "outputs": [], "source": [ "residuals = line_Yhat - train_data['Y'] " ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "
\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Using a Smoothing Model \n", "\n", "To better visualize the pattern we could apply another regression package. In the following plotting code we call a more sophisticated regression package `sklearn.kernel_ridge` to estimate a smoothed approximation to the residuals. " ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from sklearn.kernel_ridge import KernelRidge\n", "\n", "# Use Kernelized Ridge Regression with Radial Basis Functions to \n", "# compute a smoothed estimator. Later in this notebook we will \n", "# actually implement part of this ...\n", "clf = KernelRidge(kernel='rbf', alpha=2)\n", "clf.fit(train_data[['X']], residuals)\n", "residual_smoothed = clf.predict(np.reshape(X_query, (len(X_query),1)))" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "
\n", "\n", "---\n", "\n", "## What does it mean to be a _linear model_\n", "\n", "\n", "Let's return to what it means to be a linear model:\n", "\n", "$$\\large\n", "f_\\theta(x) = x^T \\theta = \\sum_{j=1}^p x_j \\theta_j\n", "$$\n", "\n", "In what sense is the above model **linear**? \n", "1. Linear in the features $x$? \n", "1. Linear in the parameters $\\theta$?\n", "1. Linear in both at the same time?\n", "\n", "---\n", "
\n", "Yes, Yes, and No. If we look at just the features or just the parameters the model is linear. However, if we look at both at the same time it is not. Why?\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Feature Functions\n", "\n", "Consider the following alternative model formulation:\n", "\n", "$$\\large\n", "f_\\theta\\left( \\phi(x) \\right) = \\phi(x)^T \\theta = \\sum_{j=1}^{k} \\phi(x)_j \\theta_j\n", "$$\n", "\n", "where $\\phi_j$ is an *arbitrary function* from $x\\in \\mathbb{R}^p$ to $\\phi(x)_j \\in \\mathbb{R}$ and we define $k$ of these functions. We often refer to these functions $\\phi_j$ as **feature functions** or **basis functions** and their design plays a critical role in both how we capture prior knowledge and our ability to fit complicated data.\n", "\n", "---\n", "\n", "
\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Capturing Domain Knowledge\n", "\n", "\n", "Feature functions can be used to capture domain knowledge by:\n", "1. Introducing additional information from other sources\n", "1. Combining related features\n", "1. Encoding non-linear patterns\n", "\n", "Suppose I had data about customer purchases and I wanted to estimate their income:\n", "\n", "\\begin{align}\n", "\\phi(\\text{date}, \\text{lat}, \\text{lon}, \\text{amount})_1 \n", "&= \\textbf{isWinter}(\\text{date}) \\\\\n", "\\phi(\\text{date}, \\text{lat}, \\text{lon}, \\text{amount})_2 \n", "&= \\cos\\left( \\frac{\\textbf{Hour}(\\text{date})}{12} \\pi \\right) \\\\\n", "\\phi(\\text{date}, \\text{lat}, \\text{lon}, \\text{amount})_3 \n", "&= \\frac{\\text{amount}}{\\textbf{avg_spend}[\\textbf{ZipCode}[\\text{lat}, \\text{lon}]]} \\\\\n", "\\phi(\\text{date}, \\text{lat}, \\text{lon}, \\text{amount})_4 \n", "&= \\exp\\left(-\\textbf{Distance}\\left((\\text{lat},\\text{lon}), \\textbf{StoreA}\\right)\\right)^2 \\\\\n", "\\phi(\\text{date}, \\text{lat}, \\text{lon}, \\text{amount})_5 \n", "&= \\exp\\left(-\\textbf{Distance}\\left((\\text{lat},\\text{lon}), \\textbf{StoreB}\\right)\\right)^2 \n", "\\end{align}\n", "\n", "**Notice:** In the above feature functions:\n", "1. The transformations are non-linear\n", "1. They pull in additional information\n", "1. They may encode implicit knowledge\n", "1. The functions $\\phi$ **do not** depend on $\\theta$\n", "\n", "---\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear in $\\theta$\n", "As a consequence, while the model $f_\\theta\\left( \\phi(x) \\right)$ is no longer linear in $x$ it is still a **linear model** because it _is linear in $\\theta$_. This means we can continue to use the normal equations to compute the optimal parameters. \n", "\n", "To apply the normal equations we define the transformed feature matrix:\n", "\n", "
![](\"phi_matrix.png\")
\n",
"\n",
"Then substituting $\\Phi$ for $X$ we obtain the **normal equation**:\n",
"\n",
"$$ \\large\n",
"\\hat{\\theta} = \\left( \\Phi^T \\Phi \\right)^{-1} \\Phi^T Y\n",
"$$\n",
"\n",
"It is worth noting that the model is also linear in $\\phi$ and that the $\\phi_j$ form a new **basis** (hence the term **basis functions**) in which the data live. As a consequence we can think of $\\phi$ as mapping the data into a new (often higher dimensional space) in which the relationship between $y$ and $\\phi(x)$ is defined by a **hyperplane**. \n",
"\n",
"---\n",
"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Transforming the data with $\\phi$\n", "\n", "In our toy data set we observed a cyclic pattern. Here we construct a $\\phi$ to capture the cyclic nature of our data and visualize the corresponding hyperplane.\n", "\n", "\n", "In the following cell we define a function $\\phi$ that maps $x\\in \\mathbb{R}$ to the vector $[x,\\sin(x)] \\in \\mathbb{R}^2$ \n", "\n", "$$ \\large\n", "\\phi(x) = [x, \\sin(x)]\n", "$$\n", "\n", "**Why not:**\n", "\n", "$$ \\large\n", "\\phi(x) = [x, \\sin(\\theta_3 x + \\theta_4)]\n", "$$\n", "\n", "This would no longer be linear $\\theta$. However, in practice we might want to consider a range of $\\sin$ basis:\n", "\n", "$$ \\large\n", "\\phi_{\\alpha,\\beta}(x) = \\sin(\\alpha x + \\beta)\n", "$$\n", "\n", "for different values of $\\alpha$ and $\\beta$. The parameters $\\alpha$ and $\\beta$ are typically called **hyperparameters** because (at least in this setting) they are not set automatically through learning.\n" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def sin_phi(x):\n", " return [x, np.sin(x)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We then compute the matrix $\\Phi$ by applying $\\phi$ to reach row (record) in the matrix $X$. " ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([[-9.88955766, 0.44822588],\n", " [-9.58831011, 0.16280424],\n", " [-9.31222958, -0.11231092],\n", " [-9.09545422, -0.32340318],\n", " [-9.07099175, -0.34645201]])" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi = np.array([sin_phi(x) for x in train_data['X']])\n", "\n", "# Look at a few examples\n", "Phi[:5,]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is worth noting that in practice we might prefer a more efficient \"vectorized\" version of the above code:\n", "```python\n", "Phi = np.vstack((train_data['X'], np.sin(train_data['X']))).T\n", "```\n", "however in this notebook we will use more explicit `for` loop notation.\n", "\n", "---\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Fit a _Linear Model_ on $\\Phi$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can again use the scikit-learn package to fit a linear model on the transformed space." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "LinearRegression(copy_X=True, fit_intercept=False, n_jobs=1, normalize=False)" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sin_reg = linear_model.LinearRegression(fit_intercept=False)\n", "sin_reg.fit(Phi, train_data['Y'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Plotting the predictions\n", "\n", "**What will the prediction from this mode look like?**\n", "\n", "$$ \\large\n", "f_\\theta(x) = \\phi(x)^T \\theta = \\theta_1 x + \\theta_2 \\sin x\n", "$$\n", "\n", "---\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# A Linear Model in Transformed Space\n", "\n", "As discussed earlier the model we just constructed, while non-linear in $x$ is actually a linear model in $\\phi(x)$ and we can visualize that linear model's structure in higher dimensions." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Computing the RMSE of the various methods so far:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def rmse(y, yhat):\n", " return np.sqrt(np.mean((yhat-y)**2))" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [], "source": [ "const_rmse = rmse(train_data['Y'], train_data['Y'].mean())\n", "line_rmse = rmse(train_data['Y'], line_Yhat)\n", "sin_rmse = rmse(train_data['Y'], sin_Yhat)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "
\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Polynomial Basis:\n", "\n", "The first set of generic feature functions we will consider is the polynomial basis:\n", "\n", "\\begin{align}\n", "\\phi(x) = [x, x^2, x^3, \\ldots, x^k]\n", "\\end{align}\n", "\n", "We can define a generic python function to implement this basis:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def poly_phi(k):\n", " return lambda x: [x ** i for i in range(1, k+1)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To simply the comparison of feature functions we define the following routine:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def evaluate_basis(phi, desc):\n", " # Apply transformation\n", " Phi = np.array([phi(x) for x in train_data['X']])\n", " \n", " # Fit a model\n", " reg_model = linear_model.LinearRegression(fit_intercept=False)\n", " reg_model.fit(Phi, train_data['Y'])\n", " \n", " # Create plot line\n", " X_test = np.linspace(-10, 10, 1000) # Fine grained test X\n", " Phi_test = np.array([phi(x) for x in X_test])\n", " Yhat_test = reg_model.predict(Phi_test)\n", " line = go.Scatter(name = desc, x=X_test, y=Yhat_test)\n", " \n", " # Compute RMSE\n", " Yhat = reg_model.predict(Phi)\n", " error = rmse(train_data['Y'], Yhat)\n", " \n", " # return results\n", " return (line, error, reg_model)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Starting with $k=5$ degree polynomials" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Gaussian Radial Basis Functions\n", "\n", "One of the more widely used generic feature functions are Gaussian radial basis functions. These feature functions take the form:\n", "\n", "$$\n", "\\phi_{(\\lambda, u_1, \\ldots, u_k)}(x) = \\left[\\exp\\left( - \\frac{\\left|\\left|x-u_1\\right|\\right|_2^2}{\\lambda} \\right), \\ldots, \n", "\\exp\\left( - \\frac{\\left|\\left| x-u_k \\right|\\right|_2^2}{\\lambda} \\right) \\right]\n", "$$\n", "\n", "The **hyper-parameters** $u_1$ through $u_k$ and $\\lambda$ are not optimized with $\\theta$ but instead are set externally. In many cases the $u_i$ may correspond to points in the training data. The term $\\lambda$ defines the spread of the basis function and determines the \"smoothness\" of the function $f_\\theta(\\phi(x))$.\n", "\n", "The following is a plot of three radial basis function centered at 2 with different values of $\\lambda$." ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def gaussian_rbf(u, lam=1):\n", " return lambda x: np.exp(-(x - u)**2 / lam**2)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "
\n", "\n", "---\n", "\n", "This is a very common occurrence in machine learning. As we increased the model complexity " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# What's happening: _Over-fitting_\n", "\n", "As we increase the expressiveness of our model we begin to **over-fit** to the variability in our training data. That is we are learning patterns that do not **generalize** beyond our training dataset\n", "\n", "**Over-fitting** is a key challenge in machine learning and statistical inference. At it's core is a fundamental trade-off between **bias** and **variance**: _the desire to explain the training data and yet be robust to variation in the training data_.\n", "\n", "We will study the **bias-variance** trade-off more in the next lecture but for now we will focus on the trade-off between under fitting and over fitting:\n", "\n", "
![](\"under_over_fitting.png\")
\n",
"\n",
"---\n",
"\n",
"" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "# Train, Test, and Validation Splitting" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To manage over-fitting it is essential to split your initial training data into a training and testing dataset. \n", "\n", "
![](\"train_test_split.png\")
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Train - Test Split\n",
"\n",
"Before running cross validation split the data into train and test subsets (typically a 90-10 split). How should you do this? You want the test data to reflect the prediction goal:\n",
"1. **Random** sample of the training data\n",
"1. **Future** examples\n",
"1. Different **stratifications**\n",
"\n",
"Ask yourself, where will I be using this model and how does that relate to my test data.\n",
"\n",
"**Do not look at the test data until after selecting your final model.** Also, it is very important to **not look at the test data until after selecting your final model.** Finally, you should **not look at the test data until after selecting your final model.** "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"# Cross Validation\n",
"\n",
"_With the **remaining training data**:_\n",
"1. Split the remaining **training dataset** k-ways as in the Figure above. The figure uses 5-Fold which is standard. You should split the data in the same way you constructed the test set (this is typically randomly)\n",
"1. For each split train the model on the training fraction and then compute the error (RMSE) on the validation fraction.\n",
"1. Average the error across each validation fraction to _estimate_ the test error.\n",
"\n",
"**Questions:**\n",
"1. What is this accomplishing?\n",
"1. What are the implication on the choice of $k$? \n",
"\n",
"---\n",
"\n",
"\n", "\n", "\n", "\n", "**Answers:**\n", "1. This is repeatedly simulating the train-test split we did earlier. We repeat this process because it is noisy.\n", "1. **Larger $k$** means we average our validation error over more instances which makes our estimate of the test error **more stable**. This typically also means that the validation set is smaller so we have more training data. However, larger $k$ also means we have to train the model more often which gets computational expensive\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# When do we use Cross Validation\n", "\n", "We use cross validation to estimate our test performance **without looking at our test data.** Remember, **do not look at the test data until after selecting your final model.**\n", "\n", "Cross validation is commonly used to tune the model complexity. In the following we will use cross validation to find the optimal number of basis functions." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Cross Validation in sklearn" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from sklearn.model_selection import KFold\n", "nsplits = 5\n", "\n", "X = train_data['X']\n", "Y = train_data['Y']\n", "\n", "\n", "nbasis_values = range(3, 20)\n", "sigma_values = np.linspace(1, 20, 50)\n", "\n", "scores = []\n", "for nbasis in nbasis_values:\n", " for sigma in sigma_values:\n", " # Define my basis for this \"experiment\"\n", " def phi(x):\n", " return (\n", " [gaussian_rbf(u, sigma)(x) for u in np.linspace(-9, 9, nbasis)]\n", " )\n", " Phi = np.array([phi(x) for x in X])\n", "\n", " # One step in k-fold cross validation\n", " def score_model(train_index, test_index):\n", " model = linear_model.LinearRegression(fit_intercept=False)\n", " model.fit(Phi[train_index,], Y[train_index])\n", " return rmse(Y[test_index], model.predict(Phi[test_index,]))\n", "\n", "\n", " # Do the actual K-Fold cross validation\n", " kfold = KFold(nsplits, shuffle=True,random_state=42)\n", " score = np.mean([score_model(tr_ind, te_ind) \n", " for (tr_ind, te_ind) in kfold.split(Phi)])\n", "\n", " # Save the results in the dictionary\n", " scores.append(dict(nbasis=nbasis,sigma=sigma,score=score))" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "nbasis 8.000000\n", "score 5.351392\n", "sigma 4.489796\n", "Name: 259, dtype: float64\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "