# -*- coding: utf-8 -*- """W6_Logistic_regression_lab Automatically generated by Colab. Original file is located at https://colab.research.google.com/drive/1MG7N2HN-Nxow9fzvc0fzxvp3WyKqtgs8 # 🚀 Logistic Regression Lab: Stock Market Prediction ## Lab Overview In this lab, we'll use logistic regression to try predicting whether the stock market goes up or down. Spoiler alert: This is intentionally a challenging prediction problem that will teach us important lessons about when logistic regression works well and when it doesn't. ## Learning Goals: - Apply logistic regression to real data - Interpret probabilities and coefficients - Understand why some prediction problems are inherently difficult - Learn proper model evaluation techniques ## The Stock Market Data In this lab we will examine the `Smarket` data, which is part of the `ISLP` library. This data set consists of percentage returns for the S&P 500 stock index over 1,250 days, from the beginning of 2001 until the end of 2005. For each date, we have recorded the percentage returns for each of the five previous trading days, `Lag1` through `Lag5`. We have also recorded `Volume` (the number of shares traded on the previous day, in billions), `Today` (the percentage return on the date in question) and `Direction` (whether the market was `Up` or `Down` on this date). ### Your Challenge **Question**: Can we predict if the S&P 500 will go up or down based on recent trading patterns? **Why This Matters:** If predictable, this would be incredibly valuable. If not predictable, we learn about market efficiency and realistic expectations for prediction models. To answer the question, **we start by importing our libraries at this top level; these are all imports we have seen in previous labs.** """ import numpy as np import pandas as pd from matplotlib.pyplot import subplots import statsmodels.api as sm from ISLP import load_data from ISLP.models import (ModelSpec as MS, summarize) """We also collect together the new imports needed for this lab.""" from ISLP import confusion_table from ISLP.models import contrast from sklearn.discriminant_analysis import \ (LinearDiscriminantAnalysis as LDA, QuadraticDiscriminantAnalysis as QDA) from sklearn.naive_bayes import GaussianNB from sklearn.neighbors import KNeighborsClassifier from sklearn.preprocessing import StandardScaler from sklearn.model_selection import train_test_split from sklearn.linear_model import LogisticRegression """Now we are ready to load the `Smarket` data.""" Smarket = load_data('Smarket') Smarket """This gives a truncated listing of the data. We can see what the variable names are. """ Smarket.columns """We compute the correlation matrix using the `corr()` method for data frames, which produces a matrix that contains all of the pairwise correlations among the variables. By instructing `pandas` to use only numeric variables, the `corr()` method does not report a correlation for the `Direction` variable because it is qualitative. ![image.png](attachment:image.png) """ Smarket.corr(numeric_only=True) """As one would expect, the correlations between the lagged return variables and today’s return are close to zero. The only substantial correlation is between `Year` and `Volume`. By plotting the data we see that `Volume` is increasing over time. In other words, the average number of shares traded daily increased from 2001 to 2005. """ Smarket.plot(y='Volume'); """## Logistic Regression Next, we will fit a logistic regression model in order to predict `Direction` using `Lag1` through `Lag5` and `Volume`. The `sm.GLM()` function fits *generalized linear models*, a class of models that includes logistic regression. Alternatively, the function `sm.Logit()` fits a logistic regression model directly. The syntax of `sm.GLM()` is similar to that of `sm.OLS()`, except that we must pass in the argument `family=sm.families.Binomial()` in order to tell `statsmodels` to run a logistic regression rather than some other type of generalized linear model. """ allvars = Smarket.columns.drop(['Today', 'Direction', 'Year']) design = MS(allvars) X = design.fit_transform(Smarket) y = Smarket.Direction == 'Up' glm = sm.GLM(y, X, family=sm.families.Binomial()) results = glm.fit() summarize(results) """The smallest *p*-value here is associated with `Lag1`. The negative coefficient for this predictor suggests that if the market had a positive return yesterday, then it is less likely to go up today. However, at a value of 0.15, the *p*-value is still relatively large, and so there is no clear evidence of a real association between `Lag1` and `Direction`. We use the `params` attribute of `results` in order to access just the coefficients for this fitted model. """ results.params """Likewise we can use the `pvalues` attribute to access the *p*-values for the coefficients. """ results.pvalues """The `predict()` method of `results` can be used to predict the probability that the market will go up, given values of the predictors. This method returns predictions on the probability scale. If no data set is supplied to the `predict()` function, then the probabilities are computed for the training data that was used to fit the logistic regression model. As with linear regression, one can pass an optional `exog` argument consistent with a design matrix if desired. Here we have printed only the first ten probabilities. """ probs = results.predict() probs[:10] """In order to make a prediction as to whether the market will go up or down on a particular day, we must convert these predicted probabilities into class labels, `Up` or `Down`. The following two commands create a vector of class predictions based on whether the predicted probability of a market increase is greater than or less than 0.5. """ labels = np.array(['Down']*1250) labels[probs>0.5] = "Up" """The `confusion_table()` function from the `ISLP` package summarizes these predictions, showing how many observations were correctly or incorrectly classified. Our function, which is adapted from a similar function in the module `sklearn.metrics`, transposes the resulting matrix and includes row and column labels. The `confusion_table()` function takes as first argument the predicted labels, and second argument the true labels. """ confusion_table(labels, Smarket.Direction) """The diagonal elements of the confusion matrix indicate correct predictions, while the off-diagonals represent incorrect predictions. Hence our model correctly predicted that the market would go up on 507 days and that it would go down on 145 days, for a total of 507 + 145 = 652 correct predictions. The `np.mean()` function can be used to compute the fraction of days for which the prediction was correct. In this case, logistic regression correctly predicted the movement of the market 52.2% of the time. """ (507+145)/1250, np.mean(labels == Smarket.Direction) """At first glance, it appears that the logistic regression model is working a little better than random guessing. However, this result is misleading because we trained and tested the model on the same set of 1,250 observations. In other words, $100-52.2=47.8%$ is the *training* error rate. As we have seen previously, the training error rate is often overly optimistic --- it tends to underestimate the test error rate. In order to better assess the accuracy of the logistic regression model in this setting, we can fit the model using part of the data, and then examine how well it predicts the *held out* data. This will yield a more realistic error rate, in the sense that in practice we will be interested in our model’s performance not on the data that we used to fit the model, but rather on days in the future for which the market’s movements are unknown. To implement this strategy, we first create a Boolean vector corresponding to the observations from 2001 through 2004. We then use this vector to create a held out data set of observations from 2005. """ train = (Smarket.Year < 2005) Smarket_train = Smarket.loc[train] Smarket_test = Smarket.loc[~train] Smarket_test.shape """The object `train` is a vector of 1,250 elements, corresponding to the observations in our data set. The elements of the vector that correspond to observations that occurred before 2005 are set to `True`, whereas those that correspond to observations in 2005 are set to `False`. Hence `train` is a *boolean* array, since its elements are `True` and `False`. Boolean arrays can be used to obtain a subset of the rows or columns of a data frame using the `loc` method. For instance, the command `Smarket.loc[train]` would pick out a submatrix of the stock market data set, corresponding only to the dates before 2005, since those are the ones for which the elements of `train` are `True`. The `~` symbol can be used to negate all of the elements of a Boolean vector. That is, `~train` is a vector similar to `train`, except that the elements that are `True` in `train` get swapped to `False` in `~train`, and vice versa. Therefore, `Smarket.loc[~train]` yields a subset of the rows of the data frame of the stock market data containing only the observations for which `train` is `False`. The output above indicates that there are 252 such observations. We now fit a logistic regression model using only the subset of the observations that correspond to dates before 2005. We then obtain predicted probabilities of the stock market going up for each of the days in our test set --- that is, for the days in 2005. """ X_train, X_test = X.loc[train], X.loc[~train] y_train, y_test = y.loc[train], y.loc[~train] glm_train = sm.GLM(y_train, X_train, family=sm.families.Binomial()) results = glm_train.fit() probs = results.predict(exog=X_test) """Notice that we have trained and tested our model on two completely separate data sets: training was performed using only the dates before 2005, and testing was performed using only the dates in 2005. Finally, we compare the predictions for 2005 to the actual movements of the market over that time period. We will first store the test and training labels (recall `y_test` is binary). """ D = Smarket.Direction L_train, L_test = D.loc[train], D.loc[~train] """Now we threshold the fitted probability at 50% to form our predicted labels. """ labels = np.array(['Down']*252) labels[probs>0.5] = 'Up' confusion_table(labels, L_test) """The test accuracy is about 48% while the error rate is about 52%""" np.mean(labels == L_test), np.mean(labels != L_test) """The `!=` notation means *not equal to*, and so the last command computes the test set error rate. The results are rather disappointing: the test error rate is 52%, which is worse than random guessing! Of course this result is not all that surprising, given that one would not generally expect to be able to use previous days’ returns to predict future market performance. (After all, if it were possible to do so, then the authors of this book would be out striking it rich rather than writing a statistics textbook.) We recall that the logistic regression model had very underwhelming *p*-values associated with all of the predictors, and that the smallest *p*-value, though not very small, corresponded to `Lag1`. Perhaps by removing the variables that appear not to be helpful in predicting `Direction`, we can obtain a more effective model. After all, using predictors that have no relationship with the response tends to cause a deterioration in the test error rate (since such predictors cause an increase in variance without a corresponding decrease in bias), and so removing such predictors may in turn yield an improvement. Below we refit the logistic regression using just `Lag1` and `Lag2`, which seemed to have the highest predictive power in the original logistic regression model. """ model = MS(['Lag1', 'Lag2']).fit(Smarket) X = model.transform(Smarket) X_train, X_test = X.loc[train], X.loc[~train] glm_train = sm.GLM(y_train, X_train, family=sm.families.Binomial()) results = glm_train.fit() probs = results.predict(exog=X_test) labels = np.array(['Down']*252) labels[probs>0.5] = 'Up' confusion_table(labels, L_test) """Let’s evaluate the overall accuracy as well as the accuracy within the days when logistic regression predicts an increase. """ (35+106)/252,106/(106+76) """Now the results appear to be a little better: 56% of the daily movements have been correctly predicted. It is worth noting that in this case, a much simpler strategy of predicting that the market will increase every day will also be correct 56% of the time! Hence, in terms of overall error rate, the logistic regression method is no better than the naive approach. However, the confusion matrix shows that on days when logistic regression predicts an increase in the market, it has a 58% accuracy rate. This suggests a possible trading strategy of buying on days when the model predicts an increasing market, and avoiding trades on days when a decrease is predicted. Of course one would need to investigate more carefully whether this small improvement was real or just due to random chance. Suppose that we want to predict the returns associated with particular values of `Lag1` and `Lag2`. In particular, we want to predict `Direction` on a day when `Lag1` and `Lag2` equal $1.2$ and $1.1$, respectively, and on a day when they equal $1.5$ and $-0.8$. We do this using the `predict()` function. """ newdata = pd.DataFrame({'Lag1':[1.2, 1.5], 'Lag2':[1.1, -0.8]}); newX = model.transform(newdata) results.predict(newX) Smarket import pandas as pd import numpy as np import matplotlib.pyplot as plt from sklearn.model_selection import train_test_split from sklearn.linear_model import LogisticRegression from sklearn.metrics import classification_report, confusion_matrix import statsmodels.api as sm # Load the dataset data = load_data('Smarket') # Display the first few rows of the dataset print(data.head()) # Prepare the data for logistic regression # Using 'Lag1' and 'Lag2' as predictors and 'Direction' as the response data['Direction'] = data['Direction'].map({'Up': 1, 'Down': 0}) X = data[['Lag1', 'Lag2']] y = data['Direction'] # Split the data into training and testing sets X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42) # Fit the logistic regression model log_reg = LogisticRegression() log_reg.fit(X_train, y_train) # Make predictions on the test set y_pred = log_reg.predict(X_test) # Print classification report and confusion matrix print(classification_report(y_test, y_pred)) print(confusion_matrix(y_test, y_pred)) # Visualize the decision boundary plt.figure(figsize=(10, 6)) # Create a mesh grid for plotting decision boundary x_min, x_max = X['Lag1'].min() - 1, X['Lag1'].max() + 1 y_min, y_max = X['Lag2'].min() - 1, X['Lag2'].max() + 1 xx, yy = np.meshgrid(np.arange(x_min, x_max, 0.01), np.arange(y_min, y_max, 0.01)) # Predict the function value for the whole grid Z = log_reg.predict(np.c_[xx.ravel(), yy.ravel()]) Z = Z.reshape(xx.shape) # Plot the decision boundary plt.contourf(xx, yy, Z, alpha=0.8) plt.scatter(X_test['Lag1'], X_test['Lag2'], c=y_test, edgecolor='k', s=20) plt.xlabel('Lag1') plt.ylabel('Lag2') plt.title('Logistic Regression Decision Boundary') plt.show()