This is a very quick run-through of some basic statistical concepts, adapted from Lab 4 in Harvard's CS109 course. Please feel free to try the original lab if you're feeling ambitious :-) The CS109 git repository also has the solutions if you're stuck.
Linear regression is used to model and predict continuous outcomes while logistic regression is used to model binary outcomes. We'll see some examples of linear regression as well as Train-test splits.
The packages we'll cover are: statsmodels, seaborn, and scikit-learn. While we don't explicitly teach statsmodels and seaborn in the Springboard workshop, those are great libraries to know.
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# special IPython command to prepare the notebook for matplotlib and other libraries
%pylab inline
import numpy as np
import pandas as pd
import scipy.stats as stats
import matplotlib.pyplot as plt
import sklearn
import seaborn as sns
# special matplotlib argument for improved plots
from matplotlib import rcParams
sns.set_style("whitegrid")
sns.set_context("poster")
Given a dataset $X$ and $Y$, linear regression can be used to:
Linear Regression is a method to model the relationship between a set of independent variables $X$ (also knowns as explanatory variables, features, predictors) and a dependent variable $Y$. This method assumes the relationship between each predictor $X$ is linearly related to the dependent variable $Y$.
$$ Y = \beta_0 + \beta_1 X + \epsilon$$where $\epsilon$ is considered as an unobservable random variable that adds noise to the linear relationship. This is the simplest form of linear regression (one variable), we'll call this the simple model.
$\beta_0$ is the intercept of the linear model
Multiple linear regression is when you have more than one independent variable
Least squares is a method that can estimate the coefficients of a linear model by minimizing the difference between the following:
$$ S = \sum_{i=1}^N r_i = \sum_{i=1}^N (y_i - (\beta_0 + \beta_1 x_i))^2 $$where $N$ is the number of observations.
The solution can be written in compact matrix notation as
$$\hat\beta = (X^T X)^{-1}X^T Y$$We wanted to show you this in case you remember linear algebra, in order for this solution to exist we need $X^T X$ to be invertible. Of course this requires a few extra assumptions, $X$ must be full rank so that $X^T X$ is invertible, etc. This is important for us because this means that having redundant features in our regression models will lead to poorly fitting (and unstable) models. We'll see an implementation of this in the extra linear regression example.
Note: The "hat" means it is an estimate of the coefficient.
The Boston Housing data set contains information about the housing values in suburbs of Boston. This dataset was originally taken from the StatLib library which is maintained at Carnegie Mellon University and is now available on the UCI Machine Learning Repository.
sklearnThis data set is available in the sklearn python module which is how we will access it today.
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from sklearn.datasets import load_boston
boston = load_boston()
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boston.keys()
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boston.data.shape
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# Print column names
print (boston.feature_names)
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# Print description of Boston housing data set
print (boston.DESCR)
Now let's explore the data set itself.
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bos = pd.DataFrame(boston.data)
bos.head()
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There are no column names in the DataFrame. Let's add those.
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bos.columns = boston.feature_names
bos.head()
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Now we have a pandas DataFrame called bos containing all the data we want to use to predict Boston Housing prices. Let's create a variable called PRICE which will contain the prices. This information is contained in the target data.
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print (boston.target.shape)
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bos['PRICE'] = boston.target
bos.head()
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bos.describe()
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plt.scatter(bos.CRIM, bos.PRICE)
plt.xlabel("Per capita crime rate by town (CRIM)")
plt.ylabel("Housing Price")
plt.title("Relationship between CRIM and Price")
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Your turn: Create scatter plots between RM and PRICE, and PTRATIO and PRICE. What do you notice?
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#your turn: scatter plot between *RM* and *PRICE*
plt.scatter(bos.RM, bos.PRICE)
plt.xlabel("average number of rooms per dwelling (RM)")
plt.ylabel("Housing Price")
plt.title("Relationship between RM and Price")
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#your turn: scatter plot between *PTRATIO* and *PRICE*
plt.scatter(bos.PTRATIO, bos.PRICE)
plt.xlabel("pupil-teacher ratio by town (PTRATIO)")
plt.ylabel("Housing Price")
plt.title("Relationship between PTRATIO and Price")
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Your turn: What are some other numeric variables of interest? Plot scatter plots with these variables and PRICE.
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#your turn: create some other scatter plots
plt.scatter(bos.AGE, bos.PRICE)
plt.xlabel("proportion of owner-occupied units built prior to 1940 (AGE)")
plt.ylabel("Housing Price")
plt.title("Relationship between House Ages and Price")
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Seaborn is a cool Python plotting library built on top of matplotlib. It provides convenient syntax and shortcuts for many common types of plots, along with better-looking defaults.
We can also use seaborn regplot for the scatterplot above. This provides automatic linear regression fits (useful for data exploration later on). Here's one example below.
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sns.regplot(y="PRICE", x="RM", data=bos, fit_reg = True)
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Histograms are a useful way to visually summarize the statistical properties of numeric variables. They can give you an idea of the mean and the spread of the variables as well as outliers.
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plt.hist(bos.CRIM)
plt.title("CRIM")
plt.xlabel("Crime rate per capita")
plt.ylabel("Frequency")
plt.show()
Your turn: Plot separate histograms and one for RM, one for PTRATIO. Any interesting observations?
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#your turn
plt.hist(bos.RM)
plt.title("RM")
plt.xlabel("average number of rooms per dwelling")
plt.ylabel("Frequency")
plt.show()
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# Histogram for pupil-teacher ratio by town
plt.hist(bos.PTRATIO)
plt.title("PTRATIO")
plt.xlabel("pupil-teacher ratio by town")
plt.ylabel("Frequency")
plt.show()
Here,
$Y$ = boston housing prices (also called "target" data in python)
and
$X$ = all the other features (or independent variables)
which we will use to fit a linear regression model and predict Boston housing prices. We will use the least squares method as the way to estimate the coefficients.
We'll use two ways of fitting a linear regression. We recommend the first but the second is also powerful in its features.
statsmodelsStatsmodels is a great Python library for a lot of basic and inferential statistics. It also provides basic regression functions using an R-like syntax, so it's commonly used by statisticians. While we don't cover statsmodels officially in the Data Science Intensive, it's a good library to have in your toolbox. Here's a quick example of what you could do with it.
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# Import regression modules
# ols - stands for Ordinary least squares, we'll use this
import statsmodels.api as sm
from statsmodels.formula.api import ols
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# statsmodels works nicely with pandas dataframes
# The thing inside the "quotes" is called a formula, a bit on that below
m = ols('PRICE ~ RM',bos).fit()
print (m.summary())
There is a ton of information in this output. But we'll concentrate on the coefficient table (middle table). We can interpret the RM coefficient (9.1021) by first noticing that the p-value (under P>|t|) is so small, basically zero. We can interpret the coefficient as, if we compare two groups of towns, one where the average number of rooms is say $5$ and the other group is the same except that they all have $6$ rooms. For these two groups the average difference in house prices is about $9.1$ (in thousands) so about $\$9,100$ difference. The confidence interval fives us a range of plausible values for this difference, about ($\$8,279, \$9,925$), deffinitely not chump change.
statsmodels formulasThis formula notation will seem familiar to R users, but will take some getting used to for people coming from other languages or are new to statistics.
The formula gives instruction for a general structure for a regression call. For statsmodels (ols or logit) calls you need to have a Pandas dataframe with column names that you will add to your formula. In the below example you need a pandas data frame that includes the columns named (Outcome, X1,X2, ...), bbut you don't need to build a new dataframe for every regression. Use the same dataframe with all these things in it. The structure is very simple:
Outcome ~ X1
But of course we want to to be able to handle more complex models, for example multiple regression is doone like this:
Outcome ~ X1 + X2 + X3
This is the very basic structure but it should be enough to get you through the homework. Things can get much more complex, for a quick run-down of further uses see the statsmodels help page.
Let's see how our model actually fit our data. We can see below that there is a ceiling effect, we should probably look into that. Also, for large values of $Y$ we get underpredictions, most predictions are below the 45-degree gridlines.
Your turn: Create a scatterpot between the predicted prices, available in m.fittedvalues and the original prices. How does the plot look?
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# your turn
plt.scatter(bos.PRICE, m.fittedvalues)
plt.xlabel("Housing Price")
plt.ylabel("Predicted Housing Price")
plt.title("Relationship between Predicted and Actual Price")
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from sklearn.linear_model import LinearRegression
X = bos.drop('PRICE', axis = 1)
# This creates a LinearRegression object
lm = LinearRegression()
lm
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Check out the scikit-learn docs here. We have listed the main functions here.
| Main functions | Description |
|---|---|
lm.fit() |
Fit a linear model |
lm.predit() |
Predict Y using the linear model with estimated coefficients |
lm.score() |
Returns the coefficient of determination (R^2). A measure of how well observed outcomes are replicated by the model, as the proportion of total variation of outcomes explained by the model |
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# Look inside lm object
#lm.<tab>
| Output | Description |
|---|---|
lm.coef_ |
Estimated coefficients |
lm.intercept_ |
Estimated intercept |
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# Use all 13 predictors to fit linear regression model
lm.fit(X, bos.PRICE)
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Your turn: How would you change the model to not fit an intercept term? Would you recommend not having an intercept?
Let's look at the estimated coefficients from the linear model using 1m.intercept_ and lm.coef_.
After we have fit our linear regression model using the least squares method, we want to see what are the estimates of our coefficients $\beta_0$, $\beta_1$, ..., $\beta_{13}$:
$$ \hat{\beta}_0, \hat{\beta}_1, \ldots, \hat{\beta}_{13} $$
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print ('Estimated intercept coefficient:', lm.intercept_)
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print ('Number of coefficients:', len(lm.coef_))
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# The coefficients
pd.DataFrame(list(zip(X.columns, lm.coef_)), columns = ['features', 'estimatedCoefficients'])
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# first five predicted prices
lm.predict(X)[0:5]
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Your turn:
statsmodels before).
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# your turn
# Plot a histogram of all the predicted prices
plt.hist(lm.predict(X))
plt.title("Predicted Prices")
plt.xlabel("Predicted Prices")
plt.ylabel("Frequency")
plt.show()
# Let's plot the true prices compared to the predicted prices to see they disagree
plt.scatter(bos.PRICE, lm.predict(X))
plt.xlabel("Housing Price")
plt.ylabel("Predicted Housing Price")
plt.title("Relationship between Predicted and Actual Price")
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print (np.sum((bos.PRICE - lm.predict(X)) ** 2))
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#your turn
print ('Mean squared error: ', np.mean((bos.PRICE - lm.predict(X)) ** 2))
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lm = LinearRegression()
lm.fit(X[['PTRATIO']], bos.PRICE)
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msePTRATIO = np.mean((bos.PRICE - lm.predict(X[['PTRATIO']])) ** 2)
print (msePTRATIO)
We can also plot the fitted linear regression line.
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plt.scatter(bos.PTRATIO, bos.PRICE)
plt.xlabel("Pupil-to-Teacher Ratio (PTRATIO)")
plt.ylabel("Housing Price")
plt.title("Relationship between PTRATIO and Price")
plt.plot(bos.PTRATIO, lm.predict(X[['PTRATIO']]), color='blue', linewidth=3)
plt.show()
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# your turn
lm.fit(X[['CRIM']], bos.PRICE)
print ('(MSE) Per capita crime rate by town: ', np.mean((bos.PRICE - lm.predict(X[['CRIM']])) ** 2))
lm.fit(X[['RM']], bos.PRICE)
print ('(MSE) Average number of rooms per dwelling: ', np.mean((bos.PRICE - lm.predict(X[['RM']])) ** 2))
lm.fit(X[['PTRATIO']], bos.PRICE)
print ('(MSE) Pupil-teacher ratio by town: ', np.mean((bos.PRICE - lm.predict(X[['PTRATIO']])) ** 2))
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sns.set(font_scale=.8)
sns.heatmap(X.corr(), vmax=.8, square=True, annot=True)
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Let's stick to the linear regression example:
One way of doing this is you can create training and testing data sets manually.
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X_train = X[:-50]
X_test = X[-50:]
Y_train = bos.PRICE[:-50]
Y_test = bos.PRICE[-50:]
print (X_train.shape)
print (X_test.shape)
print (Y_train.shape)
print (Y_test.shape)
Another way, is to split the data into random train and test subsets using the function train_test_split in sklearn.cross_validation. Here's the documentation.
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X_train, X_test, Y_train, Y_test = sklearn.cross_validation.train_test_split(
X, bos.PRICE, test_size=0.33, random_state = 5)
print (X_train.shape)
print (X_test.shape)
print (Y_train.shape)
print (Y_test.shape)
Your turn: Let's build a linear regression model using our new training data sets.
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# your turn
# Fit a linear regression model to the training set
lm.fit(X_train, Y_train)
lm.predict(X_test)
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Your turn:
Calculate the mean squared error
Are they pretty similar or very different? What does that mean?
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# your turn
# Calculate MSE using just the test data
print ('(MSE) using just the test data: ', np.mean((Y_test - lm.predict(X_test)) ** 2))
# Calculate MSE using just the training data
print ('(MSE) using just the training data: ', np.mean((Y_train - lm.predict(X_train)) ** 2))
Are they pretty similar or very different? What does that mean? -> They are very different because the model us based on training data so it will be accurate compared to the test data. The model is not exposed to test data so it will give a greater mean square error. It means there are data in test data which are different with the training data.
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plt.scatter(lm.predict(X_train), lm.predict(X_train) - Y_train, c='b', s=40, alpha=0.5)
plt.scatter(lm.predict(X_test), lm.predict(X_test) - Y_test, c='g', s=40)
plt.hlines(y = 0, xmin=0, xmax = 50)
plt.title('Residual Plot using training (blue) and test (green) data')
plt.ylabel('Residuals')
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Your turn: Do you think this linear regression model generalizes well on the test data? -> No, the scatter points are not close to zero so the model needs improvements. Check the features to see highly correlated predictors and remove one of them or check the parameters of the model and do fine-tuning.
A simple extension of the Test/train split is called K-fold cross-validation.
Here's the procedure:
Luckily you don't have to do this entire process all by hand (for loops, etc.) every single time, sci-kit learn has a very nice implementation of this, have a look at the documentation.
Your turn (extra credit): Implement K-Fold cross-validation using the procedure above and Boston Housing data set using $K=4$. How does the average prediction error compare to the train-test split above?
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from sklearn import cross_validation, linear_model
# If the estimator is a classifier and y is either binary or multiclass, StratifiedKFold is used.
# In all other cases, KFold is used
scores = cross_validation.cross_val_score(lm, X, bos.PRICE, scoring='mean_squared_error', cv=4)
# This will print metric for evaluation
print ('(MSE) Using k-fold: ', np.mean(scores))
print ('The K-fold cross-validation is not performaing well compared to the previous train-test split above')