version 0.2, May 2016
This notebook is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. Special thanks goes to Kevin Markham
Why are we learning about ensembling?
Students will be able to:
Ensemble learning is a widely studied topic in the machine learning community. The main idea behind the ensemble methodology is to combine several individual base classifiers in order to have a classifier that outperforms each of them.
Nowadays, ensemble methods are one of the most popular and well studied machine learning techniques, and it can be noted that since 2009 all the first-place and second-place winners of the KDD-Cup https://www.sigkdd.org/kddcup/ used ensemble methods. The core principle in ensemble learning, is to induce random perturbations into the learning procedure in order to produce several different base classifiers from a single training set, then combining the base classifiers in order to make the final prediction. In order to induce the random permutations and therefore create the different base classifiers, several methods have been proposed, in particular:
Finally, after the base classifiers are trained, they are typically combined using either:
There are three main reasons regarding why ensemble methods perform better than single models: statistical, computational and representational . First, from a statistical point of view, when the learning set is too small, an algorithm can find several good models within the search space, that arise to the same performance on the training set $\mathcal{S}$. Nevertheless, without a validation set, there is a risk of choosing the wrong model. The second reason is computational; in general, algorithms rely on some local search optimization and may get stuck in a local optima. Then, an ensemble may solve this by focusing different algorithms to different spaces across the training set. The last reason is representational. In most cases, for a learning set of finite size, the true function $f$ cannot be represented by any of the candidate models. By combining several models in an ensemble, it may be possible to obtain a model with a larger coverage across the space of representable functions.
Let's pretend that instead of building a single model to solve a binary classification problem, you created five independent models, and each model was correct about 70% of the time. If you combined these models into an "ensemble" and used their majority vote as a prediction, how often would the ensemble be correct?
In [1]:
import numpy as np
# set a seed for reproducibility
np.random.seed(1234)
# generate 1000 random numbers (between 0 and 1) for each model, representing 1000 observations
mod1 = np.random.rand(1000)
mod2 = np.random.rand(1000)
mod3 = np.random.rand(1000)
mod4 = np.random.rand(1000)
mod5 = np.random.rand(1000)
# each model independently predicts 1 (the "correct response") if random number was at least 0.3
preds1 = np.where(mod1 > 0.3, 1, 0)
preds2 = np.where(mod2 > 0.3, 1, 0)
preds3 = np.where(mod3 > 0.3, 1, 0)
preds4 = np.where(mod4 > 0.3, 1, 0)
preds5 = np.where(mod5 > 0.3, 1, 0)
# print the first 20 predictions from each model
print(preds1[:20])
print(preds2[:20])
print(preds3[:20])
print(preds4[:20])
print(preds5[:20])
In [2]:
# average the predictions and then round to 0 or 1
ensemble_preds = np.round((preds1 + preds2 + preds3 + preds4 + preds5)/5.0).astype(int)
# print the ensemble's first 20 predictions
print(ensemble_preds[:20])
In [3]:
# how accurate was each individual model?
print(preds1.mean())
print(preds2.mean())
print(preds3.mean())
print(preds4.mean())
print(preds5.mean())
In [4]:
# how accurate was the ensemble?
print(ensemble_preds.mean())
Note: As you add more models to the voting process, the probability of error decreases, which is known as Condorcet's Jury Theorem.
Ensemble learning (or "ensembling") is the process of combining several predictive models in order to produce a combined model that is more accurate than any individual model.
For ensembling to work well, the models must have the following characteristics:
The big idea: If you have a collection of individually imperfect (and independent) models, the "one-off" mistakes made by each model are probably not going to be made by the rest of the models, and thus the mistakes will be discarded when averaging the models.
There are two basic methods for ensembling:
If we assume that each one of the $T$ base classifiers has a probability $\rho$ of being correct, the probability of an ensemble making the correct decision, assuming independence, denoted by $P_c$, can be calculated using the binomial distribution
$$P_c = \sum_{j>T/2}^{T} {{T}\choose{j}} \rho^j(1-\rho)^{T-j}.$$Furthermore, as shown, if $T\ge3$ then:
$$ \lim_{T \to \infty} P_c= \begin{cases} 1 &\mbox{if } \rho>0.5 \\ 0 &\mbox{if } \rho<0.5 \\ 0.5 &\mbox{if } \rho=0.5 , \end{cases} $$leading to the conclusion that
Machine learning flowchart created by the winner of Kaggle's CrowdFlower competition
In [7]:
# read in and prepare the vehicle training data
import zipfile
import pandas as pd
with zipfile.ZipFile('../datasets/vehicles_train.csv.zip', 'r') as z:
f = z.open('vehicles_train.csv')
train = pd.io.parsers.read_table(f, index_col=False, sep=',')
with zipfile.ZipFile('../datasets/vehicles_test.csv.zip', 'r') as z:
f = z.open('vehicles_test.csv')
test = pd.io.parsers.read_table(f, index_col=False, sep=',')
train['vtype'] = train.vtype.map({'car':0, 'truck':1})
# read in and prepare the vehicle testing data
test['vtype'] = test.vtype.map({'car':0, 'truck':1})
In [8]:
train.head()
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In [9]:
from sklearn.linear_model import LinearRegression
from sklearn.tree import DecisionTreeRegressor
from sklearn.naive_bayes import GaussianNB
from sklearn.neighbors import KNeighborsRegressor
models = {'lr': LinearRegression(),
'dt': DecisionTreeRegressor(),
'nb': GaussianNB(),
'nn': KNeighborsRegressor()}
In [10]:
# Train all the models
X_train = train.iloc[:, 1:]
X_test = test.iloc[:, 1:]
y_train = train.price
y_test = test.price
for model in models.keys():
models[model].fit(X_train, y_train)
In [11]:
# predict test for each model
y_pred = pd.DataFrame(index=test.index, columns=models.keys())
for model in models.keys():
y_pred[model] = models[model].predict(X_test)
In [12]:
# Evaluate each model
from sklearn.metrics import mean_squared_error
for model in models.keys():
print(model,np.sqrt(mean_squared_error(y_pred[model], y_test)))
In [13]:
np.sqrt(mean_squared_error(y_pred.mean(axis=1), y_test))
Out[13]:
Advantages of manual ensembling:
Disadvantages of manual ensembling:
The primary weakness of decision trees is that they don't tend to have the best predictive accuracy. This is partially due to high variance, meaning that different splits in the training data can lead to very different trees.
Bagging is a general purpose procedure for reducing the variance of a machine learning method, but is particularly useful for decision trees. Bagging is short for bootstrap aggregation, meaning the aggregation of bootstrap samples.
What is a bootstrap sample? A random sample with replacement:
In [14]:
# set a seed for reproducibility
np.random.seed(1)
# create an array of 1 through 20
nums = np.arange(1, 21)
print(nums)
# sample that array 20 times with replacement
print(np.random.choice(a=nums, size=20, replace=True))
How does bagging work (for decision trees)?
Notes:
Bagging increases predictive accuracy by reducing the variance, similar to how cross-validation reduces the variance associated with train/test split (for estimating out-of-sample error) by splitting many times an averaging the results.
In [15]:
# set a seed for reproducibility
np.random.seed(123)
n_samples = train.shape[0]
n_B = 10
# create ten bootstrap samples (will be used to select rows from the DataFrame)
samples = [np.random.choice(a=n_samples, size=n_samples, replace=True) for _ in range(1, n_B +1 )]
samples
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In [16]:
# show the rows for the first decision tree
train.iloc[samples[0], :]
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Build one tree for each sample
In [17]:
from sklearn.tree import DecisionTreeRegressor
# grow each tree deep
treereg = DecisionTreeRegressor(max_depth=None, random_state=123)
# DataFrame for storing predicted price from each tree
y_pred = pd.DataFrame(index=test.index, columns=[list(range(n_B))])
# grow one tree for each bootstrap sample and make predictions on testing data
for i, sample in enumerate(samples):
X_train = train.iloc[sample, 1:]
y_train = train.iloc[sample, 0]
treereg.fit(X_train, y_train)
y_pred[i] = treereg.predict(X_test)
In [18]:
y_pred
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Results of each tree
In [19]:
for i in range(n_B):
print(i, np.sqrt(mean_squared_error(y_pred[i], y_test)))
Results of the ensemble
In [20]:
y_pred.mean(axis=1)
Out[20]:
In [21]:
np.sqrt(mean_squared_error(y_test, y_pred.mean(axis=1)))
Out[21]:
In [22]:
# define the training and testing sets
X_train = train.iloc[:, 1:]
y_train = train.iloc[:, 0]
X_test = test.iloc[:, 1:]
y_test = test.iloc[:, 0]
In [23]:
# instruct BaggingRegressor to use DecisionTreeRegressor as the "base estimator"
from sklearn.ensemble import BaggingRegressor
bagreg = BaggingRegressor(DecisionTreeRegressor(), n_estimators=500,
bootstrap=True, oob_score=True, random_state=1)
In [24]:
# fit and predict
bagreg.fit(X_train, y_train)
y_pred = bagreg.predict(X_test)
y_pred
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In [25]:
# calculate RMSE
np.sqrt(mean_squared_error(y_test, y_pred))
Out[25]:
In [26]:
# show the first bootstrap sample
samples[0]
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In [27]:
# show the "in-bag" observations for each sample
for sample in samples:
print(set(sample))
In [28]:
# show the "out-of-bag" observations for each sample
for sample in samples:
print(sorted(set(range(n_samples)) - set(sample)))
How to calculate "out-of-bag error":
When B is sufficiently large, the out-of-bag error is an accurate estimate of out-of-sample error.
In [29]:
# compute the out-of-bag R-squared score (not MSE, unfortunately!) for B=500
bagreg.oob_score_
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Bagging increases predictive accuracy, but decreases model interpretability because it's no longer possible to visualize the tree to understand the importance of each feature.
However, we can still obtain an overall summary of feature importance from bagged models:
Random Forests is a slight variation of bagged trees that has even better performance:
What's the point?
In [30]:
# read in the data
with zipfile.ZipFile('../datasets/hitters.csv.zip', 'r') as z:
f = z.open('hitters.csv')
hitters = pd.read_csv(f, sep=',', index_col=False)
# remove rows with missing values
hitters.dropna(inplace=True)
hitters.head()
Out[30]:
In [31]:
# encode categorical variables as integers
hitters['League'] = pd.factorize(hitters.League)[0]
hitters['Division'] = pd.factorize(hitters.Division)[0]
hitters['NewLeague'] = pd.factorize(hitters.NewLeague)[0]
hitters.head()
Out[31]:
In [32]:
# allow plots to appear in the notebook
%matplotlib inline
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')
In [33]:
# scatter plot of Years versus Hits colored by Salary
hitters.plot(kind='scatter', x='Years', y='Hits', c='Salary', colormap='jet', xlim=(0, 25), ylim=(0, 250))
Out[33]:
In [34]:
# define features: exclude career statistics (which start with "C") and the response (Salary)
feature_cols = hitters.columns[hitters.columns.str.startswith('C') == False].drop('Salary')
feature_cols
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In [35]:
# define X and y
X = hitters[feature_cols]
y = hitters.Salary
In [36]:
# list of values to try for max_depth
max_depth_range = range(1, 21)
# list to store the average RMSE for each value of max_depth
RMSE_scores = []
# use 10-fold cross-validation with each value of max_depth
from sklearn.cross_validation import cross_val_score
for depth in max_depth_range:
treereg = DecisionTreeRegressor(max_depth=depth, random_state=1)
MSE_scores = cross_val_score(treereg, X, y, cv=10, scoring='mean_squared_error')
RMSE_scores.append(np.mean(np.sqrt(-MSE_scores)))
In [37]:
# plot max_depth (x-axis) versus RMSE (y-axis)
plt.plot(max_depth_range, RMSE_scores)
plt.xlabel('max_depth')
plt.ylabel('RMSE (lower is better)')
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In [38]:
# show the best RMSE and the corresponding max_depth
sorted(zip(RMSE_scores, max_depth_range))[0]
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In [39]:
# max_depth=2 was best, so fit a tree using that parameter
treereg = DecisionTreeRegressor(max_depth=2, random_state=1)
treereg.fit(X, y)
Out[39]:
In [40]:
# compute feature importances
pd.DataFrame({'feature':feature_cols, 'importance':treereg.feature_importances_}).sort_values('importance')
Out[40]:
In [41]:
from sklearn.ensemble import RandomForestRegressor
rfreg = RandomForestRegressor()
rfreg
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In [42]:
# list of values to try for n_estimators
estimator_range = range(10, 310, 10)
# list to store the average RMSE for each value of n_estimators
RMSE_scores = []
# use 5-fold cross-validation with each value of n_estimators (WARNING: SLOW!)
for estimator in estimator_range:
rfreg = RandomForestRegressor(n_estimators=estimator, random_state=1, n_jobs=-1)
MSE_scores = cross_val_score(rfreg, X, y, cv=5, scoring='mean_squared_error')
RMSE_scores.append(np.mean(np.sqrt(-MSE_scores)))
In [43]:
# plot n_estimators (x-axis) versus RMSE (y-axis)
plt.plot(estimator_range, RMSE_scores)
plt.xlabel('n_estimators')
plt.ylabel('RMSE (lower is better)')
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In [44]:
# list of values to try for max_features
feature_range = range(1, len(feature_cols)+1)
# list to store the average RMSE for each value of max_features
RMSE_scores = []
# use 10-fold cross-validation with each value of max_features (WARNING: SLOW!)
for feature in feature_range:
rfreg = RandomForestRegressor(n_estimators=150, max_features=feature, random_state=1, n_jobs=-1)
MSE_scores = cross_val_score(rfreg, X, y, cv=10, scoring='mean_squared_error')
RMSE_scores.append(np.mean(np.sqrt(-MSE_scores)))
In [45]:
# plot max_features (x-axis) versus RMSE (y-axis)
plt.plot(feature_range, RMSE_scores)
plt.xlabel('max_features')
plt.ylabel('RMSE (lower is better)')
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In [46]:
# show the best RMSE and the corresponding max_features
sorted(zip(RMSE_scores, feature_range))[0]
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In [47]:
# max_features=8 is best and n_estimators=150 is sufficiently large
rfreg = RandomForestRegressor(n_estimators=150, max_features=8, oob_score=True, random_state=1)
rfreg.fit(X, y)
Out[47]:
In [48]:
# compute feature importances
pd.DataFrame({'feature':feature_cols, 'importance':rfreg.feature_importances_}).sort_values('importance')
Out[48]:
In [49]:
# compute the out-of-bag R-squared score
rfreg.oob_score_
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In [50]:
# check the shape of X
X.shape
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In [51]:
# set a threshold for which features to include
print(rfreg.transform(X, threshold=0.1).shape)
print(rfreg.transform(X, threshold='mean').shape)
print(rfreg.transform(X, threshold='median').shape)
In [52]:
# create a new feature matrix that only includes important features
X_important = rfreg.transform(X, threshold='mean')
In [53]:
# check the RMSE for a Random Forest that only includes important features
rfreg = RandomForestRegressor(n_estimators=150, max_features=3, random_state=1)
scores = cross_val_score(rfreg, X_important, y, cv=10, scoring='mean_squared_error')
np.mean(np.sqrt(-scores))
Out[53]:
Advantages of Random Forests:
Disadvantages of Random Forests:
Machine learning flowchart created by the second place finisher of Kaggle's Driver Telematics competition
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