by Alejandro Correa Bahnsen & Iván Torroledo
version 1.2, Feb 2018
This notebook is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. Special thanks goes to Kevin Markham
Unbalanced classification problems cause problems to many learning algorithms. These problems are characterized by the uneven proportion of cases that are available for each class of the problem.
Most classification algorithms will only perform optimally when the number of samples of each class is roughly the same. Highly skewed datasets, where the minority heavily outnumbered by one or more classes, haven proven to be a challenge while at the same time becoming more and more common.
One way of addresing this issue is by resampling the dataset as to offset this imbalance with the hope of arriving and a more robust and fair decision boundary than you would otherwise.
Resampling techniques are divided in two categories:
In [1]:
%matplotlib inline
import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import make_classification
from sklearn.decomposition import PCA
plt.style.use('ggplot')
In [2]:
# Generate some data
X, y = make_classification(n_classes=2, class_sep=2, weights=[0.9, 0.1],
n_informative=3, n_redundant=1, flip_y=0,
n_features=20, n_clusters_per_class=1,
n_samples=1000, random_state=10)
# Instanciate a PCA object for the sake of easy visualisation
pca = PCA(n_components = 2)
# Fit and transform x to visualise inside a 2D feature space
x_vis = pca.fit_transform(X)
# Plot the original data
def plot_two_classes(X, y, subplot=False, size=(10, 10)):
# Plot the two classes
if subplot == False:
fig, subplot = plt.subplots(nrows=1, ncols=1, figsize=size)
subplot.scatter(X[y==0, 0], X[y==0, 1], label="Class #0",
alpha=0.5, s=70)
subplot.scatter(X[y==1, 0], X[y==1, 1], label="Class #1",
alpha=0.5, s=70)
subplot.legend()
return subplot
plot_two_classes(x_vis, y)
Out[2]:
In [3]:
y.mean()
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In [4]:
n_samples = y.shape[0]
print(n_samples)
In [5]:
n_samples_0 = (y == 0).sum()
n_samples_0
Out[5]:
In [6]:
n_samples_1 = (y == 1).sum()
n_samples_1
Out[6]:
In [7]:
n_samples_1 / n_samples
Out[7]:
How many negatives cases should I select if I want a new dataset with 50% of positives?
0.5 = n_samples_1 / (n_samples_1 + n_samples_0_new)
(n_samples_1 + n_samples_0_new) = n_samples_1 / 0.5
In [8]:
n_samples_0_new = n_samples_1 / 0.5 - n_samples_1
n_samples_0_new
Out[8]:
In [9]:
n_samples_0_new_per = n_samples_0_new / n_samples_0
n_samples_0_new_per
Out[9]:
Create a filter to select n_samples_0_new_per from the negative class
In [10]:
# Select all negatives
filter_ = y == 0
# Random sample
np.random.seed(42)
rand_1 = np.random.binomial(n=1, p=n_samples_0_new_per, size=n_samples)
# Combine
filter_ = filter_ & rand_1
In [11]:
filter_.sum()
Out[11]:
Also select all the positives
In [12]:
filter_ = filter_ | (y == 1)
In [13]:
filter_ = filter_.astype(bool)
plot
In [14]:
plot_two_classes(x_vis[filter_], y[filter_])
Out[14]:
Convert into a function
In [15]:
def UnderSampling(X, y, target_percentage=0.5, seed=None):
# Assuming minority class is the positive
n_samples = y.shape[0]
n_samples_0 = (y == 0).sum()
n_samples_1 = (y == 1).sum()
n_samples_0_new = n_samples_1 / target_percentage - n_samples_1
n_samples_0_new_per = n_samples_0_new / n_samples_0
filter_ = y == 0
np.random.seed(seed)
rand_1 = np.random.binomial(n=1, p=n_samples_0_new_per, size=n_samples)
filter_ = filter_ & rand_1
filter_ = filter_ | (y == 1)
filter_ = filter_.astype(bool)
return X[filter_], y[filter_]
In [16]:
for target_percentage in [0.1, 0.2, 0.3, 0.4, 0.5]:
X_u, y_u = UnderSampling(x_vis, y, target_percentage, 1)
print('Target percentage', target_percentage)
print('y.shape = ',y_u.shape[0], 'y.mean() = ', y_u.mean())
plot_two_classes(X_u, y_u, size=(5, 5))
plt.show()
Find the nearest neighbour of every point
In [17]:
from sklearn.neighbors import NearestNeighbors
nn = NearestNeighbors(n_neighbors=2)
nn.fit(x_vis)
nns = nn.kneighbors(x_vis, return_distance=False)[:, 1]
In [18]:
nns[0:10]
Out[18]:
Find if tomek. Use the target vector and the first neighbour of every sample point and looks for Tomek pairs. Returning a boolean vector with True for majority Tomek links.
In [19]:
# Initialize the boolean result as false, and also a counter
links = np.zeros(len(y), dtype=bool)
# Loop through each sample of the majority class then we
# look at its first neighbour. If its closest neighbour also has the
# current sample as its closest neighbour, the two form a Tomek link.
for ind, ele in enumerate(y):
if ele == 1 | links[ind] == True: # Keep all from the minority class
continue
if y[nns[ind]] == 1:
# If they form a tomek link, put a True marker on this
# sample, and increase counter by one.
if nns[nns[ind]] == ind:
links[ind] = True
In [20]:
links.sum()
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In [21]:
filter_ = np.logical_not(links)
In [22]:
print('y.shape = ',y[filter_].shape[0], 'y.mean() = ', y[filter_].mean())
plot_two_classes(x_vis[filter_], y[filter_])
Out[22]:
The goal is to eliminate examples from the majority class that are much further away from the border
In [23]:
# Import the K-NN classifier
from sklearn.neighbors import KNeighborsClassifier
In [24]:
# Number of samples to extract in order to build the set S.
n_seeds_S = 51
# Size of the neighbourhood to consider to compute the
# average distance to the minority point samples.
size_ngh = 100
# Randomly get one sample from the majority class
np.random.seed(42)
maj_sample = np.random.choice(x_vis[y == 0].shape[0], n_seeds_S)
maj_sample = x_vis[y == 0][maj_sample]
In [25]:
# Create the set C
# Select all positive and the randomly selected negatives
C_x = np.append(x_vis[y == 1], maj_sample, axis=0)
C_y = np.append(y[y == 1], [0] * n_seeds_S)
In [26]:
# Create the set S
S_x = x_vis[y == 0]
S_y = y[y == 0]
In [27]:
knn = KNeighborsClassifier(n_neighbors=size_ngh)
# Fit C into the knn
knn.fit(C_x, C_y)
# Classify on S
pred_S_y = knn.predict(S_x)
In [28]:
# Find the misclassified S_y
idx_tmp = np.nonzero(y == 0)[0][np.nonzero(pred_S_y != S_y)]
In [29]:
filter_ = np.nonzero(y == 1)[0]
filter_ = np.concatenate((filter_, idx_tmp), axis=0)
In [30]:
print('y.shape = ',y[filter_].shape[0], 'y.mean() = ', y[filter_].mean())
plot_two_classes(x_vis[filter_], y[filter_])
Out[30]:
In [31]:
# Import the K-NN classifier
from sklearn.neighbors import KNeighborsClassifier
def CondensedNearestNeighbor(X, y, n_seeds_S=1, size_ngh=1, seed=None):
# Randomly get one sample from the majority class
np.random.seed(seed)
maj_sample = np.random.choice(X[y == 0].shape[0], n_seeds_S)
maj_sample = X[y == 0][maj_sample]
# Create the set C
# Select all positive and the randomly selected negatives
C_x = np.append(X[y == 1], maj_sample, axis=0)
C_y = np.append(y[y == 1], [0] * n_seeds_S)
# Create the set S
S_x = X[y == 0]
S_y = y[y == 0]
knn = KNeighborsClassifier(n_neighbors=size_ngh)
# Fit C into the knn
knn.fit(C_x, C_y)
# Classify on S
pred_S_y = knn.predict(S_x)
# Find the misclassified S_y
idx_tmp = np.nonzero(y == 0)[0][np.nonzero(pred_S_y != S_y)]
filter_ = np.nonzero(y == 1)[0]
filter_ = np.concatenate((filter_, idx_tmp), axis=0)
return X[filter_], y[filter_]
In [32]:
for n_seeds_S, size_ngh in [(1, 1), (100, 100), (50, 50), (100, 50), (50, 100)]:
X_u, y_u = CondensedNearestNeighbor(x_vis, y, n_seeds_S, size_ngh, 1)
print('n_seeds_S ', n_seeds_S, 'size_ngh ', size_ngh)
print('y.shape = ',y_u.shape[0], 'y.mean() = ', y_u.mean())
plot_two_classes(X_u, y_u, size=(5, 5))
plt.show()
In [33]:
import random
def OverSampling(X, y, target_percentage=0.5, seed=None):
# Assuming minority class is the positive
n_samples = y.shape[0]
n_samples_0 = (y == 0).sum()
n_samples_1 = (y == 1).sum()
n_samples_1_new = -target_percentage * n_samples_0 / (target_percentage- 1)
np.random.seed(seed)
filter_ = np.random.choice(X[y == 1].shape[0], int(n_samples_1_new))
# filter_ is within the positives, change to be of all
filter_ = np.nonzero(y == 1)[0][filter_]
filter_ = np.concatenate((filter_, np.nonzero(y == 0)[0]), axis=0)
return X[filter_], y[filter_]
In [34]:
for target_percentage in [0.1, 0.2, 0.3, 0.4, 0.5]:
X_u, y_u = OverSampling(x_vis, y, target_percentage, 1)
print('Target percentage', target_percentage)
print('y.shape = ',y_u.shape[0], 'y.mean() = ', y_u.mean())
plot_two_classes(X_u, y_u, size=(5, 5))
plt.show()
Advantages of Over-Sampling :
Disadvantages of Over-Sampling :
SMOTE (Chawla et. al. 2002) is a well-known algorithm to fight this problem. The general idea of this method is to artificially generate new examples of the minority class using the nearest neighbors of these cases. Furthermore, the majority class examples are also under-sampled, leading to a more balanced dataset.
In [35]:
# Number of nearest neighbours to used to construct synthetic samples.
k = 5
from sklearn.neighbors import NearestNeighbors
nearest_neighbour_ = NearestNeighbors(n_neighbors=k + 1)
In [36]:
# Look for k-th nearest neighbours, excluding, of course, the
# point itself.#
nearest_neighbour_.fit(x_vis[y==1])
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In [37]:
# Matrix with k-th nearest neighbours indexes for each minority
# element.#
nns = nearest_neighbour_.kneighbors(x_vis[y==1],
return_distance=False)[:, 1:]
Example of the nns of two cases
In [38]:
def base_smote_plot(sel, nns):
fig, subplot = plt.subplots(nrows=1, ncols=1, figsize=(10, 10))
for i in range(len(sel)):
# plot the select sample
subplot.scatter(x_vis[y==1, 0][sel[i]], x_vis[y==1, 1][sel[i]],
alpha=1., s=70)
# plot the neighbors
subplot.scatter(x_vis[y==1, 0][nns[sel[i]]],
x_vis[y==1, 1][nns[sel[i]]],
alpha=0.5, s=70)
# plot the lines
for nn in nns[sel[i]]:
plt.plot([x_vis[y==1, 0][sel[i]], x_vis[y==1, 0][nn]],
[x_vis[y==1, 1][sel[i]], x_vis[y==1, 1][nn]],
'r--')
xlim = subplot.get_xlim()
ylim = subplot.get_ylim()
subplot.scatter(x_vis[y==1, 0], x_vis[y==1, 1], alpha=0.1, s=70)
subplot.set_xlim(xlim)
subplot.set_ylim(ylim)
return subplot
base_smote_plot([12],nns)
Out[38]:
In [39]:
# Create syntetic sample for 12
# Select one random neighbor
np.random.seed(3)
nn_ = np.random.choice(nns[12])
x_vis[y==1][nn_]
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In [40]:
# Take a step of random size (0,1) in the direction of the
# n nearest neighbours
np.random.seed(5)
step = np.random.uniform()
# Construct synthetic sample
new = x_vis[y==1][12] - step * (x_vis[y==1][12] - x_vis[y==1][nn_])
new
Out[40]:
In [41]:
plot_ = base_smote_plot([12],nns)
plot_.scatter(new[0], new[1], alpha=1., s=400, marker="*")
Out[41]:
In [42]:
# Select one random neighbor
np.random.seed(5)
nn_2 = np.random.choice(nns[12])
np.random.seed(1)
step = np.random.uniform()
# Construct synthetic sample
new2 = x_vis[y==1][12] - step * (x_vis[y==1][12] - x_vis[y==1][nn_2])
plot_ = base_smote_plot([12],nns)
plot_.scatter(new[0], new[1], alpha=1., s=400, marker="*")
plot_.scatter(new2[0], new2[1], alpha=1., s=400, marker="*")
Out[42]:
Apply to all cases
Estimate number of synthetic cases to create
In [43]:
target_percentage = 0.5
n_samples_1_new = -target_percentage * n_samples_0 / (target_percentage- 1) - n_samples_1
n_samples_1_new = int(n_samples_1_new)
n_samples_1_new
Out[43]:
In [44]:
# A matrix to store the synthetic samples
new = np.zeros((n_samples_1_new, x_vis.shape[1]))
In [45]:
# Select examples to use as base
np.random.seed(34)
sel_ = np.random.choice(y[y==1].shape[0], n_samples_1_new)
In [46]:
# Define random seeds (2 per example)
np.random.seed(64)
nn__ = np.random.choice(k, n_samples_1_new)
np.random.seed(65)
steps = np.random.uniform(size=n_samples_1_new)
In [47]:
# For each selected examples create one synthetic case
for i, sel in enumerate(sel_):
# Select neighbor
nn_ = nn__[i]
step = steps[i]
new[i, :] = x_vis[y==1][sel] - step * (x_vis[y==1][sel] - x_vis[y==1][nn_])
In [48]:
new
Out[48]:
In [49]:
plot_ = plot_two_classes(x_vis, y, size=(10, 10))
plot_.scatter(new[:, 0], new[:, 1], alpha=0.5, s=150, marker='*')
Out[49]:
Create function
In [50]:
def SMOTE(X, y, target_percentage=0.5, k=5, seed=None):
# New samples
n_samples_1_new = int(-target_percentage * n_samples_0 / (target_percentage- 1) - n_samples_1)
# A matrix to store the synthetic samples
new = np.zeros((n_samples_1_new, x_vis.shape[1]))
# Create seeds
np.random.seed(seed)
seeds = np.random.randint(1, 1000000, 3)
# Select examples to use as base
np.random.seed(seeds[0])
sel_ = np.random.choice(y[y==1].shape[0], n_samples_1_new)
# Define random seeds (2 per example)
np.random.seed(seeds[1])
nn__ = np.random.choice(k, n_samples_1_new)
np.random.seed(seeds[2])
steps = np.random.uniform(size=n_samples_1_new)
# For each selected examples create one synthetic case
for i, sel in enumerate(sel_):
# Select neighbor
nn_ = nn__[i]
step = steps[i]
# Create new sample
new[i, :] = X[y==1][sel] - step * (X[y==1][sel] - X[y==1][nn_])
X = np.vstack((X, new))
y = np.append(y, np.ones(n_samples_1_new))
return X, y
In [51]:
for target_percentage in [0.25, 0.5]:
for k in [5, 15]:
X_u, y_u = SMOTE(x_vis, y, target_percentage, k, seed=3)
print('Target percentage', target_percentage, 'k ', k)
print('y.shape = ',y_u.shape[0], 'y.mean() = ', y_u.mean())
plot_two_classes(X_u, y_u, size=(5, 5))
plt.show()