Artificial Intelligence

L2 International, Univ. Bordeaux

Lab #3, Supervised Learning (1)

Supervised learning is where you have input variables (x) and an output variable (Y) and you use an algorithm to learn the mapping function $f$ from the input to the output.

$Y = f(X)$

The goal is to approximate the mapping function so well that when you have new input data (x) that you can predict the output variables (Y) for that data.

It is called supervised learning because the process of an algorithm learning from the training dataset can be thought of as a teacher supervising the learning process. We know the correct answers, the algorithm iteratively makes predictions on the training data and is corrected by the teacher. Learning stops when the algorithm achieves an acceptable level of performance.

$k$-Nearest Neighbors ($k$-NN)

The $k$-Nearest Neighbors algorithm ($k$-NN for short) is a very simple technique. The entire training dataset is stored. When a prediction is required, the $k$-most similar records to a new record from the training dataset are then located. From these neighbors, a summarized prediction is made. Once the neighbors are discovered, the summary prediction can be made by returning the most common outcome (for classification problems) or taking the average (for regression problems).

As usual, we need to import necessary python modules:

In [3]:
import numpy as np
import pandas as pa

Two additional modules for data visualisation:

In [4]:
from matplotlib import pyplot as plt
import seaborn as sns
sns.set()

We will illustrate our first supervised learning algorithm using it for breast cancer prediction. For this purpose, we will load a related datset which is included in the standard datasets of sklearn. Later, we will discuss how we can work with our own dataset.

In [5]:
from sklearn.datasets import load_breast_cancer

sklearn datasets are well formated. They come with a set of functions that can be used to get information about them or to visualise them:

In [6]:
breast_cancer = load_breast_cancer()
print(breast_cancer.DESCR)
.. _breast_cancer_dataset:

Breast cancer wisconsin (diagnostic) dataset
--------------------------------------------

**Data Set Characteristics:**

    :Number of Instances: 569

    :Number of Attributes: 30 numeric, predictive attributes and the class

    :Attribute Information:
        - radius (mean of distances from center to points on the perimeter)
        - texture (standard deviation of gray-scale values)
        - perimeter
        - area
        - smoothness (local variation in radius lengths)
        - compactness (perimeter^2 / area - 1.0)
        - concavity (severity of concave portions of the contour)
        - concave points (number of concave portions of the contour)
        - symmetry 
        - fractal dimension ("coastline approximation" - 1)

        The mean, standard error, and "worst" or largest (mean of the three
        largest values) of these features were computed for each image,
        resulting in 30 features.  For instance, field 3 is Mean Radius, field
        13 is Radius SE, field 23 is Worst Radius.

        - class:
                - WDBC-Malignant
                - WDBC-Benign

    :Summary Statistics:

    ===================================== ====== ======
                                           Min    Max
    ===================================== ====== ======
    radius (mean):                        6.981  28.11
    texture (mean):                       9.71   39.28
    perimeter (mean):                     43.79  188.5
    area (mean):                          143.5  2501.0
    smoothness (mean):                    0.053  0.163
    compactness (mean):                   0.019  0.345
    concavity (mean):                     0.0    0.427
    concave points (mean):                0.0    0.201
    symmetry (mean):                      0.106  0.304
    fractal dimension (mean):             0.05   0.097
    radius (standard error):              0.112  2.873
    texture (standard error):             0.36   4.885
    perimeter (standard error):           0.757  21.98
    area (standard error):                6.802  542.2
    smoothness (standard error):          0.002  0.031
    compactness (standard error):         0.002  0.135
    concavity (standard error):           0.0    0.396
    concave points (standard error):      0.0    0.053
    symmetry (standard error):            0.008  0.079
    fractal dimension (standard error):   0.001  0.03
    radius (worst):                       7.93   36.04
    texture (worst):                      12.02  49.54
    perimeter (worst):                    50.41  251.2
    area (worst):                         185.2  4254.0
    smoothness (worst):                   0.071  0.223
    compactness (worst):                  0.027  1.058
    concavity (worst):                    0.0    1.252
    concave points (worst):               0.0    0.291
    symmetry (worst):                     0.156  0.664
    fractal dimension (worst):            0.055  0.208
    ===================================== ====== ======

    :Missing Attribute Values: None

    :Class Distribution: 212 - Malignant, 357 - Benign

    :Creator:  Dr. William H. Wolberg, W. Nick Street, Olvi L. Mangasarian

    :Donor: Nick Street

    :Date: November, 1995

This is a copy of UCI ML Breast Cancer Wisconsin (Diagnostic) datasets.
https://goo.gl/U2Uwz2

Features are computed from a digitized image of a fine needle
aspirate (FNA) of a breast mass.  They describe
characteristics of the cell nuclei present in the image.

Separating plane described above was obtained using
Multisurface Method-Tree (MSM-T) [K. P. Bennett, "Decision Tree
Construction Via Linear Programming." Proceedings of the 4th
Midwest Artificial Intelligence and Cognitive Science Society,
pp. 97-101, 1992], a classification method which uses linear
programming to construct a decision tree.  Relevant features
were selected using an exhaustive search in the space of 1-4
features and 1-3 separating planes.

The actual linear program used to obtain the separating plane
in the 3-dimensional space is that described in:
[K. P. Bennett and O. L. Mangasarian: "Robust Linear
Programming Discrimination of Two Linearly Inseparable Sets",
Optimization Methods and Software 1, 1992, 23-34].

This database is also available through the UW CS ftp server:

ftp ftp.cs.wisc.edu
cd math-prog/cpo-dataset/machine-learn/WDBC/

.. topic:: References

   - W.N. Street, W.H. Wolberg and O.L. Mangasarian. Nuclear feature extraction 
     for breast tumor diagnosis. IS&T/SPIE 1993 International Symposium on 
     Electronic Imaging: Science and Technology, volume 1905, pages 861-870,
     San Jose, CA, 1993.
   - O.L. Mangasarian, W.N. Street and W.H. Wolberg. Breast cancer diagnosis and 
     prognosis via linear programming. Operations Research, 43(4), pages 570-577, 
     July-August 1995.
   - W.H. Wolberg, W.N. Street, and O.L. Mangasarian. Machine learning techniques
     to diagnose breast cancer from fine-needle aspirates. Cancer Letters 77 (1994) 
     163-171.
In [7]:
print(breast_cancer.feature_names)
['mean radius' 'mean texture' 'mean perimeter' 'mean area'
 'mean smoothness' 'mean compactness' 'mean concavity'
 'mean concave points' 'mean symmetry' 'mean fractal dimension'
 'radius error' 'texture error' 'perimeter error' 'area error'
 'smoothness error' 'compactness error' 'concavity error'
 'concave points error' 'symmetry error' 'fractal dimension error'
 'worst radius' 'worst texture' 'worst perimeter' 'worst area'
 'worst smoothness' 'worst compactness' 'worst concavity'
 'worst concave points' 'worst symmetry' 'worst fractal dimension']

In the sequel, we will not consider all the dataset. We will explain concepts and $k$-NN algorithm using only two columns (together with predicted class):

Note that function Categorical encodes the two classes as 0 and 1.

In [8]:
X = pa.DataFrame(breast_cancer.data, columns=breast_cancer.feature_names)
X = X[['mean area', 'mean compactness']]
y = pa.Categorical.from_codes(breast_cancer.target, breast_cancer.target_names)
y = pa.get_dummies(y, drop_first=True)

Now, we will split the dataset into two subsets: one for the training and the other for the test. For this, we will import the necessary function:

In [9]:
from sklearn.model_selection import train_test_split

We split the dataset into two subsets. The default ration for the test subset is 25%. However, we can modify it using parameter test_size.

In [10]:
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)

Now, we can train our supervised learning model using $k$-NN:

In [11]:
import warnings
warnings.filterwarnings("ignore")

from sklearn.neighbors import KNeighborsClassifier
k = 5
knn = KNeighborsClassifier(n_neighbors=k, metric='euclidean')
knn.fit(X_train, y_train)
Out[11]:
KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='euclidean',
                     metric_params=None, n_jobs=None, n_neighbors=5, p=2,
                     weights='uniform')

Once the model is trained, we can use it to predict the values for the test subset:

In [12]:
y_pred = knn.predict(X_test)
In [13]:
sns.scatterplot(
    x='mean area',
    y='mean compactness',
    hue='benign',
    data=X_test.join(y_test, how='outer')
)
Out[13]:
<matplotlib.axes._subplots.AxesSubplot at 0x1a19a4b7d0>
In [14]:
plt.scatter(
    X_test['mean area'],
    X_test['mean compactness'],
    c=y_pred,
    cmap='coolwarm',
    alpha=0.7
)
Out[14]:
<matplotlib.collections.PathCollection at 0x1a1aff15d0>

We can compute the confusion matrix:

In [15]:
from sklearn.metrics import confusion_matrix
cm = confusion_matrix(y_test, y_pred)
print(cm)
[[42 13]
 [ 9 79]]

End then, compute the accuracy of the model:

In [16]:
acc = np.sum(np.diag(cm))/np.sum(cm)
print('{:.2%}'.format(acc))
84.62%

k-nn is among the simplest supervised learning algorithms. Indeed, the only hyper-parameter to define k. Thus, we can write a very simple algorithm to find the best value for k:

In [17]:
def best_model_search(X_train, X_test, y_train, y_test, n=10, patience=3):
    k = 1
    best_acc = 0
    best_model = None
    while k <n and patience >0 :
        knn = KNeighborsClassifier(n_neighbors=k, metric='euclidean')
        knn.fit(X_train, y_train)
        y_pred = knn.predict(X_test)
        cm = confusion_matrix(y_test, y_pred)
        acc = np.sum(np.diag(cm))/np.sum(cm)
        print('{:.2%}'.format(acc))
        if acc > best_acc:
            best_acc = acc
            best_model = knn
        patience = patience -1
        k = k+1
    return best_model,k, best_acc
In [18]:
knn, k, acc = best_model_search(X_train, X_test, y_train, y_test, patience=10)
79.02%
79.02%
83.22%
80.42%
84.62%
83.22%
84.62%
82.52%
85.31%
In [19]:
print(k, '{:.2%}'.format(acc))
10 85.31%
In [ ]: