# DATA SCIENCE WITH PYTHON

In the concluding sessions of this course, I have shifted from talking about the data pipeline, to the functions at the end of the tunnel, our Machine Learning algorithms, which I've also likened to a stable of horses, in terms of how we "race" them to find the best. Choosing the best horse for your application takes experience. Don't expect to become a data scientist overnight.

In our sequence below, I start with a famous, oft used dataset, made of 28 by 28 numpy arrays, representing grayscale images of the numerals 0 through 9, quite a few specimens of each. They're labeled rows. We know the digits. Lets take a look.

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In [1]:

import numpy as np
print(digits.data.shape)

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(1797, 64)

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In [2]:

print(digits.DESCR)

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Optical Recognition of Handwritten Digits Data Set
===================================================

Notes
-----
Data Set Characteristics:
:Number of Instances: 5620
:Number of Attributes: 64
:Attribute Information: 8x8 image of integer pixels in the range 0..16.
:Missing Attribute Values: None
:Creator: E. Alpaydin (alpaydin '@' boun.edu.tr)
:Date: July; 1998

This is a copy of the test set of the UCI ML hand-written digits datasets
http://archive.ics.uci.edu/ml/datasets/Optical+Recognition+of+Handwritten+Digits

The data set contains images of hand-written digits: 10 classes where
each class refers to a digit.

Preprocessing programs made available by NIST were used to extract
normalized bitmaps of handwritten digits from a preprinted form. From a
total of 43 people, 30 contributed to the training set and different 13
to the test set. 32x32 bitmaps are divided into nonoverlapping blocks of
4x4 and the number of on pixels are counted in each block. This generates
an input matrix of 8x8 where each element is an integer in the range
0..16. This reduces dimensionality and gives invariance to small
distortions.

For info on NIST preprocessing routines, see M. D. Garris, J. L. Blue, G.
T. Candela, D. L. Dimmick, J. Geist, P. J. Grother, S. A. Janet, and C.
L. Wilson, NIST Form-Based Handprint Recognition System, NISTIR 5469,
1994.

References
----------
- C. Kaynak (1995) Methods of Combining Multiple Classifiers and Their
Applications to Handwritten Digit Recognition, MSc Thesis, Institute of
Graduate Studies in Science and Engineering, Bogazici University.
- E. Alpaydin, C. Kaynak (1998) Cascading Classifiers, Kybernetika.
- Ken Tang and Ponnuthurai N. Suganthan and Xi Yao and A. Kai Qin.
Linear dimensionalityreduction using relevance weighted LDA. School of
Electrical and Electronic Engineering Nanyang Technological University.
2005.
- Claudio Gentile. A New Approximate Maximal Margin Classification
Algorithm. NIPS. 2000.

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In [3]:

import matplotlib.pyplot as plt
% matplotlib inline

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In [4]:

plt.gray()  # gray reversed shown below
_ = plt.matshow(digits.images[0])

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<matplotlib.figure.Figure at 0x1a1233c5c0>

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In [5]:

plt.figure(1, figsize=(3, 3))
plt.imshow(digits.images[0], cmap=plt.cm.gray_r, interpolation='nearest')
plt.show()

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In [6]:

_ = plt.matshow(digits.images[108])

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In [7]:

digits.data[108]

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Out[7]:

array([ 0.,  0.,  2., 11., 16.,  4.,  0.,  0.,  0.,  0., 12.,  9., 11.,
15.,  1.,  0.,  0.,  0.,  2.,  0.,  4., 16.,  0.,  0.,  0.,  0.,
0.,  2.,  8., 15.,  1.,  0.,  0.,  4., 16., 16., 16., 15.,  7.,
0.,  0.,  3.,  6.,  4., 16.,  3.,  0.,  0.,  0.,  0.,  0.,  6.,
11.,  0.,  0.,  0.,  0.,  0.,  0., 12.,  7.,  0.,  0.,  0.])

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Remember how we think in machine learning. We have a multifaceted (multi-featured) set of samples, rows with many columns, and then a single column of correct results, an "answer key" if you will.

We often call this answer key column the "target" and then measure "error" as divergence between guesses and target.

Decreasing divergence bespeaks of a learning rate as the model trains on, or fits the training data. Whether we control this learning rate as a hyperparameter, or leave it to the algorithm to work at some built-in speed, depends on which machine learner type we've selected. Below we're looking at KNN and then a neural net.

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In [8]:

digits.target[108]

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Out[8]:

7

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That's a very poor rendering of the numeral 7 and we're immediately forgiving if our Machine Learning algorithm gets some wrong, with training data of such abysmal quality. As seen from `digits.data`, the 64 bits used to represent a digit are hardly enough. Other datasets come with at least 28 x 28 bits for each numeral. We're truly at the low end with this skimpy number of bits per digit.

Neverthesless, we press on... I'm making only minor changes to this open source script on Github, by Fabiosato.

Remember how KNN works:

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In [9]:

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Out[9]:

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Remember to distinguish KNN from K-Means. You might use the latter to create the clusters whereby you could then fit the former. Here's a paper on LinkedIn suggesting doing that. Once you have the clusters (voters), a new data point is "claimed" by one or more clusters.

Hierarchical clustering algorithms compete with K-Means. The latter does better for spherical or globular clusters.

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In [10]:

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Out[10]:

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In [11]:

from sklearn.metrics import confusion_matrix, accuracy_score
from sklearn.model_selection import train_test_split
from sklearn import neighbors # http://scikit-learn.org/stable/modules/classes.html#module-sklearn.neighbors

# prepare datasets from training and for validation
X_train, X_test, y_train, y_test = train_test_split(digits.data, digits.target,
test_size=0.4, random_state=0)

# runs the kNN classifier for even number of neighbors from 1 to 10
for n in range(1, 10, 2):
clf = neighbors.KNeighborsClassifier(n)

# instance based learning
clf.fit(X_train, y_train)

# our 'ground truth'
y_true = y_test

# predict
y_pred = clf.predict(X_test)

# learning metrics
cm = confusion_matrix(y_true, y_pred)
acc = accuracy_score(y_true, y_pred)

print ("Neighbors: %d" % n)
print ("Confusion Matrix")
print (cm)

print ("Accuracy score: %f" % accuracy_score(y_true, y_pred))
print ()

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Neighbors: 1
Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 73  0  0  0  0  0  0  0  0]
[ 0  0 71  0  0  0  0  0  0  0]
[ 0  0  0 70  0  0  0  0  0  0]
[ 0  0  0  0 63  0  0  0  0  0]
[ 0  0  0  0  0 87  1  0  0  1]
[ 0  0  0  0  0  0 76  0  0  0]
[ 0  0  0  0  0  0  0 65  0  0]
[ 0  2  0  1  0  0  0  0 74  1]
[ 0  0  0  2  0  1  0  0  0 71]]
Accuracy score: 0.987483

Neighbors: 3
Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 73  0  0  0  0  0  0  0  0]
[ 0  0 71  0  0  0  0  0  0  0]
[ 0  0  1 69  0  0  0  0  0  0]
[ 0  0  0  0 62  0  0  1  0  0]
[ 0  1  0  0  0 86  1  0  0  1]
[ 0  0  0  0  0  0 76  0  0  0]
[ 0  0  0  0  0  0  0 65  0  0]
[ 0  2  0  2  0  0  0  0 73  1]
[ 0  0  0  1  0  0  0  0  0 73]]
Accuracy score: 0.984701

Neighbors: 5
Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 73  0  0  0  0  0  0  0  0]
[ 0  0 70  0  0  0  0  1  0  0]
[ 0  0  1 69  0  0  0  0  0  0]
[ 0  0  0  0 62  0  0  1  0  0]
[ 0  0  0  0  0 87  1  0  0  1]
[ 0  0  0  0  0  0 76  0  0  0]
[ 0  0  0  0  0  0  0 65  0  0]
[ 0  5  0  2  0  0  0  0 70  1]
[ 0  0  0  1  0  1  0  0  0 72]]
Accuracy score: 0.979138

Neighbors: 7
Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 73  0  0  0  0  0  0  0  0]
[ 0  1 69  0  0  0  0  1  0  0]
[ 0  0  0 70  0  0  0  0  0  0]
[ 0  0  0  0 61  0  0  2  0  0]
[ 0  0  0  0  1 86  1  0  0  1]
[ 0  0  0  0  0  0 76  0  0  0]
[ 0  0  0  0  0  0  0 65  0  0]
[ 0  5  0  2  0  0  0  0 70  1]
[ 0  0  0  0  0  1  0  0  0 73]]
Accuracy score: 0.977747

Neighbors: 9
Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 72  0  0  0  0  1  0  0  0]
[ 0  1 69  0  0  0  0  1  0  0]
[ 0  0  0 70  0  0  0  0  0  0]
[ 0  0  0  0 61  0  0  2  0  0]
[ 0  0  0  0  1 86  1  0  0  1]
[ 0  0  0  0  0  0 76  0  0  0]
[ 0  0  0  0  0  0  0 65  0  0]
[ 0  5  0  2  0  0  0  1 69  1]
[ 0  0  0  1  0  1  0  0  0 72]]
Accuracy score: 0.973574

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Discerning digits within a blizzard of data points streaming in, or other patterns, may be described as a process of identifying clusters or neighborhoods. Even before we name the clusters we claim to find, we need to find them, and this is where dimensionality reduction comes in handy, as if we can get the dimensions down to three, we have some axes we might use.

"Dimensionality reduction" involves finding eigenvectors, the most efficient at singling out cells in not containing redundant info, forming a basis. An idea of ranking eigenvectors, in the sense of "most significant digits", allows us to cluster data by just the first few eigenvector coordinates.

One might usefully compare this process to discovering the desmomap, or binary tree resulting from bottom-up progressive agglomeration into larger groups. One may then place a threshold cut through the data to vary the number of clusters one wishes to regard as separate. There's a sense of binning and/or pigeon-holing, where the hyperparameter is the degree of subdivisioning.

Does a neural network fare better? Let's admit, the KNN machine learner did a great job. Fast horse!

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In [12]:

from sklearn.neural_network import MLPClassifier

# runs the MLP classifier, all with same hyperparameters
for n in range(1, 10, 2):
clf = MLPClassifier()

# instance based learning
clf.fit(X_train, y_train)

# our 'ground truth'
y_true = y_test

# predict
y_pred = clf.predict(X_test)

# learning metrics
cm = confusion_matrix(y_true, y_pred)
acc = accuracy_score(y_true, y_pred)

# print ("Neighbors: %d" % n)
print ("Confusion Matrix")
print (cm)

print ("Accuracy score: %f" % accuracy_score(y_true, y_pred))
print ()

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Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 72  0  0  0  0  0  0  1  0]
[ 0  1 68  0  0  0  0  0  2  0]
[ 0  0  0 68  0  0  0  0  1  1]
[ 0  0  0  0 63  0  0  0  0  0]
[ 0  0  0  2  1 84  1  0  0  1]
[ 0  1  0  0  0  1 74  0  0  0]
[ 0  0  0  0  0  0  0 65  0  0]
[ 0  5  1  1  0  1  0  0 68  2]
[ 1  0  0  2  0  1  0  0  0 70]]
Accuracy score: 0.962448

Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 72  0  0  0  0  0  0  1  0]
[ 0  1 68  0  0  0  0  1  1  0]
[ 0  0  1 68  0  1  0  0  0  0]
[ 0  0  0  0 63  0  0  0  0  0]
[ 0  0  0  0  0 84  1  0  0  4]
[ 0  1  0  0  0  1 74  0  0  0]
[ 0  0  0  0  1  0  0 64  0  0]
[ 0  4  0  1  0  0  0  0 73  0]
[ 0  0  1  1  0  1  0  0  0 71]]
Accuracy score: 0.969402

Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 69  0  0  0  0  1  0  2  1]
[ 0  1 70  0  0  0  0  0  0  0]
[ 0  0  1 68  0  1  0  0  0  0]
[ 0  0  0  0 62  0  0  1  0  0]
[ 0  0  0  0  1 86  1  0  0  1]
[ 0  1  0  0  0  1 74  0  0  0]
[ 0  1  0  0  0  0  0 64  0  0]
[ 0  3  1  0  1  1  0  0 71  1]
[ 0  0  0  1  0  1  0  0  0 72]]
Accuracy score: 0.968011

Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 71  0  0  0  0  0  0  1  1]
[ 0  2 68  0  0  0  0  0  1  0]
[ 0  0  0 67  0  1  0  0  2  0]
[ 0  0  0  0 61  0  0  2  0  0]
[ 0  0  0  1  0 85  2  0  0  1]
[ 1  1  0  0  0  0 74  0  0  0]
[ 0  0  0  0  2  0  0 63  0  0]
[ 0  1  0  0  1  0  0  0 74  2]
[ 0  0  0  1  0  1  0  1  0 71]]
Accuracy score: 0.965229

Confusion Matrix
[[60  0  0  0  0  0  0  0  0  0]
[ 0 70  0  0  0  0  0  0  2  1]
[ 0  1 68  1  0  0  0  0  1  0]
[ 0  0  0 68  0  1  0  0  0  1]
[ 0  0  0  0 61  0  0  2  0  0]
[ 0  1  0  0  0 85  2  0  0  1]
[ 0  1  0  0  0  1 74  0  0  0]
[ 0  0  0  0  0  0  0 65  0  0]
[ 0  3  0  0  0  0  0  1 71  3]
[ 0  0  0  1  0  2  0  1  0 70]]
Accuracy score: 0.962448

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I'd say these two are competitive, but award KNN first prize in this case. On the other hand, I did not try varying the hyperparameters available to me with the MLP classifier. Lets say the results so far are inconclusive. More research needed.