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# FYS-STK3155/4155 Applied Data Analysis and Machine Learning, http://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/index-eng.html
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This site contains all material relevant for the course on Data Analysis and Machine Learning FYS-STK3155/4155.
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## Course content
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Probability theory and statistical methods play a central role in science. Nowadays we are
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surrounded by huge amounts of data. For example, there are about one trillion web pages; more than one
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hour of video is uploaded to YouTube every second, amounting to 10 years of content every
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day; the genomes of 1000s of people, each of which has a length of more than a billion base pairs, have
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been sequenced by various labs and so on.
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This deluge of data calls for automated methods of data analysis,
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which is exactly what machine
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learning provides. In this course the approach is to define machine learning as a set of methods that can
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automatically detect patterns in data, and then use the uncovered patterns to predict future
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data, or to perform other kinds of decision making under uncertainty. Since many of these problems can be studied using
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tools of probability theory, the aim of this course is to expose you to central methods in probability theory linked with machine learning.
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This course covers thus topics like Monte Carlo methods and Markov chains, Bayesian statistics, error estimates, various regression methods, optimization of data and error analysis and central algorithms in machine learning.
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The course has several numerical projects and numerical exercises that are meant to illustrate the theory.
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## Learning outcomes
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The course introduces a variety of central algorithms and methods
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essential for studies of data analysis and machine learning. The course is project based and through the various projects, normally three, the students will be exposed to fundamental research problems in these fields, with the aim to reproduce state of the art scientific results. Both supervised and unsupervised methods will be covered. You will learn to develop and structure large codes for studying these systems, get acquainted with computing facilities and learn to handle large scientific projects. A good scientific and ethical conduct is emphasized throughout the course. More specifically, after this course you will
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- Learn about basic data analysis, statistical analysis, Monte Carlo sampling, data optimization and machine learning;
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- Be capable of extending the acquired knowledge to other systems and cases;
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- Have an understanding of central algorithms used in data analysis and machine learning;
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- Gain knowledge of central aspects of Monte Carlo methods, Markov chains, Gibbs samplers and their possible applications;
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- Understand linear methods for regression and classification, from ordinary least squares, via Lasso and Ridge to Logistic regression;
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- Learn about various neural networks and deep learning methods for supervised and unsupervised learning;
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- Learn about about decision trees and random forests
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- Learn about support vector machines and kernel transformations
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- Reduction of data sets, from PCA to clustering, supervised and unsupervided methods
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- Work on numerical projects to illustrate the theory. The projects play a central role and students are expected to know modern programming languages like Python or C++.
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## Prerequisites
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Basic knowledge in programming and mathematics, with an emphasis on linear algebra. Knowledge of Python or/and C++ as programming languages is required and experience with Jupiter notebook is recommended. Required courses are the equivalents to the University of Oslo mathematics courses MAT1100, MAT1110, MAT1120 and at least one of the corresponding computing and programming courses INF1000/INF1110 or MAT-INF1100/MAT-INF1100L/BIOS1100/KJM-INF1100.
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## The course has two central parts
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1. Statistical analysis and optimization of data
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2. Machine learning
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### Statistical analysis and optimization of data
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The following topics will be covered
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- Basic concepts, expectation values, variance, covariance, correlation functions and errors;
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- Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;
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- Central elements of Bayesian statistics and modeling;
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- Central elements from linear algebra
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- Gradient methods for data optimization
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- Monte Carlo methods, Markov chains, Metropolis-Hastings algorithm;
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- Linear methods for regression and classification;
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- Estimation of errors using cross-validation, blocking, bootstrapping and jackknife methods;
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- Practical optimization using Singular-value decomposition and least squares for parameterizing data.
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### Machine learning, mainly supervised learning
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The following topics will be covered
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- Linear Regression and Logistic Regression;
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- Neural networks and deep learning;
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- Decisions trees and nearest neighbor algorithms
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- Support vector machines
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All the above topics will be supported by examples, hands-on exercises and project work.
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Computational aspects play a central role and the students are
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expected to work on numerical examples and projects which illustrate
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the theory and methods. Some of the projects can be coordinated with the high-performance programming course IN4200.
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## Practicalities
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1. Four lectures per week, Fall semester, 10 ECTS;
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2. Four hours of laboratory sessions for work on computational projects;
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3. Three projects which are graded and count 1/3 each of the final grade;
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4. A selected number of weekly assignments;
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6. The course is part of the CS Master of Science program, but is open to other bachelor and Master of Science students at the University of Oslo;
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7. Grading scale: Grades are awarded on a scale from A to F, where A is the best grade and F is a fail;
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8. The course will be offered as a FYS-MAT4155 (Master of Science level) and a FYS-MAT3155 (senior undergraduate) course.
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## Possible textbooks
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_Recommended textbooks_:
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- Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer
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- Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly
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_General learning book on statistical analysis_:
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- Christian Robert and George Casella, Monte Carlo Statistical Methods, Springer
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- Peter Hoff, A first course in Bayesian statistical models, Springer
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_General Machine Learning Books_:
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- Kevin Murphy, Machine Learning: A Probabilistic Perspective, MIT Press
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- Christopher M. Bishop, Pattern Recognition and Machine Learning, Springer
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- David J.C. MacKay, Information Theory, Inference, and Learning Algorithms, Cambridge University Press
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- David Barber, Bayesian Reasoning and Machine Learning, Cambridge University Press
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## Links to relevant courses at the University of Oslo
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The link here https://www.mn.uio.no/english/research/about/centre-focus/innovation/data-science/studies/ gives an excellent overview of courses on Machine learning at UiO.
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- _STK2100 Machine learning and statistical methods for prediction and classification_ http://www.uio.no/studier/emner/matnat/math/STK2100/index-eng.html.
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- _IN3050 Introduction to Artificial Intelligence and Machine Learning_ https://www.uio.no/studier/emner/matnat/ifi/IN3050/index-eng.html. Introductory course in machine learning and AI with an algorithmic approach.
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- _STK-INF3000/4000 Selected Topics in Data Science_ http://www.uio.no/studier/emner/matnat/math/STK-INF3000/index-eng.html. The course provides insight into selected contemporary relevant topics within Data Science.
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- _IN4080 Natural Language Processing_ https://www.uio.no/studier/emner/matnat/ifi/IN4080/index.html. Probabilistic and machine learning techniques applied to natural language processing.
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- _STK-IN4300 Statistical learning methods in Data Science_ https://www.uio.no/studier/emner/matnat/math/STK-IN4300/index-eng.html. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.
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- _INF4490 Biologically Inspired Computing_ http://www.uio.no/studier/emner/matnat/ifi/INF4490/. An introduction to self-adapting methods also called artificial intelligence or machine learning.
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- _IN-STK5000 Adaptive Methods for Data-Based Decision Making_ https://www.uio.no/studier/emner/matnat/ifi/IN-STK5000/index-eng.html. Methods for adaptive collection and processing of data based on machine learning techniques.
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- _IN5400/INF5860 Machine Learning for Image Analysis_ https://www.uio.no/studier/emner/matnat/ifi/IN5400/. An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.
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- _TEK5040 Deep learning for autonomous systems_ https://www.uio.no/studier/emner/matnat/its/TEK5040/. The course addresses advanced algorithms and architectures for deep learning with neural networks. The course provides an introduction to how deep-learning techniques can be used in the construction of key parts of advanced autonomous systems that exist in physical environments and cyber environments.
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- _STK4051 Computational Statistics_ https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html
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- _STK4021 Applied Bayesian Analysis and Numerical Methods_ https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html
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|
||||
lobster,0,0,1,0,0,1,1,0,0,0,0,0,6,0,0,0,7
|
||||
lynx,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
|
||||
mink,1,0,0,1,0,1,1,1,1,1,0,0,4,1,0,1,1
|
||||
mole,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,0,1
|
||||
mongoose,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
|
||||
moth,1,0,1,0,1,0,0,0,0,1,0,0,6,0,0,0,6
|
||||
newt,0,0,1,0,0,1,1,1,1,1,0,0,4,1,0,0,5
|
||||
octopus,0,0,1,0,0,1,1,0,0,0,0,0,8,0,0,1,7
|
||||
opossum,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,0,1
|
||||
oryx,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,1,1
|
||||
ostrich,0,1,1,0,0,0,0,0,1,1,0,0,2,1,0,1,2
|
||||
parakeet,0,1,1,0,1,0,0,0,1,1,0,0,2,1,1,0,2
|
||||
penguin,0,1,1,0,0,1,1,0,1,1,0,0,2,1,0,1,2
|
||||
pheasant,0,1,1,0,1,0,0,0,1,1,0,0,2,1,0,0,2
|
||||
pike,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,1,4
|
||||
piranha,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,0,4
|
||||
pitviper,0,0,1,0,0,0,1,1,1,1,1,0,0,1,0,0,3
|
||||
platypus,1,0,1,1,0,1,1,0,1,1,0,0,4,1,0,1,1
|
||||
polecat,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
|
||||
pony,1,0,0,1,0,0,0,1,1,1,0,0,4,1,1,1,1
|
||||
porpoise,0,0,0,1,0,1,1,1,1,1,0,1,0,1,0,1,1
|
||||
puma,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
|
||||
pussycat,1,0,0,1,0,0,1,1,1,1,0,0,4,1,1,1,1
|
||||
raccoon,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
|
||||
reindeer,1,0,0,1,0,0,0,1,1,1,0,0,4,1,1,1,1
|
||||
rhea,0,1,1,0,0,0,1,0,1,1,0,0,2,1,0,1,2
|
||||
scorpion,0,0,0,0,0,0,1,0,0,1,1,0,8,1,0,0,7
|
||||
seahorse,0,0,1,0,0,1,0,1,1,0,0,1,0,1,0,0,4
|
||||
seal,1,0,0,1,0,1,1,1,1,1,0,1,0,0,0,1,1
|
||||
sealion,1,0,0,1,0,1,1,1,1,1,0,1,2,1,0,1,1
|
||||
seasnake,0,0,0,0,0,1,1,1,1,0,1,0,0,1,0,0,3
|
||||
seawasp,0,0,1,0,0,1,1,0,0,0,1,0,0,0,0,0,7
|
||||
skimmer,0,1,1,0,1,1,1,0,1,1,0,0,2,1,0,0,2
|
||||
skua,0,1,1,0,1,1,1,0,1,1,0,0,2,1,0,0,2
|
||||
slowworm,0,0,1,0,0,0,1,1,1,1,0,0,0,1,0,0,3
|
||||
slug,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7
|
||||
sole,0,0,1,0,0,1,0,1,1,0,0,1,0,1,0,0,4
|
||||
sparrow,0,1,1,0,1,0,0,0,1,1,0,0,2,1,0,0,2
|
||||
squirrel,1,0,0,1,0,0,0,1,1,1,0,0,2,1,0,0,1
|
||||
starfish,0,0,1,0,0,1,1,0,0,0,0,0,5,0,0,0,7
|
||||
stingray,0,0,1,0,0,1,1,1,1,0,1,1,0,1,0,1,4
|
||||
swan,0,1,1,0,1,1,0,0,1,1,0,0,2,1,0,1,2
|
||||
termite,0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,0,6
|
||||
toad,0,0,1,0,0,1,0,1,1,1,0,0,4,0,0,0,5
|
||||
tortoise,0,0,1,0,0,0,0,0,1,1,0,0,4,1,0,1,3
|
||||
tuatara,0,0,1,0,0,0,1,1,1,1,0,0,4,1,0,0,3
|
||||
tuna,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,1,4
|
||||
vampire,1,0,0,1,1,0,0,1,1,1,0,0,2,1,0,0,1
|
||||
vole,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,0,1
|
||||
vulture,0,1,1,0,1,0,1,0,1,1,0,0,2,1,0,1,2
|
||||
wallaby,1,0,0,1,0,0,0,1,1,1,0,0,2,1,0,1,1
|
||||
wasp,1,0,1,0,1,0,0,0,0,1,1,0,6,0,0,0,6
|
||||
wolf,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
|
||||
worm,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7
|
||||
wren,0,1,1,0,1,0,0,0,1,1,0,0,2,1,0,0,2
|
||||
|
@@ -0,0 +1,78 @@
|
||||
import numpy as np
|
||||
|
||||
class DecisionStump:
|
||||
def fit(self, X, y, weights):
|
||||
m, n = X.shape
|
||||
self.alpha = 0
|
||||
self.threshold = None
|
||||
self.polarity = 1
|
||||
|
||||
min_error = float('inf')
|
||||
|
||||
for feature in range(n):
|
||||
feature_values = np.unique(X[:, feature])
|
||||
|
||||
for threshold in feature_values:
|
||||
for polarity in [1, -1]:
|
||||
predictions = np.ones(m)
|
||||
predictions[X[:, feature] < threshold] = -1
|
||||
predictions *= polarity
|
||||
|
||||
error = sum(weights[predictions != y])
|
||||
|
||||
if error < min_error:
|
||||
min_error = error
|
||||
self.alpha = 0.5 * np.log((1 - error) / (error + 1e-10))
|
||||
self.threshold = threshold
|
||||
self.feature_index = feature
|
||||
self.polarity = polarity
|
||||
|
||||
def predict(self, X):
|
||||
m = X.shape[0]
|
||||
predictions = np.ones(m)
|
||||
if self.polarity == 1:
|
||||
predictions[X[:, self.feature_index] < self.threshold] = -1
|
||||
else:
|
||||
predictions[X[:, self.feature_index] >= self.threshold] = -1
|
||||
return predictions
|
||||
|
||||
class AdaBoost:
|
||||
def fit(self, X, y, n_estimators):
|
||||
m = X.shape[0]
|
||||
self.alphas = []
|
||||
self.models = []
|
||||
|
||||
weights = np.ones(m) / m
|
||||
|
||||
for _ in range(n_estimators):
|
||||
stump = DecisionStump()
|
||||
stump.fit(X, y, weights)
|
||||
predictions = stump.predict(X)
|
||||
|
||||
error = sum(weights[predictions != y])
|
||||
if error == 0:
|
||||
break
|
||||
|
||||
self.models.append(stump)
|
||||
self.alphas.append(stump.alpha)
|
||||
|
||||
weights *= np.exp(-stump.alpha * y * predictions)
|
||||
weights /= np.sum(weights)
|
||||
|
||||
def predict(self, X):
|
||||
final_predictions = np.zeros(X.shape[0])
|
||||
for alpha, model in zip(self.alphas, self.models):
|
||||
final_predictions += alpha * model.predict(X)
|
||||
return np.sign(final_predictions)
|
||||
|
||||
# Example dataset (X, y)
|
||||
X = np.array([[1], [2], [3], [4], [5], [6], [7], [8], [9], [10]])
|
||||
y = np.array([-1, -1, -1, -1, 1, 1, 1, 1, 1, 1]) # Labels must be -1 or 1
|
||||
|
||||
# Train AdaBoost
|
||||
ada = AdaBoost()
|
||||
ada.fit(X, y, n_estimators=10)
|
||||
|
||||
# Predictions
|
||||
predictions = ada.predict(X)
|
||||
print("Predictions:", predictions)
|
||||
@@ -1,65 +1,65 @@
|
||||
import matplotlib.pyplot as plt
|
||||
import numpy as np
|
||||
from sklearn.model_selection import train_test_split
|
||||
from sklearn.datasets import load_breast_cancer
|
||||
from sklearn.svm import SVC
|
||||
from sklearn.linear_model import LogisticRegression
|
||||
from sklearn.tree import DecisionTreeClassifier
|
||||
from sklearn.ensemble import AdaBoostClassifier
|
||||
e:
|
||||
return self.Node(value=self._most_common_label(y))
|
||||
left_indices = X[:, best_feature] < best_threshold
|
||||
right_indices = X[:, best_feature] >= best_threshold
|
||||
left_subtree = self._grow_tree(X[left_indices], y[left_indices], depth + 1)
|
||||
right_subtree = self._grow_tree(X[right_indices], y[right_indices], depth + 1)
|
||||
return self.Node(feature=best_feature, threshold=best_threshold, left=left_subtree, right=right_subtree)
|
||||
def _best_split(self, X, y, num_features):
|
||||
best_gain = -1
|
||||
best_feature, best_threshold = None, None
|
||||
|
||||
def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.5, -1, 1.5], alpha=0.5, contour=True):
|
||||
x1s = np.linspace(axes[0], axes[1], 100)
|
||||
x2s = np.linspace(axes[2], axes[3], 100)
|
||||
x1, x2 = np.meshgrid(x1s, x2s)
|
||||
X_new = np.c_[x1.ravel(), x2.ravel()]
|
||||
y_pred = clf.predict(X_new).reshape(x1.shape)
|
||||
custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
|
||||
plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
|
||||
if contour:
|
||||
custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
|
||||
plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
|
||||
plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", alpha=alpha)
|
||||
plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", alpha=alpha)
|
||||
plt.axis(axes)
|
||||
plt.xlabel(r"$x_1$", fontsize=18)
|
||||
plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
|
||||
for feature in range(num_features):
|
||||
thresholds, classes = zip(*sorted(zip(X[:, feature], y)))
|
||||
num_samples = len(y)
|
||||
for i in range(1, num_samples):
|
||||
if classes[i] == classes[i - 1]:
|
||||
continue
|
||||
|
||||
# Load the data
|
||||
cancer = load_breast_cancer()
|
||||
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
|
||||
#now scale the data
|
||||
from sklearn.preprocessing import StandardScaler
|
||||
scaler = StandardScaler()
|
||||
scaler.fit(X_train)
|
||||
X_train_scaled = scaler.transform(X_train)
|
||||
X_test_scaled = scaler.transform(X_test)
|
||||
threshold = (thresholds[i] + thresholds[i - 1]) / 2
|
||||
left_indices = X[:, feature] < threshold
|
||||
right_indices = X[:, feature] >= threshold
|
||||
|
||||
ada_clf = AdaBoostClassifier(
|
||||
DecisionTreeClassifier(max_depth=1), n_estimators=200,
|
||||
algorithm="SAMME.R", learning_rate=0.5, random_state=42)
|
||||
|
||||
ada_clf.fit(X_train_scaled, y_train)
|
||||
plot_decision_boundary(ada_clf, cancer.data,cancer.target)
|
||||
|
||||
m = len(X_train_scaled)
|
||||
|
||||
plt.figure(figsize=(11, 4))
|
||||
for subplot, learning_rate in ((121, 1), (122, 0.5)):
|
||||
sample_weights = np.ones(m)
|
||||
plt.subplot(subplot)
|
||||
for i in range(5):
|
||||
svm_clf = SVC(kernel="rbf", C=0.05, gamma="auto", random_state=42)
|
||||
svm_clf.fit(X_train_scaled, y_train, sample_weight=sample_weights)
|
||||
y_pred = svm_clf.predict(X_train_scaled)
|
||||
sample_weights[y_pred != y_train] *= (1 + learning_rate)
|
||||
plot_decision_boundary(svm_clf, cancer.data,cancer.target, alpha=0.2)
|
||||
plt.title("learning_rate = {}".format(learning_rate), fontsize=16)
|
||||
if subplot == 121:
|
||||
plt.text(-0.7, -0.65, "1", fontsize=14)
|
||||
plt.text(-0.6, -0.10, "2", fontsize=14)
|
||||
plt.text(-0.5, 0.10, "3", fontsize=14)
|
||||
plt.text(-0.4, 0.55, "4", fontsize=14)
|
||||
plt.text(-0.3, 0.90, "5", fontsize=14)
|
||||
|
||||
plt.show()
|
||||
gain = self._information_gain(y, y[left_indices], y[right_indices])
|
||||
if gain > best_gain:
|
||||
best_gain = gain
|
||||
best_feature = feature
|
||||
best_threshold = threshold
|
||||
return best_feature, best_threshold
|
||||
def _information_gain(self, parent, left, right):
|
||||
total_samples = len(parent)
|
||||
if len(left) == 0 or len(right) == 0:
|
||||
return 0
|
||||
|
||||
parent_entropy = self._entropy(parent)
|
||||
left_entropy = self._entropy(left)
|
||||
right_entropy = self._entropy(right)
|
||||
weighted_entropy = (len(left) / total_samples) * left_entropy + (len(right) / total_samples) * right_entropy
|
||||
return parent_entropy - weighted_entropy
|
||||
def _entropy(self, y):
|
||||
class_counts = np.bincount(y)
|
||||
probabilities = class_counts / len(y)
|
||||
return -np.sum(probabilities * np.log(probabilities + 1e-10))
|
||||
def _most_common_label(self, y):
|
||||
return np.bincount(y).argmax()
|
||||
def predict(self, X):
|
||||
return np.array([self._predict(inputs) for inputs in X])
|
||||
def _predict(self, inputs):
|
||||
node = self.tree
|
||||
while node.value is None:
|
||||
if inputs[node.feature] < node.threshold:
|
||||
node = node.left
|
||||
else:
|
||||
node = node.right
|
||||
return node.value
|
||||
# Example usage
|
||||
if __name__ == "__main__":
|
||||
# Example dataset
|
||||
X = np.array([[2.5], [1.0], [1.5], [3.0], [3.5], [2.0], [4.0], [2.2]])
|
||||
y = np.array([0, 0, 0, 1, 1, 0, 1, 0]) # Binary labels
|
||||
# Train decision tree
|
||||
tree = DecisionTree(max_depth=3)
|
||||
tree.fit(X, y)
|
||||
# Predictions
|
||||
predictions = tree.predict(X)
|
||||
print("Predictions:", predictions)~
|
||||
|
||||
@@ -0,0 +1,88 @@
|
||||
import numpy as np
|
||||
class DecisionTree:
|
||||
def __init__(self, min_samples_split=2, max_depth=float("inf")):
|
||||
self.min_samples_split = min_samples_split
|
||||
self.max_depth = max_depth
|
||||
self.tree = None
|
||||
class Node:
|
||||
def __init__(self, feature=None, threshold=None, left=None, right=None, value=None):
|
||||
self.feature = feature
|
||||
self.threshold = threshold
|
||||
self.left = left
|
||||
self.right = right
|
||||
self.value = value
|
||||
def fit(self, X, y):
|
||||
self.tree = self._grow_tree(X, y)
|
||||
def _grow_tree(self, X, y, depth=0):
|
||||
num_samples, num_features = X.shape
|
||||
unique_classes = np.unique(y)
|
||||
# Check stopping criteria
|
||||
if (num_samples < self.min_samples_split or
|
||||
depth >= self.max_depth or
|
||||
len(unique_classes) == 1):
|
||||
return self.Node(value=self._most_common_label(y))
|
||||
best_feature, best_threshold = self._best_split(X, y, num_features)
|
||||
if best_feature is None:
|
||||
return self.Node(value=self._most_common_label(y))
|
||||
left_indices = X[:, best_feature] < best_threshold
|
||||
right_indices = X[:, best_feature] >= best_threshold
|
||||
left_subtree = self._grow_tree(X[left_indices], y[left_indices], depth + 1)
|
||||
right_subtree = self._grow_tree(X[right_indices], y[right_indices], depth + 1)
|
||||
return self.Node(feature=best_feature, threshold=best_threshold, left=left_subtree, right=right_subtree)
|
||||
def _best_split(self, X, y, num_features):
|
||||
best_gain = -1
|
||||
best_feature, best_threshold = None, None
|
||||
|
||||
for feature in range(num_features):
|
||||
thresholds, classes = zip(*sorted(zip(X[:, feature], y)))
|
||||
num_samples = len(y)
|
||||
for i in range(1, num_samples):
|
||||
if classes[i] == classes[i - 1]:
|
||||
continue
|
||||
|
||||
threshold = (thresholds[i] + thresholds[i - 1]) / 2
|
||||
left_indices = X[:, feature] < threshold
|
||||
right_indices = X[:, feature] >= threshold
|
||||
gain = self._information_gain(y, y[left_indices], y[right_indices])
|
||||
if gain > best_gain:
|
||||
best_gain = gain
|
||||
best_feature = feature
|
||||
best_threshold = threshold
|
||||
return best_feature, best_threshold
|
||||
def _information_gain(self, parent, left, right):
|
||||
total_samples = len(parent)
|
||||
if len(left) == 0 or len(right) == 0:
|
||||
return 0
|
||||
|
||||
parent_entropy = self._entropy(parent)
|
||||
left_entropy = self._entropy(left)
|
||||
right_entropy = self._entropy(right)
|
||||
weighted_entropy = (len(left) / total_samples) * left_entropy + (len(right) / total_samples) * right_entropy
|
||||
return parent_entropy - weighted_entropy
|
||||
def _entropy(self, y):
|
||||
class_counts = np.bincount(y)
|
||||
probabilities = class_counts / len(y)
|
||||
return -np.sum(probabilities * np.log(probabilities + 1e-10))
|
||||
def _most_common_label(self, y):
|
||||
return np.bincount(y).argmax()
|
||||
def predict(self, X):
|
||||
return np.array([self._predict(inputs) for inputs in X])
|
||||
def _predict(self, inputs):
|
||||
node = self.tree
|
||||
while node.value is None:
|
||||
if inputs[node.feature] < node.threshold:
|
||||
node = node.left
|
||||
else:
|
||||
node = node.right
|
||||
return node.value
|
||||
# Example usage
|
||||
if __name__ == "__main__":
|
||||
# Example dataset
|
||||
X = np.array([[2.5], [1.0], [1.5], [3.0], [3.5], [2.0], [4.0], [2.2]])
|
||||
y = np.array([0, 0, 0, 1, 1, 0, 1, 0]) # Binary labels
|
||||
# Train decision tree
|
||||
tree = DecisionTree(max_depth=3)
|
||||
tree.fit(X, y)
|
||||
# Predictions
|
||||
predictions = tree.predict(X)
|
||||
print("Predictions:", predictions)
|
||||
@@ -0,0 +1,61 @@
|
||||
import tensorflow as tf
|
||||
|
||||
from tensorflow.keras import datasets, layers, models
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
# We import the data set
|
||||
(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()
|
||||
|
||||
|
||||
train_images, test_images = train_images / 255.0, test_images / 255.0
|
||||
|
||||
class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
|
||||
'dog', 'frog', 'horse', 'ship', 'truck']
|
||||
plt.figure(figsize=(10,10))
|
||||
for i in range(25):
|
||||
plt.subplot(5,5,i+1)
|
||||
plt.xticks([])
|
||||
plt.yticks([])
|
||||
plt.grid(False)
|
||||
plt.imshow(train_images[i], cmap=plt.cm.binary)
|
||||
# The CIFAR labels happen to be arrays,
|
||||
# which is why you need the extra index
|
||||
plt.xlabel(class_names[train_labels[i][0]])
|
||||
plt.show()
|
||||
|
||||
model = models.Sequential()
|
||||
model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
|
||||
model.add(layers.MaxPooling2D((2, 2)))
|
||||
model.add(layers.Conv2D(64, (3, 3), activation='relu'))
|
||||
model.add(layers.MaxPooling2D((2, 2)))
|
||||
model.add(layers.Conv2D(64, (3, 3), activation='relu'))
|
||||
|
||||
# Let's display the architecture of our model so far.
|
||||
|
||||
model.summary()
|
||||
|
||||
model.add(layers.Flatten())
|
||||
model.add(layers.Dense(64, activation='relu'))
|
||||
model.add(layers.Dense(10))
|
||||
# Here's the complete architecture of our model
|
||||
|
||||
model.summary()
|
||||
|
||||
model.compile(optimizer='adam',
|
||||
loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
|
||||
metrics=['accuracy'])
|
||||
|
||||
history = model.fit(train_images, train_labels, epochs=10,
|
||||
validation_data=(test_images, test_labels))
|
||||
|
||||
|
||||
plt.plot(history.history['accuracy'], label='accuracy')
|
||||
plt.plot(history.history['val_accuracy'], label = 'val_accuracy')
|
||||
plt.xlabel('Epoch')
|
||||
plt.ylabel('Accuracy')
|
||||
plt.ylim([0.5, 1])
|
||||
plt.legend(loc='lower right')
|
||||
|
||||
test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2)
|
||||
|
||||
print(test_acc)
|
||||
@@ -0,0 +1,134 @@
|
||||
|
||||
# import necessary packages
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
from sklearn import datasets
|
||||
|
||||
|
||||
# ensure the same random numbers appear every time
|
||||
np.random.seed(0)
|
||||
|
||||
# display images in notebook
|
||||
plt.rcParams['figure.figsize'] = (12,12)
|
||||
|
||||
|
||||
# download MNIST dataset
|
||||
digits = datasets.load_digits()
|
||||
|
||||
# define inputs and labels
|
||||
inputs = digits.images
|
||||
labels = digits.target
|
||||
|
||||
# RGB images have a depth of 3
|
||||
# our images are grayscale so they should have a depth of 1
|
||||
inputs = inputs[:,:,:,np.newaxis]
|
||||
|
||||
print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
|
||||
print("labels = (n_inputs) = " + str(labels.shape))
|
||||
|
||||
|
||||
# choose some random images to display
|
||||
n_inputs = len(inputs)
|
||||
indices = np.arange(n_inputs)
|
||||
random_indices = np.random.choice(indices, size=5)
|
||||
|
||||
for i, image in enumerate(digits.images[random_indices]):
|
||||
plt.subplot(1, 5, i+1)
|
||||
plt.axis('off')
|
||||
plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
|
||||
plt.title("Label: %d" % digits.target[random_indices[i]])
|
||||
plt.show()
|
||||
|
||||
from tensorflow.keras import datasets, layers, models
|
||||
from tensorflow.keras.layers import Input
|
||||
from tensorflow.keras.models import Sequential #This allows appending layers to existing models
|
||||
from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer
|
||||
from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)
|
||||
from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)
|
||||
from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function
|
||||
|
||||
from sklearn.model_selection import train_test_split
|
||||
|
||||
# representation of labels
|
||||
labels = to_categorical(labels)
|
||||
|
||||
# split into train and test data
|
||||
# one-liner from scikit-learn library
|
||||
train_size = 0.8
|
||||
test_size = 1 - train_size
|
||||
X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
|
||||
test_size=test_size)
|
||||
|
||||
def create_convolutional_neural_network_keras(input_shape, receptive_field,
|
||||
n_filters, n_neurons_connected, n_categories,
|
||||
eta, lmbd):
|
||||
model = Sequential()
|
||||
model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
|
||||
activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
|
||||
model.add(layers.MaxPooling2D(pool_size=(2, 2)))
|
||||
model.add(layers.Flatten())
|
||||
model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
|
||||
model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
|
||||
|
||||
sgd = optimizers.SGD(learning_rate=eta)
|
||||
model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
|
||||
|
||||
return model
|
||||
|
||||
epochs = 100
|
||||
batch_size = 100
|
||||
input_shape = X_train.shape[1:4]
|
||||
receptive_field = 3
|
||||
n_filters = 10
|
||||
n_neurons_connected = 50
|
||||
n_categories = 10
|
||||
|
||||
eta_vals = np.logspace(-5, 1, 7)
|
||||
lmbd_vals = np.logspace(-5, 1, 7)
|
||||
|
||||
CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
|
||||
|
||||
for i, eta in enumerate(eta_vals):
|
||||
for j, lmbd in enumerate(lmbd_vals):
|
||||
CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
|
||||
n_filters, n_neurons_connected, n_categories,
|
||||
eta, lmbd)
|
||||
CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
|
||||
scores = CNN.evaluate(X_test, Y_test)
|
||||
|
||||
CNN_keras[i][j] = CNN
|
||||
|
||||
print("Learning rate = ", eta)
|
||||
print("Lambda = ", lmbd)
|
||||
print("Test accuracy: %.3f" % scores[1])
|
||||
print()
|
||||
# visual representation of grid search
|
||||
# uses seaborn heatmap, could probably do this in matplotlib
|
||||
import seaborn as sns
|
||||
|
||||
sns.set()
|
||||
|
||||
train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
||||
test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
||||
|
||||
for i in range(len(eta_vals)):
|
||||
for j in range(len(lmbd_vals)):
|
||||
CNN = CNN_keras[i][j]
|
||||
|
||||
train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
|
||||
test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
|
||||
|
||||
|
||||
fig, ax = plt.subplots(figsize = (10, 10))
|
||||
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
|
||||
ax.set_title("Training Accuracy")
|
||||
ax.set_ylabel("$\eta$")
|
||||
ax.set_xlabel("$\lambda$")
|
||||
plt.show()
|
||||
|
||||
fig, ax = plt.subplots(figsize = (10, 10))
|
||||
sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
|
||||
ax.set_title("Test Accuracy")
|
||||
ax.set_ylabel("$\eta$")
|
||||
ax.set_xlabel("$\lambda$")
|
||||
plt.show()
|
||||
Reference in New Issue
Block a user