辅导案例-STAT5003
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STAT5003
Week 13
Review and Final Exam
Presented by
Dr. Justin Wishart
The University of Sydney Page 2
Exam format
– Two hour written exam
– 20 Multiple Choice questions
– Questions can have one or two correct answers. You need to select the
exact correct answer(s) to get a mark
– Some short answer questions
– Two longer answer questions
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Topics covered
– Everything in the lectures/tutorials from Weeks 1 to 12 (except
any topic that was marked as not examinable)
– Writing R code is not tested, but there could be questions on
interpreting R outputs
– You should understand how the algorithms work and be able to
sketch out the key steps in pseudo code
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Methods we have learnt
– Regression
– Multivariate linear regression
– Clustering
– Hierarchical clustering
– K-means clustering
– Classification
– Logistic regression
– LDA
– KNN
– SVM
– Random Forest
– Decision trees
– Boosted trees (Adaboost, XGBoost, GBM)
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Multiple Regression
= 0 + 1 1 + 2 2 + …+ +
– Find coefficients to minimise the total sum of squares of the
residuals
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Local regression (smoothing)
A typical model in this case is
= +
– The function f is some smooth function (differentiable).
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Density estimation
– Maximum Likelihood approach
– Reformulate as
(1, 2, … , |) Probability of observing 1, 2, … , given parameter(s)
= ς=1
→ln = σ=1
ln
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Kernel density estimation
– Smooths the data with a chosen hyperparameter (bandwidth)
to estimate the density.
መ =
1


=1




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Hierarchical Clustering
– Bottom-up clustering approach.
– Each point is its own cluster
– Clustering tuned by merging
close values
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K-means algorithm
– 1. Data randomly allocated
– 2. Centres computed.
– Data matched to closest
centre.
– Repeat.
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Principal Components Analysis (PCA)
– Find linear combinations of variables that maximum the
variability.
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PCA and t-SNE
PCA tSNE
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Logistic Regression
Logistic regression model:
= log

1 −
= 0 + 11 +⋯+ =

= Pr( = 1|) = ℎ =
=
1
1 + −

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P
ro
ba
bi
lit
y
-5 -4 -3 -2 -1 0 1 2 3 4 5
X
1
0.5

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Linear Discriminant Analysis (LDA)
: Probability of coming from class k (prior probability)
: Density function for X given that X is an observation from
class k


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Cross validation
– Fitting model to entire dataset can overfit the data and not
perform well on new data
– Split data into training and tests sets to alleviate this and find
the right bias/variance trade-off.
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Bootstrap
– Simulate related data (sampling with replacement) and
examine statistical performance on all the re-sampled data.
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Support Vector Machines (SVM)
– Find the best hyperplane or boundary to separate data into
classes.
– Image taken from
https://en.wikipedia.org/wiki/Support_vector_machine

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Missing Data
– Remove missing data (complete cases)
– Single Imputation
– Multiple imputation
– Expert knowledge of reasons for missing data.
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Basic decision trees
– Partition space into rectangular regions that minimise outcome
deviation.
Millions
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Bagging trees and random forests
– Use bootstrap technique to
create resampled trees and
average the result.
– መ =
1

σ=1
መ∗()
– Random forests do further
sampling to improve model.
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Boosting
– Fit tree to residuals and learn slowly
– Slowly improve the fit in areas where the model doesn’t
perform well.
– Some boosting algorithms discussed
– AdaBoost
– Stochastic gradient boosting
– XGBoost
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Feature Selection
– Filter selection via fold changes.
– Best subset selection.
– Forward selection.
– Backward selection.
– Choose model that minimises test error
– Directly via test set
– Indirectly via penalised criterion.
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Ridge Regression and Lasso
– Constrained optimisation techniques that minimise the squares
with different constraints.
– Lasso has the extra benefit of feature selection as a free
bonus.
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Monte Carlo Methods
– Repeated simulation to estimate the full distribution and
summary values.
– Exploits law of large numbers.
– Can sample from f if inverse of exists, then we can
generate as: = −1
– Acceptance rejection method to handle more difficult
distributions.
= න() ∙ ≈
1


=1

()
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Markov Chain Monte Carlo
– Big use in modelling Bayesian methods.
– Simulates a process (random variable that changes over time)
– Simulate new point based off the current point.
– Can estimate even more complex distributions that in Monte
Carlo methods.
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Methods and metrics to evaluate models
– Sensitivity and specificity
– Accuracy
– Residual sum of squares (for regression)
– ROC curves and AUC
– K-fold cross-validation
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Example multiple choice question
Which of the following method(s) is/are unsupervised learning
methods?
A. K means clustering
B. Logistic regression
C. Random forest
D. Support vector machines
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Example short answer question
a. Explain how the parameters are estimated in simple least
squares regression.
b. Explain a scenario where simple linear regression is not
appropriate.
c. Compute the predicted weight for a person that is 160cm tall
and compute the residual of the first person in the table below.
Sample X : Height (cm) Y: Weight (kg)
1 160 60
2 170.2 77
3 172 62
෠ = 50.412 + 0.0634
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Example long answer question
– Describe the Markov Chain Monte Carlo procedure. You may
use pseudo code as part of your answer.
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