Lieve Göbbels
Statistical Computing (JBM050)
Semester 2, 2020-2021
Statistical Computing
Introduction 3
Recap mathematical models (distributions) 3
Multinomial coef cients 3
Basic Concepts 5
The Normal (Gaussian) distribution 5
Sample statistics as estimators of population parameters 5
Sampling distribution 5
Accuracy of Sample Statistics 6
Accuracy of Sample Statistics 6
MSE and BVT 6
t-statistic and Student-t distribution 6
Sampling distribution of the proportion for binary data 7
Monte Carlo Simulation 8
Quality control 8
General form of MC simulation 8
Simulations for properties of hypothesis tests 9
Optimization I and Statistical Learning 10
Introduction 10
Notation 10
Work ow of statistical modeling 11
Optimization II 13
Optimization basics 13
(Simple) Linear Regression 14
A quick recap 14
Univariate regression models 14
Estimating model parameters 15
Limits of (simple) linear regression 15
Multiple linear regression 16
Estimating the regression model 16
Predictive Model Accuracy 17
Model accuracy 17
Predictive modeling 17
Test set 18
The collinearity paradox 19
Model selection 19
Cross-Validation 20
Validation set approach 20
, LOOCV 20
K-fold CV 21
Train-Validation-Test 21
CV with non-CV model selection 21
Nested CV 22
Logistic Regression 23
The data generating model 23
Estimation 23
Iterative Optimization 24
De ning properties 24
Convergence 24
Bisection method 25
Newton’s method 26
Constrained Optimization 28
Equality constraints 28
Inequality constraints 29
Penalized Optimization 30
Penalized optimization 30
Ridge regression 30
Lasso regression 31
,Introduction
In short:
• Recap mathematical models (distributions)
• Multinomial coef cients
Recap mathematical models (distributions)
There are di erent kinds of mathematical models (distributions):
• Bernoulli distribution
X ∼ Ber n(π) where π ≠ 3.14... but p
P(X = 1) = π ; P(X = 0) = 1 − π
• Binomial distribution
X ∼ Bin(n, π)
P(X = k) = ( ) π k (1 − π)n−k
n
k
• Hypergeometric distribution
X ∼ hypergeom et r ic(N, K, n)
( k )( n − k )
K N−K
P(X = k) = where N= total elements to draw
(n)
N
K= nr. of elements with desired feature
n= nr. of draws
k = nr. of elements with feature in draws
๏ this formula holds when drawn without replacement
Note: this is just a brief recap of the di erent distributions, more elaborate descriptions can be found
in the Data Statistics summary. It is advised to look back into the Data Statistics material if subjects
in this summary are unclear.
Multinomial coef cients
This is used when a set of n distinct items needs to be divided into r distinct groups of
respective sizes n1, n 2, …, nr where ∑i=1 ni = n.
( n2 )
n − n1
So, there are (n ) choices for the rst group,
n
for the second group and so on. It
1
n!
then follows that there are possible divisions.
n1 !n 2 !…nr !
, Example 1:
10 children are to be divided in an A team and a B team of 5 each. They will play in a di erent
league.
Solution:
10!
There are
5!5!
= 252 possible divisions.
Example 2:
10 children divide themselves into two teams of 5 each.
Solution:
10!/(5!5!)
There are
2!
= 126 possible divisions, because the order is irrelevant; it is just a division,
no A and B team.
Statistical Computing (JBM050)
Semester 2, 2020-2021
Statistical Computing
Introduction 3
Recap mathematical models (distributions) 3
Multinomial coef cients 3
Basic Concepts 5
The Normal (Gaussian) distribution 5
Sample statistics as estimators of population parameters 5
Sampling distribution 5
Accuracy of Sample Statistics 6
Accuracy of Sample Statistics 6
MSE and BVT 6
t-statistic and Student-t distribution 6
Sampling distribution of the proportion for binary data 7
Monte Carlo Simulation 8
Quality control 8
General form of MC simulation 8
Simulations for properties of hypothesis tests 9
Optimization I and Statistical Learning 10
Introduction 10
Notation 10
Work ow of statistical modeling 11
Optimization II 13
Optimization basics 13
(Simple) Linear Regression 14
A quick recap 14
Univariate regression models 14
Estimating model parameters 15
Limits of (simple) linear regression 15
Multiple linear regression 16
Estimating the regression model 16
Predictive Model Accuracy 17
Model accuracy 17
Predictive modeling 17
Test set 18
The collinearity paradox 19
Model selection 19
Cross-Validation 20
Validation set approach 20
, LOOCV 20
K-fold CV 21
Train-Validation-Test 21
CV with non-CV model selection 21
Nested CV 22
Logistic Regression 23
The data generating model 23
Estimation 23
Iterative Optimization 24
De ning properties 24
Convergence 24
Bisection method 25
Newton’s method 26
Constrained Optimization 28
Equality constraints 28
Inequality constraints 29
Penalized Optimization 30
Penalized optimization 30
Ridge regression 30
Lasso regression 31
,Introduction
In short:
• Recap mathematical models (distributions)
• Multinomial coef cients
Recap mathematical models (distributions)
There are di erent kinds of mathematical models (distributions):
• Bernoulli distribution
X ∼ Ber n(π) where π ≠ 3.14... but p
P(X = 1) = π ; P(X = 0) = 1 − π
• Binomial distribution
X ∼ Bin(n, π)
P(X = k) = ( ) π k (1 − π)n−k
n
k
• Hypergeometric distribution
X ∼ hypergeom et r ic(N, K, n)
( k )( n − k )
K N−K
P(X = k) = where N= total elements to draw
(n)
N
K= nr. of elements with desired feature
n= nr. of draws
k = nr. of elements with feature in draws
๏ this formula holds when drawn without replacement
Note: this is just a brief recap of the di erent distributions, more elaborate descriptions can be found
in the Data Statistics summary. It is advised to look back into the Data Statistics material if subjects
in this summary are unclear.
Multinomial coef cients
This is used when a set of n distinct items needs to be divided into r distinct groups of
respective sizes n1, n 2, …, nr where ∑i=1 ni = n.
( n2 )
n − n1
So, there are (n ) choices for the rst group,
n
for the second group and so on. It
1
n!
then follows that there are possible divisions.
n1 !n 2 !…nr !
, Example 1:
10 children are to be divided in an A team and a B team of 5 each. They will play in a di erent
league.
Solution:
10!
There are
5!5!
= 252 possible divisions.
Example 2:
10 children divide themselves into two teams of 5 each.
Solution:
10!/(5!5!)
There are
2!
= 126 possible divisions, because the order is irrelevant; it is just a division,
no A and B team.
