Bernoulli distribution (from http:/www.math.wm.edu/leemis/chart/UDR/UDR.html)
The shorthand X Bernoulli(p) is used to indicate that the random variable X has the Bernoulli
distribution with parameter p, where 0 < p < 1. A Bernoulli random variable X with
Stat 588
FSRM 588 SOLUTIONS
HW1
1. Problem 1
QR decomposition procedure for forward stepwise
denote (x, y) as the inner product of x and y, suppose the orthogonal basis for the column
space spanned by X1 is z1 , z2 , .zq ,in the additional pq predictors i
FSRM 588 SOLUTIONS
HW3
Problem1
Define X = (X1T , X2T , .XpT )
= V ar(X)
Denote the eigen values of as 1 , 2 , .p , and they are arraged in decreasing way.
1
0
= (v1 , .vp )
.
0
0
2
.
0
0 . 0
0 . 0
(v , .vp )0
. . . 1
. 0 p
Now lets show this propos
FSRM 588 SOLUTIONS
HW4
1000
800
600
number of stocks
1200
1400
Problem1
a.
Note: the return here is scaled in percentage
Here is the stock numbers with desired properties for each year.
1975
1980
1985
1990
1995
2000
2005
year
Figure 1. number of stocks wi
FSRM 588 SOLUTIONS
HW2
Problem1:
(a) First, lets derive the exact solution of scad estimate in the cononical setting: Define
1 ls
Q(j ) = (j j )2 p (|j |)
2
In order to maximize Q(j ),take derivarive w.r.t j , we could get the following equation:
0
ls
j +
FSRM 588 SOLUTIONS
HW6
Problem1:
a
comment:
Without weight decay=0, overfitting becomes more severe for large nnumber of hidden units.
No Weight Decay
105
100
95
Mean Square Test Error
110
115
90
1
2
3
4
5
6
7
8
9
10
11
12
Number of Hidden Units
b
comment
FSRM 563
Homework #1
Due Thursday, Sept 29, 2016
(I) Correlation
1. Simulate 2,000 independent sets of i.i.d. standard Normal random numbers, with 50 observations in each set. Find the maximum absolute correlation coecient among all pairs of sets
of your
Columbia University: Financial Engineering and Risk Management
M. Haugh & G. Iyengar
Assignment 1 Solutions
Assignment #1 Solutions
1. Lottery Payments
A major lottery advertises that it pays the winner $10 million. However, this prize
money is paid at th
Columbia University: Financial Engineering and Risk Management
M. Haugh & G. Iyengar
Assignment #1 Solutions
1. Lottery Payments
A major lottery advertises that it pays the winner $10 million. However, this prize
money is paid at the rate of $500,000 each
Financial Data Mining
FSRM588
Lecture 01: Introduction and Linear Regression
Department of Statistics & Biostatistics
Rutgers University
September 03 2013
Outline
1
Introduction
2
Linear Regression
What is Data Mining and Machine Learning?
Background: rap
Wharton Research Data Services
http:/wrds-web.wharton.upenn.edu/wrds/
Class account.
Username: fsrm2012
Password: Fsrm112358
Financial Data Mining
FSRM588
Lecture 02: Shrinkage Methods
Department of Statistics & Biostatistics
Rutgers University
September
NOTES 02
HAN XIAO
1. Review of Convex Analysis
Denition 1.1. Let C be a subset of Rn . We say C is convex if
x + (1 )y C,
x, y C, [0, 1].
n
Denition 1.2. Let C be a convex subset of R . A function f : C R is called convex if
f (x + (1 )y ) f (x) + (1 )f
Financial Data Mining
FSRM588
Lecture 03: Penalized Least Squares and Penalized Likelihood
Department of Statistics & Biostatistics
Rutgers University
September 17 2013
Lasso
Lasso solves the following optimization problem,
2
p
p
1N
yi 0
lasso = arg max
PENALIZED LEAST SQUARES AND PENALIZED LIKELIHOOD
HAN XIAO
1. Penalized Least Squares
Lasso solves the following optimization problem,
2
p
p
1N
lasso = arg max
yi 0
xij j
| j |
Rp+1 2N
i=1
j =1
j =1
(1.1)
for some 0. We can use some other penalty on th
Financial Data Mining
FSRM588
Lecture 04: Penalized Least Squares & Principal Component Analysis
Department of Statistics & Biostatistics
Rutgers University
October 01 2013
Outline
1
Penalized Least Squares
2
Principal Component Analysis
Lasso
Lasso solve
PRINCIPAL COMPONENT ANALYSIS AND FACTOR ANALYSIS
HAN XIAO
1. Principal Component Analysis
1.1. Singular value decomposition. Let A Rmn be a m n matrix with m n. It can be written as
A = U DV ;
mn
(1.1)
nn
where U R
is an orthogonal matrix, i.e. its column
FSRM588
Homework 1 (Due in class on Oct 01)
Fall 2013
Notes. For the simulation and data analysis problems, keep the code you develop as you may be asked
to present your work later.
1. Forward stepwise regression. Suppose we have the QR decomposition for