Chapter 3
Spectral Representation theorem for discrete
time stationary processes
Spectral analysis is a study of the frequency domain characteristics of a process, and
describes the contribution of each frequency to the variance of the process.
Let us def
Chapter 5
A nave non-parametric spectral estimator the periodogram
Suppose the zero mean discrete stationary process cfw_Xt has a purely continuous
spectrum with sdf S(f ). We have,
1
X
S(f ) =
s e
1
|f | .
2
i2f
= 1
With = 0, we can use the biased esti
S8 Time Series Analysis
Andrew Walden
Room: 531 Huxley
email: a.walden@imperial.ac.uk
Check and use Blackboard to obtain all course resources.
Department of Mathematics
Imperial College London
180 Queens Gate, London SW7 2BZ
1
2
Chapter 1
Introduction
Tim
(3) Either of the following definitions. are appropriate for full marks. A counting
Process. {Nthzm is a Poisson Process of rate A > 0 if
1. N9 = 0.
2. The increments are independent, that is. for any khkg 6 2+. 0 < s < t
P({N¢ — N5 =Iki}l{N, = k2,0 g r S
M3/M4 j
54
Imperial College -
London
BSc, MSci and MSc EXAMINATIONS (MATHEMATICS)
May - June 2011
This paperl is also taken for the relevant examination for the Associateship of the
Royal College of Science.
Applied Probability
Date: Friday, 20 May 2011.
(i) (a) I: P(exactly 1 event occurs in (t,t + 6t]) = Mt + 0(6t). seen u
[o(6t)/§t -) 0 as 6t —> 0].
II: P(2 or more events occur in (t,t+ 6t]) = 0(6t).
III: Occurrence of events after time t is independent of occurrence of events
before t. E
(b) Let puff.
'.'._.d._._ _._ ._L._._.'._
Marks 84
seen / unseen
(i) Either of the following deﬁnitions, are appropriate for full marks. A counting
Process, {N;}120, is a Poisson Process of rate A > 0 if
(a) N0 = D.
(b) The increments are independent, that is,
BSc and MSci EXAMINATIONS (MATHEMATICS)
May-June 2009
This paper is also taken for the relevant examination for the Associateship.
M3S4/M4S4
Applied Probability
Date:
Wednesday, 3rd June 2009
Time:
10 am 12 pm
Credit will be given for all questions attemp
M1M1: The Riemann Integral and the Fundamental Theorem of Calculus
This sheet can be found on http:/www.ma.ic.ac.uk/ajm8/M1M1
Given an interval [a, b] we define a partition to be a set of n points x1 , x2 . . . xn such that
a x0 < x1 < x2 < x3 < . . . < x
M1M1 Handout 1: Properties of the Trigonometric Power Series
This sheet can be found on http:/www.ma.ic.ac.uk/ajm8/M1M1
Proof that cos x has a zero in the interval (1.4, 1.6).
Let us denote by coz x and zin x the functions defined by the infinite power se
M1F Foundations of Analysis
Problem Sheet 5
1. Professor Mestel is furious. That quadratic equation was on his desk when he
went to lunch, and now he cant find it anywhere. He can remember its roots
and but not the equation. He tries to cheer himself up
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 9
The question for discussion with the tutor is Question 6.
1. Quaternions are 4-dimensional vectors (x1 , x2 , x3 , x4 ) R4 , with the
usual addition and multiplication by scalars. We write such a vec
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 8
Notation: when A, B are square matrices of the same size we write [A, B]
for AB BA. The trace of a square matrix A is dened as the sum of
diagonal entries tr(A) = a11 + . . . + ann . An n n matrix A
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 7
1. (i) Find the (vector) equation of the line joining the points (3, 1, 2)
and (6, 1, 8). Find the equation of the plane containing this line and the
point (0, 2, 1).
(ii) Find the equation of the pl
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 6
The question for discussion with the personal tutor is Question 2.
1. Decide which type of conic is given by each of the following equations
by reducing the equation to standard form.
(i) x1 x2 = 2
(
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 3
The question for the discussion with the personal tutor is Question 5
2
3
1. Let A =
1
(i) Solve the
1 3
x1
and x = x2 .
1 4
x3
1 2
system Ax = 0.
1
(ii) Let b = 2 . Find all solutions of the sy
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 5
The question for discussion
1 2 3
0 1 2
1. (a) Calculate det
3 0 1
2 3 0
with the personal tutor is Question 6
0
3
.
2
1
t1
3
3
(b) Solve for t the equation det 3 t + 5 3 = 0.
6
6
t4
a
bx cx
(c) Sol
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 4
The question for discussion with the personal tutor is Question 1
1*. Let A and B be n n matrices with real entries. For each of the
following statements, either give a proof or nd a counterexample.
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 1
1. Let a, b be the points (1, 2), (2, 5). Find
(i) in vector form, the line L through a and b
(ii) a vector perpendicular to L
(iii) a scalar R such that (, 6) lies on L
(iv) the point of intersectio
Formalities, tips and reporting description for Introduction to Statistics 2nd project related to
Skive fjord.
In continuation of project 1 data from Skive fjord is used for a linear regression analysis in the second
project.
The assignment is formulated