关键词 > ORBS7260/MFFM7040
ORBS7260/MFFM7040 Applied Time Series / Times Series Analysis Semester One, 2022/23
发布时间：2022-10-19
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Department of Mathematics
Assignment 1
Applied Time Series / Times Series Analysis
ORBS7260/MFFM7040
Semester One, 2022/23
Question 1
The following table shows the annual sales of a small yacht company (Xt) of a certain city from year 2010 to 2019.
|
|
One-step ahead forecast |
Squared Error |
||||||
|
Year |
t |
Xt |
3MA |
SES |
Naïve |
3MA |
SES |
Naïve |
|
|
0 |
|
|
|
|
|
|
|
|
2010 |
1 |
15 |
|
|
|
|
|
|
|
2011 |
2 |
18 |
|
|
|
|
|
|
|
2012 |
3 |
19 |
|
|
|
|
|
|
|
2013 |
4 |
21 |
|
|
|
|
|
|
|
2014 |
5 |
a |
|
|
|
|
|
|
|
2015 |
6 |
b |
19 |
18.1622 |
|
|
|
|
|
2016 |
7 |
c |
19 |
g |
|
|
|
|
|
2017 |
8 |
d |
f |
h |
|
|
|
|
|
2018 |
9 |
e |
18.3333 |
18.1310 |
|
|
|
|
|
2019 |
10 |
19 |
18 |
18.1048 |
|
|
|
|
The owner of the yacht company is going to perform an analysis for the demand in the past decade. The owner believes that the three-point moving average (3MA) model and the Simple Exponential Smoothing (SES) method with a smoothing parameter α are appropriate to
calculate the one-step ahead forecast for years 2015 to 2019.
(a) Determine the unknown values a, b, and e.
(b) Determine the value of the smoothing parameter α for the Simple Exponential Smoothing (SES) method. Round your answer in two decimal places.
(c) Using the result obtained in (b), determine the values of c, d,f, g and h.
(d) By using Theil’s U statistics for the period 2015 to 2019 only, compare the performance for these one-step ahead forecasts of the two methods.
Question 2
Consider a linear trend process,
yt 
b1 
b2 t 
t , where 
N 
0, 
2 
, b1 
0, b2 
0
It is known that DES model with exponential smoothing parameter α is used to forecast the process. The DES algorithm is as follows:
St 
Xt 
(1 
)St
1 , St[ 2 ] 
St 
(1 
)S
a(t) 
2St 
St[ 2 ] , b(t) 
St 
St[ 2 ] 
.
Show that E(ST [2] ) 
E(ST ) 
b2 .
(Note: You can make use of the result E 
ST 
E 
yT 
1 
b2 if necessary)
Question 3
Assume 
at 
is white noise process, which is a sequence of identical independent distributed
random variables with zero mean and constant variance 
.
(a) Evaluate the mean, variance and covariance function for the following processes. Determine the stationarity for each process.
(i) Yt 
tat
(ii) Wt 
Yt 
Yt 
Yt 
1
(b) Consider a stationary moving average process, Xt 
2 
at 
at
1 where 
. Show
that the maximum value of the first order autocorrelation coefficient for the process is 0.5.
