Leung Hoang (lghoang@gmail) Graham Laverty ([email protected])
Statistics 349.3(02) Analysis of Time Series. Course Information 1.Instructor: W. H. Laverty 235...
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Transcript of Statistics 349.3(02) Analysis of Time Series. Course Information 1.Instructor: W. H. Laverty 235...
Statistics 349.3(02)
Analysis of Time Series
Course Information
1. Instructor:• W. H. Laverty
• 235 McLean Hall
• Tel: 966-6096 Email: [email protected]
2. Class Times• MWF 12:30-1:20pm AGRI 1E69
3. Mark break-up• Final Exam – 60%
• Term Tests and Assignments – 40%
Course Outline
1. Introduction and Review of Probability Theory
– Probability distributions, Expectation, Variance, correlation
– Sampling distributions
– Estimation Theory
– Hypothesis Testing
Course Outline -continued
2. Fundamental concepts in Time Series Analysis– Stationarity– Autocovariance function, autocorrelation and partial
autocorrelation function
3. Models for Stationary Time series– Autoregressive (AR), Moving average (MA), mixed
Autoregressive-Moving average (ARMA) models
4. Models for Non-stationary Time series– ARIMA (Integrated Autoregressive-Moving average
models)
Course Outline -continued
5. Forecasting ARIMA Processes
6. Model Identification and Estimation
7. Models for seasonal Time series
8. Spectral Analysis of Time series
Course Outline -continued
9. State-Space modeling of time series, Hidden Markov Model (HMM)
– Kalman filtering
10. Multivariate (Multi-channel) time series analysis
11. Linear filtering
Comments
• The Text Book– Time Series Analysis: Univariate And
Multivariate Methods Wei (Paperback - 2006 Ed. 02) Recommended
• I will present the material in power point slides that will be available on my web site.
• Hopefully this makes purchasing the text book optional. It is useful for students to purchase text books and build up a library.
Some examples
of Time series data
IBM Stock Price (closing)day price
1 4602 4573 4524 4595 4626 4597 4638 4799 493
10 49011 49212 49813 49914 49715 49616 49017 48918 47819 48720 49121 48722 48223 48724 48225 47926 47827 47928 47729 479
340 364341 369342 355343 350344 353345 340346 350347 349348 358349 360350 360351 366352 359353 356354 355355 367356 357357 361358 355359 348360 343361 330362 340363 339364 331365 345366 352367 346368 352369 357
Time Sequence plot
price
0
100
200
300
400
500
600
700
0 100 200 300 400
Concentration of residue after production of a chemical
run conc1 172 16.63 16.34 16.15 17.16 16.97 16.88 17.49 17.1
10 1711 16.712 17.413 17.214 17.415 17.416 1717 17.318 17.219 17.420 16.821 17.122 17.423 17.424 17.525 17.426 17.627 17.428 17.3
168 17169 16.7170 16.9171 17.3172 17.8173 17.8174 17.6175 17.5176 17177 16.9178 17.1179 17.2180 17.4181 17.5182 17.9183 17184 17185 17186 17.2187 17.3188 17.4189 17.4190 17191 18192 18.2193 17.6194 17.8195 17.7196 17.2197 17.4
Time Sequence plot
conc
15.5
16
16.5
17
17.5
18
18.5
0 50 100 150 200
Yearly sunspot activity (1770 -1869)Year no
1770 1011771 821772 661773 351774 311775 71776 201777 921778 1541779 1251780 851781 681782 381783 231784 101785 241786 831787 1321788 1311789 1181790 901791 67
1847 981848 1241849 961850 661851 641852 541853 391854 211855 71856 41857 231858 551859 941860 961861 771862 591863 441864 471865 301866 161867 71868 371869 74
Time Sequence plot
0
20
40
60
80
100
120
140
160
180
1770 1790 1810 1830 1850 1870
Time Sequence Plots
ExampleMeasuring brain activity in an insect as an object is approaching. Time t = 0 is at the point of impact.
0
0.5
1
1.5
2
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Simulation
To aid in the understanding of time series it is useful to simulate data
Generating observations from a distribution
Generating a random number from a distribution
Let
1. f(x) denote the density function
2. F(x) denote the cumulative distribution function = P[X ≤ x]
3. F-1(x) denote the inverse cumulative distribution function
F(x) denote the cumulative distribution function
F(x)
f(x)
x
F(x) denote the cumulative distribution function
u
f(x)
F-1(u)
If u is chosen at random from 0 to 1 then x = F-1(u) is chosen at random from the density f(x).
In EXCEL the following function generates a random observation from a normal distribution.
= NORMINV(RAND(), mean, standard deviation)
ExampleRandom walkA random walk is a sequence of random variables {xt} satisfying:
xt = xt – 1 + ut
where {ut} is a sequence of independent random variables having mean 0, standard deviation . (usually normally distributed)
The excel functions 1. NORMINV(prob, mean, standard deviation) computes F-1(prob) for the
normal distribution.2. RAND() computes and random number from the Uniform distribution
from 0 to 1.3. NORMINV(RAND(), mean, standard deviation) computes and random
number from the Normal distribution with a given mean and standard deviation.