introduction to time series analysis. lecture 1
TRANSCRIPT
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Introduction to Time Series Analysis. Lecture 1.Peter Bartlett
1. Organizational issues.
2. Objectives of time series analysis. Examples.
3. Overview of the course.
4. Time series models.
5. Time series modelling: Chasing stationarity.
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Organizational Issues
• Peter Bartlett. bartlett@stat. Office hours: Tue 11-12, Thu10-11
(Evans 399).
• Joe Neeman. jneeman@stat. Office hours: Wed 1:30–2:30, Fri 2-3
(Evans ???).
• http://www.stat.berkeley.edu/∼bartlett/courses/153-fall2010/
Check it for announcements, assignments, slides, ...
• Text: Time Series Analysis and its Applications. With R Examples,
Shumway and Stoffer. 2nd Edition. 2006.
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Organizational Issues
Classroom and Computer Lab Section: Friday 9–11, in 344 Evans.
Starting tomorrow, August 27:
Sign up for computer accounts. Introduction to R.
Assessment:Lab/Homework Assignments (25%): posted on the website.
These involve a mix of pen-and-paper and computer exercises. You may use
any programming language you choose (R, Splus, Matlab, python).
Midterm Exams (30%): scheduled for October 7 and November 9,at the
lecture.
Project (10%): Analysis of a data set that you choose.
Final Exam (35%): scheduled for Friday, December 17.
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A Time Series
0 1000 2000 3000 4000 5000 6000 70000
50
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A Time Series
1960 1965 1970 1975 1980 1985 19900
50
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150
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350
400
year
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A Time Series
1960 1965 1970 1975 1980 1985 19900
50
100
150
200
250
300
350
400
year
$
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A Time Series
1960 1965 1970 1975 1980 1985 19900
50
100
150
200
250
300
350
400
year
$
SP500: 1960−1990
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A Time Series
1987 1987.05 1987.1 1987.15 1987.2 1987.25 1987.3 1987.35 1987.4 1987.45 1987.5220
240
260
280
300
320
340
year
$
SP500: Jan−Jun 1987
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A Time Series
240 250 260 270 280 290 300 3100
5
10
15
20
25
30
$
SP500 Jan−Jun 1987. Histogram
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A Time Series
0 20 40 60 80 100 120220
240
260
280
300
320
340
$
SP500: Jan−Jun 1987. Permuted.
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Objectives of Time Series Analysis
1. Compact description of data.
2. Interpretation.
3. Forecasting.
4. Control.
5. Hypothesis testing.
6. Simulation.
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Classical decomposition: An example
Monthly sales for a souvenir shop at a beach resort town in Queensland.
(Makridakis, Wheelwright and Hyndman, 1998)
0 10 20 30 40 50 60 70 80 900
2
4
6
8
10
12x 10
4
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Transformed data
0 10 20 30 40 50 60 70 80 907
7.5
8
8.5
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10.5
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11.5
12
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Trend
0 10 20 30 40 50 60 70 80 907
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10.5
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Residuals
0 10 20 30 40 50 60 70 80 90−1
−0.5
0
0.5
1
1.5
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Trend and seasonal variation
0 10 20 30 40 50 60 70 80 907
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
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Objectives of Time Series Analysis
1. Compact description of data.
Example: Classical decomposition: Xt = Tt + St + Yt.
2. Interpretation. Example: Seasonal adjustment.
3. Forecasting. Example: Predict sales.
4. Control.
5. Hypothesis testing.
6. Simulation.
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Unemployment data
Monthly number of unemployed people in Australia.(Hipel and McLeod, 1994)
1983 1984 1985 1986 1987 1988 1989 19904
4.5
5
5.5
6
6.5
7
7.5
8x 10
5
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Trend
1983 1984 1985 1986 1987 1988 1989 19904
4.5
5
5.5
6
6.5
7
7.5
8x 10
5
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Trend plus seasonal variation
1983 1984 1985 1986 1987 1988 1989 19904
4.5
5
5.5
6
6.5
7
7.5
8x 10
5
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Residuals
1983 1984 1985 1986 1987 1988 1989 1990−6
−4
−2
0
2
4
6
8x 10
4
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Predictions based on a (simulated) variable
1983 1984 1985 1986 1987 1988 1989 19904
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5
5.5
6
6.5
7
7.5
8x 10
5
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Objectives of Time Series Analysis
1. Compact description of data:
Xt = Tt + St + f(Yt) + Wt.
2. Interpretation. Example: Seasonal adjustment.
3. Forecasting. Example: Predict unemployment.
4. Control. Example: Impact of monetary policy on unemployment.
5. Hypothesis testing. Example: Global warming.
6. Simulation. Example: Estimate probability of catastrophic events.
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Overview of the Course
1. Time series models
2. Time domain methods
3. Spectral analysis
4. State space models(?)
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Overview of the Course
1. Time series models
(a) Stationarity.
(b) Autocorrelation function.
(c) Transforming to stationarity.
2. Time domain methods
3. Spectral analysis
4. State space models(?)
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Overview of the Course
1. Time series models
2. Time domain methods
(a) AR/MA/ARMA models.
(b) ACF and partial autocorrelation function.
(c) Forecasting
(d) Parameter estimation
(e) ARIMA models/seasonal ARIMA models
3. Spectral analysis
4. State space models(?)
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Overview of the Course
1. Time series models
2. Time domain methods
3. Spectral analysis
(a) Spectral density
(b) Periodogram
(c) Spectral estimation
4. State space models(?)
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Overview of the Course
1. Time series models
2. Time domain methods
3. Spectral analysis
4. State space models(?)
(a) ARMAX models.
(b) Forecasting, Kalman filter.
(c) Parameter estimation.
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Time Series Models
A time series model specifies the joint distribution of the se-
quence{Xt} of random variables.
For example:
P [X1 ≤ x1, . . . , Xt ≤ xt] for all t andx1, . . . , xt.
Notation:
X1, X2, . . . is a stochastic process.
x1, x2, . . . is a single realization.
We’ll mostly restrict our attention tosecond-order properties only:
EXt, E(Xt1 , Xt2).
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Time Series Models
Example: White noise:Xt ∼ WN(0, σ2).
i.e.,{Xt} uncorrelated, EXt = 0, VarXt = σ2.
Example: i.i.d. noise:{Xt} independent and identically distributed.
P [X1 ≤ x1, . . . , Xt ≤ xt] = P [X1 ≤ x1] · · ·P [Xt ≤ xt].
Not interesting for forecasting:
P [Xt ≤ xt|X1, . . . , Xt−1] = P [Xt ≤ xt].
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Gaussian white noise
P [Xt ≤ xt] = Φ(xt) =1√2π
∫ xt
−∞
e−x2/2dx.
0 5 10 15 20 25 30 35 40 45 50−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
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Gaussian white noise
0 5 10 15 20 25 30 35 40 45 50−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
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Time Series Models
Example: Binary i.i.d. P [Xt = 1] = P [Xt = −1] = 1/2.
0 5 10 15 20 25 30 35 40 45 50
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
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Random walk
St =∑t
i=1Xi. Differences:∇St = St − St−1 = Xt.
0 5 10 15 20 25 30 35 40 45 50−4
−2
0
2
4
6
8
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Random walk
ESt? VarSt?
0 5 10 15 20 25 30 35 40 45 50−15
−10
−5
0
5
10
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Random Walk
Recall S&P500 data. (Notice that it’s smooth)
1987 1987.05 1987.1 1987.15 1987.2 1987.25 1987.3 1987.35 1987.4 1987.45 1987.5220
240
260
280
300
320
340
year
$
SP500: Jan−Jun 1987
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Random Walk
Differences: ∇St = St − St−1 = Xt.
1987 1987.05 1987.1 1987.15 1987.2 1987.25 1987.3 1987.35 1987.4 1987.45 1987.5−10
−8
−6
−4
−2
0
2
4
6
8
10
year
$
SP500, Jan−Jun 1987. first differences
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Trend and Seasonal Models
Xt = Tt + St + Et = β0 + β1t +∑
i (βi cos(λit) + γi sin(λit)) + Et
0 50 100 150 200 2502.5
3
3.5
4
4.5
5
5.5
6
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Trend and Seasonal Models
Xt = Tt + Et = β0 + β1t + Et
0 50 100 150 200 2502.5
3
3.5
4
4.5
5
5.5
6
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Trend and Seasonal Models
Xt = Tt + St + Et = β0 + β1t +∑
i (βi cos(λit) + γi sin(λit)) + Et
0 50 100 150 200 2502.5
3
3.5
4
4.5
5
5.5
6
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Trend and Seasonal Models: Residuals
0 50 100 150 200 250−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
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Time Series Modelling
1. Plot the time series.
Look for trends, seasonal components, step changes, outliers.
2. Transform data so that residuals arestationary.
(a) Estimate and subtractTt, St.
(b) Differencing.
(c) Nonlinear transformations (log,√·).
3. Fit model to residuals.
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Nonlinear transformations
Recall: Monthly sales.(Makridakis, Wheelwright and Hyndman, 1998)
0 10 20 30 40 50 60 70 80 900
2
4
6
8
10
12x 10
4
0 10 20 30 40 50 60 70 80 907
7.5
8
8.5
9
9.5
10
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11
11.5
12
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Time Series Modelling
1. Plot the time series.
Look for trends, seasonal components, step changes, outliers.
2. Transform data so that residuals arestationary.
(a) Estimate and subtractTt, St.
(b) Differencing.
(c) Nonlinear transformations (log,√·).
3. Fit model to residuals.
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Differencing
Recall: S&P 500 data.
1987 1987.05 1987.1 1987.15 1987.2 1987.25 1987.3 1987.35 1987.4 1987.45 1987.5220
240
260
280
300
320
340
year
$
SP500: Jan−Jun 1987
1987 1987.05 1987.1 1987.15 1987.2 1987.25 1987.3 1987.35 1987.4 1987.45 1987.5−10
−8
−6
−4
−2
0
2
4
6
8
10
year
$
SP500, Jan−Jun 1987. first differences
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Differencing and Trend
Define the lag-1difference operator, (think ‘first derivative’)
∇Xt = Xt − Xt−1 = (1 − B)Xt,
whereB is thebackshift operator,BXt = Xt−1.
• If Xt = β0 + β1t + Yt, then
∇Xt = β1 + ∇Yt.
• If Xt =∑k
i=0βit
i + Yt, then
∇kXt = k!βk + ∇kYt,
where∇kXt = ∇(∇k−1Xt) and∇1Xt = ∇Xt.
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Differencing and Seasonal Variation
Define the lag-sdifference operator,
∇sXt = Xt − Xt−s = (1 − Bs)Xt,
whereBs is the backshift operator applieds times,BsXt = B(Bs−1Xt)
andB1Xt = BXt.
If Xt = Tt + St + Yt, andSt has periods (that is,St = St−s for all t), then
∇sXt = Tt − Tt−s + ∇sYt.
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Time Series Modelling
1. Plot the time series.
Look for trends, seasonal components, step changes, outliers.
2. Transform data so that residuals arestationary.
(a) Estimate and subtractTt, St.
(b) Differencing.
(c) Nonlinear transformations (log,√·).
3. Fit model to residuals.
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Outline
1. Objectives of time series analysis. Examples.
2. Overview of the course.
3. Time series models.
4. Time series modelling: Chasing stationarity.
49