advanced engineering mathematics – mathematics

Look for trends, seasonal components, step changes, outliers... Objectives of time series analysis.[r]

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Introduction to Time Series Analysis Lecture 1.

Peter Bartlett

1 Organizational issues

2 Objectives of time series analysis Examples Overview of the course

4 Time series models

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Organizational Issues

• Peter Bartlett bartlett@stat Office hours: Tue 11-12, Thu 10-11

(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,

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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 and November 9, at the lecture

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A Time Series

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A Time Series

19600 1965 1970 1975 1980 1985 1990 50

100 150 200 250 300 350 400

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A Time Series

100 150 200 250 300 350 400

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A Time Series

19600 1965 1970 1975 1980 1985 1990 50

100 150 200 250 300 350 400

year

$

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A Time Series

260 280 300 320 340

$

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A Time Series

240 250 260 270 280 290 300 310

5 10 15 20 25 30

$

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A Time Series

260 280 300 320 340

$

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Objectives of Time Series Analysis

1 Compact description of data Interpretation

3 Forecasting Control

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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)

4 10 12x 10

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Transformed data

0 10 20 30 40 50 60 70 80 90

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Trend

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Residuals

0 10 20 30 40 50 60 70 80 90

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Trend and seasonal variation

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1 Compact description of data

Example: Classical decomposition: Xt = Tt + St + Yt

2 Interpretation Example: Seasonal adjustment

3 Forecasting Example: Predict sales

4 Control

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Unemployment data

Monthly number of unemployed people in Australia (Hipel and McLeod, 1994)

5.5 6.5 7.5

8x 10

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Trend

19834 1984 1985 1986 1987 1988 1989 1990 4.5

5 5.5 6.5 7.5

8x 10

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Trend plus seasonal variation

5 5.5 6.5 7.5

8x 10

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Residuals

1983 1984 1985 1986 1987 1988 1989 1990 −6

−4 −2 8x 10

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Predictions based on a (simulated) variable

5 5.5 6.5 7.5

8x 10

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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

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Overview of the Course

1 Time series models Time domain methods Spectral analysis

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1 Time series models (a) Stationarity

(b) Autocorrelation function (c) Transforming to stationarity Time domain methods

3 Spectral analysis

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1 Time series models Time domain methods

(a) AR/MA/ARMA models

(b) ACF and partial autocorrelation function (c) Forecasting

(d) Parameter estimation

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(a) Spectral density (b) Periodogram

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4 State space models(?) (a) ARMAX models

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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 and x1, , xt

Notation:

X1, X2, is a stochastic process

x1, x2, is a single realization

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Example: White noise: Xt ∼ W N(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]

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Gaussian white noise

P[Xt ≤ xt] = Φ(xt) = √1 2π

Z xt

−∞

e−x2/2 dx

0 10 15 20 25 30 35 40 45 50

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Gaussian white noise

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Example: Binary i.i.d P[Xt = 1] = P[Xt = −1] = 1/2

0 10 15 20 25 30 35 40 45 50

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Random walk

St = Pti=1 Xi Differences: ∇St = St − St−1 = Xt

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Random walk

ESt? VarSt?

0 10 15 20 25 30 35 40 45 50

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Random Walk

Recall S&P500 data (Notice that it’s smooth)

260 280 300 320 340

$

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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 10 year $

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Trend and Seasonal Models

Xt = Tt + St + Et = β0 + β1t +

P

i (βi cos(λit) + γi sin(λit)) + Et

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Xt = Tt + Et = β0 + β1t + Et

0 50 100 150 200 250

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Xt = Tt + St + Et = β0 + β1t +

P

i (βi cos(λit) + γi sin(λit)) + Et

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Trend and Seasonal Models: Residuals

0 50 100 150 200 250

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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 are stationary.

(a) Estimate and subtract Tt, St

(b) Differencing

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Nonlinear transformations

Recall: Monthly sales (Makridakis, Wheelwright and Hyndman, 1998)

0 10 20 30 40 50 60 70 80 90 10 12x 10

4

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(b) Differencing

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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 1987.5 220 240 260 280 300 320 340 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 10 year $

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Differencing and Trend

Define the lag-1 difference operator, (think ‘first derivative’)

∇Xt = Xt − Xt−1 = (1 − B)Xt,

where B is the backshift operator, BXt = Xt−1

• If Xt = β0 + β1t + Yt, then

∇Xt = β1 + ∇Yt

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Differencing and Seasonal Variation

Define the lag-s difference operator,

∇sXt = Xt − Xt−s = (1 − B

s)X

t,

where Bs is the backshift operator applied s times, BsXt = B(Bs−1

Xt)

and B1Xt = BXt

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(b) Differencing

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Outline

1 Objectives of time series analysis Examples Overview of the course

3 Time series models

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