Time Series Analysis
-- An Introduction --
AMS 586
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Objectives of time series analysis
Data description
Data interpretation
Modeling
Control
Prediction & Forecasting
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Time-Series Data
• Numerical data obtained at regular time
intervals
• The time intervals can be annually, quarterly,
monthly, weekly, daily, hourly, etc.
• Example:
Year: 2005 2006 2007 2008 2009
Sales: 75.3 74.2 78.5 79.7 80.2
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Time Plot
A time-series plot (time plot) is a two-
dimensional plot of time series data
• the vertical axis U.S. Inflation Rate
measures the variable of 16.00
interest 14.00
Inflation Rate (%)
12.00
10.00
• the horizontal axis
8.00
6.00
corresponds to the time 4.00
2.00
periods 0.00
1975
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
Year
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Time-Series Components
Time Series
Trend Component Seasonal Cyclical Irregular /Random
Component Component Component
Overall, Regular periodic Repeating swings Erratic or residual
persistent, long- fluctuations, or movements fluctuations
term movement usually within a over more than
12-month period one year
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Trend Component
• Long-run increase or decrease over time
(overall upward or downward movement)
• Data taken over a long period of time
Sales
6 Time
Trend Component
(continued)
• Trend can be upward or downward
• Trend can be linear or non-linear
Sales Sales
Time Time
Downward linear trend Upward nonlinear trend
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Seasonal Component
• Short-term regular wave-like patterns
• Observed within 1 year
• Often monthly or quarterly
Sales
Summer
Winter
Summer
Winter Spring Fall
Spring Fall
Time (Quarterly)
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Cyclical Component
• Long-term wave-like patterns
• Regularly occur but may vary in length
• Often measured peak to peak or trough to
trough 1 Cycle
Sales
Year
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Irregular/Random Component
• Unpredictable, random, “residual” fluctuations
• “Noise” in the time series
• The truly irregular component may not be
estimated – however, the more predictable
random component can be estimated – and is
usually the emphasis of time series analysis via
the usual stationary time series models such as
AR, MA, ARMA etc after we filter out the trend,
seasonal and other cyclical components
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Two simplified time series models
• In the following, we present two classes of
simplified time series models
1. Non-seasonal Model with Trend
2. Classical Decomposition Model with Trend and
Seasonal Components
• The usual procedure is to first filter out the trend
and seasonal component – then fit the random
component with a stationary time series model to
capture the correlation structure in the time series
• If necessary, the entire time series (with seasonal,
trend, and random components) can be re-analyzed
for better estimation, modeling and prediction.
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Non-seasonal Models
with Trend
Xt = mt + Yt
Stochastic trend random
process noise
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Classical Decomposition Model
with Trend and Season
Xt = mt + st + Yt
Stochastic trend seasonal random
process component noise
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Non-seasonal Models
with Trend
There are two basic methods for
estimating/eliminating trend:
Method 1: Trend estimation
(first we estimate the trend either by
moving average smoothing or regression analysis –
then we remove it)
Method 2: Trend elimination by differencing
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Method 1: Trend Estimation by
Regression Analysis
Estimate a trend line using regression analysis
Time Use time (t) as the
Period Sales independent variable:
Year (X)
(t)
2004 0 20
2005 1 40
2006 2 30 In least squares linear, non-linear, and
2007 3 50 exponential modeling, time periods are
numbered starting with 0 and increasing
2008 4 70 by 1 for each time period.
2009 5 65
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Least Squares Regression
Without knowing the exact time series random error correlation structure,
one often resorts to the ordinary least squares regression method, not
optimal but practical. The estimated linear trend equation is:
Time
Year Period Sales
(t) (X)
Sales trend
2004 0 20
80
2005 1 40 70
60
2006 2 30 50
sales
40
2007 3 50 30
20
2008 4 70 10
0
2009 5 65
0 1 2 3 4 5 6
Year
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Linear Trend Forecasting
One can even performs trend forecasting at this point – but bear in mind
that the forecasting may not be optimal.
• Forecast for time period 6 (2010):
Time
Year Period Sales
(t) (X)
2004 0 20 Sales trend
2005 1 40 80
70
2006 2 30
60
2007 3 50 50
sales
40
2008 4 70 30
20
2009 5 65
10
2010 6 ?? 0
0 1 2 3 4 5 6
Year
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Method 2: Trend Elimination by
Differencing
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Trend Elimination by Differencing
If the operator ∇ is applied to a linear trend function:
Then we obtain the constant function:
In the same way any polynomial trend of degree k can
be removed by the operator:
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Classical Decomposition Model
(Seasonal Model)
with trend and season
where
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Classical Decomposition Model
Method 1: Filtering: First we estimate and remove the
trend component by using moving average method; then
we estimate and remove the seasonal component by
using suitable periodic averages.
Method 2: Differencing: First we remove the seasonal
component by differencing. We then remove the
trend by differencing as well.
Method 3: Joint-fit method: Alternatively, we can fit a
combined polynomial linear regression and harmonic
functions to estimate and then remove the trend and
seasonal component simultaneously as the following:
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Method 1: Filtering
(1). We first estimate the trend by the moving average:
• If d = 2q (even), we use:
• If d = 2q+1 (odd), we use:
(2). Then we estimate the seasonal component by using the average
, k = 1, …, d, of the de-trended data:
To ensure:
we further subtract the mean of
(3). One can also re-analyze the trend from the de-seasonalized data in
order to obtain a polynomial linear regression equation for modeling and
prediction
22 purposes.
Method 2: Differencing
Define the lag-d differencing operator as:
We can transform a seasonal model to a non-seasonal model:
Differencing method can then be further applied to
eliminate the trend component.
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Method 3: Joint Modeling
As shown before, one can also fit a joint model
to analyze both components simultaneously:
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Detrended series
25 P. J. Brockwell, R. A. Davis, Introduction to Time Series and Forecasting, Springer, 1987
Time series – Realization of a
stochastic process
{Xt} is a stochastic time series if each component
takes a value according to a certain probability
distribution function.
A time series model specifies the joint distribution
of the sequence of random variables.
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White noise - example of a time
series model
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Gaussian white noise
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Stochastic properties of the process
STATIONARITY
Once we have removed the seasonal and trend .1
components of a time series (as in the classical
decomposition model), the remainder (random)
component – the residual, can often be modeled by a
stationary time series.
* System does not change its properties in time
* Well-developed analytical methods of signal analysis
and stochastic processes
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WHEN A STOCHASTIC PROCESS IS STATIONARY?
{Xt} is a strictly stationary time series if
f(X1,...,Xn)= f(X1+h,...,Xn+h),
where n1, h – integer
Properties:
* The random variables are identically distributed.
* An idependent identically distributed (iid) sequence
is strictly stationary.
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Weak stationarity
{Xt} is a weakly stationary time series if
EXt = and Var(Xt) = 2 are independent of time t •
Cov(Xs, Xr) depends on (s-r) only, independent of t •
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Autocorrelation function (ACF)
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ACF for Gaussian WN
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ARMA models
Time series is an ARMA(p,q) process if Xt is stationary and if
for every t:
Xt 1Xt-1 ... pXt-p= Zt + 1Zt-1 +...+ pZt-p
where Zt represents white noise with mean 0 and variance 2
The Left side of the equation represents the Autoregressive
AR(p) part, and the right side the Moving Average MA(q)
component.
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Examples
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Exponential decay of ACF
MA(1) AR(1)
sample ACF
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More examples of ACF
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Reference
Box, George and Jenkins, Gwilym (1970) Time series analysis:
Forecasting and control, San Francisco: Holden-Day.
Brockwell, Peter J. and Davis, Richard A. (1991). Time Series:
Theory and Methods. Springer-Verlag.
Brockwell, Peter J. and Davis, Richard A. (1987, 2002).
Introduction to Time Series and Forecasting. Springer.
We also thank various on-line open resources for time series
analysis.
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