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Time Series Analysis -- An Introduction --
1
AMS 586
Objectives of time series analysis
Data description
Data interpretation
Modeling
Control
Prediction & Forecasting
2
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
3
Time Plot
• the vertical axis measures the variable of interest
• the horizontal axis corresponds to the time periods
U.S. Inflation Rate
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
19
75
19
77
19
79
19
81
19
83
19
85
19
87
19
89
19
91
19
93
19
95
19
97
19
99
20
01
Year
Infl
ati
on
Rate
(%
)
A time-series plot (time plot) is a two-dimensional plot of time series data
4
Time-Series Components
Time Series
Cyclical Component
Irregular /Random Component
Trend Component Seasonal Component
Overall, persistent, long-term movement
Regular periodic fluctuations,
usually within a 12-month period
Repeating swings or movements over more than
one year
Erratic or residual fluctuations
5
Trend Component
• Long-run increase or decrease over time (overall upward or downward movement)
• Data taken over a long period of time
Sales
Time 6
Downward linear trend
Trend Component
• Trend can be upward or downward
• Trend can be linear or non-linear
Sales
Time Upward nonlinear trend
Sales
Time
(continued)
7
Seasonal Component
• Short-term regular wave-like patterns • Observed within 1 year • Often monthly or quarterly
Sales
Time (Quarterly)
Winter
Spring
Summer
Fall
Winter
Spring
Summer
Fall
8
Cyclical Component
• Long-term wave-like patterns
• Regularly occur but may vary in length
• Often measured peak to peak or trough to trough
Sales
1 Cycle
Year 9
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
10
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.
11
odelsMeasonal s-Non with Trend
trend Stochastic process
random noise
Xt = mt + Yt
12
odelMecomposition DClassical with Trend and Season
seasonal component
trend Stochastic process
random noise
Xt = mt + st + Yt
13
odelsMeasonal s-Non with Trend
14
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
Method 1: Trend Estimation by Regression Analysis
Estimate a trend line using regression analysis
Year
Time Period
(t)
Sales (X)
2004
2005
2006
2007
2008
2009
0
1
2
3
4
5
20
40
30
50
70
65
Use time (t) as the independent variable:
In least squares linear, non-linear, and exponential modeling, time periods are numbered starting with 0 and increasing by 1 for each time period.
15
Least Squares Regression
The estimated linear trend equation is:
Sales trend
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6
Year
sale
s
Year
Time
Period
(t)
Sales
(X)
2004
2005
2006
2007
2008
2009
0
1
2
3
4
5
20
40
30
50
70
65
Without knowing the exact time series random error correlation structure, one often resorts to the ordinary least squares regression method, not optimal but practical.
16
Linear Trend Forecasting
• Forecast for time period 6 (2010):
Year
Time
Period
(t)
Sales
(X)
2004
2005
2006
2007
2008
2009
2010
0
1
2
3
4
5
6
20
40
30
50
70
65
??
One can even performs trend forecasting at this point – but bear in mind that the forecasting may not be optimal.
Sales trend
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6
Year
sale
s
17
Method 2: Trend Elimination by Differencing
18
Trend Elimination by Differencing
19
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:
odelMecomposition DClassical (Seasonal Model)
with trend and season
20
where
odel Mecomposition DClassical
21
we remove the seasonal First Differencing:: 2Method component by differencing. We then remove the trend by differencing as well.
can fit a Alternatively, wefit method: -Joint: 3Method combined polynomial linear regression and harmonic functions to estimate and then remove the trend and seasonal component simultaneously as the following:
we estimate and remove the First Filtering:: 1Method trend component by using moving average method; then we estimate and remove the seasonal component by using suitable periodic averages.
Method 1: Filtering
22
(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 purposes.
Method 2: Differencing
23
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.
Method 3: Joint Modeling
24
As shown before, one can also fit a joint model to analyze both components simultaneously:
Detrended series
P. J. Brockwell, R. A. Davis, Introduction to Time Series and Forecasting, Springer, 1987 25
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.
26
White noise - example of a time series model
27
Gaussian white noise
28
Stochastic properties of the process
STATIONARITY
.1Once we have removed the seasonal and trend 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
29
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.
30
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
31
Autocorrelation function (ACF)
32
ACF for Gaussian WN
33
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.
34
Examples
35
Exponential decay of ACF
MA(1) sample ACF
AR(1)
36
More examples of ACF
37
Reference
38
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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