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Time Series
Presented by
Vikas Kumar vidyarthi
Ph.D Scholar (10203069),CE
Instructor
Dr. L. D. Behera
Department of Electrical Engineering
Indian institute of Technology Kanpur
Contents:-
• Correlation and Regression• What is Time Series?• Field of its Applications• Methods:
Autoregressive (AR) processMoving average (MA) processARMA process
• Example of input variable selection by ACF, CCF and PACF.
• Understanding
Correlation and Regression
• Correlation:
Measures the degree of association between two variable or two series and with what extent. It is measured by the correlation coefficient r.
• Regression:
Discovering how a dependent variable (y) is related to one or more independent variable (x). So we get y= f(x) and in this way we can forecast the dependent variables for the future.
What is a Time Series?• An ordered sequence of values of a variable at equally spaced
time intervals. i.e, Collection of observations indexed by the date of each observation
• In any time series plot we generally get these four components:
Trend:
Season:
Tyyy ,,, 21
What is a Time Series? Cont….. Cycle: these are generally sinusoidal type of curve
Random:
Field of its Application• The usage of time series models is two fold:
– Obtain an understanding of the underlying forces and structure that produced the observed data.– Fit a model and proceed to forecasting, monitoring or even feedback and feedforward control.
• Time Series Analysis is used for many applications such as: Economic Forecasting Sales Forecasting Budgetary Analysis Stock Market Analysis Yield Projections Process and Quality Control Inventory Studies Workload Projections Utility Studies Census Analysis Weather data analysis Climate data analysis Tide levels analysis Seismic waves analysis
Methods:Autoregressive (AR) Processes• AR(1): First order autoregression
εt is noise.
• Stationarity: We will assume• Can be written as
ttt YcY 1
1
22
1
22
1
1 ttt
tttt
c
cccY
Properties of AR(1)
2
2
242
2
22
1
20
1
1
1
ttt
t
E
YE
c
Properties of AR(1), cont……….
jj
j
j
j
jjj
jtjtjtjtj
ttt
jttj
E
YYE
0
22
242
242
22
122
1
1
1
Autocorrelation Function for AR(1): ttt YY 18.0
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Lag
Autocorrelation
Autocorrelation Function for AR(1): ttt YY 18.0
-0.5
0.0
0.5
1.0
0 5 10 15 20
Lag
Autocorrelation
0 20 40 60 80 100
-3-2
-10
12
5.00 20 40 60 80 100
-20
24
9.0
0 20 40 60 80 100
-4-2
02
4
9.0
Autoregressive Processes of higher order
• pth order autoregression: AR(p)
• Stationarity: We will assume that the roots of the following all lie outside the unit circle.
tptpttt YYYcY 2211
01 221 p
pzzz
Properties of AR(p)
• Can solve for Autocovariances / Autocorrelations using Yule-Walker equations
pc
211
Moving Average Processes
• MA(1): First Order MA process
• “moving average”– Yt is constructed from a weighted sum of the two
most recent values of .
1 tttY
Properties of MA(1)
0
1
2
2
212
22
11
2111
22
21
21
2
21
2
jtt
ttttttt
tttttt
tttt
ttt
t
YYE
E
EYYE
E
EYE
YE
for j>1
MA(1)
• Covariance stationary– Mean and autocovariances are not functions of time
• Autocorrelation of a covariance-stationary process
• MA(1)0
jj
222
2
1 11
Autocorrelation Function for White Noise:
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Lag
Autocorrelation
ttY
Autocorrelation Function for MA(1): 18.0 tttY
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Lag
Autocorrelation
Mixed Autoregressive Moving Average (ARMA) Processes
• ARMA(p,q) includes both autoregressive and moving average terms
qtqtt
tptpttt YYYcY
2211
2211
Thank you!
White Noise Process• Basic building block for time series processes
• Independent White Noise Process– Slightly stronger condition that εt and εζ are independent
0
022
t
t
t
tt
E
E
E
Autocovariance
• Covariance of Yt with its own lagged value
• Example: Calculate autocovariances for:
jtjtttjt YYE
jttjttjt
tt
EYYE
Y
Stationarity
• Covariance-stationary or weakly stationary process– Neither the mean nor the autocovariances depend
on the date t
jjtt
t
YYE
YE
Stationarity, cont.
• Covariance stationary processes– Covariance between Yt and Yt-j depends only on j
(length of time separating the observations) and not on t (date of the observation)
jj
Stationarity, cont.
• Strict stationarity– For any values of j1, j2, …, jn, the joint distribution
of (Yt, Yt+j1, Yt+j2
, ..., Yt+jn) depends only on the
intervals separating the dates and not on the date itself
Table 1: Correlation coefficients of Q (t) for Bird Creek
Auto Correlation coefficients Cross Correlation coefficientsFlow Value Rainfall ValueQ (t) 1.0000 P (t) 0.2021Q (t-1) 0.7633 P (t-1) 0.4906Q (t-2) 0.5296 P (t-2) 0.3361Q (t-3) 0.4631 P (t-3) 0.1813Q (t-4) 0.4265 P (t-4) 0.1380Q (t-5) 0.4041 P (t-5) 0.1270Q (t-6) 0.4001 P (t-6) 0.1258Q (t-7) 0.3948 P (t-7) 0.1225Q (t-8) 0.3842 P (t-8) 0.1202Q (t-9) 0.3705 P (t-9) 0.1190Q (t-10) 0.3371 P (t-10) 0.1187
Auto correlation plot of Q (t) Cross correlation plot of Q (t)
Partial Auto Correlation Coefficient
Rainfall Value
Q (t) 1.0000
Q (t-1) 0.7633
Q (t-2) -0.1269
Q (t-3) 0.2541
Q (t-4) 0.0057
Q (t-5) 0.1222
Q (t-6) 0.0698
Q (t-7) 0.0673
Q (t-8) 0.0514
Q (t-9) 0.0400
Q (t-10) -0.0187