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