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Time Series

Time Series . Instructor Dr. L. D. Behera Department of Electrical Engineering Indian institute of Technology Kanpur. Presented by Vikas Kumar vidyarthi Ph.D Scholar (10203069),CE. Contents:-. Correlation and Regression What is Time Series? Field of its Applications Methods:

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Time Series

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  1. Time Series Instructor Dr. L. D. Behera Department of Electrical Engineering Indian institute of Technology Kanpur Presented by Vikas Kumar vidyarthi Ph.D Scholar (10203069),CE

  2. Contents:- • Correlation and Regression • What is Time Series? • Field of its Applications • Methods: • Autoregressive (AR) process • Moving average (MA) process • ARMA process • Example of input variable selection by ACF, CCF and PACF. • Understanding

  3. 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.

  4. 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:

  5. What is a Time Series? Cont….. • Cycle: these are generally sinusoidal type of curve • Random:

  6. 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

  7. Methods:Autoregressive (AR) Processes • AR(1): First order autoregression εt is noise. • Stationarity: We will assume • Can be written as

  8. Properties of AR(1)

  9. Properties of AR(1), cont……….

  10. Autocorrelation Function for AR(1):

  11. Autocorrelation Function for AR(1):

  12. 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.

  13. Properties of AR(p) • Can solve for Autocovariances/ Autocorrelations using Yule-Walker equations

  14. 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 .

  15. Properties of MA(1) for j>1

  16. MA(1) • Covariance stationary • Mean and autocovariances are not functions of time • Autocorrelation of a covariance-stationary process • MA(1)

  17. Autocorrelation Function for White Noise:

  18. Autocorrelation Function for MA(1):

  19. Mixed Autoregressive Moving Average (ARMA) Processes • ARMA(p,q) includes both autoregressive and moving average terms

  20. Thankyou!

  21. White Noise Process • Basic building block for time series processes • Independent White Noise Process • Slightly stronger condition that εt and εζ are independent

  22. Autocovariance • Covariance of Yt with its own lagged value • Example: Calculate autocovariances for:

  23. Stationarity • Covariance-stationary or weakly stationary process • Neither the mean nor the autocovariances depend on the date t

  24. 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)

  25. 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

  26. Table 1: Correlation coefficients of Q (t) for Bird Creek Auto Correlation coefficientsCross Correlation coefficients Flow Value Rainfall Value Q (t) 1.0000 P (t) 0.2021 Q (t-1) 0.7633P (t-1) 0.4906 Q (t-2) 0.5296P (t-2) 0.3361 Q (t-3) 0.4631 P (t-3) 0.1813 Q (t-4) 0.4265 P (t-4) 0.1380 Q (t-5) 0.4041P (t-5) 0.1270 Q (t-6) 0.4001P (t-6) 0.1258 Q (t-7) 0.3948P (t-7) 0.1225 Q (t-8) 0.3842 P (t-8) 0.1202 Q (t-9) 0.3705 P (t-9) 0.1190 Q (t-10) 0.3371 P (t-10) 0.1187

  27. Auto correlation plot of Q (t) Cross correlation plot of Q (t)

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