Time Series Presented by Vikas Kumar vidyarthi Ph.D Scholar ( ),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) process Moving average (MA) process ARMA 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:

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 ε t is noise. Stationarity: We will assume Can be written as

Properties of AR(1)

Properties of AR(1), cont……….

Autocorrelation Function for AR(1):

Autoregressive Processes of higher order p th order autoregression: AR(p) Stationarity: We will assume that the roots of the following all lie outside the unit circle.

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

Moving Average Processes MA(1): First Order MA process moving average – Y t is constructed from a weighted sum of the two most recent values of.

Properties of MA(1) for j>1

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

Autocorrelation Function for White Noise:

Autocorrelation Function for MA(1):

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

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

Autocovariance Covariance of Y t with its own lagged value Example: Calculate autocovariances for:

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

Stationarity, cont. Covariance stationary processes – Covariance between Y t and Y t-j depends only on j (length of time separating the observations) and not on t (date of the observation)

Stationarity, cont. Strict stationarity – For any values of j 1, j 2, …, j n, the joint distribution of (Y t, Y t+j 1, Y t+j 2,..., Y t+j n ) 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 coefficients Flow ValueRainfall Value Q (t) P (t) Q (t-1) P (t-1) Q (t-2) P (t-2) Q (t-3) P (t-3) Q (t-4) P (t-4) Q (t-5) P (t-5) Q (t-6) P (t-6) Q (t-7) P (t-7) Q (t-8) P (t-8) Q (t-9) P (t-9) Q (t-10) P (t-10)

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

Partial Auto Correlation Coefficient RainfallValue Q (t) Q (t-1) Q (t-2) Q (t-3) Q (t-4) Q (t-5) Q (t-6) Q (t-7) Q (t-8) Q (t-9) Q (t-10)