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Introduction to Time Series

Time Series Analysis with Application in R<br>Written By Jonathan D. Cryer and Kung-Sik Chan

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Introduction to Time Series

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  1. Time Series Analysis Book 1: Time Series Analysis with Application in R Jonathan D. Cryer & Kung-Sik Chan Book 2: Introductory Time Series with R Paul S.P.Cowpertwait & Andrew V.Metacalfe Instructor:Muhammad Ali, Assistant Professor of Statistics GPGC Mardan June 29, 2020 Book 1: Time Series Analysis with Application in R Jonathan D. Cryer &Kung-Sik ChanBook 2: Introductory Time Series with R Paul S.P.Cowpertwait &Andrew V.Metacalfe Muhammad Ali, Assistant Professor of Statistics

  2. Chapter 1 Introduction 1.1 Definition of Time Series Data When a variable is measured sequentially in time over or at a fixed interval, it is known as time series data.The main fea- tures of many time series are trends and seasonal variations that can be modelled deterministically with mathematical functions of time. But, another important feature of most time series is that observations close together in time tend to be correlated,some examples of time series data are given below: Interest rates and exchange rates each day Gross domestic product (GDP) of a country on yearly basis Daily minimum, and maximum temperatures of capital cities around the world Monthly rainfall at different cities of Pakistan Hourly crude oil prices Book 1: Time Series Analysis with Application in R Jonathan D. Cryer &Kung-Sik ChanBook 2: Introductory Time Series with R Paul S.P.Cowpertwait &Andrew V.Metacalfe Muhammad Ali, Assistant Professor of Statistics

  3. Introduction 1.2 Features of Air Passenger bookings example Trend Component Number of passengers traveling on the airline is increasing with time.In general, a systematic change in a time series that does not appear to be periodic is known as a trend. The simplest model for a trend is a linear increase or decrease, and this is often an adequate approximation. Seasonal Variation A repeating pattern within each year is known as seasonal variation, although the term is applied more generally to re- peating patterns within any fixed period, such as restaurant bookings on different days of the week. There is clear sea- sonal variation in the air passenger time series. At the time, bookings were highest during the summer... Book 1: Time Series Analysis with Application in R Jonathan D. Cryer &Kung-Sik ChanBook 2: Introductory Time Series with R Paul S.P.Cowpertwait &Andrew V.Metacalfe Muhammad Ali, Assistant Professor of Statistics

  4. Introduction 1.2 Features of Air Passenger booking example (Continued...) of June, July, and August and lowest during the autumn month of November and winter month of February.An understanding of the likely causes of the features in the plot helps us formulate an appropriate time series model. In this case, possible causes of the increasing trend include rising prosperity in the aftermath of the Second World War, greater availability of aircraft, cheaper flights due to competition between airlines, and an increasing population.A time series plot not only emphasises patterns and features of the data but can also expose outliers and erroneous values. One cause of the latter is that missing data are some- times coded using a negative value. Such values need to be handled differently in the analysis and must not be included as observations when fitting a model to data. Outlying values that cannot be attributed to some coding should be checked care- fully. If they are correct, they are likely to be of particular interest and should not be excluded from the analysis. Book 1: Time Series Analysis with Application in R Jonathan D. Cryer &Kung-Sik ChanBook 2: Introductory Time Series with R Paul S.P.Cowpertwait &Andrew V.Metacalfe Muhammad Ali, Assistant Professor of Statistics

  5. Introduction 1.3 Notation A time series of length n is denoted xt The ‘hat’ notation will be used to represent a prediction or forecast. For example ˆ xtis a forecast made at time t for a future value at time t+k. 1.4 Time Series Components If we assume an additive decomposition, then we can write xt= mt+ st+ zt (1) where, at time t, xtis the observed series, mtis the trend, stis the seasonal effect, and ztis the remainder component, all at period t. Alternatively, a multiplicative decomposition would be written as: Book 1: Time Series Analysis with Application in R Jonathan D. Cryer &Kung-Sik ChanBook 2: Introductory Time Series with R Paul S.P.Cowpertwait &Andrew V.Metacalfe Muhammad Ali, Assistant Professor of Statistics

  6. Introduction 1.4 Time Series Components (Continued...) xt= mt.st+ zt (2) The additive decomposition is the most appropriate if the mag- nitude of the seasonal fluctuations, or the variation around the trend-cycle, does not vary with the level of the time series. When the variation in the seasonal pattern, or the variation around the trend-cycle, appears to be proportional to the level of the time series, then a multiplicative decomposition is more appropriate. 1.5 Moving Average Method The classical method of time series decomposition originated in the 1920s and was widely used until the 1950s. It still forms the basis of many time series decomposition methods, so it is important to understand how it works... Book 1: Time Series Analysis with Application in R Jonathan D. Cryer &Kung-Sik ChanBook 2: Introductory Time Series with R Paul S.P.Cowpertwait &Andrew V.Metacalfe Muhammad Ali, Assistant Professor of Statistics

  7. Introduction 1.5 Moving Average Method (Continued) A moving average method of order m can be written as k X ˆTt=1 yt+j, (3) m k=−k where m = 2k+1, i.e. the estimate of the trend-cycle at time t is obtained by averaging values of time series within k periods of t. Observations that are nearby in time are also likely to be close in value. Therefore, the average eliminates some of the random- ness in the data, leaving a smooth trend-cycle component. We call this an m-MA, meaning a moving average of order m. It is possible to apply a moving average to a moving average. One reason for doing this is to make an even-order moving average symmetric. For example, we might take a moving average of or- der 4, and then apply another moving average of order 2 to the results Book 1: Time Series Analysis with Application in R Jonathan D. Cryer &Kung-Sik ChanBook 2: Introductory Time Series with R Paul S.P.Cowpertwait &Andrew V.Metacalfe Muhammad Ali, Assistant Professor of Statistics

  8. Introduction 1.6 Decompostion Following are the most widely used techniques for decomposing the time series. Classical decomposition X11 decomposition SEATS decomposition STL For detail study of these https://otexts.com/fpp2/decomposition.html methods please visit: Book 1: Time Series Analysis with Application in R Jonathan D. Cryer &Kung-Sik ChanBook 2: Introductory Time Series with R Paul S.P.Cowpertwait &Andrew V.Metacalfe Muhammad Ali, Assistant Professor of Statistics

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