Topic outline

• Introduction

• The Purpose and Scope of  Time Series Analysis
• The Nature of Time Series
• Model Building Strategy

Reading Assignment: Hyndman, Rob J.- Athana­sopou­los, George(2017:"Getting Started"); Shumway, Robert H.-Stoffer, David S. (2011:1-5); Cryer,Jonathan D.  - Chan, Kung-Sik (2008:1-8)

• The Nature of Time Series Data

Some of the problems and questions of interest to the prospective time series analyst can best be exposed by considering real experimental data taken from different subject areas.

Shumway, Robert H.-Stoffer, David S. (2011:1-5)

• Time Series Models

The primary objective of time series analysis is to develop mathematical models that provide plausible descriptions for sample data, like that encountered in the previous section. In order to provide a statistical setting for describing the character of data that seemingly fluctuate in a random fashion over time, we assume a time series can be defined as a collection of random variables indexed according to the order they are obtained in time.

Shumway, Robert H.-Stoffer, David S. (2011:5-10)

• Measures of Dependence

In this section, we introduce various theoretical measures used for describing how time series behave. As is usual in statistics, the complete description involves the multivariate distribution function of the jointly sampled values $$x_1, x_2, ... , x_n$$, whereas more economical descriptions can be had in terms of the mean and autocorrelation functions. Because correlation is an essential feature of time series analysis, the most useful descriptive measures are those expressed in terms of covariance and correlation functions.

Shumway, Robert H.-Stoffer, David S. (2011:10-14)

• Stationary Time Series

The notion, called weak stationarity, when the mean is constant, is fundamental in allowing us to analyze sample time series data when only a single series is available.

Shumway, Robert H.-Stoffer, David S. (2011:14-19)

• Estimation of Correlation

Although the theoretical autocorrelation and cross-correlation functions are useful for describing the properties of certain hypothesized models, most of the analyses must be performed using sampled data. This limitation means the sampled points $$x_1, x_2, ... , x_n$$,  only are available for estimating the mean, autocovariance, and autocorrelation functions. From the point of view of classical statistics, this poses a problem because we will typically not have iid copies of  $$x_t$$ that are available for estimating the covariance and correlation functions. In the usual situation with only one realization, however, the assumption of stationarity becomes critical. Somehow, we must use averages over this single realization to estimate the population means and covariance functions.

Shumway, Robert H.-Stoffer, David S. (2011:19-28)

• Classical Regression in the Time Series Context

In this session linear regression will be discussed in the time series context by assuming some output or dependent time series, say, $$x_t$$ , for $$t = 1, ... , n$$, is being influenced by a collection of possible inputs or independent series, say, $$z_{t_1}, z_{t_2},... , z_{t_q}$$, where we first regard the inputs as fixed and known. This assumption, necessary for applying conventional linear regression, will be relaxed later on.

Shumway, Robert H.-Stoffer, David S. (2011:46-55)

• Exploratory Data Analysis

In general, it is necessary for time series data to be stationary, so averaging lagged products over time, as in the previous section, will be a sensible thing to do.With time series data, it is the dependence between the values of the series that is important to measure; we must, at least, be able to estimate autocorrelations with precision. It would be difficult to measure that dependence if the dependence structure is not regular or is changing at every time point. Hence, to achieve any meaningful statistical analysis of time series data, it will be crucial that, if nothing else, the mean and the autocovariance functions satisfy the conditions of stationarity. Often, this is not the case, and we will mention some methods in this section for playing down the effects of nonstationarity so the stationary properties of the series may be studied.

Shumway, Robert H.-Stoffer, David S. (2011:55-68)

• Smoothing in the Time Series Context

Filtering or smoothing a time series is useful in discovering certain traits in a time series, such as long-term trend and seasonal components.

Shumway, Robert H.-Stoffer, David S. (2011:68-73)

Ata, Mustafa Y. (2017) Dedecting and Modelling Signal in Time Series

• This topic

Autoregressive Moving Average Models

The classical regression model was developed for the static case, namely, we only allow the dependent variable to be influenced by current values of the independent variables. In the time series case, it is desirable to allow the dependent variable to be influenced by the past values of the  independent variables and possibly by its own past values. If the present can be plausibly modeled in terms of only the past values of the independent inputs, we have the enticing prospect that forecasting will be possible.

Shumway, Robert H.-Stoffer, David S. (2011:73-93)

Hyndman, Rob J.  and Athana­sopou­los, George(2013)  Forecasting: principles and practice ARIMA Models

• Autocorrelation and Partial Autocorrelation

Theoretical autocorrelation and partial autocorrelation functions of linear stationary processes are the major tools of identifying a model for a time series.

Shumway, Robert H.-Stoffer, David S. (2011:98-105)

• Forecasting

In forecasting, the goal is to predict future values of a time series, $$x_{n+m}, m = 1,2, ...$$, based on the data collected to the present, $$x = \{x_n, x_{n-1},...,x_1\}$$. Throughout this section, we will assume $$x_t$$ is stationary and the model parameters are known.

Throughout this section, we assume we have $$n$$ observations,$$x_1,...,x_n$$, from a causal and invertible Gaussian ARMA(p, q) process in which, initially, the order parameters, p and q, are known. Our goal is to estimate the parameters, $$\phi_1,..., \phi_p$$, $$\theta_1,..., \theta_q$$ and $$\sigma^2_w$$ . We will discuss the problem of determining p and q later in this section..