Analysis of Time Series Data

Abstract

This chapter introduces several methods to analyze time series data in the time domain; an area rich in theoretical development and in practical applications. What constitutes time series data and some of the common trends encountered are first presented. This is followed by a description of three types of time-domain modeling and forecasting models. The first general class of models involves moving average smoothing techniques which are methods for removing rapid fluctuations in time series so that the general secular trend can be seen. The second class of models is similar to classical OLS regression models, but here the time variable appears as a regressor, thereby allowing the trend and the seasonal behavior in the data series to be captured by the model. Its strength lies in its ability to model the deterministic or structural trend of the data in a relatively simple manner. The third class of models called ARIMA models allows separating and modeling the systematic component of the model residuals from the purely random white noise element, thereby enhancing the prediction accuracy of the overall model. ARMAX models, which are extensions of the univariate ARIMA models to multivariate problems, and their ability to model dynamic systems are also discussed with illustrative examples. Finally, an overview is provided of a practical application involving control chart techniques which are extensively used for process and condition monitoring of engineered systems.

Problems

Pr. 9.1

Consider the time series data given in Table 9.1 which was used to illustrate various concepts throughout this chapter. Example 9.4.2 revealed that the model was still not satisfactory since the residuals still has a distinct trend. You will investigate alternative models (such as a second order linear or an exponential) in an effort to improve the residual behavior

Perform the same types of analyses as illustrated in the text. This involves determining whether the model is more accurate when fit to the first 48 data points, whether the residuals show less pronounced patterns, and whether the forecasts of the four quarters for 1986 have become more accurate. Document your findings in a succinct manner along with your conclusions.

Pr. 9.2

Use the following time series models for forecasting purposes:

• (a) ${Z_t} = 20 + {\varepsilon _t} + 0.45{\varepsilon _{t - 1}} - 0.35{\varepsilon _{t - 2}}$. Given the latest four observations:

  {17.50, 21.36, 18.24, 16.91}, compute forecasts for the next two periods

• (b) ${Z_t} = 15 + 0.86{Z_{t - 1}} - 0.32{Z_{t - 2}} + {\varepsilon _t}$. Given the latest two values of Z {32, 30), determine the next four forecasts.

Pr. 9.3.

Section 9.5.3 describes the manner in which various types of ARMA series can be synthetically generated as shown in Figs. 9.21, 9.22 and 9.23, and how one could verify different recommendations on model identification. These are useful aids for acquiring insights and confidence in the use of ARMA. You are asked to synthetically generate 50 data points using the following models and then use these data sequences to re-identify the models (because of the addition of random noise, there will be some differences in model parameters identified);

• (a) ${Z_t} = 5 + {\varepsilon _t} + 0.7{\varepsilon _{t - 1}}\;{\rm{ with }}\;N(0,0.5)$

• (b) ${Z_t} = 5 + {\varepsilon _t} + 0.7{\varepsilon _{t - 1}}\;{\rm{ with }}\;N(0,1)$

• (c) ${Z_t} = 20 + 0.6{Z_{t - 1}} + {\varepsilon _t}\;{\rm{ with }}\;N(0,1)$

• (d) ${Z_t} = 20 + 0.8{Z_{t - 1}} - 0.2{Z_{t - 1}} + {\varepsilon _t}\;{\rm{ with }}\;N(0,1)$

• (e) ${Z_t} = 20 + 0.8{Z_{t - 1}} + {\varepsilon _t} + 0.7{\varepsilon _{t - 1}}\;{\rm{ with }}\;N(0,1)$

Pr. 9.4.

Time series analysis of sun spot frequency per year from 1770–1869

Data assembled in Table B.5 (in Appendix B) represents the so-called Wolf number of sunspots per year (n) over many years (from Montgomery and Johnson 1976 by permission of McGraw-Hill).

• (a) First plot the data and visually note underlying patterns

• (b) You will develop at least 2 alternative models using data from years 1770–1859. The models should include different trend and/or seasonal OLS models, as well as sub-classes of the ARIMA models (where the trends have been removed by OLS models or by differencing). Note that you will have to compute the ACF and PACF for model identification purposes

• (c) Evaluate these models using the expost approach where the data for years 1860–1869 are assumed to be known with certainty (as done in Example 9.6.1).

Pr. 9.5.

Time series of yearly atmospheric CO ₂ concentrations from 1979–2005

Table B.6 (refer to Appendix B) assembles data of yearly carbon-dioxide (CO₂) concentrations (in ppm) in the atmosphere and the temperature difference with respect to a base year (in °C) (from Andrews and Jelley 2007 by permission of Oxford University Press).

• (a) Plot the data both as time series as well as scatter plots and look for underlying trends

• (b) Using data from years 1979–1999, develop at least two models for the Temp. difference variable. These could be trend and /or seasonal or ARIMA type models

• (c) Repeat step (b) but for the CO₂ concentration variable

• (d) Using the same data from 1979–1999, develop a model for CO₂ where Temp.diff is one of the regressor variables

• (e) Evaluate the models developed in (c) and (d) using data from 2000–2005 assumed known (this is the expost conditional case)

• (f) Compare the above results for the exante unconditional situation. In this case, future values of temperature difference are not know, and so model developed in step (b) will be used to first predict this variable, which will then be used as an input to the model developed in step (d)

• (g) Using the final model, forecast the CO₂ concentration for 2006 along with 95% CL.

Pr. 9.6

Time series of monthly atmospheric CO ₂ concentrations from 2002–2006

Figure 9.33 represents global CO₂ levels but at monthly levels. Clearly there is both a long-term trend and a cyclic seasonal variation. The corresponding data is shown in Table B.7 (and can be found in Appendix B). You will use the first four years of data (2002–2005) to identify different moving average smoothing techniques, trend+seasonal OLS models, as well as ARIMA models as illustrated through several examples in the text. Subsequently, evaluate these models in terms of how well they predict the monthly values of the last year i.e., year 2006).

Fig. 9.33

Monthly mean global CO₂ concentration for the period 2002–2007. The smoothened line is a moving average over 10 adjacent months. (Downloaded from NOAA website http://www.cmdl.noaa.gov/ccgg/trends/index.php#mlo,2006)

Pr. 9.7

Transfer function analysis of unsteady state heat transfer through a wall

You will use the conduction transfer function coefficients given in Example 9.6.1 to calculate the hourly heat gains (Q_cond) through the wall for a constant room temperature of 24°C and the hourly solar-air temperatures for a day given in Table 9.13 (adapted from Kreider et al. 2009). You will assume guess values to start the calculation, and repeat the diurnal calculation over as many days as needed to achieve convergence assuming the same T_solair values for successive days. This problem is conveniently solved on a spreadsheet.

Table 9.13 Data table for Problem 9.7

Pr. 9.8

Transfer function analysis using simulated hourly loads in a commercial building

The hourly loads (total electrical, thermal cooling and thermal heating) for a large hotel in Chicago, IL have been generated for three days in August using a detailed building energy simulation program. The data shown in Table B.8 (given in Appendix B) consists of outdoor dry-bulb (T_db) and wet-bulb (T_wb) temperatures in °F as well as the internal electric loads of the building Q_int (these are the three regressor variables). The response variables are the total building electric power use (kWh) and the cooling and heating thermal loads (Btu/h).

• (a) Plot the various variables as time series plots and note underlying patterns.

• (b) Use OLS to identify a trend and seasonal model using indicator variables but with no lagged terms for Total Building electric power.

• (c) For the same response variable, evaluate whether the seasonal differencing approach, i.e., ∇₂₄Y_t  = Y_t − Y_t-24 is as good as the trend and seasonal model in detrending the data series.

• (d) Identify ARMAX models for all three response variables separately by using two days for model identification and the last day for model evaluation

• (e) Report all pertinent statistics and compare the results of different models. Provide reasons as to why the particular model was selected as the best one for each of the three response variables.

Pr. 9.9

Example 9.7.1 illustrated the use of Shewhart charts for variables.

• (a) Reproduce the analysis results in order to gain confidence

• (b) Repeat the analysis but using the Cusum and EWMA (with λ = 0.4) and compare results.

Pr. 9.10

Example 9.7.1 illustrated the use of Shewhart charts for attributes variables.

• (a) Reproduce the analysis results in order to gain confidence

• (b) Repeat the analysis but using the Cusum and EWMA (with λ = 0.4) and compare results.