The Practice of Time Series Analysis

This chapter introduces a successful application of the optimal regulator to steam temperature control of thermal electric power plants, in which the state equation for the control system design is derived based on a statistical model.

In the fields of medical or biological science, time series analysis including the analysis of fluctuation become popular in recent years. In spite of the fact, the analysis of the univariate system is mostly used and the analysis of the multivariate system is very few. Of course, there are some handbooks for multivariate time series analysis for medical data. However even in these handbooks, only the coherency among mutual variables, i.e. correlation, although it is classified according to the frequency, is referred to as a pivotal factor and lacks the consideration on the effect of feedback. It should be recognized that such a situation is quite regrettable considering from medical researchers’ viewpoint. Perhaps it is due to the fact that, in addition to the difficulty in taking in the multivariate data for time series analysis, the difficulty in the analysis or absence of methods usable for analysis of actual feedback systems.

In analyzing factors of economic time series fluctuations, it is important to compare, first of all, the characteristics between a macro-econometric model and a time series model in order to explain to what extent an economic theory should be utilized or how deeply a statistical theory should be taken in. With the macro-econometric model, we make much of the fact that movement of a variety of factors mainly concerning business fluctuations should be described deterministically with the aid of the economic theory. At the same time, the macro-econometric model is often used for the procedures including simulation for the purpose of measuring the effect of the fiscal and monetary policies, and no attention is paid to the time series structure of economic variables. On the other hand, with a general time series model, we need none of economic theories to analyze the time series structure of the data. Therefore the model is described stochastically and we make much of the prediction performance.¹⁾ Needless to say, there might actually be neither an econometric model ignoring the time series characteristics nor economic time series analysis neglecting the economic theories. The purpose of this chapter resides in the improvement of the time series model by taking in the economic knowledge expressed by an econometric model in analyzing economic data.

In the identification of ship motion on ocean, it is especially required to adopt a statistical model, since external disturbances caused by wind and wave and furthermore the motion of the hulls itself responding to such an oceanic disturbances are intrinsically irregular. Moreover, the range of the strength of the external disturbances is very wide from a mirror-like quiet sea to a rough sea where violent storm is raging. Thus the ship motions under wide range of disturbances are so largely changed as can never be imagined in the case of the other transport facilities. A method of practical analysis of such irregular phenomena have earlier been established in a frequency domain (Isobe 1960) and then, the ship motions under disturbances has been dealt with as a stochastic process also in the field of ship-building engineering. Nowadays, it can be said that statistical methods to analyze the time series gained from model-tests in irregular waves or records of actual-sea tests in a frequency domain have been established (Yamanouchi 1961a, 1961b). However reliable modeling method in a time domain for irregular time series is quite difficult, and no attempt to use the obtained analytical results for control has been made.

When an earthquake occurs, vibrations propagate from the source in all directions. For large earthquakes, vibrations are soon observed even at sites far from the source. Generally, such a vibration is called a seismic wave. The seismic waves which propagate through the interior of the earth are comprised of two types: longitudinal P waves and transverse S waves. Additionally, surface waves (Rayleigh waves, Love waves, etc.) which propagate close to the surface of the earth also exist. These body waves and surface waves propagate with velocities which depend on the physical properties of the earth interior. Generally the velocity of the seismic wave increases with depth in the earth. Furthermore, the velocity of P waves is approximately 1.73 times than that of S waves, while the velocity of surface waves is 0.92 times that of the S waves. Therefore, depending on the distance of the observation point from the hypocenter and the depth of the hypocenter, there are differences in the arrival times of the individual waves. At present a table (travel time table) showing the relation between the distance, depth, and arrival times is available, and may be used to estimate the hypocenter. Observation of the arrival times at many observation points together with the use of the table allows the time and location of the earthquake to be estimated. Also, in contrast, a detailed velocity structures of the earth can be estimated by using many of the observation data of accurate arrival times. This results in the generation of a more accurate travel time table, thus more accurate hypocenter locations.

An automobile can never be driven into motion until the driver manipulates an accelerator pedal, brake pedal or steering wheel. Therefore when we analyze the motion characteristics of an automobile, it becomes very important not only just to pay attention to the vehicle itself in its operation characteristics but also to identify it as a driver-vehicle system including the driver’s operation characteristics.

In rough seas, it is very important for mariners to grasp the sea state around the ship and to secure the safety of the hull and cargo by suitable maneuvers, such as changing the course, slowing down the speed and so on. The difficulty of decision making for these maneuvers arises from the relation between the safety and the cost of the voyage. Therefore the mariners are required to have profound knowledge concerning the sea-keeping characteristics of the ship.

Although longer than 1,000m of a cocoon filament can be reeled from a cocoon obtained from a silkworm, the filament is thin enough just to weigh no more than 3g per 10,000m and textural raw threads called raw silk are produced by binding several filaments and by adding other filaments when the reeling filaments are broken. Since the property and style of the reeled cocoon filaments have greatly influence on the quality, production efficiency, yield, etc. of the raw silk, many of important control factors of the process depend on the reeling time. Therefore in settling a filature controlling criterion, a result obtained through time series analysis plays a very important role.

Once a drug is administered to the human body, it undergoes the processes of absorption, distribution, metabolism and excretion. The area to study the kinetics of these processes is called pharmacokinetics.

In various fields of scientific research, it is possible to find time series data that cannot be adequately dealt with by using an ordinary linear Gaussian model. As seen in examples such as the concentration of hormone in the blood, electric potential of the nerve, and river-flow rate, such data exhibiting distinct pulse-shaped patterns can be taken as typical exemplification. Analysis using an input-output model can be made, provided that the input data including pulses is known and induces the pulses in the output. For example, Ozaki (1985) identified the changes in river-flow as the output of a non-linear system with the amount of rainfall as the input, and has been successful in predicting the river-flow rate. However, when the input data are not available, an alternative method is required.

Autoregressive (AR) models are very useful for time series analysis. As shown in Figure 11.1, there is correspondence between the AR model and the autocovariance function and the power spectrum of an univariate stationary time series. Therefore, if an AR model is estimated from a time series, then the estimates of the autocovariance function and the power spectrum are obtained immediately. For the analysis of multivariate time series, Figure 11.2 shows the relation between the vector autoregressive (VAR) model and the cross-covariance function, the cross-spectrum, and the relative power contribution. Akaike and Nakagawa (1988), and Kitagawa (1993) developed procedures and programs for the analysis of stationary time series by using the AR and VAR models.

One of the most important processes in a cement plant is a kiln process. The kiln process is a multiple input and output system having many internal noise sources and internal feedback loops, and the process is, viewing from the physical characteristics, a distributed parameter system which has an extension of the behavior over space and time. Its model is given by the partial differential equations concerning the gas temperature T _G and solid temperature T _M derived from the heat balance and material balance in the kiln. At the initial stage of the development (1962) when the first computer for process control was installed to Kumagaya Plant of Chichibu Cement Co. Ltd., this mathematical model was used for control. However the mathematical model was in reality not satisfactory enough to be used as the model for stabilizing control because of the fact that none of noise sources existing in the actual process were considered.

For bicycles or motorcycles that are running straight, the following conditions should be satisfied. (1) Balancing stability should be retained. (2) Directional stability (route retention) should be retained. Why can a 2-wheeled vehicle run without capsizing? This has long been a problem stimulating people’s keen interest, and the cause was once said to be gyroscopic effect. However computation reveals instantly that the gyroscopic effect is almost negligible. A 2-wheeled vehicle is intrinsically of static instability, and is liable to be overturned without control. This is because the running bicycle is continuously under the influence of many kinds of disturbances. Even the bicycle’s rider him/herself is often a cause of the disturbance. Hereafter, the disturbance shall be just called noise.

Roads have generally random surface undulations, which cause the vertical vibration of automobiles. Those road inputs act on their bodies, suspensions and axles and then have severe effects on the durability of the components and the riding comforts. The concept of the random data process for automobiles has been applied for a long time. The stress occurrence analysis for durability study was previously carried out with scales on pen-oscillograms. The analysis is now automatically executed with advances of electronic devices. Spectral analysis for study on frequency characteristics of vibration and stress has been popular since 1960’s and now becomes one of routine works at engineering divisions with the spread of FFT equipment. ISO/DIS 8608 Mechanical Vibration-Road Surface Profiles-Reporting Measured Data which is the accumulation of measured road surface undulations for the standard property presentation was published. We can evaluate wear of the component materials and the riding comfort through the DIS data. Most of time series data are currently processed through FFT methods for ease of computation. It is still the problem whether the instantaneous values of road surface undulation during running can be obtained with enough accuracy. The frequency response analyses among accelerations and stresses of components during running are usually carried out in practice. The power contribution method is also used in case of feedback system. When the system has human factors, auto-regressive models for short time data are often in consideration of the performance change with time.

A fetus’s heart rate variation is regulated by the heart rate control center existing in the brainstem, and its rhythmic movement is a sharp information reflecting the ever-changing autonomic nervous activity state (for example, antagonistic regulation of vagus nerves and sympathetic nerves). However, no report describing the continuity of the sway of the heart rate variation ranging from a fetus to a newborn infant has been available until now. With such a situation in mind, analysis is made in this chapter with the maturity process of the autonomic nervous activity ranging from a human fetus to a newborn infant by applying an auto-regressive model. Thus, studies have been carried out with the purpose of estimating the physiological significance lurking in the background of the above process.

The mammalian brain can be regarded as a huge and complicated dynamical information processing system composed of single units called neurons or nerve cells. The mechanisms of brain function have, traditionally, been elucidated with the aid of single microelectrodes to measure the responses in single neurons. This approach, however, seems to be insufficient for identifying the complex dynamical system as the brain. An optical recording method, on the other hand, has made possible real-time multipoint measurement of the evoked neural activities distributed in the brain. This new recording method can be used to explore new mechanisms responsible for the dynamical neural processing activities of the brain. Such neural activities always exhibit nonlinear and nonstationary characteristics, and so straight forward application of any system identification theory to the neural system is inappropriate. On the other hand, many industrial dynamical systems, which involve nonstationary and nonlinear dynamical phenomena, exquisitely are modeled and controlled by using the extensive linear theory regarding to system identification and control. From this fact, there is a possibility that a linear identification theory such as time series analysis could be used in exploring the functioning of a nonlinear and nonstationary brain.

With the appreciation of stock prices in the late 1980s and subsequent crash in the 1990s, financial institutions and institutional investors have once again realized that asset management by holding large positions in volatile high-risk assets such as stocks carries the danger of incurring huge losses. This has focused attention on the need to improve risk management and asset management efficiency.

Controversy is liable to be evoked on the matter to what extent economics is of use for enhancement of people’s economic welfare. Economics not only should possess the double sides as normative science and positive science but also should make judgment from normative viewpoint on the actual economy, playing a role of providing political suggestions. Meanwhile the empirical evidence of economic activities based on the actual data is laid much stress on as positive science. In consideration of the fact that the economic theory is forced to make development with confrontation to the economic phenomena most characteristic of a period, a high possibility is also pointed out that necessity of the traditional theory being replaced with a new one. That is the reason why the empirical analysis has important role and the economics is called empirical science.

Mankind, who are full of intelligent curiosity, are always anxious to satisfy such curiosity and are being engaged in development of new observation instruments. The observation utilizing spacecraft such as rockets, balloons, etc. has aimed at exceeding the limitations of the information that can be measured and obtained exclusively on the ground. The human desire to make probe in a direction of height has been expanded and promoted to the outer space, and furthermore to the universe. Thus commencement of such observation methods as to dispatch observation equipment and apparatus has been seen, that is to say, observation by means of artificial satellites, called satellites simply, has been started.

The sun and the moon existing near the earth generate tidal forces on the earth. The force is produced because of the fact that the attraction from the heavenly bodies are slightly different from each other on the center and on the surface of the earth. Also the force is in proportion to the gradient of the attraction, and becomes larger in proportion to the size of the body to which the attraction is affected. The magnitude of the attraction is in proportion to the mass of the heavenly body and in inverse proportion to the square of the distance according to the law of gravitation. The tide-generating force is obtained by differentiating the gravitational attraction in space. Consequently, the force is in proportion to the mass of the heavenly body and in inverse proportion to the cube of the distance between the heavenly body and the earth. From this reason, the tide-generating force by the moon is more than twice larger than that by the sun, while the gravitation of the sun is by far larger than that of the moon.

An earthquake is a waveform representing catastrophic release of strain energy that was gradually accumulated in the crust. Crustal strain is considered to be changed by some causes immediately before the huge destruction, and leveling, tilt, volumetric strain, groundwater level and radon concentration in groundwater are continuously observed with an aim of detecting precursory changes mainly in the South Kanto and the Tokai districts. The most famous example of crustal deformation preceding a large earthquake is the change of level in Kakegawa, Shizuoka Prefecture before the 1944 To-Nankai Earthquake (M7.9; Mogi, 1985).

Thanks to rapid and remarkable development of electronic devices such as computers and measurement instruments, a variety of time series has been available automatically and continuously, resulting in minute-by-minute accumulation of a large amount of data. However when time series is observed for a long period, part of the time series is sometimes unable to be observed owing to some accidental causes such as malfunction of instrument or some physical restrictions of the observation system. In such a case, unobserved data is called missing observations. Even with only several percent of the missing observations, if they are studded with the data, the length of actually available data may become very short if only a part observed continuously is taken out.

The philosophical or psychological frame of the researcher is reflected on the way of the development of a scientific research. Since the statistical method is adopted in case where the structure of the objective is inherently undetermined, particular adjustment of the mental framework is required in its application. In ordinary textbooks of statistics, this aspect is not usually discussed. The examples of real time series analysis included in this book provide precious clues to find out how fruitful outcomes have been obtained by approaching concrete problems with particular frames of mind.