PHILOSOPHICAL AND STATISTICAL FOUNDATION OF TIME AND SYSTEM AGE MODELLING

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1 PHILOSOPHICAL AND STATISTICAL FOUNDATION OF TIME AND SYSTEM AGE MODELLING R. Guo 1 and T. Dunne 2 1 Department of Statistical Sciences University of Cape Town Private Bag, Rondebosch 7700 Cape Town, South Africa Corresponding author s rguo@stats.uct.ac.za 2 Department of Statistical Sciences University of Cape Town Abstract: As Levich (1995) noted, In today s natural science time is an initial, un-definable concept. Nevertheless, the scientists intuitive idea of time essentially affects the experimental and theoretical results of scientific work. Time as a concept has always been of vital importance in many fields of investigation. In most studies, time is an element that has a great bearing on the outcome, or on the interpretation of results. Instances of this may be, for example, when predictions of future changes are being made; in studies of system dynamics and evolution; the aging processes in complex systems; in the search for the underlying mechanism of dynamics. In reliability engineering, we use time and its extensions, for instance, random time, stopping time, censoring time, and fuzzy time, etc, in a seemingly natural and conventional way without any examination. The well-accepted notion of time is actually a concept of Newtonian mechanics and behaves like a geometric straight line. In this paper, we begin with an examination of the variability nature of time with a sequence examples in physics, biology, probability. The foundations of the concept of time both philosophically and mathematical are examined. Furthermore, the contradiction between engineering practices and reliability modeling exercises allows us to argue that the variable characteristic of time should be a basic component for defining both the system operating time and system age in reliability engineering modeling and maintenance policy decision-making. We also discuss the multivariate nature of system operating time and system age. Pre-time (event-ordering) and parametric time concepts are explored in the reliability engineering context and the application areas, including system substitutional time and system entropy time are discussed. 1. INTRODUCTION Natural scientists are not always pleased with the physical context of the concept of time measured by physical clocks and imaged as points of the real axis. Physics "specifies" time by excluding the formation, i.e., the property of time described using the concepts of past, present and future rather then terms "before" and "after". In natural science where all the studied objects are frail, where the beginning and the end of each reality is so essentially inevitable, where the reversibility of phenomena is an exception rather than a rule, the "specific" image of time can dramatically narrow the possibility of extrapolating the physical concept of time beyond the frames of special relativity. The interpretation of time as an intrinsic property of a physical system goes beyond the frames of the traditional physical description (Prigogine 1984). The question of whether the time in physics is the one in natural science is so far unsolved, though it is repeatedly discussed. In natural science, it is a common understanding that time is an explanatory variable with no taste, no colour, no direction and a non-reversible flow of real-number system. Time as perceived in the natural world flows uniformly. Researchers such as Prigogine (1996) have recently begun to question nature of time, particularly, the direction of time. 2. TIME IS A MEASURE OF VARIABILITY Levich (1995) pointed out that time is the variability of natural systems and furthermore stressed "to make the difference between time as variability of the objects and time as a tool for measuring this variability." It is obvious that it is impossible to find the laws of variability without having the correct causal and parametric descriptions of time. For instance, in Newtonian mechanics the elementary objects are abstracted as material points with their positions and velocities in the physical space. The variability is determined by the trajectories of the points. The introduction of clocks and parametric time is just an object description of the variability. 986

2 Time is most closely linked to changes. Time can sometimes replace distances, energy, money (a payment for distance) and helps one to specify causes and effects in the event chains. it would be a derivative, secondary concept rather a basic, non-definable one. (Armand 1995). It might be worth mentioning that ancient Chinese scholars in the classic philosophy book Yi Jing ( 易经 ), ( The Book of Changes in western translation), had a systematic exploration on the variability characteristic of time and space. Yi Jing uses the two imagery symbols to represent Yang and Yin respectively, which signify and illustrate the rigid-soft and Figure 1: Yang yao and Yin yao in Yi-Jing dynamic-static characteristic and interaction of changing phenomena of everything in the universe. In the 64-gua Fuxi square-circle graph, it is clear that time and space (location and direction) are fully specified by the Yin-Yang changing evolutions. The circle graph shows the time evolution rule which represents the law of time change in Solar system. The square graph represents the location and direction. Figure 2: Fuxi 64-gua Square-Circle Graph In the Fuxi sense, time is nothing but a measure of change or variability. 3. A RANDOM CLOCK MODEL IN PHYSICS Aristov (1995) denoted as the theoretical time (or model time) and the value of time interval which is the function of differences of spatial positions (for all particles). Then the particle has a displacement and thus the time interval 1 where is the mean square of velocity of all particles. In other words, a time interval between two experiments in which space positions are fixed, is defined as the mean square of differentials of all particles radius vectors. Furthermore, it is assumed that an "ideal physical clock" is related to a model clock as following 2 Then the relative difference between them is 3 987

3 4 When,, then classical clock model is accurate, otherwise, the errors are of order. 4. ANCIENT CHINESE TIME SYSTEM The Chinese Lunar calendar is expressed in terms of the heavenly stem and earthly branch, in total sixty stem-branch combinations. Stem and branch are not only symbols recording year, month, day and time but also have rich connotations, life and astronomical significance. The heavenly stems use upper case letters, to distinguish them from the earthly branches. Stem Interpretation Comparative life Yin-Yang Wu Xing Direction stage JIA Yang Wood East Yang Qi begins to move. Birth YI Yin Wood East Yin Qi is still strong. Suckling BING Yang Fire South Yin Qi will start to decline as Yang Qi stirs. Toddler DING Yin Fire South Everything flourishes. Youth WU Yang Earth Centre The middle palace, image of five dragons intertwined. Prime years JI Yin Earth Centre The middle palace. Middle age GENG Yang Metal West Crops are maturing. Ageing XIN Yin Metal West Everything has matured. Twilight years REN Yang Water North Yin reaches its apex, and Yang is Death born. GUI Yin Water North The land is flat, image of water flowing into the ground from all directions. Decomposition TABLE 1. The Ten Heavenly Stems There are twelve earthly branches, indicated with lower case letters, which similarly represent the myriad routes which life can take as it proceeds through the regenerating cycle of fragile youth to powerful maturity, to frail age and death. FIGURE 3: The Correspondence Of Stem, Branch, Wu-xing, Four Seasons, Lunar Months, Times And Directions In the Northern Hemisphere 988

4 The associations that have been so long established between Stem and Branch and the various times and directions mean that all years, months, days and even hours can be identified by their Stem and Branch, and then associated with an eight trigram (Ba Gua), direction and number. A Chinese lunar month begins with the New Moon and a Chinese lunar year starts from the Starting of Spring. The Chinese time system richly reveals the life changes associated with seasons and therefore illustrates the general variability (life) nature of time of planet of Earth. Time being a component of the Universe, it possesses Yin-Yang and Wu-xing characteristics and the flow of time is associated with variation 5. BACKMAN'S ORGANIC TIME Growth is the basis of life and is a sure expression of the inner source of life... The possibility to foresee the events of life's run is based on the knowledge of that fact that organisms have their `own time', what I called organic time. Maurins (1995) pointed out that the concept basis of Backman's function of growth is the postulate that the logarithm of growth rate is negatively proportional to the square of the time's logarithm where constant k 2 <0. The detection of the growth curve 5 where C 0 is the constant of the proportionality of quantitative growth values according to a standard integral, x is the measure of organic time 6 with scalar constant C 1 of organic time and constant C 2 being defined as the logarithm value of the physical measure, at the time necessary to reach half of the whole size t 0.5 is measured. Backman further noticed that values 7 have an especially important biological significance. The difference 8 9 which is named as the quantum of life. Backman stated that x 2 of an organism at a given instant reflects the number of the already gone through and forth coming quanta of life. On the basis of this concept, he derived a formula of detecting the time of the most greatly expressed potencies ("strength of life"), t v, the transition time from quick to slow down aging, t a, and the general life duration, t d. Backman s organic life is clearly a reflection of that time is just the changing state of an organism. 6. QUALITATIVE PROPERTIES OF TIME The qualitative or topological properties of time are more fundamental, in that they hold independently of specific procedures of measurement and remain unchanged even if the forms of measuring time are varied. (Reichenbach 1956) 989

5 Time flows from the past to the future and the present divides the past from the future. Furthermore, it is noticed that the past is expired, it cannot be changed by us at the present and it never returns. However, what may be changed is the future subject to a great limitation. Ordering is a fundamental characteristic of time and thus any mathematical or statistical modeling of a system must clearly reflect such a characteristic. Any modeling techniques, say, histograms, fitting an empirical distribution, Q-Q plot, etc, require re-examination of their foundations. On the other hand, time-plots and time-series analysis, and stochastic process related methodology are logically solid. Knowing or unknowing is not the criterion to differentiate the past and future. Some future astronomical events are well-known facts but some historical event, for instance, the Tomb of Qin Shi Huang (The first Emperor of Qin Dynasty) may remain unknown, although the clay soldiers and horses have already attracted a great popularity. Our knowledge of the past is based on records. In statistical language, any revelation of the past event is dependent upon the past data information and the correct data processing and inferences. The future event can only be forecasted only based on the comprehensive information covering the total occurrence. Therefore, the past can be recorded as data but not the future. The past events are established facts while the future events are not. From such a viewpoint, the past events are deterministic but future event uncertain. However, the uncertainty aspect, particularly, fuzzy uncertainty exists all the way through no matter the past event or a future one. The suitable explication of time is rooted in the relationship of causality. There is a close connection between time order and causal sequences. It is obvious that time order is reducible to causal order. Causal connection is a mapping between physical (or chemical, bio-chemical etc) events, which can be formulated in objective terms. If we define time order in terms of causal connection, we have shown which specific features of physical reality are reflected in the structure of time, and we have given an explication of the vague concept of time order. The relation of cause and effect could be assumed as a primitive term and the temporal order, and even spatial order, could be reduced to this primitive relation. Order and direction are two related concepts but there is distinction between them. Time does not only have an order but also a direction. Pimenov (1995) systematically investigated the mathematical temporal constructions. 7. SUBSTITUTIONAL TIME AND ENTROPY TIME The substitutional time has been defined as the number of elements replaced in the system. The entropy time is connected with the number of transformations of the system. Since (according to the variability unification principle) the transformations consist in just element replacements, at a qualitative level a relation between metabolic and entropy time is evident. However, there is an unambiguous quantitative relation between them. For example, for the variation of species structure in ecology of communities (Levich 1995), the entropy is where 10 by taking into account that 11 The entropy is converted into where the Lagrange multipliers are also -dependent. 990

6 8. STATISTICAL MODELING OF TIME AND AGE System age is actually a general variation process. The variability of the system is therefore conditional, multidimensional, directional, ordered, locally cyclic, and even random (and fuzzy) in nature. Also, it should emphasize that time-age processes, i.e. the general variation processes, are the reflections of the knowledge of the causes of effects in the event chains. Therefore, in order to match an age model to a system, the knowledge, i.e. the information records about the system should play a critical role. Unfortunately, it is often the exercise of the reliability analyst to impose a model frame on a collected data, rather to understand the variability process of the system. This may be partially addressing the dilemma in reliability modeling practices up to today: on the hand, there is a scarcity of system data for modeling purposes and on the other hand, the huge volume of system operation information records collected by system control panel automatically are never attracting attention to system analysts. The understanding of time and age should be always reflected in the statistical analysis. In particular, some statistical methodology will destroy certain critical features of time and age, say, ordinality. Then the statistical analysis will no longer make sense in the reliability engineering context but constitutes a data manipulations game. 10. CONCLUDING REMARKS In this paper, we are trying to explore the nature of time, particular, the changing-variability characteristic from the large time research literature. Although Newtonian metric time is deeply imprinted, it is not the full picture. Time, as constituted by variability processes, is conditional, multidimensional, directional, ordered, locally cyclic, and even random (and fuzzy) in nature. Time should not be separated from the particular evolving material environment. As Vladimirov s (1995) pointed out, "time and space form a single whole". 11. REFERENCES 1. Armand, A. D. (1995). Time in the Earth Sciences. On the way to understanding the time phenomenon: The construction of time in natural science, Part One Editor Levich A. P. World Scientific Publishing Co. Pte. LtD, pp Aristov, V. V. (1995). Relative Statistical Model of Clocks and Physical Properties of Time. On the way to understanding the time phenomenon: The construction of time in natural science, Part One Editor Levich A. P. World Scientific Publishing Co. Pte. LtD, pp Levich, A. P. (1995). System Theory. On the way to understanding the time phenomenon: The construction of time in natural science, Part One Editor Levich A. P. World Scientific Publishing Co. Pte. LtD, pp Maurins, A. M. (1995), Backman s Conception of organic time and the experience of its applications. On the way to understanding the time phenomenon: The construction of time in natural science, Part One Editor Levich A. P. World Scientific Publishing Co. Pte. LtD, pp Pimenov, R. I. (1995). Mathematical Temporal Constructions. On the way to understanding the time phenomenon: The construction of time in natural science, Part One Editor Levich A. P. World Scientific Publishing Co. Pte. LtD, pp Prigogine, I. and Stengers. I. (1984). Order out of Chaos. Heinmann, London 7. Prigogine, I. (1996). The End of Uncertainty. The Free Press, New York 8. Reichenbach, H., (1956). The Direction of Time. Edited by Reichenbach, M. University of California Press, Berkeley. 9. Sharov, A. A. (1995). Analysis of Meyen s Typological Concept of Time. On the way to understanding the time phenomenon: The construction of time in natural science, Part One Editor Levich A. P. World Scientific Publishing Co. Pte. LtD, pp Vladimirov, V. M. (1995). Time Structure of the World.. On the way to understanding the time phenomenon: The construction of time in natural science, Part One Editor Levich A. P. World Scientific Publishing Co. Pte. LtD, pp