As Ross Ashby noted long ago, no system (neither computer nor organism) can produce something. new if this system does not contain some source of randomness. In a computer, this would be a random number generator, thanks to which the "search" of the machine by trial and error eventually exhausts all the possibilities of the study area.

In other words, everyone who creates something new, that is, creative systems are, in the language of Chapter 2, divergent; on the contrary, sequences of events that are predictable are, ipso facto*, converging.

By the way, this does not mean that all divergent systems are stochastic. This process requires not only access to randomness, but also a built-in comparator, called in evolution "natural selection" and in thinking "preference" or "reinforcement".

It is quite possible that from the point of view of eternity, that is, in a cosmic and eternal context, all sequences of events become stochastic. From this point of view, or even from the point of view of a calmly sympathetic Taoist saint, it may be clear that there is no need for any final preference to guide the whole system. But we live in a limited area of ​​the universe, and each of us exists in a limited time. For us, divergence is real and a potential source of confusion or innovation.

Sometimes I even suspect that we, though bound by illusion, are doing this work of choice and preference for the Taoist who is watching from the outside. (I remember a certain poet who refused military service. He allegedly stated: “I am the civilization for which these guys are fighting.” Maybe he was right in some sense?).

One way or another, apparently, we exist in a limited biosphere, where the main direction is determined by two linked stochastic processes. Such a system cannot remain unchanged for long. But speed change is limited by three factors:

a. The Weismann barrier separating somatic from genetic change, discussed in Section 1 of this chapter, ensures that somatic adjustment does not become recklessly irreversible.

b. In each generation, sexual reproduction ensures that the DNA blueprint of the new cell does not come into sharp conflict with that of the old one, that is, with a form of natural selection acting at the DNA level, no matter what this deviant new blueprint may mean for the phenotype. .

in. Epigenesis acts as a convergent and conserved system; the development of the embryo, in itself, constitutes a selection context that favors conservatism.

The fact that natural selection there is a conservative process, first realized Alfred Russell Wallace. We mentioned earlier, on a different occasion, the related quasi-cybernetic model from his letter to Darwin explaining his idea:

“This principle operates exactly like that of the centrifugal governor of a steam engine, which checks and corrects all deviations almost before they become apparent; similarly, in the animal kingdom, no deviation from the balance can ever reach any noticeable magnitude, since it would become felt at the very first step, making existence difficult and making subsequent extinction almost inevitable.

9. comparison and combination of both stochastic systems

In this section, I will attempt to refine the description of both systems, explore the functions of each, and finally explore the nature larger system universal evolution, which is a combination of these two subsystems.

Each subsystem has two components (as follows from the word stochastic) (see Glossary): a random component and a selection process acting on the products of the random component.

In the stochastic system to which the Darwinists gave most attention, the random component is genetic change, by mutation or by rearrangement of genes between members of a population. I assume that the mutation does not respond to environmental demands or to internal stresses organism. But at the same time, I assume that the mechanism of selection, acting on a randomly changing organism, includes both the internal tensions of each being, and, further, the conditions of the environment acting on this being.

First of all, it should be noted that, since the embryos are protected by the egg or the mother's body, external environment does not have a strong selective influence on genetic innovation before epigenesis has passed through a series of stages. In the past, as still, external natural selection favored changes that protected the embryo and young from external dangers. The result was an ever-increasing separation of the two stochastic systems.

An alternative method to ensure the survival of at least some of the offspring is to multiply them by a large number. If in each cycle of reproduction an individual produces millions embryos, then the rising generation can endure accidental killing, especially leaving alive only a few individuals out of a million. This means a probabilistic attitude to the external causes of death, without any attempt to adapt to their private nature. With this strategy, internal selection also gains the ability to freely control change.

Thus, thanks to the protection of immature offspring, or thanks to the astronomical multiplication of their number, it turned out that in our time, for many organisms, a new form should primarily subject to the constraints of internal conditions. Will the new form be viable in this environment? Will the developing embryo be able to tolerate the new form, or will the change cause lethal abnormalities in the development of the embryo? The answer will depend on the somatic flexibility of the embryo.

Moreover, in sexual reproduction, the combination of chromosomes during fertilization inevitably leads to a process of comparison. Whatever is new in the egg or in the sperm has to meet the old in its partner, and this test favors conformity and immutability. Too abrupt innovation will be eliminated as incompatible.

The process of fusion in reproduction is followed by all the complexities of development, and here the combinatorial aspect of embryology, emphasized by the term epigenesis*, requires further conformity testing. As we know, in the status quo ante, all compatibility requirements have been met to produce a sexually mature phenotype. If this were not the case, then the status quo ante would never exist.

It is very easy to fall into the delusion that the viability of the new means that something was wrong with the old. This view, to which organisms already suffering from the pathologies of too rapid, reckless social change are inevitably inclined, is, of course, for the most part erroneous. Is always need to be sure that the new no worse old. We still don't have confidence that a society with internal combustion engines is viable, or that electronic means of communication like the television are compatible with the aggressive intraspecific competition generated by the Industrial Revolution. Other things being equal (which rarely happens), the old, somewhat tested, can be considered more viable than the new, which has not been tested at all.

Thus, internal selection is the first series of trials for any new genetic component or combination.

In contrast, the second stochastic system has its direct roots in external adjustment (i.e., in the interaction between phenotype and environment). The random component is delivered here by a system consisting of a phenotype interacting with the environment.

Particularly acquired traits, caused by a reaction to some given environmental change, can be predictable. If food delivery is reduced, then the body is likely to lose weight, mainly due to the metabolism of its own fats. Exercise and non-exercise cause changes in the development or underdevelopment of individual organs, and so on. Similarly, individual changes in the environment can often be predicted: a colder climate change can be predicted to reduce local biomass, and thereby reduce food delivery to many species of organisms. But together the phenotype and the organism produce something unpredictable. Neither the organism nor environment has no information about what the partner will do in the next step. But this subsystem already has a selection component, to the extent that the somatic changes caused by the habit and the environment (including the habit itself) are adaptive. (A broad class of environmental and experiential changes that are neither adaptive nor conducive to survival is known as addictions).

Environment and physiology together offer somatic changes that may or may not be viable, and their viability is determined by the current state of the organism, which determines genetics. As I explained in Section 4, the limits that somatic change or learning can reach are ultimately determined by genetics.

As a result, the combination of phenotype and environment constitutes a random component of the stochastic system, which offers change; and the genetic state has, allowing some changes and disallowing others. Lamarckists would like somatic changes to control genetics, but the opposite is true. It is genetics that limits somatic changes, making some possible and others impossible.

Moreover, the genome of an individual organism, where the possibilities of change reside, is what computer engineers would call data bank– it delivers a supply of available alternative ways of accommodation. In a given individual, most of these alternatives remain unused and therefore invisible.

Similarly, in another stochastic system, the genome populations is now thought to be extremely heterogeneous. All possible genetic combinations, even rare ones, are created by the rearrangement of genes in sexual reproduction. Thus, there is a vast supply of alternative genetic pathways that a natural population may choose under selection pressure, as shown by Waddington's research on genetic assimilation (discussed in Section 3).

If this picture is correct, then both the population and the individual are ready for change. It can be assumed that there is no need to wait for the proper mutations, and this is of some historical interest. As you know, Darwin wavered in his views on Lamarckism, believing that geological time was not enough for the process of evolution to operate without Lamarckian heredity. Therefore, in later editions of the Origin of Species, he adopted Lamarck's position. Theodosius Dobzhansky's discovery that the unit of evolution is the population, and that the population is a heterogeneous repository of genetic possibilities, greatly shortens the time required by evolutionary theory. The population is able to immediately respond to environmental pressure. The individual organism has the capacity for adaptive somatic change, but it is the population, through the selective elimination of individuals, that brings about the change that is passed on to future generations. The subject of selection is possibility somatic change. The selection carried out by the environment acts on populations.

We now turn to the study of individual contributions to the overall process of evolution of each of these two stochastic systems. It is clear that in each case the direction of the changes that ultimately enter the overall picture is determined by the selective component.

The time structure of the two stochastic processes is necessarily different. In random genetic change, a new state of DNA exists from the moment of fertilization, but may only contribute to external adjustment much later. In other words, the first test of genetic change is a test conservatism. Consequently, it is precisely this internal stochastic system that guarantees the formal similarity of internal relations between parts (ie, homology), which is so noticeable in all cases. In addition, it is possible to predict which of the many kinds of homology will be most preferred for internal selection; answer primarily- cytological: this is a striking similarity that connects the whole world of cellular organisms. Wherever we look, we find comparable forms and processes in cells. The dance of chromosomes, the mitochondria and other organelles of the cytoplasm, the uniform ultramicroscopic structure of flagella wherever they occur, in both plants and animals, all these profound formal similarities are the result of internal selection insisting on conservatism at this elementary level.

The question of the further fate of the changes that survived the first cytological tests leads us to a similar conclusion. The change that affected earlier stage of the life of the embryo, should disrupt a longer and, accordingly, more complex chain of further events. It is difficult or impossible to indicate any quantitative estimates of the distribution of homologies in the history of organisms. When they say that homology is most pronounced on the most early stages gamete production, fertilization, and so on, this means some quantitative statement about degrees homology, giving importance to characteristics such as chromosome number, mitotic patterns, bilateral symmetry, five-toed limbs, central nervous system with spinal cord, and so on. Such estimates are, of course, highly artificial in a world where (as noted in Chapter 2) quantity never determines a pattern. But the intuition still remains. The only the formal patterns shared by all cellular organisms—both plants and animals—are at the cellular level.

An interesting conclusion follows from this line of thought: After all the controversy and doubt, the theory of repetition deserves support. There is a priori reason to expect that embryos will, in their formal patterns, more closely resemble the embryonic forms of their ancestors than adults do the forms of their adult ancestors. This is far from what Haeckel and Herbert Spencer dreamed about, who imagined that embryology should follow the path of phylogeny. The modern formulation is more negative: Departure from the beginning of the path is more difficult (less likely) than deviating from the later stages. If we, as evolutionary engineers, were faced with the task of choosing the path of phylogeny from free-swimming, tadpole-like organisms to sessile, worm-like, mud-dwelling Balanoglossus, then we would find that the easiest way of evolution would be to avoid too early or too abrupt disturbances in the embryonic stage. Maybe we would even find that evolutionary the process is simplified by subdividing epigenesis by delimiting the individual stages. We would then arrive at an organism with free-swimming, tadpole-like embryos that at some point metamorphose into worm-like, sessile adults.

The mechanism of variability doesn't just allow, and it doesn't just create. It has a continuous determinism, where possible changes are Class changes suitable for this mechanism. The system of random genetic changes, filtered by the selective process of internal viability, gives phylogenesis the character of ubiquitous homology.

If we now consider the second stochastic system, we will come to a completely different picture. While no amount of learning or somatic change can directly affect DNA, the obvious way is that somatic changes (i.e., the proverbial acquired traits) are usually adaptive. In terms of individual survival and/or reproduction and/or simple convenience and stress reduction, adaptation to environmental change is beneficial. This adjustment occurs at many levels, but at each level there is a real or perceived advantage. A good idea- rapid breathing when you get to a high altitude; It's also a good idea to learn how to manage without getting short of breath if you have to stay long in the mountains. It is a good idea to have a physiological system that can adapt to physiological stress, although such adaptation leads to acclimatization, and acclimatization can become an addiction.

In other words, somatic adjustment always creates a context for genetic change, but whether such genetic change then occurs is quite another matter. I will leave this question aside for now and consider what range of genetic changes may be suggested by somatic change. Of course, this spectrum or this set of possibilities sets an outer limit to what a given stochastic component of evolution can achieve.

One common feature somatic variability is immediately obvious: all such changes are quantitative or - as computer engineers would say - analog. In the animal body, the central nervous system and DNA are largely (maybe completely) discrete, but the rest of the physiology is analog.

Thus, comparing the random genetic changes of the first stochastic system with the reactive somatic changes of the second, we again meet the generalization emphasized in Chapter 2: The quantity does not determine the pattern. Genetic changes may be highly abstract, may operate at a distance of many steps from their final phenotypic expression, and, undoubtedly, in their final expression they may be both quantitative and qualitative. But the somatic changes are much more immediate and, I believe, purely quantitative. As far as I know, descriptive sentences that introduce patterns in common with other species (i.e., homologies) into the description of a species are never disturbed by somatic changes that habit and environment can produce.

In other words, the contrast demonstrated by D'Arcy Thompson (see Fig. 9) seems to have roots in (i.e. follows from) the contrast between the two great stochastic systems.

Finally, I must compare thought processes with the dual stochastic system of biological evolution. Is such a dual system also inherent in thinking? (If this is not the case, then the whole structure of this book becomes questionable.)

First of all, it is important to note that "Platonism," as I called it in Chapter 1, is made possible in our day by arguments that are almost the opposite of those that dualistic theology would prefer. The parallelism between biological evolution and the mind (mind) is created not by postulating an Engineer or a Master hiding in the mechanism of the evolutionary process, but, on the contrary, by postulating stochastic thinking. Nineteenth-century critics of Darwin (especially Samuel Butler) wanted to introduce into the biosphere what they called "mind" (i.e. supernatural entelechy*). Nowadays, I would emphasize that creative thought always contains a random component. The research process is an endless process trial and error psychic (mental) progress - can achieve new only by entering on randomly arising paths; some of them, when tested, are somehow selected for something like survival.

If we allow for the fundamentally stochastic nature of creative thinking, then a positive analogy arises with several aspects of the human mental process. We are looking for a binary separation of the thought process, stochastic in both halves and such that the random component of one half must be discrete, and the random component of the other half must be analog.

The simplest way to approach this problem seems to be to consider first the selection processes that determine and limit its outcomes. Here we meet with two main ways of testing thoughts or ideas.

The first of these is a test of logical coherence: does the new idea make sense in light of what is already known or what we believe? Although there are many kinds of meaning, and although “logic”, as we have seen, is only a poor model of how things are in the world, yet the thinker’s first requirement for the concepts that arise in his mind remains something like coherence or coherence - strict or imaginary. On the contrary, the generation of new concepts depends almost entirely (though perhaps not entirely) on the rearrangement and recombination of existing ideas.

Indeed, there is a remarkably close parallel between the stochastic process that takes place inside the brain and another stochastic process, the genesis of random genetic changes, on the results of which the process of internal selection works, providing some correspondence between the old and the new. And with a closer study of this subject, the formal similarity seems to increase.

In discussing the contrast between epigenesis and creative evolution, I pointed out that in epigenesis all new information must be left aside, and that this process is more like deriving theorems within the framework of some initial tautology. As I noted in this chapter, the entire process of epigenesis can be seen as a filter that explicitly and unconditionally requires the growing individual to conform to certain standards.

Now, we notice that there is a similar filter in the intracranial thought process, which, like epigenesis in the individual organism, demands obedience and imposes this demand through a process more or less reminiscent of logic (i.e. similar to the construction of a tautology to create theorems ). In the process of thinking severity similar internal connectivity in evolution.

In summary, the intracranial stochastic thinking or learning system closely resembles that component of evolution in which random genetic changes are selected by epigenesis. Finally, the cultural historian has at his disposal a world where formal resemblance persists through many generations of cultural history, so that he can look for appropriate patterns there in the same way that a zoologist looks for homology.

Turning now to another process of learning or creative thinking, involving not only the brain of the individual, but also the world around the organism, we find an analogue of this process in evolution, where experience creates that relationship between the organism and the environment, which we call adaptation by imposing on the body changes in habits and soma.

Every action of a living organism involves a certain amount of trial and error, and for a trial to be new, it must be somewhat random. Even if the new action is only an element of some well-studied class action, yet, since it is new, it must become, to some extent, an acknowledgment or study of the proposition "this is how it is done."

But in learning, as in somatic change, there are limits and facilitation that take away what can be learned. Some of them are external to the body, others are internal. In the first case, what can be learned at the moment is limited or facilitated by what has been learned before. In fact, there is still learning about how to learn - with a finite limit determined by the genetic makeup - that can be immediately changed in response to the demands of the environment. And at every step, ultimately, genetic control is at work (as was pointed out in the discussion of somatic variability in Section 4).

Finally, it is necessary to compare both stochastic processes, which I have separated for the purpose of analysis. What formal relationship exists between them?

As I understand it, the essence of the matter lies in the contrast between discrete and analog, or, in another language, between name and named process.

But naming is a process in itself, and one that takes place not only in our analysis, but also, in a profound and significant way, in the very systems we are trying to analyze. Whatever the coding and mechanical relationship between DNA and the phenotype, DNA is still in some way a commanding organ prescribing—and in this sense naming—the relationships that must manifest themselves in the phenotype.

But if we admit that naming is a phenomenon that occurs in and organizes the phenomena we study, then we admit ipso facto that we expect to find in this phenomenon a hierarchy of logical types.

Up to this point we can get by with Russell and Principia.¦ But now we are no longer in the Russellian world of abstract logic and mathematics, and we cannot accept an empty hierarchy of names or classes. Mathematician is easy to talk about names of names of names or about classes classes classes. But for a scientist this empty world is not enough.+ We are trying to understand the interweaving or interaction of discrete stages (ie names) with analog stages. The naming process itself is naming, and this fact forces us to replace alternation simple ladder of logical types offered by Principia.

In other words, in order to reunite the two stochastic systems into which I have subdivided both evolution and mental process for the purpose of analysis, I will have to consider both in alternating order. What's in Principia appears as a ladder of steps of one kind (names of names of names, and so on), becomes an alternation of steps of two kinds. To come from name to name name we need to go through process name naming. There must always be a generation process that creates classes before they can be named.

This very broad and complex subject will be dealt with in Chapter 7.

Time series. A time series is a set of observations generated sequentially over time. If time is continuous, the time series is said to be continuous. If time changes discretely, the time series is discrete. Discrete time series observations made at time points can be denoted by . This book deals only with discrete time series in which observations are made at a fixed interval. When there are consecutive values ​​of such a series available for analysis, we write , denoting observations made at equally spaced points in time . In many cases, the values ​​of and are not important, but if you need to accurately determine the times of observations, you need to specify these two values. If we take it as the beginning and as the unit of time, we can consider it as an observation at time .

Discrete time series can appear in two ways.

1) A selection from continuous time series, for example, in the situation shown in fig. 1.2, where the values ​​of the continuous input and output of the gas furnace are read at intervals of 9 s.

2) Accumulation of a variable over a certain period of time; examples are rainfall, which typically accumulates over periods such as a day or a month, or the output of batches of product that accumulates over a cycle. For example, in fig. Figure 2.1 shows a time series of yields from 70 consecutive batches of chemical process product.

Rice. 2.1 Yield of 70 consecutive batches of chemical process product .

Deterministic and random time series. If the future values ​​of the time series are precisely determined by some mathematical function, such as

,

the time series is called deterministic. If future values ​​can only be described using a probability distribution, the time series is said to be non-deterministic, or simply random. Data on batches of a product on fig. 2.1 is an example of a random time series. Although there is a distinct up-and-down trend in this row, it is impossible to accurately predict the output of the next batch. In this book, we will explore just such random time series.

Stochastic processes. A static phenomenon that develops in time according to the laws of probability theory is called a stochastic process. We will often refer to it simply as a process, omitting the word "stochastic". The time series to be analyzed can be considered as one particular implementation of the system under study, generated by a hidden probabilistic mechanism. In other words, when analyzing a time series, we consider it as a realization of a stochastic process.

Rice. 2.2 Observed time series (bold line) and other time series that are realizations of the same stochastic series.

Rice. 2. 3. Isolines of the density of the two-dimensional probability distribution describing stochastic process at times and , there is also a marginal distribution at time .

For example, when analyzing the yield data of a batch of product in Figure 2.1, we can imagine other sets of observations (other realizations of the stochastic process that generates these observations) that can be generated by the same chemical system, in the same cycles. So, for example, in Fig. Figure 2.2 shows the yields of product batches from to (bold line) along with other time series that could be obtained from a population of time series defined by the same stochastic process. It follows that we can consider an observation at a given time , say, as a realization of a random variable with a probability density . with a probability density .

Consider a variable that obeys a Markov stochastic process. Let's assume that its current value is 10, and the change during the year is described by the function 0(0, 1), where a) is a normal probability distribution with mathematical expectation // and standard deviation o. What probability distribution describes the change in this variable over two years?
The change in the variable after two years is described by the sum of two normal distributions with zero mathematical expectations and unit standard deviations. Because the variable is Markovian, these distributions are independent of each other. By adding two independent normal distributions, we get a normal distribution, the mathematical expectation of which is equal to the sum of the mathematical expectations of each of the terms, and the variance is the sum of their variances. Thus, the mathematical expectation of changes in the variable under consideration over two years is zero, and the variance is 2.0. Therefore, the change in the value of the variable after two years is a random variable with a probability distribution φ(0, %/2).
Consider next the change in the variable over six months. The variance of changes in this variable during one year is equal to the sum of the variances of these changes during the first and second six months. We assume that these variances are the same. Then the variance of changes in the variable over six months is 0.5, and the standard deviation is 1/0.5. Therefore, the probability distribution of the change in the variable over the course of six months is φ(0, \DW)
Similar reasoning allows us to prove that the change in the variable over three months has a distribution of 0(0, ^/0.25). Generally speaking, the change of a variable over a time period of length T is described by the probability distribution φ(0, \[T) ).
The square roots in these expressions may seem strange. They arise from the fact that, in the analysis of a Markov process, the variances of changes in a variable at successive points in time add up, but standard deviations do not. In our example, the variance of changes in a variable during one year is 1.0, so the variance of changes in this variable for two years is 2.0, and after three years it is 3.0. At the same time, the standard deviation
of changes in variables after two and three years are \/2 and \/3, respectively. Strictly speaking, we should not say that the standard deviation of changes in a variable in one year is 1.0 per year. It should be said that it is equal to the "square root of unity per year." This explains why the amount of uncertainty is often considered to be proportional to the square root of time.
Wiener processes
The process to which the variable discussed above is subject is called the Wiener process. It is a special case of the Markov stochastic process, when the expectation of changes in the variable is zero, and their variance is 1.0. This process is widely used in physics to describe the motion of a particle involved in a large number of collisions with molecules (this phenomenon is called brownian motion(Brownian motion)).
Formally speaking, a variable z obeys a Wiener process if it has the following properties.
PROPERTY 1. The change in Az over a small time interval At satisfies the equality
Az = ey/At, (12.1)
where e is a random variable obeying the standardized normal distribution φ(0.1).
Property 2. The values ​​Az on two small time intervals At are independent.
It follows from the first property that the quantity Az has a normal distribution, in which the mathematical expectation is equal to zero, the standard deviation is equal to VAt, and the variance is equal to At. The second property means that the quantity 2 obeys a Markov process.
Consider an increase in the variable z over a relatively long period of time T. This change can be denoted as z(T) - z(0). It can be represented as the sum of the increase in the variable r over N relatively small time intervals of length At. Here
Consequently,
z(t)z(o) = J2?^t' (12.2)
r=1
where?r,r = 1,2,...,LG are random variables having a probability distribution φ(0,1). From the second property of the Wiener process it follows that the quantities?
?; are independent of each other. It follows from expression (12.2) that the random variable z(T) - z(0) has a normal distribution, the mathematical expectation of which is zero, the variance is NAt = T, and the standard deviation is y/T. These conclusions are consistent with the results indicated above. Example 12.1
Suppose that the value of r of a random variable obeying the Wiener process at the initial moment of time is 25, and time is measured in years. At the end of the first year, the value of the variable is normally distributed with an expected value of 25 and a standard deviation of 1.0. At the end of the fifth year, the value of the variable has a normal distribution with a mean of 25 and a standard deviation of n/5, i.e. 2.236. The uncertainty of a variable's value at some point in the future, as measured by its standard deviation, increases as Square root from the length of the predicted interval. ?
In mathematical analysis, the passage to the limit is widely used, when the value of small changes tends to zero. For example, when At -> 0, the quantity Ax = aAt turns into the quantity dx = adt. In the analysis of stochastic processes, similar notation is used. For example, as At -> 0, the process Az described above tends to the Wiener process dz.
On fig. Figure 12.1 shows how the trajectory of the variable z changes as At -> 0. Note that this graph is jagged. This is because the change in the variable z over time At is proportional to the value of v^Af, and when the value of At becomes small, the number \/At is much larger than At. Because of this, the Wiener process has two intriguing properties.
1. The expected length of the trajectory that the variable z travels during any period of time is infinite.
2. The expected number of matches of the variable z with any particular value in any period of time is infinite.
Generalized Wiener Process
The drift rate, or drift coefficient, of a stochastic process is the average change in a variable per unit time, and the variance rate, or diffusion coefficient, is the amount of fluctuation per unit time. The drift rate of the main Wiener process dz discussed above is zero and the variance is 1.0. Zero drift means that the expected value of the variable z at any given time is equal to its current value. The unit variance of the process means that the variance of the change in the variable z in the time interval T is equal to its length.
Rice. 12.1. Change in stock price in the example
The generalized Wiener process for x can be defined in terms of dz as follows.
dx - adt + bdz, (12.3)
where a and b are constants.
To understand the meaning of equation (12.3), it is useful to consider the two terms on the right side separately. The term a dt means that the expected drift velocity of the variable x is 0 units per unit time. Without the second term, equation (12.3) turns into the equation
dx=adt,
whence it follows that
dx
Integrating this equation over time, we get
x = xo + a?,
where xo is the value of the variable x at zero time. Thus, over a period of time T, the variable x increases by the value of ee. The term b dz can be thought of as noise, or variability in the trajectory that the variable x travels. The magnitude of this noise is b times greater than the value of the Wiener process. The standard deviation of the Wiener process is 1.0. It follows that the standard deviation of b dz is equal to b. On small time intervals AL, the change in the variable x is determined by equations (12.1) and (12.3).
Ax \u003d aAb ​​+ bEY / Ab,
where e, as before, is a random variable with a standardized normal distribution. So, the quantity Ax has a normal distribution, the mathematical expectation of which is equal to aAb, the standard deviation is 6n/D7, and the variance is b2D/. Similar reasoning can show that the change in the variable x during an arbitrary time interval T has a normal distribution with the mathematical expectation c.T, the standard deviation by/T and the variance b2T. Thus, the expected drift rate of the generalized Wiener process (12.3) (i.e., the average change in drift per unit time) is equal to a, and the variance (i.e., the variance of the variable per unit time) is b2. This process is shown in Fig. 12.2. Let's illustrate the download with the following example.
Example 12.2
Consider a situation in which the share of a company's assets invested in short-term cash positions (cash position) measured in thousands of dollars is subject to a generalized Wiener process with a drift rate of $20,000 per year and a variance of $900,000 per year. year. At the first moment of time, the share of assets is $50,000. After a year, this share of assets will have a normal distribution with a mathematical expectation of $70,000 and a standard deviation of l/900, i.e. $30. Six months later, it will be normally distributed with an expectation of $60,000 and a standard deviation of $30\DC >= $21.21. The uncertainty associated with the share of assets invested in short-term cash equivalents measured using the standard deviation increases as the square root of the length of the predicted interval. Note that this share of assets can become negative (when the company borrows). ?
Ito process
The Ito stochastic process is a generalized Wiener process in which the parameters a and b are functions depending on the variable x and the time t. The Ito process can be expressed by the following formula.
dx = a(x, t)dt + b(x, t)d,z,?
Both the expected drift rate and the variance of this process change over time. In a short period of time from t to At, the variable changes from
x to x + ah where
Ax = a(x, t) At + b(x, t)e\fAt.
This relationship contains a bit of a stretch. It is related to the fact that we consider the drift and variance of the variable x constants, which on the time interval from t to At are equal to a(x, t) and b(x, t)2, respectively.

Stochasticity (ancient Greek στόχος - goal, assumption) means randomness. A stochastic process is a process whose behavior is not deterministic, and the subsequent state of such a system is described both by quantities that can be predicted and random. However, according to M. Katz and E. Nelson, any development of a process in time (whether deterministic or probabilistic) when analyzed in terms of probabilities will be a stochastic process (in other words, all processes that develop in time, from the point of view of probability theory, are stochastic ).

An example of a real stochastic process in our world is the modeling of gas pressure using the Wiener process. Despite the fact that each gas molecule moves along its own strictly defined path (in this model, and not in a real gas), the movement of a set of such molecules is practically impossible to calculate and predict. A sufficiently large set of molecules will have stochastic properties, such as filling the vessel, equalizing pressure, moving towards a smaller concentration gradient, etc. Thus, the emergence of the system is manifested.

The Monte Carlo method gained popularity thanks to the physicists Stanislaw Ulam, Enrico Fermi, John von Neumann and Nicholas Metropolis. The name comes from a casino in Monte Carlo, Monaco, where Uncle Ulama borrowed money to play. Using the nature of chance and repetition to study processes is analogous to the activities that take place in a casino.

Methods for performing calculations and experiments based on random processes as a form of stochastic modeling were used at the dawn of the development of probability theory (for example, the Buffon problem and works on the estimation of small samples by William Gosset), but most developed in the pre-computer era. hallmark Monte Carlo simulation methods is that first there is a search for a probabilistic analogue (see the annealing simulation algorithm). Prior to this, simulation methods went in the opposite direction: simulation was used to test the outcome of a previously determined problem. And although such approaches existed before, they were not common and popular until the Monte Carlo method appeared.

Perhaps the most famous of the early applications of these methods is due to Enrico Fermi, who in 1930 used stochastic methods to calculate the properties of the newly discovered neutron. Monte Carlo methods were widely used during the work on the Manhattan project, despite the fact that the capabilities of computers were severely limited. For this reason, it was only with the advent of computers that Monte Carlo methods began to become widespread. In the 1950s, Los Alamos National Laboratory uses them to create a hydrogen bomb. Methods are widely used in such fields as Physics, Physical Chemistry and Operations Research.

The use of Monte Carlo methods requires a large number random variables, which consequently led to the development of pseudo-random number generators that were much faster than the tabular generation methods previously used for statistical sampling.

The study of statistical patterns is the most important cognitive task of statistics, which it solves with the help of special methods that change depending on the nature of the initial information and the goals of knowledge. Knowing the nature and strength of ties makes it possible to manage socio-economic processes and predict their development.

Among the many forms of connections, the most important is the causal one, which determines all other forms. The essence of causality is the generation of one phenomenon by another. At the same time, the cause itself does not yet determine the effect; it also depends on the conditions in which the action of the cause proceeds. For the occurrence of the effect, all the factors that determine it are needed - the cause and conditions. The necessary conditionality of phenomena by a multitude of factors is called determinism.

The objects of study in the statistical measurement of relationships are, as a rule, the determinism of the effect by factors (cause and conditions). Signs according to their importance for the study of the relationship are divided into two classes. Signs that cause changes in other related signs are called factorial, or simply factors. Signs that change under the influence of factor signs are called effective.

Relationships between phenomena and their features are classified according to the degree of tightness of the connection, direction and analytical expression.

Between various phenomena and their features, it is necessary, first of all, to distinguish two types of relationships: functional (rigidly determined) and statistical (stochastically determined).

The relationship of the attribute "y" with the attribute "x" is called functional if each possible value of the independent attribute "x" corresponds to one or more strictly defined values ​​of the dependent attribute "y". The definition of a functional connection can be easily generalized for the case of many attributes x 1 ,x 2 ,...,x n .

A characteristic feature of functional relationships is that in each individual case, the full list of factors that determine the value of the dependent (resultant) attribute is known, as well as the exact mechanism of their influence, expressed by a certain equation.

The functional relationship can be represented by the equation: y i =f(x i), where y i is the effective feature (i = 1, ..., n); f(x i) is a known function of the connection between the effective and factor characteristics; x i - factor sign.

Most often, functional connections are observed in the phenomena described by mathematics, physics and other exact sciences. Functional connections also take place in socio-economic processes, but quite rarely (they reflect the interconnection of only individual aspects of complex phenomena). public life). In economics, an example of a functional relationship is the relationship between wages y and the number of manufactured parts x for simple piecework wages.

In real social life, due to the incompleteness of the information of a rigidly determined system, uncertainty may arise, due to which this system by its nature must be considered as a probabilistic one, while the relationship between the features becomes stochastic.

A stochastic connection is a connection between quantities in which one of them, a random variable y, reacts to a change in another value x or other values ​​x 1 , x 2 ,..., x n , (random or non-random) by changing the distribution law. This is due to the fact that the dependent variable (resultant sign), in addition to the considered independent ones, is subject to the influence of a number of unaccounted for or uncontrolled (random) factors, as well as some inevitable errors in the measurement of variables. Since the values ​​of the dependent variable are subject to random variation, they cannot be predicted with sufficient accuracy, but only indicated with a certain probability.

A characteristic feature of stochastic relationships is that they appear in the entire population, and not in each of its units (and neither the complete list of factors that determine the value of the effective feature, nor the exact mechanism of their functioning and interaction with the effective feature is known).

The stochastic connection model can be represented in a general form by the equation: ŷ i = f(x i) + ε i , where ŷ i- calculated value of the effective feature; f(x i) - a part of the effective feature, formed under the influence of the considered known factor features (one or many), which are in a stochastic relationship with the feature; ε i- a part of the effective feature that has arisen as a result of the action of uncontrolled or unaccounted for factors, as well as the measurement of features, which is inevitably accompanied by some random errors.

The manifestation of stochastic connections is subject to the action of the law of large numbers: only in a sufficiently large number of units will individual characteristics be smoothed out, chances will cancel each other out, and the dependence, if it has a significant force, will manifest itself quite clearly.

In socio-economic life one has to deal with many phenomena of a probabilistic nature. For example, the level of labor productivity of workers is stochastically related to a whole range of factors: qualifications, work experience, the level of mechanization and automation of production, labor intensity, downtime, the health status of the worker, his mood, atmospheric pressure, and others. Full list factors are almost impossible to determine.

A special case of a stochastic connection is a correlation, in which the average value (expectation) of a random variable of the effective feature y naturally changes depending on the change in another value x or other random variables x 1 ,x 2 ,...,x n . Correlation does not appear in each individual case, but in the entire population as a whole. Only with a sufficiently large number of cases, each value of the random feature x will correspond to the distribution of the average values ​​of the random feature y. The presence of correlations is inherent in many social phenomena.

Depending on the direction of action, functional and stochastic relationships can be direct and reverse. With a direct connection, the direction of change in the resulting attribute coincides with the direction of change in the attribute-factor, i.e. with an increase in the factor attribute, the effective attribute also increases, and vice versa, with a decrease in the factor attribute, the effective attribute also decreases. Otherwise, there are feedbacks between the considered quantities. For example, the higher the qualification of the worker (rank), the higher the level of labor productivity - a direct relationship. And the higher the productivity of labor, the lower the unit cost of production - feedback.

According to the analytical expression (form), the connections can be rectilinear and non-linear (curvilinear). With a straight-line relationship with an increase in the value of the factor attribute, there is a continuous increase (or decrease) in the values ​​of the resulting attribute. Mathematically, such a relationship is represented by the equation of a straight line, and graphically - by a straight line. Hence its more short name- linear connection.

With curvilinear relationships with an increase in the value of the factor attribute, the increase (or decrease) of the effective attribute occurs unevenly, or the direction of its change is reversed. Geometrically, such connections are represented by curved lines (hyperbola, parabola, etc.).

According to the number of factors acting on the resultant attribute, relationships differ between single-factor (one factor) and multi-factor (two or more factors). One-factor (simple) relationships are usually called paired (since a pair of features is considered). For example, the correlation between profit and labor productivity. In the case of a multifactorial (multiple) connection, it is understood that all factors act in a complex manner, i.e. at the same time and in interconnection, for example, the correlation between labor productivity and the level of labor organization, production automation, worker qualifications, work experience, downtime and other factor characteristics.

With the help of multiple correlation, it is possible to cover the whole complex of factor characteristics and objectively reflect the existing multiple relationships.

The word stochastic is used by mathematicians and physicists to describe processes in which there is an element of chance. It comes directly from the Greek word "atoaaizeoa". In Aristotle's ethics, the word is used in the sense of "the ability to guess." Mathematicians have used the word apparently on the grounds that there is an element of chance in having to guess. In Webster's New International Dictionary, the word stochastic is defined as conjectural. We thus notice that the technical meaning of this word is not exactly in line with its lexical (dictionary) definition. In the same sense as "stochastic process", some authors use the expression "random process". In the future, we will talk about processes and signals that are not purely random, but contain randomness to one degree or another. For this reason, we prefer the word "stochastic".

Rice. 3.1-1. Comparison of typical stochastic and predictable signals.

On fig. 3.1-1 compares simple waveforms of stochastic and regular signals. If we repeat the experiment on measuring a stochastic signal, then we will get oscillations of a new form, different from the previous one, but still showing some similarity in characteristic features. Recording ocean waves

is another example of a stochastic signal. Why is it necessary to talk about these rather unusual stochastic signals? The answer to this question is based on the fact that the input signals of automation systems are often not completely predictable like a sine wave or a simple transient. In fact, stochastic signals are found in research automatic systems more often than predictable signals. However, the fact that predictable signals have great importance so far, is not a serious omission. Quite often, one can arrive at an acceptable technique by selecting signals from a class of predictable signals in such a way as to display characteristics true signal, which is stochastic in nature. An example of this kind is the use of several suitably selected sinusoids to represent stochastic changes in the moments that cause roll in a ship stability problem. On the other hand, we encounter problems in which it is very difficult to represent a true stochastic signal using a predictable function. As a first example, consider a diagram of an automatic target tracking and fire control system. Here, the pointing radar device does not measure the pointing error exactly, but only approximately. The difference between the true pointing error and what the radar measures is often referred to as radar noise. It is usually very difficult to approximate radar noise with a few sinusoids or other simple functions. Another example is the weaving of textile fibers. In the process of weaving, a thread is drawn from randomly tangled bundles of fiber (called yarn). The thickness of the thread, in a sense, can be considered as an input signal in the regulation of the weaving process. Variations in this process are due to variations in the number and thickness of individual fibers in the various interlacing sections of the yarn. Obviously, this type of deviation is stochastic in nature, and it is difficult to approximate it with any regular functions.

The previous considerations show that stochastic signals play an important role in the study of control systems. So far, we have been talking about stochastic signals as signals caused by processes containing some element of randomness. To proceed further, we must clarify the concepts of such signals. Modern physics, and especially quantum mechanics, teaches that all physical processes, when examined in detail,

are discontinuous and non-deterministic. The laws of classical mechanics are replaced by statistical laws based on the probability of events. For example, we usually consider the voltage of oscillations that occur on the screen of an oscilloscope vacuum tube to be a smooth function. However, we know that if these oscillations are examined under a microscope, they will not look as smooth due to the shot noise in the tube that accompanies the excitation of the oscillations. After some reflection, it is not difficult to incline to the conclusion that all signals in nature are stochastic. Although at first we assumed that compared to a sinusoid or a unit hop function, a stochastic signal is a relatively abstract concept, but in reality the opposite is true: a sine wave, a unit hop function, and generally regular signals represent an abstraction. However, like Euclidean geometry, it is a useful abstraction.

A stochastic signal cannot be represented graphically in a predetermined way, since it is due to a process containing an element of chance. We cannot say what the magnitude of the stochastic signal is at a future point in time. About a stochastic signal at a future moment of time, one can only say what is the probability that its value falls within a certain interval. Thus, we see that the concepts of a function for a stochastic signal and for a regular signal are completely different. For a regular variable, the idea of ​​a function implies a certain dependence of the variable on its argument. With each argument value, we associate one or more variable values. In the case of a stochastic function, we cannot uniquely relate the value of a variable to some particular value of the argument. All we can do is associate some probability distributions with the particular values ​​of the argument. In a certain sense, all regular signals are the limiting case of stochastic signals, when the probability distributions have high peaks, so that the uncertainty of the position of the variable for a particular value of the argument is zero. At first glance, a stochastic variable may seem so uncertain that its analytical treatment is impossible. However, we will see that the analysis of stochastic signals can be carried out using probability density functions and other statistical characteristics such as means, rms and correlation functions. In view of the statistical nature of stochastic signals, it is often convenient to consider them as elements of a set of signals, each of which is due to the same process. This set of signals is called an ensemble. The concept of an ensemble for stochastic signals corresponds to the concept of a population in statistics. Characteristics of a stochastic signal

refer usually to the ensemble, and not to a partial signal of the ensemble. Thus, when we talk about certain properties of a stochastic signal, we usually mean that the ensemble has these properties. In general, it is impossible to consider that a single stochastic signal has arbitrary properties (with the possible exception of non-essential properties). In the next section, we will discuss an important exception to this general rule.