The process of bringing the system to the best possible state

First letter "o"

Second letter "p"

Third letter "t"

The last beech is the letter "I"

Answer for the clue "The process of bringing a system to its best condition", 11 letters:
optimization

Alternative questions in crossword puzzles for the word optimization

The process of bringing the system to the best (optimal) state

The process of choosing the best possible option

Word definitions for optimization in dictionaries

Wikipedia The meaning of the word in the Wikipedia dictionary
Optimization is the modification of a system to improve its efficiency. The system can be single computer program, a digital device, a set of computers, or even an entire network such as the Internet. Although the goal of optimization is to obtain the optimal...

Economic dictionary terms The meaning of the word in the dictionary Economic dictionary of terms
determination of the values ​​of economic indicators at which the optimum is achieved, that is, the optimal, best state of the system. Most often, the optimum corresponds to achieving the highest score given the cost of resources or achieving a given result ...

Encyclopedic Dictionary, 1998 The meaning of the word in the dictionary Encyclopedic Dictionary, 1998
the process of choosing the best possible option. The process of bringing the system to the best (optimal) state.

Big Soviet Encyclopedia The meaning of the word in the dictionary Great Soviet Encyclopedia
(from Latin optimum ≈ the best), the process of finding an extremum (global maximum or minimum) of a certain function or choosing the best (optimal) option from a variety of possible ones. The most reliable way to find the best option is...

Examples of the use of the word optimization in the literature.

Additional optimization is achieved by filtering the table before it is sent, i.e.

The purposes for which mirroring is used are high availability and optimization read operations.

But even in this case, on many machines it is possible optimization, if you put such an array in write-protected memory.

With his direct participation and support, the main scientific directions: network planning and management, scheduling theory and scheduling, nonlinear and stochastic programming, differential games, dynamic models of the economy, methods of discrete optimization etc.

Quite complex methods are used optimization operators, which we will discuss in detail in the following lectures.

By tradition, England is considered the classic "land of ghosts", almost their homeland, if such a concept is applicable to this kind of intangible phenomena. After all, according to the Dictionary of the Russian Language, a ghost is “the ghost of a person who is absent or dead, appearing to people with a sick imagination.” In turn, a ghost is defined as “anything that only seems real; what is imagining." And the Encyclopedic Dictionary does not mention such phenomena at all.

However, in England the attitude towards ghosts is very serious. And a clear evidence of this is the existence of such unusual at first glance, but quite respectable organizations, as the commercial company "Ghostbusters" and the "Ghost Experts Club".

The headquarters of the first is located in the city of Grimsby, near the east coast of Central England. It is led by Robin Farman, who has been involved in research activities in the field of paranormal phenomena and lecturing on parapsychology. The staff of the company is small: Farmen's wife Sheila, a clairvoyant nicknamed Immortal Aisha; his son Andy; two afterlife experts, Janice Paterson and Rodney Mitchell, and Newfoundland Ben, the Farmen's dog with a phenomenal flair for parapsychological manifestations.

All employees of the company wear the same half-breeches with the inscription "Ghostbusters", decorated with the corresponding pattern and the Grimsby coat of arms. They go to the challenges in an aristocratic black 1959 Austin Princess limousine, formerly owned by the mayor of the city. At the same time, these catchy attributes, outwardly looking like a clowning act, are in fact a deeply thought-out result of many years of serious work and are intended primarily to reassure frightened people by their very appearance. Because it is this category that makes up the vast majority of Farmen's clients.

When many of us read about ghosts making blood-curdling moans or noisy poltergeists wreaking havoc in the house, we probably think, "This will never happen to me," says Farman. - But if, unexpectedly waking up at night in a cold sweat, you see a ghostly luminous figure in the twilight and this is repeated over and over again, and visits to a psychoanalyst do not give results, all your skepticism will surely evaporate. When the furniture begins to inexplicably move around the apartment, and household items fly through the air and you hear the slamming of doors and the steps of some invisible person, then you, no doubt, will passionately want to call for help from specialists from our company ...

"Ghostbusters" does not refuse anyone, no matter where the call comes from and no matter how fantastic it may seem. One of the company's employees immediately leaves the place - as a rule, it is Farmen himself - to meet with a person asking for help. According to Farman, the first thing to do is to establish the nature of what is happening: is it a physical phenomenon or a psychological one; an optical illusion, a play of light, or the result of a special sensitivity of certain unusual people. Here you need a professional. The first person you meet cannot be a ghost hunter. After all, it often happens that someone was confused by a dress thrown on the back of a chair, and someone was reflected in a mirror or even on a polished closet door.

Once we went to one extremely serious person, - says Farman, - and during the preliminary investigation, we noticed that in his house there was a strong smell of some kind of disinfectant, as we thought. When we asked what was going on with him, the owner replied that he was being attacked by some incorporeal beings. I really consider such a phenomenon to be possible in principle, and therefore I was ready to believe in his story.

We had already begun to agree on the upcoming visit of our team, when suddenly my counterpart jumped to his feet. “Look, there, there it is! he shouted, pointing his finger somewhere into the void. - There it is! Now I will finish him! - Having snatched out from behind the chair a huge bottle with fly agaric tincture (as it turned out later), the owner of the house began to fiercely puff out of it in all directions. - Here, finished it! he finally declared triumphantly, wiping the profuse sweat from his forehead. - With them, you understand, only like this ... ". It turned out that he had bottles of fly agaric prepared everywhere in the house and even in the car. It became absolutely clear to me that the person was hallucinating ...

More often, however, it happens that the expert comes to the conclusion that a detailed study is necessary. Then the whole group leaves for the “haunted house” with their unique equipment for detecting ghosts. The kit includes a temperature sensor connected by a long cord to a unit that gives a digital display of temperature fluctuations in the room, since, as the "ghostbusters" have established, as a rule, a sharp decrease is observed during mysterious phenomena.

To measure light radiation, "Robogost" is used - Personal Computer"Aikorn", capable of capturing even the most insignificant light emissions of living and inanimate objects. In addition, it has two more special sensors - one registers sound waves, the other - vibration. The acoustic output is equipped with an amplifier, which makes it possible to hear even the "inaudible" flight of a piece of paper. In this case, if necessary, the speaker itself can be turned off, limiting itself to observing the screen, where the luminous ray shows the strength of the sound that occurs in the room with the sensor.

I must warn you, ghost hunting is a very boring business. After the equipment is installed, it remains only to wait for the ghost to deign to appear. And because their behavior is unpredictable, the wait can be delayed indefinitely. In order not to experience the hospitality of the owners, it is necessary to arrange a temporary headquarters in a local hotel, only from time to time visiting the “haunted house” to take tapes with records of instrument readings.

And yet, if paranormal manifestations do take place, in the vast majority of cases, sooner or later it is possible to catch the perpetrator of the disturbance red-handed, the head of an unusual company assures. “Despite the name of our company, we don't hunt ghosts in the literal sense of the word, we don't shoot them with some kind of neutron gun,” he continues. -

This is nonsense invented by journalists in pursuit of sensationalism. It all depends on what happens in the client's home. It could be a ghost, a ghost, a poltergeist, his own mania, anything. And yet - we act depending on the attitude of the owner to the observed phenomenon. Far from always people want to get rid of their ghosts, it is not so rare that they just want to understand. We also come across those in whose houses ghosts appear from time to time, but the owners keep with them on a friendly footing - the spirits become almost family members.

If someone perceives it as a threat, we can use pentagrams or pyramids - they help. We can invite a medium, and he will ask the ghost to state what is troubling him. After that, appropriate measures should be taken, and the unwanted visitor, most likely, will "move out of the apartment." It happens that the rooms themselves create a special atmosphere. In such cases, it is enough to simply re-paint the walls, open windows and doors, and ventilate the room well to change this atmosphere and reassure the owners.

Another thing is the poltergeist phenomenon, says Farmen. - You do not see any ghost, but suddenly there is a knock on the walls, some steps are heard, trinkets fall from the mantelpiece, otherwise things start flying around the room. It scares people insanely. The poltergeist has a tendency to show up in places of misfortune and frustration, most often in the neighborhood of young people, much less often with the elderly. It is impossible to expel him - he does not give in. But in order to help someone possessed by it, it is possible to use the services of a priest. In addition, poltergeists are very "shy." Sometimes it is enough to start an instrumental investigation to get rid of it.

For many years, Farmen and his comrades have been dealing with mysterious phenomena, but they prefer not to talk too much about their essence.

When we are asked: “Do you believe in ghosts?” - this is a question with a double meaning, because the word "ghost" usually means some kind of deceased who suddenly appears to terrorize the living. I don't believe in such things. But a lot of people observe ghosts of one kind or another, and I'm interested to find out what it is - something from some other, otherworldly world or some anomalies of time. There are a lot of hypotheses in this regard, so there is a rather extensive field for research here ...

That is what the Ghost Experts Club, led by its president, Peter Underwood, is doing. Evidence of ghosts is as old as the world, and examples of this can be found in every civilization, culture, habitat around the world, he argues. Plutarch, Pliny, Socrates, Cicero - they all took the existence of ghosts for granted, and many other prominent personalities in the history of mankind thought the same way.

The most reliable evidence, both ancient and modern, describes these phenomena not at all as something ephemeral, transparent, but as ordinary people of flesh and blood, whose immaterial nature betrays itself when they suddenly disappear under the most mysterious circumstances.

Ghosts are not only in human form. Often people see animals, especially horses and dogs, perhaps because they have long lived next to humans. Ghosts and inanimate objects have been observed, including something like; (on one road in Scotland there is a dangerous bend, where, according to evidence different people, a small blue car appears and disappears); ghost trains (Abraham Lincoln's funeral train is said to have been seen for several years, each time in April, on the New York Central railway, slowly and sadly returning to Illinois). After all, there is a lot of evidence of "ephemeral" furniture, such as armchairs, in which some ghosts like to sit.

Over the years of research, club experts have revealed interesting facts about the "localization" of ghosts. It turned out that, for example, among public buildings they prefer churches, and among residential buildings - houses of priests. It is not surprising, therefore, that most often ghosts appear in the form of a monk or nun, although they have many other images.

There is an interesting photograph of a ghost on the Tulip Stairs in the Queen's House in Greenwich. It was taken in 1966 by Reverend Hardy and his wife, who had repeatedly seen someone's ghost at the site and finally decided to try and photograph it, says Peter Underwood.

At first glance, it seems that the lens captures one figure, covered with a hood, sneaking up the stairs of the Tulip. But if you take a closer look, you can see that there are two figures in the photo, and one almost completely obscures the other. One can clearly distinguish the left hands with the same wedding rings on the same finger. Such a ring could belong to the unfortunate Queen Henrietta, whose house was built by order of her husband, King Charles I. It seems that both figures are the same person, captured in motion.

So how to explain that people are ghosts? According to Underwood, there is no one overarching reason for this. As for individual special cases, he put forward an interesting hypothesis to explain them. It is now recognized that under tragic circumstances or when severely traumatized, people acquire heightened susceptibility. In addition, in an extreme situation, the hidden reserves of the human body, primarily in the field of the psyche, can be released, as well as other unusual effects. But in order to confirm or refute this hypothesis, parapsychologists and psychoanalysts must study and systematize precisely the conditions under which people can see - and see - ghosts.

There are other types of ghosts, for example, crisis or dying visions, while they appear no more than 4 days. Such phantoms usually appear to their relatives, friends or relatives, and later it becomes known that during these 4 days the person whose phantom they saw died or that he was ill and experienced a crisis at that time.

This kind of phenomenon was quite common during both world wars, when people were deeply concerned for the fate of their loved ones, especially if they fought somewhere thousands of kilometers away. There are many testimonies of credible, mentally perfectly healthy individuals who clearly saw a relative who visited them for a moment and then disappeared. Later, they learned that the one they saw died at the very moment when his phantom appeared to them.

However, according to Peter Underwood, lengthy haunted encounters are far more rare than many people think, and far less frightening than we're used to thinking. Of all the cases of "meetings" with them that people talk about, 98% are due to natural causes. And only for 2% it is impossible to find them. Moreover, recently it was possible to discover an objective criterion that makes it possible to reliably judge the authenticity of a particular ghost.

When it appears, a decrease in temperature by 8-9 °C is invariably recorded in the immediate vicinity of the phantom compared to the temperature of the surrounding space. A large number of checks with the help of sensitive equipment revealed another interesting phenomenon: in "haunted houses" in the usual places of their appearance, "cold spots" are formed that persist for a long time. At the same time, none of the club's experts undertake to explain why this is happening.

Underwood himself believes that this phenomenon is most likely associated with the "withdrawal" of energy from our space for the "materialization" of a ghost in it. Perhaps its physical nature can be unraveled if we turn to the latest work in the field of quantum theory. It's about about the so-called quantum inseparability, which implies "non-force", "extra-spatial" and "timeless" connection of particles.

According to this hypothesis, interacting with each other elementary particles can no longer be considered as separate objects, even if they go beyond the zone of conditional interaction. They seem to always "remember" each other. Such "quantum inseparability" essentially means that all objects that once interacted with each other, in a sense, remain connected forever.

Moreover, this fixation of quantum connections has a "non-local", "extra-spatial" character. Unlike gravity and electromagnetism, it arises not due to some fields, but absolutely independently. Since, in this case, nothing real, tangible is transferred through the space separating the particles, the matter located between them cannot slow down such a connection.

Extra-spatial connections do not stretch anywhere, and, therefore, they are not afraid of distances. They are as powerful a million miles away as they are a millimeter away. And because these connections "ignore" space, they also do not care about time. For an outside observer, in accordance with the theory of relativity, this effect - occurring instantly, ahead of light, the mutual influence of particles - should seem like a movement backwards in time. It is possible to assume that it is in this way, literally passing through the walls, that ghosts arise, taking energy from our space for their “materialization”. And evidence of this is the drop in temperature at the place of their appearance.

There are no hard and fast rules in "ghostology", and we in our club are always looking for amendments to our hypotheses and ideas. We always try to keep our minds free and admit that in the world of mechanization, standardization and automation there are things that we cannot explain, says Peter Underwood. - And Dr. Samuel Johnson, a medieval English writer, author of the philosophical treatise "Rasselas, Prince of the Abyssinian", was not the only person who believed that the ghost phenomenon is "one of the most important issues, which can appear before the human mind ... This question, even now, after millennia of the existence of the human race, is still not resolved.

It seems more and more likely that the border between that world and our world, so familiar and comfortable, is just one step away from us. The data we have collected suggests that in the course of each exploration of the Ghost Experts Club, we may be constantly walking along the border of the unknown ...

Although Dr. Donald Carpenter, an American theoretical physicist, was not a member of the English "Ghost Experts Club", it was the work of this society that played a major role in his decision to enter into such a mystical field as ghosts. At the same time, it was not at all in the physical hypotheses of Peter Underwood, although they probably contained a rational grain. Carpenter was much more interested in the statistics of temperature fluctuations given in the works, obtained with the help of modern measuring equipment.

However, to be completely frank, it should be noted that interest in the research of English ghostwriters, accidentally seen on the shelf of the university library, was aroused, nevertheless, by one childhood memory of Carpenter himself. At the age of 9, he and his mother were at home when the ghost of his father suddenly appeared in the room, who had gone on business to another city.

They later learned that he had died that very night. As an adult, Carpenter once again encountered a ghost. 1971 - He spoke with a railroad worker who, as Carpenter later found out, had died in an accident a few hours earlier.

The task that this physicist has set for himself, at first glance, may seem absurd - to calculate the mass of a ghost.

In order to appreciate the courage of an American, one must take into account that, with rare exceptions, scientists, and especially physicists, a priori consider ghosts, spirits, ghosts to be complete nonsense. For their relationship, the well-known formulation is perfect: "This cannot be because it can never be." In fact, they refuse in advance to consider everything that is beyond the scope of modern scientific knowledge besides, it is difficult to reproduce with the help of experiments. And this applies not only to ghosts, but to a wide range of strange phenomena, such as UFOs, telepathy, poltergeists.

By the way, it can be recalled that at one time the French Academy of Sciences did not allow the possibility of meteorites falling to Earth, and back in the 30s it was considered an illusion.

Because the problem of the other world and everything connected with it is too extensive and contradictory, before embarking on the solution of the task, Carpenter tried to clearly define the conditions necessary for this. Studying works English club took a lot of time. But in the end, he had several starting points that facilitated the upcoming work.

Firstly, only “pure” ghosts were subject to consideration, leaving out spirits, poltergeists, ghosts of animals and inanimate objects. Further, it was necessary to proceed from the fact that, since the phenomena under consideration are homogeneous, they must have the same main characteristics.

After all, since ghosts in at least one aspect - the temperature jump at their appearance - certainly interact with our world, all known physical laws must apply to them and no reference to magic and miracles is allowed. And the last. Based on the fact that, according to the descriptions of eyewitnesses, the ghosts looked like ordinary people from blood and flesh, the size of phantoms in three-dimensional space should be on average 0.07 cubic meters (this is exactly the volume of a person weighing 70 kg).

It would seem that formulating the above 4 points is not so difficult. But this is only at first glance. Carpenter had to read hundreds of pages of eyewitness accounts, tabulating recurring points before reaching final conclusions about appearance ghosts.

As a rule, it appears before a person face to face, and only in those cases when there are several people in the room, someone sees it from the side. A weak white glow emanates from the phantom, rarely bluish or greenish, sometimes with a reddish tinge, with a power of one to 20 watts. In some cases, a faint smell may be felt near the ghost.

Behind the impossibility of comparison, eyewitnesses define it with one word: "Strange." The sounds that the ghost makes are often limited to groans, although sometimes howls, screams, ringing of chains, some kind of "grave" laughter are heard, very rarely - individual words. The duration of the phantom phenomenon ranges from a few seconds to 10 minutes or more, with short-term visits being most frequent.

So, the generalized portrait-robot of the ghost is ready. And then the question arises: how does a person see a phantom? From a scientific point of view, three options are possible: the ghost excites electrochemical processes in the retina of the eye, which are transmitted along the nerves to the visual center in the cerebral cortex, causing it to be seen; he can achieve the same result by acting directly on this center where the visual sensation arises; in the end, a ghost can give rise to a stream of photons in the space bordering on him, and a person sees it in the same way as an image on a TV screen.

Without categorically rejecting the first two options, I still tend to the third, says Carpenter. - And that's why. When a ghost is observed by several people, each sees it from its own angle of view, depending on where it is, as if it were looking at a television screen, albeit three-dimensional. But that's not all. Since the ghost seems to be hanging in the air, it is the latter that is the substance that emits photons that “draw” the phantom. And since we ruled out magic, the only possible explanation for this process looks like this: the photon flux is the result of a change electronic structure atoms and molecules of air. In turn, this requires a certain amount of energy. Without going into details, suffice it to say that it is precisely these costs of hers that apparently cause the temperature to drop at the place where the ghosts appear.

To further prove the correctness of this conclusion, Carpenter did simple, but rather convincing calculations. A ghost with a maximum luminous power of 20 watts, drawing energy from a decrease in ambient air temperature, should cause it to fall at a rate of 14.5 ° C per minute until it reaches the dew point. That is, the temperature at which the air is cooled to such an extent that the water vapor contained in it is saturated and begins to condense into separate droplets.

In a normal climate, this would take just over a minute. At the same time, the ghost would have to be shrouded in mist. However, this does not happen, because the air has thermal conductivity and manages to partially compensate for the energy costs at the phantom boundary due to the influx of heat from the surrounding space. But where its humidity is high - on the sea coast and even in England in general - and, consequently, the dew point temperature is higher, ghosts can often be observed as if surrounded by a light fog. A similar effect occurs when the ghost emits, for example, loud moans. After all, to cause sound waves, energy is also needed.

Among other things, one very important conclusion follows from the above, Carpenter believes. - Namely, that ghosts are able to manipulate energy - turn heat into electromagnetic or sound waves. But this is possible only if they are composed of matter or energy. Because none of the thousands of observations contains any indication of the material nature of the phantom, it is possible to assume that its essence is energy. Although what kind of energy, we do not yet know. Something else is important. According to the principle of equivalence of Einstein's theory of relativity, a certain amount of energy must correspond to a certain mass, the so-called mass equivalent.

On the other hand, energy does not come from nothing. She only changes her shape. Therefore, the energy essence of ghosts is inherited by them from human body in . This is what in religion and everyday life is traditionally called the "soul of man." Therefore, the expression: "The soul left the body" - in physical sense means that this material object has lost its energy essence. As a consequence, according to the principle of equivalence, its mass should also decrease. The difference between its initial and final value will be the mass of the ghost.

Let's move on, - continues Carpenter. - Because we proceeded from the fact that all ghosts obey the same laws, and all energy entities have the same mass equivalent, initially embedded in material objects no matter who - God, higher intelligence or someone else - this means that the mass of the human body decreases by the same constant value, which does not depend on its size. In other words, it makes no difference whether it is a child or an adult...

Unfortunately, Carpenter was unable to find reliable data obtained from the relevant measurements at the time of death. The only thing that the American managed to firmly establish concerned babies and boiled down to the following: it was never noted that the surface of the cradle was even slightly straightened. As a result of the experiments, the scientist was able to find out that the latter is possible only if the change in the mass of the object does not exceed 1% of the original.

For babies, this figure is 22.7 grams. So the mass of the ghost cannot be more than this figure. As for the lower limit, then through complex calculations, Carpenter managed to calculate it too - 6 grams. In other words, the mass equivalent of a phantom lies in the range from 6 to 22.7 grams. To clarify and confirm Carpenter's conclusions, in his opinion, further experiments using the latest highly sensitive measuring equipment and involving, for example, thousands of hopeless adult patients and elderly volunteers, would help.

In addition, there are other questions that need to be answered. In what specific place does the energy essence of a person, or his “soul”, leave the body and in what direction does it move? When does it enter the body - at the moment of birth or even earlier? Do animals have such an energy essence? If yes, which ones? What forces determine the interaction of entities with each other? How are they affected by radiation or penetration through matter?

The difficulties here are enormous. But if unbiased scientists, experimenters and observers manage to overcome them, an important breakthrough will be made in the disclosure of many mysteries of nature, man, religious beliefs, and even in the knowledge of the Universe.

It is not yet possible to fully appreciate the significance of the results, says Carpenter. - Another thing is unclear. Namely, in what direction can further research go in the future. And yet I believe that an important step has been taken: it is now possible to consider as scientifically proven the fact that the mysterious “something” that gives rise to the phenomenon of ghosts does not belong to the realm of the human psyche at all, but to the world of physical phenomena.

The future will show the validity of the hypothesis proposed by Dr. Donald Carpenter. At the same time, if it is confirmed, a reassessment of much of what is considered an indisputable truth is inevitable. But even now, the clear and precise logic of Carpenter's conclusions convincingly testifies that there is nothing supernatural, magical in the phenomenon of ghosts, since the natural laws of nature apply to them. Simply, not all of them are known to us yet.

To understand how to bring the initial data series to a stationary form, let us first consider what types of non-stationary processes are distinguished by classical econometrics. And there are only two of them:

• 1) a process that is stationary in finite differences;
• 2) a process that is not stationary in finite differences.

The first may include various processes described by trend models, as well as processes with the transition of an indicator from one level to another. They are shown in fig. 8.5.

The second type of processes includes all other non-stationary processes.

The upper left graph in Fig. 8.5 is a non-stationary process described by a linear trend

This process will be stationary in the first differences, since taking them in such a situation entails getting rid of the slope angle:

which brings us to the permanent mathematical expectation and dispersion.

The lower left graph in Fig. 8.5 shows a non-stationary process described by a parabola.

Rice. 8.5.

By analogy with a linear trend, it can be shown that this process will be stationary in the second differences.

In the upper right graph of Fig. 8.5 shows a non-stationary process with a change in the level of the series. It is noteworthy that until approximately the 20th observation, the value at t oscillates around one value of the constant, and after it around another. This series also becomes stationary in the case of taking the first differences, since in this case the constants are eliminated and all fluctuations occur already at the zero level. Note that in the difference series, in this case, an outlier appears corresponding to the transition to a new level after the 20th observation.

Finally, in the lower right graph of Fig. 8.5 shows a non-stationary process with a change in the angle of inclination. Approximately before the 20th observation, the dynamics of the process is calmer than after it. Taking the second differences in this case also brings the process to a stationary form.

It is obvious that economic processes are not limited to these types (we cited only the basic ones as an example) and taking the differences, unfortunately, does not always improve the accuracy of the forecast. And in the case of evolutionary processes in which there are constant changes in all statistical parameters, taking differences can, on the contrary, reduce the accuracy of the forecast: the series can indeed become "stationary" from a statistical point of view, but there will be little benefit from this, because already in the forecasting period the trend may change. Therefore, reducing the series to a stationary form cannot be considered a "panacea".

To include differences in an ARMA model, the lag operator is usually used. For example, first-order differences can be written as follows:

The second differences in the same entry take the form

Applying now to (8.40) the lag operator, we obtain

In general, order differences d using the lag operator are written as

A moving average autoregressive model based on differences is denoted as ARIMA( p,d,q), where the letter 1 responsible for order d integration (taken difference), and is written in a compact form

Usually in case d> 0, the constant from (8.43) is removed, since when passing from the differences to the original data, a trend appears in the model due to it c t, and trend components in ARIMA should be described by autoregressive elements. However, sometimes a constant is left, and then such a model is called an ARIMA model with drift ("with drift").

Consider now, for example, what the ARIMA(2,1,1) model will represent. It is written in the form

To understand what such a model corresponds to, let's expand this entry:

Let's rewrite the difference in terms of the difference operator:

and expand the brackets in (8.45):

We transfer everything except the value of the difference to the observation t, to the right side:

Now to get the final value y t, formula (8.39) must be used. Get

or, equivalently,

Using model (8.47), one can easily get a prediction one step ahead:

Unfortunately, in order to get a forecast for more steps forward, we need to use the recursive procedure we described in Section 8.1.

Note that usually when building an ARIMA model, the order of differences is limited by the number d= 2. This is due to the fact that taking the second differences usually makes it possible to bring almost any non-stationary data series to a stationary form.

As you can see, the difference tool is quite convenient and fits perfectly into the ARMA model, but this, of course, is not the only tool for reducing non-stationary series to a stationary view. Let's talk briefly about other methods.

One of the most simple methods bringing to stationarity is building a trend model based on the initial data series. Having built a model of the selected trend, the researcher calculates the residuals according to the model and builds the ARMA model based on them.

This method has one significant drawback - inflexibility. The trends themselves in no way take into account new incoming information, so the models built on their basis will be frozen. However, the use of trends in bringing the series to a stationary form allows, in some cases, to obtain more accurate forecasts in the long term.

A rather promising, but not very common alternative to taking differences is the method proposed in 1982 by E. Parzen. Its essence is to describe the original data series non-stationary model AR, and then build the ARMA model based on the resulting residuals. The resulting model is called ARARMA.

Another option for reducing a series of data to a stationary form is taking non-integer differences(when d becomes a non-integer), which is achieved by expanding (l-B)d into Taylor series. The essence of the method is that taking integer differences may be redundant for some data series (stationarity may lie somewhere between d= 0 and d= one). Order difference d in this case it is selected automatically. According to the transformed series, ARMA is again built. The model resulting from this is called ARFIMA (AutoRegressive Fractionally Integrated Moving Average) .

In addition, there is a method used to obtain a constant variance (usually problem-solving heteroscedasticity in many cases), – logarithm of the original data series. It allows you to get a series of constant dispersion when the error in the model is multiplicative.

Seasonal data series can also be considered non-stationary (since the variance, for example, at the beginning of the year may differ from the variance in the middle). To get rid of seasonality, you can either use one of the seasonal decomposition methods, or take seasonal differences. The first solution to this problem was considered by us in paragraph 6.1, and we will turn to the second in paragraph 8.4.

• Makridakis Spyros, Hibon Michele. ARMA models and the Box-Jenkins Methodology // Journal of Forecasting. 1997 Vol. 16. P. 147–163.
• Parzen E. ARARMA Models for Time Series Analysis and Forecasting // Journal of Forecasting. 1982 Vol. 1. P. 67–82.
• Granger C.W.J., Joyeux Roselyne. An Introduction to Long-memory Time Series Models and Fractional Differencing // Journal of Time Series Analysis. 1980 Vol. 1. Issue 1. P. 15-29.

attention as it will reinforce and enrich your initial understanding of your beliefs and values.

9. Bring your vision, identification, beliefs, and values ​​into the "ability" space. Notice how it enhances, changes, or enriches your sense of your own abilities.

10. Bring your vision, identity, beliefs, values, and abilities into the space of "action." Please note that even the most seemingly independent actions are a reflection and manifestation of your experience at all higher levels.

11. Transfer all the experience gained to the “environment” space and note how it has changed and enriched.

12. Remember the feeling of being in alignment. Imagine yourself in this state at that moment and in that place when and where you will most need it.

This procedure can be carried out alone or together with another person who will act as a consultant or "guide". Below is a recording showing how one person can guide another through the process of creating a state of conformity.

R.D. I would like to demonstrate the leveling process with Jay. To begin with, we will designate a physical space for each of the levels: environment, actions, abilities, beliefs. a values, identification and spirituality. I usually prefer to lay them out in a row, so that you move backward from the environment level to the spiritual level, and move forward from the spiritual level to the environment level.

Jay. Good.

R. D. And what I want to ask you first of all is to take the position of the environment and think about the conditions under which you would like to be in a state of greater personal correspondence.

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Jay. Well, I am a manager and I would like to be more in line with my work.

R.D. Can you describe the conditions in which you act as a manager? Answer the question: where and when do you function as a manager? Tell us when and where you do your work.

Jay. I am usually in my office at our headquarters. I can also be in the meeting room, which is similar to this one, but a little smaller. Lots of people come and go. I interact with about a dozen key people a day. This usually happens throughout the week, sometimes even on weekends.

R. D. So. Very well. (Audience) Note that Jay speaks exclusively in the language of facts, which is quite consistent with the description at the level of the environment. Your environment is just what you see and hear around you. People often try to make judgments or interpretations about their environment instead of simply describing what they see and hear. If someone says: "I work in a hostile environment," this is his judgment about the environment. In our exercise, it is very important at this level to describe only on the basis of the information that your senses provide, as Jay does.

(to Jay) Now I want you to move to the “action” position and describe what actions you perform when you effectively manage people in the place and time you specified. Answer the questions: what do you want to do more in the "where" and "when" you specified? What activities do you perform when you function in this place and at this time as an effective manager?

Jay. I can stand, sit or walk around. Sometimes I talk, sometimes I offer ideas, sometimes I negotiate with people. Sometimes I try to support the person by asking questions, speaking or writing something down. So I listen, I ask questions, I try to get an idea of ​​what the person is saying and doing, and then I try to guide or keep them on the right path.

R.D. Good. (Audience) Note again that Jay is simply describing certain actions without giving any interpretation. If he said, "I am behaving well or badly," that would be a judgment regarding his actions. In this exercise, at the action level, you should describe what could be

From vision to action

capture a video camera, if it is at your workplace. The camcorder does not judge or offer interpretations, it simply records.

(to Jay) I would like you to stay at this level for a little longer and try to identify some "micro-actions" that can accompany your state of "greater alignment". Pay attention to your body and try to understand what postures correspond to the state of "fitness" when you are sitting, standing or walking around in your office. How do you keep your body?

Jay(straightens up and assumes a collected pose). I stand or sit more upright and upright. I think my head is thrown back and slightly raised.

R. D. And what do you do with your hands? What gestures do you make when you are in a "fit" state?

Jay(making appropriate gestures). I think that I stretch my hands in the direction of the interlocutor more often and look into his eyes more often.

R. D. Do you notice any differences or peculiarities in your breathing when you are in a state of “correspondence”?

Jay. It is slower and deeper.

R.D. Fine. Now I want you to move into the ability position. Here I invite you to answer how you manage to do these "what" in the conditions of these "where" and "when". In other words, what know-how or mental strategies and skills do you need to listen, ask questions, and guide people in a balanced, direct, clear, and even manner?

Jay. Well, I have to use the ability to structure and organize information. I need my professional knowledge to translate this structure into words and actions. I also need the ability to put myself in the other person's position to see things from their point of view. And I also need the ability to take the position of an outside observer in order to see the relationship between me and other people. Also I have to use the ability to create a picture of what I am going to do.

^SHSH Chapter 2 _____________________________________________

R.D. Good. (Audience) Notice how Jay's physiology changes every time he moves to a new level. The number of gestures and movements is increasing. Each next level involves more and more of his personality and, therefore, the nervous system. At the level of the environment, he practically did not move at all. On the action level, he began to gesticulate and turn slightly. At the ability level, you could see his eye movements and the way he looked around as he listed his abilities and skills. You can also note the change in the timbre of his voice to various levels. He now speaks somewhat more slowly and in a lower voice. The vocabulary has also changed. All this points to the participation of more and more diverse personal processes. at

(to Jay) What mental processes do you need to maintain the physical manifestations of your state of "correspondence"? What internal abilities allow you to be balanced and direct, to make gestures towards the interlocutor, to breathe more deeply and speak more softly and clearly?

Jay. This is the ability to clearly see the goal in front of you and realize how it fits the situation. It is also the ability to maintain feedback with oneself and with other people. I believe that first of all it is the ability to be aware of myself and what I want and the quality of my relationships with employees.

R. D. Great. Now I ask you to move to the position of beliefs and answer the question: why do you use the listed abilities to "act in this way and in such conditions? Why did you decide to use your ability to structure information, put structure into words, look at things from different points vision, create a picture of what you are going to do, be aware of yourself and others and receive feedback? What beliefs and values ​​led you to use this know-how and act in this way at that time and place?

Jay. Well, I believe in respect, respect for other people. I also believe in a good relationship. I believe that the support I give to others and to myself will allow us to achieve more together. I believe that it is very important to achieve integrity and that it is good to have many skills in order to do something.

R. D. So, you value respect, relationships, support, skill, and integrity. These are important and weighty values. (Audience) On this

From vision to action

level we explore the question: what beliefs do you have about yourself, people and your business? What beliefs guide you? Note that Jay's voice has become even lower. As if you began to hear his words better. He speaks from the heart.

(to Jay) Now I invite you to move into a position of identification and answer the questions: “Who am I, at my core, am I, someone who respects the lives of other people, wants to have good relationships, support people, who values ​​skill and believes in integrity? Who am I if I use my mind to create structure, to put it into words, to look at things from different points of view, to clearly imagine the future, and to maintain awareness of myself and my goal? Who am I if I have to develop the strength to come up with ideas to sit, stand, walk around, listen, ask questions and answer people in those rooms during the week and sometimes weekends in the part of the planet where my office is located? Who am I as a manager?

What type of manager are you? What kind of person are you? What metaphor can you give for yourself and your mission?

Jay (pause). Well, one metaphor could be this: I am something of a lighthouse. I am a beacon that shows people the way and helps them reach their goal.

R.D. So, you are like a beacon that supports people and points them in the direction. It is a metaphor or symbol for how you come to realize your identity or mission.

Now I want to ask you to take the last step, to go beyond being a beacon and move to a higher position, into what can be called "spirituality". While in this position, think about your vision. Imagine being able to briefly see what is outside of your life and your identity. What is your goal? How do you serve your identity as a beacon? What does this lighthouse show? What vision does your mission support?

The question here is: “What kind of vision is associated for me with the image of a lighthouse and a manager who values ​​respect, support, skill and integrity? What vision do I have as someone who structures and expresses the goals that he sees in his thoughts, who looks at things from different points of view, maintains awareness and receives feedback from what is happening inside him and between him and those around him? What vision do I have when I sit, walk

eV &STEP. Chapter 2

and listen, breathe deeply, and speak softly but clearly, stand straight and straight, addressing a group of twelve or so people in that office, in that meeting room?”

Jay (long pause). It's something to do with congruence and integrity, but I don't know how to put it into words.

R. D. Take your time. These words are very important. There is no need to customize them. Let them appear or emerge from your subconscious. Just describe what happens as you experience it. Maybe it's just a color or some kind of image.

Jay (pause). I see something that looks like a big ball or planet. Most of the planet is lit, but there are dark areas. Some areas are in complete darkness, while others are only slightly shaded. But in those places that are in darkness at this time, people need beacons that help them find their way out of the darkness. I'm not the only lighthouse. There are many of them, but since they are very far apart to help travelers better, they are not always aware that there are other lighthouses somewhere. But I hear some noise, it is like those beeps that give in the fog, and this allows the beacon to understand that there are other beacons somewhere.

R. D. (audience). You can tell by the way Jay says that he is going through a very deep experience. Just look how much his breathing has changed. We rarely open ourselves up to such experiences, except perhaps when we are close to death or in deadly love. And yet in our nervous system always hidden resources for this kind of experience. We are completely absorbed in "experiencing" our daily life and forget that this life takes place within a much larger frame that is difficult to describe in words. Each of us has probably had an experience like this but found it difficult to hold on to or relate to the experience of "everyday" life. Yet it is this level of experience that gives our "everyday" experience meaning and purpose. I am sure that it is possible to transfer this quality of awareness into our everyday life and work.

(to Jay) Jay, I would like you to stand for a moment, continuing to fully maintain the state of awareness of this planet with light and dark areas and beacons that will recognize the presence of another beacon by beeping. Be proud of it, hold it, anchor it. And then go into the identity space and transfer this state from

From vision to action

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yourself. That's right, take a step forward into the position of identification and bring that vision with you. Notice how awareness of the larger vision enhances and enriches your experience of who you are. If you like, you can make changes or additions to your description and metaphor for identity and mission.

Jay. I can feel it, but it's hard for me to put it into words. It was as if some excitement and awareness of being a part of something bigger had been added.

R. D. You are no longer just a lighthouse, it is an awareness of yourself as a particle of light and a whole network of lighthouses that help travelers find their way. And that's the excitement that this vision gave you as you brought it into your sense of identity and mission.

Jay. It is also a feeling of gratitude... an experience of gratitude. And at the same time it gives more energy.

R. D. This vision mobilized your nervous system and filled it with energy and strength. Now I want to ask you to accept this feeling of energy and gratitude, the feeling of being part of a large planet and a network of beacons that shine in the dark, and move forward into the space of beliefs. Take the feeling of your vision and your mission into your heart. How have your beliefs and values ​​been strengthened and enriched? Maybe some new beliefs have come from the depths of your heart?

Jay. Well, on one level, I can really feel the integrity of the whole system and the importance of relationships and support. It is also a belief in the importance of growth for its own sake. As if growth were the natural result of integrity and support. It may have a purpose, but I feel that growing is a good thing in itself, regardless of other purposes. This is first. And then there is the realization that I am here at this time, that I have also received support and can relax, enjoy what I am doing and appreciate it more.

R.D. I want to suggest that you focus again on feeling like a vast network of lighthouses. For example, you can mentally hear those beeps when you feel the value of respect, support, skill, and integrity. Also, don't lose the feeling that growth is good, even if it doesn't serve a specific purpose, and that you belong in the present moment and can appreciate what you're doing even more. I want you to bring all this into the ability space. Link your vision, your mission, your sense of integrity and

<m>J?im:- Chapter 2

your heart with your mind. Realize that your skills and abilities are a reflection and expression of your beliefs and values, your integrity, mission and vision. Your mind is the way these deeper structures manifest themselves. When you are fully aware of these levels of your personality, how does it enhance and enrich the perception of the abilities you have to realize your values, beliefs, integrity and vision?

Jay. I am more able to be myself and respect my own point of view. It is also easy for me to see other points of view and to look at them all from a wider and wider perspective. It became easier to understand other cards and other styles of thinking. It's as if I had fewer restrictions. I can better sense where the flow of people and events is going and give direction to that flow.

R. D. Now I invite you to move forward into the space of action and re-experience the actions that you spoke about: listening, asking questions, leading other people, being balanced and direct, gesturing towards the interlocutor, breathing more deeply, speaking more soft and clear. Bring a sense of your abilities, beliefs, identifications and representations into your actions. Link your strengths as a manager to your mind, heart, mission and vision. Has anything changed in your experience of these actions? How has your holistic image enriched the specific actions you perform? Perhaps the quality of what you do has changed.

Jay. It has changed a lot. I am more aware of what is happening around me, what I do and hear. If I explain something, I know why it should be done. I have a much stronger sense of confidence and significance in my work.

R. D. Finally, I invite you to return to the space of the environment, to those specific “where” and “when”, to your headquarters, to your office, to the meeting room, to those twelve people with whom you interact daily . Align your vision and sense of purpose with your identity, your heart and mind with your actions in the environment. How does your perception and experience of the environment change if you bring into it the awareness of yourself as part of a network of beacons that indicate to other people

From vision to action

dyam direction in this environment? Also bring here your belief in the importance of relationships, that growth is beautiful in itself, your sense of belonging, your belief in the value of respect, support, skill, and integrity. Take your ability to receive feedback from yourself and others, understand other people's cards and see a broader perspective, and that you have fewer restrictions and are able to direct the flow of your experience. Also transfer here your physiology in a state of “fit”, your soft but clear voice and the feeling of awareness, confidence and significance that is associated with what you are doing. Notice how your experiences change and become richer.

Jay. I feel that the work can be less effortful, that I can be more in touch with the environment and my co-workers. I am more aware of my belonging. I also feel that I can be more creative with my work.

R.D. You may also note that this environment is one that is associated with respect, support, skill, and integrity. It is a place connected to a larger perspective, where it is possible to understand other people's cards and where growth is valued, whether it has an obvious goal or not. This is the place where you belong and where you can go beyond old boundaries and limitations. This is the place where you can let your vision and your values ​​shine through.

Jay. Yes, I think it's possible. Thank you.

Applause.

R. D. I find the leveling process to be extremely important. In fact, I do this every day. For example, in the morning, while preparing for an effective seminar, I imagine a seminar room and how I want to behave in that room. I also consider what abilities I will need to be an effective leader, and what style, beliefs, and values ​​will support me the most. I also make sure that I have an understanding of my identity and mission, i.e. the purpose, the reason why I am here. Then I reaffirm my vision for my life's work. And then I bring all these aspects of my experience into line. This usually takes 5-10 minutes, but it's a very effective way to get ready to do my best.

^W M-Chapter 2 ___________

Bringing into compliance state by levels in a group

The level matching procedure can also be used as an effective team or group building tool. When people share a common vision, mission, values, and capabilities, they are able to work together much more effectively. Undoubtedly, this is the basis of what is known as team spirit.

Alignment across levels within a group is performed as an extended version of the individual procedure. After each participant has come into a state of "correspondence" (and served as a guide for the other participant), group members can come together and exchange the answers they gave at various levels. By listening to each other, participants can find common ground in what they say about their environment, activities, abilities, beliefs, values, role, mission, and vision. (It makes sense to choose one person who will write down the general answers in a notebook.)

Once participants have shared their responses at all levels and found common themes, they can synthesize personal responses to find a "group" vision and mission. If the group was originally formed to achieve some common goal, the participants may focus on identifying the group's values, abilities, etc. 1 .

I even used this procedure to help the team form a group identity: we identified common values ​​and then tried to determine what kind of identification these values ​​could be "added" to or an expression of what kind of identification they were. Just like an individual, a group can define its identity through a symbol or a metaphor.

This procedure has been successfully used in companies such as Lufthansa, IBM and"Fiat".

From vision to action