Talk:Stochastic process/Archive 1

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Physicist!). Mathematician’s task is to prove that there EXISTS a solution for a specific problem. Engineer’s task is to FIND a solution for a specific problem The relative story goes like this: Question - What is the value of pi (π) ? Mathematician’s answer: 3,1415926535897932384626433832795 ........ Physicist’s answer: 3,14159 engineer’s answer : about 3 (!!!) 2)The story: The wise old man of the village is talking with the local judge when some neighbor is approaching them and says to wise old man, ignoring totally the judge. "My neighbor did to me this and I forced to do to him this and this. Wasn’t I right?". The wise old man look firmly at him and with the most assuring voice says, "Yes, you were right". The man leaves satisfied. In a while the other neighbor appears and goes straight to the wise old man, ignoring totally the judge, and says to him: "my neighbor did to me this and so I forced to do to him this and this. Wasn’t I right?". The wise old man look also firmly at him and with the most assuring voice says, "Yes, you were right". The judge watching the whole scene with an out bursting voice says to him "Look wise old man, they can not be both right". The wise old man stares patiently at him and with the most assuring voice says: "You are right to". 3)The Book passage: (Unfortunately I could not find the original one, someone might help with this. The passage is from Lewis Carroll book "Through the Looking-Glass", so the following is what I recall) "When I say something", said the Humty Dumpty, "I mean not more not less from what I mean". "Yes, but what counts is the meaning you are giving to your words", said Alice. "No", said Humty Dumpty, "What counts is who is the BOSS" -- October 25th, 2006 -- diam@duth.gr

905:. Roughly speaking, the "probability space" refers to three things: 1) a set of things that we may think of as "primitive" events, 2) a collection of subsets of the primitive events and 3) a function or algorithm that is able to assign a probablity to each of these subsets. In the simple examples above, we can only give a partial description of the probability space. In example 1, the primitive events can be described as "all possible 6 to 7 AM time periods at the airport". In example 2, the primitive events can be described as "all possible pictures that people might draw on 8.5" by 11" paper". These descriptions dodge the question of which subsets of primitive events can be assigned a probability and how this might be done. A primitive event in the probability space should determine all the values of the random phenomenon. For instance, in example 1 it would not be correct to say that a pressure reading of 1013.25 millibars at 6:03 AM is a primitive event. Giving the air pressure at a single time would not determine it at the others. In example 2, a primitive event would not be "all the colors a person might draw at some location on the paper" or "all the images a person might draw in the upper left hand corner of the picture". A primitive event should determine the whole picture. 672:

Of course, when dealing with deep measure theory this point of view might be the proper setting, but it does not make much sense in an introductory essay. I understand probability theory not as structure theory like algebra or differential geometry for which (but not even there!) there exists a broad common agreement about the 'right' formal framework. As Miguel, I suggest to take the plain and simple definition of a random process as an (indexed) family of measurable functions on defined on a probability space. This definition should be broad enough for most articles in this enceclopedia, and if some articles need more specific definitions they can still introduce them. If the others here agree, I will do the changes. --

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how the process might evolve under time (as it is the case for solutions of an ordinary differential equation, just as an example), in a random process there is some indeterminacy in its future evolution described by propability distributions. This means that even if the intial condition (or starting point) is known, there are more possibilities the process might go to, but some paths are more probable and others less... And now one could mention prominent examples like Brownian motion (and maybe other diffusions), Markov processes, Poisson processes and that they have important applications in physics, biology, finance, etc. After all, one could proceed with the formal definition.

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who is not a little bit into mathematics will look up the Kolmogorov extension theorem. Here it makes sense to be more technical. But with such general notions as a stochastic process it is more likely (and in my opinion also the purpose of a wikipedia article) to meet the interest of a general (non-specialist) audience, as many people have heard of such processes and would like to know a little bit about it. Their needs should be served first, and afterwards we can go more into math. I think it is possible to write an article which satisfies both mathematicians and those who are not. It will not be so easy but worth to try it.

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wondering, "And what was the point of this again?" Sometimes the relevance of the technique gets lost in the carrying out of the technique. Could you perhaps expand the beginning a bit to explain why this is so important? For example, could you put it into context by saying what statisticians were limited to doing prior to the idea of stochastics, who invented it and what problem they were trying to solve that led them to the technique, and conceptually why it is so powerful? I would start it, but I need to learn how this wiki thing works a bit more first. I just got a username to post this comment! Thanks! --

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the algebraic approach, one starts by defining an algebra of integrable functions and the integral as a positive linear functional on that algebra. Then, measurable sets are those whose characteristic function is integrable and their measure is the integral of their characteristic function. The geometric approach gets harder and harder when vector measures on high-dimension measurable sets are involved, while the algebraic approach is no harder for Banach-space valued measures on infinite-dimensional spaces than it is for real random variables on .

1132:, has a range that is the set of real numbers that describe a color intensity ( instead of the set of 3 dimensional vectors of such numbers). This somewhat goes against the idea of having each datum in the range of the random variables be "all of the same format", since we might not consider "red color intensity" information to have the same physical meaning as "blue color intensity". However, if we decide to think of all these data as random variables whose range is the set of real numbers, then we may do this. 2266: 31: 1320:

30 tosses before getting the first occurrence of 'heads'? " Then we must consider the situation where a coin is tossed over and over again an unlimited number of times. The usual way to view this is to consider the repeated tossing of the coin to be a stochastic process. Each toss is a datum that is either "heads" or "tails". A realization of the process is one particular infinite sequence of such data.

789:" includes those cases where the amount of data produced is finite. Suppose instead of the continuous readings of example 1, the sample of pressure readings might be recorded by a device that records one temperature reading per second Then each realization (recording for an hour) produces 3600 readings. We may consider this situation to be a stochastic process. (We might also consider it to be a 1335:' must be a primitive event in the probability space. Since an infinite sequence of coin tosses is a conceptual experiment rather than an acutual one, we don't define a primitive event in the probability space to be an event like "Tosses begun by Lula Mumshelter on January 3, 2002 9:42 AM". The customary way to define the primitive events for coin tossing is to say that it is the set 1061:, etc. However the concept of "index set" in a stochastic process is more general than this. The "index set" can be any set at all.) The pressure at a specific time, such as 6:03 AM can be viewed as a single random variable since the pressure at this time will be different on different days. We think of the stochastic process as an infinite family of random variables 625:, bith of which start off with a clear definition. I suppose that what I am really saying is that I find the definition given unclear as well as un-rigorous, perhaps with the first as a consequence of the second. As an alternative suggestion how about calling the 'definition' a 'description' (although I still feel that it needs work), then placing a proper definition later on..? 1570:

field. You might want to remove stock markets and heart rate from the list of processes. They show characteristics of chaotic systems. See this article A Multifractal Walk down Wall Street; February 1999; by Mandelbrot; I also have heard that one reason people like Bach's Brandenburg concertos are that they are fractal and even mimic the heart rate. --

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variables and doing certain computations with the results to arrive at the realization of the stochastic process. In such a case, the primitive events are the set of all possible realizations of the random variables employed by the algorithm. The probabilities involved are computed from the joint distribution of these random variables.

2235:"One solution to this problem is to require that the stochastic process be separable. In other words, that there be some countable set of coordinates {f(xi)} whose values determine the whole random function f." — Rather rough. Consider for instance the process whose values at all rational points are the same (random) 1169:

In the "indexed collection of random variables" view of stochastic process , all the variables have the same domain and range. The fact they have the same range implements the idea that the phenomena produces data that is "all of the same format". The fact they have the same domain indicates

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Example 2: Suppose we hand out 8.5" by 11" sheets of white paper to each member of an audience and ask them to draw a picture. Let us take the simplistic view that underlying process that creates the picture is the same for each member of the audience. Each picture we receive is a realization

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Concerning the definition of a stochastic process, I agree with Miguel's point of view in the general discussion above. In my opinion the present definition as L^0-valued function is a little bit too technical, moreover it does not explain why considering a sample path as an equivalence class mod 0.

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There is some inconsistency with the way books treat the term "stochastic process". Some (such as Gardiner) restrict it to a process in time. This appears to be in the same spirit as author's who say, for a vector, "In this book W will be a finite dimensional vector space" in order to set a context

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I think intuitive descriptions are very useful for everyone - also for mathematicians. This is a general problem of formal sciences: inspiration and a deeper feeling for the subject is usually far away from its precise 'crystalline' description. It is not so easy to find a good balance between both

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I definitely agree with the comment of diam@duth.gr, and also with Michael Hardy's one below. There should be a rough description of the concept of a random process understandable for non-mathematicians. In my opinion it is not so good to start with a formal 'dry' definition which seems to be written

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of infinite sequences of heads or tails. ( The domain is not merely the set of two things {heads, tails} . Remember that an event in the domain must determines the entire realization of the random process, which is all the data for the entire collection of random variables) As remarked above,

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vectors occur. Most of the image that we would create this way would be a cloudy mess. They would lack the images of people, houses, flowers and dogs that appear in pictures drawn by human beings. Stochastic processes like example 2, whose realizations typically contain a high degree of

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It is not correct to think each variable in the "indexed collection of random variables" as necessarily being the "same" random variable realized over and over again. Two random variable in the collection must be "the same" only in two respects: they must have the same range and same domain.

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Example 4: Suppose we perform an experiment where we measure the weight, height and temperature of a randomly selected person. This is usually viewed as a realization of a multivariate random variable which has three components. It is also possible to view this as a stochastic process that

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OK, I pasted the draft into the beginning of the article. I confess that I don't know html or the wikipedia editing conventions very well. My method is compose the pages with Mozilla, view source, cut and paste into the Wiki editing window. Some spurious blank lines have been introduced and people

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Wow, that's pretty impressive. Why don't you add the content to the article a bit at a time? That way you will get around the 33Kb limit, and if you use sectioning (see the editing help about this) appropriately you will never hit the limit. Also, the rest of us will be able to work on your edits as

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For example I would start as follows: In the mathematics of probability, a stochastic process or random process is a process that can be described by a probability distribution. Roughly speaking it is the counterpart of a deterministic process; instead of dealing only with one possible 'reality' of

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Mathematical literature contains variations of the definitions that we have sketched above. Many authors (e.g. ) do not explicitly say that each member of the "indexed collection of random variables" must have the same probability space. However, in studying specific stochastic processes

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If we think of example 6 as an "indexed collection of random variables" then the index set can be defined as the set of integers {1,2,3...} As mentioned above, the index set for a process need not be a set of integers, but in this particular example it is. We can define the random variable

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describe the results of a particular infinite sequence of coin tosses. To meet all the requirements a probability space we must be able to compute probabilities. The technical details will not be given here. Probability texts give examples of computing various subsets of primitive events.

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is to create terminology that is broad enough to describe a random phenomenon that produces an infinite amount of data each time it occurs. Another goal is to have terminology that is narrow enough to describe situations where the data is , in a manner of speaking, all of the same format (e.g.

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Again, the big question is for which audience the article should be written. I prefer not to be too technical (also from the point of view of a pure mathematician), there is always enough space and time to do this later on whenever it seems reasonable. What I mean is the following: probably no one

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Remove it, as it is half-cooked. However, this is what's going on. There are two approaches to measure theory and integration. In the geometric approach, one defines first what is meant by measurable sets and measures on them, and then uses measures to define integrals and integrable functions. In

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need not be 0.5 ) A single toss of a coin is usually viewed as a realization of a random variable. A fixed number of tosses of a coin can be viewed as a multivariate random variable. Suppose we wish to consider a question such as "What is the probability that we must make more than

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Example 1: Consider air pressures in millibars at the local airport airport from 6:00AM to 7:00AM. Assuming that time is a continuum, there are an infinite number of times between 6:00AM and 7:00AM. A single occurrence ("realization") of the process is this infinite set of pressures that

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Hello! I have a small suggestion if you are still editing. I'm a student in Econ getting my MA and I haven't really been able to get an answer to this one question: Why is stochastic processes so important? As if often the case in grad school, I find myself learning all of this stuff and then

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Also the paragraph claims "One of the important features of the algebraic approach is that apparently infinite-dimensional probability distributions are not harder to formalize than finite-dimensional ones." Can anybody make sense of this? The approach given seems to be only for one-dimensional

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I have now changed the "definition" section to "Common stochastic processes" and removed some of what I felt was rather confusing language. I've put in an example, probably in the wrong place, which I hope gives a feel for the index set and the domain of the random variables. I plan to rework the

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As I already mentioned in my latter comments, I am not very happy with the article in its present form. Please don't get me wrong, I appreciate all the work that had been done! But I believe that such a prominent notion as a stochastic process should deserve a more extensive treatise. I think an

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Kolmogorov's axioms for probability theory are analogous to the geometric approach to measure theory, and the Kolmogorov extension theorem constructs a sigma-algebra of measurable sets and a probability measure on it given the finite-dimensional distributions of a stochastic process. There is an

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need not represent time. The word "Field" does not necessarily imply that a Random Field represents something like an electric or magnetic field, although there are many publications that do apply the theory of stochastic processes to such topics. Example 2 can be regarded as a random

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The definition given on this page for "stochastic process" as a random function is an elegant mathematical definition if one takes the viewpoint that there is a family of functions on a common domain and range and a probability measure on a sigma algebra of subsets of that family. But might a

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I’m sorry to interrupt your nice discussion with tree (3) sort sentences (comment, story, book passage). It might be that it seams irrelevant. I think not. 1)The comment: There is a totally different approach of facing or treating THINGS by a Mathematician and by an Engineer (not to mention a

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Whatever the final verdict on the above point is, it would help to explain how the often seen definition of a stochastic process as "an indexed family of random variables" agrees with the idea of "random function". This would involve explaining that an indexing set can be finite, countable or

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In practical applications of stochastic processes , there is often a quantitative description of the probability space. For example, one may assume a specific formula or algorithm generates the occurrence of the process. The algorithm will usually involve taking realizations of random

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I have moved your discussion to the end of the article because it does not really have the structure of an Encyclopedia article, although there is very good content in it. I will work on merging your discussion into the existing content. The first thing I'll do is section your discussion. --

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1662, "pertaining to conjecture," from Gk. stokhastikos "able to guess, conjecturing," from stokhazesthai "guess," from stokhos "a guess, aim, target, mark," lit. "pointed stick set up for archers to shoot at" (see sting). The sense of "randomly determined" is first recorded 1934, from Ger.

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Once again I was not precise enough with my language - I meant 'further down' when I wrote 'later on', so I think we are in agreement. Thus I propose putting the definition further down. This necessitates renaming the existing 'definition' (which I don't think is appropriate anyway) - any

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the codomain D, often the real number R. If the D could be something other than R, can somebody please give an example? In the above example 2, the D is vector (r, g, b), any way, r,g,b are still real number. Is there any example, that D has nothing to do with real number? Thanks.

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is either "heads" or "tails". Notice that the general concept of a stochastic process has no requirement that a realization of the process contains exactly the same information as a primitive event. However, this is special feature of example 6. In coin tossing both

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they make additional definitions that do require this. Some authors (e.g. ) say the index set must represent time. Some authors say that the random variables must be real valued (e.g. ). The definition we will give is consistent with the above informal discussions.

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Example 3: Suppose we measure the height of a randomly selected person, we may think of this as a process that produces a single datum, the persons height, each time we perform it The most common mathematical treatment of this situation is to view it as a realization of a

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that is more specific than the general definition of "vector". There are books which use "Random Field" to include processes that take place in time. These are arbitrary conventions. It would help the reader to mention that he may encounter these inconsistencies.

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I'm putting it here. for the time being. This will allow us to fix the formatting problems, and also to move content gradually from here to the main text. I don't think a new article is necessary, but maybe the need for it will become apparent in the process. --

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The section titled "The Algebraic Approach" seems to contain an attempt to axiomatically define random variables (only for the complex valued case) and expectations for these. This is largely unrelated to the topic of the page, i.e. to "stochastic processes".

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I forgot to mention: to my knowledge L^0 is the space of all measurable functions (mod 0), not necessarily bounded. Bounded functions are usually (but not always, I agree) understood as uniformly bounded, which means that their (essential) supremum is finite.

720:" in mathematical literature. This is not particularly scandalous. Some mathematical definitions describe things which have many specific properties that are crucial in writing mathematical proofs. (For an example of such a definition, see the entry for 1405:

In example 6, a reader may think of the process as "realizing the same random variable over and over again". As we pointed out earlier, this is not a correct view of the general stochastic process. However in example 6, random variables such as

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It is important to understand that the same practical problem can be described in mathematics in different ways. Mathematics itself does not specify a unique translation of a physical situation into mathematical terms. We give some examples of such

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It's not clear to me how a modification of a stochastic process is relevant right at the beginning. At each point of time the process has a random variable which must conincide almost everywhere with the one form the modified process. Did I say that right?

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is not a simple task, even for a person experienced in probability theory. However a reader who is familiar with computer programming should be able to write a Monte Carlo simulation of this example and investigate the dependence of the two

1093:). The set of all locations on the image is the "index set". (The index set in this example does not consist of integers. A location such as ( 1.345, 4.019) is considered to be an index.) Each location may be viewed as a random variable 1052:

is a time and the index set is the set of all times between 6:00 AM and 7:00 AM (One should not assume that an "index set " must be a set of integers. Indexing via a set of integers is often used computer programming, as when we index an

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above "modified definition" to remove some stuff that is probably redundant due to overlap with the reworked first section of the article, then include it towards the end of the article. I'd like to hear opinions and comments on both parts.

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I can't yet expound upon the QM approach to stochastic processes. Below is what I propose for the traditional approach. If I put it all in at the top of the page, I get a warning about a 33 Kb limit, so I didn't edit the entry yet.

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Example 5: For the sake of having a simple example, assume that nature generates the air pressures of example 1 according to the following scheme. Pick two air pressures in millibars by selecting a two random numbers number

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begin with a definition? Sometimes clarity might be better served by beginning with an intuitive description and putting the rigorous definition after some discussion that enables the reader to understand the definition.

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is "heads". To assign probabilities to the events "heads" and "tails" we need a distribution whose domain is set of two events {heads, tails}. As remarked above, this set of two events is not the domain of the

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2) A list of beautiful examples (maybe one/some explained in detail, like a Markov chain, even if it seems to be redundant). Pictures (like a generic sample path of a Brownian motion, for example) would be great!

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I don't find the given definition very rigourous - it is more of a description than a definition. Heres the beginings of a new version. I still need to do some work on this as it tails off a bit towards the end...

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Example 5 can be interpreted as an "indexed" collection of random variables in the same way as example 1, except that the primitive events in the probability space are given by the set of possible values for

1871:(random variables with values in X), whereas a modification is a random variable which attains values in X^T. Therefore, a modification is a stronger notion (not every process has a (measurable) modification). -- 892:

The values of most of the variables in the above examples are familiar mathematical quantites, such as numbers or vectors. But the reader may it have difficulty conceptualizing the nature of the variable

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I think it is best to give the broadest possible definition to which the basic techniques can be applied, and list special cases or narrower definitions later in the article. If you can add other definitions

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Consider an attempt to model the process of example 2 by using a single random variable. Suppose we scan a large number of sample pictures into pixels. Then we create a histogram for how often each

732:", may treat the definition rather casually. Such texts chose to add details later by defining special cases of the general concept. In order to explain the distinctions among the various defintiions of " 79:

just for the sake of correctness, I think it would be better to start with a somewhat 'philosophical' discourse on random processes first - and then afterwards one presents a rigorous formal definition.

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it might be all be prices in dollars or it might be all be measurements of air pressure in millibars ). We give some examples of some physical situations that can be viewed as a stochastic process

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I'll attempt to do that, but first I must study how the Knowledge does things. Is there a way to use standard html editors like Netscape Composer or Mozilla's composer to create Knowledge pages?

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reader interpret this to mean that there could be a random choice of a function from a set of functions with different domains and ranges? (And would that fall within the intended definition?)

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Many authors ( [ Karlin and Tayor]) define a stochastic process as an "indexed collection of random variables". The idea of "random function" can be reconciled with this viewpoint. If

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is the probability space for the phenomena. In this example, the domain is a datum that would describe a specific date at the airport such as "January 3, 2002 6:00AM to 7:00AM"

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What is the root or where did the word Stochastic originate? Latin 'Sto' may be translated as 'place' or 'casue to stand' but 'chastic' does not have a common definition. Does anyone know?

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Moved the Brownian motion example into the following section, which I renamed "examples". I then removed the empty examples section that was already there. Forgot to add an edit summary...

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For example, texts show how to answer questions like "What is the probability that there are at least 30 tails before the first head". This is the probability of the subset of

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are independent. And they do have the same marginal distributions. ( The phrase "marginal distribution" is required if we wish to express the idea that the probability that

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The following situation is discussed in most probability texts. Certain aspects of it are misleading if the reader erroneously assumes they apply to all stochastic processes.

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on the interval -0.2.to 0.2. Let the resulting pressure readings be given by a pressure-vs-time graph that is a straight line connecting the points ( 6:00 AM, 1013.25 +

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who use the Wiki editor may not like the symbols introduced by Mozilla. Is there a consensus about whether using html editors to compose Wiki pages is a good or bad thing?

1459:" by memorizing only one example, it would be best not to choose example 6. Coin tossing has many features that are not typical of more general stochastic processes. 520: 2147: 1398:

this example is unusual in that one realization of the indexed collection of random variables can be identified with an event in probability space associated with

2060: 1967: 487: 2062:(which is a stronger notion; the difference is esp. important when T is "large"). I am a bit wary of fixing it, since every time it returns to some incorrect form. 1870:

The definition that is written in the article is simply incorrect (I have fixed it once already, but someone has changed it back). A process is a function T --: -->

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denote the pressure readings at two different times. A realization of the process consists of picking a particular day at the airport. The realization of

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Some authors view a stochastic process as a "random function" in the following manner. A stochastic process is considered to be a function of two variables

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link you can follow to get instructions on how to format the content and include mathematics. And remember the wikipedia policy: be bold in updating pages. --

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of this random phenomena. Each picture contains an infinite amout of data if we take the idea of the space as a continuum seriously. At each location

2243:. The countable set of coordinates does determine, but not in the way required in the definition of separable process. The definition is more technical, see 782:) that measure the red,blue and green intensities. Since the data has this form at all locations, we can think of it as being "all of the same format". 617:

I quite agree, but in a mathematical article such as this anything that claims to be a definition should be rigourous; compare for example the articles on

1969:, which is implicitly assumed to be predefined, but if you look better, it is only a "flat" set that doesn't describe the whole stochastic process. 820:. The first variable describes which datum we wish to examine from all the data produced by one realization of the process. The second variable 1073:) is a single real number. So we may say its range is the set of numbers that are possible pressure readings. The domain of the random variable 809:. It is also possible to view this as a stochastic process that produces 1 datum on each realization. However it is not customary to do this. 1602:

alternative algebraic axiomatization of probability theory in terms of algebras, and an alternative "extension theorem" within that framework.

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My developmental neurobiology teacher once defined "stochastic process" as "a process we don't understand yet" (compare to "random process").

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be a datum that describes a specific picture , such as "picture by Wilbur Semismith, completed January 3, 2003 9:42 AM". We consider

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gives the pressure readings at those times on that one date. We do not think of a realization of the process as measuring a value for

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In practical applications, a theoretical infinity of data is often approximated by a large but finite amount of data. The concept of "

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In the example 1, the stochastic process can be express as f(t, w). What does it mean for the derivative over t, and the integral over t?

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2) A formal description which is not too technical but broead enough to serve the needs of most articles which deal with such processes.

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on a sheet of 8.5" by 11" paper. To describe a color we may need more than a single number. Suppose the color data at location

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It also suggests that their marginal probability distributions are not the same. To write the formula for the joint distribution of

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as providing a way to index a specific datum in all the data that have been produced by one realization of the stochastic process.

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In example 2, once the random picture is realized, we can refer to a single datum by giving its location as a 2 dimensional vector

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Good start! Are you going to follow it up with simplifying the main body of the article, as you outlined in your earlier comments?

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with 3600 components.) Likewise the images in example 2 might be recorded with a scanner. This would produce a file which had

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The definition of s.p. is OK; what is wrong is the definition of modification. A modification is a random variable with values in

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produces 1 datum , the triplet of (weight, height , temperature), on each realization. However it is not customary to do this.

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in the picture there is a certain color value that is part of the realized data. There are an infinite number of such locations

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1) First of all, and very importantly, an intuitive description of a random process (as I proposed in one of my latter comments)

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A particular stochastic process is determined by specifying the joint probability distributions of the various random variables

204:. The point of wiki is that ho HTML is necessary to create a page, you just write text and add very little wiki-specific markup. 724:

in the Knowledge.) Other mathematical definitions do not provide much specific information. The traditional definitions of "

1546:. The use of the word "Series" does not imply that the data must necessarily be indexed by integers. Both the case where 1170:

that they are all realized when a single primitive event in the probability space is realized. For instance, in example 2,

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is a datum that describes a specific picture such as "picture by Wilbur Semismith, completed January 3, 2003 9:42 AM".

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What do others think about? Doing all alone is a lot of work, I would be happy for suggestions, and help of course! --

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You are right! I apologize for my inadequate formulation. This comes from not knowing how wikipedia articles grow. --

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is 1,2 or 3 depending on whether we wish to index the red,green or blue datum. Then an individual random variable

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Stochastic processes can then be discrete or continuous: a discrete stochastic process is an indexed collection of

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is 6:03 AM. From this point of view, the realization of the process consists of picking a specific value of

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consisting of all sequences whose first 30 terms are "tails" and whose other terms contain at least one "head".

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Actually, not the set is the stochastic process, but that implicitly predefined family of random variables

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I have thought another minute about what you wrote. You are right, but this is standard notation. Writing

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The probabilities of various subsets of primitive events can be computed from the joint distribution of

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Example 6: Consider tossing a coin and suppose the probability it lands head is a known probability

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has the same domain. Saying the marginal distributions are the same correctly expresses the idea.)

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is a datum that specifies information such as "January 3, 2002 6:00AM to 7:00AM". The variable

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One way to deal with varying domains is by taking direct sums of the function spaces involved. -- ]

1939:

According to that, a stochastic process is just a set of random variables, without any indexing.

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The Kolmogorov extension theorem is a topic which serves enough material for a seperate article.

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that is the set of two things {heads,tails}. The domain of this random variable is the the set

639: 618: 610: 260: 226: 2159: 1872: 445: 315: 1315:. (The example of tossing a thumbtack is also used by authors who wish to make it clear that 885:

to be a vector valued function whose range is the set of 3 dimensional vectors that give the

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I just tried to be bold, hopefully not too much, and changed a little bit the introduction. --

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are near the maximum (i.e. unless the linear graph is high at both ends). This suggests tha

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value for each pixel from this histogram according to the frequency with which the various

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This article needs input from otehr people, please add this definition if you'd like. --

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datum occurred in their pixels. To realize a random picture, we randomly select an

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Technically a stochastic process is not a function, only a particular realisation of it,

1605:

But, like I said, the current section is half-baked, so it would be fair to remove it.

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My suggestion would be to simply remove this section titled "The Algebraic Approch". --

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random variables. And what might be the meaning of "apparently infinite-dimensional"?

1069:. A random variable has a domain and a range. A realization of a variable like 664: 656: 648: 626: 602: 1554:

is the set of real numbers are studied. Example 1 can be regarded as a Time Series.

1198:

However they need not be independent of each other. In example 5, the measurement

647:

suggestions? Also I'd like comment on the definition above - is it completely correct?

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you add them to the article more easily than if you added the whole thing at once. --

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uncountable. That fact may surprise someone who hasn't studied abstract mathematics.

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Yes, I would like to do that! I hope I will find enough time the following days. --

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and stating exactly what possible values it make have. The possible values of

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The last section is too long. How about spinning it off as a seperate article? -

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If you wish to start a new discussion or revive an old one, please do so on the

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would be used to indicate a specific time. For example, one possible value of

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If you wish to start a new discussion or revive an old one, please do so on the

1704:

The suggestions are all very prudent, but perhaps, you shouldn't use the title

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Another way of looking at example 2 is to let the index set be the set of all

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are independent. For example, we can compute the probability of the subset "

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new version of the article - such things are better accomplished in stages.

254: 995:, which is the product of two uniform distributions since we have assumed 858:

alone. So a single realization of the process is a specific function of

87:

qualities - intuition and rigour. Nevertheless it is worth to try it. --

1443:

are any values whatsoever. So we may say the marginal distribution of

1158:, determines all the pressure readings.) This is the domain of each 2244: 2018:. I think it shoud be explicitly defined that way. What do you think? 751:

occurs on a particular day. Each of these pressures is a datum that

609:

In an encyclopedia article, clarity should come first and rigor later.

289:

assigning one of these random variables to each member of an index set

1299:

organization and structure, are poorly approximated by making each

1190:

on one day and then picking a different day to measure the value of

1565:", especially when an author wishes to emphasize that the index set 394:

corresponds to a value for each random variable and hence a function

1629:

00 what do you know about the random function Z=50%*f(Z)+50%*g(Z)?--

1455:

If we reader wishes to be reminded of the general definition of a "

2209:

Has John Horton Conway anything to do with stochastic processes??

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A stochastic process whose index set represents time is called a

865:

To view example 2 as a "random function" , we consider it to be

824:

represents which specific realization of the process occurred.

968:. A primitive event is the selection of specific values for 728:" fall in this latter category. Even a book whose subject is " 2231:

Separability, or what the Kolmogorov extension does not provide

1101:

is the set of all 3 dimensional vectors of color information

2260: 2116:, people usually mean a function from T to the set of objects 1942:

This definition seems to work at first glance because it uses

1507:

be a collection of random variables indexed by some index set

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is the set of numbers that give a single pressure reading.

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We may view stochastic process of example 5 as a function

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th toss of the coin. The range of each random variable

1519:

is random variable whose domain is the probability space

309:. All of the random variables share the same codomain (a 1303:

an independent realization of the same random variable.

1202:

is completely determined by the choice of a value for

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If you know the limits of X=f(X) and Y=g(Y) as n-: -->

1620:

and created links from five other pages to that page.

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has domain {heads,tails}. The marginal distribution

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can be near the maximum pressure of 1013.25 + 0.2 if

2122: 2076: 2041: 1978: 1948: 1897: 1891:

I think that the definition of stochastic processes (

1239:

is near its maximum of 0.2. But the measurement at

538: 499: 475: 448: 400: 374: 350: 318: 295: 263: 235: 1077:is not the set of times, even though the notation 736:" it is best to begin with an informal discussion. 2149:(and not just the set). Is this what bothers you? 2141: 2108: 2054: 2010: 1961: 1928: 566: 514: 481: 461: 425: 386: 356: 336: 301: 281: 241: 1323:We may think of example 6 as a random function 1041:are viewed as an "index set". One may think of 1037:is viewed as an index and the possible values of 489:can be described as such. Despite this, the term 1418:is "heads" is the same as the probability that 1339:of all possible infinite sequences of the form 877:that give a location on the picture. We let 797:data for a large but finite number of pixels. 1839:The Latin Sto/Stare is probably unrelated. -- 716:Several different definitions are found for " 8: 1923: 1898: 2239:, and at all irrational points the value 2 133:that will greatly improve the article. -- 2130: 2121: 2094: 2084: 2075: 2046: 2040: 1996: 1986: 1977: 1953: 1947: 1905: 1896: 1677:introduction to this topic should include 1081:makes it appear this way. The domain of 543: 537: 498: 474: 453: 447: 405: 399: 373: 349: 317: 294: 262: 234: 1803:Where did the word Stochastic come from? 1550:is a set of integers and the case where 1243:cannot be near the maximum unless both 161:. Under the editing window there is an 2279:Do not edit the contents of this page. 2179:My favorite definition of 'stochastic' 1473:be a probability space consisting of 1097:. The range of each random variable 1033:is a "random function", the variable 44:Do not edit the contents of this page. 1672:Entirely new version of this article? 1435:is used to find the probability that 1120:defines a location on the image and 7: 1850:modification of a stochastic process 1493:is a probability measure defined on 754:is in the same format as the others. 778:is given by a triplet of numbers ( 567:{\displaystyle S_{i}:\Omega \to D,} 1511:and having the property that each 1150:. (Assigning specific values to 739:One purpose of defining the term 580:runs over the countable index set 552: 426:{\displaystyle f_{\omega }:I\to D} 387:{\displaystyle \omega \in \Omega } 381: 322: 24: 1485:is a sigma algebra of subsets of 1057:by integers in order to refer to 601:Comments and opinions please...! 131:with references to the literature 2264: 2205:Conway in the "See also" section 2109:{\displaystyle (F_{t})_{t\in T}} 2011:{\displaystyle (F_{t})_{t\in T}} 1929:{\displaystyle \{F_{t}:t\in T\}} 1561:is often used as a synonym for " 29: 1655:Calculus on stochastic process 1558: 1543: 1214:depends both on the choice of 790: 2257:07:15, 26 September 2011 (UTC) 2136: 2123: 2091: 2077: 1993: 1979: 1616:I have moved that material to 1065:that are indexed by the time 555: 509: 503: 417: 331: 319: 273: 1: 2199:02:20, 26 November 2008 (UTC) 1811:09:28, 11 July 2007 (UTC)Evan 1667:00:32, 1 September 2006 (UTC) 1427:The marginal distribution of 522:are convenient abbreviations. 1210:However the measurement at 791:multivariate random variable 282:{\displaystyle \eta :I\to S} 200:You really have to read the 2172:17:43, 21 August 2008 (UTC) 1936:) used here is not exact. 1844:20:18, 8 October 2007 (UTC) 1827:Online Etymology dictionary 1650:17:49, 31 August 2006 (UTC) 1618:algebra of random variables 850:being fixed, the function 462:{\displaystyle f_{\omega }} 337:{\displaystyle (\Omega ,P)} 2320: 2225:22:12, 23 March 2010 (UTC) 1782:10:34, 24 April 2007 (UTC) 1769:23:11, 23 April 2007 (UTC) 1756:14:28, 22 April 2007 (UTC) 1743:23:42, 21 April 2007 (UTC) 1726:16:50, 21 April 2007 (UTC) 1713:08:29, 21 April 2007 (UTC) 1699:09:44, 13 April 2007 (UTC) 1531:is stochastic process on 687:09:03, 13 April 2007 (UTC) 677:08:41, 13 April 2007 (UTC) 437:of the stochastic process. 92:13:44, 12 April 2007 (UTC) 2178: 2030:13:54, 14 July 2008 (UTC) 1865:11:12, 19 June 2008 (UTC) 1816:09:28, 11 July 2007 (UTC) 1738:and improve the article. 1633:01:05, 25 Jun 2005 (UTC) 1612:02:02, 2004 Nov 28 (UTC) 1594:00:45, 28 Nov 2004 (UTC) 938:) with (7:00, 1013.25 + 708:17:17, 14 Nov 2003 (UTC) 700:03:07, 14 Nov 2003 (UTC) 659:20:30, 15 Dec 2004 (UTC) 651:01:01, 14 Dec 2004 (UTC) 629:13:37, 13 Dec 2004 (UTC) 605:15:21, 12 Dec 2004 (UTC) 1881:17:08, 6 July 2008 (UTC) 1624:02:29, 28 Nov 2004 (UTC) 1574:12:57, 3 Aug 2004 (UTC) 1439:is heads and the other 1385:to be the result of the 1105:. The domain of each 929:probability distribution 854:becomes a function of 667:20:51, 15 Dec 2004 (UTC) 642:22:33, 13 Dec 2004 (UTC) 613:19:56, 12 Dec 2004 (UTC) 515:{\displaystyle \eta (t)} 2142:{\displaystyle (F_{t})} 1797:15:22, 2 May 2007 (UTC) 1265:) since both depend on 1261:) is not independent of 873:be a vector of numbers 18:Talk:Stochastic process 2143: 2110: 2056: 2012: 1963: 1930: 1578:The algebraic approach 568: 516: 493:and notations such as 483: 463: 427: 388: 358: 344:) and the same domain 338: 303: 283: 243: 2277:of past discussions. 2144: 2111: 2057: 2055:{\displaystyle X^{T}} 2013: 1964: 1962:{\displaystyle F_{t}} 1931: 569: 517: 484: 482:{\displaystyle \eta } 464: 428: 389: 359: 339: 304: 284: 244: 42:of past discussions. 2120: 2074: 2039: 1976: 1946: 1895: 1228:. The measurement 741:"stochastic process" 730:stochastic processes 536: 497: 473: 446: 398: 372: 348: 316: 293: 261: 233: 1657:Stochastic calculus 1563:stochastic process 1523:and whose range is 787:stochastic process 712:Informal Discussion 368:). Thus each point 216:Modified Definition 2139: 2106: 2052: 2008: 1959: 1926: 1457:stochastic process 1327:. The variable ' 903:probability space 788: 742: 734:stochastic process 726:stochastic process 718:stochastic process 564: 512: 479: 459: 423: 384: 354: 334: 299: 279: 239: 229:consists of a set 227:stochastic process 225:A one dimensional 157:Just go ahead and 2307: 2306: 2289: 2288: 2283:current talk page 2215:comment added by 2189:comment added by 2174: 2158:comment added by 1162:. The range of 901:are taken from a 786: 740: 357:{\displaystyle D} 311:probability space 302:{\displaystyle I} 242:{\displaystyle S} 72: 71: 54: 53: 48:current talk page 2311: 2298: 2291: 2290: 2268: 2267: 2261: 2227: 2201: 2153: 2148: 2146: 2145: 2140: 2135: 2134: 2115: 2113: 2112: 2107: 2105: 2104: 2089: 2088: 2061: 2059: 2058: 2053: 2051: 2050: 2017: 2015: 2014: 2009: 2007: 2006: 1991: 1990: 1968: 1966: 1965: 1960: 1958: 1957: 1935: 1933: 1932: 1927: 1910: 1909: 1560: 1545: 830:We may think of 807:random variable 792: 692:The last section 573: 571: 570: 565: 548: 547: 529:random variables 521: 519: 518: 513: 488: 486: 485: 480: 469:and the mapping 468: 466: 465: 460: 458: 457: 432: 430: 429: 424: 410: 409: 393: 391: 390: 385: 366:measurable space 363: 361: 360: 355: 343: 341: 340: 335: 308: 306: 305: 300: 288: 286: 285: 280: 253:together with a 251:random variables 248: 246: 245: 240: 63: 56: 55: 33: 32: 26: 2319: 2318: 2314: 2313: 2312: 2310: 2309: 2308: 2294: 2265: 2249:Boris Tsirelson 2233: 2210: 2207: 2184: 2181: 2126: 2118: 2117: 2090: 2080: 2072: 2071: 2042: 2037: 2036: 2022:132.231.132.110 1992: 1982: 1974: 1973: 1949: 1944: 1943: 1901: 1893: 1892: 1889: 1887:Bad definition? 1852: 1805: 1674: 1660: 1642: 1580: 1535:with index set 1360: 1352: 1348: 1344: 1270: 1255: 1248: 1237: 1226: 1219: 1207: 1148: 1141: 1022: 1014: 1007: 1000: 992: 988: 980: 973: 965: 961: 943: 936: 926: 919: 846:. Then, with 714: 694: 633:But should one 539: 534: 533: 495: 494: 491:random function 471: 470: 449: 444: 443: 401: 396: 395: 370: 369: 346: 345: 314: 313: 291: 290: 259: 258: 231: 230: 218: 145:Stephen Tashiro 59: 30: 22: 21: 20: 12: 11: 5: 2317: 2315: 2305: 2304: 2299: 2287: 2286: 2269: 2232: 2229: 2206: 2203: 2191:71.172.120.144 2180: 2177: 2176: 2175: 2138: 2133: 2129: 2125: 2103: 2100: 2097: 2093: 2087: 2083: 2079: 2067: 2066: 2049: 2045: 2005: 2002: 1999: 1995: 1989: 1985: 1981: 1956: 1952: 1925: 1922: 1919: 1916: 1913: 1908: 1904: 1900: 1888: 1885: 1884: 1883: 1851: 1848: 1847: 1846: 1837: 1836: 1835: 1823:Dictionary.com 1804: 1801: 1800: 1799: 1788: 1787: 1786: 1785: 1784: 1772: 1771: 1761: 1760: 1759: 1758: 1746: 1745: 1731: 1730: 1729: 1728: 1716: 1715: 1693: 1685: 1682: 1680: 1678: 1673: 1670: 1659: 1653: 1641: 1638: 1636: 1631:SurrealWarrior 1626: 1625: 1579: 1576: 1556: 1555: 1541: 1540: 1502: 1468: 1467: 1465: 1464: 1461: 1460: 1454: 1453: 1452: 1404: 1403: 1379: 1378: 1377: 1358: 1350: 1346: 1342: 1322: 1321: 1310: 1309: 1307: 1306: 1305: 1304: 1284: 1283: 1268: 1253: 1246: 1235: 1224: 1217: 1205: 1196: 1195: 1168: 1167: 1146: 1139: 1134: 1133: 1111: 1110: 1088: 1048:In example 1, 1028: 1020: 1012: 1005: 998: 990: 986: 978: 971: 963: 959: 941: 934: 924: 917: 864: 863: 784: 783: 713: 710: 693: 690: 669: 668: 644: 643: 623:sigma-algebras 615: 614: 599: 598: 586: 585: 574: 563: 560: 557: 554: 551: 546: 542: 531: 524: 523: 511: 508: 505: 502: 478: 456: 452: 439: 438: 422: 419: 416: 413: 408: 404: 383: 380: 377: 353: 333: 330: 327: 324: 321: 298: 278: 275: 272: 269: 266: 238: 217: 214: 213: 212: 205: 190: 189: 174: 173: 172: 171: 170: 169: 150: 149: 148: 147: 138: 137: 120: 118: 117: 106: 105: 96: 70: 69: 64: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 2316: 2303: 2300: 2297: 2293: 2292: 2284: 2280: 2276: 2275: 2270: 2263: 2262: 2259: 2258: 2254: 2250: 2246: 2242: 2238: 2230: 2228: 2226: 2222: 2218: 2214: 2204: 2202: 2200: 2196: 2192: 2188: 2173: 2169: 2165: 2161: 2157: 2152: 2131: 2127: 2101: 2098: 2095: 2085: 2081: 2069: 2068: 2065: 2047: 2043: 2034: 2033: 2032: 2031: 2027: 2023: 2019: 2003: 2000: 1997: 1987: 1983: 1970: 1954: 1950: 1940: 1937: 1920: 1917: 1914: 1911: 1906: 1902: 1886: 1882: 1878: 1874: 1869: 1868: 1867: 1866: 1862: 1858: 1849: 1845: 1842: 1838: 1834: 1830: 1829: 1828: 1824: 1821:According to 1820: 1819: 1818: 1817: 1814: 1810: 1802: 1798: 1795: 1790: 1789: 1783: 1780: 1776: 1775: 1774: 1773: 1770: 1767: 1763: 1762: 1757: 1754: 1750: 1749: 1748: 1747: 1744: 1741: 1737: 1733: 1732: 1727: 1724: 1720: 1719: 1718: 1717: 1714: 1711: 1707: 1703: 1702: 1701: 1700: 1697: 1689: 1686: 1671: 1669: 1668: 1665: 1658: 1654: 1652: 1651: 1648: 1639: 1637: 1634: 1632: 1623: 1622:Michael Hardy 1619: 1615: 1614: 1613: 1611: 1606: 1603: 1599: 1595: 1593: 1588: 1584: 1577: 1575: 1573: 1572:66.44.104.246 1568: 1564: 1559:Random Field 1553: 1549: 1538: 1534: 1530: 1526: 1522: 1518: 1514: 1510: 1506: 1500: 1496: 1492: 1488: 1484: 1480: 1476: 1472: 1458: 1450: 1446: 1442: 1438: 1434: 1430: 1426: 1421: 1417: 1413: 1409: 1401: 1396: 1392: 1388: 1384: 1375: 1370: 1366: 1361: 1354: 1338: 1334: 1330: 1326: 1318: 1314: 1302: 1297: 1293: 1289: 1282:measurements. 1280: 1276: 1272: 1264: 1260: 1256: 1249: 1242: 1238: 1231: 1227: 1220: 1213: 1209: 1201: 1193: 1189: 1185: 1181: 1177: 1173: 1165: 1161: 1157: 1153: 1149: 1142: 1131: 1127: 1123: 1119: 1115: 1108: 1104: 1100: 1096: 1092: 1086: 1084: 1080: 1076: 1072: 1068: 1064: 1060: 1056: 1051: 1046: 1044: 1040: 1036: 1032: 1026: 1023: 1015: 1008: 1001: 994: 982: 974: 967: 956:is the vector 955: 951: 946: 944: 937: 930: 923: 916: 910: 906: 904: 900: 896: 890: 888: 884: 880: 876: 872: 868: 861: 857: 853: 849: 845: 841: 837: 833: 829: 827:In example 1, 825: 823: 819: 814: 810: 808: 802: 798: 796: 781: 777: 773: 769: 768: 764: 757: 755: 752: 748: 747: 737: 735: 731: 727: 723: 719: 711: 709: 707: 701: 699: 691: 689: 688: 685: 679: 678: 675: 666: 662: 661: 660: 658: 652: 650: 641: 640:Michael Hardy 636: 632: 631: 630: 628: 624: 620: 612: 611:Michael Hardy 608: 607: 606: 604: 596: 592: 588: 587: 583: 579: 575: 561: 558: 549: 544: 540: 532: 530: 526: 525: 506: 500: 492: 476: 454: 450: 441: 440: 436: 433:, known as a 420: 414: 411: 406: 402: 378: 375: 367: 351: 328: 325: 312: 296: 276: 270: 267: 264: 256: 252: 236: 228: 224: 223: 222: 215: 211: 206: 203: 199: 198: 197: 196: 188: 183: 182: 181: 180: 168: 164: 160: 159:edit the page 156: 155: 154: 153: 152: 151: 146: 142: 141: 140: 139: 136: 132: 127: 126: 125: 121: 116: 112: 111: 110: 103: 102: 101: 97: 94: 93: 90: 84: 80: 76: 68: 65: 62: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 2295: 2278: 2272: 2240: 2236: 2234: 2217:139.82.27.31 2208: 2182: 2020: 1971: 1941: 1938: 1890: 1853: 1831: 1813:80.249.76.34 1809:80.249.76.34 1806: 1734:No problem! 1705: 1690: 1687: 1675: 1661: 1643: 1635: 1627: 1607: 1604: 1600: 1596: 1589: 1585: 1581: 1566: 1562: 1551: 1547: 1544:Time Series 1536: 1532: 1528: 1524: 1520: 1516: 1512: 1508: 1504: 1498: 1494: 1490: 1486: 1482: 1478: 1474: 1470: 1456: 1448: 1444: 1440: 1436: 1432: 1428: 1424: 1419: 1415: 1411: 1407: 1399: 1394: 1390: 1386: 1382: 1373: 1368: 1364: 1356: 1340: 1336: 1332: 1328: 1324: 1316: 1312: 1300: 1295: 1291: 1287: 1278: 1274: 1266: 1262: 1258: 1251: 1244: 1240: 1233: 1229: 1222: 1215: 1211: 1203: 1199: 1191: 1187: 1183: 1179: 1175: 1171: 1163: 1159: 1155: 1151: 1144: 1137: 1129: 1125: 1121: 1117: 1113: 1106: 1102: 1098: 1094: 1090: 1087: 1082: 1078: 1074: 1070: 1066: 1062: 1058: 1054: 1049: 1047: 1042: 1038: 1034: 1030: 1027: 1018: 1010: 1003: 996: 984: 976: 969: 957: 953: 949: 947: 939: 932: 921: 914: 911: 907: 898: 894: 891: 886: 882: 878: 874: 870: 866: 859: 855: 851: 847: 843: 839: 835: 831: 828: 826: 821: 817: 815: 811: 803: 799: 794: 779: 775: 771: 766: 762: 761: 758: 756: 753: 749: 745: 738: 733: 729: 725: 722:vector space 717: 715: 702: 695: 680: 670: 653: 645: 634: 616: 600: 594: 590: 581: 577: 490: 434: 219: 202:Editing Help 191: 175: 163:editing help 162: 158: 130: 122: 119: 107: 98: 95: 85: 81: 77: 73: 60: 43: 37: 2271:This is an 2211:—Preceding 2185:—Preceding 2154:—Preceding 1833:Stochastik. 1481:is a set, 1130:s = (x,y,k) 1075:X(603 AM) ) 869:. We let 435:realisation 36:This is an 1640:Codomain D 1501:be a set. 1466:Definition 1355:where each 1279:X(6:30 AM) 1275:X(6:00 AM) 1263:X(6:00 AM 1259:t X(6:30AM 1241:X(6:30 AM) 1230:X(6:00 AM) 1212:X(6:30 AM) 1192:X(6:30 AM) 1188:X(6:00 AM) 1180:X(6:00 AM) 1176:X(6:30 AM) 1172:X(6:00 AM) 1083:X(6:03 AM) 801:ambiguity. 2302:Archive 2 2296:Archive 1 1779:Uli.loewe 1753:Uli.loewe 1723:Uli.loewe 1696:Uli.loewe 1557:The term 1200:X(6:00AM) 1091:s = (x,y 1071:X(6:03 AM 1025:0.1". 684:Uli.loewe 674:Uli.loewe 665:SgtThroat 657:SgtThroat 649:SgtThroat 627:SgtThroat 603:SgtThroat 255:bijective 89:Uli.loewe 67:Archive 2 61:Archive 1 2245:Springer 2213:unsigned 2187:unsigned 2168:contribs 2156:unsigned 1841:Mcorazao 1706:entirely 1497:. Let 1477:) where 1017:0.0 and 927:from a 2274:archive 1736:Be bold 1664:Jackzhp 1647:Jackzhp 1527:. Then 1425:X(i). 1296:(r,g,b) 1292:(r,g,b) 1288:(r,g,b) 1184:X(6:30) 1128:, with 1116:where 1114:(x,y,k) 1103:(r,g,b) 1031:f(t,w) 952:where 887:(r,g,b) 795:(r,g,b) 195:Tashiro 179:Tashiro 39:archive 1857:MarSch 1794:Smarzo 1766:Arcfrk 1740:Arcfrk 1710:Arcfrk 1610:Miguel 1592:Jochen 1475:(S,E,m 1431:over 1369:f(t,w) 1325:f(t,w) 950:f(t,w) 883:f(s,w) 867:f(s,w) 852:f(t,w) 818:f(t,w) 706:Miguel 635:always 576:where 210:Miguel 187:Miguel 167:Miguel 135:Miguel 115:Miguel 2160:Sodin 2151:Sasha 2064:Sasha 1873:Sodin 1469:Let 1449:X(2) 1445:X(1) 1437:X(1) 1429:X(1) 1118:(x,y) 1079:X(t) 1059:A, A 1053:array 1024:: --> 1016:: --> 889:data. 875:(x,y) 780:r,g,b 776:(x,y) 772:(x,y) 698:wshun 619:rings 16:< 2253:talk 2221:talk 2195:talk 2164:talk 2026:talk 1877:talk 1861:talk 1825:and 1513:X(T) 1503:Let 1489:and 1441:X(i) 1420:X(2) 1416:X(1) 1412:X(2) 1410:and 1408:X(1) 1391:X(k) 1383:X(k) 1367:and 1353:...} 1301:X(t) 1250:and 1221:and 1182:and 1174:and 1164:X(t) 1160:X(t) 1143:and 1126:X(s) 1107:X(s) 1099:X(s) 1095:X(s) 1063:X(t) 1002:and 981:. 975:and 257:map 1515:in 1402:. 1277:and 1154:and 1122:k 920:and 765:x,y 621:or 364:(a 249:of 2255:) 2247:. 2223:) 2197:) 2170:) 2166:• 2099:∈ 2028:) 2001:∈ 1918:∈ 1879:) 1863:) 1855:-- 1608:— 1483:E 1395:S 1349:,r 1345:,r 1341:{r 1218:0 1156:p1 1152:p0 1050:t 989:,p 985:(p 962:,p 958:(p 945:). 856:t 836:t 682:-- 597:). 556:→ 553:Ω 501:η 477:η 455:ω 418:→ 407:ω 382:Ω 379:∈ 376:ω 323:Ω 274:→ 265:η 2285:. 2251:( 2241:X 2237:X 2219:( 2193:( 2162:( 2137:) 2132:t 2128:F 2124:( 2102:T 2096:t 2092:) 2086:t 2082:F 2078:( 2048:T 2044:X 2024:( 2004:T 1998:t 1994:) 1988:t 1984:F 1980:( 1955:t 1951:F 1924:} 1921:T 1915:t 1912:: 1907:t 1903:F 1899:{ 1875:( 1859:( 1567:T 1552:T 1548:T 1539:. 1537:T 1533:P 1529:X 1525:R 1521:P 1517:X 1509:T 1505:X 1499:R 1495:E 1491:m 1487:E 1479:S 1471:P 1433:S 1400:w 1387:k 1374:S 1365:w 1359:i 1357:r 1351:3 1347:2 1343:1 1337:S 1333:w 1329:t 1317:p 1313:p 1271:. 1269:0 1267:p 1254:1 1252:p 1247:0 1245:p 1236:0 1234:p 1225:1 1223:p 1216:p 1208:. 1206:0 1204:p 1194:. 1147:1 1145:p 1140:0 1138:p 1067:t 1055:A 1043:t 1039:t 1035:t 1021:2 1019:p 1013:1 1011:p 1006:1 1004:p 999:0 997:p 993:) 991:1 987:0 979:1 977:p 972:0 970:p 966:) 964:1 960:0 954:w 942:1 940:p 935:0 933:p 925:1 922:p 918:0 915:p 899:w 895:w 879:w 871:s 862:. 860:t 848:w 844:w 840:t 832:w 822:w 767:) 763:( 746:. 595:x 593:( 591:f 582:I 578:i 562:, 559:D 550:: 545:i 541:S 510:) 507:t 504:( 451:f 421:D 415:I 412:: 403:f 352:D 332:) 329:P 326:, 320:( 297:I 277:S 271:I 268:: 237:S 50:.

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