Week up to: probability theory and stochastic processes (martingale) and Monte Carlo methods to track _Geoinformatics

Do not write for a long time. This period of time has undergone a lot of things, it does not take into account this effort and thought "100 article" story. Recently been to various forms of finishing graduate several years of experience, feel that the simple combination of their own experiences, lessons learned and to receive from the teachers and students to a variety of information, chat about probability of learning and research proposals must surely be a good thing. To be honest, if I start learning path probability theory, its results and now there is little difference. This distinction is not a resource sense, because I still feel that regardless of my undergraduate or graduate in four years, five years, there are sufficient resources available, the key issue of how to use. I wanted to share some time with us to talk to the "how to" questions, many of which are themselves through detours, and there is an urgent want to fill the vacancy. Of course, different people learn probability or the purpose of random process. I am here for all to engage in the probability of friends. Today on an undergraduate preparation. Since science probability is math, an undergraduate will lay the foundation of mathematics. Courses of nature do not have to say, I want to talk about the focus is on the probability of course how you want to achieve. Generally, undergraduate courses on the basis of probability is elementary probability theory and mathematical statistics, two-door, if any, will also have the opportunity to learn the application of stochastic processes. Of course, there are also courses in statistics, such as multivariate time series, and so on, but I'll focus on the first three classes: elementary probability: since no measure on the basis, so this period the most important tasks have two: through specific probability model and model, intuitive understanding of the "probability" and the basic concepts of meaning; in addition, to master some basic probability calculations because of undergraduate is essentially based on probability calculus, so for discrete and continuous random variables, operations must have a certain amount of training. Dispensable. On the one hand, recognizing the concept of probability theory related to background meaning does not fall into the death count without direction; on the other hand, the probability is very particular about the mathematical calculation of a door, just stay in concept, there is no sufficient probability-aware, so I built, probability theory engfish classmates is the base of the good students calculus, for series and integrals of operation is very skilled. Statistics: statistics of style with probability theory and different, it's the same with two tasks: first, to understand the idea behind statistical theory, because of the statistical methods, in many cases is based on the subjective experience of people, such as maximum likelihood estimation. So you want to begin to understand, most statistics and is not a "right or wrong", but "there is no reason", but this is the probability of some principles to support (such as consistency). This is our way of thinking is a challenge. Interpretation based on statistical theory, the next step is to be considered, mathematical statistics calculation more probability calculations, so these operations can better help you understand the concepts of probability theory and the theorem. Applications of stochastic processes: a lot of friends will be misunderstood undergraduate course should be "application", but in fact the case. Application by the Crown "," main or distinguished from the graduate of specialty stochastic process, because the measure on yet. Therefore, the application of the stochastic processes can be understood as "avoidance measure theory of stochastic processes". Of course, some textbooks did indeed is a prominent part of the "application", will teach the application of some random process. But in my opinion very directly, in application of the stochastic process of course, you do one thing: learn a feature called "Markov chain" model. Although in General, the teachers speak in addition to other things, such as Brown, but my personal opinion is that one semester of course, impossible to grasp more, if you do not consider the exam stress, more energy in the Markov chain model, it's relatively simple mathematical description, easier to understand, intuitive for training, as well as future probabilities, are very necessary. One problem is that an undergraduate should learn measure on? of course North of undergraduate students demanding, in large measure on three will learn. I think this varies from person to person. So, measure on the probability of a modern language, or that it is the probability of mathematical form, it is not in the spirit of probability theory. Personal recommendations, undergraduate, or multiple visual culture probability, as part of the mathematics of probability to specific computing training. In addition, there are two classes I to separate out the one course is ordinary differential equations, another course is physical. For a probability of learning, the two are often neglected or insufficient training. On the one hand, the probability of random processes as "version of the dynamic systems", in research often need to speak to "determine the edition of dynamical systems" for comparison, compare the similarities and differences between them is something very important, also the birth of many valuable; on the other hand, with regard to physical, I think that while many specific physics time difficult to feedback to the probabilistic learning, we need physical intuitive accumulation and physical training methodology, because once the probability from physical intuitive, it is just a second, and so on. Probability has never been formalized, the value of it precisely in its always good physical contact. So physical learning not only important, but is a constant accumulation process. Introduction to probability theory training and other http://www.sciencenet.cn/u/zhouda1112/ statistical due to talk about the next semester to do statistical assistant, so the time to get down to review basic statistics. Reading is a wonderful thing, once per read more, there will be new harvest. Especially when they work in probability theory in the world for a long time after, re-learn statistics is an interesting process. Relation between probability and statistics, but to see if they wanna be's perspective is very different. The previously own lengbuding blows on the blog, but the statistics of cattle to be self-aware, try to keep the distance with statistics. Today, while rereading statistics rehu, honest chat with members. In General, statistics or facing real-world problems. Even if theThe involvement of a large number of theorists make statistics sometimes could be frustrating, but for most people, the task is processing data, this subject is very much about usability. Furthermore, the story of the statistics is not always together with probability theory. The reason why modern probability became the basis of the statistics, it is only because it is statistics provides a reasonable and effective basis. For example, statistics are most concerned about is the data behind the "law", what is the rule, it should be how to characterize, in fact, plagued by many people. And probability theory completely solved the problem: the observed data as sample value, and the data behind the rule as "distribution", and subject to the overall distribution of a random variable to represent the overall. From the "data" up to "statistical model", this is the history of human thought and a great leap. Therefore, statistics provides a data model of the ideas and methods. Actual operation, people will begin according to the actual experience (this kind of experience is important, and not in vain), on the overall distribution for some restrictions. In other words, in our experience, determine a reasonable "distributions", and we assume that the real distribution of cover in the family. Very often, the distributions are more deterministic. For example, "life questions" can generally be attributed to the exponential distribution, only parameter value cannot be determined, in General, parameters are values in a subset of the European space. This is actually very easy, the rest of the work is based on the data to the parameter. Of course, it is foreseeable that so easy examples. But even so, the statistics of these classic content also strongly contributed to scientific progress. The basic contents of classical statistics about three: parameter estimation, hypothesis testing, and regression analysis. Of these, parameter estimation's main task is to find a reasonable statistics on the unknown parameters of regression analysis for inferred; the task is to discuss the different statistical relationships between variables; hypothesis testing task somewhat more difficult to explain, for example, for example, a newly developed drugs through double blind experiment to test the validity of the validity of the judgment is based on the work of the hypothesis testing. Easy to think that these three work whether statistical analysis, or depend on the extent of the model itself. After a number of statistics that the 20th century, many of the classic problem of operation has largely been standardized. The most prominent example is the medicine clinical trials and production control. As for the modern statistical development out how tools to deal with complex issues, I do not quite understand. To biological science-oriented disciplines, new development, the most obvious feature is the amount of data that increasingly large scale. Processing the data of statistics, as its mission to serve the modern science, is both a challenge and an opportunity. You can imagine is that today's statistics of a major task is the huge data compression and reduced dimensions, making it into the modern computer can handle. While these efforts, a considerable number have not probability theory can help, but more and more other mathematics (such as geometry, topology) sent Vilanculos. But as I said, the theater is an application of the subject, even if the theory does not follow, as long as they can have an effect on the actual problems, get everyone's attention. Also, I am very optimistic about the future of science in statistics. Because it might make up for the traditional mathematical sciences in some short Board, thereby creating new thinking in human science. On random depiction of the phenomenon in probability theory, depicting the amount of random phenomena known as "random elements". Of course, General textbook barely use this concept. This concept covers the familiar "random", "random vector" and "random". So, how to characterize the "probability theory random elements?" the most important concept is the "probability distribution". Probability distribution told us the random elements at different values of the "possibility", "statistical nature." It can be said that the probability distribution once determined, in principle we appropriate random phenomena for a complete description. Of course, there are prompts you that for multiple random variables, and stochastic processes, we refer to "probability distribution" is the joint probability distribution. But the truth is not so smooth as imagined. Generally you want to get the probability distribution is not easy. At this time, we are taking a step back, hoping to find some "rough" point characteristics to characterize our concern random phenomena. The most important of course is expected, or you can called average. Expectations of importance was obvious, however, for different situations, expected utility is not the same. For "normal" in such a "unimodal" distribution, expect a very good description of the distribution of overall average nature, that is, we can assert that obey normal distribution of overall, most individuals are concentrated in the vicinity of the "mean"; but if you type "au", or the distribution of the double peak, expectations while also reveals the average behavior of the system, but at this point we cannot say that the "mean" reflects the overall distribution of the "trend", but merely a mathematical average. Like we said two class average is 80, but A majority of 80 classmates; B classmates are 100 and 60. For these two classes, the significance of 80 is not the same. Therefore, it is not enough light there is hope, we also care about the distribution of deviations. So introduce the concepts of variance. Use very fashionable words, we are not only concerned about average "income", are also concerned about "risk". Variance is a measure of risk. It should be noted that, if there is a random variable, we also should characterize the relationship between the different components, then a covariance matrix concept. Mathematically speaking, expectation and variance just reveals random variables "moments" and "second-order moment" in nature. That we may ask, "is more higher moments" covers how new information? if so as expected variance has quite intuitive meaning? on this issue, I think for different practical issues should have a different interpretation, at least in the textbooks, the higher moments of physical meaning few uniform interpretation. However, I would like to share with you another very interesting question--"the moment". Earlier we mentioned that, if on probability distribution difficult completely theSolutions that can only be achieved through the "expected variance" these characteristics for rough depiction. So, if I have to be "arbitrary moments" of information, whether it be approximation for complete information on the distribution of it in probability theory circles are very famous. Here to give you a good conclusion: If the value of a random variable is bound, the moment is feasible. In other words, through the many moments of measurement, you can grasp the asymptotic distribution of information. Probability, statistics, is a problem of perspective, but also the wisdom a coin is not hypothetical, random events? just suppose, in a vacuum of-gravity environments, predetermined good coin, strength, and the key elements of the projectile, and so on, then it can be predicted by the analysis of the dynamics of General Physics, coins appear negative results can be determined. In other words, if you can control with a coin to all factors related to "," coin has nothing to do with randomness seems to be. But in fact, who guarantee can get things done on the share? or, even if it did, the cost will be to what extent? so what to study on a coin to such acts? one perspective: we might as well start with intuitive easy control on the part of the extracted as "variables" that are difficult to grasp in part as "random noise". Of course this one important aspect is, what parts as noise is required to see the actual situation. Such as air flow smoothly, we put the atmospheric environment as identification factors, human factors such as aerosol propellant for random noise, but for air volatile environment, atmospheric environment should also be regarded as random noise. Since it considered the core of noise, is to "measure" noise. Probability theory (or statistics) way is by a large number of experiments, the noise of statistic properties. Perspective 2: this method is actually reflects the probability theory in dealing with the question wisdom: that is, it is difficult to characterize factor is for random noise. Further, we also see the angle of encounter a problem is how to select noise. So, why don't we apply all of the factors are treated as noise! simply, to coin the activities directly as "pure" random, not to consider it behind the "determine the force" and "random force". Directly through a large number of coin, inferred from the experimental results of statistical law. From the above exposition, you will find something "randomization", sometimes is a subjective thing. Or, is a world perspective. How to easily and effectively deal with problems, is our greatest concern. Therefore, the modern statistical analysis in the study of weather problems. On the one hand, of course, we cannot do without the "weather forecast", it cannot substitute; in addition, even if there is no high-tech, we can rely on low-cost monitoring and statistical analysis on the weather "crap", even if it's just a statistical properties, but still can serve the community. So, I think the probability (including statistics) wisdom mainly reflected in the "observations". I also introduced the same processing functions as a general point of view of mathematical processing function is "from the arguments to the variable"; often, "observations" is actually just a variable, since the variables and the function of the form it is very difficult for us to watch, what we do for variable value, and mastering it statistics of distribution of nature, which, in turn, to construct reasonable probability space model analysis. This is from the "coin" example obviously. We do not care about the initial value of a coin, mechanical formula and the environment, we only pass on the results of observation, analysis of problems in search of the rule to. Stochastic processes track classification has just finished Memorial Xu Bao Lu, the centenary of the birth of probability and statistics workshop. So the idea, combined with a Conference, talk to you about some probability of some specific issues. We all know that, in General, physical world, the campaign has two types of forms: one continuous track movement; the other which is not continuous rail movement, also is to say in the process with jumps. We are putting mathematics inside delicate thing, using a simple point of view, the two movements are very natural. Sometimes, a kind of sports as a consecutive or jumps, have to rely on "observation" conditions. Such as the classical physics, we often bring movement into continuous rail movement processed; but in the micro-physics, restrictions based on observation, in theory, always put the motion processing into discrete jumps. Of course, this does not mean that micro-movement must be processed into discrete, macro movement must be continuous. Very often, to see the characteristics of the problem itself. In probability theory, research, most have two basic model. One is familiar to Brownian motion, it is typical of continuous process, just saying in random factors, its orbit looks is not smooth, but some rambling, but its orbit is still continuous. Probability based on Brownian motion, scientists developed a comprehensive theory, stochastic analysis we call it. Another process is called Poisson process, I would like to know a lot of friends. Generally, we introduced the Poisson process from counting. For example, we said that the hospital's reception desk, the number of patients will normally be assumed that it obeys a Poisson process. This is, of course, "counting process with jump", skip a step, is to count time plus one. Therefore, it is recommended that you want to learn the friends of random processes, from the process ofPlay. And then gradually learning more in-depth stochastic processes. From research, we have just talked about the analysis based on random Brownian motion theory is better. Correspondingly, the jump relative difficulty of some of the process. Give an inappropriate examples, such as members of ordinary differential equations will find easier, but on difference equations will be difficult for some. Truth is similar. (Of course don't misunderstand that random analysis is easy to do) in this commemorative meeting, a number of teacher's report entitled introduction to "jump" stochastic processes. From one side, people eager to know about with jump random process more information. Here we should not underestimate the "Skip", the characterization of the jump in mathematics can be very complex, or even not possible to give a visual image. In the process, with jumps, a class called Levy process is the most adequate. Levy process is a large class of procedure, it not only includes many jump processes (such as Poisson process), but also Brownian motion, just now probability scientist is in with the theory of the Levy with jumps. I think, be familiar with the financial engineering friends contact with Levy processes, its application is very wide. Summing up, we will conduct random process follow the track, the classification is based on very simple observation. From research techniques, this classification is justified. Or would like to reiterate that a movement like what track is processed depends on the object, but sometimes it is a subjective thing. For example, in financial mathematics, studies of multiple asset pricing is based on random Brownian motion and analysis; and the insurance industry is more of a jump. Here, I just want to learn and understand the probability of a friend, from orbit level on stochastic processes do differently, and recommendations from the Brownian motion and POISSON process started. Martingale ? "asked some friends to" martingale "concept. I always have avoided because this concept is too large, and their understanding of it is not very thorough, not daring to comments. But travelled to see the ladies tea ", think it's important to be able to use popular language a little explanation. Martingale English is the martingale. We check Baidu dictionary, there are two explanations: 1. Mr Chin habenular; 2. doubling the bet. Explains the seemingly not taking sides with mathematics. But in fact, when the Levy "martingale" as the name is very neat. Hope that through this Exchange, we can understand the "horse Chin habenular" and the "mathematical" martingale exquisite relationship. Martingale should call the martingale process because it is a large class of stochastic processes. Therefore, martingale is not mysterious, it is a kind of process, but such a procedure has a very good character. This kind of "good" nature, many Markov processes, including the Brownian motion, simple random walks; even many seemingly did not have the "nature" of the martingale, slightly modified, can easily join the "family of martingale process". That this "good character" is. I'm from Chin habenular. Ma-throated habenular is France farmer sets of horses a device, let the horse bow not dumped backwards. In the device's control, the horse's head can be active, but the horse's next most likely location is its location. In other words, the Chin, the reins of the "Tau movement" a random process with the "current best estimate of the future" in nature. If you still do not quite understand, we then use the "double the stakes" for a little explanation. We assume that gambling is fair, that is, the probability of winning or losing. At this point, the next step in the loss of the expected value is equal to the current losses. gambler Therefore, it was simply "fair gambling" intuitive interpretation "martingale". Generally, the more fashionable financial terminology: martingale is "based on current information available on a future best expected asset prices are current prices of assets. "Martingale" represents the validity of the financial market. In other words, the efficient market hypothesis, the stock could not have been manipulated, where market information free, equal opportunity. Almost the entire premise of financial mathematics are "efficient market hypothesis", in other words, the stock price is a martingale process. Finally, the mathematics portion of the return of martingale. If Xn is the martingale process, E (X (n + 1) | Fn )=X( n )�� Mean, known n all information before, X (n + 1) expectations is X (n). Martingale theory can be said to have become the most mature of probability theory in the development of a branch. Probability of scientists concerned with the martingale, of course, not only because of the practical value of martingale, also because the martingale with very good mathematical in nature. Very often, we want to put the question to be processed into a martingale, then use the mathematical properties of martingale wonderful. Therefore, to reflect the wide range of martingale physical facts, and a wonderful mathematical structure, such good things, must be the darling of the academic world. MonteCarlo method MonteCarlo method is also known as random sampling techniques or methods of statistical experiment, belongs to a branch of mathematics calculation, it is in the last century, in the mid-1940s, in order to meet the needs of the Manhattan project at that time and in the United States LosAlamos laboratory, blunt is made before the United States in order to make atomic bombs. MonteCarlo method and calculation methods in general are quite different from the General calculation method for solving the cube or factors complicated problem is very difficult, and the MonteCarlo method to resolve such problems are relatively simple, MonteCarlo method since it's birth date is a rapid development, has now developed into computational mathematics in an indispensable component. MonteCarlo method founder mainly these four: Stanislaw Marcin Ulam, Enrico Fermi, John vonNeumann (learn computer must know the cow) and Nicholas Metropolis. Stanislaw MarcinUlam is American mathematician, Poland is the study of topology of his early years, because of the interest involved in the Manhattan project, then goThe applied mathematics, he first proposed by MonteCarlo method to solve computational mathematics in some problems, and then apply it to solve chain theory, can be said to be the founder of MC method; EnricoFermi is a physical Daniel, theory and experiments are Daniel, which is rare in the physical world, in the "physical Daniel gossip," the article refers to the man many times that person for such cattle can only be a great (not to mention my mouth lesions, Ah, God is jealous!); Johnvon Neumann is the computer industry of Newton, Pacific cattle, and Fermi, God is jealous; American mathematician NicholasMetropolis, Greece, physicists, computer scientists, the man on the contribution of MonteCarlo method is considerable, he made the formal result of what algorithms (forgot the name) that makes the MonteCarlo method to get such a broad range of applications, this person is still alive, unlike previous bit cattle were different, the Metropolis is single-minded, his life is the main contribution of MonteCarlo method. MonteCarol method the name quite interesting, why is it called Monte Carlo? this is the history from MonteCarlo method. The original Lee Lee told me brag b when mentioned a place called Monaco, in France (not Morocco Oh, this in North Africa), is one of the Monaco MonteCarlo-names, some say there are only two places in Monaco, in addition to the Royal Palace is the MonteCarlo ... MonteCarlo and Las Vegas, Macau and is known as the world's largest casino. Ulam said reason is called the MonteCarlo method, is to commemorate his uncle, and his uncle was a gambler, often went to Monte Carlo in the pool, so it is called the MonteCarlo method, for this reason you really compare uncommon, but if you understand the principle of the Monte Carlo method, we will find the MonteCarlo method is actually a gambling game, that name is still very relevant. Now take a look at why MC method is gambling game, this is the basic idea from its start. The basic idea of MC method is when the solution of the problem is the probability of an event occurring or the expectations of a random variable, you can pass a test method by the probability of an event occurs, or the average of this random variable, and use them as the solution of the problem. Application of MC method to solve practical problems, not like the usual statistical methods that pass a real experiment, but to grasp things in motion on the number and geometry, mathematical methods to simulate, that is, a digital simulation, simulation experiment, the more, the simulation results will be close to the real value. For specific mathematical or physical problems, to get more accurate simulation results, often require more time, or even hundreds of thousands, millions of digital simulation experiment, its computation is enormous, so the computer has not invented, MC method itself isn't much greater development and application. In the electronic computer and rapid development, the use of a large amount of MC method of digital simulation experiment, MC method finally ushered in the development of spring. This shows that the MC is the development of electronic computer closely linked is the mathematical statistics and electronic computer product of the combination. The last paragraph of the abstract, the following can give a simple example: Let's say you want to calculate pi, MC method to resolve this problem idea is as follows: x = (random #) * r y = (random #) * r dist = sqrt (x ^ 2 + y ^ 2) ifdist.from.origin (less.than.or.equal.to) r lethits = hits + 1 above this paragraph you can think about it this way: in a side-r square in uniform plumbing, and then determine whether the point falls and this square inscribed in the circle of RADIUS r, if point falls in the circle, the records are not logged. So, if the vote cast by point enough and sufficiently uniform, fall inside the circle the number of points divided by the total vote count both round and square area ratio, know the value, we can conclude that the value of pi. The above example just to illustrate the principle, MC method does not reflect the superiority of MC method, let's think about this slightly more complex example: Poker points are placed uncomfortable everyone played? if the operation is correct, ask a poker pendulum awkward chance of EgyptAir? this as long as the rules for good use MC method solutions is very easy, other methods will not be able to imagine. This shows that the MC method plainly is actually a brute force method, use our words is called "simple and crude"! so someone MC method is called the "last resort". But often it is this simple and crude approach in handling complex problems is very effective, it's just a simple example, similar, we can also be calculated by MC complex metamorphosis of definite integral, methods and means of the above calculation of pi. Now MC method has been applied to many areas, in physical, through MC method to simulate interaction of particles and substancesRole of the various reactions, and we are interested in physical quantities for statistics, there are many physical quantity cannot be measured by means of experiments; in the finance field, many large corporations use MC method risk investment risk coefficient; see also an article the other day Egypt's article is MC method simulation of floods, and Egypt on how to repair a dam ... In short, as long as the establishment of model good enough, MC concluded very credible, but also can emulate the statistical fluctuations, too realistic. 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