Markov Ketten Homogene Markov-Kette
Eine Markow-Kette (englisch Markov chain; auch Markow-Prozess, nach Andrei Andrejewitsch Markow; andere Schreibweisen Markov-Kette, Markoff-Kette. Eine Markow-Kette ist ein spezieller stochastischer Prozess. Ziel bei der Anwendung von Markow-Ketten ist es, Wahrscheinlichkeiten für das Eintreten zukünftiger Ereignisse anzugeben. Zur Motivation der Einführung von Markov-Ketten betrachte folgendes Beispiel: Beispiel. Wir wollen die folgende Situation mathematisch formalisieren: Eine. Handelt es sich um einen zeitdiskreten Prozess, wenn also X(t) nur abzählbar viele Werte annehmen kann, so heißt Dein Prozess Markov-Kette. mit deren Hilfe viele Probleme, die als absorbierende Markov-Kette gesehen Mit sogenannten Markow-Ketten können bestimmte stochastische Prozesse.
Also, in Go we can define methods on any named type not just structs , so we can add methods that operate on Prefix if we need to.
The String method. The first method we define on Prefix is String. It returns a string representation of a Prefix by joining the slice elements together with spaces.
We will use this method to generate keys when working with the chain map. Building the chain. The Build method reads text from an io. Reader and parses it into prefixes and suffixes that are stored in the Chain.
The io. Reader is an interface type that is widely used by the standard library and other Go code.
Our code uses the fmt. Fscan function, which reads space-separated values from an io. The Build method returns once the Reader 's Read method returns io.
EOF end of file or some other read error occurs. Buffering the input. This function does many small reads, which can be inefficient for some Readers.
For efficiency we wrap the provided io. Reader with bufio. NewReader to create a new io. Reader that provides buffering.
The Prefix variable. At the top of the function we make a Prefix slice p using the Chain 's prefixLen field as its length.
We'll use this variable to hold the current prefix and mutate it with each new word we encounter. Scanning words.
In our loop we read words from the Reader into a string variable s using fmt. Since Fscan uses space to separate each input value, each call will yield just one word including punctuation , which is exactly what we need.
Fscan returns an error if it encounters a read error io. EOF , for example or if it can't scan the requested value in our case, a single string.
In either case we just want to stop scanning, so we break out of the loop. Adding a prefix and suffix to the chain.
The word stored in s is a new suffix. String and appending the suffix to the slice stored under that key. The built-in append function appends elements to a slice and allocates new storage when necessary.
When the provided slice is nil , append allocates a new slice. This behavior conveniently ties in with the semantics of our map: retrieving an unset key returns the zero value of the value type and the zero value of []string is nil.
When our program encounters a new prefix yielding a nil value in the map append will allocate a new slice.
While MCMC methods were created to address multi-dimensional problems better than generic Monte Carlo algorithms, when the number of dimensions rises they too tend to suffer the curse of dimensionality : regions of higher probability tend to stretch and get lost in an increasing volume of space that contributes little to the integral.
One way to address this problem could be shortening the steps of the walker, so that it doesn't continuously try to exit the highest probability region, though this way the process would be highly autocorrelated and expensive i.
More sophisticated methods such as Hamiltonian Monte Carlo and the Wang and Landau algorithm use various ways of reducing this autocorrelation, while managing to keep the process in the regions that give a higher contribution to the integral.
These algorithms usually rely on a more complicated theory and are harder to implement, but they usually converge faster.
Interacting MCMC methodologies are a class of mean field particle methods for obtaining random samples from a sequence of probability distributions with an increasing level of sampling complexity.
In principle, any Markov chain Monte Carlo sampler can be turned into an interacting Markov chain Monte Carlo sampler. These interacting Markov chain Monte Carlo samplers can be interpreted as a way to run in parallel a sequence of Markov chain Monte Carlo samplers.
For instance, interacting simulated annealing algorithms are based on independent Metropolis-Hastings moves interacting sequentially with a selection-resampling type mechanism.
In contrast to traditional Markov chain Monte Carlo methods, the precision parameter of this class of interacting Markov chain Monte Carlo samplers is only related to the number of interacting Markov chain Monte Carlo samplers.
These advanced particle methodologies belong to the class of Feynman-Kac particle models, [14] [15] also called Sequential Monte Carlo or particle filter methods in Bayesian inference and signal processing communities.
The advantage of low-discrepancy sequences in lieu of random numbers for simple independent Monte Carlo sampling is well known. Empirically it allows the reduction of both estimation error and convergence time by an order of magnitude.
Usually it is not hard to construct a Markov chain with the desired properties. The more difficult problem is to determine how many steps are needed to converge to the stationary distribution within an acceptable error.
A standard empirical method to assess convergence is to run several independent simulated Markov chains and check that the ratio of inter-chain to intra-chain variances for all the parameters sampled is close to 1.
Typically, Markov chain Monte Carlo sampling can only approximate the target distribution, as there is always some residual effect of the starting position.
More sophisticated Markov chain Monte Carlo-based algorithms such as coupling from the past can produce exact samples, at the cost of additional computation and an unbounded though finite in expectation running time.
Many random walk Monte Carlo methods move around the equilibrium distribution in relatively small steps, with no tendency for the steps to proceed in the same direction.
These methods are easy to implement and analyze, but unfortunately it can take a long time for the walker to explore all of the space.
The walker will often double back and cover ground already covered. Further consideration of convergence is at Markov chain central limit theorem.
See [25] for a discussion of the theory related to convergence and stationarity of the Metropolis-Hastings algorithm. From Wikipedia, the free encyclopedia.
Class of dependent sampling algorithms. September Physical Review E. April AIChE Journal. CRC Press. Journal of the Royal Statistical Society.
Series C Applied Statistics. Bibcode : ITSP Journal of the American Statistical Association. Learn More in these related Britannica articles:.
A stochastic process is called Markovian after the Russian mathematician Andrey Andreyevich Markov if at any time t the conditional probability of an arbitrary future event given the entire past of the process—i.
Andrey Nikolayevich Kolmogorov: Mathematical research. Kolmogorov invented a pair of functions to characterize the transition probabilities for a Markov process and….
Andrey Andreyevich Markov , Russian mathematician who helped to develop the theory of stochastic processes, especially those called Markov chains.
Based on the study of the probability of mutually dependent events, his work has been developed and widely…. History at your fingertips.
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Since Fscan uses space to separate each input value, each call will yield just one word including punctuation , which is exactly what we need.
Fscan returns an error if it encounters a read error io. EOF , for example or if it can't scan the requested value in our case, a single string.
In either case we just want to stop scanning, so we break out of the loop. Adding a prefix and suffix to the chain. The word stored in s is a new suffix.
String and appending the suffix to the slice stored under that key. The built-in append function appends elements to a slice and allocates new storage when necessary.
When the provided slice is nil , append allocates a new slice. This behavior conveniently ties in with the semantics of our map: retrieving an unset key returns the zero value of the value type and the zero value of []string is nil.
When our program encounters a new prefix yielding a nil value in the map append will allocate a new slice.
For more information about the append function and slices in general see the Slices: usage and internals article. Pushing the suffix onto the prefix.
Before reading the next word our algorithm requires us to drop the first word from the prefix and push the current suffix onto the prefix.
The Shift method. The Shift method uses the built-in copy function to copy the last len p -1 elements of p to the start of the slice, effectively moving the elements one index to the left if you consider zero as the leftmost index.
Generating text. The Generate method is similar to Build except that instead of reading words from a Reader and storing them in a map, it reads words from the map and appends them to a slice words.
Generate uses a conditional for loop to generate up to n words. Getting potential suffixes. At each iteration of the loop we retrieve a list of potential suffixes for the current prefix.
We access the chain map at key p. String and assign its contents to choices. If len choices is zero we break out of the loop as there are no potential suffixes for that prefix.
This test also works if the key isn't present in the map at all: in that case, choices will be nil and the length of a nil slice is zero.
Choosing a suffix at random. To choose a suffix we use the rand. Intn function. It returns a random integer up to but not including the provided value.
Passing in len choices gives us a random index into the full length of the list. We use that index to pick our new suffix, assign it to next and append it to the words slice.
Next, we Shift the new suffix onto the prefix just as we did in the Build method. Returning the generated text. Before returning the generated text as a string, we use the strings.
Join function to join the elements of the words slice together, separated by spaces. Command-line flags. Markov chains have many applications as statistical models of real-world processes, [1] [4] [5] [6] such as studying cruise control systems in motor vehicles , queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics.
Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo , which are used for simulating sampling from complex probability distributions, and have found application in Bayesian statistics and artificial intelligence.
The adjective Markovian is used to describe something that is related to a Markov process. A Markov process is a stochastic process that satisfies the Markov property [1] sometimes characterized as " memorylessness ".
In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history.
A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set often representing time , but the precise definition of a Markov chain varies.
The system's state space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v.
Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes.
Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain DTMC , [1] [17] [17] but a few authors use the term "Markov process" to refer to a continuous-time Markov chain CTMC without explicit mention.
Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs.
Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term.
While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space.
Besides time-index and state-space parameters, there are many other variations, extensions and generalizations see Variations.
For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise.
The changes of state of the system are called transitions. The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state or initial distribution across the state space.
By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate.
A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. Formally, the steps are the integers or natural numbers , and the random process is a mapping of these to states.
Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future.
Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest in , and a branching process, introduced by Francis Galton and Henry William Watson in , preceding the work of Markov.
Andrei Kolmogorov developed in a paper a large part of the early theory of continuous-time Markov processes.
Random walks based on integers and the gambler's ruin problem are examples of Markov processes. From any position there are two possible transitions, to the next or previous integer.
The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0.
These probabilities are independent of whether the system was previously in 4 or 6. Another example is the dietary habits of a creature who eats only grapes, cheese, or lettuce, and whose dietary habits conform to the following rules:.
This creature's eating habits can be modeled with a Markov chain since its choice tomorrow depends solely on what it ate today, not what it ate yesterday or any other time in the past.
One statistical property that could be calculated is the expected percentage, over a long period, of the days on which the creature will eat grapes.
A series of independent events for example, a series of coin flips satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next step depends non-trivially on the current state.
To see why this is the case, suppose that in the first six draws, all five nickels and a quarter are drawn. However, it is possible to model this scenario as a Markov process.
This new model would be represented by possible states that is, 6x6x6 states, since each of the three coin types could have zero to five coins on the table by the end of the 6 draws.
After the second draw, the third draw depends on which coins have so far been drawn, but no longer only on the coins that were drawn for the first state since probabilistically important information has since been added to the scenario.
A discrete-time Markov chain is a sequence of random variables X 1 , X 2 , X 3 , The possible values of X i form a countable set S called the state space of the chain.
The elements q ii are chosen such that each row of the transition rate matrix sums to zero, while the row-sums of a probability transition matrix in a discrete Markov chain are all equal to one.
There are three equivalent definitions of the process. Define a discrete-time Markov chain Y n to describe the n th jump of the process and variables S 1 , S 2 , S 3 , If the state space is finite , the transition probability distribution can be represented by a matrix , called the transition matrix, with the i , j th element of P equal to.
Since each row of P sums to one and all elements are non-negative, P is a right stochastic matrix. By comparing this definition with that of an eigenvector we see that the two concepts are related and that.
If there is more than one unit eigenvector then a weighted sum of the corresponding stationary states is also a stationary state.
But for a Markov chain one is usually more interested in a stationary state that is the limit of the sequence of distributions for some initial distribution.
If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k -step transition probability can be computed as the k -th power of the transition matrix, P k.
This is stated by the Perron—Frobenius theorem. Because there are a number of different special cases to consider, the process of finding this limit if it exists can be a lengthy task.
However, there are many techniques that can assist in finding this limit. Multiplying together stochastic matrices always yields another stochastic matrix, so Q must be a stochastic matrix see the definition above.
It is sometimes sufficient to use the matrix equation above and the fact that Q is a stochastic matrix to solve for Q. Here is one method for doing so: first, define the function f A to return the matrix A with its right-most column replaced with all 1's.
One thing to notice is that if P has an element P i , i on its main diagonal that is equal to 1 and the i th row or column is otherwise filled with 0's, then that row or column will remain unchanged in all of the subsequent powers P k.
Hence, the i th row or column of Q will have the 1 and the 0's in the same positions as in P. Then assuming that P is diagonalizable or equivalently that P has n linearly independent eigenvectors, speed of convergence is elaborated as follows.
For non-diagonalizable, that is, defective matrices , one may start with the Jordan normal form of P and proceed with a bit more involved set of arguments in a similar way.
Then by eigendecomposition. Since P is a row stochastic matrix, its largest left eigenvalue is 1. That means. Many results for Markov chains with finite state space can be generalized to chains with uncountable state space through Harris chains.
The main idea is to see if there is a point in the state space that the chain hits with probability one.
Lastly, the collection of Harris chains is a comfortable level of generality, which is broad enough to contain a large number of interesting examples, yet restrictive enough to allow for a rich theory.
The use of Markov chains in Markov chain Monte Carlo methods covers cases where the process follows a continuous state space.
Considering a collection of Markov chains whose evolution takes in account the state of other Markov chains, is related to the notion of locally interacting Markov chains.
This corresponds to the situation when the state space has a Cartesian- product form. See interacting particle system and stochastic cellular automata probabilistic cellular automata.
See for instance Interaction of Markov Processes [53] or [54]. Two states communicate with each other if both are reachable from one another by a sequence of transitions that have positive probability.
This is an equivalence relation which yields a set of communicating classes. A class is closed if the probability of leaving the class is zero.
A Markov chain is irreducible if there is one communicating class, the state space. That is:. A state i is said to be transient if, starting from i , there is a non-zero probability that the chain will never return to i.
It is recurrent otherwise. For a recurrent state i , the mean hitting time is defined as:. Periodicity, transience, recurrence and positive and null recurrence are class properties—that is, if one state has the property then all states in its communicating class have the property.
A state i is said to be ergodic if it is aperiodic and positive recurrent. In other words, a state i is ergodic if it is recurrent, has a period of 1 , and has finite mean recurrence time.
If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state.
More generally, a Markov chain is ergodic if there is a number N such that any state can be reached from any other state in any number of steps less or equal to a number N.
A Markov chain with more than one state and just one out-going transition per state is either not irreducible or not aperiodic, hence cannot be ergodic.
In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the 'current' and 'future' states.
For example, let X be a non-Markovian process. Then define a process Y , such that each state of Y represents a time-interval of states of X.
Mathematically, this takes the form:. An example of a non-Markovian process with a Markovian representation is an autoregressive time series of order greater than one.
The hitting time is the time, starting in a given set of states until the chain arrives in a given state or set of states. The distribution of such a time period has a phase type distribution.
The simplest such distribution is that of a single exponentially distributed transition. By Kelly's lemma this process has the same stationary distribution as the forward process.
A chain is said to be reversible if the reversed process is the same as the forward process. Kolmogorov's criterion states that the necessary and sufficient condition for a process to be reversible is that the product of transition rates around a closed loop must be the same in both directions.
Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as a jump process. Each element of the one-step transition probability matrix of the EMC, S , is denoted by s ij , and represents the conditional probability of transitioning from state i into state j.
These conditional probabilities may be found by. S may be periodic, even if Q is not. Markov models are used to model changing systems.
There are 4 main types of models, that generalize Markov chains depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made:.
A Bernoulli scheme is a special case of a Markov chain where the transition probability matrix has identical rows, which means that the next state is even independent of the current state in addition to being independent of the past states.
A Bernoulli scheme with only two possible states is known as a Bernoulli process. Research has reported the application and usefulness of Markov chains in a wide range of topics such as physics, chemistry, biology, medicine, music, game theory and sports.
Markovian systems appear extensively in thermodynamics and statistical mechanics , whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description.
Therefore, Markov Chain Monte Carlo method can be used to draw samples randomly from a black-box to approximate the probability distribution of attributes over a range of objects.
The paths, in the path integral formulation of quantum mechanics, are Markov chains. Markov chains are used in lattice QCD simulations.
A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain.
For example, imagine a large number n of molecules in solution in state A, each of which can undergo a chemical reaction to state B with a certain average rate.
Perhaps the molecule is an enzyme, and the states refer to how it is folded. The state of any single enzyme follows a Markov chain, and since the molecules are essentially independent of each other, the number of molecules in state A or B at a time is n times the probability a given molecule is in that state.
The classical model of enzyme activity, Michaelis—Menten kinetics , can be viewed as a Markov chain, where at each time step the reaction proceeds in some direction.
Markov chains, named after Andrey Markov , are mathematical systems that hop from one "state" a situation or set of values to another.
For example, if you made a Markov chain model of a baby's behavior, you might include "playing," "eating", "sleeping," and "crying" as states, which together with other behaviors could form a 'state space': a list of all possible states.
In addition, on top of the state space, a Markov chain tells you the probabilitiy of hopping, or "transitioning," from one state to any other statee.
With two states A and B in our state space, there are 4 possible transitions not 2, because a state can transition back into itself.
If we're at 'A' we could transition to 'B' or stay at 'A'. If we're at 'B' we could transition to 'A' or stay at 'B'. In this two state diagram, the probability of transitioning from any state to any other state is 0.
Of course, real modelers don't always draw out Markov chain diagrams. Instead they use a "transition matrix" to tally the transition probabilities.
Every state in the state space is included once as a row and again as a column, and each cell in the matrix tells you the probability of transitioning from its row's state to its column's state.
So, in the matrix, the cells do the same job that the arrows do in the diagram.
und es hat das Analogon?
ich beglГјckwГјnsche, Ihr Gedanke ist sehr gut
Jetzt kann ich an der Diskussion nicht teilnehmen - es gibt keine freie Zeit. Sehr werde ich bald die Meinung unbedingt aussprechen.
Einem Gott ist es bekannt!