Markov Processes for Stochastic Modeling


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Background

Then above I trimmed the pairs down even further into something very interesting. Every key is matched with an array of possible tokens that could follow that key. Every key has possible words that could follow it.

If we were to give this structure from above to someone they could potentially recreate our original sentence! Being that there is only key that follows we have to pick it. But seriously…think about it. We used the current state current key to determine our next state. Further our next state could only be a key that follows the current key.

Sounds cool, but it gets even cooler! Look closely, each oval with a word inside it represents a key with the arrows pointing to potential keys that can follow it! But wait it gets even cooler:. Each arrow has a probability that it will be selected to be the path that the current state will follow to the next state. In summary, we now understand and have illustrated a Markov Model by using the Dr.

Seuss starter sentence. Full Example Summary You made it! But guess what!

A methodology for stochastic analysis of share prices as Markov chains with finite states

Larger Example Keeping in the spirit of Dr. Seuss quotes I went ahead and found four quotes that Theodor Seuss Geisel has immortalized:. The biggest difference between the original starter sentence and our new sentence is the fact that some keys follow different keys a variable amount of times. So what will this additional complexity do to our Markov Model construction? The inner dictionary is severing as a histogram - it is soley keeping track of keys and their occurrences! Well, we will get different distribution of words which is great and will impact the entire structure, but in the larger scope of generating natural unique generated sentences you should aim to have at minimum 20, tokens.

It would be better if you would have at least ,, tokens. But lets chat about how the distribution of words are in a one key window with this larger example. Very nice! Want to know a little secret?


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Lets look at a real example from our data:. Make sense? Bigger Windows Currently, we have only been looking at markov models with windows of size one. By more accurate I mean there will be less randomness in the generated sentences by the model because they will be closer and closer to the original corpus sentences. The current examples we have worked with have been first order markov models. If we use a second order Markov Model our window size would be two!

The window is the data in the current state of the Markov Model and is what is used for decision making. If there is a bigger window in a smaller data set it is unlikely that there will be large unique distributions for the possible outcomes from one window therefore it could only recreate the same sentences.

Very interesting! Any observations? This reveals a potential issue you can face with Markov Models…if you do not have a large enough corpus you will likely only generate sentences within the corpus which is not generating anything unique. Basically it is a histogram built using a dictionary because dictionaries has the unique property of having constant lookup time O 1! The dictogram class can be created with an iterable data set, such as a list of words or entire books.

It is also good to note that I made two functions to return a random word. One just picks a random key and the other function takes into account the amount of occurrences for each word and then returns a weighted random word! In my implementation I have a dictionary that stores windows as the key in the key-value pair and then the value for each key is a dictogram.


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We do this because a tuple is a great way to represent a single list. Parse Markov Model Yay!! Otherwise, you start the generated data with a starting state which I generate from valid starts , then you just keep looking at the possible keys by going into the dictogram for that key that could follow the current state and make a decision based on probability and randomness weighted probability. We keep repeating this until we do it length times!

Applications Some classic examples of Markov models include peoples actions based on weather, the stock market, and tweet generators! Think about what would change? Hint: Not too much, if you have a solid understanding of what, why, and how Markov Models work and can be created the only difference will be how you parse the Markov Model and if you add any unique restrictions. Further Reading Now that you have a good understanding of what a Markov Model is maybe you could explore how a Hidden Markov Model works.

Stochastic shielding and edge importance for Markov chains with timescale separation

Or maybe if you are more inclined to build something using your new found knowledge you could read my artcile on building a HBO Silicon Valley Tweet Generator using a markov model coming soon! Alexander Dejeu alexdejeu. Tweet This.

Operations Research 13A: Stochastic Process & Markov Chain

Typically, a Markov decision process is used to compute a policy of actions that will maximize some utility with respect to expected rewards. It is closely related to reinforcement learning , and can be solved with value iteration and related methods. A partially observable Markov decision process POMDP is a Markov decision process in which the state of the system is only partially observed.

POMDPs are known to be NP complete , but recent approximation techniques have made them useful for a variety of applications, such as controlling simple agents or robots. A Markov random field , or Markov network, may be considered to be a generalization of a Markov chain in multiple dimensions. In a Markov chain, state depends only on the previous state in time, whereas in a Markov random field, each state depends on its neighbors in any of multiple directions. A Markov random field may be visualized as a field or graph of random variables, where the distribution of each random variable depends on the neighboring variables with which it is connected.

More specifically, the joint distribution for any random variable in the graph can be computed as the product of the "clique potentials" of all the cliques in the graph that contain that random variable.

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Modeling a problem as a Markov random field is useful because it implies that the joint distributions at each vertex in the graph may be computed in this manner. Hierarchical Markov models can be applied to categorize human behavior at various levels of abstraction. For example, a series of simple observations, such as a person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing.

A TMM can model three different natures: substitutions, additions or deletions. Successful applications have been efficiently implemented in DNA sequences compression. Markov-chains have been used as a forecasting methods for several topics, for example price trends [9] , wind power [10] and solar irradiance [11]. The Markov-chain forecasting models utilize a variety of different settings, from discretizing the time-series [10] to hidden Markov-models combined with wavelets [9] and the Markov-chain mixture distribution model MCM [11].

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Main article: Markov chain. Main article: Hidden Markov model. Main article: Markov decision process. Artificial Intelligence. Machine Learning. Journal of Artificial Intelligence Research. Michigan State University.

Markov Processes for Stochastic Modeling Markov Processes for Stochastic Modeling
Markov Processes for Stochastic Modeling Markov Processes for Stochastic Modeling
Markov Processes for Stochastic Modeling Markov Processes for Stochastic Modeling
Markov Processes for Stochastic Modeling Markov Processes for Stochastic Modeling
Markov Processes for Stochastic Modeling Markov Processes for Stochastic Modeling
Markov Processes for Stochastic Modeling Markov Processes for Stochastic Modeling
Markov Processes for Stochastic Modeling Markov Processes for Stochastic Modeling
Markov Processes for Stochastic Modeling Markov Processes for Stochastic Modeling

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