Markov chain geostatistics
Encyclopedia
Markov chain
geostatistics
refer to the Markov chain
models, simulation
algorithms and associated spatial correlation
measures (e.g., transiogram
) based on the Markov chain random field
theory, which extends a single Markov chain
into a multi-dimensional field for geostatistical modeling. A Markov chain random field is still a single spatial Markov chain. The spatial Markov chain moves or jumps in a space and decides its state at a location through interactions with its nearest known neighbors in different directions, including its last stay location. Because single-step transition probability matrices
are difficult to estimate from sparse sample
data and are impractical in representing the complex spatial heterogeneity of states, the transiogram
, which is defined as a transition probability function
over the distance lag
, is proposed as the accompanying spatial measure of Markov chain
random fields.
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
geostatistics
Geostatistics
Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology,...
refer to the Markov chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
models, simulation
Simulation
Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....
algorithms and associated spatial correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
measures (e.g., transiogram
Transiogram
Transiogram is the accompanying spatial correlation measure of Markov chain random fields and an important part of Markov chain geostatistics. It is defined as a transition probability function over the distance lag. Simply, a transiogram refers to a transition probability diagram. Transiograms...
) based on the Markov chain random field
Markov network
A Markov random field, Markov network or undirected graphical model is a set of variables having a Markov property described by an undirected graph. A Markov random field is similar to a Bayesian network in its representation of dependencies...
theory, which extends a single Markov chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
into a multi-dimensional field for geostatistical modeling. A Markov chain random field is still a single spatial Markov chain. The spatial Markov chain moves or jumps in a space and decides its state at a location through interactions with its nearest known neighbors in different directions, including its last stay location. Because single-step transition probability matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
are difficult to estimate from sparse sample
Sample (statistics)
In statistics, a sample is a subset of a population. Typically, the population is very large, making a census or a complete enumeration of all the values in the population impractical or impossible. The sample represents a subset of manageable size...
data and are impractical in representing the complex spatial heterogeneity of states, the transiogram
Transiogram
Transiogram is the accompanying spatial correlation measure of Markov chain random fields and an important part of Markov chain geostatistics. It is defined as a transition probability function over the distance lag. Simply, a transiogram refers to a transition probability diagram. Transiograms...
, which is defined as a transition probability function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
over the distance lag
Lag
Lag is a common word meaning to fail to keep up or to fall behind. In real-time applications, the term is used when the application fails to respond in a timely fashion to inputs...
, is proposed as the accompanying spatial measure of Markov chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
random fields.