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Forward–backward Algorithm
The forward–backward algorithm is an inference algorithm for hidden Markov models which computes the posterior marginals of all hidden state variables given a sequence of observations/emissions o_:= o_1,\dots,o_T, i.e. it computes, for all hidden state variables X_t \in \, the distribution P(X_t\ , \ o_). This inference task is usually called ''smoothing''. The algorithm makes use of the principle of dynamic programming to efficiently compute the values that are required to obtain the posterior marginal distributions in two passes. The first pass goes forward in time while the second goes backward in time; hence the name ''forward–backward algorithm''. The term ''forward–backward algorithm'' is also used to refer to any algorithm belonging to the general class of algorithms that operate on sequence models in a forward–backward manner. In this sense, the descriptions in the remainder of this article refer but to one specific instance of this class. Overview In the firs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Inference
Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. In machine learning, the term ''inference'' is sometimes used instead to mean "make a prediction, by evaluating an already trained model"; in this context inferring properties of the model is referred to as ''training'' or ''learning'' (rather than ''inference''), and using a model for prediction is referred to as ''inference'' (instead of ''prediction''); ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Island Algorithm
The island algorithm is an algorithm for performing inference on hidden Markov models, or their generalization, dynamic Bayesian networks. It calculates the marginal distribution for each unobserved node, conditional on any observed nodes. The island algorithm is a modification of belief propagation. It trades smaller memory usage for longer running time: while belief propagation takes O(n) time and O(n) memory, the island algorithm takes O(n log n) time and O(log n) memory. On a computer with an unlimited number of processors, this can be reduced to O(n) total time, while still taking only O(log n) memory.J. Binder, K. Murphy and S. Russell.Space-Efficient Inference in Dynamic Probabilistic Networks Int'l, Joint Conf. on Artificial Intelligence, 1997. The algorithm For simplicity, we describe the algorithm on hidden Markov models. It can be easily generalized to dynamic Bayesian networks by using a junction tree. Belief propagation involves sending a message from the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Detection And Correction
In information theory and coding theory with applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of digital data over unreliable communication channels. Many communication channels are subject to channel noise, and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases. Definitions ''Error detection'' is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver. ''Error correction'' is the detection of errors and reconstruction of the original, error-free data. History In classical antiquity, copyists of the Hebrew Bible were paid for their work according to the number of stichs (lines of verse). As the prose books of the Bible were hardly ever ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have ''optimal substructure''. If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.Cormen, T. H.; Leiserson, C. E.; Rives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE
The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operations center in Piscataway, New Jersey. The mission of the IEEE is ''advancing technology for the benefit of humanity''. The IEEE was formed from the amalgamation of the American Institute of Electrical Engineers and the Institute of Radio Engineers in 1963. Due to its expansion of scope into so many related fields, it is simply referred to by the letters I-E-E-E (pronounced I-triple-E), except on legal business documents. , it is the world's largest association of technical professionals with more than 423,000 members in over 160 countries around the world. Its objectives are the educational and technical advancement of electrical and electronic engineering, telecommunications, computer engineering and similar disciplines. History Origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lawrence Rabiner
Lawrence R. Rabiner (born 28 September 1943) is an electrical engineer working in the fields of digital signal processing and speech processing; in particular in digital signal processing for automatic speech recognition. He has worked on systems for AT&T Corporation for speech recognition. He holds a joint academic appointment between Rutgers University and the University of California, Santa Barbara. Education * B.S. in electrical engineering, Massachusetts Institute of Technology, 1964 * M.S. in electrical engineering, Massachusetts Institute of Technology, 1964 * Ph.D. in electrical engineering, Massachusetts Institute of Technology, 1967 Research interests * Digital signal processing * Speech processing * Multimodal user interfaces * Multimedia communications * Shared collaboration systems for tele-collaboration Life Rabiner was born in Brooklyn, New York, in 1943. During his studies at MIT, he participated in the cooperative program at AT&T Bell Laboratories, during w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BCJR Algorithm
The BCJR algorithm is an algorithm for maximum a posteriori decoding of error correcting codes defined on trellises (principally convolutional codes). The algorithm is named after its inventors: Bahl, Cocke, Frederick Jelinek, Jelinek and Raviv.L.Bahl, J.Cocke, F.Jelinek, and J.Raviv, "Optimal Decoding of Linear Codes for minimizing symbol error rate", IEEE Transactions on Information Theory, vol. IT-20(2), pp. 284-287, March 1974. This algorithm is critical to modern iteratively-decoded error-correcting codes, including turbo codes and low-density parity-check codes. Steps involved Based on the convolutional code, trellis: * Compute forward probabilities \alpha * Compute backward probabilities \beta * Compute smoothed probabilities based on other information (i.e. noise variance for AWGN, bit crossover probability for binary symmetric channel) Variations SBGT BCJR Berrou, Glavieux and Thitimajshima simplification. Log-Map BCJR P. Robertson, P. Hoeher and E. Villebrun, "Optim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baum–Welch Algorithm
In electrical engineering, statistical computing and bioinformatics, the Baum–Welch algorithm is a special case of the expectation–maximization algorithm used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm to compute the statistics for the expectation step. History The Baum–Welch algorithm was named after its inventors Leonard E. Baum and Lloyd R. Welch. The algorithm and the Hidden Markov models were first described in a series of articles by Baum and his peers at the IDA Center for Communications Research, Princeton in the late 1960s and early 1970s. One of the first major applications of HMMs was to the field of speech processing. In the 1980s, HMMs were emerging as a useful tool in the analysis of biological systems and information, and in particular genetic information. They have since become an important tool in the probabilistic modeling of genomic sequences. Description A hidden Markov model describes th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python Programming Language
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language and first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000 and introduced new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 3.0, released in 2008, was a major revision that is not completely backward-compatible with earlier versions. Python 2 was discontinued with version 2.7.18 in 2020. Python consistently ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ill-conditioned
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given f(x) = y, one is solving for ''x,'' and thus the condition number of the (local) inverse must be used. In linear regression the condition number of the moment matrix can be used as a diagnostic for multicollinearity. The condition number is an application of the derivative, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Complexity
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
