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Importance Sampling
Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally attributed to a paper by Teun Kloek and Herman K. van Dijk in 1978, but its precursors can be found in statistical physics as early as 1949. Importance sampling is also related to umbrella sampling in computational physics. Depending on the application, the term may refer to the process of sampling from this alternative distribution, the process of inference, or both. Basic theory Let X\colon \Omega\to \mathbb be a random variable in some probability space (\Omega,\mathcal,P). We wish to estimate the expected value of ''X'' under ''P'', denoted E 'X;P'' If we have statistically independent random samples x_1, \ldots, x_n, generated according to ''P'', then an empirical estimate of E 'X;P''is : \widehat_ ;P= \frac \sum_^n x_i \quad \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of ris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Important
Importance is a property of entities that matter or make a difference. For example, World War II was an important event and Albert Einstein was an important person because of how they affected the world. There are disagreements in the academic literature about what type of difference is required. According to the causal impact view, something is important if it has a big causal impact on the world. This view is rejected by various theorists, who insist that an additional aspect is required: that the impact in question makes a value difference. This is often understood in terms of how the important thing affects the well-being of people. So on this view, World War II was important, not just because it brought about many wide-ranging changes but because these changes had severe negative impacts on the well-being of the people involved. The difference in question is usually understood counterfactually as the contrast between how the world actually is and how the world would have bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Confidence Interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 90% or 99%, are sometimes used. The confidence level represents the long-run proportion of corresponding CIs that contain the true value of the parameter. For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter's true value. Factors affecting the width of the CI include the sample size, the variability in the sample, and the confidence level. All else being the same, a larger sample produces a narrower confidence interval, greater variability in the sample produces a wider confidence interval, and a higher confidence level produces a wider confidence interval. Definition Let be a random sample from a probability distribution with statistical parameter , which is a quantity to be estimate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loss Function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viterbi Decoder
A Viterbi decoder uses the Viterbi algorithm for decoding a bitstream that has been encoded using a convolutional code or trellis code. There are other algorithms for decoding a convolutionally encoded stream (for example, the Fano algorithm). The Viterbi algorithm is the most resource-consuming, but it does the maximum likelihood decoding. It is most often used for decoding convolutional codes with constraint lengths k≤3, but values up to k=15 are used in practice. Viterbi decoding was developed by Andrew J. Viterbi and published in the paper There are both hardware (in modems) and software implementations of a Viterbi decoder. Viterbi decoding is used in the iterative Viterbi decoding algorithm. Hardware implementation A hardware Viterbi decoder for basic (not punctured) code usually consists of the following major blocks: *Branch metric unit (BMU) *Path metric unit (PMU) *Traceback unit (TBU) Branch metric unit (BMU) A branch metric unit's function is to calcu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersymbol Interference
In telecommunication, intersymbol interference (ISI) is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have a similar effect as noise, thus making the communication less reliable. The spreading of the pulse beyond its allotted time interval causes it to interfere with neighboring pulses. ISI is usually caused by multipath propagation or the inherent linear or non-linear frequency response of a communication channel causing successive symbols to blur together. The presence of ISI in the system introduces errors in the decision device at the receiver output. Therefore, in the design of the transmitting and receiving filters, the objective is to minimize the effects of ISI, and thereby deliver the digital data to its destination with the smallest error rate possible. Ways to alleviate intersymbol interference include adaptive equalization and error correcting codes. Causes Multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Communication
Data transmission and data reception or, more broadly, data communication or digital communications is the transfer and reception of data in the form of a digital bitstream or a digitized analog signal transmitted over a point-to-point or point-to-multipoint communication channel. Examples of such channels are copper wires, optical fibers, wireless communication using radio spectrum, storage media and computer buses. The data are represented as an electromagnetic signal, such as an electrical voltage, radiowave, microwave, or infrared signal. Analog transmission is a method of conveying voice, data, image, signal or video information using a continuous signal which varies in amplitude, phase, or some other property in proportion to that of a variable. The messages are either represented by a sequence of pulses by means of a line code (''baseband transmission''), or by a limited set of continuously varying waveforms (''passband transmission''), using a digital modulation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: ''success'' (with probability ''p'') or ''failure'' (with probability q=1-p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., ''n'' = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size ''n'' drawn with replacement from a population of size ''N''. If the sampling is carried out without replacement, the draws are not independent and so the resulting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent And Identically Distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as ''i.i.d.'', ''iid'', or ''IID''. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirk Kroese
Dirk Pieter Kroese (born 1963) is a Dutch-Australian mathematician and statistician, and Professor at the University of Queensland. He is known for several contributions to applied probability, kernel density estimation, Monte Carlo methods and Rare event sampling, rare event simulation. He is, with Reuven Rubinstein, a pioneer of the Cross-entropy method, Cross-Entropy (CE) method. Biography Born in Wapenveld (municipality of Heerde), Dirk Kroese received his MSc (Engineer's degree, Netherlands Ingenieur (ir) degree) in 1986 and his Ph.D. (cum laude) in 1990, both from the Department of Applied Mathematics at the University of Twente. His dissertation was entitled ''Stochastic Models in Reliability''. His PhD advisors were Joseph H. A. de Smit and Wilbert C. M. Kallenberg. Part of his PhD research was carried out at Princeton University under the guidance of Erhan Çinlar. He has held teaching and research positions at University of Texas at Austin (1986), Princeton Universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
