# least squares estimator derivationrobert smithson entropy

", Nie, Yu & Zhang, H.M. & Recker, W.W., 2005. Im confused with Least Squares Regression Derivation (Linear Algebra) Hot Network Questions A maximum entropy-least squares estimator for elastic origin-destination trip matrix estimation. Journal of Statistics. ", LeBlanc, Larry J. & Farhangian, Keyvan, 1982. The underlying assumption is that each cell of the subnetwork O–D flow table contains an elastic demand function rather than a fixed demand rate and the demand function can capture all traffic diversion effect under various network changes. ", Kumar, Anshuman Anjani & Kang, Jee Eun & Kwon, Changhyun & Nikolaev, Alexander, 2016. ", Chao Sun & Yulin Chang & Yuji Shi & Lin Cheng & Jie Ma, 2019. Finally, the high-resolution or aperture-compensated velocity gather is used to ex-trapolate near- and far-offset traces. Minimum mean-square estimation suppose x ∈ Rn and y ∈ Rm are random vectors (not necessarily Gaussian) we seek to estimate x given y thus we seek a function φ : Rm → Rn such that xˆ = φ(y) is near x one common measure of nearness: mean-square error, Ekφ(y)−xk2 minimum mean-square estimator (MMSE) φmmse minimizes this quantity And so on. Motivated by recent work of Joe (1989,Ann. Here, as usual, the entropy of a distribution p is deﬁned as H(p) = p[ln(1=p)] and the relative entropy, or Kullback-Leibler divergence, as D(p k q) = p[ln(p=q)]. If qk is not None, then compute the Kullback-Leibler divergence S = sum(pk * log(pk / qk), axis=axis).. Alternatively, the latter are also characterized by a postulate of composition consistency. When requesting a correction, please mention this item's handle: RePEc:eee:transb:v:45:y:2011:i:9:p:1465-1482. ... How to find the closed form formula for $\hat{\beta}$ while using ordinary least squares estimation? This illustrates under what circumstances entropy estimation is likely to be preferable to traditional econometric estimators based on the characteristic of the available data and … least-squares solution. The consequent estimator of entropy pro-posed by Correa (1995) is given by HCmn = 1 n Xn i=1 log 0 B B B @ i+P m j = i m (X (j ) X i)(j i) n i+Pm j = i m (X(j ) X (i))2 1 C C C A; Downloaded from jirss.irstat.ir at … Again, the di erential entropy provides the rule of thumb D(Q ) ˇ(1=12)22[H(Q ) H(f)]for small . This online calculator computes Shannon entropy for a given event probability table and for a given message. ", Chen, Anthony & Chootinan, Piya & Recker, Will, 2009. We study the effects of tail behaviour, distribution smoothness and dimensionality on convergence properties. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation. ", Yang, Hai & Iida, Yasunori & Sasaki, Tsuna, 1991. (4) In order to estimate we need to minimize . http://www.sciencedirect.com/science/article/pii/S0191261511000683, A maximum entropy-least squares estimator for elastic origin–destination trip matrix estimation, Transportation Research Part B: Methodological, The equilibrium-based origin-destination matrix estimation problem, Most likely origin-destination link uses from equilibrium assignment, Selection of a trip table which reproduces observed link flows, Inferences on trip matrices from observations on link volumes: A Bayesian statistical approach, Estimation of trip matrices from traffic counts and survey data: A generalized least squares estimator, A maximum likelihood model for estimating origin-destination matrices, A Relaxation Approach for Estimating Origin–Destination Trip Tables, On combining maximum entropy trip matrix estimation with user optimal assignment, An analysis of the reliability of an origin-destination trip matrix estimated from traffic counts, Variances and covariances for origin-destination flows when estimated by log-linear models, Estimation of an origin-destination matrix with random link choice proportions: A statistical approach, Inferring origin-destination trip matrices with a decoupled GLS path flow estimator, Estimation of origin-destination matrices from link traffic counts on congested networks, A linear programming approach for synthesizing origin-destination trip tables from link traffic volumes, Norm approximation method for handling traffic count inconsistencies in path flow estimator, The most likely trip matrix estimated from traffic counts, Subnetwork Origin-Destination Matrix Estimation Under Travel Demand Constraints, A decomposition approach to the static traffic assignment problem, Inferring origin-destination pairs and utility-based travel preferences of shared mobility system users in a multi-modal environment, User-equilibrium route flows and the condition of proportionality, An Excess-Demand Dynamic Traffic Assignment Approach for Inferring Origin-Destination Trip Matrices, Estimating the geographic distribution of originating air travel demand using a bi-level optimization model, Transportation Research Part E: Logistics and Transportation Review, Path Flow Estimator in an Entropy Model Using a Nonlinear L-Shaped Algorithm, http://www.elsevier.com/wps/find/journaldescription.cws_home/548/description#description, Xie, Chi & Kockelman, Kara M. & Waller, S. Travis, 2011. Histogram estimator. This can be related to cross-entropy in two steps: 1) convert into a likelihood, 2) con- H(Q ) + 1 2 log(12D(Q )) = H(f): (24) Here f is assumed to satisfy some smoothness and tail conditions. ", Yang, Hai & Iida, Yasunori & Sasaki, Tsuna, 1994. While the estimator is powered by the classic convex combination algorithm, computational difficulties emerge within the algorithm implementation until we incorporate partial optimality conditions and a column generation procedure into the algorithmic framework. You can help correct errors and omissions. ", Lo, H. P. & Zhang, N. & Lam, W. H. K., 1996. The simple way of evaluation of a probability distribution () of biological variable with the entropy normalized by its maximum value (= ), = − ∑ = ()demonstrates advantages over standard physiological indices in the estimation of functional status of cardiovascular, nervous and immune systems.. Another approach uses the idea that the differential entropy, 0. If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. condentropy, mutinformation, natstobits. Thus, the maximum entropy principle entropy; Examples In the case of linear Gaussian case, a very mature TLS parameter estimation algorithm has been developed. My context is mainly of a practical nature: When collecting entropy to seed a CSPRNG, I want the CSPRNG to be available as soon as possible, but not until at least n bits (say 128 bits) of entropy (unpredictable data) has been collected and fed to the CSPRNG. The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication". Statist. ", Jafari, Ehsan & Pandey, Venktesh & Boyles, Stephen D., 2017. eracy of a Bayesian estimator, section 8.2 gives a consistency result for a potentially more powerful regularization method than the one examined in depth here, and section 8.3 attempts to place our results in the context of estimation of more general functionals of the probability distribution (that is, not just entropy and mutual information). Mathematically this means that in order to estimate the we have to minimize which in matrix notation is nothing else than . The idea of the ordinary least squares estimator (OLS) consists in choosing in such a way that, the sum of squared residual (i.e. ) As a special case, a derivation of the method of maximum entropy from a small set of natural axioms is obtained. @NetranjitBorgohain that's a different method, but again it expects a different set of parameters entropy_joint(X, base=2, fill_value=-1, estimator='ML', Alphabet_X=None, keep_dims=False) see documentation for details – nickthefreak Mar 28 '19 at 15:21 Numerical results from applying the combined estimator to a couple of subnetwork examples show that an elastic O–D flow table, when used as input for subnetwork flow evaluations, reflects network flow changes significantly better than its fixed counterpart. but high entropy as described by Smithson. Copyright © 2020 Elsevier B.V. or its licensors or contributors. $\begingroup$ This was informative. In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent in the variable's possible outcomes. We propose a combined maximum entropy-least squares estimator, by which O–D flows are distributed over the subnetwork in terms of the maximum entropy principle, while demand function parameters are estimated for achieving the least sum of squared estimation errors. Start with least squares, min y k X k (y k x k)2 (1) where x kare the given data and y kare the corresponding points estimated by the model. Properties of Least Squares Estimators Each ^ iis an unbiased estimator of i: E[ ^ i] = i; V( ^ i) = c ii˙2, where c ii is the element in the ith row and ith column of (X0X) 1; Cov( ^ i; ^ i) = c ij˙2; The estimator S2 = SSE n (k+ 1) = Y0Y ^0X0Y n (k+ 1) is an unbiased estimator of ˙2. in the sample is as small as possible. In transportation subnetwork–supernetwork analysis, it is well known that the origin–destination (O–D) flow table of a subnetwork is not only determined by trip generation and distribution, but also a result from traffic routing and diversion, due to the existence of internal–external, external–internal and external–external flows. If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form . The entropy estimator using plug-in values under -estimates the true entropy value In fact: = + (n−1)/2T is a better estimator of the entropy (MM=Miller-Madow) No unbiased estimator of entropy … Robust least-squares estimation with a relative entropy constraint Abstract: Given a nominal statistical model, we consider the minimax estimation problem consisting of finding the best least-squares estimator for the least favorable statistical model within a … Computer Science, University of A Coruna, 15071 A Coruna, Spain Abstract.Minimum MSE plays an indispensable role in learning and Dept., University of Florida, Gainesville, FL 32611, USA 2 Dept. In a mathematical frame, the given information used in the principle of maximum entropy, is expressed as a set of constraints formed as expectations of functions g General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/548/description#description . 11 Master thesis of the National Institute of Applied Sciences of Lyon. We propose a combined maximum entropy-least squares (ME-LS) estimator, by which O-D flows are distributed over the subnetwork so as to maximize the trip distribution entropy, while demand function parameters are estimated for achieving the least sum of squared estimation errors. Please note that corrections may take a couple of weeks to filter through Shannon Entropy. (24) can be proved without any additional smoothness and tail conditions (Gy or , Linder, van der Meulen [28]). tity, and derive least squares as a special case. When q0 is uniform this is the same as maximizing the entropy. Recursive Least Squares for an Entropy Regularized MSE Cost Function Deniz Erdogmus1, Yadunandana N. Rao1, Jose C. Principe1 Oscar Fontenla-Romero2, Amparo Alonso-Betanzos2 1 Electrical Eng. Nonparametric entropy estimation : An overview. We use cookies to help provide and enhance our service and tailor content and ads. Estimator: autocorrelation, maximum entropy (Burg), least-squares [...] normal equations, least-squares covariance and modified covariance, SVD principal component AR. Apply the entropy formula considering only sunny entropy. In particular, we argue that root-n consistency of entropy estimation requires appropriate assumptions about each of these three features. person_outlineTimurschedule 2013-06-04 15:04:43. Copyright © 2011 Published by Elsevier Ltd. Procedia - Social and Behavioral Sciences, https://doi.org/10.1016/j.sbspro.2011.04.514. As the access to this document is restricted, you may want to search for a different version of it. This result indicates the variable nature of subnetwork O–D flows. GME Estimation in Linear Regression Model GME Command with User Supplied Parameter Support Matrix Sign and Cross-Parameter Restrictions Conclusion Generalized Maximum Entropy Estimation GME estimator developed by Golan, Judge, and Miller (1996) Campbell and Hill (2006) impose inequality restrictions on GME estimator in a linear regression model Numerical results from applying the combined estimator to a couple of subnetwork examples show that an elastic O-D flow table, when used as input for subnetwork flow evaluations, reflects network flow changes significantly better than its fixed counterpart. A Maximum Entropy-least Squares Estimator for Elastic Origin-Destination Trip Matrix Estimation In transportation subnetwork-supernetwork analysis, it is well known that the origin-destination (O-D) flow table of a subnetwork is not only determined by trip generation and distribution, but also by traffic routing and diversion, due to the existence of internal-external, external-internal and external-external flows. The plugin estimator uses empirical estimates of the frequencies ^p j= 1 n P n i=1 1[X i= j] to obtain an estimate of the entropy as follows: H^ n= Xd j=1 p^ jlog 2 ( ^p j) LP Estimator The LP Estimator works by transforming the samples fX ign i=1 into a ngerprint, which is the vector f= (f 1;f 2;:::) for which f In information theory, entropy is a measure of the uncertainty in a random variable. & Willumsen, Luis G., 1980. The entropy of a substance is influenced by structure of the particles (atoms or molecules) that comprise the substance. Theres 3 sunny instances divided into 2 classes being 2 sunny related with Tennis and 1 related to Cinema. the various RePEc services. ". I estimate that you could get to the top with as few as thirty-five to fort y- ... which are proportionnal to the square root of text length. This allows to link your profile to this item. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. A maximum entropy-least squares estimator for elastic origin-destination trip matrix estimation. ", Bar-Gera, Hillel & Boyce, David & Nie, Yu (Marco), 2012. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Improving entropy estimation and the inference of genetic regulatory networks. ", Van Zuylen, Henk J. The underlying assumption is that each cell of the subnetwork O-D flow table contains an elastic demand function rather than a fixed demand rate and the demand function can capture all traffic diversion effect under various network changes. +kbuk2 SSE +SSR; (2) where SST, SSE and SSR mean the total sum of squares, the explained sum of squares, and the residual sum of squares (or the sum of squared residuals), respectively. Public profiles for Economics researchers, Various rankings of research in Economics & related fields, Curated articles & papers on various economics topics, Upload your paper to be listed on RePEc and IDEAS, RePEc working paper series dedicated to the job market, Pretend you are at the helm of an economics department, Data, research, apps & more from the St. Louis Fed, Initiative for open bibliographies in Economics, Have your institution's/publisher's output listed on RePEc. For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Haili He). If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. Math.,41, 683–697), we introduce estimators of entropy and describe their properties. While the estimator is powered by the classic convex combination algorithm, computational difficulties emerge within the algorithm implementation until we incorporate partial optimality conditions and a column generation procedure into the algorithmic framework. We propose a combined maximum entropy-least squares (ME-LS) estimator, by which O-D flows are distributed over the subnetwork so as to maximize the trip distribution entropy, while demand function parameters are estimated for achieving the least sum of squared estimation errors. choose the distribution that minimizes entropy relative to the default estimate q0. Hausser J. it, the resulted maximum entropy distribution “is the least biased estimate possible on the given information; i.e., it is maximally noncommittal with regard to missing information”. How was the formula for Ordinary Least Squares Linear Regression arrived at? This result indicates the variable nature of subnetwork O-D flows. Note I am not only looking for the proof, but also the derivation. The entropy estimator is then given by ... via least square method. See general information about how to correct material in RePEc. ", Yang, Hai & Sasaki, Tsuna & Iida, Yasunori & Asakura, Yasuo, 1992. ", Maryam Abareshi & Mehdi Zaferanieh & Bagher Keramati, 2017. scipy.stats.entropy¶ scipy.stats.entropy (pk, qk = None, base = None, axis = 0) [source] ¶ Calculate the entropy of a distribution for given probability values. ", Sherali, Hanif D. & Sivanandan, R. & Hobeika, Antoine G., 1994. This note is for people who are familiar with least squares but less so with entropy. By continuing you agree to the use of cookies. We propose a combined maximum entropy-least squares estimator, by which O–D flows are distributed over the subnetwork in terms of the maximum entropy principle, while demand function parameters are estimated for achieving the least sum of squared estimation errors. All material on this site has been provided by the respective publishers and authors. If only probabilities pk are given, the entropy is calculated as S =-sum(pk * log(pk), axis=axis).. This paper discusses an elastic O–D flow table estimation problem for subnetwork analysis. Inst. It also allows you to accept potential citations to this item that we are uncertain about. In transportation subnetwork-supernetwork analysis, it is well known that the origin-destination (O-D) flow table of a subnetwork is not only determined by trip generation and distribution, but also by traffic routing and diversion, due to the existence of internal-external, external-internal and external-external flows. (2006). Aliases. See Also. The total least square (TLS) estimation problem of random systems is widely found in many fields of engineering and science, such as signal processing, automatic control, system theory and so on. This paper discusses an elastic O-D flow table estimation problem for subnetwork analysis. INTRODUCTION dow sometimes cause a poor velocity resolution when using Conventional velocity analysis is performed by measuring energy along hyperbolic paths for a set of tentative veloci-ties. distributions of ordinary least squares and entropy estimators when data are limited. So the entropy formula for sunny gets something like this: -2/3 log2(2/3) - 1/3 log2(1/3) = 0.918. As corollaries, axiomatic characterizations of the methods of least squares and minimum discrimination information are arrived at.

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