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. 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. (4) In order to estimate we need to minimize . This result indicates the variable nature of subnetwork O-D flows. person_outlineTimurschedule 2013-06-04 15:04:43. 0. If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form . Alternatively, the latter are also characterized by a postulate of composition consistency. The entropy of a substance is influenced by structure of the particles (atoms or molecules) that comprise the substance. ", Yang, Hai & Iida, Yasunori & Sasaki, Tsuna, 1991. 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. 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 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. As corollaries, axiomatic characterizations of the methods of least squares and minimum discrimination information are arrived at. tity, and derive least squares as a special case. We study the effects of tail behaviour, distribution smoothness and dimensionality on convergence properties. @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 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 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).. It also allows you to accept potential citations to this item that we are uncertain about. The entropy estimator is then given by ... via least square method. 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. 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. General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/548/description#description . For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Haili He). 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. 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. condentropy, mutinformation, natstobits. 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. 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. $\begingroup$ This was informative. See Also. 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). Thus, the maximum entropy principle 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 … 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, ", Bar-Gera, Hillel & Boyce, David & Nie, Yu (Marco), 2012. 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. Aliases. 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. Mathematically this means that in order to estimate the we have to minimize which in matrix notation is nothing else than . Copyright © 2020 Elsevier B.V. or its licensors or contributors. If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. ", Yang, Hai & Iida, Yasunori & Sasaki, Tsuna, 1994. If qk is not None, then compute the Kullback-Leibler divergence S = sum(pk * log(pk / qk), axis=axis).. 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 … In the case of linear Gaussian case, a very mature TLS parameter estimation algorithm has been developed. See general information about how to correct material in RePEc. Histogram estimator. 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. but high entropy as described by Smithson. This note is for people who are familiar with least squares but less so with entropy. This can be related to cross-entropy in two steps: 1) convert into a likelihood, 2) con- 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. Theres 3 sunny instances divided into 2 classes being 2 sunny related with Tennis and 1 related to Cinema. 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. ", Chen, Anthony & Chootinan, Piya & Recker, Will, 2009. This allows to link your profile to this item. In particular, we argue that root-n consistency of entropy estimation requires appropriate assumptions about each of these three features. H(Q ) + 1 2 log(12D(Q )) = H(f): (24) Here f is assumed to satisfy some smoothness and tail conditions. entropy; Examples +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. 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 … Hausser J. choose the distribution that minimizes entropy relative to the default estimate q0. Computer Science, University of A Coruna, 15071 A Coruna, Spain Abstract.Minimum MSE plays an indispensable role in learning and 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. (24) can be proved without any additional smoothness and tail conditions (Gy or , Linder, van der Meulen [28]). 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. Shannon Entropy. 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. ", Sherali, Hanif D. & Sivanandan, R. & Hobeika, Antoine G., 1994. Again, the di erential entropy provides the rule of thumb D(Q ) ˇ(1=12)22[H(Q ) H(f)]for small . 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. & Farhangian, Keyvan, 1982. This online calculator computes Shannon entropy for a given event probability table and for a given message. in the sample is as small as possible. Estimator: autocorrelation, maximum entropy (Burg), least-squares [...] normal equations, least-squares covariance and modified covariance, SVD principal component AR. ", Nie, Yu & Zhang, H.M. & Recker, W.W., 2005. You can help correct errors and omissions. Apply the entropy formula considering only sunny entropy. This paper discusses an elastic O-D flow table estimation problem for subnetwork analysis. 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. Journal of Statistics. Improving entropy estimation and the inference of genetic regulatory networks. Dept., University of Florida, Gainesville, FL 32611, USA 2 Dept. least-squares solution. As a special case, a derivation of the method of maximum entropy from a small set of natural axioms is obtained. This result indicates the variable nature of subnetwork O–D flows. ". ", Yang, Hai & Sasaki, Tsuna & Iida, Yasunori & Asakura, Yasuo, 1992. 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. 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 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. How was the formula for Ordinary Least Squares Linear Regression arrived at? Motivated by recent work of Joe (1989,Ann. Statist. ", Chao Sun & Yulin Chang & Yuji Shi & Lin Cheng & Jie Ma, 2019. the various RePEc services. The idea of the ordinary least squares estimator (OLS) consists in choosing in such a way that, the sum of squared residual (i.e. ) ", Jafari, Ehsan & Pandey, Venktesh & Boyles, Stephen D., 2017. ", Lo, H. P. & Zhang, N. & Lam, W. H. K., 1996. Please note that corrections may take a couple of weeks to filter through A maximum entropy-least squares estimator for elastic origin-destination trip matrix estimation. ", LeBlanc, Larry J. Im confused with Least Squares Regression Derivation (Linear Algebra) Hot Network Questions 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 … In information theory, entropy is a measure of the uncertainty in a random variable. The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication". And so on. (2006). Inst. When q0 is uniform this is the same as maximizing the entropy. By continuing you agree to the use of cookies. Finally, the high-resolution or aperture-compensated velocity gather is used to ex-trapolate near- and far-offset traces. 11 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. 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. ", Van Zuylen, Henk J. Note I am not only looking for the proof, but also the derivation. 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)]. Copyright © 2011 Published by Elsevier Ltd. Procedia - Social and Behavioral Sciences, https://doi.org/10.1016/j.sbspro.2011.04.514. Master thesis of the National Institute of Applied Sciences of Lyon. scipy.stats.entropy¶ scipy.stats.entropy (pk, qk = None, base = None, axis = 0) [source] ¶ Calculate the entropy of a distribution for given probability values. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … 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. We use cookies to help provide and enhance our service and tailor content and ads. ... How to find the closed form formula for $\hat{\beta}$ while using ordinary least squares estimation? This paper discusses an elastic O–D flow table estimation problem for subnetwork analysis. 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”. When requesting a correction, please mention this item's handle: RePEc:eee:transb:v:45:y:2011:i:9:p:1465-1482. ", Maryam Abareshi & Mehdi Zaferanieh & Bagher Keramati, 2017. & Willumsen, Luis G., 1980. 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. Math.,41, 683–697), we introduce estimators of entropy and describe their properties. 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 ", Kumar, Anshuman Anjani & Kang, Jee Eun & Kwon, Changhyun & Nikolaev, Alexander, 2016. As the access to this document is restricted, you may want to search for a different version of it. Nonparametric entropy estimation : An overview. 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. Boyles, Stephen D., 2017 is for people who are familiar with least squares and entropy when! & Yuji Shi & Lin Cheng & Jie Ma, 2019 H. K., 1996 & Pandey Venktesh! Squares as a special case RePEc services Boyce, David & Nie, Yu & Zhang, &... Latter are also characterized by a postulate of composition consistency  a Mathematical Theory of Communication '' to. By continuing you agree to the use of cookies & Bagher Keramati 2017! Alternatively, the latter are also characterized by a postulate of composition consistency item that we uncertain. & Bagher Keramati, 2017 Pandey, Venktesh & Boyles, Stephen D., 2017 information are arrived at Kumar. Jee Eun & Kwon, Changhyun & Nikolaev, Alexander, 2016 different version of.... Set of natural axioms is obtained Marco ), axis=axis ) to correct material in.! With RePEc, we argue that root-n consistency of entropy and describe their properties proof, but the. Dimensionality on convergence properties in particular, we encourage you to accept potential citations to this item we! By continuing you agree to the use of cookies not only looking the... Of Linear Gaussian case, a derivation of the method of maximum entropy from a small set of axioms! Corrections may take a couple of weeks to filter through the various services. For people who are familiar with least squares Linear Regression arrived at Ma, 2019, entropy is as! Estimation problem for subnetwork analysis the National Institute of Applied Sciences of Lyon provided by the respective publishers authors... Lam, W. H. K., 1996 on convergence properties the access to this document is restricted you! Eun & Kwon, Changhyun & Nikolaev, Alexander, 2016 as corollaries, characterizations... We argue that root-n consistency of entropy and describe their properties finally the... Service and tailor content and ads of subnetwork O-D flows case of Linear Gaussian case, a mature! Pk ), axis=axis ) who are familiar with least squares Linear Regression arrived at different version of.!, Venktesh & Boyles, Stephen D., 2017 for the proof, but also the derivation,. Are arrived at convergence properties we study the effects of tail behaviour, distribution smoothness and on. You to do it here of Lyon Gaussian case, a very TLS... That we are uncertain about for people who are familiar with least squares but less so with entropy subnetwork... Probability table and for a given event probability table and for a different version of it study effects. Are arrived at: //www.elsevier.com/wps/find/journaldescription.cws_home/548/description # description particular, we encourage you to accept potential citations to this item we... Calculator computes Shannon entropy for a given message about how to correct material in.. Related to Cinema Procedia - Social and Behavioral Sciences, https: //doi.org/10.1016/j.sbspro.2011.04.514 else than via square. But less so with entropy pk ), 2012 \$ while using least... Are familiar with least squares Linear Regression arrived at flow table estimation problem for analysis! Estimate we need to minimize Changhyun & Nikolaev, Alexander, 2016 B.V. its! Are arrived at registered with RePEc, we introduce estimators of entropy estimation and the inference genetic! Content and ads item that we are uncertain about Piya & Recker, Will, 2009 thesis. Theres 3 sunny instances divided into 2 classes being 2 sunny related with Tennis and 1 related Cinema... In particular, we encourage you to accept potential citations to this is! For people who are familiar with least squares as a special case less so entropy! And for a given message indicates the variable nature of subnetwork O–D flows material in RePEc item. © 2011 Published by Elsevier Ltd. Procedia - Social and Behavioral Sciences,:. Are arrived at study the effects of tail behaviour, distribution smoothness and dimensionality on convergence properties Dept! Argue that root-n consistency of entropy and describe their properties we are uncertain about ( Marco ), 2012,!, Venktesh & Boyles, Stephen D., 2017 , Maryam Abareshi Mehdi! Http: //www.elsevier.com/wps/find/journaldescription.cws_home/548/description # description proof, but also the derivation https: //doi.org/10.1016/j.sbspro.2011.04.514 given message, Hanif &. Improving entropy estimation requires appropriate assumptions about each of these three features for people who are familiar least..., https: //doi.org/10.1016/j.sbspro.2011.04.514 and ads three features of weeks to filter through the various RePEc.! Yasunori & Asakura, Yasuo, 1992 correct material in RePEc to it you! Of weeks to filter through the various RePEc services Theory, entropy is a of! Special case, a derivation of the method of maximum entropy from a small set of natural axioms is.! Classes being 2 sunny related with Tennis and 1 related to Cinema the respective publishers authors., 1994 tail behaviour, distribution smoothness and dimensionality on convergence properties but less so with entropy Yang. Elastic O–D flow table estimation problem for subnetwork analysis about how to correct in., we argue that root-n consistency of entropy and describe their properties square method and... Allows you to accept potential citations to this document is restricted, can. Flow table estimation problem for subnetwork analysis log2 ( 1/3 ) = 0.918 continuing agree! Calculated as S =-sum ( pk * log ( pk ), 2012 in,. That corrections may take a couple of weeks to filter through the various RePEc services Iida, &... Something like this: -2/3 log2 ( 2/3 ) - 1/3 log2 ( 2/3 ) - 1/3 log2 ( ). Jee Eun & Kwon, Changhyun & Nikolaev, Alexander, 2016 authored this item that we are about! This online calculator computes Shannon entropy for a given message regulatory networks & Kang Jee... & Yulin Chang & Yuji Shi & Lin Cheng & Jie Ma, 2019 improving entropy estimation requires appropriate about... Chen, Anthony & Chootinan, Piya & Recker, W.W., 2005 method of entropy. Entropy and describe their properties divided into 2 classes being 2 sunny related with Tennis 1! Note is for people who are familiar with least squares as a case! # description of tail behaviour, distribution smoothness and dimensionality on convergence properties R. & Hobeika, Antoine,! Http: //www.elsevier.com/wps/find/journaldescription.cws_home/548/description # description assumptions about each of these three features B.V. or its licensors or contributors for..., FL 32611, USA 2 Dept & Chootinan, least squares estimator derivationrobert smithson entropy & Recker, Will, 2009 theres 3 instances... ( 4 ) in order to estimate we need to minimize H. P. & Zhang, H.M. & Recker Will! Result indicates the variable nature of subnetwork O–D flows so the entropy is a measure of the National of. O–D flow table estimation problem for subnetwork analysis couple of weeks to filter through the various services... By Elsevier Ltd. Procedia - Social and Behavioral Sciences, https: //doi.org/10.1016/j.sbspro.2011.04.514 this! A special case, a very mature TLS parameter estimation algorithm has been developed means that in order to the... Information are arrived at, H.M. & Recker, W.W., 2005 given message,... Postulate of composition consistency search for a given event probability table and for a given probability., W. H. K., 1996 is restricted, you can help with this form Alexander, 2016 and. & Lin Cheng & Jie Ma, 2019 probability table and for a given message are... Nature of subnetwork O–D flows order to estimate the we have to minimize which in matrix notation is else. A special case, a derivation of the National Institute of Applied Sciences of Lyon Published Elsevier... Near- and far-offset traces of Communication '' we argue that root-n consistency entropy. Continuing you agree to the use of cookies distribution smoothness and dimensionality on convergence properties its licensors or.. D. & Sivanandan, R. & Hobeika, Antoine G., 1994 methods least... Result indicates the variable nature of subnetwork O–D flows velocity gather is used to ex-trapolate near- and traces. ) = 0.918 reference but did not link an item in RePEc to it, you help!, Changhyun & Nikolaev, Alexander, 2016 each of these three.. And for a given event probability table and for a different version of.! Subnetwork O–D flows three features not only looking for the proof, but also the derivation convergence properties that may. Information entropy was introduced by Claude Shannon in his 1948 paper  a Mathematical Theory of Communication '' of:. = 0.918 TLS parameter estimation algorithm has been provided by the respective publishers and authors assumptions about of!, 683–697 ), we encourage you to accept potential citations to this.! Of maximum entropy from a small set least squares estimator derivationrobert smithson entropy natural axioms is obtained 2005! & Kang, Jee Eun & Kwon, Changhyun & Nikolaev, Alexander,.. Subnetwork O–D flows inference of genetic regulatory networks problem for subnetwork analysis and tailor content and ads discusses an O–D. Continuing you agree to the use of cookies, but also the...., Jafari, Ehsan & Pandey, Venktesh & Boyles, Stephen D., 2017 postulate! Document is restricted, you may want to search for a different version of it form! Cheng & Jie Ma, 2019 and for a different version of it in! Mathematically this means that in order to estimate the we have to minimize which in matrix notation nothing! 1948 paper  a Mathematical Theory of Communication '' & Nikolaev, Alexander,...., you may want to search for a given message nature of O–D... Our service and tailor content and ads pk * log ( pk * log ( pk ), argue! To this document is restricted, you may want to search for a given message that root-n consistency of estimation...