Information Entropy and Correlations in Prime Numbers
Redundancy and synergy arising from correlations in large ensembles
a rX iv:c ond-ma t/12119v1[c ond-m at.stat-m ech]7Dec2Redundancy and synergy arising from correlations in large ensembles Michele Bezzi,Mathew E.Diamond and Alessandro Treves SISSA -Programme in Neuroscience via Beirut 4,34014Trieste,Italy February 1,2008Abstract Multielectrode arrays allow recording of the activity of many single neurons,from which correlations can be calculated.The functional roles of correlations can be revealed by the measures of the information con-veyed by neuronal activity;a simple formula has been shown to discrimi-nate the information transmitted by individual spikes from the positive or negative contributions due to correlations (Panzeri et al,Proc.Roy.Soc.B.,266:1001–1012(1999)).The formula quantifies the corrections to the single-unit instantaneous information rate which result from correlations in spike emission between pairs of neurons.Positive corrections imply synergy,while negative corrections indicate redundancy.Here,this anal-ysis,previously applied to recordings from small ensembles,is developed further by considering a model of a large ensemble,in which correlations among the signal and noise components of neuronal firing are small in absolute value and entirely random in origin.Even such small random correlations are shown to lead to large possible synergy or redundancy,whenever the time window for extracting information from neuronal fir-ing extends to the order of the mean interspike interval.In addition,a sample of recordings from rat barrel cortex illustrates the mean time win-dow at which such ‘corrections’dominate when correlations are,as often in the real brain,neither random nor small.The presence of this kind of correlations for a large ensemble of cells restricts further the time of validity of the expansion,unless what is decodable by the receiver is also taken into account.11Do correlations convey more information than do rates alone?Our intuition often brings us to regard neurons as independent actors in the business of information processing.We are then reminded of the potential for intricate mutual dependence in their activity,stemming from common inputs and from interconnections,and arefinally brought to consider correlations as sources of much richer,although somewhat hidden,information about what a neural ensemble is really doing.Now that the recording of multiple single units is common practice in many laboratories,correlations in their activity can be measured and their role in information processing can be elucidated case by case.Is the information conveyed by the activity of an ensemble of neurons determined solely by the number of spikesfired by each cell as could be quantified also with non-simultaneous recordings[1];or do correlations in the emission of action potentials also play a significant role?Experimental evidence on the role of correlations in neural coding of sensory events,or of internal states,has been largely confined to ensembles of very few cells.Their contribution has been said to be positive,i.e.the information contained in the ensemble response is greater than the sum of contributions of single cells(synergy)[2,3,4,5,6],or negative(redundancy)[7,8,9].Thus,2specific examples can be found of correlations that limit thefidelity of signal transmission,and others that carry additional information.Another view is that usually correlations do not make much of a difference in either direction[10,11, 12]and that their presence can be regarded as a kind of random noise.In this paper,we show that even when correlations are of a completely random nature they may contribute very substantially,and to some extent predictably, to information transmission.To discuss this point we mustfirst quantify the amount of information con-tained in the neural rmation theory[13]provides one framework for describing mathematically the process of information transmission,and it has been applied successfully to the analysis of neuronal recordings[12,14,15,16]. Consider a stimulus taken from afinite discrete set S with S elements,each stimulus s occurring with probability P(s).The probability of response r(the ensemble activity,imagined as thefiring rate vector)is P(r),and the joint probability distribution is P(s,r).The mutual information between the set of stimuli S and the response isI(t)= s∈S r P(s,r)log2P(s,r)In the t→0limit,the mutual information can be broken down into afiring rates and correlations components,as shown by Panzeri et al.[17]and summarized in the next section.The correlation-dependent part can be further expanded by considering“small”correlation coefficients(see Section3).In this(additional) limit approximation the effects of correlations can be analyzed and it will be seen that even if they are random they give large contributions to the total information.The number of second-order(pairwise)correlation terms in the information expansion in fact grows as C2,where C is the number of cells, while contributions that depend only on individual cellfiring rates of course grow linearly with C.As a result,as shown by Panzeri et al.[24],the time window to which the expansion is applicable shrinks as the number of cells increases,and conversely the overall effect of correlation grows.We complement this derivation by analysing(see Section4)the response of cells in the rat somatosensory barrel cortex during the response to deflections of the vibrassae.Conclusions about the general applicability of correlation measures to information transmission are drawn in the last section.42The short time expansionIn the limit t→0,following Ref.[17],the information carried by the population response can be expanded in a Taylor seriest2I(t)=t I t+r i(s)t(1+γij(s))(3) (where theγij(s)coefficient quantifies correlations)the expansion(2)becomes an expansion in the total number of spikes emitted by an assembly of neurons (see Ref.[17]for details).Briefly,the procedure is the following:the expression for P(n)(3)is inserted in the Shannon formula for information,Eq.(1),whose logarithm is then expanded in a power series.All terms,with the same power of t,are grouped and compared to Eq.(2),to extractfirst and second order derivatives.Thefirst time derivative(i.e.the information rate)depends only on the5firing rates averaged over trials with the same stimulus,denoted asr i(s)log2r i(s)r j(s)r i(s)sion,given that the same cell has alreadyfired in the same time window;i.e.γii(s)=r i(s)r2i(s)−1The relationship with alternative cross-correlation coefficients,like the Pearson correlation,is discussed in Ref.[17].‘Signal’correlations measure the tendency of pairs of cells to respond more (or less)to the same stimuli in the stimulus set.As in the previous case we introduce the signal cross-correlation coefficientνij,νij=<r j(s)>s r i(s)>s<ln2Ci=1C j=i r j(s) s νij+(1+νij)ln(1 ln2Ci=1C j=i r j(s)γij(s) s ln(1ln2Ci=1C j=i r j(s)(1+γij(s))ln r j(s′) s′(1+γij(s))r i(s′)The sum I t t+1r i(s),(rate only contribution)and itsfirst term is always greater than or equal to zero,while I(1)tt is always less than or equal to zero.In the presence of correlations,i.e.non zeroγij andνij,more information may be available when observing simultaneously the responses of many cells, than when observing them separately:synergy.For two cells,it can happen due to positive correlations in the variability,if the mean rates to different stimuli are anticorrelated,or vice-versa.If the signs of signal and noise correlations are the same,the result is always redundancy.Quantitatively,the impact of correlations is minimal when the mean responses are only weakly correlated across the stimulus set.The time range of validity of the expansion(2)is limited by the requirement that second order terms be small with respect tofirst order ones,and successive orders be negligible.Since at order n there are C n terms with C cells,the applicability of the short time limit contracts for larger populations.3Large number of cellsLet us investigate the role of correlations in the transmission of information by large populations of cells.For a few cells,all cases of synergy or redundancy are possible if the correlations are properly engineered,in simulations,or if,8in experiments,the appropriate special case is recorded.The outcome of the information analysis simply reflects the peculiarity of each case.With large populations,one may hope to have a better grasp of generic,or typical,cases, more indicative of conditions prevailing at the level of,say,a given cortical module,such as a column.Consider a‘null’hypothesis model of a large population:purely random correlations;i.e.correlations that were not designed to play any special role in the system being analyzed.In this null hypothesis,signal correlationsνij can be thought of as arising from a random walk with S steps(the number of stimuli).Such a random√walk of positive and negative steps typically spans a range of sizeνij andδγij(s),i.e.assuming|νij|<<1and|δγij(s)|<<1.Consider,first,the expansion of I(1)tt,that does not depend onγij(s).Ex-panding in powers ofνij and neglecting terms of order3or higher,we easily get:I(1)tt=−1r i(s)sln2Ci=1C j=i (1+γij) r j(s) sν2ij+(6)+(1+γij) r j(s)s νij+ r j(s)δγij(s)sνij (7)The third contribution,I(3)tt is more complicated,as an expansion inδγij(s) is required as well.Expanding the logarithm in these small parameters up to second order we get:I(3)tt=11+γij r j(s)δγij(s)2 s− r j(s)δγij(s)2sr i(s)r i(s)r i(s)SSi=1r j(s)r i(s)ln2Ci=1C j=i(νij+1) r j(s) sthat is a non-negative quantity,i.e.a synergetic contribution to information.In case of random “noise”correlations,with zero weighted average over the set of stimuli,i.e. δγij (s ) {i,j },s =0,this equation can be re-written,I (3)tt =11+γij r j (s ) s δγij (s )2{i,j },s ≡C (C +1)C (C +1)/2C i =1C j =i νij +1r i (s ) sln 2C i =1C j =i (1+γij r i (s ) s ln 2 ν2 C(10)where we have introduced (in a similar way as for δγ;these two definitions coincide for νij and γij →0):ν2 C =12) r j (s ) s ν2ijThis contribution (Eq.10)to information is always negative (redundancy ).Thus the leading contributions of the new Taylor expansion are of two types,both coming as C (C +1)/2terms proportional to r j (s ) s .The first one,Eq.(10),is a redundancy term proportional to ν2 ;the second one,Eq.(9),is11a synergy term roughly proportional to δγ2 .These leading contributions to I tt can be compared tofirst order contribu-tions to the original Taylor expansion in t(i.e.,to the C terms in I t)in different time ranges.For times t≈ISI/C,that is tr ≈1,first order terms are of order C,while second order ones are of order C2 ν2 C(with a minus sign,signifying redundancy) and C2 δγ2 C(with a plus sign,signifying synergy)respectively.If ν2 C and δγ2 C are not sufficiently small to counteract the additional C factor,these “random”redundancy and synergy contributions will be substantial.Moreover, over the same time ranges leading contributions to I ttt and to the next terms in the Taylor expansion in time may be expected to be substantial.The expansion already will be of limited use by the time most cells havefired just a single spike.If this bleak conclusion comes from a model with small and random correla-tions,what is the time range of applicability of the expansion when several real12cells are recorded simultaneously?4Measuring correlations in rat barrel cortex We have analyzed many sets of data recorded from rat cortex.Part of the primary somatosensory cortex of the rat(the barrel cortex)is organized in a grid of columns,with each column anatomically and functionally associated with one homologous whisker(vibrissa)on the controlateral side:the column’s neurons respond maximally,on average,to the deflection of this“principal”whisker.In our experiments,in a urethane-anesthetized rat,one whisker was stimulated at 1Hz,and each deflection lasted for100ms.The latency(time delay between stimulus onset and the evoked response)in this fast sensory system is usually around5−10ms.We present here the complete analysis of a single typical dataset.The physiological methods are described in Ref.[23].For each stimulus site there were50trials and in our analysis we have con-sidered up to6stimulus sites,(i.e.different whiskers)with12cells recorded simultaneously.In Fig.1we report thefiring distributions of9of the12cells for each of the6stimuli.One can immediately note that several cells are most strongly activated by a single whisker,while responding more weakly or not at all to the others.Other cells have less sharply tuned receptivefields.A mixture of sharply tuned and more broadly tuned receptivefields is characteristic of a13given population of barrel cortex neurons.We have computed the distribution ofνij andδγij(s)for different time windows.In Fig.2we have plotted the distribution of allνij.In thefirstfigure (Fig.2,top,left)we have considered2stimuli,taken from the set of6stimuli, and averaged over all the possible pairs.In the followingfigure(Fig.2,top, right),where we take all the6stimuli,the distribution is broader(σ=1.2vs.σ=0.5in the previous case),and the maximum value ofνis2.8(vs.1.0).This larger spread ofνvalues can be explained by the fact that most cells have a greater response for one stimulus,and weaker for the other.This can be seen by considering a limit case:two cells i and jfire just to a single stimulus s′,i.e.r j(s′)=0and r i(s)=0for s=s′.From Eq.(4),we haveνij=1r i(s)1r i(s)1r j(s)−1=1r i(s′)1r i(s′)1r j(s′)−1=S−1As the total number of stimuli S increases,νvalues of the order of S appear, and broaden the distribution.The distribution does not change qualitatively when the time window lengthens,(Fig.2,bottom,left)at least from30ms to 40ms,except for a somewhat narrower width with the longer time-window.For very short time windows(≤20ms)we observe instead a peak atν=0and ν=−1due to the prevalence of cases of zero spikes:when the mean rates of at least one of the two cells are zero to all stimuliν=0,and when the stimuli,to14which each gives a non-zero response,are mismatched,thenν=−1.In the last panel(Fig.2,bottom,right)we have taken a more limited sample(20trials), which in this case does not significantly change the moments of the distribution.The distribution ofδγis illustrated in Fig.3.This distribution has0average by definition;we can observe that the spread becomes larger when increasing the number of stimuli from2(Fig.3,top left)to6(Fig.3,top right).This derives from having rates r i(s)that differ from zero and only for one or a few stimuli. In this case increasing the number of stimuli thefluctuations in the distribution ofγij(and hence ofδγij(s))become larger,broadening the distribution.For longer time windows(Fig.3,bottom,left),there are more spikes and a better sampling of the rates,so the spread of the distribution decreases(σ=4.5for 40ms vs.σ=5.7for30ms).The effect offinite sampling(20trials)illustrated in the last plot(Fig.3,bottom,left),is now a substantial reduction in width.In Fig.4we have plotted,for the same experiment,the values of the in-formation and of single terms of the second order expansion discussed above. The full curve represents the information I(t)up to the second order,i.e. I(t)=t I t+t2I(1)tt t2,i.e.the rate2only contribution,as it depends only on averagefiring rates,I(2)tt t2,i.e.the contribution of correlations to information215even if they are stimulus independent.The last second-order term,1relations in a large ensembles,that for times t≃ISI(inter-spike interval),the expansion would already begin to break down.The overall contribution offirst order terms is in fact of order C,while second order ones are of order−C2 ν2 C (redundancy)and C2 δγ2 C(synergy).These‘random’redundancy and syn-ergy contributions will normally be substantial,unless a specific mechanism minimizes the values of ν2 C and δγ2 C well below order1/C.Further,data from the somatosensory cortex of the rat indicate that the assumption of‘small’correlations may be far too optimistic in the real brain situation;the expansion may then break down even sooner,although one should consider that the rat somatosensory cortex is a“fast”system,with short-latency responses and high instantaneousfiring rates.Our data show(see Fig.4)that the range of validity of the second-order expansion decreases approximatively as1/C.The length of the time interval over which the expansion is valid is roughly10−15ms for9or12cells,in agreement with Panzeri and Schultz[24].They have found,analyzing a large amount of cells recorded from the somatosensory barrel cortex of an adult rat, that for single cells the expansion works well up to100ms.In its range of validity this expansion constitutes an efficient tool for measuring information even in the presence of limited sampling of data,when a direct approach using the full17Shannon formula,Eq.(1),turns out to be impossible[17].When its limits are respected,the expansion can be used to address fundamental questions,such as extra information in timing and the relevance of correlations contribution.It is important that if second order terms are comparable in size tofirst order terms,all successive orders in the expansion are also likely to be relevant.The breakdown of the expansion is then not merely a failure of the mathematical formalism,but an indication that this particular attempt to quantify,in absolute terms,the information conveyed by a large ensemble is intrinsically ill-posed in that time range.There might be other expansions,or other ways to measure mutual information e.g.the reconstruction method[25],that lead to better results.A pessimistic conclusion is then that the expansion should be applied only to very brief times,of the order of t≈ISI/C.In this range the information rates of different cells add up independently,even if cells are highly correlated, but the total information conveyed,no matter how large the ensemble,remains of order1bit.A more optimistic interpretation stresses the importance of considering infor-mation decoding along with information encoding.In this vein,not all pairwise correlations are taken into account on the same footing,and similarly not all18correlations to higher orders;rather,appropriate models of neuronal decoding prescribe which variables can affect the activity of neurons downstream,and it is only a limited number of such variables that are included as corrections into the evaluation of the information conveyed by the ensembles.This embodies the assumption that real neurons may not be influenced by the information(and the synergy and redundancy)encoded in a multitude of variables that cannot be decoded.In an ideal world,it would be preferable to characterize the quan-tity of information present in population activity and to assume that the target neurons can conserve all such information.In real life,such an assumption does not seem to be justified,and considerable further work is now needed to explore different models of neuronal decoding,and their implementation in estimating information,in order to make full use of the potential offered by the availability of large scale multiple single-unit recording techniques. AcknowledgmentsThis work was supported in part by HFSP grant RG0110/1998-B,and is a follow-up to the analyses of the short time limit,in which Stefano Panzeri has played a leading role.We are grateful to him,and to Simon Schultz,also for the information extraction software,and to Misha Lebedev and Rasmus Petersen for their help with the experimental data.The physiology experiments were19supported by NIH grant NS32647to M.E.D.20References[1]E.T.Rolls,A.Treves and M.J.Tov´e e(1997)The representational capacityof the distributed encoding of information provided by populations of neu-rons in primate temporal visual cortex.Experimental Brain Research114: 149-162.[2]E.Vaadia,I.Haalman,M.Abeles,H.Bergaman,Y.Prut,H.Slovin andA.Aertsen,(1995)Dynamics of neural interactions in monkey cortex inrelation to behavioural events,Nature373,515-518.[3]R.C.deCharms,M.M.Merzenich,(1996),Primary cortical representationof sounds by the coordination of action potential,Nature381,610-613. 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[25]F.Rieke,D.Warland,R.R.de Ruyter van Steveninck and W.Bialek,(1999)Spikes,MIT Press.24Figure1:Firing rate distributions(probability of observing a given number of spikes emitted in the time window of40ms),for9cells and6different stimuli (that is,6whisker sites).The bars,from left to right,represent the probability to have(0not shown)1,2,3,or more than3(black bar)spikes during a single stimulus presentation.Maximum y-axis set to0.5.25Figure2:The distribution P(ν),considering theνij for each cell pair i and j. Computed after30ms with2stimuli,50trials per stimulus(averaged over all possible pairs from among6stimuli,top left),with all the6stimuli(top right), with6stimuli and after40ms(bottom left)and with6stimuli and after40ms but with only20trials per stimulus(bottom right).26Figure3:The distribution P(δγ),considering allδγij for each cell pair i and j and for each stimulus puted after30ms with2stimuli,50trials per stimulus(averaged over all possible pairs of6stimuli,top left),with all the6 stimuli(top right),with6stimuli and after40ms(bottom left)and with with 6stimuli and after40ms but with only20trials per stimulus(bottom right).27Figure4:The short time limit expansion breaks down sooner when the larger population is considered.Cells in rat somatosensory barrel cortex for2stimulus ponents of the transmitted information(see text for details)with3 (top,left),6(top,right),9(bottom,left)and12cells(bottom,right).The initial slope(i.e.I t)is roughly proportional to the number of cells.The effects of the second order terms,quadratic in t,are visible over the brief times between the linear regime and the break-down of the rmation is estimated taking into accountfinite sampling effects[22].Time window starts5ms after the stimulus onset.28。
图像的分割和配准中英文翻译
外文文献资料翻译：李睿钦指导老师：刘文军Medical image registration with partial dataSenthil Periaswamy，Hany FaridThe goal of image registration is to find a transformation that aligns one image to another. Medical image registration has emerged from this broad area of research as a particularly active field. This activity is due in part to the many clinical applications including diagnosis, longitudinal studies, and surgical planning, and to the need for registration across different imaging modalities (e.g., MRI, CT, PET, X-ray, etc.). Medical image registration, however, still presents many challenges. Several notable difficulties are (1) the transformation between images can vary widely and be highly non-rigid in nature; (2) images acquired from different modalities may differ significantly in overall appearance and resolution; (3) there may not be a one-to-one correspondence between the images (missing/partial data); and (4) each imaging modality introduces its own unique challenges, making it difficult to develop a single generic registration algorithm.In estimating the transformation that aligns two images we must choose: (1) to estimate the transformation between a small number of extracted features, or between the complete unprocessed intensity images; (2) a model that describes the geometric transformation; (3) whether to and how to explicitly model intensity changes; (4) an error metric that incorporates the previous three choices; and (5) a minimization technique for minimizing the error metric, yielding the desired transformation.Feature-based approaches extract a (typically small) number of corresponding landmarks or features between the pair of images to be registered. The overall transformation is estimated from these features. Common features include corresponding points, edges, contours or surfaces. These features may be specified manually or extracted automatically. Fiducial markers may also be used as features;these markers are usually selected to be visible in different modalities. Feature-based approaches have the advantage of greatly reducing computational complexity. Depending on the feature extraction process, these approaches may also be more robust to intensity variations that arise during, for example, cross modality registration. Also, features may be chosen to help reduce sensor noise. These approaches can be, however, highly sensitive to the accuracy of the feature extraction. Intensity-based approaches, on the other hand, estimate the transformation between the entire intensity images. Such an approach is typically more computationally demanding, but avoids the difficulties of a feature extraction stage.Independent of the choice of a feature- or intensity-based technique, a model describing the geometric transform is required. A common and straightforward choice is a model that embodies a single global transformation. The problem of estimating a global translation and rotation parameter has been studied in detail, and a closed form solution was proposed by Schonemann. Other closed-form solutions include methods based on singular value decomposition (SVD), eigenvalue-eigenvector decomposition and unit quaternions. One idea for a global transformation model is to use polynomials. For example, a zeroth-order polynomial limits the transformation to simple translations, a first-order polynomial allows for an affine transformation, and, of course, higher-order polynomials can be employed yielding progressively more flexible transformations. For example, the registration package Automated Image Registration (AIR) can employ (as an option) a fifth-order polynomial consisting of 168 parameters (for 3-D registration). The global approach has the advantage that the model consists of a relatively small number of parameters to be estimated, and the global nature of the model ensures a consistent transformation across the entire image. The disadvantage of this approach is that estimation of higher-order polynomials can lead to an unstable transformation, especially near the image boundaries. In addition, a relatively small and local perturbation can cause disproportionate and unpredictable changes in the overall transformation. An alternative to these global approaches are techniques that model the global transformation as a piecewise collection of local transformations. For example, the transformation between each local region may bemodeled with a low-order polynomial, and global consistency is enforced via some form of a smoothness constraint. The advantage of such an approach is that it is capable of modeling highly nonlinear transformations without the numerical instability of high-order global models. The disadvantage is one of computational inefficiency due to the significantly larger number of model parameters that need to be estimated, and the need to guarantee global consistency. Low-order polynomials are, of course, only one of many possible local models that may be employed. Other local models include B-splines, thin-plate splines, and a multitude of related techniques. The package Statistical Parametric Mapping (SPM) uses the low-frequency discrete cosine basis functions, where a bending-energy function is used to ensure global consistency. Physics-based techniques that compute a local geometric transform include those based on the Navier–Stokes equilibrium equations for linear elastici and those based on viscous fluid approaches.Under certain conditions a purely geometric transformation is sufficient to model the transformation between a pair of images. Under many real-world conditions, however, the images undergo changes in both geometry and intensity (e.g., brightness and contrast). Many registration techniques attempt to remove these intensity differences with a pre-processing stage, such as histogram matching or homomorphic filtering. The issues involved with modeling intensity differences are similar to those involved in choosing a geometric model. Because the simultaneous estimation of geometric and intensity changes can be difficult, few techniques build explicit models of intensity differences. A few notable exceptions include AIR, in which global intensity differences are modeled with a single multiplicative contrast term, and SPM in which local intensity differences are modeled with a basis function approach.Having decided upon a transformation model, the task of estimating the model parameters begins. As a first step, an error function in the model parameters must be chosen. This error function should embody some notion of what is meant for a pair of images to be registered. Perhaps the most common choice is a mean square error (MSE), defined as the mean of the square of the differences (in either feature distance or intensity) between the pair of images. This metric is easy to compute and oftenaffords simple minimization techniques. A variation of this metric is the unnormalized correlation coefficient applicable to intensity-based techniques. This error metric is defined as the sum of the point-wise products of the image intensities, and can be efficiently computed using Fourier techniques. A disadvantage of these error metrics is that images that would qualitatively be considered to be in good registration may still have large errors due to, for example, intensity variations, or slight misalignments. Another error metric (included in AIR) is the ratio of image uniformity (RIU) defined as the normalized standard deviation of the ratio of image intensities. Such a metric is invariant to overall intensity scale differences, but typically leads to nonlinear minimization schemes. Mutual information, entropy and the Pearson product moment cross correlation are just a few examples of other possible error functions. Such error metrics are often adopted to deal with the lack of an explicit model of intensity transformations .In the final step of registration, the chosen error function is minimized yielding the desired model parameters. In the most straightforward case, least-squares estimation is used when the error function is linear in the unknown model parameters. This closed-form solution is attractive as it avoids the pitfalls of iterative minimization schemes such as gradient-descent or simulated annealing. Such nonlinear minimization schemes are, however, necessary due to an often nonlinear error function. A reasonable compromise between these approaches is to begin with a linear error function, solve using least-squares, and use this solution as a starting point for a nonlinear minimization.译文：部分信息的医学图像配准Senthil Periaswamy，Hany Farid图像配准的目的是找到一种能把一副图像对准另外一副图像的变换算法。
英语,汉语信源信源熵研究
英语信源，汉语信源的信源熵的研究吴斌伟2902102020【摘要】信息是个很抽象的概念。
人们常常说信息很多，或者信息较少，但却很难说清楚信息到底有多少。
比如一本五十万字的中文书到底有多少信息量。
直到1948年，香农提出了“信息熵”的概念，才解决了对信息的量化度量问题。
因此，信源的信息熵是衡量一个信源所含信息多少的度量。
信息的基本作用就是消除人们对事物了解的不确定性。
一个信源所发出的编码的不确定性越大，那么这个信源所含的信息量越大。
若一个信源发出某个码字的概率为一，那么该信源可传达的信息量为零。
美国信息论创始人香农发现任何信息都存在冗余，冗余大小与信息中每个符号（数字、字母或单词）的出现概率或者说不确定性有关。
香农借鉴了热力学的概念，把信息中排除了冗余后的平均信息量称为“信息熵”。
信源熵是信息论中用来衡量信源信息量有序化程度的一个概念，定义为信源各个离散消息的自信息量的数学期望（即概率加权的统计平均值）。
根据定义，信源熵值与信源有序化程度成反比；有序度越高，信源熵值越低，反之亦成立。
不同的语言，如汉语，英语，德语，法语等，所含的信息熵各不相同。
具体数据如下：按字母计算：英文的平均信息熵是4.03 比特，法文的平均信息熵是3.98，西班牙文的平均信息熵是4.01，德文的平均信息熵是4.10，俄文的平均信息熵是4.8，中文的平均信息熵是9.65比特由上述数据可知，法语的信息熵最小，而中文的信息熵最大。
因此有人说汉语这种语言不如其他语言，汉语是落后的。
显然这样的答案是否定的。
平均信息熵并不是语言文字效率的基本公式，而是在通讯中编码的码长的效率！提出这公式，申农是用以研究信息编码的。
说得通俗一点，就是要（在可能有噪音的情况下）把已方（信息源）的信息进行标准化编码（比如，0－1化），然后传送出去，对方接收，解码，恢复成原来的信息。
研究的重点，是多长的一组码为合理——如果太短，无法正确还原，如果太长，就有冗余。
AI术语
人工智能专业重要词汇表1、A开头的词汇：Artificial General Intelligence/AGI通用人工智能Artificial Intelligence/AI人工智能Association analysis关联分析Attention mechanism注意力机制Attribute conditional independence assumption属性条件独立性假设Attribute space属性空间Attribute value属性值Autoencoder自编码器Automatic speech recognition自动语音识别Automatic summarization自动摘要Average gradient平均梯度Average-Pooling平均池化Accumulated error backpropagation累积误差逆传播Activation Function激活函数Adaptive Resonance Theory/ART自适应谐振理论Addictive model加性学习Adversarial Networks对抗网络Affine Layer仿射层Affinity matrix亲和矩阵Agent代理/ 智能体Algorithm算法Alpha-beta pruningα-β剪枝Anomaly detection异常检测Approximation近似Area Under ROC Curve／AUC R oc 曲线下面积2、B开头的词汇Backpropagation Through Time通过时间的反向传播Backpropagation/BP反向传播Base learner基学习器Base learning algorithm基学习算法Batch Normalization/BN批量归一化Bayes decision rule贝叶斯判定准则Bayes Model Averaging／BMA贝叶斯模型平均Bayes optimal classifier贝叶斯最优分类器Bayesian decision theory贝叶斯决策论Bayesian network贝叶斯网络Between-class scatter matrix类间散度矩阵Bias偏置/ 偏差Bias-variance decomposition偏差-方差分解Bias-Variance Dilemma偏差–方差困境Bi-directional Long-Short Term Memory/Bi-LSTM双向长短期记忆Binary classification二分类Binomial test二项检验Bi-partition二分法Boltzmann machine玻尔兹曼机Bootstrap sampling自助采样法／可重复采样／有放回采样Bootstrapping自助法Break-Event Point／BEP平衡点3、C开头的词汇Calibration校准Cascade-Correlation级联相关Categorical attribute离散属性Class-conditional probability类条件概率Classification and regression tree/CART分类与回归树Classifier分类器Class-imbalance类别不平衡Closed -form闭式Cluster簇/类/集群Cluster analysis聚类分析Clustering聚类Clustering ensemble聚类集成Co-adapting共适应Coding matrix编码矩阵COLT国际学习理论会议Committee-based learning基于委员会的学习Competitive learning竞争型学习Component learner组件学习器Comprehensibility可解释性Computation Cost计算成本Computational Linguistics计算语言学Computer vision计算机视觉Concept drift概念漂移Concept Learning System /CLS概念学习系统Conditional entropy条件熵Conditional mutual information条件互信息Conditional Probability Table／CPT条件概率表Conditional random field/CRF条件随机场Conditional risk条件风险Confidence置信度Confusion matrix混淆矩阵Connection weight连接权Connectionism连结主义Consistency一致性／相合性Contingency table列联表Continuous attribute连续属性Convergence收敛Conversational agent会话智能体Convex quadratic programming凸二次规划Convexity凸性Convolutional neural network/CNN卷积神经网络Co-occurrence同现Correlation coefficient相关系数Cosine similarity余弦相似度Cost curve成本曲线Cost Function成本函数Cost matrix成本矩阵Cost-sensitive成本敏感Cross entropy交叉熵Cross validation交叉验证Crowdsourcing众包Curse of dimensionality维数灾难Cut point截断点Cutting plane algorithm割平面法4、D开头的词汇Data mining数据挖掘Data set数据集Decision Boundary决策边界Decision stump决策树桩Decision tree决策树／判定树Deduction演绎Deep Belief Network深度信念网络Deep Convolutional Generative Adversarial Network/DCGAN深度卷积生成对抗网络Deep learning深度学习Deep neural network/DNN深度神经网络Deep Q-Learning深度Q 学习Deep Q-Network深度Q 网络Density estimation密度估计Density-based clustering密度聚类Differentiable neural computer可微分神经计算机Dimensionality reduction algorithm降维算法Directed edge有向边Disagreement measure不合度量Discriminative model判别模型Discriminator判别器Distance measure距离度量Distance metric learning距离度量学习Distribution分布Divergence散度Diversity measure多样性度量／差异性度量Domain adaption领域自适应Downsampling下采样D-separation （Directed separation）有向分离Dual problem对偶问题Dummy node哑结点Dynamic Fusion动态融合Dynamic programming动态规划5、E开头的词汇Eigenvalue decomposition特征值分解Embedding嵌入Emotional analysis情绪分析Empirical conditional entropy经验条件熵Empirical entropy经验熵Empirical error经验误差Empirical risk经验风险End-to-End端到端Energy-based model基于能量的模型Ensemble learning集成学习Ensemble pruning集成修剪Error Correcting Output Codes／ECOC纠错输出码Error rate错误率Error-ambiguity decomposition误差-分歧分解Euclidean distance欧氏距离Evolutionary computation演化计算Expectation-Maximization期望最大化Expected loss期望损失Exploding Gradient Problem梯度爆炸问题Exponential loss function指数损失函数Extreme Learning Machine/ELM超限学习机6、F开头的词汇Factorization因子分解False negative假负类False positive假正类False Positive Rate/FPR假正例率Feature engineering特征工程Feature selection特征选择Feature vector特征向量Featured Learning特征学习Feedforward Neural Networks/FNN前馈神经网络Fine-tuning微调Flipping output翻转法Fluctuation震荡Forward stagewise algorithm前向分步算法Frequentist频率主义学派Full-rank matrix满秩矩阵Functional neuron功能神经元7、G开头的词汇Gain ratio增益率Game theory博弈论Gaussian kernel function高斯核函数Gaussian Mixture Model高斯混合模型General Problem Solving通用问题求解Generalization泛化Generalization error泛化误差Generalization error bound泛化误差上界Generalized Lagrange function广义拉格朗日函数Generalized linear model广义线性模型Generalized Rayleigh quotient广义瑞利商Generative Adversarial Networks/GAN生成对抗网络Generative Model生成模型Generator生成器Genetic Algorithm/GA遗传算法Gibbs sampling吉布斯采样Gini index基尼指数Global minimum全局最小Global Optimization全局优化Gradient boosting梯度提升Gradient Descent梯度下降Graph theory图论Ground-truth真相／真实8、H开头的词汇Hard margin硬间隔Hard voting硬投票Harmonic mean调和平均Hesse matrix海塞矩阵Hidden dynamic model隐动态模型Hidden layer隐藏层Hidden Markov Model/HMM隐马尔可夫模型Hierarchical clustering层次聚类Hilbert space希尔伯特空间Hinge loss function合页损失函数Hold-out留出法Homogeneous同质Hybrid computing混合计算Hyperparameter超参数Hypothesis假设Hypothesis test假设验证9、I开头的词汇ICML国际机器学习会议Improved iterative scaling/IIS改进的迭代尺度法Incremental learning增量学习Independent and identically distributed/i.i.d.独立同分布Independent Component Analysis/ICA独立成分分析Indicator function指示函数Individual learner个体学习器Induction归纳Inductive bias归纳偏好Inductive learning归纳学习Inductive Logic Programming／ILP归纳逻辑程序设计Information entropy信息熵Information gain信息增益Input layer输入层Insensitive loss不敏感损失Inter-cluster similarity簇间相似度International Conference for Machine Learning/ICML国际机器学习大会Intra-cluster similarity簇内相似度Intrinsic value固有值Isometric Mapping/Isomap等度量映射Isotonic regression等分回归Iterative Dichotomiser迭代二分器10、K开头的词汇Kernel method核方法Kernel trick核技巧Kernelized Linear Discriminant Analysis／KLDA核线性判别分析K-fold cross validation k 折交叉验证／k 倍交叉验证K-Means Clustering K –均值聚类K-Nearest Neighbours Algorithm/KNN K近邻算法Knowledge base知识库Knowledge Representation知识表征11、L开头的词汇Label space标记空间Lagrange duality拉格朗日对偶性Lagrange multiplier拉格朗日乘子Laplace smoothing拉普拉斯平滑Laplacian correction拉普拉斯修正Latent Dirichlet Allocation隐狄利克雷分布Latent semantic analysis潜在语义分析Latent variable隐变量Lazy learning懒惰学习Learner学习器Learning by analogy类比学习Learning rate学习率Learning Vector Quantization/LVQ学习向量量化Least squares regression tree最小二乘回归树Leave-One-Out/LOO留一法linear chain conditional random field线性链条件随机场Linear Discriminant Analysis／LDA线性判别分析Linear model线性模型Linear Regression线性回归Link function联系函数Local Markov property局部马尔可夫性Local minimum局部最小Log likelihood对数似然Log odds／logit对数几率Logistic Regression Logistic 回归Log-likelihood对数似然Log-linear regression对数线性回归Long-Short Term Memory/LSTM长短期记忆Loss function损失函数12、M开头的词汇Machine translation/MT机器翻译Macron-P宏查准率Macron-R宏查全率Majority voting绝对多数投票法Manifold assumption流形假设Manifold learning流形学习Margin theory间隔理论Marginal distribution边际分布Marginal independence边际独立性Marginalization边际化Markov Chain Monte Carlo/MCMC马尔可夫链蒙特卡罗方法Markov Random Field马尔可夫随机场Maximal clique最大团Maximum Likelihood Estimation/MLE极大似然估计／极大似然法Maximum margin最大间隔Maximum weighted spanning tree最大带权生成树Max-Pooling最大池化Mean squared error均方误差Meta-learner元学习器Metric learning度量学习Micro-P微查准率Micro-R微查全率Minimal Description Length/MDL最小描述长度Minimax game极小极大博弈Misclassification cost误分类成本Mixture of experts混合专家Momentum动量Moral graph道德图／端正图Multi-class classification多分类Multi-document summarization多文档摘要Multi-layer feedforward neural networks多层前馈神经网络Multilayer Perceptron/MLP多层感知器Multimodal learning多模态学习Multiple Dimensional Scaling多维缩放Multiple linear regression多元线性回归Multi-response Linear Regression ／MLR多响应线性回归Mutual information互信息13、N开头的词汇Naive bayes朴素贝叶斯Naive Bayes Classifier朴素贝叶斯分类器Named entity recognition命名实体识别Nash equilibrium纳什均衡Natural language generation/NLG自然语言生成Natural language processing自然语言处理Negative class负类Negative correlation负相关法Negative Log Likelihood负对数似然Neighbourhood Component Analysis/NCA近邻成分分析Neural Machine Translation神经机器翻译Neural Turing Machine神经图灵机Newton method牛顿法NIPS国际神经信息处理系统会议No Free Lunch Theorem／NFL没有免费的午餐定理Noise-contrastive estimation噪音对比估计Nominal attribute列名属性Non-convex optimization非凸优化Nonlinear model非线性模型Non-metric distance非度量距离Non-negative matrix factorization非负矩阵分解Non-ordinal attribute无序属性Non-Saturating Game非饱和博弈Norm范数Normalization归一化Nuclear norm核范数Numerical attribute数值属性14、O开头的词汇Objective function目标函数Oblique decision tree斜决策树Occam’s razor奥卡姆剃刀Odds几率Off-Policy离策略One shot learning一次性学习One-Dependent Estimator／ODE独依赖估计On-Policy在策略Ordinal attribute有序属性Out-of-bag estimate包外估计Output layer输出层Output smearing输出调制法Overfitting过拟合／过配Oversampling过采样15、P开头的词汇Paired t-test成对t 检验Pairwise成对型Pairwise Markov property成对马尔可夫性Parameter参数Parameter estimation参数估计Parameter tuning调参Parse tree解析树Particle Swarm Optimization/PSO粒子群优化算法Part-of-speech tagging词性标注Perceptron感知机Performance measure性能度量Plug and Play Generative Network即插即用生成网络Plurality voting相对多数投票法Polarity detection极性检测Polynomial kernel function多项式核函数Pooling池化Positive class正类Positive definite matrix正定矩阵Post-hoc test后续检验Post-pruning后剪枝potential function势函数Precision查准率／准确率Prepruning预剪枝Principal component analysis/PCA主成分分析Principle of multiple explanations多释原则Prior先验Probability Graphical Model概率图模型Proximal Gradient Descent/PGD近端梯度下降Pruning剪枝Pseudo-label伪标记16、Q开头的词汇Quantized Neural Network量子化神经网络Quantum computer量子计算机Quantum Computing量子计算Quasi Newton method拟牛顿法17、R开头的词汇Radial Basis Function／RBF径向基函数Random Forest Algorithm随机森林算法Random walk随机漫步Recall查全率／召回率Receiver Operating Characteristic/ROC受试者工作特征Rectified Linear Unit/ReLU线性修正单元Recurrent Neural Network循环神经网络Recursive neural network递归神经网络Reference model参考模型Regression回归Regularization正则化Reinforcement learning/RL强化学习Representation learning表征学习Representer theorem表示定理reproducing kernel Hilbert space/RKHS再生核希尔伯特空间Re-sampling重采样法Rescaling再缩放Residual Mapping残差映射Residual Network残差网络Restricted Boltzmann Machine/RBM受限玻尔兹曼机Restricted Isometry Property/RIP限定等距性Re-weighting重赋权法Robustness稳健性/鲁棒性Root node根结点Rule Engine规则引擎Rule learning规则学习18、S开头的词汇Saddle point鞍点Sample space样本空间Sampling采样Score function评分函数Self-Driving自动驾驶Self-Organizing Map／SOM自组织映射Semi-naive Bayes classifiers半朴素贝叶斯分类器Semi-Supervised Learning半监督学习semi-Supervised Support Vector Machine半监督支持向量机Sentiment analysis情感分析Separating hyperplane分离超平面Sigmoid function Sigmoid 函数Similarity measure相似度度量Simulated annealing模拟退火Simultaneous localization and mapping同步定位与地图构建Singular Value Decomposition奇异值分解Slack variables松弛变量Smoothing平滑Soft margin软间隔Soft margin maximization软间隔最大化Soft voting软投票Sparse representation稀疏表征Sparsity稀疏性Specialization特化Spectral Clustering谱聚类Speech Recognition语音识别Splitting variable切分变量Squashing function挤压函数Stability-plasticity dilemma可塑性-稳定性困境Statistical learning统计学习Status feature function状态特征函Stochastic gradient descent随机梯度下降Stratified sampling分层采样Structural risk结构风险Structural risk minimization/SRM结构风险最小化Subspace子空间Supervised learning监督学习／有导师学习support vector expansion支持向量展式Support Vector Machine/SVM支持向量机Surrogat loss替代损失Surrogate function替代函数Symbolic learning符号学习Symbolism符号主义Synset同义词集19、T开头的词汇T-Distribution Stochastic Neighbour Embedding/t-SNE T–分布随机近邻嵌入Tensor张量Tensor Processing Units/TPU张量处理单元The least square method最小二乘法Threshold阈值Threshold logic unit阈值逻辑单元Threshold-moving阈值移动Time Step时间步骤Tokenization标记化Training error训练误差Training instance训练示例／训练例Transductive learning直推学习Transfer learning迁移学习Treebank树库Tria-by-error试错法True negative真负类True positive真正类True Positive Rate/TPR真正例率Turing Machine图灵机Twice-learning二次学习20、U开头的词汇Underfitting欠拟合／欠配Undersampling欠采样Understandability可理解性Unequal cost非均等代价Unit-step function单位阶跃函数Univariate decision tree单变量决策树Unsupervised learning无监督学习／无导师学习Unsupervised layer-wise training无监督逐层训练Upsampling上采样21、V开头的词汇Vanishing Gradient Problem梯度消失问题Variational inference变分推断VC Theory VC维理论Version space版本空间Viterbi algorithm维特比算法Von Neumann architecture冯·诺伊曼架构22、W开头的词汇Wasserstein GAN/WGAN Wasserstein生成对抗网络Weak learner弱学习器Weight权重Weight sharing权共享Weighted voting加权投票法Within-class scatter matrix类内散度矩阵Word embedding词嵌入Word sense disambiguation词义消歧23、Z开头的词汇Zero-data learning零数据学习Zero-shot learning零次学习。
统计学专业英语词汇
Averageconfidenceintervallength,平均置信区间长度
Averagegrowthrate,平均增长率
B
Barchart,条形图
Bargraph,条形图
Baseperiod,基期
Bayestheorem,贝叶斯定理
Bell-shapedcurve,钟形曲线
Boxplots,箱线图/箱尾图
Breakdownbound,崩溃界/崩溃点
C
Canonicalcorrelation,典型相关
Caption,纵标目
Case-controlstudy,病例对照研究
Categoricalvariable,分类变量
Catenary,悬链线
Cauchydistribution,柯西分布
Chernofffaces,切尔诺夫脸谱图
Chi-squaretest,卡方检验/χ2检验
Choleskeydecomposition,乔洛斯基分解
Circlechart,圆图
Classinterval,组距
Classmid-value,组中值
Classupperlimit,组上限
Classifiedvariable,分类变量
Consistencycheck,一致性检验
Consistentasymptoticallynormalestimate,相合渐近正态估计
Consistentestimate,相合估计
Constrainednonlinearregression,受约束非线性回归
Constraint,约束
Contaminateddistribution,污染分布
Chance,机遇
L02 Entropy, Relative Entropy, and Mutual Information(1)信息论课件第二章
DUT
应用信息论基础
金明录 教授
Definitions
DUT
应用信息论基础
金明录 教授
Definitions
DUT
应用信息论基础
金明录 教授
Example
Consider a blood testing equipment which detects the presence of a
disease 99% of the cases, if presented with infected samples. Thus 1% of the infected escape undetected. On the other hand, the test gives false results for 2% of healthy patients too. Suppose, on average, 1 out of 1000 people are infected. If the machine gives a positive test, what is the chance of the blood sample being actually infected.
random variable (``self-information'') of a single
Conditional entropy H(X|Y) is the entropy of one random variable
conditional upon knowledge of another.
DUT
应用信息论基础
金明录 教授
Definitions
A discrete random variable X takes on values x from the discrete
各种距离（欧氏距离、曼哈顿距离、切比雪夫距离、马氏距离等）
各种距离（欧⽒距离、曼哈顿距离、切⽐雪夫距离、马⽒距离等）在做分类时常常需要估算不同样本之间的相似性度量(SimilarityMeasurement)，这时通常采⽤的⽅法就是计算样本间的“距离”(Distance)。
采⽤什么样的⽅法计算距离是很讲究，甚⾄关系到分类的正确与否。
本⽂的⽬的就是对常⽤的相似性度量作⼀个总结。
本⽂⽬录：1.欧⽒距离2.曼哈顿距离3. 切⽐雪夫距离4. 闵可夫斯基距离5.标准化欧⽒距离6.马⽒距离7.夹⾓余弦8.汉明距离9.杰卡德距离& 杰卡德相似系数10.相关系数& 相关距离11.信息熵1. 欧⽒距离(EuclideanDistance)欧⽒距离是最易于理解的⼀种距离计算⽅法，源⾃欧⽒空间中两点间的距离公式。
(1)⼆维平⾯上两点a(x1,y1)与b(x2,y2)间的欧⽒距离：(2)三维空间两点a(x1,y1,z1)与b(x2,y2,z2)间的欧⽒距离：(3)两个n维向量a(x11,x12,…,x1n)与 b(x21,x22,…,x2n)间的欧⽒距离： 也可以⽤表⽰成向量运算的形式：(4)Matlab计算欧⽒距离Matlab计算距离主要使⽤pdist函数。
若X是⼀个M×N的矩阵，则pdist(X)将X矩阵M⾏的每⼀⾏作为⼀个N维向量，然后计算这M个向量两两间的距离。
例⼦：计算向量(0,0)、(1,0)、(0,2)两两间的欧式距离X= [0 0 ; 1 0 ; 0 2]D= pdist(X,'euclidean')结果：D=1.00002.0000 2.23612. 曼哈顿距离(ManhattanDistance)从名字就可以猜出这种距离的计算⽅法了。
想象你在曼哈顿要从⼀个⼗字路⼝开车到另外⼀个⼗字路⼝，驾驶距离是两点间的直线距离吗？显然不是，除⾮你能穿越⼤楼。
实际驾驶距离就是这个“曼哈顿距离”。
⽽这也是曼哈顿距离名称的来源，曼哈顿距离也称为城市街区距离(CityBlock distance)。
Information theory
Notice that we can think of the entropy as average length of message needed to transmit the outcome of the random variable.
6
An example
Uniform distribution: pi=1/8. Uniform distribution has higher entropy.
expectation.
10
Mutual information (互信息)
We showed: H(X,Y) = H(X) + H(Y|X) = H(Y) + H(X|Y), which implies: H(X) - H(X|Y) = H(Y) - H(Y|X)
This difference is called the mutual information between X and Y and is denoted I(X;Y):
–
On the other hand: P{X = x,Y = y} = P{X = x} P{Y = y} = pq Therefore, the function I should satisfy the identity I(pq) = I(p) + I(q)
4
–
Information: Mathematical Form
2
Data Transmission Intuition
Intuition: a ‘Good’ compression algorithm (e.g., Huffman coding) encodes more frequent events with shorter codes;
数据仓库与数据挖掘教程（第2版）课后习题答案第七章
数据仓库与数据挖掘教程（第2版）课后习题答案第七章第七章作业1.信息论的基本原理是什么？一个传递信息的系统是由发送端(信源)和接收端(信宿)以及连接两者的通道(信道)组成的。
信息论把通信过程看做是在随机干扰的环境中传递信息的过程。
在这个通信模型中，信息源和干扰(噪声)都被理解为某种随机过程或随机序列。
在进行实际的通信之前，收信者(信宿)不可能确切了解信源究竟会发出什么样的具体信息，也不可能判断信源会处于什么样的状态。
这种情形就称为信宿对于信源状态具有不确定性，而且这种不确定性是存在于通信之前的，因而又叫做先验不确定性。
在通信后，信宿收到了信源发来的信息，这种先验不确定性才会被消除或者被减少。
如果干扰很小，不会对传递的信息产生任何可察觉的影响，信源发出的信息能够被信宿全部收到，在这种情况下，信宿的先验不确定性就会被完全消除。
但是，在一般情况下，干扰总会对信源发出的信息造成某种破坏，使信宿收到的信息不完全。
因此，先验不确定性不能全部被消除, 只能部分地消除。
换句话说，通信结束之后，信宿仍具有一定程度的不确定性。
这就是后验不确定性。
2.学习信道模型是什么？学习信道模型是信息模型应用于机器学习和数据挖掘的具体化。
学习信道模型的信源是实体的类别，采用简单“是”、“非”两类，令实体类别U 的值域为{u1，u2}，U 取u1表示取“是”类中任一例子，取u2表示取“非”类中任一例子。
信宿是实体的特征（属性）取值。
实体中某个特征属性V ，他的值域为{v1，v2……vq}。
3.为什么机器学习和数据挖掘的分类问题可以利用信息论原理？信息论原理是数据挖掘的理论基础之一。
一般用于分类问题，即从大量数据中获取分类知识。
具体来说，就是在已知各实例的类别的数据中，找出确定类别的关键的条件属性。
求关键属性的方法，即先计算各条件属性的信息量，再从中选出信息量最大的属性，信息量的计算是利用信息论原理中的公式。
4自信息：单个消息ui 发出前的不确定性（随机性）称为自信息。
信息熵与图像熵的计算
实验一信息熵与图像熵计算一、实验目的1．复习MATLAB 的基本命令，熟悉MATLAB 下的基本函数。
2．复习信息熵基本定义, 能够自学图像熵定义和基本概念。
二、实验仪器、设备1．计算机－系统最低配置 256M 内存、P4 CPU。
2．Matlab 仿真软件－ 7.0 / 7.1 / 2006a 等版本Matlab 软件。
三、实验内容与原理（1）内容：1．能够写出MATLAB 源代码，求信源的信息熵。
2．根据图像熵基本知识，综合设计出MATLAB 程序，求出给定图像的图像熵。
（2）原理1． MATLAB 中数据类型、矩阵运算、图像文件输入与输出知识复习。
2．利用信息论中信息熵概念，求出任意一个离散信源的熵（平均自信息量）。
自信息是一个随机变量,它是指某一信源发出某一消息所含有的信息量。
所发出的消息不同，它们所含有的信息量也就不同。
任何一个消息的自信息量都代表不了信源所包含的平均自信息量。
不能作为整个信源的信息测度，因此定义自信息量的数学期望为信源的平均自信息量：信息熵的意义：信源的信息熵H是从整个信源的统计特性来考虑的。
它是从平均意义上来表征信源的总体特性的。
对于某特定的信源，其信息熵只有一个。
不同的信源因统计特性不同，其熵也不同。
3．学习图像熵基本概念，能够求出图像一维熵和二维熵。
图像熵是一种特征的统计形式，它反映了图像中平均信息量的多少。
图像的一维熵表示图像中灰度分布的聚集特征所包含的信息量，令Pi 表示图像中灰度值为i的像素所占的比例，则定义灰度图像的一元灰度熵为：255log i iip p ==∑H图像的一维熵可以表示图像灰度分布的聚集特征，却不能反映图像灰度分布的空间特征，为了表征这种空间特征，可以在一维熵的基础上引入能够反映灰度分布空间特征的特征量来组成图像的二维熵。
选择图像的邻域灰度均值作为灰度分布的空间特征量，与图像的像素灰度组成特征二元组，记为( i, j )，其中i 表示像素的灰度值(0 <= i <= 255)，j 表示邻域灰度(0 <= j <= 255),2(,)/ijP f i j N =上式能反应某像素位置上的灰度值与其周围像素灰度分布的综合特征，其中f(i, j) 为特征二元组(i, j)出现的频数，N 为图像的尺度，定义离散的图像二维熵为：255logij ijip p ==∑H构造的图像二维熵可以在图像所包含信息量的前提下，突出反映图像中像素位置的灰度信息和像素邻域内灰度分布的综合特征.四、实验步骤1．求解信息熵过程：1) 输入一个离散信源，并检查该信源是否是完备集。
基于邻域互信息最大相关性最小冗余度的特征选择
基于邻域互信息最大相关性最小冗余度的特征选择林培榕【摘要】Feature selection is an important data preprocessing technique, where mutual information has been widely studied in information measure. However, mutual information cannot directly calculate relevancy among numeric features. In this paper, we first introduce neighborhood entropy and neighborhood mutual information. Then, we propose neighborhood mutual information based max relevance and min redundancy feature selection. Finally, experimental results show that the proposed method can effectively select a discriminative feature subset, and outperform or equalto other popular feature selection algorithms in classification performance.%特征选择是一种重要的数据预处理步骤，其中互信息是一类重要的信息度量方法。
本文针对互信息不能很好地处理数值型的特征，介绍了邻域信息熵与邻域互信息。
其次，设计了基于邻域互信息的最大相关性最小冗余度的特征排序算法。
最后，用此算法选择前若干特征进行分类并与其它算法比较分类精度。
实验结果表明本文提出算法在分类精度方面且优于或相当于其它流行特征选择算法。
对偶犹豫模糊集的相关系数及其应用
对偶犹豫模糊集的相关系数及其应用吴婉莹;金飞飞;郭甦;陈华友;周礼刚【摘要】Dual hesitant fuzzy set has become a hot issue in fuzzy decision making problem as a result of providing more decision information for decision makers. The correlation index can be used to measure the relationship between two fuzzy information, and entropy is a measurement of the degree of uncertainty for fuzzy information. This paper presents a fuzzy multiple attribute group decision making method based on the dual hesitant fuzzy correlation coefficient and entropy. The definition of thedual hesitant fuzzy correlation coefficient is proposed. It discusses some basic properties. Two kinds of entropy for dual hesitant fuzzy set are put forward, based on which, it presents the method of determining weight for fuzzy multiple attribute group decision making. On the basis of the dual hesitant fuzzy set correlation coefficient and entropy, a new method for dual hesitant fuzzy multi-attribute group decision making problem with completely unknown attribute weight information is proposed. It demonstrates that the method is practically and effective through an analysis of a case.%对偶犹豫模糊集因其可以给决策者提供更多的决策信息成为模糊决策的热点研究问题，相关性指标可以用来度量两个模糊信息之间的相关关系，熵可以用来度量模糊信息的不确定程度。
信息论和生命科学——历史和前沿
妖踪初现
• 分子信息棘齿
Davis AP (2007) Nature Nanotechnology
Serreli V et al (2007), A molecular information ratchet, Nature
General Calculus of Biology ?
• 薛定谔《生命是什么》: 寻找生命科学的量化定律
Information content of DNA
Haeseleer PD (2006) Nature Biotechnology
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Schneider TD (2006) IEEE Engineering in Medicine and Biology Magazine
现代信息论的先导
• 1944 薛定谔: 生命以负熵为生
– 生命从环境中摄取负熵，保持自身结构高度有序
– 生命遗传信息载体”非周期性晶体”
• 1953 克里克/沃森
– 提出DNA双螺旋结构现代生命科学
Schrödinger, Erwin (1944). What is Life - the Physical Aspect of the Living Cell. Cambridge University Press
ATAATGCAACAAGGCTTGGAAGGCTAACCTGGGGTGAGGCCGGGTTGGGGCCGGGCTGGGGGTGGGAGGAGTCCTCACTGGCGGTTGATTGACAGTTTCTCC TTCCCCAGACTGGCCAATCACAGGCAGGAAGATGAAGGTTCTGTGGGCTGCGTTGCTGGTCACATTCCTGGCAGGTATGGGGGCGGGGCTTGCTCGGTTCCCC CCGCTCCTCCCCCTCTCATCCTCACCTCAACCTCCTGGCCCCATTCAGGCAGACCCTGGGCCCCCTCTTCTGAGGCTTCTGTGCTGCTTCCTGGCTCTGAACAGCGA TTTGACGCTCTCTGGGCCTCGGTTTCCCCCATCCTTGAGATAGGAGTTAGAAGTTGTTTTGTTGTTGTTGTTTGTTGTTGTTGTTTTGTTTTTTTGAGATGAAGTCT CGCTCTGTCGCCCAGGCTGGAGTGCAGTGGCGGGATCTCGGCTCACTGCAAGCTCCGCCTCCCAGGTCCACGCCATTCTCCTGCCTCAGCCTCCCAAGTAGCTG GGACTACAGGCACATGCCACCACACCCGACTAACTTTTTTGTATTTTCAGTAGAGACGGGGTTTCACCATGTTGGCCAGGCTGGTCTGGAACTCCTGACCTCAGG TGATCTGCCCGTTTCGATCTCCCAAAGTGCTGGGATTACAGGCGTGAGCCACCGCACCTGGCTGGGAGTTAGAGGTTTCTAATGCATTGCAGGCAGATAGTGAAT ACCAGACACGGGGCAGCTGTGATCTTTATTCTCCATCACCCCCACACAGCCCTGCCTGGGGCACACAAGGACACTCAATACATGCTTTTCCGCTGGGCGCGGTGG CTCACCCCTGTAATCCCAGCACTTTGGGAGGCCAAGGTGGGAGGATCACTTGAGCCCAGGAGTTCAACACCAGCCTGGGCAACATAGTGAGACCCTGTCTCTAC TAAAAATACAAAAATTAGCCAGGCATGGTGCCACACACCTGTGCTCTCAGCTACTCAGGAGGCTGAGGCAGGAGGATCGCTTGAGCCCAGAAGGTCAAGGTTG CAGTGAACCATGTTCAGGCCGCTGCACTCCAGCCTGGGTGACAGAGCAAGACCCTGTTTATAAATACATAATGCTTTCCAAGTGATTAAACCGACTCCCCCCTCAC CCTGCCCACCATGGCTCCAAAGAAGCATTTGTGGAGCACCTTCTGTGTGCCCCTAGGTACTAGATGCCTGGACGGGGTCAGAAGGACCCTGACCCACCTTGAAC TTGTTCCACACAGGATGCCAGGCCAAGGTGGAGCAAGCGGTGGAGACAGAGCCGGAGCCCGAGCTGCGCCAGCAGACCGAGTGGCAGAGCGGCCAGCGCT GGGAACTGGCACTGGGTCGCTTTTGGGATTACCTGCGCTGGGTGCAGACACTGTCTGAGCAGGTGCAGGAGGAGCTGCTCAGCTCCCAGGTCACCCAGGAAC TGAGGTGAGTGTCCCCATCCTGGCCCTTGACCCTCCTGGTGGGCGGCTATACCTCCCCAGGTCCAGGTTTCATTCTGCCCCTGTCGCTAAGTCTTGGGGGGCCTG GGTCTCTGCTGGTTCTAGCTTCCTCTTCCCATTTCTGACTCCTGGCTTTAGCTCTCTGGAATTCTCTCTCTCAGCTTTGTCTCTCTCTCTTCCCTTCTGACTCAGTCTCT CACACTCGTCCTGGCTCTGTCTCTGTCCTTCCCTAGCTCTTTTATATAGAGACAGAGAGATGGGGTCTCACTGTGTTGCCCAGGCTGGTCTTGAACTTCTGGGCTC AAGCGATCCTCCCGCCTCGGCCTCCCAAAGTGCTGGGATTAGAGGCATGAGCCACCTTGCCCGGCCTCCTAGCTCCTTCTTCGTCTCTGCCTCTGCCCTCTGCATC TGCTCTCTGCATCTGTCTCTGTCTCCTTCTCTCGGCCTCTGCCCCGTTCCTTCTCTCCCTCTTGGGTCTCTCTGGCTCATCCCCATCTCGCCCGCCCCATCCCAGCCCT TCTCCCCGCCTCCCACTGTGCGACACCCTCCCGCCCTCTCGGCCGCAGGGCGCTGATGGACGAGACCATGAAGGAGTTGAAGGCCTACAAATCGGAACTGGAG GAACAACTGACCCCGGTGGCGGAGGAGACGCGGGCACGGCTGTCCAAGGAGCTGCAGGCGGCGCAGGCCCGGCTGGGCGCGGACATGGAGGACGTGTGCG GCCGCCTGGTGCAGTACCGCGGCGAGGTGCAGGCCATGCTCGGCCAGAGCACCGAGGAGCTGCGGGTGCGCCTCGCCTCCCACCTGCGCAAGCTGCGTAAGC GGCTCCTCCGCGATGCCGATGACCTGCAGAAGCGCCTGGCAGTGTACCAGGCCGGGGCCCGCGAGGGCGCCGAGCGCGGCCTCAGCGCCATCCGCGAGCGCC TGGGGCCCCTGGTGGAACAGGGCCGCGTGCGGGCCGCCACTGTGGGCTCCCTGGCCGGCCAGCCGCTACAGGAGCGGGCCCAGGCCTGGGGCGAGCGGCT GCGCGCGCGGATGGAGGAGATGGGCAGCCGGACCCGCGACCGCCTGGACGAGGTGAAGGAGCAGGTGGCGGAGGTGCGCGCCAAGCTGGAGGAGCAGGC CCAGCAGATACGCCTGCAGGCCGAGGCCTTCCAGGCCCGCCTCAAGAGCTGGTTCGAGCCCCTGGTGGAAGACATGCAGCGCCAGTGGGCCGGGCTGGTGGA GAAGGTGCAGGCTGCCGTGGGCACCAGCGCCGCCCCTGTGCCCAGCGACAATCACTGAACGCCGAAGCCTGCAGCCATGCGACCCCACGCCACCCCGTGCCTC CTGCCTCCGCGCAGCCTGCAGCGGGAGACCCTGTCCCCGCCCCAGCCGTCCTCCTGGGGTGGACCCTAGTTTAATAAAGATTCACCAAGTTTCACGCA
犹豫模糊多属性决策的折中比值法
犹豫模糊多属性决策的折中比值法李兰平【摘要】针对属性值为犹豫模糊元的多属性决策问题，提出了一种新的多属性决策方法——折中比值法。
折中比值法方法是通过定义能同时反映出备选方案尽可能地接近正理想点又同时尽可能地远离负理想点，并且把决策者的主观态度也包含在内的排序指标对备选方案进行排序和择优。
最后，通过应用实例说明了所提出的方法的有效性和可行性。
%For the multiple attribute decision making problem with attribute value expressed as hesitant fuzzy elements, a new multi-attribute decision making method, and named compromise ratio method is proposed. Compromise ratio method is developed by introducing the ranking index based on the concept that the chosen alternative should be as aloes as possible to the ideal solution and as far away from the negative ideal solution as possible simultaneously, while the decision maker's subjective attitude are also included. Finally, a practical example is presented to demonstrate the effectiveness and feasibility of the proposed method.【期刊名称】《齐齐哈尔大学学报（自然科学版）》【年(卷),期】2015(000)001【总页数】4页(P57-60)【关键词】犹豫模糊集;多属性决策;折中比值法;理想点【作者】李兰平【作者单位】湖南财政经济学院基础课部，长沙 410205【正文语种】中文【中图分类】C934自Zadeh提出模糊集[1]以来，模糊集理论已经被应用到各个领域。
Homework
Discussion TopicsLiang Hao 142107201811. What the basic functional blocks are included in a digital communication system?Source encoder, channel encoder, digital modulator, digital demodulator, channel demodulator, source decoder.2. What are the average mutual information, entropy, and conditional entropy? What is the relationship of these three concepts?For two discrete random variables X and Y, The mutual information is defined as()()()()()()()j i j i n i mj j i n i mj j i j i y P x P y x P y x P y x I y x P Y X I ,log,;,;1111∑∑∑∑======Where p(x, y) is the joint probability distribution function of X and Y, and p(x) and p(y) are the marginal probability distribution functions of X and Y respectively. An important characteristic of the average mutual information is that()0;≥Y X I .When X and Y are statistically independent, we define average self-information as:()()()()()i nii i n i i x P x P x I x P X H log 11∑∑==-==When X represents the alphabet of possible output letters from a source, represents the average self-information per source letter, and it is called the entropy of the source.An average conditional self-information is called conditional entropy. It’s def ined as:()()()j i n i mjj i y x P y x P Y X H |1log ,|11∑∑===The relationship between the mutual information, entropy and conditional entropy is:()()()Y X H X H Y X I \;-=3. What is the Shannon source coding theorem?In information theory, Shannon's source coding theorem (or noiseless coding theorem ) establishes the limits to possible data compression, and the operationalmeaning of the Shannon entropy.The source coding theorem shows that (in the limit, as the length of a stream of independent and identically-distributed random variable (i.i.d.) data tends to infinity) it is impossible to compress the data such that the code rate (average number of bits per symbol) is less than the Shannon entropy of the source, without it being virtually certain that information will be lost. However it is possible to get the code rate arbitrarily close to the Shannon entropy, with negligible probability of loss.4. How to represent the band-pass signal using low-pass signal?Firstly develop a mathematical representation of such signals construct a signal that contains only the positive frequencies in s (t). ()()()f S f u f S 2=+ The signal S t (t) is called the analytic or the pre-envelope of s(t) :()()()()()t s t j t s t s t j t t s *1*ππδ+=⎥⎦⎤⎢⎣⎡+=+ Define ()t s ˆ as ()()()τττππd t s t s tt s ⎰∞∞--==1*1ˆ the signal may be viewed as the output of the filter with impulse response h (t).The analytic ()t s + is a band-pass signal ，we may obtain an equivalent low-pass representation by performing a frequency translation ：Define S l (f ) as ()()c l f f S f S +=+. The equivalent time-domain relation is()()()()[]t f j t f j l c c e t s j t s e t s t s ππ22ˆ--++==. In general ，the signal ()t s l is complex-valued ，and may be expressed as:()()()t jy t x t s l +=. The expression is the desired form for the representationof a band-pass signals.5. How to express the energy in the band-pass signal in terms of the equivalent low-pass signal?The energy in signal is defined as ()()[]{}⎰⎰∞∞-∞∞-==dt et s dtt s E t f j lc 222Re π. Theenergy in band-pass signal is expressed in terms of equivalent low-pass signal is()⎰∞∞-=dt t s E l 221.6. What is the AWGN noise? Make discussion in details.Additive white Gaussian noise (AWGN) is a basic noise model used in Information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:(1) Additive because it is added to any noise that might be intrinsic to the information system.(2) White refers to the idea that it has uniform power across the frequency band for the information system. It is an analogy to the color white which has uniform emissions at all frequencies in the visible spectrum.(3) Gaussian because it has a normal distribution in the time domain with an average time domain value of zero.7. What is the modulated carrier signal of the minimum-shift keying modulation?Minimum Shift Keying (MSK) is a special from of binary CPFSK where the modulation index h = 1/2. The carrier-modulated signal corresponding may beexpressed as ()()[]0;2cos 2φφπ++=I t f TEt s c8. What is the expression of the power density spectrum if the sequence is real and mutual uncorrelated?When the information symbols in the sequence are real and mutually uncorrelated. The autocorrelation function can be expressed as()()()⎩⎨⎧≠=+=00222m m m ii i ii μμσφ. We represent the()m ii φ in()()∑∞-∞=-=ΦmfmTj ii ii e m f πφ2 we can get the result()∑∞-∞=-+=ΦmfmTj ii ii e f πμσ222.9. Discussion on the optimal receiver configuration over AWGN channels?The optimal receiver means we can minimize the probability of making error. So we can conveniently subdivided the receiver into two parts: signal demodulator and detector. There are two ways to realize the signal demodulator, one is based on the use of signal correlators, and the other is based on the use of matched filters. Normally we have two demodulator: one is correlation type demodulator and the other is matched-filter type demodulator. Both the two demodulator can produce the vector which contains all the necessary information. The optimum detector that follows the signal demodulator is designed to minimize the probability of error. To achieve this, we can apply the MAP rule or ML rule on our detector.10. What are the optimum decision criterions of the optimal receiver over AWGN channels?There are two optimum decision criterions of the optimal receiver over AWGN channels, one is the MAP criterion-minimum distance detection. The decision criterion is based on selecting the signal corresponding to the maximum of the set of posterior probabilities such a criterion is called as maximum a posteriori probability (MAP) criterion. The other is maximum-likelihood (ML) criterion.11. What is the Shannon capacity of the band-limited AWGN waveform channel with a band-limited and average power-limited input? And make the discussion in detail about this Shannon capacity .The Shannon capacity is ⎪⎪⎭⎫⎝⎛+=01log WN P W C av . Where the average received power is av P and the noise power spectral density is 0N .is known as received signal-to-noise ratio (SNR). When the SNR is large (SNR >>0dB), the capacity C ≈⎪⎪⎭⎫⎝⎛=0log WN P W C av is logarithmic in power and approximatelylinear in bandwidth. This is called the bandwidth-limited regime. When the SNR is small (SNR << 0dB), the capacity C ≈ e N P C av log 2=is linear in power butinsensitive to bandwidth. This is called the power-limited regime.。
信息熵(informationentropy)百科物理
信息熵（informationentropy）百科物理
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一起来阅读信息熵(informationentropy)百科物理吧~
信息熵〔informationentropy〕
信息熵(informationentropy)
是信息论中信息量的统计表述。
香农(Shannon)定义信息量为：
`I=-Ksum_ip_ilnp_i`，表示信息所消除的不确定性(系统有序程度)的量度，K为待定常数，pi为事件出现的概率，$sump_i=1$。
对于N
个等概率事件，pi=1/N，系统的信息量为I=-Klnpi=KlnN。
平衡态
时系统热力学函数熵的最大值为$S=-
ksum_iW_ilnW_i=klnOmega$，k为玻尔兹曼常数，Wi=1/为系统各状
态的概率，$sum_iW_i=1$，为系统状态数，熵是无序程度的量度。
信息量I与熵S具有相同的统计意义。
设K为玻尔兹曼常数k，那
么信息量I可称信息熵，为$H=-ksum_ip_ilnp_i$，信息给系统带
来负熵。
如取K=1，对数底取2，熵的单位为比特(bit);取底为e，
那么称尼特。
信息熵是生命系统(作为非平衡系统)在形成有序结构
耗散结构时，所接受的负熵的一部分。
由查字典物理网独家提供信息熵(informationentropy)百科物理，
希望给大家提供帮助。
干扰效果在线评估参数筛选与特征表示方法
子战［４５］项目进行干扰效果在线评估方法研 究，例 如 行 为 学 习 型 自 适 应 电 子 战 项 目［６］、认 知 干 扰 机 项 目 和 ［７］ 自 适 应 雷 达对 抗 （ａｄａｐｔｉｖｅｒａｄａｒｃｏｕｎｔｅｒｍｅａｓｕｒｅ，ＡＲＣ）项 目［８］等。 目前，美国在干扰效 果 在 线 评 估 领 域 的 研 究 正 在 由 技 术 开 发转向装备应 用，着 手 进 行 基 于 Ｆ１８、Ｆ３５ 平 台 的 初 期 装 备列装测试，但因研究方向敏感，很少 有 公 开 文 献 报 道。 按 照此思路国内外 学 者 也 展 开 了 相 关 研 究，文 献［９］分析了雷 达 受 扰 行 为 及 参 数 变 化 特 点 ，通 过 参 数 威 胁 等 级 的 变 化 结 合 支持向量机（ｓｕｐｐｏｒｔｖｅｃｔｏｒｍａｃｈｉｎｅ，ＳＶＭ）进 行 干 扰 效 果 评 估 。该 方 法 提 供 了 基 于 干 扰 方 进 行 评 估 的 思 路 ，但 未 说 明 如
关 键 词 ：干 扰 效 果 ；在 线 评 估 ；信 息 熵 ；盒 维 数 ；皮 尔 逊 相 关 系 数 中 图 分 类 号 ：ＴＮ９７４ 文 献 标 志 码 ：Ａ 犇犗犐：１０．３９６９／ｊ．ｉｓｓｎ．１００１５０６Ｘ．２０２０．１２．１１
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（犇犲狆犪狉狋犿犲狀狋狅犳犐狀犳狅狉犿犪狋犻狅狀犛狔狊狋犲犿 ，犇犪犾犻犪狀犖犪狏犪犾犃犮犪犱犲犿狔，犇犪犾犻犪狀１１６０１８，犆犺犻狀犪）
关于熵的英语作文
关于熵的英语作文Entropy is a fundamental concept in physics, information theory, and other fields. It is a measure of the disorder or randomness in a system, and plays a key role in understanding the behavior of complex systems.In thermodynamics, entropy is often described as a measure of the degree of disorder in a system. The second law of thermodynamics states that the total entropy of a closed system always increases over time, meaning that the system becomes more disordered or random. For example, when heat is added to a system, it increases the disorder of the system, and this leads to an increase in entropy.In information theory, entropy is used to measure the amount of uncertainty or randomness in a message. The more uncertain or random a message is, the higher its entropy. For example, a message with only one possible outcome has zero entropy, while a message with many possible outcomes has higher entropy.Entropy is also important in the study of complex systems, such as biological systems or social systems. In these systems, there are many different components that interact in complex ways, and understanding the entropy of these systems can provide insights into their behavior.One example of this is the study of protein folding. Proteins are long chains of amino acids that must fold into specific three-dimensional shapes in order to function properly. The process of protein folding is highly complex and can be affected by many different factors, including temperature and pressure. Understanding the entropy of the system can help scientists predict how a protein will fold, and this can have important implications for drug design and other areas of biotechnology.In summary, entropy is a fundamental concept that plays a key role in understanding the behavior of complex systems. It is a measure of the disorder or randomness in a system, and is used in thermodynamics, information theory, and other fields. By studying entropy, scientists can gain insights into the behavior of complex systems, and this can have important practical applications in fields such as biotechnology and materials science.译文：熵是物理学、信息论和其他领域中的一个基本概念。
信息熵的计算及实现
认知实习报告题目 __信息熵的计算及实现_________ _ （院）系数理系 ___________ 专业 _______信息与计算科学__________________ 班级_ _ 学号_ 20081001 _学生姓名 _ _导师姓名_ ___ ________完成日期 ________2011年12月23日___________信息熵的计算及实现信息与计算科学专业：指 导 教 师：摘要：信息的销毁是一个不可逆过程，一般而言，当一种信息出现概率更高的时候，表明它被传播得更广泛，或者说，被引用的程度更高。
我们可以认为，从信息传播的角度来看，信息熵可以表示信息的价值。
这样我们就有一个衡量信息价值高低的标准，可以做出关于知识流通问题的更多推论。
本文讨论了一维几种熵的计算方法：离散信源的熵、图像熵的一维熵和二维熵、基于信息熵的Web 页面主题信息计算方法，并给出一定的理论分析和数值实验以及数值实验结果。
关键字：离散信源的熵、图像熵、Web 页面主题信息1 引言信息论之父 C. E. Shannon 在 1948 年发表的论文“通信的数学理论（ AMathematical Theory of Communication ）”中， Shannon 指出，任何信息都存在冗余，冗余大小与信息中每个符号（数字、字母或单词）的出现概率或者说不确定性有关。
Shannon 借鉴了热力学的概念，把信息中排除了冗余后的平均信息量称为“信息熵”，并给出了计算信息熵的数学表达式。
2 问题提出信源的平均不定度。
在信息论中信源输出是随机量，因而其不定度可以用概率分布来度量。
记 H(X)＝H(P1，P2，…，Pn)＝P(xi)logP(xi)，这里P(xi)，i ＝1，2，…，n 为信源取第i 个符号的概率。
P(xi)=1，H(X)称为信源的信息熵。
2.1 离散信源的熵利用信息论中信息熵概念，求出任意一个离散信源的熵（平均自信息量）。
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a r X i v :c o n d -m a t /0303110v 4 [c o n d -m a t .s t a t -m e c h ] 8 A p r 2003Information Entropy and Correlations in Prime NumbersPradeep Kumar,Plamen Ch.Ivanov,H.Eugene Stanley Center for Polymer Studies and Department of Physics,Boston University,Boston,MA 02215(Dated:February 2,2008)Abstract The difference between two consecutive prime numbers is called the distance between the primes.We study the statistical properties of the distances and their increments (the difference between two consecutive distances)for a sequence comprising the first 5×107prime numbers.We find that the histogram of the increments follows an exponential distribution with superposed periodic behavior of period three,similar to previously-reported period six oscillations for the distances.Recent reports indicate that many physical and biological systems exhibit patterns where prime numbers play an important role.Examples range from the periodic orbits of a system in quantum chaos to the life cycles of species[1,2,3,4,5,6,7].Recent work reports on a potential for which the quantum energy levels of a particle can be mapped onto the sequence of primes[8].Furthermore,it has been shown that a gas of independent bosons with energies equal to the logarithm of consecutive primes possesses a canonical partition function coinciding with the Riemann function[9].The partition function of a system with energies equal to the distances between two consecutive prime numbers behaves like a set of non-interacting harmonic oscillators[10].Most recently,power-law behavior in the distribution of primes and correlations in prime numbers have been found[11],along with multifractal features in the distances between consecutive primes[12].Previous work thus further motivates studies of prime numbers using methods of statistical physics.Here,we focus on the statistical properties of the distances between consecutive prime numbers and the increments in these distances[Fig.1].Since the distribution of distances is well-studied,we discuss the occurrence frequency of increments between consecutive distances.Wefind that the distribution of increments [Fig.2(a)]exhibits large peaks for given values of the increments and medium and small peaks for other values,and that these peaks follow period-three oscillation.Specifically,we find that the increments with values of6k+2(k=0,1,2,3...)have the highest occurrence frequency,followed by increments with values of6k+4.Values of6k are relatively rare and correspond to the small peaks in the distribution.This regularity is present for both positive and negative increments and does not depend on the length N p of the sequence. We alsofind that the occurrence frequency of increments decreases exponentially and that this exponential behavior is well pronounced for both large and small peaks,forming a “double-tent”shape[Fig.2(b)].Wefind exponential behavior with superposed periodic behavior with period-three oscilla-tion for the distribution of increments similar to the period-six oscillation for the distribution of distances[10].Further,wefind that the occurrence frequency of a positive increment is almost the same as the occurrence frequency of its negative counterpart for a given sequence length N p[Fig.2(c)].In summary,wefind a new statistical feature in the sequence of increments between con-secutive prime distances.Wefind a period-three oscillation in the distribution of incrementsand this distribution follows an exponential form.This empirical observation may be of importance in further understanding the nature of prime numbers as well as those physical and biological processes where prime numbers play a role.AcknowledgmentsWe thank M.Wolf,S.Havlin and M.Taqqu for helpful discussions.[1]N.Argaman,F.-M.Dittes,E.Doron,J.P.Keating,A.Yu.Kitaev,M.Sieber,and U.Smi-lansky,Phys.Rev.Lett.71,4326(1993).[2] E.Goles,O.Schulz,and M.Markus,Complexity5,33(2001).[3]M.Planat,Fluctuation and Noise Letters1,R65[4] C.M.Ko,Chaos,Solitons,and Fractals13,1295(2001).[5]S.R.Dahmen,S.D.Prado,and T.Stuermer-Daitx,Physica A296,523(2001).[6]R.L.Liboffand M.Wong,Int.J.Theor.Phys.37,3109(1998).[7]J.Toha and M.A.Soto,Medical Hypothesis53,361(1999).[8]G.Mussardo,Preprint ISAP/EP/97/153(1997),available as cond-mat/9712010.[9] B.Julia,Statistical Theory of Numbers(Springer,Berlin,1990).[10]M.Wolf,Physica A274,149(1999).[11]M.Wolf,Physica A241,493(1997).[12]M.Wolf,Physica A160,24(1989).[13]L.Brillouin,Science and Information Theory(Academic Press,New York,1962).[14]H.E.Stanley,Rev.Mod.Phys.71,S358(1999).[15] B.B.Mandelbrot,The Fractal Geometry of Nature(W.H.Freeman,San Francisco,1983).[16] A.Bunde,and S.Havlin,Fractals in Science(Springer,1995).[17]T.Viscek,Fractal Growth Phenomena,2nd Ed.(World Scientific,Singapore,1993).[18]J.B.Bassingthwaighte,L.S.Liebovitch,and B.J.West,Fractal Physiology(Oxford Univer-sity Press,New York,1994).[19] A.L.Barabasi and H.E.Stanley,Fractal Concepts in Surface Growth(Cambridge UniversityPress,Cambridge,1995).[20]P.Meakin,Fractals,Scaling,and Growth Far from Equilibrim(Cambridge University Press,Cambridge,1997).[21] C.K.Peng et al.,Chaos5,82(1995).[22]S.V.Buldyrev et al.,Biophys.J.65,2673(1993).[23]J.W.Kantelhardt,E.Koscielny-Bunde,H.H.A.Rego,S.Havlin,and A.Bunde,Physica A294,441(2001).[24]K.Hu et al.,Phys.Rev.E64,011114(2001).[25]Z.Chen et al.,Phys.Rev.E65,041107(2002).[26]M.S.Taqqu,V.Teverovsky,and W.Willinger,Fractals3,185(1996).[27]S.M.Ossadnik et al.,Biophys.J.67,64(1994).[28]J.M.Hausdorff,C.-K.Peng,din,J.Wei,and A.L.Goldberger,J.Applied Physiol.78,349(1995).[29]Y.Liu et al.,Physica A245,437(1997).[30]P.Ch.Ivanov et al.,Europhys.Lett.48,594(1999).[31]K.Ivanova and M.Ausloos,Physica A274,349(1999).[32]Y.Liu et al.,Phys.Rev.E60,1390(1999).[33]P.Talkner and R.O.Weber,Phys.Rev.E62,150(2000).[34]P.Ch.Ivanov et al.,Chaos11,641(2001).[35]S.Bahar,J.W.Kantelhardt,A.Neiman,H.H.H Rego,D.F.Russell,L.Wilkens,A.Bunde,and F.Moss,Europhys.Lett.56,454(2001).[36]indan,D.Vyushin,and A.Bunde,Phys.Rev.Lett.89,028501(2002).[37]J.W.Kantelhardt et al.,Phys.Rev.E65,051908(2002).[38]Y.Ashkenazy et al.,Phys.Rev.Lett.86,1900(2001).244002450024600Index 020406080 240002450025000020406080D i s t a n c e s 01000020000300004000050000050100(a)244002450024600Index −100−5050240002450025000−90−401060I n c r e m e n t s 01000020000300004000050000−1000100(b)FIG.1:Distances between consecutive prime numbers (indexed sequentially)and their increments.(a)The first 5×104distances between consecutive prime numbers.(b)The first 5×104increments.−50050Increments 02x1044x1046x1048x104H i s t o g r a m o f I n c r e m e n t s (a)024810−8−10−4−2−12126−6−100−50050100Increments 101102103104105H i s t o g r a m o f I n c r e m e n ts 020406080100Increments 101102103104105H i s t o g r a m o f I n c r e m e n t sFIG.2:(a)Histogram of increments in the distances between consecutive prime numbers for the sequence of the first N p =106primes.The bin width is 1.The occurrence frequency of increments with given values exhibits a robust period-three oscillation.Increments with values ±(6k +2)(k =0,1,2,3,....)occur most often,increments with values ±(6k +4)occur less often,。