• <ins id="pjuwb"></ins>
    <blockquote id="pjuwb"><pre id="pjuwb"></pre></blockquote>
    <noscript id="pjuwb"></noscript>
          <sup id="pjuwb"><pre id="pjuwb"></pre></sup>
            <dd id="pjuwb"></dd>
            <abbr id="pjuwb"></abbr>

            O(1) 的小樂

            Job Hunting

            公告

            記錄我的生活和工作。。。
            <2025年7月>
            293012345
            6789101112
            13141516171819
            20212223242526
            272829303112
            3456789

            統(tǒng)計(jì)

            • 隨筆 - 182
            • 文章 - 1
            • 評(píng)論 - 41
            • 引用 - 0

            留言簿(10)

            隨筆分類(70)

            隨筆檔案(182)

            文章檔案(1)

            如影隨形

            搜索

            •  

            最新隨筆

            最新評(píng)論

            閱讀排行榜

            評(píng)論排行榜

            Kullback–Leibler divergence KL散度

            In probability theory and information theory, the Kullback–Leibler divergence[1][2][3] (also information divergence,information gain, relative entropy, or KLIC) is a non-symmetric measure of the difference between two probability distributions P and Q. KL measures the expected number of extra bits required to code samples from P when using a code based on Q, rather than using a code based on P. Typically P represents the "true" distribution of data, observations, or a precise calculated theoretical distribution. The measure Q typically represents a theory, model, description, or approximation of P.

            Although it is often intuited as a distance metric, the KL divergence is not a true metric – for example, the KL from P to Q is not necessarily the same as the KL from Q to P.

            KL divergence is a special case of a broader class of divergences called f-divergences. Originally introduced by Solomon Kullbackand Richard Leibler in 1951 as the directed divergence between two distributions, it is not the same as a divergence incalculus. However, the KL divergence can be derived from the Bregman divergence.

             

             

            注意P通常指數(shù)據(jù)集,我們已有的數(shù)據(jù)集,Q表示理論結(jié)果,所以KL divergence 的物理含義就是當(dāng)用Q來編碼P中的采樣時(shí),比用P來編碼P中的采用需要多用的位數(shù)!

             

            KL散度,也有人稱為KL距離,但是它并不是嚴(yán)格的距離概念,其不滿足三角不等式

             

            KL散度是不對(duì)稱的,當(dāng)然,如果希望把它變對(duì)稱,

            Ds(p1, p2) = [D(p1, p2) + D(p2, p1)] / 2

             

            下面是KL散度的離散和連續(xù)定義!

            D_{\mathrm{KL}}(P\|Q) = \sum_i P(i) \log \frac{P(i)}{Q(i)}. \!

            D_{\mathrm{KL}}(P\|Q) = \int_{-\infty}^\infty p(x) \log \frac{p(x)}{q(x)} \; dx, \!

            注意的一點(diǎn)是p(x) 和q(x)分別是pq兩個(gè)隨機(jī)變量的PDF,D(P||Q)是一個(gè)數(shù)值,而不是一個(gè)函數(shù),看下圖!

             

            注意:KL Area to be Integrated!

             

            File:KL-Gauss-Example.png

             

            KL 散度一個(gè)很強(qiáng)大的性質(zhì):

            The Kullback–Leibler divergence is always non-negative,

            D_{\mathrm{KL}}(P\|Q) \geq 0, \,

            a result known as , with DKL(P||Q) zero if and only if P = Q.

             

            計(jì)算KL散度的時(shí)候,注意問題是在稀疏數(shù)據(jù)集上KL散度計(jì)算通常會(huì)出現(xiàn)分母為零的情況!

             

             

            Matlab中的函數(shù):KLDIV給出了兩個(gè)分布的KL散度

            Description

            KLDIV Kullback-Leibler or Jensen-Shannon divergence between two distributions.

            KLDIV(X,P1,P2) returns the Kullback-Leibler divergence between two distributions specified over the M variable values in vector X. P1 is a length-M vector of probabilities representing distribution 1, and P2 is a length-M vector of probabilities representing distribution 2. Thus, the probability of value X(i) is P1(i) for distribution 1 and P2(i) for distribution 2. The Kullback-Leibler divergence is given by:

               KL(P1(x),P2(x)) = sum[P1(x).log(P1(x)/P2(x))]

            If X contains duplicate values, there will be an warning message, and these values will be treated as distinct values. (I.e., the actual values do not enter into the computation, but the probabilities for the two duplicate values will be considered as probabilities corresponding to two unique values.) The elements of probability vectors P1 and P2 must each sum to 1 +/- .00001.

            A "log of zero" warning will be thrown for zero-valued probabilities. Handle this however you wish. Adding 'eps' or some other small value to all probabilities seems reasonable. (Renormalize if necessary.)

            KLDIV(X,P1,P2,'sym') returns a symmetric variant of the Kullback-Leibler divergence, given by [KL(P1,P2)+KL(P2,P1)]/2. See Johnson and Sinanovic (2001).

            KLDIV(X,P1,P2,'js') returns the Jensen-Shannon divergence, given by [KL(P1,Q)+KL(P2,Q)]/2, where Q = (P1+P2)/2. See the Wikipedia article for "Kullback–Leibler divergence". This is equal to 1/2 the so-called "Jeffrey divergence." See Rubner et al. (2000).

            EXAMPLE: Let the event set and probability sets be as follow:
               X = [1 2 3 3 4]';
               P1 = ones(5,1)/5;
               P2 = [0 0 .5 .2 .3]' + eps;
            Note that the event set here has duplicate values (two 3's). These will be treated as DISTINCT events by KLDIV. If you want these to be treated as the SAME event, you will need to collapse their probabilities together before running KLDIV. One way to do this is to use UNIQUE to find the set of unique events, and then iterate over that set, summing probabilities for each instance of each unique event. Here, we just leave the duplicate values to be treated independently (the default):
               KL = kldiv(X,P1,P2);
               KL =
                    19.4899

            Note also that we avoided the log-of-zero warning by adding 'eps' to all probability values in P2. We didn't need to renormalize because we're still within the sum-to-one tolerance.

            REFERENCES:
            1) Cover, T.M. and J.A. Thomas. "Elements of Information Theory," Wiley, 1991.
            2) Johnson, D.H. and S. Sinanovic. "Symmetrizing the Kullback-Leibler distance." IEEE Transactions on Information Theory (Submitted).
            3) Rubner, Y., Tomasi, C., and Guibas, L. J., 2000. "The Earth Mover's distance as a metric for image retrieval." International Journal of Computer Vision, 40(2): 99-121.
            4) <a href="
            http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence"&gt;Kullback–Leibler divergence</a>. Wikipedia, The Free Encyclopedia.

            posted on 2010-10-16 15:04 Sosi 閱讀(10020) 評(píng)論(2)  編輯 收藏 引用 所屬分類: Taps in Research

            評(píng)論

            # re: Kullback&ndash;Leibler divergence KL散度 2010-11-30 16:17 tintin0324

            博主,本人的研究方向需要了解kl距離,有些問題想請(qǐng)教下,怎么聯(lián)系呢?

            # re: Kullback&ndash;Leibler divergence KL散度 2010-12-05 22:37 Sosi

            @tintin0324
            KL 距離本身很簡單,如果就是那樣子定義的,意義也如上面所說。。如果你想深入了解的話,可以讀以下相關(guān)文獻(xiàn)
            統(tǒng)計(jì)系統(tǒng)
            欧美日韩精品久久久免费观看| 国内精品综合久久久40p| 国内精品久久久久影院日本| 2020久久精品亚洲热综合一本 | 精品伊人久久久| 久久无码AV一区二区三区| 久久精品国产男包| 国产精品99久久精品| 亚洲va久久久噜噜噜久久男同| 午夜精品久久影院蜜桃| 国产精品热久久无码av| 蜜臀av性久久久久蜜臀aⅴ| 四虎国产精品免费久久| 日产精品99久久久久久| 久久久久亚洲AV无码专区桃色| 久久综合九色综合精品| 久久久亚洲欧洲日产国码二区| 怡红院日本一道日本久久 | 久久久WWW成人免费毛片| 国内精品久久久人妻中文字幕| 久久久久这里只有精品| 久久夜色精品国产噜噜噜亚洲AV | 久久99热这里只有精品66| 狠狠狠色丁香婷婷综合久久俺| 久久99精品国产麻豆宅宅| 国产成人99久久亚洲综合精品| 久久久久亚洲AV片无码下载蜜桃| 久久久久亚洲AV无码去区首| 久久精品一区二区| 99久久精品毛片免费播放| 久久国产精品99精品国产| 日韩人妻无码一区二区三区久久 | 久久99热这里只有精品国产| 国产精品久久成人影院| 久久精品国产亚洲AV蜜臀色欲 | 精品久久久久久无码人妻热 | 久久久久免费看成人影片| 2021国产精品午夜久久| 久久精品综合网| 伊人久久大香线焦AV综合影院| 亚洲精品国产综合久久一线|