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\frac{\omega_j^{(k)} \cdot f_j(x_i ~|~ \theta_j^{(k)})}{\sum_{j=1}^k \omega_j^{(k)} \cdot f_j(x_i ~|~ \theta_j^{(k)})}\] + +Здесь $\gamma_{ij}$ означает вероятность того, что $i$-ый элемент соответствует $j$-ой компоненте. +Главной идеей стратегии моментов является сопоставление эмпирических моментов истинным и решение системы уравнений относительно параметров компоненты. + +\section{Взвешенные эмпирические моменты} + +Обновление весов смеси происходит как и в стратегии Q-функции: + +\[\omega_j^{(k+1)} = \frac{\sum_{i=1}^n \gamma^{(k)}_{ij}}{n}\] + +Положим эмпирический начальный момент $r$-ого порядка для $j$-ой компоненты следующим образом: + +\[\hat{\mu}_{r,j}^{(k)} = \frac{\sum_{i=1}^n \gamma^{(k)}_{ij} \cdot x_i}{\sum_{i=1}^n \gamma^{(k)}_{ij} }\] + +\section{Покомпонентная оптимизация} + +Пусть вектор параметров $j$-ой компоненты имеет $M_j$ неизвестных величин. Теоретический момент $r$-ого порядка для $j$-ой компоненты имеет вид: +\[\mu'_{r, j}(\theta_j) = \int_{-\infty}^\infty x^r f_j(x ~|~ \theta_j)dx\] + +Стратегия оптимизации заключается в решении $k$ независимых систем уравнений (по одной для каждой компоненты). Системы получаются путем приравнивания теоретических моментов к соответствующим взвешенным эмпирическим: +\[\begin{cases} \mu'_{1, j}(\theta_j^{(k+1)}) = \hat{\mu}^{(k)}_{1,j} \\ \mu'_{2, j}(\theta_j^{(k+1)}) = \hat{\mu}^{(k)}_{2,j} \\ \vdots \\ \mu'_{M_j, j}(\theta_j^{(k+1)}) = \hat{\mu}^{(k)}_{M_j,j}\end{cases}\] + +Решая эту систему относительно параметров $\theta_j^{(k+1)}$ получаем новую оценку для следующей итерации алгоритма. + + +\section{Пример: смесь двух нормальных распределений} + +Пусть мы имеем смесь двух нормальных распределений имеющую плотность: + +\[f(x ~|~ \mu_1, \mu_2, \sigma_1, \sigma_2) = \omega_1 \cdot f_1(x~|~ \mu_1, \sigma_1) + (1 - \omega_1) \cdot f_2(x~|~ \mu_2, \sigma_2)\] + +Для нормального распределения параметры связаны с теоретическими начальными моментами следующим образом: + +\[\begin{cases} \mu'_1 = \mu \\ \mu'_2 = \mu^2 + \sigma^2 \end{cases}\] + + +Таким образом, для $i$-ой компоненты ($i = 1,2$) на $k$-ом шаге алгоритма мы последовательно решаем две системы уравнений: + +\[\begin{cases} \hat{\mu}^{(k)}_{1, i} = \mu_i \\ \hat{\mu}^{(k)}_{2, i} = \mu_i^2 + \sigma_i^2 \end{cases}\] \end{document} From 01cddaed6b1440cc30c1e3b24e1820e57f0558ac Mon Sep 17 00:00:00 2001 From: iraedeus Date: Mon, 13 Oct 2025 19:08:50 +0300 Subject: [PATCH 2/3] docs: add a book citation Finite mixture models McLachlan 2000 --- docs/math/moments.pdf | Bin 28715 -> 36975 bytes docs/math/source/moments.tex | 12 ++++++++---- docs/math/source/ref.bib | 12 +++++++----- 3 files changed, 15 insertions(+), 9 deletions(-) diff --git a/docs/math/moments.pdf b/docs/math/moments.pdf index f6369b332e5d4913c35056776bda75f51f266800..0285e23fbce8970bd9ed639cdadfd5a3d0fbf2e1 100644 GIT binary patch delta 35056 zcmZsCQ+OrZvTetwP`C(j;gjX*1-p1Ae48JmVVUJqJAp(@O^};kfY+}kksSjM?Ezk7veD`3}-ah^pUhv*!o_S*=<2(JWRjMTdJwh8>lh^Hj|Lp&f>PSSdG zLxku$4VI~b9XFgbynOKY)YuI|_Ddyqs)cdx1sD;0N)|Fc&BE zgm4BYauSCEVkpc53AuW>7>=@J=0-l>Q%o5KX*NgP{MdCYAJBMK8n)_nD(8=R>VkBP 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a/docs/math/source/moments.tex +++ b/docs/math/source/moments.tex @@ -5,6 +5,7 @@ \input{./styles.tex} \input{./commands.tex} + \title{Обоснование стратегии оценки параметров методом моментов} \begin{document} @@ -14,7 +15,7 @@ \section{Контекст: E-шаг и взвешенные данные} -Работаем с матрицей ответственностей посчитанной на прошлом шаге: +Работаем с матрицей ответственностей посчитанной на прошлом шаге \cite{mclachlan2000finite}{ (см. равенства 1.19 и 2.23)}: \[\gamma_{ij} = \frac{\omega_j^{(k)} \cdot f_j(x_i ~|~ \theta_j^{(k)})}{\sum_{j=1}^k \omega_j^{(k)} \cdot f_j(x_i ~|~ \theta_j^{(k)})}\] @@ -23,7 +24,7 @@ \section{Контекст: E-шаг и взвешенные данные} \section{Взвешенные эмпирические моменты} -Обновление весов смеси происходит как и в стратегии Q-функции: +Обновление весов смеси происходит как и в стратегии Q-функции \cite{mclachlan2000finite}{ (см. равенство 2.21)}: \[\omega_j^{(k+1)} = \frac{\sum_{i=1}^n \gamma^{(k)}_{ij}}{n}\] @@ -36,7 +37,9 @@ \section{Покомпонентная оптимизация} Пусть вектор параметров $j$-ой компоненты имеет $M_j$ неизвестных величин. Теоретический момент $r$-ого порядка для $j$-ой компоненты имеет вид: \[\mu'_{r, j}(\theta_j) = \int_{-\infty}^\infty x^r f_j(x ~|~ \theta_j)dx\] -Стратегия оптимизации заключается в решении $k$ независимых систем уравнений (по одной для каждой компоненты). Системы получаются путем приравнивания теоретических моментов к соответствующим взвешенным эмпирическим: +Известно, что эмпирический момент $r$-ого порядка для $j$-ой компоненты при увеличении $n$ стремится к истинным моментам компоненты (я не нашел ни одной статьи с доказательством для смесей). + +Поэтому стратегия оптимизации заключается в решении $k$ независимых систем уравнений (по одной для каждой компоненты). Системы получаются путем приравнивания теоретических моментов к соответствующим взвешенным эмпирическим: \[\begin{cases} \mu'_{1, j}(\theta_j^{(k+1)}) = \hat{\mu}^{(k)}_{1,j} \\ \mu'_{2, j}(\theta_j^{(k+1)}) = \hat{\mu}^{(k)}_{2,j} \\ \vdots \\ \mu'_{M_j, j}(\theta_j^{(k+1)}) = \hat{\mu}^{(k)}_{M_j,j}\end{cases}\] Решая эту систему относительно параметров $\theta_j^{(k+1)}$ получаем новую оценку для следующей итерации алгоритма. @@ -48,7 +51,7 @@ \section{Пример: смесь двух нормальных распреде \[f(x ~|~ \mu_1, \mu_2, \sigma_1, \sigma_2) = \omega_1 \cdot f_1(x~|~ \mu_1, \sigma_1) + (1 - \omega_1) \cdot f_2(x~|~ \mu_2, \sigma_2)\] -Для нормального распределения параметры связаны с теоретическими начальными моментами следующим образом: +Для нормального распределения параметры связаны с теоретическими начальными моментами следующим образом \href{https://faculty.washington.edu/yenchic/20A_stat512/Lec6_MLE.pdf}{[2]}: \[\begin{cases} \mu'_1 = \mu \\ \mu'_2 = \mu^2 + \sigma^2 \end{cases}\] @@ -57,4 +60,5 @@ \section{Пример: смесь двух нормальных распреде \[\begin{cases} \hat{\mu}^{(k)}_{1, i} = \mu_i \\ \hat{\mu}^{(k)}_{2, i} = \mu_i^2 + \sigma_i^2 \end{cases}\] +\printbibliography \end{document} diff --git a/docs/math/source/ref.bib b/docs/math/source/ref.bib index fcb45c32..5f4ff4ab 100644 --- a/docs/math/source/ref.bib +++ b/docs/math/source/ref.bib @@ -1,8 +1,10 @@ -@book{McLachlan2000, - author = {McLachlan, Geoffrey J. and Peel, David}, +@book{mclachlan2000finite, + author = {Geoffrey J. McLachlan and David Peel}, title = {Finite Mixture Models}, - publisher = {Wiley}, - year = {2000}, + series = {Wiley Series in Probability and Statistics}, + publisher = {John Wiley \& Sons, Inc.}, address = {New York}, - isbn = {978-0471006268}, + year = {2000}, + pages = {464}, + isbn = {978-0-471-00626-8}, } From 55c2497c5b4b5778888ab41b4f0216e03b83fe2d Mon Sep 17 00:00:00 2001 From: iraedeus Date: Mon, 13 Oct 2025 19:46:36 +0300 Subject: [PATCH 3/3] docs: add citations to q_function strategy note --- docs/math/q-function.pdf | Bin 33831 -> 0 bytes docs/math/q_function.pdf | Bin 0 -> 49066 bytes docs/math/source/packages.tex | 1 + docs/math/source/q_function.tex | 12 +++++++----- docs/math/source/ref.bib | 28 ++++++++++++++++++++++++++++ 5 files changed, 36 insertions(+), 5 deletions(-) delete mode 100644 docs/math/q-function.pdf create mode 100644 docs/math/q_function.pdf diff --git a/docs/math/q-function.pdf b/docs/math/q-function.pdf deleted file mode 100644 index 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zHEe*~YKVSFkU+{*?A~Mu0Jjg}RJ^Y&?`(+iun+Pug1Ni|bQk@pu>=U%9>CllCQT6} zL=k92fhu3NOB)P!CkWeKfQ3~EnN^6>Jr9Ie4#2)Fgu5(Esv=13B8=TE4DKR?9y37j zB*YIhKoJwcZp+_AZ*1wkik_bmN0bfxnx)M2X68Ixg_+WyJX0X;tW|1nGyBP6PU2$7 z{8hwBjo3G%8h@HoeZ3L$vzME$F@CxlZ6?v$XLIAHedzvSBl)t_iAIt*-$E6suWKl| zddZc3{Ch|GBZ?f_Hv@MAHJFWJBr_W%4YoLvl^Ts)ji T&7fE~SQ$8=NJ&NI#i0H#{tF%f literal 0 HcmV?d00001 diff --git a/docs/math/source/packages.tex b/docs/math/source/packages.tex index e40d2249..9f5f14cc 100644 --- a/docs/math/source/packages.tex +++ b/docs/math/source/packages.tex @@ -11,6 +11,7 @@ \setmainlanguage{russian} \setotherlanguage{english} \setmainfont{CMU Serif} +\newfontfamily{\cyrillicfonttt}{CMU Typewriter Text} \usepackage{color} \usepackage{hyperref} diff --git a/docs/math/source/q_function.tex b/docs/math/source/q_function.tex index f6982661..1f8e0e42 100644 --- a/docs/math/source/q_function.tex +++ b/docs/math/source/q_function.tex @@ -22,11 +22,11 @@ \section{Правдоподобие смеси} где $\Psi$ --- вектор параметров смеси $\omega_i, \theta_i$, $f_i$ --- $i$-ая компонента смеси. -Правдоподобием смеси распределений при наблюдаемых данных будет называться функция: +Правдоподобием смеси распределений при наблюдаемых данных будет называться функция \cite{mclachlan2000finite}{ (см. уравнение 2.2)}: \[\mathcal{L}(x ~|~ \Psi) = \prod_{i=1}^n f(x_i ~|~ \Psi) = \prod_{i=1}^n \left(\sum_{j=1}^k \omega_j \cdot f_j(x_i ~|~ \theta_j)\right)\] -Для оценки параметров достаточно максимизировать правдоподобие. В силу монотонности логарифма можно оптимизировать логарифм правдоподобия: +Для оценки параметров достаточно максимизировать правдоподобие. В силу монотонности логарифма можно оптимизировать логарифм правдоподобия \cite{mclachlan2000finite}{ (см. уравнение 2.19)}: \[\ln \mathcal{L}(x ~|~ \Psi) = \sum_{i=1}^n \ln \left(\sum_{j=1}^k \omega_j \cdot f_j(x_i~|~ \theta_j)\right)\] @@ -59,14 +59,14 @@ \section{Правдоподобие полных данных} \[\mathcal{L}_c(x, z ~|~ \Psi) = \prod_{i=1}^n \prod_{j=1}^k (\omega_j \cdot f_j(x_i~|~ \theta_j))^{z_{ij}}\] -Стандартно, работаем с логарифмом правдоподобия: +Стандартно, работаем с логарифмом правдоподобия \cite{mclachlan2000finite}{ (см. уравнение 2.26)}: \[\ln \mathcal{L}_c (x, z ~|~ \Psi) = \sum_{i=1}^n \sum_{j=1}^k z_{ij} \cdot (\ln \omega_j + \ln f_j(x_i ~|~ \theta_j))\] \section{Q-функция} -Однако в обычных условиях мы не знаем $z_{ij}$. Пусть $Z_{ij}$ --- случайные величины соответствующие $z_{ij}$. Тогда введём Q-функцию: +Однако в обычных условиях мы не знаем $z_{ij}$. Пусть $Z_{ij}$ --- случайные величины соответствующие $z_{ij}$. Тогда введём Q-функцию \cite{mclachlan2000finite}{ (см. уравнение 2.27)}: \[Q(\Psi ~|~ \Psi^{(k)}) = \mathbb{E}_{Z | x, \Psi^{(k)}}[\ln \mathcal{L}_c(x, Z ~|~ \Psi)]\] @@ -86,6 +86,8 @@ \section{Q-функция} \[Q(\Psi ~|~ \Psi^{(k)}) = \sum_{i=1}^n \sum_{j=1}^k \gamma_{ij}^{(k)} \cdot (\ln \omega_j + \ln f_j(x_i ~|~ \theta_j))\] +Стоит отметить, что итерация EM алгоритма путём оптимизации Q-функции эквивалентна оптимизации логарифма правдоподобия \cite{Dempster}. + \section{Покомпонентная оптимизация Q-функции} Имеем: @@ -101,5 +103,5 @@ \section{Покомпонентная оптимизация Q-функции} \[\sum_{i=1}^n \sum_{j=1}^k \gamma_{ij}^{(k)} \ln f_j(x_i ~|~ \theta_j) = \sum_{j=1}^k \sum_{i=1}^n \gamma_{ij}^{(k)}\ln f_j(x_i ~|~ \theta_j)\] Очевидно что если рассмотреть для каждого индекса $j$ то получаем функции независящие друг от друга. - +\printbibliography \end{document} diff --git a/docs/math/source/ref.bib b/docs/math/source/ref.bib index 5f4ff4ab..f3ab0566 100644 --- a/docs/math/source/ref.bib +++ b/docs/math/source/ref.bib @@ -8,3 +8,31 @@ @book{mclachlan2000finite pages = {464}, isbn = {978-0-471-00626-8}, } + + +@article{Dempster, + author = {Dempster, A. P. and Laird, N. M. and Rubin, D. B.}, + title = {Maximum Likelihood from Incomplete Data Via the EM Algorithm}, + journal = {Journal of the Royal Statistical Society: Series B + (Methodological)}, + volume = {39}, + number = {1}, + pages = {1-22}, + year = {2018}, + month = {12}, + abstract = {A broadly applicable algorithm for computing maximum likelihood + estimates from incomplete data is presented at various levels of + generality. Theory showing the monotone behaviour of the + likelihood and convergence of the algorithm is derived. Many + examples are sketched, including missing value situations, + applications to grouped, censored or truncated data, finite + mixture models, variance component estimation, hyperparameter + estimation, iteratively reweighted least squares and factor + analysis.}, + issn = {0035-9246}, + doi = {10.1111/j.2517-6161.1977.tb01600.x}, + url = {https://doi.org/10.1111/j.2517-6161.1977.tb01600.x}, + eprint = { + https://academic.oup.com/jrsssb/article-pdf/39/1/1/49117094/jrsssb_39_1_1.pdf + }, +}