From be4e4ace4c56ed9baaaaddb2ab99800b6a8f2dea Mon Sep 17 00:00:00 2001 From: Cezary Pukownik Date: Fri, 29 May 2020 11:49:23 +0200 Subject: [PATCH] =?UTF-8?q?Poprawiono=20prac=C4=99=20zgodnie=20z=20wytyczn?= =?UTF-8?q?ymi=20promotora?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- docs/document.aux | 27 ++-- docs/document.fdb_latexmk | 54 +++++-- docs/document.fls | 55 +++++-- docs/document.log | 206 +++++++++++++----------- docs/document.pdf | Bin 569967 -> 636582 bytes docs/document.synctex.gz | Bin 241740 -> 243750 bytes docs/document.tex | 193 ++++++++++++---------- docs/document.toc | 10 +- docs/images/naural_model_multi_ver2.png | Bin 0 -> 60625 bytes docs/images/naural_model_one_ver2.png | Bin 0 -> 38478 bytes docs/images/neural_net_1_ver2.png | Bin 0 -> 101593 bytes docs/images/plots.py | 108 +++++++++++++ docs/images/tanh.png | Bin 0 -> 11044 bytes 13 files changed, 425 insertions(+), 228 deletions(-) create mode 100644 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sobie, w pracy dyplomow \section{Regresja liniowa} Podstawą wszystkich sieci neuronowych jest regresja liniowa. W statystyce wykorzystywana, aby wyjaśnić liniowe zalezności między zmiennymi. - Prosty model regresji liniowej dla jednej zmiennej można opisać wzorem~\cite{statystyka} + Prosty model regresji liniowej dla jednej zmiennej można opisać wzorem % \footnote{Statystyka, Mieczysław Sobczyk s.179} \[y = ax+b+\epsilon,\] @@ -197,7 +199,7 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \item $x$ jest to zmienna objaśniająca, \item $a$ jest parametrem modelu, \item $b$ jest wyrazem wolnym modelu, - \item $\epsilon$ jest składnikiem losowym. + \item $\epsilon$ jest składnikiem losowym \cite{statystyka}. \end{itemize} \medskip @@ -213,46 +215,47 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \label{fig:linreg} \end{figure} - Wartość zmiennej objaśnianej $y$ można również opisać za pomocą wielu zmiennych objaśniająych. Wtedy dla zmiennych objaśniającej $x_1, x_2, ... , x_n \in \mathbb{R}$ szukamy parametrów $\theta_1, \theta_2, ... ,\theta_n \in \mathbb{R}$. Otrzymany w ten sposób model nazywany jest również hipotezą i oznaczamy go $h(x)$. + Wartość zmiennej objaśnianej $y$ można również opisać za pomocą wielu zmiennych objaśniająych. Wtedy dla zmiennych objaśniającej $x_1, x_2, ... , x_p \in \mathbb{R}$ szukamy parametrów $\theta_1, \theta_2, ... ,\theta_p \in \mathbb{R}$, gdzi $p \in \mathbb{N}$ jest liczbą cech. Otrzymany w ten sposób model nazywany jest również hipotezą i oznaczamy go $h(x)$. - \[h(x) = b + \theta_1x_2 + \theta_2x_2 + ... + \theta_nx_n + \epsilon = b + \sum_{i=1}^{n} \theta_ix_i + \epsilon.\] + \[h(x) = b + \theta_1x_2 + \theta_2x_2 + ... + \theta_px_p + \epsilon = b + \sum_{i=1}^{p} \theta_ix_i + \epsilon.\] Rysunek~\ref{fig:linreg} przedstawia przykładowy model regresji liniowej jednej zmiennej, dopasowany do zbioru. \section{Uczenie modelu} - Celem uczenia modelu jest znalezienie ogólnych parametrów, aby model dla wartości wejściowych $x$ zwracał wartości predykcji $\hat{y}$ najlepiej opisujące całe zjawisko według pewnego kryterium. Formalnie, aby suma wszystkich różnic między predykcją a rzeczywistością była najmniejsza. + Celem uczenia modelu jest znalezienie ogólnych parametrów, aby model dla wartości wejściowych $x$ zwracał wartości predykcji $\hat{y}$ najlepiej opisującej całe zjawisko według pewnego kryterium. Formalnie, aby suma wszystkich różnic między predykcją, a rzeczywistością była najmniejsza. \[ - \textoperatorname{błąd} = \sum_{i=1}^m | \textoperatorname{predykcja} - \textoperatorname{rzeczywistość} | - \] , gdzie $m \in \mathbb{N}$ jest wielkością zbioru danych jakim dysponujemy. Minimalizując bład dla modelu jesteśmy wstanie znaleźć przybliżenie funkcji $h(x)$. + \textoperatorname{błąd} = \sum_{i=1}^m | \textoperatorname{predykcja} - \textoperatorname{rzeczywistość} |, + \] + gdzie $n \in \mathbb{N}$ jest wielkością zbioru danych jakim dysponujemy. Minimalizując bład dla modelu jesteśmy wstanie znaleźć przybliżenie funkcji $h(x)$. \subsection{Funkcja kosztu} - W tym celu używa się funkcji $J_\theta(h)$, która zwraca wartość błędu między wartościami $h(x)$ oraz $y$ dla wszystkich obserwacji. Taka funkcja nazywana jest funkcją kosztu (\textit{cost function}). + W tym celu używa się funkcji $J_\theta(h)$, która zwraca wartość błędu między wartościami $h(x)$ oraz $y$ dla wszystkich obserwacji. Taka funkcja nazywana jest funkcją kosztu (\textit{ang. cost function}). - Dla przykładu regresji liniowej funkcją kosztu może być błąd średniokwadratowy (\textit{mean squared error}). Wtedy funkcja kosztu przyjmuje postać: - \[ J_\theta(h) = \frac{1}{m}\sum_{i=1}^{m}(y_i-h(x_i))^2 \]. + Dla przykładu regresji liniowej funkcją kosztu może być błąd średniokwadratowy (\textit{ang. mean squared error}). Wtedy funkcja kosztu przyjmuje postać: + \[ J_\theta(h) = \frac{1}{m}\sum_{i=1}^{m}(y_i-h(x_i))^2.\] Przy zdefiniowanej funkcji kosztu proces uczenia sprowadza się do znalezienia takich parametrów funkcji $h(x)$, aby funkcja kosztu była najmniejsza. Jest to problem optymalizacyjny sprowadzający się do znalezienia globalnego minimum funkcji. \subsection{Znajdowanie minimum funkcji} - Aby znaleźć minimum funkcji $f$ możemy skorzystać z analizy matematycznej. Wiemy, że jeśli funkcja $f$ jest różniczkowalna to funkcja może przyjmować minimum lokalne, gdy $f'(x_0) = 0$ dla pewnego $x_0$ z dzieniny funkcji $f$. Dodatkowo jeśli istanieje otoczenie punktu $x_0$, że dla wszystkich punktów z tego otoczenia spełniona jest nierówność: - \[ f(x)>f(x_0) \] + Aby znaleźć minimum funkcji $f$ możemy skorzystać z analizy matematycznej. Wiemy, że jeśli funkcja $f$ jest różniczkowalna, to funkcja może przyjmować minimum lokalne, gdy $f'(x_0) = 0$ dla pewnego $x_0$ z dzieniny funkcji $f$. Dodatkowo jeśli istanieje otoczenie punktu $x_0$, że dla wszystkich punktów z tego otoczenia spełniona jest nierówność: + \[ f(x)>f(x_0), \] to znaleziony punkt $x_0$ jest minimum lokalnym. - W teorii należałoby zatem wybrać taką funkcję kosztu, aby była różniczkowalna. Obliczyć równanie $J_\theta'(h)=0$, następnie dla otrzymanych wyników sprawdzić powyższą nierówność oraz wybrać najmniejszy wynik ze wszystkich. \cite{analiza_mat} W praktyce rozwiązanie takie równania ze względu na jego złożoność może się okazać niewykonalne. Aby rozwiązać ten problem powstały inne metody, które pozwalają szukać ekstremów funkcji, jednak nigdy nie będziemy mieli pewności, że otrzymany wynik jest minimum globalnym funkcji kosztu. + W teorii należałoby zatem wybrać taką funkcję kosztu, aby była różniczkowalna. Rozwiązać równanie $J_\theta'(h)=0$, następnie dla otrzymanych wyników sprawdzić powyższą nierówność oraz wybrać najmniejszy wynik ze wszystkich \cite{analiza_mat}. W praktyce rozwiązanie takiego równania ze względu na jego złożoność może się okazać niewykonalne. Aby rozwiązać ten problem powstały inne metody, które pozwalają szukać ekstremów funkcji, jednak nigdy nie będziemy mieli pewności, że otrzymany wynik jest minimum globalnym funkcji kosztu. % \footnote{źródlo: Analiza matematyczna, Krysicki Włodarski, s.187 }. \subsection{Metody gradientowe} - Metody gradientowe (\textit{gradient descent}) są to iteracyjne algorytmy służące do znajdowania minimum funkcji. Aby móc skorzystać z metod gradientowych analizowana funkcja musi być ciągła oraz różniczkowalna. Sposób działania ich można intuicyjnie opisać w nastepujących krokach. + Metody gradientowe (\textit{ang. gradient descent}) są to iteracyjne algorytmy służące do znajdowania minimum funkcji. Aby móc skorzystać z metod gradientowych analizowana funkcja musi być ciągła oraz różniczkowalna. Sposób ich działania można intuicyjnie opisać w nastepujących krokach. 1. Wybierz punkt początkowy. 2. Oblicz kierunek, w którym funkcja maleje. - 3. Przejdź do kolejnego punktu zgodnie obliczonym kierunkiem o pewną małą odległość. + 3. Przejdź do kolejnego punktu zgodnie obliczonym kierunkiem o pewną małą wartość. 4. Powtarzamy, aż osiągniemy minimum funkcji. @@ -268,13 +271,13 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \end{figure} \medskip - Dla funkcji $h(x)$ należy ustalić wartość początkową $\Theta_0$ dla wszystkich parametrów $\theta_1$ ... $\theta_n$. + Dla funkcji $h(x)$ należy ustalić wartość początkową $\Theta_0$ dla wszystkich parametrów $\theta_1$ ... $\theta_p$. - \[ \Theta_0 = \left[ \theta_1, \theta_2, ... ,\theta_n \right] \] + \[ \Theta_0 = \mathbf{\boldsymbol{\left[ \theta_1, \theta_2, ... ,\theta_n \right]}}. \] Następnie policzyć wszystkie pochodne cześciowe $\frac{\partial J_\theta(h)}{\partial \theta_i}$. Otrzymamy w ten sposób gradient $\nabla J_\theta(h)$, gdzie \[ - \nabla J_\theta(h) = \left[ \frac{\partial J_\theta(h)}{\partial \theta_1}, \frac{\partial J_\theta(h)}{\partial \theta_2}, ... , \frac{\partial J_\theta(h)}{\partial \theta_n} \right] + \nabla J_\theta(h) = \mathbf{\boldsymbol{\left[ \frac{\partial J_\theta(h)}{\partial \theta_1}, \frac{\partial J_\theta(h)}{\partial \theta_2}, ... , \frac{\partial J_\theta(h)}{\partial \theta_n} \right]}}. \] Następnie obliczyć element $\Theta_{k+1}$ ze wzoru @@ -282,10 +285,10 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \[ \Theta_{k+1} = \Theta_{k} - \alpha\nabla J_\theta(h), \] - gdzie $\alpha \in \mathbb{R}$ jest współczynnikiem uczenia (\textit{learning rate}). Proces ten należy powtarzać do pewnego momentu. Najczęśćiej z góry określoną liczbę razy lub do momentu, gdy uzysk funkcji kosztu spowodowany nastepną iteracją jest mniejszy niż ustalona wartość. Otrzymany w ten sposób wektor parametrów $\Theta_k$ jest wynikiem algorytmu. \cite{survay} + gdzie $\alpha \in \mathbb{R}$ jest współczynnikiem uczenia (\textit{ang. learning rate}), a $k \in \mathbb{N}$ jest kolejną iteracją algorytmu. Proces ten należy powtarzać do pewnego momentu. Najczęściej z góry określoną liczbę razy lub do momentu, gdy uzysk funkcji kosztu spowodowany nastepną iteracją jest mniejszy niż ustalona wartość. Otrzymany w ten sposób wektor parametrów $\Theta_k$ jest wynikiem algorytmu \cite{survay}. % \footnote{Deep Learning techniques for music geneation - A survay s.44} - Wykorzystując metody gradientowe otrzymujemy wyuczony model. Parametry $\theta_i$ modelu $h(x)$ zostały ustalone w taki sposób, aby błąd między predykcją a rzeczywistością był najmniejszy. + Wykorzystując metody gradientowe otrzymujemy wyuczony model. Parametry $\theta_i$ modelu $h(x)$ zostały ustalone w taki sposób, aby błąd między predykcją, a rzeczywistością był najmniejszy. \section{Regresja liniowa jako model sieci neuronowej} \label{section:linreg} @@ -294,72 +297,74 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \begin{figure}[!htb] \centering - \includegraphics[width=8cm]{images/naural_model_one.png} + \includegraphics[width=8cm]{images/naural_model_one_ver2.png} \caption{Regresja liniowa jako model sieci neuronowej} \label{fig:neural_model_one} \end{figure} - Każdy węzeł z lewej strony reprezentuje zmienną objaśniającą $x_i$. Połączenia nazywane są wagami i reprezentują one parametry $\theta_i$. Węzeł z prawej strony oznaczony jako $\hat{y}$ jest sumą iloczynów wag oraz wartości węzłów z prawej strony. Wtedy + Każdy węzeł z lewej strony reprezentuje zmienną objaśniającą $x_i$. Połączenia nazywane są wagami (\textit{ang. weights}) i reprezentują one parametry $\theta_i$. Węzeł z prawej strony oznaczony jako $\hat{y}$ jest sumą iloczynów wag oraz wartości węzłów z prawej strony. Wtedy \[ \hat{y} = - \begin{bmatrix} + \mathbb{\boldsymbol{\begin{bmatrix} 1 \\ x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \begin{bmatrix} - b & \theta_1 & \theta_2 & \dots & \theta_n - \end{bmatrix} + b & \theta_1 & \theta_2 & \dots & \theta_p + \end{bmatrix} }} = b + x_1\theta_1 + x_2\theta_2 + \dots + x_n\theta_n = b + \sum_{i=1}^n x_i\theta_i \] - co jest równoważne omawianemu modelowi regresji liniowej. Węzły sieci nazywane są neuronami, a wyraz wolny modelu $b$ nazywany jest biasem (\textit{bias}). + co jest równoważne omawianemu modelowi regresji liniowej. Węzły sieci nazywane są neuronami, a wyraz wolny modelu $b$ nazywany jest biasem (\textit{ang. bias}). - W łatwy sposób możemy rozbudować ten model do regresji liniowej wielu zmiennych. Predykcją modelu nie będzie jak do tej pory jedna wartość $\hat{y}$, tylko wektor wartości $\hat{y_1}, \hat{y_2}, \dots , \hat{y_m}$, który oznaczać bedziemy jako $\hat{Y}$. Model ten zostal przedstawiony na rysunku~\ref{fig:neural_model_multi}. + W łatwy sposób możemy rozbudować ten model do regresji liniowej wielu zmiennych. Predykcją modelu nie będzie jak do tej pory jedna wartość $\hat{y}$, tylko wektor wartości $\hat{y_1}, \hat{y_2}, \dots , \hat{y_q}$, który oznaczać bedziemy jako $\hat{Y}$. Model ten został przedstawiony na rysunku~\ref{fig:neural_model_multi}. \begin{figure}[!htb] \centering - \includegraphics[width=8cm]{images/naural_model_multi.png} - \caption{Regresja liniowa wielu zmiennych jako model sieci nauronowe.j} + \includegraphics[width=8cm]{images/naural_model_multi_ver2.png} + \caption{Regresja liniowa wielu zmiennych jako model sieci nauronowej} \label{fig:neural_model_multi} \end{figure} - Dla uogólnienia pojedyncze wagi modelu zapisywać będę jako $w_{nm}$, natomiast macierz wag jako $W$. Algebricznie zapisalibyśmy ten model jako + Dla uogólnienia pojedyncze wagi modelu zapisywać będę jako $w_{pq}$, natomiast macierz wag jako $W$. Algebricznie zapisalibyśmy ten model jako \[ - \begin{bmatrix} + \mathbb{\boldsymbol{\begin{bmatrix} 1 & 1 & \dots & 1\\ - x_{11} & x_{12} & \dots & x_{1m} \\ - x_{21} & x_{22} & \dots & x_{2m} \\ + x_{11} & x_{12} & \dots & x_{1q} \\ + x_{21} & x_{22} & \dots & x_{2q} \\ \vdots & \vdots & \ddots & \vdots \\ - x_{n1} & x_{12} & \dots & x_{nm} \\ + x_{p1} & x_{p2} & \dots & x_{pq} \\ \end{bmatrix} \begin{bmatrix} - b_1 & w_{11} & w_{12} & \dots & w_{1n} \\ - b_2 & w_{21} & w_{22} & \dots & w_{2n} \\ + b_1 & w_{11} & w_{12} & \dots & w_{1p} \\ + b_2 & w_{21} & w_{22} & \dots & w_{2p} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ - b_m & w_{m1} & w_{m2} & \dots & w_{mn} \\ - \end{bmatrix} + b_q & w_{q1} & w_{q2} & \dots & w_{qp} \\ + \end{bmatrix} }} = + \mathbb{\boldsymbol{ \begin{bmatrix} - h_1(x) \\ h_2(x) \\ \vdots \\ h_m(x) - \end{bmatrix} + h_1(x) \\ h_2(x) \\ \vdots \\ h_q(x) + \end{bmatrix} }} = + \mathbb{\boldsymbol{ \begin{bmatrix} - \hat{y_1} \\ \hat{y_2} \\ \vdots \\ \hat{y_m} - \end{bmatrix} + \hat{y_1} \\ \hat{y_2} \\ \vdots \\ \hat{y_q} + \end{bmatrix} }} \] \[ - b+XW = \hat{Y} + b+XW = \hat{Y}, \] - gdzie $n$ jest liczbą zmiennych niezależnych, $m$ jest liczbą zmiennych zaleznych, $X$ jest rozszerzonym do macierzy o rozmiarach $m$ x $n$ wektorem zmiennych objaśniających, w taki sposób że $x_{i1} = x_{i2} = \dots = x_{in}$ dla $i = 1, 2, ..., m$, $W$ jest macierzą wag o rozmiarach $n$ x $m$, natomiast $b$ jest sumą wyrazów wolnych $b_1, ... ,b_m$. Możemy zauważyć, że model dla wielu zmiennych jest wieloma modelami dla jednej zmiennej, gdzie każdy model operuje na tych samych danych wejściowych. Taki model może być uznany za sieć neuronową i nazywany jest perceptronem. + gdzie $p$ jest liczbą zmiennych niezależnych, $q$ jest liczbą zmiennych zaleznych, $X$ jest rozszerzonym do macierzy o rozmiarach $q$ x $p$ wektorem zmiennych objaśniających, w taki sposób że $x_{i1} = x_{i2} = \dots = x_{ip}$ dla $i = 1, 2, ..., q$, $W$ jest macierzą wag o rozmiarach $p$ x $q$, natomiast $b$ jest sumą wyrazów wolnych $b_1, ... ,b_q$. Możemy zauważyć, że model dla wielu zmiennych jest wieloma modelami dla jednej zmiennej, gdzie każdy model operuje na tych samych danych wejściowych. Taki model może być uznany za sieć neuronową i nazywany jest perceptronem. \section{Funkcje aktywacji} - Omawiany model służy rozwiązywaniu problemu regresji, ponieważ wartości predykcji nie są uregulowane i mogą przyjmować wartości z $\mathbb{R}$. W celu przekształcenia tego modelu, aby móc go wykorzystać do rozwiązania problemu klasyfikacji, należy dodatkowo na otrzymanym wektorze $\hat{Y}$ wykonać pewną funkcję, która przekształci wynik. W tym celu używamy funkcji aktywacji (\textit{activation function}). Istnieje wiele różnych funkcji aktywacji, a każda posiada inną charakterystykę i wpływ na model. Najpopularnijeszą grupą fukcji są funkcje sigmoidalne (\textit{sigmoid functions}). Jedną z nich jest funkcja logistyczna (\textit{logistic curve}) o wzrorze + Omawiany model służy rozwiązywaniu problemu regresji, ponieważ wartości predykcji nie są uregulowane i mogą przyjmować wartości z $\mathbb{R}$. W celu przekształcenia tego modelu, aby móc go wykorzystać do rozwiązania problemu klasyfikacji, należy dodatkowo na otrzymanym wektorze $\hat{Y}$ wykonać pewną funkcję, która przekształci wynik. W tym celu używamy funkcji aktywacji (\textit{ang. activation function}). Istnieje wiele różnych funkcji aktywacji, a każda posiada inną charakterystykę i wpływ na model. Najpopularnijeszą grupą fukcji są funkcje sigmoidalne (\textit{ang. sigmoid functions}). Jedną z nich jest funkcja logistyczna (\textit{ang. logistic curve}) \[ \sigma(x) = \frac{1}{1+e^{-x}} \] @@ -372,57 +377,58 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \label{fig:sigmoid} \end{figure} - Funkcja logistyczna ma pewne użyteczne właściwości, które pozwolą kontrolować wartości węzłów oraz rzutować wartości z całego $\mathbb{R}$ do wartości z przedziału $(0,1)$. Dzięki tej właściwości funkcja logistyczna jest często używana, aby otrzymać prawdopodobieństwo wystąpienia pewnego zdarzenia. Dodatkowo funkcja logistyczna szybko przyjmuje wartości skrajne, co oznacza że dla bardzo dużych wartości ujemnych i bardzo dużych wartości dodatnich funkcja staje się mało wrażliwa na zmiany wartości wraz ze zmianą wartości argumentu. \cite{deep_learning} % \footnote{Deep Learning Book, s.66} + Funkcja logistyczna ma pewne użyteczne właściwości, które pozwolą kontrolować wartości węzłów oraz rzutować wartości z całego $\mathbb{R}$ do wartości z przedziału $(0,1)$. Dzięki tej właściwości funkcja logistyczna jest często używana, aby otrzymać prawdopodobieństwo wystąpienia pewnego zdarzenia. Dodatkowo funkcja logistyczna szybko przyjmuje wartości skrajne, co oznacza że dla bardzo dużych wartości ujemnych i bardzo dużych wartości dodatnich funkcja staje się mało wrażliwa na zmiany wartości wraz ze zmianą wartości argumentu + \cite{deep_learning}. + + % \footnote{Deep Learning Book, s.66} W ten sposób możemy w łatwy sposób zmienić model regresji liniowej na model regresji logistycznej. - \[ \sigma(b+XW) = \hat{Y} \] - W dalszych cześciach pracy, kiedy bedę używał funckji aktywacji nie wskazując na konkretną funckcję, będe wykorzystywał oznaczenie $AF(x)$. + W dalszych cześciach pracy, kiedy bedę używał funkcji aktywacji nie wskazując na konkretną funkcję, będe wykorzystywał oznaczenie $AF(x)$. - \section{Głębokie sieci neuronowe} + \section{Wielowarstwowe sieci neuronowe} - Model omawiany wcześniej może posłużyć jako podstawowy element do budowania bardziej skomplikowanych modeli. Aby to zrobić, należy potraktować otrzymany wektor $\hat{Y}$ jako wektor wejściowy do następngo podstawowego modelu. Składając ze sobą wiele perceptronów w jeden model, tworzymy warstwy (\textit{layers}) sieci neuronowej. + Model omawiany wcześniej może posłużyć jako podstawowy element do budowania bardziej skomplikowanych modeli. Aby to zrobić, należy potraktować otrzymany wektor $\hat{Y}$ jako wektor wejściowy do następngo podstawowego modelu. Składając ze sobą wiele perceptronów w jeden model, tworzymy warstwy (\textit{ang. layers}) sieci neuronowej. Wyróżniamy trzy rodzaje warstw: \begin{itemize} - \item warstwę wejściową (\textit{input layer}), która jest pierwszą warstwą modelu, - \item warstwę wyjściową (\textit{output layer}), która jest ostatnią wartstwą modelu, - \item wartwy ukryte (\textit{hidden layer}), które są warstwami pomiędzy warstwą wejsciową oraz wyjściową. + \item warstwę wejściową (\textit{ang. input layer}), która jest pierwszą warstwą modelu, + \item warstwę wyjściową (\textit{ang. output layer}), która jest ostatnią wartstwą modelu, + \item wartwy ukryte (\textit{ang. hidden layer}), które są warstwami pomiędzy warstwą wejsciową oraz wyjściową. \end{itemize} - Na rysunku~\ref{fig:neural_net_1} przedstawiono przykład posiadający warswę wejściową, dwie wartswy ukryte oraz wartswę wyjsciową. + Na rysunku~\ref{fig:neural_net_1} przedstawiono sieć posiadającą warstwę wejściową, dwie wartswy ukryte oraz wartswę wyjsciową. \begin{figure}[!htb] \centering - \includegraphics[width=8cm]{images/neural_net_1.png} - \caption{Przykład modelu sieci neuronowej} + \includegraphics[width=8cm]{images/neural_net_1_ver2.png} + \caption{Przykład modelu wielowarstwowej sieci neuronowej} \label{fig:neural_net_1} \end{figure} - Tego typu modele są głębokimi sieciami neuronowymi (\textit{deep neural networks}). Istnieje wiele różnich architektur głębokich sieci neuronowych, które wykorzystują te podstawowe koncepcje i rozszerzają je o dodatkowe warstwy, połącznia, funkcje aktywacji czy neurony o specjalnych właściwościach. + Tego typu modele są głębokimi sieciami neuronowymi (\textit{ang. deep neural networks}). Istnieje wiele różnich architektur głębokich sieci neuronowych, które wykorzystują te podstawowe koncepcje i rozszerzają je o dodatkowe warstwy, połącznia, funkcje aktywacji czy neurony o specjalnych właściwościach. \subsection{Jednokierunkowe sieci neuronowe} \label{section:feedforeward} - Jednokierunkowe sieci neuronowe (\textit{feedforward neural networks}) są to najprostrze sieci neuronowe, które wprost czerpią z omówionych wcześniej podstawowych wartsw. Możemy się również spotkać z nazwą wielowarstwowy perceptron (\textit{multi layer perceptron - MLP}) ze względu na fakt, że jest zbudowany z wielu perceptronów zaprezentoanych w części~\ref{section:linreg}. Działają one w taki sposób, że zasila się je danymi do warstwy wejściowej, następnie sukcesywnie wykonuje się obliczenia do momentu dotarcia do końca sieci. + Jednokierunkowe sieci neuronowe (\textit{ang. feedforward neural networks}) są to najprostrze sieci neuronowe, które wprost czerpią z omówionych wcześniej podstawowych wartsw. Możemy się również spotkać z nazwą wielowarstwowy perceptron (\textit{ang. multi layer perceptron - MLP}) ze względu na fakt, że jest zbudowany z wielu perceptronów zaprezentoanych w rozdziale~\ref{section:linreg}. Działają one w taki sposób, że zasila się je danymi do warstwy wejściowej, następnie sukcesywnie wykonuje się obliczenia do momentu dotarcia do końca sieci. Każdy krok z warstwy $k-1$ do warstwy $k$ obliczany jest zgodnie ze wzorem \cite{survay} %\footnote{Deep Learning techniques for music geneation - A survay s.63} - \[ - X_k = AF(b_k + W_kX_{k-1}) + X_k = AF(b_k + W_kX_{k-1}). \] \subsubsection{Propagacja wsteczna błędu} \label{section:backpropagation} - Kiedy używamy jednokierunkowych sieci neuronowych, zasilamy je danymi wejściowymi $x$ ostatecznie otrzymując predykcję $\hat{y}$. Taki sposób działania nazywa się propagcją wprzód (\textit{foreward propagation}). Podczas uczenia sieci kontynuuje się ten proces obliczając koszt $J(h)$. Propagacja wsteczna (\textit{back-propagation}) pozwala na przepływ informacji od funkcji kosztu wstecz sieci neuronowej, aby ostatecznie obliczyć gradient. Zasada działania algorytmu propagacji wsteczniej błędu polega na sukcesywnym aktualizowaniu wag i biasów oraz przesyłaniu wstecz po warstwach sieci. Dzięki temu jesteśmy wstanie wyuczyć sieć oraz obliczyć optymalne wagi i biasy dla całej sieci neuronowej. + Kiedy używamy jednokierunkowych sieci neuronowych, zasilamy je danymi wejściowymi $x$ ostatecznie otrzymując predykcję $\hat{y}$. Taki sposób działania nazywa się propagcją wprzód (\textit{ang. foreward propagation}). Podczas uczenia sieci kontynuuje się ten proces obliczając koszt $J(h)$. Propagacja wsteczna (\textit{ang. back-propagation}) pozwala na przepływ informacji od funkcji kosztu wstecz sieci neuronowej, aby ostatecznie obliczyć gradient. Zasada działania algorytmu propagacji wsteczniej błędu polega na sukcesywnym aktualizowaniu wag i biasów oraz przesyłaniu wstecz po warstwach sieci. Dzięki temu jesteśmy wstanie wyuczyć sieć oraz obliczyć optymalne wagi i biasy dla całej sieci neuronowej. \subsection{Autoencodery} \label{section:autoencoder} - Autoencoder jest szczególnym przypadkiem sieci neuronowej. Posiada jedną warstwę ukrytą, a rozmiar wartwy wejściowej musi być równy rozmiarowi wartwy wyjściowej, tworząc w ten sposób symetryczną sieć, której kształt przypomina klepsydrę. Przykład autoencodera przedstawiono na rysunku ~\ref{fig:autoencoder}. + Autoencoder jest szczególnym przypadkiem sieci neuronowej. Posiada jedną warstwę ukrytą, a rozmiar wartwy wejściowej musi być równy rozmiarowi wartwy wyjściowej, tworząc w ten sposób symetryczną sieć, której kształt przypomina klepsydrę. Przykład autoencodera przedstawiono na rysunku~\ref{fig:autoencoder}. \begin{figure}[!htb] \centering @@ -431,17 +437,17 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \label{fig:autoencoder} \end{figure} - Podczas uczenia autoencodera przedstawia się dane wejściowe jako cel. W ten sposób ta architektura stara się odtworzyć funckje identyczności. Zadanie nie jest trywialne jak mogło by się zdawać, ponieważ zazwyczaj ukryta warstwa jest mniejszego rozmiaru niż dane wejściowe. Z tego względu autoencoder jest zmuszony do wydobycia istotnych cech danych wejściowych, skompresowania, a następnie jak najwierniejszego ich odtworzenia. Część kompresująca dane nazywana jest encoderem, natomiast część dekompresująca decoderem. - Wektor cech, które zostały odkryte przez autoencoder nazywane są zmiennymi utajnionymi (\textit{latent variables}). Zarówno encoder jak i dekoder można wyodrębnić z autoencodera i wykorzystywać go jako osobną sieć neuronową. + Podczas uczenia autoencodera przedstawia się dane wejściowe jako cel. W ten sposób ta architektura stara się odtworzyć funkcje identycznościowe. Zadanie nie jest trywialne jak mogło by się zdawać, ponieważ zazwyczaj ukryta warstwa jest mniejszego rozmiaru niż dane wejściowe. Z tego względu autoencoder jest zmuszony do wydobycia istotnych cech danych wejściowych, skompresowania, a następnie jak najwierniejszego ich odtworzenia. Część kompresująca dane nazywana jest encoderem, natomiast część dekompresująca decoderem. + Cechy, które zostały odkryte przez autoencoder nazywane są zmiennymi utajnionymi (\textit{ang. latent variables}). Zarówno encoder jak i dekoder można wyodrębnić z autoencodera i wykorzystywać go jako osobną sieć neuronową. Ciekawą cechą decodera jest jego generatywny charakter, ponieważ dostarczając zupełnie nowe informacje jako zmienne wejściowe, decoder odtworzy je na podobieństwo danych, na których został nauczony. \subsection{Rekurencyjne sieci neuronowe} \label{section:rnn} - Rekurencyjne sieci neuronowe (\textit{recurrent neural networks - RNN}) w uproszczeniu są to MLP posiadające pamięć. Wykorzystywane są do analizawania i przewidywania sekwencji wartości uporządkowanych w czasie. Rekurencyjnych sieci neuronowe znalazły zastosowanie w przetwarzaniu języka natutralnego, np. tłumaczenia na różne języki świata. Potrafią poradziac sobie z różnej długości sekwencjami od krótkich zawierających kilka elementów do bardzo długich jak próbki audio, czy tekst zawierający dziesiątki tysięcy kroków czasu. + Rekurencyjne sieci neuronowe (\textit{ang. recurrent neural networks; RNN}) w uproszczeniu są to MLP posiadające pamięć. Wykorzystywane są do analizawania i przewidywania sekwencji wartości uporządkowanych w czasie. Rekurencyjne sieci neuronowe znalazły zastosowanie w przetwarzaniu języka natutralnego, np. tłumaczenia na różne języki świata. Potrafią poradzić sobie z różnej długości sekwencjami od krótkich zawierających kilka elementów do bardzo długich jak próbki audio, czy tekst zawierający dziesiątki tysięcy kroków czasu. - Rekurencyjne sieci neuronowe działają podobnie do omawianych w sekcji~\ref{section:feedforeward} sieci jednokierunkowych z tym wyjątkiem, że kierunek przepływu informacji płynie również wstecz sieci. Jeden neuron sieci RNN otrzymuje dane wejściowe $x_{(t)}$, wytwarza dane wyjściowe $y_{(t)}$, a następnie wysyła te dane wyjściowe z powrotem do samego siebie. W ten sposób neuron RNN posiada dwa wejścia $x_{(t)}$ oraz $y_{(t-1)}$. Możemy również zaprezentować sieć RNN w postaci udwiniętej w czasie (\textit{unrolled through time}). + Rekurencyjne sieci neuronowe działają podobnie do omawianych w sekcji~\ref{section:feedforeward} sieci jednokierunkowych z tym wyjątkiem, że kierunek przepływu informacji płynie również wstecz sieci. Jeden neuron sieci RNN otrzymuje dane wejściowe $x_{(t)}$, wytwarza dane wyjściowe $y_{(t)}$, a następnie wysyła te dane wyjściowe z powrotem do samego siebie. W ten sposób neuron RNN posiada dwa wejścia $x_{(t)}$ oraz $y_{(t-1)}$. Możemy również zaprezentować sieć RNN w postaci odwiniętej w czasie (\textit{ang. unrolled through time}). \begin{figure}[!htb] \centering @@ -457,12 +463,12 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \] - Aby wytrenować sieć neuronową stosuje się propagację wsteczną w czasie (\textit{backpropagation through time - BPTT}). Polega ona na odwinięciu sieci RNN, a następnie zastosowania zwykłej metody wstecznej propagacji. \cite{handson} %\footnote{Hands-on machine learning with scikit-learn, keras and TensorFlow s.497} + Aby wytrenować sieć neuronową stosuje się propagację wsteczną w czasie (\textit{ang. backpropagation through time; BPTT}). Polega ona na odwinięciu sieci RNN, a następnie zastosowania zwykłej metody wstecznej propagacji\cite{handson}. %\footnote{Hands-on machine learning with scikit-learn, keras and TensorFlow s.497} \subsection{LSTM} - Komórki LSTM (\textit{long-short term memory}) są rozszerzeniem neuronów sieci rekurencyjnych. Pozwalają wykrywać zależności w danych w długim okresie. Posiadają dwa wektory opisujące stan neuronu. Wektor $h_{(t)}$ określa stan krótkookresowy i wektor $c_{(t)}$ określa stan długookresowy. + Komórki LSTM (\textit{ang. long-short term memory}) są rozszerzeniem neuronów sieci rekurencyjnych. Pozwalają wykrywać zależności w danych w długim okresie. Posiadają dwa wektory opisujące stan neuronu. Wektor $h_{(t)}$ określa stan krótkookresowy i wektor $c_{(t)}$ określa stan długookresowy. \begin{figure}[!htb] \centering @@ -471,7 +477,7 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \label{fig:lstm} \end{figure} - Główny pomysł na funcjonowanie komórek LSTM był taki, aby sieć sama mogła się nauczyć jakie informacje są istotne i je przechować, a które informacje można pominąć, zapomnieć. Schemat komórki LSTM przedstawiono na rysunku~\ref{fig:lstm}. Aby to osiągnąć powstała idea bramek (\textit{gates}), oraz kontrolerów bramek (\textit{gate controllers}). W komórce LSTM wyróżniamy trzy bramki. Bramkę zapomnienia (\textit{forget gate}) sterowaną przez $f_{(t)}$, bramkę wejściową(\textit{input gate}) sterowaną przez $i_{(t)}$, oraz bramkę wyjściową (\textit{output gate}), sterowaną przez $o_{(t)}$. Przepływ danych w komórce LSTM zaczyna w miejscu gdzie wektor wejściowy $x_{(t)}$ i poprzedni którkoterminowy stan $h_{(t-1)}$ trafiają do czterech warstw. Główną warstwą jest ta zwracająca $g_{(t)}$. W podstawowej komórce RNN jest tylko ta warstwa. Pozostałe trzy wartswy po przejściu przez funkcje logistyczne trafiają do bramek. Bramka zapomnienia kontroluje, które informacje z długookresowego stanu $c_{(t-1)}$ powinny zostać wykasowane. Bramka wejściowa kontroluje jakie informacje z $g_{(t)}$ powinny zostać przekazane dalej i dodane do nastepnego stanu długookresowego $c_{(t)}$. Bramka wyjściowa odpowiada za wybranie odpowiednich elementów z stanu długookresowego i przekazanie ich następnych kroku. Wynik komórki zostaje przekazany do wyjścia komórki $y_{(t)}$ oraz jako następny stan krótkoterminowy $h_{(t)}$. + Główny pomysł na funcjonowanie komórek LSTM był taki, aby sieć sama mogła się nauczyć jakie informacje są istotne i je przechować, a które informacje można pominąć, zapomnieć. Schemat komórki LSTM przedstawiono na rysunku~\ref{fig:lstm}. Aby to osiągnąć powstała idea bramek (\textit{ang. gates}), oraz kontrolerów bramek (\textit{ang. gate controllers}). W komórce LSTM wyróżniamy trzy bramki. Bramkę zapomnienia (\textit{ang. forget gate}) sterowaną przez $f_{(t)}$, bramkę wejściową(\textit{ang. input gate}) sterowaną przez $i_{(t)}$, oraz bramkę wyjściową (\textit{ang. output gate}), sterowaną przez $o_{(t)}$. Przepływ danych w komórce LSTM zaczyna w miejscu gdzie wektor wejściowy $x_{(t)}$ i poprzedni którkoterminowy stan $h_{(t-1)}$ trafiają do czterech warstw. Główną warstwą jest ta zwracająca $g_{(t)}$. W podstawowej komórce RNN jest tylko ta warstwa. Pozostałe trzy wartswy po przejściu przez funkcje logistyczne trafiają do bramek. Bramka zapomnienia kontroluje, które informacje z długookresowego stanu $c_{(t-1)}$ powinny zostać wykasowane. Bramka wejściowa kontroluje jakie informacje z $g_{(t)}$ powinny zostać przekazane dalej i dodane do nastepnego stanu długookresowego $c_{(t)}$. Bramka wyjściowa odpowiada za wybranie odpowiednich elementów z stanu długookresowego i przekazanie ich następnych kroku. Wynik komórki zostaje przekazany do wyjścia komórki $y_{(t)}$ oraz jako następny stan krótkoterminowy $h_{(t)}$. Kolejne etapy komórki LSTM obliczane są zgodnie z poniższymy wzorami: \[ @@ -484,7 +490,7 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow o_{(t)} = \sigma(W_{xo}^{\top}x_{(t)} + W_{ho}^{\top}h_{(t-1)} + b_o) \] \[ - g_{(t)} = \sigma(W_{xg}^{\top}x_{(t)} + W_{hg}^{\top}h_{(t-1)} + b_g) + g_{(t)} = tanh(W_{xg}^{\top}x_{(t)} + W_{hg}^{\top}h_{(t-1)} + b_g) \] \[ c_{(t)} = f_{(t)}\tens{} c_{(t-1)} + i_{(t)}\tens{} g_{(t)} @@ -493,17 +499,29 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow y_{(t)} = h_{(t)} = o_{(t)} \tens{} tanh \left( c_{(t)} \right), \] - gdzie $W_{xi}$, $W_{xf}$, $W_{xo}$, $W_{xg}$ są to macierze wag dla każdej w czterech warstw połączonych z wektorem wejściowym $x_{(t)}$, $W_{hi}$, $W_{hf}$, $W_{ho}$, $W_{hg}$ są to macierze wag dla każdej w czterech warstw połączonych z poprzednim krótkookresowym stanem $h_{(t-1)}$ a $b_i$, $b_f$, $b_o$, $b_g$to biasy dla każdej z tych warstw. \cite{handson} %\footnote{Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow s.517} + gdzie $W_{xi}$, $W_{xf}$, $W_{xo}$, $W_{xg}$ są to macierze wag dla każdej w czterech warstw połączonych z wektorem wejściowym $x_{(t)}$, $W_{hi}$, $W_{hf}$, $W_{ho}$, $W_{hg}$ są to macierze wag dla każdej w czterech warstw połączonych z poprzednim krótkookresowym stanem $h_{(t-1)}$, a $b_i$, $b_f$, $b_o$, $b_g$ to biasy dla każdej z tych warstw \cite{handson}. %\footnote{Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow s.517} - Funckja $tanh$ to tangens hiperboliczny, jedna z funkcji sigmoidalnych. To co różni funkcję $tanh$ od $\sigma$ to zakres przyjmowanych wartości. Tangens hiperboliczny przyjmuje wartości z przedziału $(-1, 1)$ + Funckja $tanh$ to tangens hiperboliczny, jedna z funkcji sigmoidalnych. Wykres funkcji $tanh$ został przedstawiony na rysunku~\ref{fig:tanh} + \[ + tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} + \] + + To co różni funkcję $tanh$ od $\sigma$ to zakres przyjmowanych wartości. Tangens hiperboliczny przyjmuje wartości z przedziału $(-1, 1)$. + + \begin{figure}[!htb] + \centering + \includegraphics[width=\linewidth]{images/tanh.png} + \caption{Tangens hiperboliczny} + \label{fig:tanh} + \end{figure} - \subsection{Sequance-to-sequance} + \subsection{Sequence-to-sequence} - Model w architekturze Sequence-to-sequance (\textit{seq2seq}) został wynaleziony z myślą o tłumaczeniu maszynowym języków, ale zastosowanie dla niego znaleziono również w rozpoznawaniu mowy, opisywaniu wideo, czy tworzeniu chatbotów. Jego główną zaletą jest przetwarzanie sekwencji elementów o różnych długościach. Jest to naturalne, ponieważ tłumacząc z języka na język często tą samą sentencję można wyrazić różną liczbę słów. Dla przykładu zdanie po Polsku "Co dzisiaj robisz?" zawiera trzy słowa, natomiast przetłumaczone na Angielski "What are you doing today?" zawiera pięć słów. Nie można tego osiągnąć zwykłą siecią LSTM, dlatego model seq2seq został zaprojektowany, aby móc go zastosować do tego typu problemów. \cite{seq2seq} %\footnote{https://towardsdatascience.com/understanding-encoder-decoder-sequence-to-sequence + Model w architekturze Sequence-to-sequance (\textit{ang. seq2seq}) został wynaleziony z myślą o tłumaczeniu maszynowym języków, ale zastosowanie dla niego znaleziono również w rozpoznawaniu mowy, opisywaniu wideo, czy tworzeniu chatbotów. Jego główną zaletą jest przetwarzanie sekwencji elementów o różnych długościach. Jest to naturalne, ponieważ tłumacząc z języka na język często tą samą sentencję można wyrazić różną liczbę słów. Dla przykładu zdanie po Polsku "Co dzisiaj robisz?" zawiera trzy słowa, natomiast przetłumaczone na Angielski "What are you doing today?" zawiera pięć słów. Nie można tego osiągnąć zwykłą siecią LSTM, dlatego model seq2seq został zaprojektowany, aby móc go zastosować do tego typu problemów \cite{seq2seq}. %\footnote{https://towardsdatascience.com/understanding-encoder-decoder-sequence-to-sequence %-model-679e04af4346 26.05.2020 14:58} - Model sequence to sequance ma dwie części, encoder i decoder. Obie czeście są w zasadzie dwiema zupełnie innymi modelami, połączonymi ze sobą w jedną wielką sieć. + Model sequence-to-sequance ma dwie części, encoder i decoder. Obie czeście są w zasadzie dwiema zupełnie osobnymi modelami, połączonymi ze sobą w jedną sieć. Zadaniem encodera, podobnie jak zostało to opisane w rozdziale~\ref{section:autoencoder} o autoencoderze, jest wydobycie z wektora wejściowego najistotniejszych informacji i skompresowanie ich. Następnie wektor stanu encodera jest przekazywany do decodera, który na jego podstawie rekonstruuje sekwencję. @@ -514,11 +532,10 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \label{fig:seq2seq} \end{figure} - Więcej szczegółów technicznych dotycząych modelu sequance-to-sequance przedstawię w dalszych rozdziałach pracy. - - + Więcej szczegółów technicznych dotycząych modelu sequence-to-sequence przedstawię w dalszych rozdziałach pracy. + -\chapter{Reprezentacja danych muzycznych} +\chapter{Wprowadzenie do teorii muzyki} W tym rozdziale przedstawię podstawowe koncepcje muzyczne, sposoby reprezentacji muzyki. @@ -528,10 +545,10 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow Drgania powietrza z otoczenia człowieka są przetwarzane w mózgu i rozumiane jako dźwięki. Takie drgania nazywamy falą dźwiękową. Dzwięk muzyczny jest to fala dźwiękowa, którą wytwarza intrument muzyczny. Dzwięk muzyczny charakteryzuje się trzema podstawowymi parametrami: \begin{itemize} - \item wysokością (\textit{pitch}) - jest to częstotliwość drgań wyrażona w hercach. Im większa częstotliwość tym dzźięk jest rozumiany jako wyższy, + \item wysokością (\textit{ang. pitch}) - jest to częstotliwość drgań wyrażona w hercach. Im większa częstotliwość tym dzźięk jest rozumiany jako wyższy, Zakres słyszalny dla człowieka wynosi od 20Hz do 20kHz. - \item głośność (\textit{velocity}) - jest to amplituda drgań fali dźwiękowej. Im większa aplituda, tym dźwięk jest odczuwany jako głosniejszy, - \item długość (\textit{duration}) - jest to czas z jakim dźwięk wybrzmiewa, np. 2 sekundy. + \item głośność (\textit{ang. velocity}) - jest to amplituda drgań fali dźwiękowej. Im większa aplituda, tym dźwięk jest odczuwany jako głosniejszy, + \item długość (\textit{ang. duration}) - jest to czas z jakim dźwięk wybrzmiewa, np. 2 sekundy. \end{itemize} % Przykład dźwięku myczynego przedstawiono na Rusynku~\ref{fig:sine} @@ -545,7 +562,7 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow \subsection{Sygnał dźwiękowy} - W rzeczywistości utwór muzyczny jest zazwyczaj kombinacją wielu fal dźwiękowych, o różnych charaketystykach i nazwywany jest sygnałem dźwiękowym. Wizualizację sygnalu dźwiękowego przedstawiono na Rysunku~\ref{fig:waveform} + W rzeczywistości utwór muzyczny jest zazwyczaj kombinacją wielu fal dźwiękowych, o różnych charaketystykach i nazwywany jest sygnałem dźwiękowym. Wizualizację sygnalu dźwiękowego przedstawiono na Rysunku ~\ref{fig:waveform} \begin{figure}[!htb] \centering @@ -559,20 +576,24 @@ Jednocześnie przyjmuję do wiadomości, że przypisanie sobie, w pracy dyplomow Reprezentacja muzyki jako sygnału dźwiękowego przechowuje informacje o dokładnym brzmieniu danego utworu tzn. jakie drgania należy wytworzyć, aby móc odtwozyć muzykę. Taki zapis nie informuje nas bezpośrednio jakie instrumenty zostały użyte, jakie wysokości i długości dźwięków zostały wykorzystane. Dlatego ludzkość na przestrzeni wieków opracowała abstrakcyjne objekty, które reprezentują utwór w czytelny dla człowieka sposób. \subsubsection{Tempo} - W muzyce symbolicznej tempo informuje nas o prędkości utworu. W muzyce klasycznej stosowało się opisowy sposób dostosowywania tempa np. Allegro - Szybko lub Adagio - wolno. Jak można szybko stwierdzić są to zwroty subiektywne i nie wyznaczają tempa jednoznacznie. Obecnie wyraża się tempo w liczbie uderzeń na minutę (\textit{beats per minute - BPM}). I tak Allegro jest to od 120 do 168 BPM a Adagio od 66 do 76 BPM. \cite{tempos} % \footnote{źródlo: http://www.classicalmusiccity.com/search/article.php?vars=446/ + W muzyce symbolicznej tempo informuje nas o prędkości utworu. W muzyce klasycznej stosowało się opisowy sposób dostosowywania tempa np. Allegro - Szybko lub Adagio - wolno. Jak można szybko stwierdzić są to zwroty subiektywne i nie wyznaczają tempa jednoznacznie. Obecnie wyraża się tempo w liczbie uderzeń na minutę (\textit{ang. beats per minute; BPM}). I tak Allegro jest to od 120 do 168 BPM a Adagio od 66 do 76 BPM \cite{tempos}. % \footnote{źródlo: http://www.classicalmusiccity.com/search/article.php?vars=446/ % Basic-Tempo-Markings.html 5 kwietnia 19:37 % } \subsubsection{Nuta} - Nuta jest to graficzna reprezentacja dźwięku muzycznego. Informuje nas ona o dwóch parametrach dźwięku, wysokości oraz długości dźwięku. Długość dźwięku nazywa się jej wartością. Podstawową wartością nuty jest ćwierćnuta, odpowiada ona jednemu uderzeniu (\textit{beat}). Ta wartość pozwala nam zrozumieć jak długo nalezy wygrywać nutę relatywnie do pozostałych nut w utworze. Jeśli obok siebie ustawimy dwie nuty o wartościach ćwirćnuty i ósemki, wiemy że tę drugą nutę powinniśmy zagrać dwa razy krócej niż pierwszą. Aby wiedzieć dokładnie jak dlugo powinna wybrzmiewać nuta musimy odwołać się do tempa utworu. Dla przykładu w tempie 60 BPM w ciągu minuty zagramy dokładnie 60 ćwierć nut. Kolejne wartości tworzone poprzez sumowanie lub podział długości ćwierćnuty. Półnuta trwa tyle co dwie ćwierćnuty, cała nuta tyle co dwie półnuty, ósemka trwa połowę czasu ćwiercnuty, a szesnastka połowę ósemki itd. + Nuta jest to graficzna reprezentacja dźwięku muzycznego. Informuje nas ona o dwóch parametrach dźwięku, wysokości oraz długości dźwięku. Długość dźwięku nazywa się jej wartością. Podstawową wartością nuty jest ćwierćnuta, odpowiada ona jednemu uderzeniu (\textit{ang. beat}). Ta wartość pozwala nam zrozumieć jak długo nalezy wygrywać nutę relatywnie do pozostałych nut w utworze. Jeśli obok siebie ustawimy dwie nuty o wartościach ćwirćnuty i ósemki, wiemy że tę drugą nutę powinniśmy zagrać dwa razy krócej niż pierwszą. Aby wiedzieć dokładnie jak dlugo powinna wybrzmiewać nuta musimy odwołać się do tempa utworu. Dla przykładu w tempie 60 BPM w ciągu minuty zagramy dokładnie 60 ćwierć nut. Kolejne wartości tworzone poprzez sumowanie lub podział długości ćwierćnuty. Półnuta trwa tyle co dwie ćwierćnuty, cała nuta tyle co dwie półnuty, ósemka trwa połowę czasu ćwiercnuty, a szesnastka połowę ósemki itd. \begin{figure}[!htb] \centering \includegraphics[width=\linewidth]{images/wartosc_nut.jpg} - \caption{źródlo: https://www.infomusic.pl/poradnik/46934,poradnik-teoria-muzyki-rytm 5 kwietnia 2020 12:46} + \caption[]{Podział wartości nut\footnotemark} \label{fig:waveform} \end{figure} + \footnotetext{źródlo: https://www.infomusic.pl/poradnik/46934,poradnik-teoria-muzyki-rytm + \newline + 5 kwietnia 2020 12:46} + Tak jak pisałem wcześniej, wysokość dźwięku jest to częstotliwość drgań fali dźwiękowej wyrażona w hercach. W muzyce symbolicznej dla uproszczenia wybrane częstotliwości zostały nazwane literami alfabetu C, D, E, F, G, A, H. Każdej literze przypisana jest częstotliwość zgodnie z Tabelą~\ref{table:dzwieki} diff --git a/docs/document.toc b/docs/document.toc index 787ff4d..34e9554 100644 --- a/docs/document.toc +++ b/docs/document.toc @@ -9,19 +9,19 @@ \contentsline {subsection}{\numberline {1.2.3\relax .\leavevmode@ifvmode \kern .5em }Metody gradientowe}{15}% \contentsline {section}{\numberline {1.3\relax .\leavevmode@ifvmode \kern .5em }Regresja liniowa jako model sieci neuronowej}{17}% \contentsline {section}{\numberline {1.4\relax .\leavevmode@ifvmode \kern .5em }Funkcje aktywacji}{19}% -\contentsline {section}{\numberline {1.5\relax .\leavevmode@ifvmode \kern .5em }G\IeC {\l }\IeC {\k e}bokie sieci neuronowe}{20}% +\contentsline {section}{\numberline {1.5\relax .\leavevmode@ifvmode \kern .5em }Wielowarstwowe sieci neuronowe}{19}% \contentsline {subsection}{\numberline {1.5.1\relax .\leavevmode@ifvmode \kern .5em }Jednokierunkowe sieci neuronowe}{20}% \contentsline {subsection}{\numberline {1.5.2\relax .\leavevmode@ifvmode \kern .5em }Autoencodery}{21}% \contentsline {subsection}{\numberline {1.5.3\relax .\leavevmode@ifvmode \kern .5em }Rekurencyjne sieci neuronowe}{22}% \contentsline {subsection}{\numberline {1.5.4\relax .\leavevmode@ifvmode \kern .5em }LSTM}{23}% -\contentsline {subsection}{\numberline {1.5.5\relax .\leavevmode@ifvmode \kern .5em }Sequance-to-sequance}{25}% -\contentsline {chapter}{Rozdzia\PlPrIeC {\l }\ 2\relax .\leavevmode@ifvmode \kern .5em Reprezentacja danych muzycznych}{27}% +\contentsline {subsection}{\numberline {1.5.5\relax .\leavevmode@ifvmode \kern .5em }Sequence-to-sequence}{25}% +\contentsline {chapter}{Rozdzia\PlPrIeC {\l }\ 2\relax .\leavevmode@ifvmode \kern .5em Wprowadzenie do teorii muzyki}{27}% \contentsline {section}{\numberline {2.1\relax .\leavevmode@ifvmode \kern .5em }Podstawowe koncepcje}{27}% \contentsline {subsection}{\numberline {2.1.1\relax .\leavevmode@ifvmode \kern .5em }D\IeC {\'z}wi\IeC {\k e}k muzyczny}{27}% \contentsline {subsection}{\numberline {2.1.2\relax .\leavevmode@ifvmode \kern .5em }Sygna\IeC {\l } d\IeC {\'z}wi\IeC {\k e}kowy}{27}% \contentsline {subsection}{\numberline {2.1.3\relax .\leavevmode@ifvmode \kern .5em }Zapis nutowy}{27}% -\contentsline {section}{\numberline {2.2\relax .\leavevmode@ifvmode \kern .5em }Cyfrowa reprezentacja muzyki symbolicznej}{30}% -\contentsline {subsection}{\numberline {2.2.1\relax .\leavevmode@ifvmode \kern .5em }Standard MIDI}{30}% +\contentsline {section}{\numberline {2.2\relax .\leavevmode@ifvmode \kern .5em }Cyfrowa reprezentacja muzyki symbolicznej}{31}% +\contentsline {subsection}{\numberline {2.2.1\relax .\leavevmode@ifvmode \kern .5em }Standard MIDI}{31}% \contentsline {chapter}{Rozdzia\PlPrIeC {\l }\ 3\relax .\leavevmode@ifvmode \kern .5em Projekt}{33}% \contentsline {section}{\numberline {3.1\relax .\leavevmode@ifvmode \kern .5em }Koncepcja}{33}% \contentsline {section}{\numberline {3.2\relax .\leavevmode@ifvmode \kern .5em }Przygotowanie danych}{34}% 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Turn this addition off with the DataScience.changeDirOnImportExport setting +# ms-python.python added +import os +try: + os.chdir(os.path.join(os.getcwd(), 'docs\\images')) + print(os.getcwd()) +except: + pass + +#%% +import music21 +from music21.midi import MidiFile + +import numpy as np +import matplotlib.pyplot as plt +import mido + +#%% [markdown] +# # midi messages + +#%% +filepath = '/home/altarin/praca-magisterska/docs/images/seq2seq_generated_midi_7.mid' +mid = mido.MidiFile(filepath) +for i, track in enumerate(mid.tracks): + for msg in track: + print(msg) + +#%% [markdown] +# # Regresja liniowa + +#%% +x = np.arange(0,10) +y = np.arange(0,10) + np.random.random(10)-0.5 +y_hat = np.arange(0,10) +plt.scatter(x, y, c='k') +plt.plot(x,y_hat, c='r') +# plt.labels +# plt.savefig('linear_reg.png') + +#%% +from math import exp + +x = np.arange(-10,10, 0.1) +# y = np.arange(0,10) + +tanh = lambda x: (exp(x) - exp(-x))/(exp(x) + exp(-x)) + +y_hat = [tanh(yy) for yy in x] +# plt.scatter(x, y, c='k') +plt.plot(x, y_hat) +# plt.show() +# plt.labels +plt.savefig('tanh.png') + +#%% +# Gradient descent + +fx = x^2 +dx = 2x + c + + + + +#%% +func = lambda x: x**2 +func_dx = lambda x:2*x + +x = np.arange(-20,20,0.1) +y = func(x) + +point_x = -6 +point_y = func(point_x) + + + +dx = styczna(x) + +learning_points_xs = np.arange(point_x, 0, 0.8) +learning_points_ys = func(learning_points_xs) + +fig, ax = plt.subplots() +ax.plot(x, y, c='k') + +for px in learning_points_xs[0:1]: + slope = func_dx(px) + intercept = -px**2 + styczna = lambda x: slope*x + intercept + dx = styczna(x) + ax.plot(x, dx, c='r', zorder=1) + +ax.scatter(x=point_x, y=point_y, c='r', zorder=6) #start +# ax.scatter(x=0, y=0, c='g', zorder=6) #min +# ax.scatter(x=learning_points_xs, y=learning_points_ys, c='y', zorder=5) +plt.ylim((-20,80)) +plt.xlim((-20,20)) +# plt.xlabel('x') +# plt.ylabel('f(x) = x^2') + +plt.savefig('gradient_descent_1.png') + +# 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