From 3d1b0956e5a3dc29a6aeb39e6adb7a8f4a706932 Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Fri, 29 Jan 2021 22:19:29 +0100 Subject: [PATCH] =?UTF-8?q?fpcs=20eingef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Acronyms.tex | 3 + .../Information and Pattern Recognition.tex | 1 + .../Fuzzy Pattern Classifier.tex | 121 ++++++++++++++++++ images/fpc_asymmetric_aproach.png | Bin 0 -> 53368 bytes images/fpc_closed.png | Bin 0 -> 25600 bytes images/fpc_example.png | Bin 0 -> 111604 bytes images/fpc_open.png | Bin 0 -> 178071 bytes images/fpc_variants.png | Bin 0 -> 139588 bytes images/fuzzy-pattern-classifier.png | Bin 0 -> 74762 bytes images/fuzzy_averaging_operator.png | Bin 0 -> 31225 bytes images/parameterized_fpc.png | Bin 0 -> 73666 bytes parts/Supervised Learning.tex | 3 +- 12 files changed, 127 insertions(+), 1 deletion(-) create mode 100644 chapters/Supervised Learning/Fuzzy Pattern Classifier.tex create mode 100644 images/fpc_asymmetric_aproach.png create mode 100644 images/fpc_closed.png create mode 100644 images/fpc_example.png create mode 100644 images/fpc_open.png create mode 100644 images/fpc_variants.png create mode 100644 images/fuzzy-pattern-classifier.png create mode 100644 images/fuzzy_averaging_operator.png create mode 100644 images/parameterized_fpc.png diff --git a/Acronyms.tex b/Acronyms.tex index 03edfa8..af9ac55 100644 --- a/Acronyms.tex +++ b/Acronyms.tex @@ -13,4 +13,7 @@ \acro{ML}{Maschinelles Lernen} \acro{CNN}{Convolutional Neural Network} \acro{RNN}{Recurrent Neural Network} + \acro{FPC}{(Bocklisch's) Fuzzy Pattern Classifier} + \acro{MFPC}{Modified-Fuzzy-Pattern-Classifier} + \acro{FPGA}{Field Programmable Gate Array} \end{acronym} \ No newline at end of file diff --git a/chapters/Basics/Information and Pattern Recognition.tex b/chapters/Basics/Information and Pattern Recognition.tex index 53685d4..ddc3926 100644 --- a/chapters/Basics/Information and Pattern Recognition.tex +++ b/chapters/Basics/Information and Pattern Recognition.tex @@ -1,4 +1,5 @@ \chapter{Information and Pattern Recognition} +\label{information and pattern regocgnition} Falls zu viele Daten vorliegen um diese manuell auszuwerten kann es möglich sein den Vorgang zu automatisieren. Hierbei werden die Daten mithilfe eines Musters klassifiziert (Pattern assignment $\rightarrow$ classification). Das Verfahren zur Auswahl der jeweiligen Klassenzuordnung wird als \say{Pattern Recognition} bezeichnet. diff --git a/chapters/Supervised Learning/Fuzzy Pattern Classifier.tex b/chapters/Supervised Learning/Fuzzy Pattern Classifier.tex new file mode 100644 index 0000000..526bc74 --- /dev/null +++ b/chapters/Supervised Learning/Fuzzy Pattern Classifier.tex @@ -0,0 +1,121 @@ +\chapter{\acf{FPC}} +\label{fuzzy-pattern-classifier} + Der \ac{FPC} wird allgemein als $h(\bm{m},\bm{p})$angegeben. + \paragraph{Beispiel:}\mbox{}\\ + \includegraphics[width = .6\textwidth]{fpc_closed.png}\\ + + Der \ac{FPC} besteht aus mehreren Bestandteilen:\\ + \includegraphics[width = .8\textwidth]{fpc_open.png} + + \section{\acf{FPC} membership function} + \label{fpc membership function} + Allgmein geht es bei den \acp{FPC} darum Zugehörigkeitsfunktionen aus der Fuzzy Logic zu berechnen und die Ergebnisse zu aggregieren. + Dies wird dazu genutzt eine Klassifikation aufzubauen, die viele Merkmale miteinbezieht. + Die \ac{FPC} membership function stellt eine Abbildung $\mu(m,\bm{p})\rightarrow I$ auf die Identität $I\in \{0,1\}$ dar: + $$ \mu(m,\bm{p}) = \frac{A}{1+d(m,\bm{p})}(\Rightarrow A \in I) \text{ mit } d(m,\bm{p}) = \left(\frac{|m-S|}{C}\right)^D$$ + \begin{itemize} + \item $\bm{p} = (S,C,D)$:\\ + der Parameter-Vektor, der die Eigenschaften der der membership function festlegt + \begin{itemize} + \item $S$: der Durchschnittswert + \item $C$: die Breite + \item $D$: die Flankensteilheit + \end{itemize} + \item $d(m,\bm{p})$:\\ + der Abstand von einem Feature-Wert $m$ von dem Durchschnittswert $S$ + \item $A$:\\ + der größte Wert der membership function + \end{itemize} + + \includegraphics[width = \textwidth]{fuzzy-pattern-classifier.png}\\ + Die einzelnen Parameter werden jeweils vorgegeben oder angelernt. + Die Form der Funktion wird dabei immer an den jeweiligen Einsatzzweck angepasst.\\ + \includegraphics[width = \textwidth]{parameterized_fpc.png} + Auch eine mehrdimensionale Implementierung oder die Implementierung von parallel laufenden Funktionen ist möglich:\\ + \includegraphics[width = \textwidth]{fpc_variants.png} + + \subsection{\acf{MFPC} membership function} + \label{mfpc membership function} + \acp{MFPC} sind hardware-optimierte Varianten von \ac{FPC}. + Da zum Zeitpunkt der Entwicklung der \ac{FPC} membership funciton (\ref{fpc membership function}) war diese aufgrund der Division nicht für die Hardware-Implementation geeignet. + Die zugehörige membership function ist definiert durch: + $$\mu_\text{MFPC}(m,\bm{p}) = 2^{-d(m,\bm{p})}\in I \text{ mit } d(m,\bm{p}) = \left(\frac{|m-S|}{C}\right)^D$$ + Diese Funktion ist sehr effizient in Hardware implementierbar. + \acp{MFPC} wurden dafür entworfen Pattern Recognition (\ref{information and pattern regocgnition}) Systeme auf \acp{FPGA} zu verwenden. + + \subsection{Bestimmung der Parameter} + \label{fpc: parameterbestimmung} + \begin{itemize} + \item Distanzfunktion $d_i$ für jedes Feature $m_i$: + $$d_i(m_i;\bm{p})\left(\frac{|m_i-S_i}{C_i}\right)^D_i$$ + \item Durchschnittswert $S$: + $$S_i = \frac{1}{N}\sum^{N-1}_{j=0}m_j$$ + oder für kompakte Datensätze: + $$S_i = \frac{1}{2}\cdot \left(\max(m_i)+\min(m_i)\right)$$ + \item Breite $C$ kann durch anlernen ermittelt werden: + $$C_i = \frac{1}{2}\cdot\left[\max(m_i)-\min(m_i)\right]+C_{E,i}$$ + $C_{E,i}$ ist definiert als $C_{E,i}=p_{C_e}\cdot\left[\max(m_i)-\min(m_i)\right]$. + Hieraus ergibt sich dann für $C_i$: + $$C_i(1+2p_{C_e})\cdot \frac{1}{2}\left[\max(m_i)-\min(m_i)\right]$$ + $p_{C_e}$ ist ein Controll-Parameter (in der Praxis meist $\approx5\%$) + \item Flankensteilheit $D$ wird empirisch festgelegt ($D=2\vee4\vee8$) oder mit dem \say{Petker, Mönks}-Ansatz erlent + \end{itemize} + + \section{Defuzzifizierung} + \label{defuzzification} + Die Rückgabewerte des \ac{FPC} werden anhand eines Schwellwertes $B$ defuzzifiziert. + Hierdurch findet dann die eindeutige Klassifizierung statt. + $$c=\begin{cases} + 1, &\mu(\bm{m};\bm{p})\ge B\\ + 0, &\mu(\bm{m};\bm{p})< B + \end{cases} + $$ + + \section{Aggregationsoperator $h$} + \label{fpc: aggregationsopperator} + Um die Ergebnisse der membership functions (\ref{fpc membership function}) der einzelnen Features zusammenzufassen wird ein Aggregationsoperator $h$ verwendet. + Hierfür werden meist \say{fuzzy averaging operators} verwendet. + Diese liegen frei zwischen einem harten \textbf{AND}-Operator und einem harten \textbf{OR}-Operator. + Man misst jeweils die \say{orness} bzw. \say{andness} des Operators.\\ + \includegraphics[width = .8\textwidth]{fuzzy_averaging_operator.png} + Die geometrische Mitte liegt hierbei bei $\approx0.666$ andness. + + \subsection{Axiome} + \label{fpc: aggregation axioms} + Um zu zeigen, dass die Leistungsfähigkeit für eine \ac{MFPC} aggregating fuzzy membership function ausreichend ist müssen folgende Axiome erfüllt sein: + \begin{itemize} + \item \textbf{Axiom H 1.} Monotonicity:\\ + Das Verhältnis von zwei Eingangsparameter-Mengen soll sich in der Ausgabe des Aggregationsoperators wiederspiegeln. + D.h. für die Parameter-Tupel $(a_1,a_2,\dots,a_n)$ und $(b_1,b_2,\dots,b_n)$ ($a_i,b_i\in I \forall i\in \mathbb{N}_n)$) soll für + $a_i\le b_i\forall i\in \mathbb{N}_n$ gelten: + $$h(a_1,a_2,\dots,a_n)\le h(b_1,b_2,\dots,b_n)$$ + \item \textbf{Axion H 2.} Continuity:\\ + Eine kleine Änderung der Eingangsparameter soll auch nur zu einer kleinen Änderung des Ergebnisses der Aggregationsoperation führen + $$\lim_{\epsilon\rightarrow0} h (a_1,\dots,a_i + \epsilon, \dots,a_n) = h(a_1,\dots,a_i,\dots,a_n)$$ + \item \textbf{Axiom H 3.} Symmetry:\\ + Das Ergebnis des Aggregationsoperators bleibt für jede Permutation $p$ auf $\mathbb{N}_n = \{1,\dots n\}$ gleich: + $$h(a_1,a_2,\dots,a_n) = h(a_{p(1)},a_{p(2)},\dots,a_{p(n)})$$ + \item \textbf{Axiom H 4.} Idempotency:\\ + Wenn $a_i = a \forall i \in \mathbb{N}_n$, dann + $$ h(a_1,a_2,\dots,a_n) = a$$ + \end{itemize} + + \subsection{\ac{MFPC} aggregate membership function} + \label{mfpc: aggregate membership function} + Der Aggregationsoperator für die \acp{MFPC} ist definiert durch: + \Large + $$\mu(\bm{m},\bm{p}) = 2 ^ {-\frac{1}{M}\sum^M_{i=1}\left(\frac{|m_i-S_i|}{C_i}\right)^{D_i}} = \left[\prod^M_{i=1} 2^{-\left(\frac{|m_i-S_i|}{C_i}\right)^{D_i}}\right]^{\frac{1}{M}}$$ + \normalsize + Hierbei ist der geometrische Mittelpunkt definiert durch: + \Large + $$\mu(\bm{m};\bm{p})= 2^{-\frac{1}{M}\sum^{M-1}_{i=0} d(m_i;\bm{p})}$$ + \normalsize + + Diese Aggregationsoperation erfüllt alle nötigen Axiome (\ref{fpc: aggregation axioms}) (Beweis: siehe Vorlesung). + + \section{Asymmetrischer Ansatz} + In der Praxis wird neben dem normalen (symmetrischen) Ansatz auch der asymmetrische Ansatz verwendet.\\ + \includegraphics[width =\textwidth]{fpc_asymmetric_aproach.png} + + \section{Anwendungsbeispiel: Banknotenüberprüfung} + \includegraphics[width = \textwidth]{fpc_example.png} \ No newline at end of file diff --git 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Learning/Linear Machines.tex} - \input{chapters/Supervised Learning/SVM.tex} \ No newline at end of file + \input{chapters/Supervised Learning/SVM.tex} + \input{chapters/Supervised Learning/Fuzzy Pattern Classifier.tex} \ No newline at end of file

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