Mathematische Mustererkennung und Hidden-Markov-Modelle

Machine recognition of different patterns is termed pattern recognition. The assignation of patterns in sensory signals to knowledge is also known as classification. Organisms constantly classify their sensory perceptions to highlight this. Mathematical pattern recognition can be demonstrated with the following situation: There are one of the coincidentally determined patterns, from a finite but unobservable set, and an observation dependent upon the pattern. The pair consisting of the pattern and the observation is called a sample. There may be further observations necessary which could be used to accomplish the following task. On the basis of these observations, the pattern should be recognized, that is the observations are assigned to a pattern with a function. Such a function is termed a classificator. Since each false classification can stand for a loss, the mean loss may be seen as a characteristic of the quality supplied by the classificator. This characteristic is termed the risk of the classificator. In this text, the model of pattern classification is further extended, so that a temporal component may be built in the model. This makes it possible to regard processes of samples in which patterns arise as mutually dependent. The extension of the model ist termed Chronometric Model. In special cases, the sample process is a Hidden-Markov-Model. In the Chronometric Model the risk of optimal classificators are examined. In the first and the second chapter, the bases of the mathematical pattern recognition and the definition of the Chronometric Model are represented. The third chapter concerns the characteristics of Hidden-Markov-Models. In the second part the risk of optimal classificators in the Chronometric Model is examined, followed by further investigation of the special case of a Hidden-Markov-Model. The cases of finite an continuous sets of observation will regarded individually. In the last part, two algorithms are given to compute the risk of an optimal classificator. Bezeichnungen Im Folgenden wird ein Reihe von Bezeichnungen zusammengestellt, die im laufenden Text ohne weitere Erklärungen verwendet werden. · Ist M eine Menge und sind m,n ∈ N,m ≤ n, so bezeichne xm ∈ Mn−m+1 den Vektor (xm, . . . , xn). Die Schreibweise y m = x n m (1) wird als Abkürzung für yi = xi für alle i ∈ {m, . . . , n} dienen. · Ist für m,n ∈ N mit m < n (Mi)i=m eine Folge von Mengen, so wird in Analogie zu (1) xm ∈M m anstelle von xi ∈Mi für alle i ∈ {m, . . . , n} geschrieben. · Die von dem Prozess (Xt, Yt)t∈Z erzeugten σ-Algebren werden wie folgt bezeichnet: F t s := σ(X t s, Y t s ) für alle s, t ∈ Z, s ≤ t . F∞ s := σ(X t s, Y t s , t ∈ Z≥s) für alle s ∈ Z . F t −∞ := σ(X t s, Y t s , s ∈ Z≤t) für alle t ∈ Z . · Seien X und Y Zufallsvariablen. Dann wird die bedingte Verteilung von X gegeben Y = y mit PX|Y=y bezeichnet. · Liegen entsprechende Zufallsvariablen vor, so wird kurz P (x|y, z) anstelle von P (X = x|Y = y, Z = z) geschrieben.