Nonsupervised sequential classification and recognition of patterns

A Bayes approach to nonsupervised pattern recognition is given wheren l-dimensional vector samplesX₁, X₂, ⋯ , Xₙare received unclassified, i.e., any one ofMpattern sourcesω₁, ω₂, ⋯, ωM, with corresponding probabilities of occurrenceQ_{1ₒ}, Q_{2ₒ} , ⋯ , Q_{Mₒ}, caused each sampleXₛ, s=1,2, ⋯ , n. The approach utilizes the fact that the cumulative distribution function (c.d.f.) ofXₛis a mixture c.d.f.,F(Xₛ)= ∑ᵢ₌₁M F(Xₛ|ωᵢ) Q_{iₒ}. It is assumed that available a priori knowledge includes knowledge ofMand the family\{F(Xₛ|ωᵢ)\}, whereF(Xₛ|ωᵢ)is characterized by a vectorB_{iₒ}. In general,B_{iₒ}andQ_{iₒ}, i = 1,2, ⋯ , Mare considered fixed but unknown, and conditional probability of error in deciding which source causedXₙis minimized. When the functional form ofF(Xₛ|ωᵢ)in terms ofB_{iₒ}is unknown, the family\{F(Xₛ|ωᵢ)\}is taken to be the family of multinomial c.d.f.'s--an application of the histogram concept to the nonsupervisory problem. Additional nonparameteric a priori knowledge about the family--such asF(Xₛ|ωᵢ)is symmetrical, and/orF(Xₛ|ωᵢ)differs fromF(Xₛ|ωⱼ)only by a translational vector--can be utilized in the Bayes solution.