Bağımsız Bileşenler Analizinin İstatistiksel Bakış Açısıyla Değerlendirilmesi ve Temel Bileşenler Analizi ile Karşılaştırılması

Öz

İstatistik ve ilgili alanlardaki en önemli problemlerden biri, çok değişkenli verinin uygun bir temsilinin bulunmasıdır. Burada temsilden kasıt; veriyi, esas yapısı daha görünür (ulaşılır) bir şekle dönüştürmektir. Bağımsız Bileşenler Analizi (BBA); çok değişkenli verinin altında yatan bileşenleri bularak esas yapısını daha görünür hale getiren istatistiksel bir yöntemdir. Bu açıdan BBA, Temel Bileşenler Analizi’nin (TBA) bir uzantısı olarak da görülebilir. Ancak BBA, TBA’nın aksine ilişkisizlik yerine istatistiksel bağımsızlığı temel alır ve istatistiksel bağımsızlık, ilişkisizliğe göre çok daha güçlü bir özelliktir. Ayrıca TBA’da elde edilen bileşenlerin normal dağılması istenirken, BBA’da tam tersi bağımsız bileşenlerin normal dağılmaması istenmektedir. Çalışmada, çok değişkenli istatistiksel bir yöntem olmasına rağmen istatistik alanında pek fazla bilinmeyen ve daha çok mühendislik alanında kullanılan BBA konusu ayrıntılı bir şekilde ele alınmış ve konuyla ilgili kısıtlı istatistik literatürüne katkıda bulunulmuştur. Uygulama bölümünde BBA, benzer bir yöntem olan TBA ile karşılaştırılmıştır. Her iki analiz yapay bir veri kümesine uygulanmış ve BBA’nın normal olmayan bileşenleri ortaya çıkarmada TBA’dan çok daha başarılı olduğu sonucuna ulaşılmıştır.

Anahtar Kelimeler

• Bağımsız bileşenler analizi
• Çok değişkenli istatistiksel yöntemler
• Kör kaynak ayrıştırma problemi
• Temel bileşenler analizi

Evaluation of Independent Components Analysis from Statistical Perspective and Its Comparison with Principal Components Analysis

Öz

One of the most important problems in statistics and related fields is that finding an appropriate representation of multivariate data. Here is meant by representation; to transform the data into a more visible (accessible) form. Independent Components Analysis (ICA) is a statistical method used to find the underlying components of multivariate data and makes its main structure more visible. In this respect, ICA can also be seen as an extension of the Principal Components Analysis (PCA). However, ICA, contrary to PCA, is based on statistical independence rather than unrelatedness and statistical independence is a much stronger feature than unrelatedness. In addition, while the normal distribution of the components obtained in PCA is desired, the independent components of ICA are requested not to distribute normally. In the study, although it is a multivariate statistical method, the subject of ICA, which is not well known in the field of statistics and which is mostly used in engineering, was discussed in detail and contributed to the limited statistical literature on the subject. In the application part, ICA was compared with a similar method, PCA. Both analyzes were applied to an artificial dataset and it was concluded that ICA was much more successful than PCA in detecting non-normal components.

Anahtar Kelimeler

• Independent components analysis
• Multivariate statistical methods
• Blind source separation problem
• Principal components analysis

Kaynakça

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