All the class frequencies of various orders are not independent of each other and any class frequency can always be expressed in terms of class frequencies of higher order.
N = (A) + (α) = (B) + (β) = (C) + (γ),etc.
Also, since each of these A’s or α’s can either be B’s or β’s, we have
Similarly (B) = (AB) + (αB) and (β) = (Aβ) + (αβ)
(AB) = (ABC) + (ABγ), (Aβ) = (AβC) + (Aβγ)
(αB) = (αBC) + (αBγ), (αβ) = (αβC) + (αβγ)
And so on. Thus
(β) = (Aβ) + (αβ) = (AβC) + (Aβγ) + (αβC) + (αβγ), etc.
The classes of highest order are called the ultimate classes and their frequencies, the ultimate class frequencies. Thus in case of n attributes, the ultimate class frequencies will be the frequencies of nth order. For example, the class frequencies (ABC), (ABγ), (AβC), (Aβγ), (αBC), (αβC), (αβγ) are the ultimate frequencies for three attributes A, B and C.
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