24.5 Difference filters

In matching polynomials to time series (xt*)t∈T a problem is the degree of the polynomial. Similar to classical time series it is also possible for fuzzy time series using so-called difference filters.

Remark 24.4:

For a polynomial f (t) = a0* ⊕ a1* · t ⊕ a2* · t2 ⊕ … ⊕ am* · tm with degree m > 0 and a0*, a1*, … , am* ∈ 2 () the difference g(t) = f (t) ~H f (t − 1) is for t > 0 a polynomial of maximal degree m − 1.

Proof:

By Remark 23.6 with

and ti − (t − 1)i ≥ 0, i = 1 (1)m we obtain

The Hukuhara difference g(t) is a polynomial of degree (m−1) which is uniquely determined by Theorem 23.1.

Definition 24.4:

Let (xt*)t∈T be a fuzzy time series. A mapping Δ, defined by

is called a difference filter of order 1. Difference filters of order m with m > 1 are recursively defined by

The difference operation reduces the degree of a polynomial by 1. Applying the difference ...

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