Yu.N. Minaev, O.Yu. Filimonova, J.I.MinaevaYu.N. Minaev, O.Yu. Filimonova, J.I.Minaeva
Èlektron. model. 2018, 38(6):45-66
https://doi.org/10.15407/emodel.38.06.045
ABSTRACT
A problem of structuring the time series (TS) (in a form of a window, fragment, segment or others structure parts) has been investigated, as well as presentation of a separate window in the form of 2D tensor with X matrix of dimensionality m x m (m · m is the number of window elements TS) with following determination of m -vectors u, v (with certain restrictions), which for the given matrix of data X minimize a criterion ||X-Kr uvT||2F +Pλ(u,v), where trace{(X - uvT)(X - uvT)T}; Pλ(u,v)— a penalty function, -Kr — a symbol of Kronecker difference. Vectors u, v are considered as a subset of ordered pairs, where vector v plays a role of
membership function, i.e. (v →[0, 1]). The expediency of using the procedure of a singular decomposition
for this purpose is shown.
A subset of ordered pairs {u, v}, considered as psevdo FS, represents 2D tensor with the matrix of dimensionality 2 x m, allows us to shorten a body of stored information (m · m > 2 · m), to obtain hidden knowledge in the form of the spectrum of singular values and to obtain new possibilities in deciding the problems of forecasting and anomaly identifications of TS anomalies as the result of using the tensor invariants.
KEYWORDS
fuzzy set, time row, tensor decomposition, singular values, Kronecker product.
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