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Therefore, the tensor rank of coincides with the model order d.

In Figure 7, the model order d is equal to 2, while in Figure 8, d = 3.

Since in practice d is not known, P denotes a candidate value for, which is our estimate of the model order d.

Hence, the problem we are solving can therefore be stated in the following fashion: given a noisy measurement tensor, we desire to estimate the model order d.

where represents an estimate of the model order d, and g(G)(P) and a(G)(P) are the geometric and arithmetic means of the P smallest global eigenvalues, respectively.

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An ARIMA model is notated as ARIMA p,d,q), where p indicates the AR order, d the differencing order and q the MA order.

The model was defined with an autoregressive part of order p, a moving average part of order q, a seasonal-autoregressive part of order P, a seasonal-moving average part of order Q, differencing and seasonal-differencing orders d and D, and periodic variable n.

Therefore, using the sequential definition of the global eigenvalues from "R-D Exponential Fitting Test (R-D EFT)", we can estimate the model order considering four modes.

In this section, the multi-dimensional model order selection schemes are proposed based on the global eigenvalues, the R-D subspace, or tensor-based data model.

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