Linearly immutable continuously time series modelled bivariate stochastic processes with vector values: Distinguishing features

Introduction

We examine the statistical properties of the linearity relationship between X(t) and Y(t) as represented by the extended finite Fourier transformation, following proposals by D.R. Brillinger (1967) [3] , M. Rosenblatt, (1967) [3] ; D.R. Brillinger (2001) [4] , Ghazal and Farag (2005) [8] , Teamah (2004) [1] and Elhassain (2014) [2] and Ghazal, et al. (2005) [8] ; For a brief overview, of the format of the article, consider the following: (Part1) is an introductory section., Part (2) explored the Approximate Attributes of process as observed, investigated the Approximating aspects of the process as unobserved in Part (3), and in Part (4) we implement our theoretic ideas into practical application, Our method was applied to a study of Average monthly energy imports and exports of the General Electric Corporation between January 2011 and December 2020.

The Observed Process's Approximate Attributes

Presume of a fixed sequence that is a vector of values (𝑖 + 𝑗) ℜ(𝑡) = [𝑋(𝑡) 𝑌(𝑡)] 𝑇 (2.1) 𝑡 = 0, ±1, ±2, . .., 𝑋(𝑡) i-valued-vector and 𝑌(𝑡) j-valued-vector.

That 𝛽 𝑎 (𝑡), 𝑎 = 1,2, . . . , 𝑖, (𝑡 ∈ 𝑅) exists for all t, which is independent on

Independent data can be used successfully without caring about the results of another. For the modified series perception, considering

Assumption

The time interval (t) is limited in the data window ℓ 𝑎 (𝑇) (𝑡) so that it has a restricted range, a finite variation, and vanishes between0 𝑎𝑛𝑑 𝑇 -1.

Then the dispersion of 𝜔 𝑎 (𝑇) (ℎ) is thus determined to be as follows: We'll clarify each one. In particular, we obtain since 𝑓 𝑎𝑏 (𝜓)is continuoue at Ψ = h , 𝑎, 𝑏 = 1, . . , 𝑚𝑖𝑛(𝑖, 𝑗), then we have

Assuming that 𝑓 𝑎𝑏 (𝜓) is continuing 𝜓 = ℎ, 𝑎, 𝑏 = 1, . . , 𝑚𝑖𝑛(𝑖, 𝑗), we've got 𝐵 2 ≤ 𝛺. Now, 𝐵 2 is very small, so any 𝛺 very small, so 𝐵 2 = 0. With a constant G that limits the value of 𝑓 𝑎𝑎 (ℎ), 𝑎 = 1, . . . , 𝑚𝑖𝑛( 𝑖, 𝑗), ℎ ∈ 𝑅 to be limited, we have

The theorem has been proved, then.

Analytical Exported of Energy

In this study, we will compare the results produced using the standard methods, in which all data is observed, with those obtained using a fixed-time process model with partial missed of data. With a fixed j-valued-vector time process,𝑌 𝑎 (𝑡), (𝑡 = (1,2, . . . , 𝑇] and a stochastically independent Bernoulli sequence, 𝛽 𝑎 (𝑡), the results can be written as 𝜛 𝑎 (𝑡) = 𝛽 𝑎 (𝑡)𝑌 𝑎 (𝑡), 𝑎 = 1,2, . . . , 𝑗, an average monthly energy exports with complete records is denoted by 𝑌 𝑎 (𝑡), where 𝑎 = 1,2, . . . , 𝑗. Table 2 compares the results for the two scenarios where some data are missing at random (𝛽 = 0) and results for the standard case(𝛽 = 0).

Regression Model Analysis of Energy Imports and Exports

In this analysis, we will evaluate the two scenarios listed in the table below using our outcomes from monthly averages of energy imports and exports are analyzed using a regression model where the outcomes from model with some missed of observations and the classical results from the same model when all observations are available is shown through table (3).

Conclusion

1. The findings of missing data time process analysis were identical to those obtained from regular time series analysis. 2. As with missed data, the results of the investigated for both the X(t) and Y(t) regression models are equivalent since they met the analytical, numerical, and least squares constraints for a regression model between standard time series.

References

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