J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
1Ae8.
Signal Processing Of The One-Factor Mean-Reverting Model In Energy System Getut Pramesti1O and Ristu Saptono2 Department of Mathematics Education .
Universitas Sebelas Maret.
Indonesia.
getutpramesti@staff.
Department of Informatics.
Universitas Sebelas Maret.
Indonesia.
saptono@staff.
Abstract.
One of the models that can be considered in the energy system is the one-factor mean-reverting process.
We propose the one-factor mean-reverting model with sinusoidal signal processing involved.
The frequency component of the model can be estimated with a high-frequency scheme.
The estimation of the frequency component is believed to produce a precise estimate.
This is because the highfrequency scheme has the potential to handle possible non-linear coefficient cases in a unified way, that is, nh Ie O, and nh2 Ie 0.
This paper shows that the frequency component estimator in the one-factor mean-reverting model is strongly consistent with the rate convergence, namely .
It is also can be shown that the estimator has a normal approximation with a mean of 0 and variance 16 .
2 ).
We applied the proposed model to the energy systems data.
Key words and Phrases:
one-factor, mean-reverting, frequency, signals, high- Kata kunci: satu-faktor, mean-reverting, frekuensi, sinyal, high-frequency INTRODUCTION Diffusion process contain drift and diffusion coefficients.
The stochastic diffusion process has two type depend on its drift coefficients, namely homogeneous diffusion process and non-homogeneous diffusion process.
The first model whose drift coefficient is a constant or not depends on time, while the second model whose drift coefficient depends on the time variable.
The one-factor mean-reverting model is a part of the non-homogeneous diffusion process.
The mean-reversion process is an interesting part of a stochastic O Corresponding author 2020 Mathematics Subject Classification: 92E24, 35B40, 62M10 Received: 21-02-2025, accepted: 12-03-2025.
differential equation since the process can be regarded as an energyAos path, which is the pattern will always tend to its mean.
shows the digital signal process can be used in approaching the periodicity term of the seasonal variation of the energy system time series data.
The theoretical background became an important thing in proposing a forecast energy model, i.
, in the case of the one-factor mean-reverting Moreover, we can apply the proposed model to the systems.
This study will provide the consistency of the least-square of the frequency component of the signal processing, and also the asymptotic normality of the estimates.
The numerical study will be examined to support the theoretical findings.
Therefore, we do literature research on signal processing .
, the harmonic sinusoidal function .
, and the previous research of .
which provides the theoretical background of the sinusoidal signal processing of the drift.
The proposed model will be applied to the energy systems.
The electric power and energy systems model is accommodated in .
Consider {Xt } is a stochastic process that has a following equation dXt = OeXt dt Edwt log Ct = AAt Xt , .
where AAt is a deterministic function of t, t OO .
O).
If we denote Yt := log Ct , .
then we can find the relation of the one-factor mean-reverting model and OrnsteinAeUhlenbeck process.
Observe that d(Yt Oe AAt ) Oe(Yt Oe AAt )dt Edwt dYt Oe dAAt (OeYt AAt ) dt Edwt dAAt dt Edwt .
OeYt AAt Assume that dAAt (E) .
with = (.
E), for any parameter E of the deterministic function AAt .
then we dYt = (OeYt bt ()) dt Edwt .
bt () := AAt (E) From .
, we can find the relation between the diffusion process, the Gompertz process, and the one-factor mean-reverting model.
If bt () = 0 then the expression .
can be defined as a diffusion process.
If bt () = c, where c is a constant then the expression .
can be classified as a time-homogeneous Gompertz diffusion process .
ee e.
, .
, .
for reference.
If bt () is defined as .
then the expression .
can be called as the one-factor mean-reverting model.
This model is part of a time-inhomogeneous OrnsteinAeUhlenbeck process .
ee e.
, .
for referenc.
Discrete-time sinusoidal signals can be expressed as .
ee i.
, .
for referenc.
= A cos(On .
, .
A n is an integer variable.
A A is the amplitude of the sinusoidal.
A O is the frequency component.
A c is the phase of the sinusoidal.
If we took continuous-time sinusoidal signals, namely AAt () =: sin.
, with t OO .
O), then we have bt () = sin.
, and the one-factor mean-reverting process with sinusoidal signal as follow dYt = (OeYt sin.
) dt dwt .
For = 1, then we have .
dYt = (OeYt sin.
) dt dwt .
Further.
Z t Z t dYt = Z t (OeYs sin.
) ds Z t Yt = Y0 (OeYs sin.
) ds wt defined on an underlying complete filtered probability space (E.
F, (Ft )tOOR .
P) with Ft = E.
s : s O .
, the parameter spaces o OC .
O) are bounded convex domains.
The true parameter value is denoted by 0 OO o.
Using EulerAeMaruyama approach, we have Z tj Ytj = YtjOe1 (OeYs sin.
) ds OIj w, .
tjOe1 with OIj w = wtj Oe wtjOe1 .
Moreover, we define OIj Y := Ytj Oe YtjOe1 , with Z tj
P 0
OIj Y = (OeYs sin.
) ds OIj w.
tjOe1 We apply least-squares estimation (LSE) for the parameters of .
based on discrete time observations, that is, we concentrate on the situation where the diffusion process is observed at discrete times 0 O t0 < t1 < A A A < tn , where tnj = tj = jh, with j O n and for some non-random discrete instant time step h := hn .
, .
for reference of the time ste.
, h Ie 0, .
such that for n Ie O.
T := T n = nh.
T n Ie O, .
nh2 Ie 0.
We define the LSE of as Cn OO argmin Qn (), .
Qn () = 2 OIj Y Oe OeYtjOe1 sin.
jOe1 ) cos.
jOe1 ) h , h j=1 with OIj Y = Ytj Oe YtjOe1 .
MAIN RESULTS
In this section, first we provide the consistency of the estimate.
The consistency of the LSE of .
First of all, we define Gn () = [Qn () Oe Qn .
)] .
Using Lemma 4.
1 of .
we show the estimator is a strongly consistent.
Observe that Gn () = [Qn () Oe Qn .
)] .
jOe1 .
) Oe gjOe1 ()] n .
jOe1 .
) Oe gjOe1 ()] OIj w op .
, n j=1 T j=1 gjOe1 () = OeYtjOe1 sin.
jOe1 ) cos.
jOe1 ).
Moreover, we find the following expression Gn () = G1,j () G2,j () op .
, .
G1,j () = n j=1 sin.
tjOe1 ) Oe sin.
jOe1 ) 0 cos.
tjOe1 ) Oe cos.
jOe1 ) , .
G2,j () = 2E X n sin( Oe sin.
cos( Oe cos.
OIj w.
jOe1 jOe1 jOe1 jOe1 T n j=1 .
From .
, we have 1 Xn .
jOe1 ) Oe sin.
tjOe1 )] .
tjOe1 ) Oe cos.
jOe1 )] G1,j () = n j=1 2 .
jOe1 ) Oe sin.
tjOe1 )] .
tjOe1 ) Oe cos.
jOe1 )] = 1 .
02 ) o Clearly, we obtain the fact lim inf G1,j () > 0.
lim sup G2,j () = 0.
T n IeO Whenever for .
T n IeO Therefore, based on .
, .
, and Lemma 4.
1 of .
we get C OeOeIe T n Ie O.
The asymptotic normality of the LSE of .
Now, we will provide the proof of asymptotic normality of the estimate.
Using TaylorAos approach, we have Qn () = Qn .
) OCQn (O )( Oe 0 ) OCQn () = OCQn .
) OC 2 Qn (O )( Oe 0 ) O is the point between and 0 .
Because of .
then OCQn () = 0.
Therefore OCQn .
) OC22 Qn (O )( Oe 0 ) OC0 Qn .
) OeOC22 Qn (O )( Oe 0 ) By Lemma 3.
12 of .
for the right hand side of the above equation, then we have OC0 Qn .
) = Oe p OC22 Qn .
) p (T n )3 ( Oe 0 ).
(T n )3 (T n )3 0 (T n )3 We will proceed the left hand side and the first of the right hand side of the equation .
to provide the following Theorem prove.
Theorem 2.
We have (T ) ( Oe 0 ) Oe Ie N 0, .
0 ) T n Ie O.
Proof.
We will prove the Theorem 2.
1 with the show the following expressions.
Oep (T n )3 OC22 Qn .
) p Oe Ie .
02 ), (T n )3 .
(T n )3 OCQn .
) Oe IeN 0, .
0 ) .
First, we will take a look .
If E0 jOe1 denotes the expectation operator under P0 conditional on FtjOe1 , then by applying Lemma 9 of .
, we can obtain E0 jOe1 .
jOe1 cos.
tjOe1 ) Oe 0 tjOe1 sin.
tjOe1 )] (T n )3 .
jOe1 cos.
tjOe1 ) Oe 0 tjOe1 sin.
tjOe1 )] |FtjOe1 (T n )3 .
jOe1 cos.
tjOe1 ) Oe 0 tjOe1 sin.
tjOe1 )] (T n )3 j=1 h X 2 tjOe1 cos2 .
tjOe1 ) 02 t2jOe1 sin2 .
tjOe1 ) Oe 2t2jOe1 0 cos.
tjOe1 ) sin.
tjOe1 ) (T ) j=1 Clearly, we can easy find the expression above using Lemma 3.
1 and Corollary 3.
Hence E0 jOe1 cos( Oe sin( Oe jOe1 0 jOe1 0 jOe1 0 jOe1 (T n )3 Therefore FtjOe1 Ie .
02 ).
jOe1 cos.
tjOe1 ) Oe 0 tjOe1 sin.
tjOe1 )] Oe
(T )
Similarly with .
, we can obtain the following E0 jOe1 .
jOe1 cos.
tjOe1 ) Oe 0 tjOe1 sin.
tjOe1 )] (T n )3 Oe Ie 0.
Next, we will observe .
E0 jOe1 p OC0 gjOe1 .
)OIj w
n )3 cos( Oe
sin( 1 02
tjOe1 h = jOe1
0 jOe1
0 jOe1
0 jOe1
(T )
ESTIMATING FREQUENCY COMPONENT IN APPLIANCES
ENERGY DATASET
The proposed model was applied to real data for the energy consumption of light fixtures in one Belgium household.
These data are available at .
and the relevant paper at .
The dataset is at 10 min for about 4.
5 months.
The dataset contains 19737 energy uses in the household with a ten-minute sampling rate over a period from January 11, to May 27, 2016.
We estimate the frequency component of .
of appliances energy use .
n W.
All calculations in the empirical results have been performed in the R program.
We visualize the appliances energy of the dataset .
with the proposed model .
Figure 1.
Ten-minute sampling rate of appliances energy (Y) over a period from January 1, to February 18, 2016.
Time in hours.
We choose the considered period, namely:
A T n = 45.
T n h = 4.
5 for period January 11, 2016 to January 14, 2016.
A T n = 100.
T n h = 2.
5 for period January 11, 2016 to February 8, 2016.
A T n = 150.
T n h = 1.
5 for period January 11, 2016 to April 24, 2016.
A T n = 165.
T n h = 1.
4 for period January 11, 2016 to May 25, 2016.
Table 1.
The performance of Cn of .
for considered T n and T n h.
The Root Mean Squares Error (RMSE) of the model is given.
T nh Cn RMSE From Table 1, we can see that the estimate get better for larger T n and smaller T n h.
the RMSE of the model seems getting smaller.
CONCLUDING REMARKS
The sinusoidal signal processing of the one-factor mean-reverting model can be considered in energy system modeling.
For a simple rate of reversion = 1, and sinusoidal signal sin.
in the one-factor mean-reverting model .
tend to Normal distribution with a mean 0, and variance 16 .
2 ).
In the future, we can extend .
for general mean reversion , and various siPK nusoidal signal processing form AAt (E), i.
AAt (E) = k=1 [Ak sin.
Bk cos.
], with E = (A.
B, ).
Data availability statement The data that support the findings of this study are available in UCI Machine Learning Repository at .
ttps://archive.
edu/ml/machine-learning-databases/00374/].
These data were derived from the following resources available in the public domain:
ttps://archive.
edu/ml/datasets/Appliances energy predictio.
REFERENCES