In this work the methods of mode decomposition and Fourier analysis of quantum fields on curved spacetimes previously available mainly for the scalar fields on Friedman-Robertson-Walker spacetimes are extended to arbitrary vector fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier analysis. Explicit constructions are performed for a variety of situations arising in homogeneous cosmology. A number of results concerning classical and quantum fields known for very restricted situations are generalized to cover almost all cosmological models.

In many situations, fluid flows can exhibit a wide range of temporal and spatial phenomena. It has become common to extract physically important features, called modes, as a first step in the analysis of flows with high complexity. One of the most prominent modal analysis techniques in the context of fluid dynamics is Proper Orthogonal Decomposition (POD), which enables extraction of energetically coherent structures present in the flow field. This method does, however, suffer from the lack of connection with the mathematical theory of dynamical systems and its utility in the analysis of arbitrarily complex flows might therefore be limited. In the present work, we instead consider application of the Koopman Mode Decomposition (KMD), which is an approach based on spectral decomposition of the Koopman operator. This technique is employed for modal analysis of the incompressible, two-dimensional ow past two side-by-side cylinders at Re = 60 and with a non-dimensional cylinder gap spacing g* = 1. This particular configuration yields a wake ow which exhibits in-phase vortex shedding during finite time, while later transforming into the so-called flip-flopping phenomena, which is characterised by a slow, periodic switching of the gap ow direction during O(10) vortex shedding cycles. The KMD approach yields modal structures which, in contrary to POD, are associated with specific oscillation frequencies. Specifically, these structures are here vorticity modes. By studying these modes, we are able to extract the ow components which are responsible for the flip-flop phenomenon. In particular, it is found that the flip-flop instability is mainly driven by three different modal structures, oscillating with Strouhal frequencies St1 = 0:023, St2 = 0:121 and St3 = 0:144, where it is noted that St3 = St1 + St2. In addition, we study the in-phase vortex shedding regime, as well as the transient regime connecting the two states of the flow. The study of the in-phase vortex shedding reveals| - not surprisingly - the presence of a single fundamental frequency, while the study of the transient reveals a Koopman spectrum which might indicate the existence of a bifurcation in the phase space of the flow field; this idea has been proposed before in Carini et al. (2015b). We conclude that the KMD offers a powerful framework for analysis of this ow case, and its range of applications might soon include even more complex flows.

No description available.

The study of nonlinear dynamical systems via the spectrum of the Koopman operator has emerged as a paradigm shift, from the Poincaré's geometric picture that centers the attention on the evolution of states, to the Koopman operator's picture that focuses on the evolution of observables. The Koopman operator-theoretic framework rests on the idea of lifting the states of a nonlinear dynamical system to a higher dimensional space; these lifted states are referred to as the Koopman eigenfunctions. To determine the Koopman eigenfunctions, one performs a nonlinear transformation of the states by relying on the so-called observables, that is, scalar-valued functions of the states. In other words, one executes a change of coordinates from the state space to another set of coordinates, which are denominated Koopman canonical coordinates. The variables defined on these intrinsic coordinates will evolve linearly in time, despite the underlying system being nonlinear. Since the Koopman operator is linear, it is natural to exploit its spectral properties. In fact, the theory surrounding the spectral properties of linear operators has well-known implications in electric power systems. Examples include small-signal stability analysis and direct methods for transient stability analysis based on the Lyapunov function. From the applications' standpoint, this framework based on the Koopman operator is attractive because it is capable of revealing linear and nonlinear modes by only applying well-established tools that have been developed for linear systems. With the challenges associated with the high-dimensionality and increasing uncertainties in the power systems models, researchers and practitioners are seeking alternative modeling approaches capable of incorporating information from measurements. This is fueled by an increasing amount of data made available by the wide-scale deployment of measuring devices such as phasor measurement units and smart meters. Along these lines, the Koopman operator theory is a promising framework for the integration of data analysis into our mathematical knowledge and is bringing an exciting perspective to the community. The present dissertation reports on the application of the Koopman operator for identification, estimation, and control of electric power systems. A dynamic state estimator based on the Koopman operator has been developed and compares favorably against model-based approaches, in particular for centralized dynamic state estimation. Also, a data-driven method to compute participation factors for nonlinear systems based on Koopman mode decomposition has been developed; it generalizes the original definition of participation factors under certain conditions. / PHD / Electric power systems are complex, large-scale, and given the bidirectional causality between economic growth and electricity consumption, they are constantly being expanded. In the U.S., some of the electric power grid facilities date back to the 1880s, and this aging system is operating at its capacity limits. In addition, the international pressure for sustainability is driving an unprecedented deployment of renewable energy sources into the grid. Unlike the case of other primary sources of electric energy such as coal and nuclear, the electricity generated from renewable energy sources is strongly influenced by the weather conditions, which are very challenging to forecast even for short periods of time. Within this context, the mathematical models that have aided engineers to design and operate electric power grids over the past decades are falling short when uncertainties are incorporated to the models of such high-dimensional systems. Consequently, researchers are investigating alternative data-driven approaches. This is not only motivated by the need to overcome the above challenges, but it is also fueled by the increasing amount of data produced by today’s powerful computational resources and experimental apparatus. In power systems, a massive amount of data will be available thanks to the deployment of measuring devices called phasor measurement units. Along these lines, the Koopman operator theory is a promising framework for the integration of data analysis into our mathematical knowledge, and is bringing an exciting perspective on the treatment of high-dimensional systems that lie in the forefront of science and technology. In the research work reported in this dissertation, the Koopman operator theory has been exploited to seek for solutions to some of the challenges that are threatening the safe, reliable, and efficient operation of electric power systems.

Le travail présenté dans cette thèse a pour objectif de proposer et d'évaluer les performances de différentes techniques de suppression de bruit impulsif de type asynchrone adaptées aux transmissions sur courants porteurs en lignes (CPL) indoor. En effet, outre les caractéristiques physiques spécifiques à ce type de canal de transmission, le bruit impulsif asynchrone reste la contrainte sévère qui pénalise les systèmes CPL en terme de QoS. Pour remédier aux dégradations dues aux bruits impulsifs asynchrones, les techniques dites de retransmission sont souvent très utilisées. Bien qu'elles soient efficaces, ces techniques de retransmission conduisent à une réduction de débit et à l’introduction de délais de traitement supplémentaires pouvant être critiques pour des applications temps réel. Par ailleurs, plusieurs solutions alternatives sont proposées dans la littérature pour minimiser l'impact du bruit impulsif sur les transmissions CPL. Cependant, le nombre de techniques, qui permettent d'obtenir un bon compromis entre capacité de correction et complexité d'implantation reste faible pour les systèmes CPL. Dans ce contexte, nous proposons dans un premier temps d'utiliser un filtre linéaire adaptatif : le filtre de Widrow, nommé aussi ADALINE (ADAptive LInear NEuron), que nous utilisons comme méthode de débruitage pour les systèmes CPL. Pour améliorer les performances du débruitage effectué à l'aide d'ADALINE, nous proposons d'utiliser un réseau de neurones (RN) non linéaire comme méthode de débruitage. Le réseau de neurones est un bon outil qui est une généralisation de la structure du filtre ADALINE. Dans un deuxième temps, pour améliorer les performances du débruitage par un réseau de neurones, nous proposons un procédé d'annulation du bruit impulsif constitué de deux algorithmes : EMD (Empirical Mode Decomposition) associé à un réseau de neurones de type perceptron multicouches. L'EMD effectue le prétraitement en décomposant le signal bruité en signaux moins complexes et donc plus facilement analysables. Après quoi le réseau de neurones effectue le débruitage. Enfin, nous proposons une méthode d'estimation du bruit impulsif utilisant la méthode GPOF (Generalized Pencil Of Function). L'efficacité des deux méthodes, EMD-RN et la technique utilisant l'algorithme GPOF, est évaluée en utilisant une chaîne de simulation de transmission numérique compatible avec le standard HPAV. / The aim of our thesis is to propose and to evaluate the performances of some asynchronous impulsive noise mitigation techniques for transmission over indoor power lines. Indeed, besides the particular physical properties that characterize this transmission channel type, asynchronous impulsive noise remains the difficult constraint to overcome on power lines communications (PLC). Usually, the impact of asynchronous impulsive disturbances over power lines is partly compensated by means of retransmission mechanisms. However, the main drawbacks of the use of retransmission solutions for impulsive noise mitigation are the bitrate loss and the induced time delays that may be prohibitive for real-time services. Although several other countering strategies are proposed in the literature, only very few of them have a good compromise between correction capability and implementing complexity for PLC systems. In this context, we proposed an adaptive linear filter, the Widrow filter, also known as ADALINE (Adaptive LInear neurons), as a denoising method for PLC systems. To improve the performance of the denoising method using ADALINE, we proposed to use a neural network (NN) as a nonlinear denoising method. The neural network is a good generalization of the ADALINE filter. In a second step, to improve the performances of denoising by NN, we proposed a combined denoising method based on EMD (Empirical Mode Decomposition) and MLPNN (Multi Layer Perceptron Neural Network). The noised signal is pre-processed by EMD which decomposes it into signals less complex and therefore more easily analyzed. Then the MLPNN denoises it. Finally, we proposed an asynchronous impulsive noise estimation method using the GPOF method (Generalized Pencil Of Function). The performances of the two methods, EMD-MLPNN and GPOF technique, are evaluated using a PLC transmission chain compatible with the HPAV standard.

The idea of time series forecasting techniques is that the past has certain information about future. So, the question of how the information is encoded in the past can be interpreted and later used to extrapolate events of future constitute the crux of time series analysis and forecasting. Several methods such as qualitative techniques (e.g., Delphi method), causal techniques (e.g., least squares regression), quantitative techniques (e.g., smoothing method, time series models) have been developed in the past in which the concept lies in establishing a model either theoretically or mathematically from past observations and estimate future from it. Of all the models, time series methods such as autoregressive moving average (ARMA) process have gained popularity because of their simplicity in implementation and accuracy in obtaining forecasts. But, these models were formulated based on certain properties that a time series is assumed to possess. Classical decomposition techniques were developed to supplement the requirements of time series models. These methods try to define a time series in terms of simple patterns called trend, cyclical and seasonal patterns along with noise. So, the idea of decomposing a time series into component patterns, later modeling each component using forecasting processes and finally combining the component forecasts to obtain actual time series predictions yielded superior performance over standard forecasting techniques. All these methods involve basic principle of moving average computation. But, the developed classical decomposition methods are disadvantageous in terms of containing fixed number of components for any time series, data independent decompositions. During moving average computation, edges of time series might not get modeled properly which affects long range forecasting. So, these issues are to be addressed by more efficient and advanced decomposition techniques such as Wavelets and Empirical Mode Decomposition (EMD). Wavelets and EMD are some of the most innovative concepts considered in time series analysis and are focused on processing nonlinear and nonstationary time series. Hence, this research has been undertaken to ascertain the predictability of nonstationary time series using wavelet and Empirical Mode Decomposition (EMD) based ARMA models. The development of wavelets has been made based on concepts of Fourier analysis and Window Fourier Transform. In accordance with this, initially, the necessity of involving the advent of wavelets has been presented. This is followed by the discussion regarding the advantages that are provided by wavelets. Primarily, the wavelets were defined in the sense of continuous time series. Later, in order to match the real world requirements, wavelets analysis has been defined in discrete scenario which is called as Discrete Wavelet Transform (DWT). The current thesis utilized DWT for performing time series decomposition. The detailed discussion regarding the theory behind time series decomposition is presented in the thesis. This is followed by description regarding mathematical viewpoint of time series decomposition using DWT, which involves decomposition algorithm. EMD also comes under same class as wavelets in the consequence of time series decomposition. EMD is developed out of the fact that most of the time series in nature contain multiple frequencies leading to existence of different scales simultaneously. This method, when compared to standard Fourier analysis and wavelet algorithms, has greater scope of adaptation in processing various nonstationary time series. The method involves decomposing any complicated time series into a very small number of finite empirical modes (IMFs-Intrinsic Mode Functions), where each mode contains information of the original time series. The algorithm of time series decomposition using EMD is presented post conceptual elucidation in the current thesis. Later, the proposed time series forecasting algorithm that couples EMD and ARMA model is presented that even considers the number of time steps ahead of which forecasting needs to be performed. In order to test the methodologies of wavelet and EMD based algorithms for prediction of time series with non stationarity, series of streamflow data from USA and rainfall data from India are used in the study. Four non-stationary streamflow sites (USGS data resources) of monthly total volumes and two non-stationary gridded rainfall sites (IMD) of monthly total rainfall are considered for the study. The predictability by the proposed algorithm is checked in two scenarios, first being six months ahead forecast and the second being twelve months ahead forecast. Normalized Root Mean Square Error (NRMSE) and Nash Sutcliffe Efficiency Index (Ef) are considered to evaluate the performance of the proposed techniques. Based on the performance measures, the results indicate that wavelet based analyses generate good variations in the case of six months ahead forecast maintaining harmony with the observed values at most of the sites. Although the methods are observed to capture the minima of the time series effectively both in the case of six and twelve months ahead predictions, better forecasts are obtained with wavelet based method over EMD based method in the case of twelve months ahead predictions. It is therefore inferred that wavelet based method has better prediction capabilities over EMD based method despite some of the limitations of time series methods and the manner in which decomposition takes place. Finally, the study concludes that the wavelet based time series algorithm could be used to model events such as droughts with reasonable accuracy. Also, some modifications that could be made in the model have been suggested which can extend the scope of applicability to other areas in the field of hydrology.

對投資者而言，投資股市的目的就是賺錢，但影響股價因素眾多，我們要如何判斷明天是漲是跌？因此如何建立一個準確的預測模型，一直是財務市場研究的課題之一，然而財務市場一直被認為是一個複雜.充滿不確定性及非線性的動態系統，這也是在建構模型上一個很大的阻礙，本篇研究中使用的EEMD方法則適合解決如金融市場或氣候等此類的非線性問題及有趨勢性的資料上。 在本研究中，我們將EEMD結合ANN建構出兩種不同形式的模型去進行台股個股的預測，也試圖改善ARMA模型使其預測效果較好；此外為了能夠達到分散風險的效果，採用了投資組合的方式，在權重的決定上，我們結合動態與靜態的方式來計算權重；至於在交易策略上，本研究也加入了移動平均線，希望能找到最適合的預測模型，本研究所使用的標的物為曾在該期間被列為注意股票的10檔股票。 另外，我們也分析了影響台股個股價格波動的因素，透過EEMD拆解，我們能夠從中得到具有不同意義的本徵模態函數(IMF)，藉由統計值分析重要的IMF其所代表的意義。例如：影響高頻波動的重要因素為新聞媒體或突發事件，影響中頻的重要因素為法人買賣及季報，而影響低頻的重要因素則為季節循環。 結果顯示，EEMD-ANN Model 1是一個穩健的模型，能夠創造出將近20%的年報酬率，其次為EEMD-ANN Model 2，在搭配移動平均線的策略後，表現與Model 1差不多，但在沒有配合移動平均線策略時，雖報酬率仍為正，但較不穩定，因此從研究結果也可以看到，EEMD-ANN的模型皆表現比ARMA的預測模型好。 / The main purpose of investing is to earn profits for an investor, but there are many factors that can influence stock price. Investments want to know the price will rise or fall tomorrow. Therefore, how to establish an accurate forecasting model is one of the important issue that researched by researchers of financial market. However, the financial market is considered of a complex, uncertainty, and non-linear dynamic systems. These characteristics are obstacles on constructing model. The measure, EEMD, used in this study is suitable to solve questions that are non-linear but have trends such as financial market, climate and so on. In this thesis, we used three models including ARMA model and two types of EEMD-ANN composite models to forecast the stock price. In addition, we tried to improve ARMA model, so a new model was proposed. Through EEMD, the fluctuation of stock price can be decomposed into several IMFs with different economical meanings. Moreover, we adopted portfolio approach to spread risks. We integrate the static weight and the dynamic weight to decide the optimal weights. Also, we added the moving average indicator to our trading strategy. The subject matters in this study are 10 attention stocks. Our results showed that EEMD-ANN Model 1 is a robust model. It is not only the best model but also can produce near 20% of 1-year return ratio. We also find that our EEMD-ANN model have better outcome than those of the traditional ARMA model. Owing to that, the increases of trading performance would be expected via the selected EEMD-ANN model.

<p>Combustion instability, a complex phenomenon observed in combustion chambers is due to the coupling between heat release and other unsteady flow processes. Combustion instability has long been a topic of interest to rocket scientists and has been extensively investigated experimentally and computationally. However, to date, there is no computational tool that can accurately predict the combustion instabilities in full-size combustors because of the amount of computational power required to perform a high-fidelity simulation of a multi-element chamber. Hence, the focus is shifted to reduced fidelity computational tools which may accurately predict the instability by using the information available from the high-fidelity simulations or experiments of single or few-element combustors. One way of developing reduced fidelity computational tools involves using a reduced fidelity solver together with the flame transfer functions that carry important information about the flame behavior from a high-fidelity simulation or experiment to a reduced fidelity simulation.</p> <p> </p> <p>To date, research has been focused mainly on premixed flames and using acoustic solvers together with the global flame transfer functions that were obtained by integrating over a region. However, in the case of rockets, the flame is non-premixed and distributed in space and time. Further, the mixing of propellants is impacted by the level of flow fluctuations and can lead to non-uniform mean properties and hence, there is a need for reduced fidelity solver that can capture the gas dynamics, nonlinearities and steep-fronted waves accurately. Nonlinear Euler equations have all the required capabilities and are at the bottom of the list in terms of the computational cost among the solvers that can solve for mean flow and allow multi-dimensional modeling of combustion instabilities. Hence, in the current work, nonlinear Euler solver together with the spatially distributed local flame transfer functions that capture the coupling between flame, acoustics, and hydrodynamics is explored.</p> <p> </p> <p>In this thesis, the approach to extract flame transfer functions from high-fidelity simulations and their integration with nonlinear Euler solver is presented. The dynamic mode decomposition (DMD) was used to extract spatially distributed flame transfer function (FTF) from high fidelity simulation of a single element non-premixed flame. Once extracted, the FTF was integrated with nonlinear Euler equations as a fluctuating source term of the energy equation. The time-averaged species destruction rates from the high-fidelity simulation were used as the mean source terms of the species equations. Following a variable gain approach, the local species destruction rates were modified to account for local cell constituents and maintain correct mean conditions at every time step of the nonlinear Euler simulation. The proposed reduced fidelity model was verified using a Rijke tube test case and to further assess the capabilities of the proposed model it was applied to a single element model rocket combustor, the Continuously Variable Resonance Combustor (CVRC), that exhibited self-excited combustion instabilities that are on the order of 10% of the mean pressure. The results showed that the proposed model could reproduce the unsteady behavior of the CVRC predicted by the high-fidelity simulation reasonably well. The effects of control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size are also studied. The reduced fidelity model could reproduce the limit cycle amplitude within a few percent of the mean pressure. The successful constraints on the model include good spatial resolution and FTF with all modes up to at least one dominant frequency higher than the frequencies of interest. Furthermore, the reduced fidelity model reproduced consistent mode shapes and linear growth rates that reasonably matched the experimental observations, although the apparent ability to match growth rates needs to be better understood. However, the presence of significant heat release near a pressure node of a higher harmonic mode was found to be an issue. This issue was rectified by expanding the pressure node of the higher frequency mode. Analysis of two-dimensional effects and coupling between the local pressure and heat release fluctuations showed that it may be necessary to use two dimensional spatially distributed local FTFs for accurate prediction of combustion instabilities in high energy devices such as rocket combustors. Hybrid RANS/LES-FTF simulation of the CVRC revealed that it might be necessary to use Flame Describing Function (FDF) to capture the growth of pressure fluctuations to limit cycle when Navier-Stokes solver is used.</p> <p> </p> <p>The main objectives of this thesis are:</p> <p>1. Extraction of spatially distributed local flame transfer function from the high fidelity simulation using dynamic mode decomposition and its integration with nonlinear Euler solver</p> <p>2. Verification of the proposed approach and its application to the Continuously Variable Resonance Combustor (CVRC).</p> <p>3. Sensitivity analysis of the reduced fidelity model to control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size.</p> <p> </p> <p>The goal of this thesis is to contribute towards a reduced fidelity computational tool which can accurately predict the combustion instabilities in practical systems using flame transfer functions, by providing a path way for reduced fidelity multi-element simulation, and by defining the limitations associated with using flame transfer functions and nonlinear Euler equations for non-premixed flames.</p> <p> </p><br>

Le contrôle d'écoulements turbulents est un enjeu majeur en aérodynamique. Cependant, la présence d'un grand nombre de degrés de libertés et d'une dynamique complexe rend délicat la modélisation dynamique de ces écoulements qui est pourtant nécessaire à la conception d'un contrôle efficace. Au cours de cette thèse, différentes directions ont été suivies afin de développer des modèles réduits dans des configurations réalistes d'écoulements et d'utiliser ces modèles pour le contrôle.Premièrement, la décomposition en modes dynamiques (DMD), et certaines de ses variantes, ont été exploitées en tant que base réduite afin d'extraire au mieux le comportement dynamique de l'écoulement. Par la suite, nous nous sommes intéressés à l'assimilation de données 4D-Var qui permet de combiner des informations inhomogènes provenant d'un modèle dynamique, d'observations et de connaissances a priori du système. Nous avons ainsi élaboré des modèles réduits POD et DMD d'un écoulement turbulent autour d'un cylindre à partir de données expérimentales PIV. Finalement, nous avons considéré le contrôle d'écoulement dans un contexte d'interaction fluide/structure. Après avoir montré que les mouvements de solides immergés dans le fluide pouvaient être représentés comme une contrainte supplémentaire dans le modèle réduit, nous avons stabilisé un écoulement de sillage de cylindre par oscillation verticale. / Control of turbulent flows is still today a challenge in aerodynamics. Indeed, the presence of a high number of active degrees of freedom and of a complex dynamics leads to the need of strong modelling efforts for an efficient control design. During this PhD, various directions have been followed in order to develop reduced-order models of flows in realistic situations and to use it for control. First, dynamic mode decomposition (DMD), and some of its variants, have been exploited as reduced basis for extracting at best the dynamical behaviour of the flow. Thereafter, we were interested in 4D-variational data assimilation which combines inhomogeneous informations coming from a dynamical model, observations and an a priori knowledge of the system. POD and DMD reduced-order models of a turbulent cylinder wake flow have been successfully derived using data assimilation of PIV measurements. Finally, we considered flow control in a fluid-structure interaction context. After showing that the immersed body motion can be represented as an additional constraint in the reduced-order model, we stabilized a cylinder wake flow by vertical oscillations.

本文有兩個研究目標，第一個是比較政大用電量與氣溫之間的相關性，第二則是分析影響黃金價格波動的因素。本文使用到的研究方法有希爾柏特黃轉換（HHT）與一些統計值。 　　本研究使用的分析數據如下：政大逐時用電量、台北逐時氣溫以及倫敦金屬交易所（London Metal Exchange）的月平均黃金價格。透過經驗模態分解法（EMD），我們可以將分析數據拆解成數個互相獨立的分量，再藉由統計值選出較重要的分量並分析其意義。逐時用電量的重要分量為日分量、週分量與趨勢；逐時氣溫的重要分量為日分量與趨勢；月平均黃金價格的重要分量則是低頻分量與趨勢。 藉由這些重要分量，我們可以更加了解原始數據震盪的特性，並且選出合理的平均週期將所有的分量分組，做更進一步的分析。逐時用電量與逐時氣溫分成高頻、中頻、低頻與趨勢四組，其中低頻與趨勢相加的組合具有最高的相關性。月平均黃金價格則是分為高頻、低頻與趨勢三組，其中高頻表現出供需以及突發事件等短週期因素，低頻與歷史上對經濟有重大影響的事件相對應，趨勢則是反應出通貨膨脹的現象。 / There are two main separated researched purposes in this thesis. First one is comparing the correlation between electricity consumption and temperature in NCCU. Another one is analyzing the properties of gold price volatility. The methods used in the study are Hilbert-Huang transform (HHT) and some statistical measures. 　　The following original data: hourly electricity consumption in NCCU, hourly temperature in Taipei, and the LME monthly gold prices are decomposed into several components by empirical mode decomposition (EMD). We can ascertain the significant components and analyze their meanings or properties by statistical measures. The significant components of each data are shown as follows: daily component, weekly component and residue for hourly electricity consumption; daily component and residue for hourly temperature; low frequency components and residue for the LME monthly gold prices. 　　We can understand more properties about these data according to the significant components, and dividing the components into several terms based on reasonable mean period. The components of hourly electricity consumption and hourly temperature are divided into high, mid, low frequency terms and trends, and the composition of low frequency terms and trends have the highest correlation between them. The components of LME monthly gold prices are divided into high, low frequency term and trend. High frequency term reveals the supply-demand and abrupt events. The low frequency term represents the significant events affecting economy seriously, and trend shows the inflation in the long run.