Modeling and Forecasting Time Series Using Transfer Function and Intervention Methods

In this study, single-input transfer function model is applied to forecast monthly mean sea surface temperature(SST) in 2010 at Yeosu in Korean coastal waters. As input series, monthly mean air temperature series for ten years(2000-2009) at Yeosu in Korea is used, and Monthly mean SST at Yeosu station in Korean coastal waters is used as output series(the same period of input). To build transfer function model, first, input time series is prewhitened, and then cross-correlation functions between prewhitened input and output series are determined. The cross-correlation functions have just two significant values at time lag at 0 and 1. The lag between input and output series, the order of denominator and the order of numerator of transfer function, (b, r, s) are identified as (0, 1, 0). The selected transfer function model shows that there does not exist the lag between monthly mean air temperature and monthly mean SST, and that transfer function has a first-order autoregressive component for monthly mean SST, and that noise model was identified as . The forecasted values by the selected transfer function model are generally higher than actual SST in 2010 and have 6.4 % mean absolute percentage error(MAPE). The error is 2 % lower than MAPE by ARIMA model. This implies that transfer function model could be more available than ARIMA model in terms of forecasting performance of SST.

In the paper the algorithm of singular structural analysis and forecasting of a multidimensional series by the MSSA method is used. The program was developed, in which the steps of the method for the selection of singular decomposition components were implemented, the analysis and forecast of real time series was carried out. Singular spectrum analysis (SSA) is increasingly used in the study of time series. Unlike other methods of statistical research of time series, this method is used to study the structure, selection of individual components and forecast of both stationary and non-stationary time series. It does not require an analytical model of the series. In fact, this approach is based on the method of principal components. It is based on the transformation of a series into a matrix and its singular decomposition. After identifying the components of the singular decomposition, their grouping takes place, which leads to the decomposition of the original series into additive components, such as trend, oscillations (periodics), and noise. The SSA method allows you to continue the structure of the time series, thus building a forecast (continuation). An important direction of the development of the SSA method as a method of time series analysis is its generalization for the analysis of multidimensional time series. The method is known as MSSA (Multi-Channel SSA) or E-EOFs (Extended Empirical Orthogonal Functions). In this case, the expected result is the simultaneous decomposition of several series into interpreted components. However, a sufficiently complete theory for MSSA does not exist.

Forecasting is considered an estimation of future values based on past data, and its importance lies in future planning. Necessity often calls for finding a model that logically describes the dynamic relationships connecting a single output series to one or more input series. In this research, daily time series data for several years of maximum and minimum temperatures were used as input variables, and evaporation as a response variable for the Baghdad governorate during the period. (2012-2022). Environmental and climatic data often suffer from problems such as heterogeneity and unreliability, which are results of the non-linearity of that type of data. It is possible to use the Transfer function model, denoted by the symbol TF, to model the causal relationship between the output variable and one or more input variables after synchronizing the data temporally to achieve its homogeneity. The aim of this research is to improve the forecasting of time series data, and one of the proposed methods for this is the hybrid model combining the TF model and the Spider Monkey Optimization (SMO) Algorithm, referred to as the TF-SMO hybrid model. The outputs of the TF model are utilized as inputs in the spider monkey algorithm to integrate linear and nonlinear hybrid effects for data processing, modeling, and improving forecasting accuracy. To improve modeling and forecasting results, the (SMO) Algorithm was used as a method for optimizing the modeling of non-linear data patterns and enhancing forecasting. The hybrid model TF-SMO was proposed, which is a hybrid model of the SMO algorithm as an optimization method with the TF model. As for the issue of data heterogeneity, the Time Stratified (TS) method was used to temporally stratify the data into hot and cold seasons. Where the data for the hot and cold seasons for maximum and minimum temperatures and evaporation were divided into two groups: the first group for the training period and the second group for the testing period, approximately 65% for training and approximately 35% for testing .In this research ,data similar to real data was generated with the same sample size for the hot and cold seasons for the transfer function. The results indicate that the hybrid model achieves better accuracy compared to the TF model. Therefore, it is possible to use the TF-SMO hybrid model structure for similar accuracy in forecasting climatic time series data.

The article will present the methodology for holistic time series analysis, based on time series transformation model REFII (REFII is an acronym for Raise-Equal-Fall and the model version is II or 2) Patel et al. (Mining motifs in massive time series databases, 2002) [1], Perng and Parker (SQL/LPP: a time series extension of SQL based on limited patience patterns, 1999) [2], Popivanov and Miller (Similarity search over time series data using wavelets, 2002) [3]. The main purpose of REFII model is to automate time series analysis through a unique transformation model of time series. The advantage of this approach to a time series analysis is the linkage of different methods for time series analysis linking traditional data mining tools in time series, and constructing new algorithms for analyzing time series. REFII model is not a closed system, which means that there is a finite set of methods. This is primarily a model used for transformation of values of time series, which prepares data used by different sets of methods based on the same model of transformation in the domain of problem space. REFII model gives a new approach in time series analysis based on a unique model of transformation, which is a base for all kind of time series analyses. In combination with elements of other methods, such as self-organizing maps or frequent-pattern trees, REFII models can make new hybrid methods for efficient time temporal data mining. Similar principle of hybridization could be used as a tool for time series temporal pattern recognition. The article describes real case study illustrating practical application of described methodology.

Chapter 4 - Bayesian nonparametric methods for financial and macroeconomic time series analysis

This paper deals with the Caterpillar»-SSA method, a novel and powerful model-free method of time series analysis and forecasting. Alongside with signal processing this method is successfully used to study the time series in many various areas: in meteorology, hydrology, sociology, economics, traffic analysis, wherever the trend or periodic behavior can present. Examples of application of the Caterpillar”-SSA technique for analysis of one-dimensional time series in Latvian economics are presented in this work. We solve the task of analysis and forecasting of following time series: agricultural crop yield, milk production and purchase, number of road traffic accidents and number of registered road vehicles, electricity consumption. The application of Caterpillar»-SSA approach in geotechnical investigation for processing of data of the static penetration test of soils are offered. This method combines the advantage of many other methods, in particular, Fourier analysis and regressive analysis. At the same time it is noted for simplicity and clearness.

Oilseeds have been the backbone of India’s agricultural economy since long. Oilseed crops play the second most important role in Indian agricultural economy, next to food grains, in terms of area and production. Oilseeds production in India has increased with time, however, the increasing demand for edible oils necessitated the imports in large quantities, leading to a substantial drain of foreign exchange. The need for addressing this deficit motivated a systematic study of the oilseeds economy to formulate appropriate strategies to bridge the demand-supply gap. In this study, an effort is made to forecast oilseeds production by using Autoregressive Integrated Moving Average (ARIMA) model, which is the most widely used model for forecasting time series. One of the main drawbacks of this model is the presumption of linearity. The Group Method of Data Handling (GMDH) model has also been applied for forecasting the oilseeds production because it contains nonlinear patterns. Both ARIMA and GMDH are mathematical models well-known for time series forecasting. The results obtained by the GMDH are compared with the results of ARIMA model. The comparison of modeling results shows that the GMDH model perform better than the ARIMA model in terms of mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE). The experimental results of both models indicate that the GMDH model is a powerful tool to handle the time series data and it provides a promising technique in time series forecasting methods.

In order to provide for the nature of the various deformations effectively and discover the deformation regularity and trend of deformation body, people propose an indirect deformation prediction method. It applies the segmented model of multidimensional time series to present an inertial measurement parameters. The method applies the MIMU measurement unit to realize data collection. Through the analysis of filter compensation, it conducts the deformation prediction. MIMU signal changes on time rate and has high noise. According to the analysis of error sources of MIMU signal, we adopt the method of time series analysis to data preprocessing and establish the AR model of the time series. By using AR-Kalman filter algorithm, we take care of random error processing. Due to deformation occurs more slowly. It’s difficult for the deformation prediction of the time series data after filtering and MIMU directly calculats posture. So this paper through the calculation of similarity of each period of time series predicts the deformation tendency of deformation. Through the acquisition of the results of the actual data of the bridge deformation, this paper introduces the deformation prediction method is correct and effective, and it can be implemented to the deformation forecast in practice.

We present a method for time series analysis of both scalar and nonscalar time-delay systems. If the dynamics of the system investigated is governed by a time-delay-induced instability, the method allows one to determine the delay time. In a second step, the time-delay differential equation can be recovered from the time series. The method is a generalization of our recently proposed method suitable for time series analysis of scalar time-delay systems. The dynamics is not required to be settled on its attractor, and this also makes transient motion accessible to the analysis. If the motion actually takes place on a chaotic attractor, the applicability of the method does not depend on the dimensionality of the chaotic attractor---one main advantage over all time series analysis methods known until now. For a demonstration, we analyze time series, which are obtained with the help of the numerical integration of a two-dimensional time-delay differential equation. After having determined the delay time, we recover the nonscalar time-delay differential equation from the time series, in agreement with the ``original'' time-delay equation. Finally, possible applications of our analysis method in such different fields as medicine, hydrodynamics, laser physics, and chemistry are discussed.

This paper introduces a novel methodology for analyzing the complexity of time series using the fractional order dispersion entropy (qDE) and fractional order reverse dispersion entropy (qRDE) plane. Building on the theoretical foundations of dispersion entropy and its transformation, this paper introduces new entropy measures, namely, qDE and qRDE, detailing the computation processes of these entropies and constructing an entropy plane. This new approach offers fresh perspectives and tools for time series complexity analysis. Through theoretical derivations and numerical experiments, including analysis of both simulated and real stock market data, this method has demonstrated strong effectiveness in distinguishing time series of varying complexities, particularly in capturing the trends of financial market outputs over different periods. The results demonstrate that the qDE-qRDE plane can distinctly differentiate time series with varying dynamics and effectively reveal the complexities and cyclical changes in financial markets, showcasing the potential and practical value of this method in time series analysis.

In the complex system of urban rail transit, passenger flow forecast is an essential basis for rail transit network planning, rail station scale construction, and rail transit operation management. Urban rail transit passenger flow characteristics embody not only the periodicity and tendency but also the mutagenicity and particularity. Despite the time series analysis method is efficient and applicable, there are serious challenges associated with producing reliable and high-quality forecasts. When it comes to the mutagenicity and particularity of time series in passenger flow especially, it fails to live up to expectations. To tackle these challenges, a time series prediction model based on automatic machine learning is proposed, which can interpret the trend, periodicity, and holiday effects of subway passenger flow. In this paper, the Prophet model is constructed to verify the robustness and accuracy of the passenger flow time series forecast, drawing on the daily time series of passenger flow in the Beijing metro Guomao Station. Our founding indicates that the MAPE of the prediction reaches 5%. The comparison illustrates that the accuracy of the Prophet model is superior to that of the ARIMA and SARIMA models.

In this paper, simulations for detecting outliers and studying their effects on the values of AR(1) time series, transfer function model with one-input variable, transfer function model with two-input variables processes, and simultaneous transfer function (STF) are conducted using the STFMODEL JOINTMDL paragraph in the Scientific Computing Associate Corporation (SCA) program. A simulation of a transfer function model is conducted to check its validity. By using the SCA program, Victor Gomez and Agustin Maravall's example for detecting outliers in time series by TRAMO program is pursued. The conclusion, which we come up with, is that the presence of outliers, depending on their nature, may have a moderate to substantial impact on the effectiveness of the standard methodology for time series analysis with respect to model identification, estimation, and forecasting. .

Big data is one of the most important phenomena of the 21st century, creating unique opportunities and challenges in its processing and interpretation. Machine learning (ML), a subset of artificial intelligence (AI), has become a foundation of data science, which enables applications ranging from computer vision, geoscience, aviation and medicine. ML becomes important when establishing mathematical models that connect explanatory variables to predicted variables is impossible due to complexity. Deep learning (DL), a subset of ML, has revolutionized fields such as speech recognition, email filtering, and time series analysis. However, DL methods face challenges such as high data demand, overfitting, and the “black box” problem.We review least-squares-based deep learning (LSBDL), a framework that combines the interpretability of linear least squares (LS) theory with the flexibility and power of deep learning (DL). LS theory, widely used in engineering and geosciences, provides powerful tools for parameter estimation, quality control, and reliability through linear models. DL, on the other hand, deals with modelling complex nonlinear relationships where the mapping between explanatory and predicted variables is unknown. LSBDL bridges these approaches by formulating DL within the LS framework: training networks to establish a design matrix, an essential element of linear models. Through this integration, LSBDL enhances DL with transparency, statistical inference, and reliability. Gradient descent methods such as steepest descent and Gauss-Newton methods are used to construct an adaptive design matrix. By combining the transparency of LS theory with the data-driven adaptability of DL, LSBDL addresses challenges in different fields including geoscience, aviation, and data science. This approach not only improves the interpretability of DL models, but also extends the applicability of LS theory to nonlinear and complex systems, offering new opportunities for innovation and research.By embedding statistical foundations in the DL workflow, LSBDL offers a three-fold advantage: i) Direct computation of covariance matrices for predicted outcomes allows for quantitative assessment of model uncertainty. ii) Well-established theories of hypothesis testing and outlier detection facilitate the identification of model misspecifications and outlying data, and iii) The covariance matrix of observations can be used to train networks with statistically correlated, inconsistent, or heterogeneous datasets. Incorporating least squares principles increases model explainability, a critical aspect of interpretable and explainable artificial intelligence, and bridges the gap between traditional statistical methods and modern DL techniques. For example, LSBDL can incorporate prior knowledge using soft and hard physics-based constraints, a technique known as physics-informed machine learning (PIML).The approach is illustrated through three illustrative examples: Surface fitting, time series forecasting, and groundwater storage downscaling. Beyond these examples, LSBDL offers opportunities for various applications including geoscience, inverse problems, aviation, data assimilation, sensor fusion, and time series analysis.

Time series analysis and short-term forecasting of solar irradiation, a new hybrid approach

This paper explores innovative approaches that combine financial risk management with time series forecasting. Financial risk management is particularly crucial in highly volatile and complex financial markets, as a core mechanism for financial institutions and investors to mitigate potential losses. Time series forecasting, a mathematical and statistical tool, offers distinct advantages in identifying and analyzing variations in data over time. This paper reviews the theoretical foundation of time series forecasting in financial risk management, analyzing its fundamental principles and applicability in financial risk assessment. The key technique and methods are discussed, including model selection and optimization, interactive analysis of time series data, and integrated risk control strategies. An innovative risk management model based on time series analysis is proposed, and its theoretical foundation and applicability are elaborated upon, providing a new theoretical framework and methodological support for financial risk management.