Linear Recurrence Formula Research Articles

State-Dependent Model Based on Singular Spectrum Analysis Vector for Modeling Structural Breaks: Forecasting Indonesian Export

Standard time-series modeling requires the stability of model parameters over time. The instability of model parameters is often caused by structural breaks, leading to the formation of nonlinear models. A state-dependent model (SDM) is a more general and flexible scheme in nonlinear modeling. On the other hand, time-series data often exhibit multiple frequency components, such as trends, seasonality, cycles, and noise. These frequency components can be optimized in forecasting using Singular Spectrum Analysis (SSA). Furthermore, the two most widely used approaches in SSA are Linear Recurrent Formula (SSAR) and Vector (SSAV). SSAV has better accuracy and robustness than SSAR, especially in handling structural breaks. Therefore, this research proposes modeling the SSAV coefficient with an SDM approach to take structural breaks called SDM-SSAV. SDM recursively updates the SSAV coefficient to adapt over time and between states using an Extended Kalman Filter (EKF). Empirical results with Indonesian Export data and simulation studies show that the accuracy of SDM-SSAV outperforms SSAR, SSAV, SDM-SSAR, hybrid ARIMA-LSTM, and VARI.

Prediction of Rainfall Trends Using Forecasting Approaches Based on Singular Spectrum Analysis

Advanced technologies such as the Internet of Things provide an integrated platform for weather focusing, including rainfall and flood prediction. Large rainfall data frequently contain noise, which can be difficult to analyze using a standard time series model due to violated assumptions. Singular spectrum analysis (SSA) is a model-free time series analysis method that is widely used. This study aims to predict the rainfall trends in the Special Region of Yogyakarta, Indonesia, using the Recurrent SSA (SSA-R) and Vector SSA (SSA-V). The SSA-R forecasts using the recurrent continuation directly with the linear recurrent formula, while the SSA-V is a modified recurrent method. This study used 50 years of monthly rainfall data (1970-2019) from 25 stations in the special region of Yogyakarta, Indonesia. The SSA steps for forecasting rainfall data include decomposition (embedding and singular value decomposition), reconstruction (grouping and diagonal averaging), and evaluating the SSA model using w-correlation (if w-correlation is close to zero, returning to the decomposition stage; otherwise, continue the process), forecasting, evaluating the forecast results using root mean square error (RMSE), mean absolute error, r, and mean forecast error, and finally selecting the best model (either the SSA-R or SSA-V model). The results showed that the SSA-R performed better than SSA-V due to the smallest RMSE in the dry, rainy, and inter-monsoon seasons. The SSA-R model’s forecast results revealed faint, constant patterns for the dry, and rainy seasons and an increasing pattern for the inter-monsoon season. The novelty of this study is to compare the performance of the SSA-R and SSA-V models in the large rainfall data in the special region of Yogyakarta, Indonesia.

On the high-th mean of one special character sums modulo a prime

<abstract><p>Using the elementary method of the classical Gauss sums and the properties of character sums, we study a linear recurrence formula about the form $ G\left(n\right) = 1+\sum_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) $ and about the mean value of $ G(n) $. This is a further exploration of Yuan and Zhang's research in 2022, which help us to better understand the character sums wide range application.</p></abstract>

On the Hybrid Power Mean of Two-Term Exponential Sums and Cubic Gauss Sums

In this paper, an interesting third-order linear recurrence formula is presented by using elementary and analytic methods. This formula is concerned with the calculating problem of the hybrid power mean of a certain two-term exponential sums and the cubic Gauss sums. As an application of this result, some exact computational formulas for one kind hybrid power mean of trigonometric sums are obtained.

A state‐dependent linear recurrent formula with application to time series with structural breaks

AbstractAn underlying assumption in Multivariate Singular Spectrum Analysis (MSSA) is that the time series are governed by a linear recurrent continuation. However, in the presence of a structural break, multiple series can be transferred from one homogeneous state to another over a comparatively short time breaking this assumption. As a consequence, forecasting performance can degrade significantly. In this paper, we propose a state‐dependent model to incorporate the movement of states in the linear recurrent formula called a State‐Dependent Multivariate SSA (SD‐MSSA) model. The proposed model is examined for its reliability in the presence of a structural break by conducting an empirical analysis covering both synthetic and real data. Comparison with standard MSSA, BVAR, VAR and VECM models shows the proposed model outperforms all three models significantly.

A Four-Order Linear Recurrence Formula Involving the Quartic Gauss Sums and One Kind Two-Term Exponential Sums

The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind hybrid power mean involving the quartic Gauss sums and two-term exponential sums and give an interesting four-order linear recurrence formula for it. As an application, we can obtain all values of this kind hybrid power mean with mathematica software.

Counting and enumerating galled networks

Galled trees are widely studied as a recombination model in population genetics. This class of phylogenetic networks is generalized into galled networks by relaxing a structural condition. In this work, a linear recurrence formula is given for counting 1-galled networks, which are galled networks satisfying the condition that each reticulate node has only one leaf descendant. Since each galled network consists of a set of 1-galled networks connected together in a tree structure, an algorithm is also presented to count and enumerate galled networks.

Time series forecasting using singular spectrum analysis, fuzzy systems and neural networks

Hybrid methodologies have become popular in many fields of research as they allow researchers to explore various methods, understand their strengths and weaknesses and combine them into new frameworks. Thus, the combination of different methods into a hybrid methodology allows to overcome the shortcomings of each singular method. This paper presents the methodology for two hybrid methods that can be used for time series forecasting. The first combines singular spectrum analysis with linear recurrent formula (SSA-LRF) and neural networks (NN), while the second combines the SSA-LRF and weighted fuzzy time series (WFTS). Some of the highlights of these proposed methodologies are:•The two hybrid methods proposed here are applicable to load data series and other time series data.•The two hybrid methods handle the deterministic and the nonlinear stochastic pattern in the data.•The two hybrid methods show a significant improvement to the single methods used separately and to other hybrid methods.

Indonesian electricity load forecasting using singular spectrum analysis, fuzzy systems and neural networks

Electricity plays a key role in human life. This study presents several methods to forecast Indonesian electricity load demand and compares the performance of the methods. The Indonesian hourly and half-hourly load series tend to have multiple seasonal patterns. Singular Spectrum Analysis (SSA) is chosen because of its capability in decomposing the series into two separable components, a combination of cyclist and seasonal series and noise (irregular) components. In this paper we propose to model time series data by obtaining the forecast values with SSA considering the Linear Recurrent Formula (LRF) and, afterwards, to model the irregular component by fuzzy systems and neural networks (NN). The forecast values obtained from SSA-LRF are then compared with the forecast values obtained from the combining methods, i.e. SSA-LRF-Fuzzy and SSA-LRF-NN. Based on RMSE and MAPE, the SSA-LRF-NN is the most appropriate method to predict the future values of electricity load series. Four Indonesian electricity load data sets were considered in this study to validate the effectiveness of the proposed hybrid methods. The results show that the proposed methods, namely the SSA-LRF-NN algorithm can reduce the RMSE for the testing data from that obtained by SSA-LRF up to 83%.

An Empirical Study of Error Evaluation in Trend and Seasonal Time Series Forecasting Based on SSA

SSA (Singular Spectrum Analysis) starts to become a popular method in decomposing time series into some separable and interpretable series. This study provides an error evaluation in the SSA-based model for trend and multiple seasonal time series forecasting. This error evaluation is obtained by means of a numerical study on the mean square error of the estimators and mean absolute percentage error of the forecast values. Four distinct types of data generating processes (DGP) with varying sample sizes are considered in this experimental study. The parameters are estimated from the component series of SSA. Each DGP is decomposed into trend, periodic and irregular components. All these components except the irregular one are fitted by appropriate deterministic function separately. Based on the numerical simulation results, the estimated parameters are closer to the true values as the sample size increases. As the illustrative example of the real data set implementation, we used the monthly atmospheric concentrations of CO2 from Moana Loa observatory for period January 1959 to June 1972. The proposed method produces better forecast values than the results of SSA-LRF (Linear Recurrent Formula) and TLSAR (Two Level Seasonal Autoregressive). The results encourage the improvement in the time series modeling on the more complex pattern.

The hybrid power mean of the quartic Gauss sums and the two-term exponential sums

In this paper, we use the analytic method and the properties of classical Gauss sums to study the computational problems of one kind hybrid power mean of quartic Gauss sums and two-term exponential sums, and give an interesting fourth-order linear recurrence formula for it.

Window Length Selection of Singular Spectrum Analysis and Application to Precipitation Time Series

Window length is a very critical tuning parameter in Singular Spectrum Analysis (SSA) technique. For finding the optimal value of window length in SSA application, Periodogram analysis method with SSA for referencing on the selection of window length and confirm that the periodogram analysis can provide a good option for window length selection in the application of SSA. Several potential periods of Florida precipitation data are firstly obtained using periodogram analysis method. The SSA technique is applied to precipitation data with different window length as the period and experiential recommendation to extract the precipitation time series, which determines the leading components for reconstructing the precipitation and forecast respectively. A regressive model linear recurrent formula (LRF) model is used to discover physically evolution with the SSA modes of precipitation variability. Precipitation forecasts are deduced from SSA patterns and compared with observed precipitation. Comparison of forecasting results with observed precipitation indicates that the forecasts with window length of L=60 have the better performance among all. Our findings successfully confirm that the periodogram analysis can provide a good option for window length selection in the application of SSA and presents a detailed physical explanation on the varying conditions of precipitation variables.

On the fourth-order linear recurrence formula related to classical Gauss sums

Abstract Let p be an odd prime with p ≡ 1 mod 4, k be any positive integer, ψ be any fourth-order character mod p. In this paper, we use the analytic method and the properties of character sums mod p to study the computational problem of G(k, p) = τk(ψ)+τk(ψ), and give an interesting fourth-order linear recurrence formula for it, where τ(ψ) denotes the classical Gauss sums.

A forecasting algorithm for Singular Spectrum Analysis based on bootstrap Linear Recurrent Formula coefficients

A novel SSA forecasting algorithm based on coefficients of LRF is proposed. The performance of the proposed method is examined in terms of a variety of window lengths and noise levels. To this end, average bootstrap SSA is used which provides precise estimation of underlying coefficients of LRF. A simulation study and an application to real data are conducted to investigate the sensitivity and accuracy of the proposed technique in comparison to basic SSA in order to show the applicability of the method. Furthermore, a forecast interval based on bootstrap SSA is established and it succeeds in producing a comparatively smoother boundary with a low level of uncertainty and less variation.

“Caterpillar”-SSA and Box-Jenkins hybrid models and methods for time series forecasting

Trend and decomposition approaches to non-stationary time series forecasting are considered in the paper. According to them, various hybrid models for non-stationary time series forecasting, as well as identification methods for these models based on the combined use of the “Caterpillar”-SSA and Box-Jenkins methods were proposed. Hybrid mathematical models of the trend approach to forecasting, based on the “Caterpillar”-SSA and Box-Jenkins methods lie in modeling the process as deviation of actual time series values with respect to the trend component, which is represented in the proposed models by the linear recurrence formula (LRF) of the “Caterpillar”-SSA method and its approximation by the SARIMA model. The main goal of the decomposition approach to forecasting based on the “Caterpillar”-SSA and Box-Jenkins methods is the decomposition of the original time series into multiple time series with a simpler structure, considered independently of each other using the “Caterpillar”-SSA method; forecasting the data of decomposition components by SARIMA models and calculating the total forecast by combining forecasts of the constructed simplified models. The proposed models were tested on the electricity and natural gas consumption time series, and their forecasting results were compared with the results, obtained by classical probabilistic SARIMA models, generalized for the case of several seasonal components. The obtained results allow to conclude that for effective forecasts, it is necessary to carry out decomposition of the studied time series and combine different models, describing both statistical and deterministic time series components that provides the best forecasting quality.

Common singular spectrum analysis of several time series

Singular spectrum analysis (SSA) is a non-parametric time series modelling technique where an observed time series is unfolded into the column vectors of a Hankel structured matrix, known as a trajectory matrix. For noise-free signals the column vectors of the trajectory matrix lie on a single R-flat. Singular value decomposition (SVD) can be used to find the orthonormal base vectors of the linear subspace parallel to this R-flat. SSA can essentially handle functions that are governed by a linear recurrent formula (LRF) and include the broad class of functions that was proposed by Buchstaber [1994. Time series analysis and Grassmannians. Amer. Math. Soc. Transl. 162 (2), 1–17]. SSA is useful to model time series with complex cyclical patterns that increase over time. Various methods have been studied to extend SSA for application to several time series, see Golyandina et al. [2003. Variants of the Caterpillar SSA-method for analysis of multidimensional time series (in Russian) 〈hhttp://www.gistatgroup.com/cat/i〉]. Prior to that Von Storch and Zwiers (1999) and Allen and Robertson (1996) (see Ghil et al. [2002. Advanced spectral methods for climatic time series. Rev. Geophys. 40 (1), 3.1–3.41]) used multi-channel SSA (M-SSA), to apply SSA to “grand” block matrices. Our approach is different from all of these by using the common principal components approaches introduced by Flury [1988. Common Principal Components and Related Multivariate Models. Wiley, New York]. In this paper SSA is extended to several time series which are similar in some respects, like cointegrated, i.e. sharing a common R-flat. By using the common principal component (CPC) approach of Flury [1988. Common Principal Components and Related Multivariate Models. Wiley, New York] the SSA method is extended to common singular spectrum analysis (CSSA) where common features of several time series can be studied. CSSA decomposes the different original time series into the sum of a common small number of components which are related to common trend and oscillatory components and noise. The determination of the most likely dimension of the supporting linear subspace is studied using a heuristic approach and a hierarchical selection procedure.

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae Z n = T ( n ) Z n - 1 + U ( n ) Z n - 2 ( n ⩾ 2 ) with sequences of integers ( T ( n ) ) n ⩾ 0 and ( U ( n ) ) n ⩾ 0 , which are ultimately periodic modulo m, e.g. ( T ( n ) mod m ) n ⩾ 0 = ( a 0 , a 1 , a 2 , … , a ρ , T 1 , T 2 , … , T w ¯ ) ( U ( n ) mod m ) n ⩾ 0 = ( b 0 , b 1 , b 2 , … , b ρ , U 1 , U 2 , … , U w ¯ ) . In a former paper of this journal the authors computed explicitly the coefficients of a linear three-term recurrence formula for z n = Z rn + i with 0 ⩽ i < r , when ( T ( n ) ) n ⩾ 0 and ( U ( n ) ) n ⩾ 0 belong to regular or non-regular Hurwitz-type continued fraction expansions. Using this result we show now that the sequence ( Z n ) n ⩾ 0 is ultimately periodic modulo m. As a consequence, for Hurwitz-type continued fraction expansions α = [ a 0 ; T 1 ( k ) , … , T r ( k ) ¯ ] k = 1 ∞ or α = [ a 0 ; a 1 , T 1 ( k ) , … , T r ( k ) ¯ ] k = 1 ∞ with polynomials T 1 ≠ const . , T 2 , … , T r we deduce for all positive integers a and m that liminf q q ‖ q α ‖ = 0 , where q ≡ a mod m and ‖ · ‖ denotes the distance from the nearest integer. Finally, we are particularly interested in the recurrence formula Z n = ( an + b ) Z n - 1 + cZ n - 2 ( n ⩾ 2 ) and compute the length of a period of the sequence ( Z n mod m ) n ⩾ 0 , when ( a , m ) divides b, and ( c , m ) = 1 . This generalizes former results of the authors dealing with regular continued fraction expansions of the numbers exp ( 1 / s ) for integers s ⩾ 1 .

The sharp constant in the ring lemma

In his paper [2] Lowell J. Hansen found a (nonlinear) recurrence formula for the (sharp constant appearing in the “Ring Lemma” of Rodin and Sullivan [3]. In the following we improve Hansen's result and replace his recurrence relation by a linear recurrence formula leading to a closed formula for the Ring Lemma constant. Moreover, we show that the Ring Lemma constant is a reciprocal of an integer for each n