Conventional Autoregressive Models Research Articles

Discrete diffusion models with Refined Language-Image Pre-trained representations for remote sensing image captioning

RS image captioning (RSIC) utilizes natural language to provide a description of image content, assisting in the comprehension of object properties and relationships. Nonetheless, RS images are characterized by variations in object scales, distributions, and quantities, which make it challenging to obtain global semantic information and object connections. To enhance the accuracy of captions produced from RS images, this paper proposes a novel method referred to as Discrete Diffusion Models with Refined Language-Image Pre-trained representations (DDM-RLIP), leveraging an advanced discrete diffusion model (DDM) for nosing and denoising text tokens. DDM-RLIP is based on an advanced DDM-based method designed for natural pictures. The primary approach for refining image representations involves fine-tuning a CLIP image encoder on RS images, followed by adapting the transformer with an additional attention module to focus on crucial image regions and relevant words. Furthermore, experiments were conducted on three datasets, Sydney-Captions, UCM-Captions, and NWPU-Captions, and the results demonstrated the superior performance of the proposed method compared to conventional autoregressive models. On the NWPU-Captions dataset, the CIDEr score improved from 116.4 to 197.7, further validating the efficacy and potential of DDM-RLIP. The implementation codes for our approach DDM-RLIP are available at https://github.com/Leng-bingo/DDM-RLIP.

Root cause diagnosis for complex industrial process faults via spatiotemporal coalescent based time series prediction and optimized Granger causality

It is essential to conduct root cause diagnosis (RCD) to guarantee the safety of operations of chemical processes and suppress fault deterioration, yet RCD has encountered challenges in the context of the increasing complexity of industrial processes. Granger causality (GC) analysis is one of the most commonly used methods to construct process causal maps and identify root causes. However, its use is subject to a number of limitations for dynamic nonlinear industrial processes. Therefore, an optimized GC analysis is proposed to perform RCD in complex industrial processes based on the spatiotemporal coalescent-based prediction model (SCPM) and causality verification algorithm (CVA). The time series prediction model SCPM is presented as an alternative to GC’s conventional autoregressive model in order to avoid spurious regressive facing nonlinear processes, which stacks dilated convolutional neural networks (DCNN) and bidirectional gated recurrent unit (BiGRU) to coalesce time series nonlinear couplings and long-term dependence information. Moreover, considering data that may deviate from normal distribution after the fault occurs, a nonparametric causality test method CVA based on the Wilcoxon signed-rank test (WSRT) is constructed combined with SCPM to eliminate false causality discovery. Empirical results on the Tennessee Eastman process and the blast furnace process demonstrate the effectiveness of the proposed RCD method.

Построение и анализ модели машинного обучения для краткосрочного прогнозирования рынка биткоина на основе рекуррентных нейронных сетей

In this article, the construction and analysis of machine learning models were performed for short-term forecasting in the cryptocurrency market on the example of bitcoin – one of the most popular cryptocurrencies in the world. The initial data for the study leads to the conclusion that over the long period of its existence, bitcoin has shown a high degree of volatility, especially evident in comparison with traditional financial instruments. The article substantiates that this market is influenced by a multitude of factors. No one can say for sure what makes up the value of a particular cryptocurrency, as it involves a range of reasons, which cannot be fully taken into account. To overcome this problem, we have considered the principle of recurrent neural network. It is described why networks with memory are better at making predictions on the time series than conventional autoregressive model and standard forward propagation networks. The initial data processing algorithm and transformation methods are defined. The sample was reduced in order to increase the speed of the network, by reducing the number of recalculations of weights. The algorithm of the family of recurrent neural networks was built and trained to test the hypothesis about their better adaptivity due to short-term and long-term memory. The model is evaluated on the test data representing the bitcoin exchange rate for 2021–2022, since this period is characterized by high volatility. It is concluded that it is reasonable to use a similar type of models for short-term forecasting of cryptocurrency rates.

Non-Autoregressive Fully Parallel Deep Convolutional Neural Speech Synthesis

Deep learning-based speech synthesis evolves by employing a sequence-to-sequence (seq2seq) structure with an attention mechanism. The seq2seq speech synthesis model consists of a pair of the encoder for delivering the linguistic features and the decoder for predicting the mel-spectrogram, and learns the alignment between text and speech through the attention mechanism. The decoder predicts the mel-spectrogram by an autoregressive flow that considers the current input and what they have learned from previous inputs. This is beneficial when processing the sequential data, as in speech synthesis. However, the recursive generation of speech typically requires extensive training time, which slows the speed of synthesis. To overcome these obstacles, we propose a non-autoregressive framework for fully parallel deep convolutional neural speech synthesis. Firstly, we design a new synthesis paradigm that integrates a time-varying metatemplate (TVMT), whose length is modeled with a separate conditional distribution, to prepare the decoder input. The decoding step converts the TVMT into spectral features, which eliminates the autoregressive flow. Secondly, we propose a structure that uses multiple decoders interconnected by up-down chains with an iterative attention mechanism. The decoder chains distribute the burden of decoding, progressively infusing the information obtained from the training target example into the chains to refine the predicted spectral features at each decoding step. For each decoder, the attention mechanism is repeatedly applied to produce the elaborated alignment between the linguistic features and the TVMT, which is gradually transformed into the spectral features. The proposed architecture substantially improves the synthesis speed, and the resulting speech quality is superior to that of a conventional autoregressive model.

StackDA: A Stacked Dual Attention Neural Network for Multivariate Time-Series Forecasting

Multivariate time-series forecasting derives key seasonality from past patterns to predict future time-series. Multi-step forecasting is crucial in the industrial sector because a continuous perspective leads to more effective decisions. However, because it depends on previous prediction values, multi-step forecasting is highly unstable. To mitigate this problem, we introduce a novel model, named stacked dual attention neural network (StackDA), based on an encoder-decoder. In dual attention, the initial attention is for the time dependency between the encoder and decoder, and the second attention is for the time dependency in the decoder time steps. We stack dual attention to stabilize the long-term dependency and multi-step forecasting problem. We add an autoregression component to resolve the lack of linear properties because our method is based on a nonlinear neural network model. Unlike the conventional autoregressive model, we propose skip autoregressive to deal with multiple seasonalities. Furthermore, we propose a denoising training method to take advantage of both the teacher forcing and without teacher forcing methods. We adopt multi-head fully connected layers for the variable-specific modeling owing to our multivariate time-series data. We add positional encoding to provide the model with time information to recognize seasonality more accurately. We compare our model performance with that of machine learning and deep learning models to verify our approach. Finally, we conduct various experiments, including an ablation study, a seasonality determination test, and a stack attention test, to demonstrate the performance of StackDA.

Predicting the fMRI Signal Fluctuation with Recurrent Neural Networks Trained on Vascular Network Dynamics.

Resting-state functional MRI (rs-fMRI) studies have revealed specific low-frequency hemodynamic signal fluctuations (<0.1 Hz) in the brain, which could be related to neuronal oscillations through the neurovascular coupling mechanism. Given the vascular origin of the fMRI signal, it remains challenging to separate the neural correlates of global rs-fMRI signal fluctuations from other confounding sources. However, the slow-oscillation detected from individual vessels by single-vessel fMRI presents strong correlation to neural oscillations. Here, we use recurrent neural networks (RNNs) to predict the future temporal evolution of the rs-fMRI slow oscillation from both rodent and human brains. The RNNs trained with vessel-specific rs-fMRI signals encode the unique brain oscillatory dynamic feature, presenting more effective prediction than the conventional autoregressive model. This RNN-based predictive modeling of rs-fMRI datasets from the Human Connectome Project (HCP) reveals brain state-specific characteristics, demonstrating an inverse relationship between the global rs-fMRI signal fluctuation with the internal default-mode network (DMN) correlation. The RNN prediction method presents a unique data-driven encoding scheme to specify potential brain state differences based on the global fMRI signal fluctuation, but not solely dependent on the global variance.

Comparative assessment of time series and artificial intelligence models to estimate monthly streamflow: A local and external data analysis approach

River flow rates are important for water resources projects. Given this, the current study explored the use of autoregressive (AR) and moving average (MA) techniques as individual time series models and compared them to the same models hybridized with an autoregressive conditional heteroscedasticity (ARCH) model to estimate monthly streamflow. In addition, two artificial intelligence (AI) approaches, namely, multivariate adaptive regression splines (MARS) and gene expression programming (GEP), were explored. The performance of each of these models in estimating monthly streamflow was compared based on local and external data analyses. Using the local data analysis approach, streamflow data at each target station was estimated using observed streamflow at the same station. The external data analysis approach used neighboring station streamflow data to estimate streamflow data for the target station. The Beinerahe Roodbar and Pole Astaneh stations on the Sefidrood River, Iran, as well as the Port Elgin and Walkerton stations on the Saugeen River, Canada, were used as study areas. Upstream and downstream monthly streamflow time series data were used. The performance of all models was compared using three error metrics, including the root mean square error, mean absolute error, and correlation coefficient. The results showed that the hybrid time series models (i.e., AR-ARCH and MA-ARCH) outperformed the conventional AR and MA models. A comparison of all applied models revealed that the hybrid AR-ARCH and MA-ARCH time series models performed better than the AI techniques, when using a local data analysis approach. The external data analysis approach was more accurate for monthly streamflow estimation than the local data analysis approach. To conclude, based on the outcomes of the AI models under the external data analysis approach, nearby data can be used to estimate streamflow of a target station when the target station streamflow data are not available.

Solar radiation synthetic series for power purchase agreements.

The present paper proposes a methodology based on the implementation and assessment of autoregressive (AR) solar radiation models for generating synthetic series and providing guidance on bidding strategies for power purchase agreements. The work considered conventional and periodic AR models with different lag orders, assessing the models against real solar radiation measurements. The synthetic series generation process developed 1000 1-year monthly solar radiation scenarios that were later employed for simulating electric energy production and power purchase agreement models. This application allowed one to evaluate the risk associated with the energy supply security, supporting bidding strategies in energy auctions. A real study case is also illustrated in detail, referring to a spot in the Brazilian best irradiance area.

Multi-scale Gaussian process experts for dynamic evolution prediction of complex systems

Predictive analytics has become an important topic in expert and intelligent systems, with broad applications across various engineering and business domains, such as the prediction of exchange rate in finance, weather and demand for energy using mixture of experts. However, selection of the number of experts and assignment of the input to individual experts remain elusive, especially for highly nonlinear and nonstationary systems. This paper presents a novel mixture of experts, namely, nonparametric multi-scale Gaussian process (MGP) experts to predict the dynamic evolution of such complex systems. Concretely, intrinsic time-scale decomposition is first used to iteratively decompose the time series generated from such complex systems into a series of proper rotation components and a baseline trend component. Those components delineate the true time-frequency-energy patterns of the complex systems at different granularity. A Gaussian process (GP) expert is then applied on each component to predict the system evolution at each scale. MGP circumvent the tedious selection and assignment problems via the nonparametric ITD. Summation of those individual forecasts represents the overall evolution of the original time series. Case studies using synthetic and real-world data elucidated that the proposed MGP model significantly outperforms conventional autoregressive models, composite GP model, and support vector regression in terms of prediction accuracy, and it is particularly effective for multi-step forecasting.

A Volterra series-based method for extracting target echoes in the seafloor mining environment

The purpose of this research was to evaluate the applicability of the Volterra adaptive method to predict the target echo of an ultrasonic signal in an underwater seafloor mining environment. There is growing interest in mining of seafloor minerals because they offer an alternative source of rare metals. Mining the minerals cause the seafloor sediments to be stirred up and suspended in sea water. In such an environment, the target signals used for seafloor mapping are unable to be detected because of the unavoidable presence of volume reverberation induced by the suspended sediments. The detection of target signals in reverberation is currently performed using a stochastic model (for example, the autoregressive (AR) model) based on the statistical characterisation of reverberation. However, we examined a new method of signal detection in volume reverberation based on the Volterra series by confirming that the reverberation is a chaotic signal and generated by a deterministic process. The advantage of this method over the stochastic model is that attributions of the specific physical process are considered in the signal detection problem. To test the Volterra series based method and its applicability to target signal detection in the volume reverberation environment derived from the seafloor mining process, we simulated the real-life conditions of seafloor mining in a water filled tank of dimensions of 5×3×1.8m. The bottom of the tank was covered with 10cm of an irregular sand layer under which 5cm of an irregular cobalt-rich crusts layer was placed. The bottom was interrogated by an acoustic wave generated as 16μs pulses of 500kHz frequency. This frequency is demonstrated to ensure a resolution on the order of one centimetre, which is adequate in exploration practice. Echo signals were collected with a data acquisition card (PCI 1714 UL, 12-bit). Detection of the target echo in these signals was performed by both the Volterra series based model and the AR model. The results obtained confirm that the Volterra series based method is more efficient in the detection of the signal in reverberation than the conventional AR model (the accuracy is 80% for the PIM-Volterra prediction model versus 40% for the AR model).

Order-disorder transitions in time-discrete mean field systems with memory: a novel approach via nonlinear autoregressive models

In a previous study strongly nonlinear autoregressive (SNAR) models have been introduced as a generalization of the widely-used time-discrete autoregressive models that are known to apply both to Markov and non-Markovian systems. In contrast to conventional autoregressive models, SNAR models depend on process mean values. So far, only linear dependences have been studied. We consider the case in which process mean values can have a nonlinear impact on the processes under consideration. It is shown that such models describe Markov and non-Markovian many-body systems with mean field forces that exhibit a nonlinear impact on single subsystems. We exemplify that such nonlinear dependences can describe order-disorder phase transitions of time-discrete Markovian and non-Markovian many-body systems. The relevant order parameter equations are derived and issues of stability and stationarity are studied.

Intraday Index Predictability and Options Trading Profitability

In this paper we study the intraday dynamics of E-mini S&P 500 index futures and the option trading strategies employing the weekly E-mini S&P 500 index futures options. We make a number of contributions to the literature in the area of intra-day equity index futures return predictability and trading profitability. As far as we know till at present, ours is one of the first studies on intraday implied moments of S&P 500 index futures return using intraday option prices. We use intra-day E-mini S&P500 European-style weekly options data from August 2009 to December 2012 and improve on existing techniques to extract the first four moments of the risk-neutral futures return distribution. Secondly we perform intraday out-of-sample forecasting or prediction, and document the intraday dynamics of the risk-neutral moments. We introduce a novel local autoregression method that allows variable windows in estimating the autoregressive parameters. This is particularly useful in situations when there may be intraday news that cause structural breaks in the otherwise smooth process. It also distinguishes itself from the conventional autoregressive model with predetermined sample lengths. Thirdly, we show profitability in the options trading strategies involving the various risk-neutral moment forecasts, particularly that involving skewness. The positive profitability after transaction costs in skewness trading indicates that the market is not as efficient as thought to be. We also use a novel technique in kurtosis trading that resulted in positive profits before cost, something new in the literature where negative profits were found in kurtosis trading. These results may explain the persistence of intraday trading activities in the market. Our intraday risk-neutral moments also suggest that forecast increases in volatility and skewness lead to an average increase in subsequent return over the next 10 minutes. On the other hand, intraday forecast increase in risk-neutral kurtosis leads to an average decrease in subsequent return. These intraday results appear to be contrary to existing studies using risk-neutral moments over daily intervals. This suggests that intraday price dynamics is different from daily price dynamics.

A strategy to overcome adverse effects of autoregressive updating of streamflow forecasts

Abstract. For streamflow forecasting, rainfall–runoff models are often augmented with updating procedures that correct forecasts based on the latest available streamflow observations of streamflow. A popular approach for updating forecasts is autoregressive (AR) models, which exploit the "memory" in hydrological model simulation errors. AR models may be applied to raw errors directly or to normalised errors. In this study, we demonstrate that AR models applied in either way can sometimes cause over-correction of forecasts. In using an AR model applied to raw errors, the over-correction usually occurs when streamflow is rapidly receding. In applying an AR model to normalised errors, the over-correction usually occurs when streamflow is rapidly rising. In addition, when parameters of a hydrological model and an AR model are estimated jointly, the AR model applied to normalised errors sometimes degrades the stand-alone performance of the base hydrological model. This is not desirable for forecasting applications, as forecasts should rely as much as possible on the base hydrological model, with updating only used to correct minor errors. To overcome the adverse effects of the conventional AR models, a restricted AR model applied to normalised errors is introduced. We show that the new model reduces over-correction and improves the performance of the base hydrological model considerably.

Prediction of Millers Ferry Dam Reservoir Level in USA Using Artificial Neural Network

Reservoir level modeling is important for the operation of dam reservoir, design of hydraulic structures, determining pollution in reservoir and the safety of dam. In this study, daily reservoir levels for Millers Ferry Dam on the Alabama River in USA were predicted using artificial neural networks (ANN). Bayesian regularization backpropagation training algorithm is employed for optimization of the network. The results of the optimal ANN models were compared with conventional auto-regressive models (AR), auto-regressive moving average (ARMA), multi-linear regression (MLR) models. The models are compared with each other according to the three criteria, namely, mean square errors, mean absolute relative error and correlation coecient. The comparison results show that the ANN models perform better than the conventional models.

FORECAST OF METEOROLOGICAL DATA UTILIZING STATE-SPACE MODEL UTILIZING METRIC-MULTI DIMENSIONAL SCALING

Based on Bayesian statistics using the expected a posterior (EAP) estimation, we forecast time evolution of meteorological data, such as the temperature, at a target point via information on a set of time-series of the temperatures at sampling points selected by the metric multi-dimensional scaling (metric-MDS), without using information on that of the target point. Using numerical calculations with respect to the climate statistics in Kanto district, we clarify that the metric-MDS can select a set of sampling points whose data are similar to that of the target. Then, we clarify that the EAP estimation succeeds in predicting time evolution on the temperature in Maebashi using the set of time-series of the temperatures at the selected sampling points around Maebashi. Also, we find that the EAP estimation predicts the time evolution of the temperature more accurately than the conventional autoregressive model.

Surface myoelectric signal classification using the AR-GARCH model

In myoelectric prostheses design, it is normally assumed that the necessary control information can be extracted from the surface myoelectric signals. In the pattern classification paradigm for controlling myoelectric prosthesis, the autoregressive (AR) model coefficients are generally considered an efficient and robust feature set. However, no formal statistical methodologies or tests are reported in the literature to analyze and model the myoelectric signal as an AR process. We analyzed the myoelectric signal as a stochastic time-series and found that the signal is heteroscedastic, i.e., the AR modeling residuals exhibit a time-varying variance. Heteroscedasticity is a major concern in statistical modeling because it can invalidate statistical tests of significance which may assume that the modeling errors are uncorrelated and that the error variances do not vary with the effects being modeled. We subsequently proposed to model the myoelectric signal as an autoregressive-generalized autoregressive conditional heteroscedastic (AR-GARCH) process and used the model parameters as a feature set for signal classification. Multiple statistical tests including the Ljung–Box Q-test, Engle's test for heteroscedasticity, the Kolmogorov–Smirnov test and the goodness of fit test were performed to show the validity of the proposed model. Our experimental results show that the proposed AR-GARCH model coefficients, when used as a feature set in two different classification schemes, significantly outperformed (p<.01) the conventional AR model coefficients.

Application of generalized regression neural network and support vector regression for monthly rainfall forecasting in western Jilin Province, China

Research Article| June 18 2014 Application of generalized regression neural network and support vector regression for monthly rainfall forecasting in western Jilin Province, China Wenxi Lu; Wenxi Lu 1College of Environment and Resources, Jilin University, Changchun 130021, China and Key Laboratory of Groundwater Resources and Environment, Ministry of Education, Jilin University, Changchun 130021, China E-mail: luwenxi@jlu.edu.cn Search for other works by this author on: This Site PubMed Google Scholar Haibo Chu; Haibo Chu 1College of Environment and Resources, Jilin University, Changchun 130021, China and Key Laboratory of Groundwater Resources and Environment, Ministry of Education, Jilin University, Changchun 130021, China Search for other works by this author on: This Site PubMed Google Scholar Zheng Zhang Zheng Zhang 1College of Environment and Resources, Jilin University, Changchun 130021, China and Key Laboratory of Groundwater Resources and Environment, Ministry of Education, Jilin University, Changchun 130021, China Search for other works by this author on: This Site PubMed Google Scholar Journal of Water Supply: Research and Technology-Aqua (2015) 64 (1): 95–104. https://doi.org/10.2166/aqua.2014.002 Article history Received: December 25 2013 Accepted: May 23 2014 Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Cite Icon Cite Permissions Search Site Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAll JournalsThis Journal Search Advanced Search Citation Wenxi Lu, Haibo Chu, Zheng Zhang; Application of generalized regression neural network and support vector regression for monthly rainfall forecasting in western Jilin Province, China. Journal of Water Supply: Research and Technology-Aqua 1 February 2015; 64 (1): 95–104. doi: https://doi.org/10.2166/aqua.2014.002 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex Rainfall is a key part of the hydrological cycle, and correct forecasting of rainfall is vital in the planning and management of water resources. Generalized regression neural network (GRNN) and support vector regression (SVR) were both applied to forecast monthly rainfall, and the conventional autoregressive model was built for comparison. Furthermore, Akaike Information Criteria were used to identify the proper inputs for the rainfall forecasting model. The data sets of monthly rainfall for a 53-year period from 1957 to 2010 in western Jilin Province, China, were used. The results indicated that the proper inputs would help in effectively improving the prediction accuracy. Furthermore, the results showed that both the SVR and the GRNN model performed better than the autoregressive model in forecasting monthly rainfall. SVR models outperformed all other models during the testing period in terms of the mean absolute error, root-mean-square error, coefficient of efficiency and R2. Therefore, SVR models were applied to forecast monthly rainfall for six cities including Baicheng, Qianguo, Fuyu, Qian'an, Changling and Tongyu. generalized regression neural network, monthly rainfall forecasting, support vector regression, western Jilin Province © IWA Publishing 2015 You do not currently have access to this content.

Generalized versus non-generalized neural network model for multi-lead inflow forecasting at Aswan High Dam

Abstract. Artificial neural networks (ANN) have been found efficient, particularly in problems where characteristics of the processes are stochastic and difficult to describe using explicit mathematical models. However, time series prediction based on ANN algorithms is fundamentally difficult and faces problems. One of the major shortcomings is the search for the optimal input pattern in order to enhance the forecasting capabilities for the output. The second challenge is the over-fitting problem during the training procedure and this occurs when ANN loses its generalization. In this research, autocorrelation and cross correlation analyses are suggested as a method for searching the optimal input pattern. On the other hand, two generalized methods namely, Regularized Neural Network (RNN) and Ensemble Neural Network (ENN) models are developed to overcome the drawbacks of classical ANN models. Using Generalized Neural Network (GNN) helped avoid over-fitting of training data which was observed as a limitation of classical ANN models. Real inflow data collected over the last 130 years at Lake Nasser was used to train, test and validate the proposed model. Results show that the proposed GNN model outperforms non-generalized neural network and conventional auto-regressive models and it could provide accurate inflow forecasting.

Threshold Cointegration and the PPP Hypothesis

Self-Exciting Threshold Autoregressive (SETAR) models are a non-linear variant ofconventional linear Autoregressive (AR) models. One advantage of SETAR modelsover conventional AR models lies in its flexible nature in dealing with possibleasymmetric behaviour of economic variables. The concept of threshold cointegrationimplies that the Error Correction Mechanism (ECM) at a particular interval isinactive as a result of adjustment costs, and active when deviations from equilibriumexceed certain thresholds. For instance, the presence of adjustment costs can, inmany circumstances, justify the fact that economic agents intervene to recalibrateback to a tolerable limit, as in the case when the benefits of adjustment are superiorto its costs. We introduce an approach that accounts for potential asymmetry andwe investigate the presence of the relative version of the purchasing power parity(PPP) hypothesis for 14 countries. Based on a threshold cointegration adaptation ofthe unit root test procedure suggested by Caner & Hansen (2001), we findevidence of an asymmetric adjustment for the relative version of PPP for eight pairsof countries.

Fast Triangular Factorization of Covariance Matrices of Differenced Time Series

An algorithm is presented for computing the triangular factors of the covariance matrix of a random vector whose elements are the successive differences of a stationary time series. This algorithm has applications in the modeling of time series using a difference operator rather than using a shift operator, as is done in the conventional autoregressive model. The difference-operator based models offer the benefit of better numerical conditioning when applied to series that are obtained by sampling continuous-time processes at rapid rates. The covariance matrix of diferenced data is not Toeplitz; however, it has a Toeplitz-like structure that is exploited in this paper to obtain an $O( n^2 )$ algorithm. This algorithm is amenable to parallelization; i.e., it requires only $O( n )$ time when using n processors in parallel.