{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T14:02:07Z","timestamp":1781618527979,"version":"3.54.5"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2017,6,17]],"date-time":"2017-06-17T00:00:00Z","timestamp":1497657600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Strong coupling between values at different times that exhibit properties of long range dependence, non-stationary, spiky signals cannot be processed by the conventional time series analysis. The autoregressive fractional integral moving average (ARFIMA) model, a fractional order signal processing technique, is the generalization of the conventional integer order models\u2014autoregressive integral moving average (ARIMA) and autoregressive moving average (ARMA) model. Therefore, it has much wider applications since it could capture both short-range dependence and long range dependence. For now, several software programs have been developed to deal with ARFIMA processes. However, it is unfortunate to see that using different numerical tools for time series analysis usually gives quite different and sometimes radically different results. Users are often puzzled about which tool is suitable for a specific application. We performed a comprehensive survey and evaluation of available ARFIMA tools in the literature in the hope of benefiting researchers with different academic backgrounds. In this paper, four aspects of ARFIMA programs concerning simulation, fractional order difference filter, estimation and forecast are compared and evaluated, respectively, in various software platforms. Our informative comments can serve as useful selection guidelines.<\/jats:p>","DOI":"10.3390\/axioms6020016","type":"journal-article","created":{"date-parts":[[2017,6,19]],"date-time":"2017-06-19T10:29:26Z","timestamp":1497868166000},"page":"16","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":50,"title":["An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs"],"prefix":"10.3390","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9822-4544","authenticated-orcid":false,"given":"Kai","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mechanical Electronic & Information Engineering, China University of Mining and Technology,Beijing, Beijing 100083, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7422-5988","authenticated-orcid":false,"given":"YangQuan","family":"Chen","sequence":"additional","affiliation":[{"name":"Mechatronics, Embedded Systems and Automation Lab, School of Engineering, University of California, Merced, CA 95343, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xi","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mechanical Electronic & Information Engineering, China University of Mining and Technology,Beijing, Beijing 100083, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,6,17]]},"reference":[{"key":"ref_1","unstructured":"Box, G.E., Hunter, W.G., and Hunter, J.S. (1978). Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, John Wiley & Sons."},{"key":"ref_2","unstructured":"Box, G.E., Jenkins, G.M., Reinsel, G.C., and Ljung, G.M. (2015). Time Series Analysis: Forecasting and Control, John Wiley & Sons."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sheng, H., Chen, Y., and Qiu, T. (2011). Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications, Springer Science & Business Media.","DOI":"10.1007\/978-1-4471-2233-3"},{"key":"ref_4","unstructured":"Mathai, A.M., and Saxena, R.K. (1978). The H-Function with Applications in Statistics and Other Disciplines, John Wiley & Sons."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1023\/A:1021175108964","article-title":"On fractional kinetic equations","volume":"282","author":"Saxena","year":"2002","journal-title":"Astrophys. Space Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1016\/j.physa.2004.06.048","article-title":"On generalized fractional kinetic equations","volume":"344","author":"Saxena","year":"2004","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Sun, R., Chen, Y., and Li, Q. (2007, January 4\u20137). Modeling and prediction of Great Salt Lake elevation time series based on ARFIMA. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Las Vegas, NV, USA.","DOI":"10.1115\/DETC2007-34905"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Li, Q., Tricaud, C., Sun, R., and Chen, Y. (2007, January 4\u20137). Great Salt Lake surface level forecasting using FIGARCH model. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Las Vegas, NV, USA.","DOI":"10.1115\/DETC2007-34909"},{"key":"ref_9","unstructured":"Sheng, H., and Chen, Y. (September, January 30). The modeling of Great Salt Lake elevation time series based on ARFIMA with stable innovations. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, San Diego, CA, USA."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2309","DOI":"10.1007\/s00180-013-0408-7","article-title":"Statistical analysis of autoregressive fractionally integrated moving average models in R","volume":"28","author":"Palma","year":"2013","journal-title":"Comput. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/S0169-2070(01)00154-6","article-title":"Modeling and forecasting from trend-stationary long memory models with applications to climatology","volume":"18","author":"Baillie","year":"2002","journal-title":"Int. J. Forecast."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/S0167-9473(02)00212-8","article-title":"Computational aspects of maximum likelihood estimation of autoregressive fractionally integrated moving average models","volume":"42","author":"Doornik","year":"2003","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_13","unstructured":"Doornik, J.A., and Ooms, M. (1999). A package for estimating, forecasting and simulating ARFIMA models: Arfima package 1.0 for Ox, Erasmus University."},{"key":"ref_14","first-page":"1208","article-title":"Inference and forecasting for ARFIMA models with an application to US and UK inflation","volume":"8","author":"Doornik","year":"2004","journal-title":"Stud. Nonlinear Dyn. Econ."},{"key":"ref_15","unstructured":"Burnecki, K. (2012). Identification, Validation and Prediction of Fractional Dynamical Systems, Oficyna Wydawnicza Politechniki Wroc\u0142awskiej."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1061\/TACEAT.0006518","article-title":"Long-term storage capacity of reservoirs","volume":"116","author":"Hurst","year":"1951","journal-title":"Trans. Am. Soc. Civ. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1049\/iet-spr.2012.0050","article-title":"Effects of trends and seasonalities on robustness of the Hurst parameter estimators","volume":"6","author":"Ye","year":"2012","journal-title":"IET Signal Process."},{"key":"ref_18","unstructured":"Samorodnitsky, G., and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, CRC Press."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Woodward, W.A., Gray, H.L., and Elliott, A.C. (2016). Applied Time Series Analysis with R, CRC Press. [2nd ed.].","DOI":"10.1201\/9781315161143"},{"key":"ref_20","first-page":"8","article-title":"Fractal analysis of time series and distribution properties of Hurst exponent","volume":"5","author":"Kale","year":"2011","journal-title":"J. Math. Sci. Math. Educ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.1109\/26.935161","article-title":"Network heavy traffic modeling using \u03b1-stable self-similar processes","volume":"49","author":"Karasaridis","year":"2001","journal-title":"IEEE Trans. Commun."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1016\/j.sigpro.2010.01.023","article-title":"FARIMA with stable innovations model of Great Salt Lake elevation time series","volume":"91","author":"Sheng","year":"2011","journal-title":"Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1111\/j.1467-9892.1980.tb00297.x","article-title":"An introduction to long-memory time series models and fractional differencing","volume":"1","author":"Granger","year":"1980","journal-title":"J. Time Ser. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1093\/biomet\/68.1.165","article-title":"Fractional differencing","volume":"68","author":"Hosking","year":"1981","journal-title":"Biometrika"},{"key":"ref_25","unstructured":"Brockwell, P.J., and Davis, R.A. (2013). Time Series: Theory and Methods, Springer Science & Business Media."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1081\/SAC-100107781","article-title":"Estimation of parameters in ARFIMA processes: A simulation study","volume":"30","author":"Reisen","year":"2001","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1111\/j.1467-9892.1994.tb00198.x","article-title":"Estimation of the fractional difference parameter in the ARIMA (p, d, q) model using the smoothed periodogram","volume":"15","author":"Reisen","year":"1994","journal-title":"J. Time Ser. Anal."},{"key":"ref_28","unstructured":"Fatichi, S. (2017, June 15). ARFIMA Simulations. Available online: https:\/\/www.mathworks.com\/matlabcentral\/fileexchange\/25611-arfima-simulations\/content\/ARFIMASIM.m."},{"key":"ref_29","unstructured":"Caballero, C.V.R. (2017, June 15). ARFIMA(p, d, q). Available online: https:\/\/www.mathworks.com\/matlabcentral\/fileexchange\/53301-arfima-p-d-q-\/content\/dgparfima.m."},{"key":"ref_30","unstructured":"Inzelt, G. (2017, June 15). ARFIMA(p, d, q) Estimator. Available online: https:\/\/www.mathworks.com\/matlabcentral\/fileexchange\/30238-arfima-p-d-q--estimator."},{"key":"ref_31","unstructured":"Constantine, W., Percival, D., Constantine, M.W., and Percival, D.B. (2017, June 15). The Fractal Package for R. Available online: https:\/\/cran.r-project.org\/web\/packages\/fractal\/fractal.pdf."},{"key":"ref_32","unstructured":"Maechler, M., Fraley, C., and Leisch, F. (2017, June 15). The Fracdiff Package for R. Available online: https:\/\/cran.r-project.org\/web\/packages\/fracdiff\/fracdiff.pdf."},{"key":"ref_33","unstructured":"Contreras-Reyes, J.E., Goerg, G.M., and Palma, W. (2017, June 15). The Afmtools Package for R. Available online: http:\/\/www2.uaem.mx\/r-mirror\/web\/packages\/afmtools\/afmtools.pdf."},{"key":"ref_34","unstructured":"Kraft, P., Weber, C., and Lebo, M. (2017, June 15). The ArfimaMLM Package for R. Available online: https:\/\/cran.r-project.org\/web\/packages\/ArfimaMLM\/ArfimaMLM.pdf."},{"key":"ref_35","unstructured":"Veenstra, J.Q., and McLeod, A. (2017, June 15). The Arfima Package for R. Available online: https:\/\/cran.r-project.org\/web\/packages\/arfima\/arfima.pdf."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Shumway, R.H., and Stoffer, D.S. (2010). Time Series Analysis and Its Applications: With R Examples, Springer Science & Business Media.","DOI":"10.1007\/978-1-4419-7865-3"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1111\/jtsa.12074","article-title":"A fast fractional difference algorithm","volume":"35","author":"Jensen","year":"2014","journal-title":"J. Time Ser. Anal."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1093\/biomet\/65.2.297","article-title":"On a measure of lack of fit in time series models","volume":"65","author":"Ljung","year":"1978","journal-title":"Biometrika"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1049\/iet-spr.2009.0241","article-title":"On the robustness of Hurst estimators","volume":"5","author":"Sheng","year":"2011","journal-title":"IET Signal Process."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Chen, Y., Sun, R., and Zhou, A. (2007, January 4\u20137). An improved Hurst parameter estimator based on fractional Fourier transform. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Las Vegas, NV, USA.","DOI":"10.1115\/DETC2007-34242"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Palma, W. (2007). Long-Memory Time Series: Theory and Methods, John Wiley & Sons.","DOI":"10.1002\/9780470131466"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2958","DOI":"10.1214\/10-AOS812","article-title":"An efficient estimator for locally stationary Gaussian long-memory processes","volume":"38","author":"Palma","year":"2010","journal-title":"Ann. Stat."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/6\/2\/16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:39:28Z","timestamp":1760207968000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/6\/2\/16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,17]]},"references-count":42,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,6]]}},"alternative-id":["axioms6020016"],"URL":"https:\/\/doi.org\/10.3390\/axioms6020016","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,6,17]]}}}