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This package contains a precision-agnostic, header-only, C++ implementation of Burg’s recursive method for estimating autoregressive model parameters. Many usability-related extensions, in particular Octave- and Python-friendly functions, have been added to permit simply obtaining autocorrelation information from the resulting estimated model.

The implementation permits extracting a sequence of AR(p) models for p from one up to some maximum order:

Estimating at most an AR(7) model using 10 samples AR RMS/N Gain Filter Coefficients -- ----- ---- ------------------- 0 5.91e+00 1.00e+00 [ 1 ] 1 1.71e-03 3.45e+03 [ 1 -0.9999 ] 2 2.01e-05 2.94e+05 [ 1 -1.994 0.9941 ] 3 2.55e-08 2.32e+08 [ 1 -2.987 2.987 -0.9994 ] 4 1.01e-09 5.83e+09 [ 1 -3.967 5.913 -3.927 0.9799 ] 5 2.61e-11 2.26e+11 [ 1 -4.934 9.789 -9.763 4.895 -0.987 ] 6 3.42e-12 1.73e+12 [ 1 -5.854 14.35 -18.86 14.02 -5.586 0.9322 ] 7 3.65e-13 1.62e+13 [ 1 -6.735 19.63 -32.12 31.85 -19.15 6.465 -0.9452 ] AIC selects model order 7 as best AICC selects model order 6 as best CIC selects model order 7 as best

A variety of finite sample model selection criteria are implemented following [Broersen2000]. In particular, the

- generalized information criterion (GIC),
- Akaike information criterion (AIC),
- consistent criterion BIC,
- minimally consistent criterion (MCC),
- asymptotically-corrected Akaike information criterion (AICC),
- finite information criterion (FIC),
- finite sample information criterion (FSIC), and
- combined information criterion (CIC)

are all implemented. An included sample program called `arsel` uses CIC to
select the best model order given data from standard input. It also estimates
the effective sample size and corresponding variance using ideas from
[Trenberth1984], [Thiebaux1984], and [vonStorch2001]. For example, `arsel
--subtract-mean < rhoe.dat` reproduces results from ARMASA [Broersen2002] on a
turbulence signal:

# absrho true # criterion CIC # eff_N 28.18777014115533 # eff_var 3.6732508957963943e-05 # gain 4249.4040527950729 # maxorder 512 # minorder 0 # mu 0.20955287956200269 # mu_sigma 0.0011415499935005066 # N 1753 # AR(p) 6 # sigma2eps 8.3374920647988362e-09 # sigma2x 3.5429372570302933e-05 # submean true # T0 62.190091348891279 # window_T0 1 +1 -2.6990334396411866 +2.8771681702855281 -1.7247852051789097 +0.75024605955486146 -0.26866837869957461 +0.06700587276734557

Also included is a Toeplitz linear equation solver for a single right hand side using O(3m^2) operations. This solver is useful for investigating the correctness and numerical stability of estimated process parameters and autocorrelation information. The algorithm is [Zohar1974]’s improvement of [Trench1967]’s work. See [Bunch1985] for a discussion of the stability of Trench-like algorithms and for faster, albeit much more complicated, variants.

Topmost row of Toeplitz matrix is: 1 2 3 5 7 11 13 17 Leftmost column of Toeplitz matrix is: 1 2 4 8 16 32 64 128 Right hand side data is: 1 2 3 4 5 6 7 8 Expected solution is: -0.62963 0.148148 3.55556 -1.66667 0 -2 -1 2 Solution computed by zohar_linear_solve is: -0.62963 0.148148 3.55556 -1.66667 7.10543e-15 -2 -1 2 Term-by-term errors are: 5.55112e-16 1.04361e-14 -2.70894e-14 9.99201e-15 -7.10543e-15 4.44089e-15 1.26565e-14 -9.32587e-15 Sum of the absolute errors is: 8.16014e-14

The automated model selection procedure exposed by *arsel.cpp* has been
extensively tested against simulated data from the Lorenz attractor as
implemented in *lorenz.cpp*. Please see [Oliver2014] for full details.

*Makefile*- Try
`make`followed by`make check`. On Linux, try`make stress`to examine the implementation’s performance when piping in plain text data. Octave and/or Python functionality also will be built in-place when possible. *ar.hpp*- The standalone header implementing all algorithms. Complete API documentation is available at http://rhysu.github.com/ar.
*arsel.cpp*- Given data on standard input, use Burg’s method to compute a hierarchy of
candidate models and select the best one using CIC. Try
`arsel --help`to see available options. This is perhaps the most useful standalone utility. *arsel-octfile.cpp*,*arcov-octfile.cpp*- Provides arsel.cpp-like capabilities for GNU Octave. This is perhaps the most feature-rich way to start using these AR tools. See appendix A (“Dynamically Linked Functions”) within [Octave] for implementation details. Also demonstrates how working storage may be reused across multiple invocations to reduce the number of allocations for processing data sets.
*ar-python.cpp*,*setup.py*- Provides some functionality as a Python extension module called ‘ar’. This is modeled after the Octave wrapper but is not yet as robust.
*test.cpp*- A test driver for testing
`ar.hpp`against benchmarks by [Bourke1998]. *example.cpp*- A test driver extracting a hierarchy of AR(p) models for a sample given by [Collomb2009].
*zohar.cpp*- A test driver solving a nonsymmetric, real-valued Toeplitz set of linear equations.
*collomb2009.cpp*,*faber1986.cpp*- For implementation testing and comparison purposes, a nearly verbatim copy
of the recursive denominator algorithmic variant presented in
[Kay1981,Faber1986] and [Collomb2009]. See comments at
*issue3.dat*regarding numerical stability. *lorenz.cpp*To aid investigating the behavior of the model selection and decorrelation routines for stationary chaotic systems, this is a flexible utility for outputting the

`(t, x, y, z)`trajectory of the Lorenz attractor to standard output. This can be directly plotted, or manipulated using`cut(1)`and piped to`arsel --subtract-mean`. Try`lorenz --help`to see the available options.For example, one can examine the long-time behavior of the Lorenz

`z`coordinate using something akin to:./lorenz --every=5 | cut -f 4 | ./arsel -ns | cut -s '-d ' -f 2-

*test*.coeff*,*test*.dat*- Sample data and exact parameters from [Bourke1998] used for
`make check`. *rhoe.coeff*,*rhoe.dat*- Sample turbulent total energy RMS fluctuation data and optimal parameters found by automatically by ARMASA [Broersen2002].
*issue3.dat*- A large dataset from Nicholas Malaya generated by the Lorenz attractor. For AR(4) and higher order models, this data tickles an instability present in [Andersen1978]’s recursive denominator variant of Burg’s algorithm. Namely, this variant will return a non-stationary process with complex poles outside the unit circle. See https://github.com/RhysU/ar/issues/3 for details.
*WuleYalker.tex*- A derivation of some equations closely connected with the Yule–Walker system. Solving these permits recovering autocorrelations from process parameters.
*FiniteSampleCriteria.tex*- A catalog of all implemented autoregressive model selection criteria.
*optionparser.h*- The Lean Mean C++ Option Parser from http://optionparser.sourceforge.net which is used to parse command line arguments within sample applications.

If you find these tools useful towards publishing research, please consider citing:

– [Oliver2014] Todd A. Oliver, Nicholas Malaya, Rhys Ulerich, and Robert D. Moser. “Estimating uncertainties in statistics computed from direct numerical simulation.” Physics of Fluids 26 (March 2014): 035101+. http://dx.doi.org/10.1063/1.4866813

– [Akaike1973] Akaike, Hirotugu. “Block Toeplitz Matrix Inversion.” SIAM Journal on Applied Mathematics 24 (March 1973): 234-241. http://dx.doi.org/10.1137/0124024

– [Andersen1978] Andersen, N. “Comments on the performance of maximum entropy algorithms.” Proceedings of the IEEE 66 (November 1978): 1581-1582. http://dx.doi.org/10.1109/PROC.1978.11160

– [Bernardo1976] Bernardo, J. M. “Algorithm AS 103: Psi (digamma) function.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 25 (1976). http://www.jstor.org/stable/2347257

– [Bourke1998] Bourke, Paul. AutoRegression Analysis, November 1998. http://paulbourke.net/miscellaneous/ar/

– [Box2008] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. Time Series Analysis : Forecasting and Control. 4 edition. John Wiley, June 2008.

– [Broersen2000] Broersen, P. M. T. “Finite sample criteria for autoregressive order selection.” IEEE Transactions on Signal Processing 48 (December 2000): 3550-3558. http://dx.doi.org/10.1109/78.887047

– [Broersen2002] Broersen, P. M. T. “Automatic spectral analysis with time series models.” IEEE Transactions on Instrumentation and Measurement 51 (April 2002): 211-216. http://dx.doi.org/10.1109/19.997814

– [Broersen2006] Broersen, P. M. T. Automatic autocorrelation and spectral analysis. Springer, 2006. http://dx.doi.org/10.1007/1-84628-329-9

– [Bunch1985] Bunch, James R. “Stability of Methods for Solving Toeplitz Systems of Equations.” SIAM Journal on Scientific and Statistical Computing 6 (1985): 349-364. http://dx.doi.org/10.1137/0906025

– [Campbell1993] Campbell, W. and D. N. Swingler. “Frequency estimation performance of several weighted Burg algorithms.” IEEE Transactions on Signal Processing 41 (March 1993): 1237-1247. http://dx.doi.org/10.1109/78.205726

– [Collomb2009] Cedrick Collomb. “Burg’s method, algorithm, and recursion”, November 2009. http://www.emptyloop.com/technotes/A%20tutorial%20on%20Burg’s%20method,%20algorithm%20and%20recursion.pdf

– [Faber1986] Faber, L. J. “Commentary on the denominator recursion for Burg’s block algorithm.” Proceedings of the IEEE 74 (July 1986): 1046-1047. http://dx.doi.org/10.1109/PROC.1986.13584

– [GalassiGSL] M. Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078. url{http://www.gnu.org/software/gsl/}

– [Hurvich1989] Hurvich, Clifford M. and Chih-Ling Tsai. “Regression and time series model selection in small samples.” Biometrika 76 (June 1989): 297-307. http://dx.doi.org/10.1093/biomet/76.2.297

– [Ibrahim1987a] Ibrahim, M. K. “Improvement in the speed of the data-adaptive weighted Burg technique.” IEEE Transactions on Acoustics, Speech, and Signal Processing 35 (October 1987): 1474–1476. http://dx.doi.org/10.1109/TASSP.1987.1165046

– [Ibrahim1987b] Ibrahim, M. K. “On line splitting in the optimum tapered Burg algorithm.” IEEE Transactions on Acoustics, Speech, and Signal Processing 35 (October 1987): 1476–1479. http://dx.doi.org/10.1109/TASSP.1987.1165047

– [Ibrahim1989] Ibrahim, M. K. “Correction to ‘Improvement in the speed of the data-adaptive weighted Burg technique’.” IEEE Transactions on Acoustics, Speech, and Signal Processing 37 (1989): 128. http://dx.doi.org/10.1109/29.17511

– [Kahan1965] Kahan, W. “Further remarks on reducing truncation errors.” Communications of the ACM 8 (January 1965): 40+. http://dx.doi.org/10.1145/363707.363723

– [Kay1981] Kay, S. M. and S. L. Marple. “Spectrum analysis- A modern perspective.” Proceedings of the IEEE 69 (November 1981): 1380-1419. http://dx.doi.org/10.1109/PROC.1981.12184

– [Merchant1982] Merchant, G. and T. Parks. “Efficient solution of a Toeplitz-plus-Hankel coefficient matrix system of equations.” IEEE Transactions on Acoustics, Speech, and Signal Processing 30 (February 1982): 40-44. http://dx.doi.org/10.1109/TASSP.1982.1163845

– [Octave] Eaton, John W., David Bateman, and Søren Hauberg. GNU Octave Manual Version 3. Network Theory Limited, 2008. http://www.octave.org/

– [Press2007] Press, William H., Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical recipes : The Art of Scientific Computing. Third edition. Cambridge University Press, September 2007.

– [Seghouane2004] Seghouane, A. K. and M. Bekara. “A Small Sample Model Selection Criterion Based on Kullback’s Symmetric Divergence.” IEEE Transactions on Signal Processing 52 (December 2004): 3314-3323. http://dx.doi.org/10.1109/TSP.2004.837416

– [vonStorch2001] Hans von Storch and Francis W. Zwiers. Statistical analysis in climate research. Cambridge University Press, March 2001. ISBN 978-0521012300.

– [Thiebaux1984] Thiébaux, H. J. and F. W. Zwiers. “The Interpretation and Estimation of Effective Sample Size.” J. Climate Appl. Meteor. 23 (May 1984): 800-811. http://dx.doi.org/10.1175/1520-0450(1984)023%253C0800:TIAEOE%253E2.0.CO;2

– [Trenberth1984] Trenberth, K. E. “Some effects of finite sample size and persistence on meteorological statistics. Part I: Autocorrelations.” Monthly Weather Review 112 (1984). http://dx.doi.org/10.1175/1520-0493(1984)112%3C2359:SEOFSS%3E2.0.CO;2

– [Trench1967] Trench, William F. Weighting coefficients for the prediction of stationary time series from the finite past. SIAM J. Appl. Math. 15, 6 (Nov. 1967), 1502-1510. http://www.jstor.org/stable/2099503

– [Vandevender1982] Vandevender, W. H. and K. H. Haskell. “The SLATEC mathematical subroutine library.” ACM SIGNUM Newsletter 17 (September 1982): 16-21. http://dx.doi.org/10.1145/1057594.1057595

– [Welford1962] Welford, B. P. “Note on a Method for Calculating Corrected Sums of Squares and Products.” Technometrics 4 (1962). http://www.jstor.org/stable/1266577

– [Zohar1974] Zohar, Shalhav. “The Solution of a Toeplitz Set of Linear Equations.” J. ACM 21 (April 1974): 272-276. http://dx.doi.org/10.1145/321812.321822

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