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资源说明：Autoregressive process modeling tools in header-only C++
Autoregressive modeling tools in header-only C++
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|BuildStatus|_
.. |BuildStatus| image:: https://app.travis-ci.com/RhysU/ar.svg?branch=master
.. _BuildStatus: https://app.travis-ci.com/RhysU/ar
Overview
--------
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 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.
Contents
--------
*Makefile*
Try ``make`` followed by ``make check``. On Linux, try ``make stress`` to
examine the implementation's performance when piping in plain text data.
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.
*ar6.cpp*
A sample program generating data from an AR(6) process specified in
section 4.1 of [Beyhaghi2018]. Try ``./ar6 -b 5000 -t 50000 | cut -f2 |
./arsel`` to see the statistical parameters estimated by ``arsel`` when
fed input from a realization of this process.
*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*
These two Octave wrappers were removed after
http://github.com/RhysU/ar/releases/tag/0.1.5 on account of licensing
considerations. The code followed appendix A ("Dynamically Linked
Functions") of [Octave].
*ar-python.cpp*, *setup.py*
Provides some functionality as a Python extension module called
'ar'. This is perhaps the easiest way to start using these AR tools.
Also demonstrates how working storage may be reused across multiple
invocations to reduce the number of allocations for processing
data sets.
*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.
Attribution
-----------
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
References
----------
-- [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
-- [Beyhaghi2018] Beyhaghi, Pooriya and Alimohammadi, Shahrouz and Bewley Thomas. "Uncertainty Quantification of the time averaging of a Statistics Computed from Numerical Simulation of Turbulent Flow." https://arxiv.org/abs/1802.01056v1
-- [Bourke1998] Bourke, Paul. AutoRegression Analysis, November 1998. http://paulbourke.net/miscellaneous/ar/
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-- [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
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-- [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
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