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ar.hpp
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ar.hpp
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// Copyright (C) 2012, 2013 Rhys Ulerich
//
// This Source Code Form is subject to the terms of the Mozilla Public
// License, v. 2.0. If a copy of the MPL was not distributed with this
// file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef AR_HPP
#define AR_HPP
#ifndef AR_SUPPRESS_DOXYGEN_MAINPAGE
/**
* @mainpage
*
* \ref ar implements \ref ar "modeling tools" for autoregressive processes in
* header-only C++.
*
* See the current <a
* href="https://github.com/RhysU/ar/blob/master/README.rst"> README</a> for a
* more detailed overview and http://github.com/RhysU/ar for project
* information.
*
* If you find these tools useful towards publishing research, please consider
* citing:
* \li 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
*/
#endif /* AR_SUPPRESS_DOXYGEN_MAINPAGE */
/** @file
* Autoregressive process modeling tools in header-only C++.
*/
#include <algorithm>
#include <cassert>
#include <cmath>
#include <functional>
#include <iterator>
#include <limits>
#include <numeric>
#include <stdexcept>
#include <string>
#include <vector>
/**
* Autoregressive process modeling tools in header-only C++.
*
* All routines estimate and/or evaluate autoregressive models of the form
* \f{align}{
* x_n + a_1 x_{n - 1} + \dots + a_p x_{n - p} &= \epsilon_n
* &
* \epsilon_n &\sim{} N\left(0, \sigma^2_\epsilon\right)
* \\
* \sigma^2_x \left(
* \rho_0 + a_1 \rho_{1} + \dots + a_p \rho_{p}
* \right) &= \sigma^2_\epsilon
* &
* \rho_0 &= 1
* \\
* \rho_k + a_1 \rho_{k-1} + \dots + a_p \rho_{k-p} &= 0
* &
* k &\geq{} p
* \f}
* where \f$x_i\f$ are the process values, \f$a_i\f$ are the model parameters,
* and \f$\rho_i\f$ are the lag \f$i\f$ autocorrelations. The white noise
* input process \f$\epsilon_n\f$ has variance \f$\sigma^2_\epsilon\f$. The
* model has output variance \f$\sigma^2_x\f$ and therefore a gain equal to
* \f$\sigma^2_x / \sigma^2_\epsilon\f$.
*/
namespace ar
{
/////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////
/**
* Preprocessor macros to simplify implementation.
*
* @{
*/
/** Helper for defining GCC version checks. */
#define AR_GCC_VERSION \
(__GNUC__ * 10000 + __GNUC_MINOR__ * 100 + __GNUC_PATCHLEVEL__)
/** Helper macro for implementing \ref AR_STRINGIFY. */
#define AR_STRINGIFY_HELPER(x) #x
/** Expand and stringify the provided argument. */
#define AR_STRINGIFY(x) AR_STRINGIFY_HELPER(x)
/**
* Ensure that \c expr evaluates to boolean \c true at runtime. If \c expr
* evaluates to boolean \c false, then an exception \c except is thrown with
* message \c msg.
*
* This macro is intended for <tt>assert</tt>-like checks which should always
* be performed regardless of whether or not \c NDEBUG is <tt>#define</tt>d.
*/
#define AR_ENSURE_MSGEXCEPT(expr, msg, except) \
if (!(expr)) throw except(msg)
/**
* Ensure that \c expr evaluates to boolean \c true at runtime. If \c expr
* evaluates to boolean \c false, then a <tt>std::logic_error</tt> is thrown
* with message \c msg.
*
* This macro is intended for <tt>assert</tt>-like checks which should always
* be performed regardless of whether or not \c NDEBUG is <tt>#define</tt>d.
*/
#define AR_ENSURE_MSG(expr, msg) \
AR_ENSURE_MSGEXCEPT(expr, msg, std::logic_error)
/**
* Ensure that \c expr evaluates to boolean \c true at runtime. If \c expr
* evaluates to boolean \c false, then a <tt>std::logic_error</tt> is thrown.
*
* This macro is intended for <tt>assert</tt>-like checks which should always
* be performed regardless of whether or not \c NDEBUG is <tt>#define</tt>d.
*/
#define AR_ENSURE(expr) \
AR_ENSURE_MSG(expr, AR_STRINGIFY(expr)" false")
/**
* Ensure that the argument-related \c expr evaluates to boolean \c true at
* runtime. If \c expr evaluates to boolean \c false, then a
* <tt>std::invalid_argument</tt> is thrown.
*
* This macro is intended for <tt>assert</tt>-like checks which should always
* be performed regardless of whether or not \c NDEBUG is <tt>#define</tt>d.
*/
#define AR_ENSURE_ARG(expr) \
AR_ENSURE_MSGEXCEPT(expr, AR_STRINGIFY(expr)" false", std::invalid_argument)
/**
* Ensure that \c expr evaluates to boolean \c true at runtime. If \c expr
* evaluates to boolean \c false, then an exception \c except is thrown.
*
* This macro is intended for <tt>assert</tt>-like checks which should always
* be performed regardless of whether or not \c NDEBUG is <tt>#define</tt>d.
*/
#define AR_ENSURE_EXCEPT(expr, except) \
AR_ENSURE_MSGEXCEPT(expr, AR_STRINGIFY(expr)" false", except)
/**
* @}
*/
/////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////
/**
* Stable, one-pass algorithms for computing variances and covariances.
*
* @{
*/
/**
* Compute the mean and the number of samples, N, times the population variance
* using Welford's algorithm. The latter quantity is effectively the centered
* sum of squares. The algorithm is found in Knuth's TAOCP volume 2 section
* 4.2.2.A on page 232. The implementation follows
* http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance.
*
* @param[in] first Beginning of the input data range.
* @param[in] last Exclusive end of the input data range.
* @param[out] mean Mean of the data in <tt>[first, last)</tt>.
* @param[out] nvar N times the variance of the data.
*
* @returns the number data values processed within <tt>[first, last)</tt>.
*/
template <typename InputIterator,
typename OutputType1,
typename OutputType2>
std::size_t welford_nvariance(InputIterator first,
InputIterator last,
OutputType1& mean,
OutputType2& nvar)
{
using std::iterator_traits;
using std::size_t;
typedef typename iterator_traits<InputIterator>::value_type value;
size_t N = 1; // Running next sample number
value m = 0; // Running mean of data thus far
value nv = 0; // Running variance times the number of samples
while (first != last)
{
value x = *first++;
value d = x - m;
m += d / N++;
nv += d*(x - m);
}
mean = m;
nvar = nv;
return N-1;
}
/**
* Compute the mean and population variance using Welford's algorithm.
*
* @param[in] first Beginning of the input data range.
* @param[in] last Exclusive end of the input data range.
* @param[out] mean Mean of the data in <tt>[first, last)</tt>.
* @param[out] var The population variance of the data.
*
* @returns N, the number data values processed within <tt>[first, last)</tt>.
*/
template <typename InputIterator,
typename OutputType1,
typename OutputType2>
std::size_t welford_variance_population(InputIterator first,
InputIterator last,
OutputType1& mean,
OutputType2& var)
{
using std::size_t;
size_t N = welford_nvariance(first, last, mean, var);
var /= N;
return N;
}
/**
* Compute the mean and sample variance using Welford's algorithm.
*
* @param[in] first Beginning of the input data range.
* @param[in] last Exclusive end of the input data range.
* @param[out] mean Mean of the data in <tt>[first, last)</tt>.
* @param[out] var The sample variance of the data.
*
* @returns N, the number data values processed within <tt>[first, last)</tt>.
*/
template <typename InputIterator,
typename OutputType1,
typename OutputType2>
std::size_t welford_variance_sample(InputIterator first,
InputIterator last,
OutputType1& mean,
OutputType2& var)
{
using std::size_t;
size_t N = welford_nvariance(first, last, mean, var);
var /= (N - 1);
return N;
}
/**
* Compute means and the number of samples, N, times the population covariance
* using Welford's algorithm. The implementation follows the covariance
* section of http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance.
*
* @param[in] first1 Beginning of the first input range.
* @param[in] last1 Exclusive end of first input range.
* @param[in] first2 Beginning of the second input range.
* @param[out] mean1 Mean of the first data set.
* @param[out] mean2 Mean of the second data set.
* @param[out] ncovar N times the covariance of the two sets.
*
* @returns the number data values processed within <tt>[first1, last1)</tt>.
*/
template <typename InputIterator1,
typename InputIterator2,
typename OutputType1,
typename OutputType2,
typename OutputType3>
std::size_t welford_ncovariance(InputIterator1 first1,
InputIterator1 last1,
InputIterator2 first2,
OutputType1& mean1,
OutputType2& mean2,
OutputType3& ncovar)
{
using std::iterator_traits;
using std::size_t;
typedef typename iterator_traits<InputIterator1>::value_type value1;
typedef typename iterator_traits<InputIterator2>::value_type value2;
size_t N = 1; // Running next sample number
value1 m1 = 0; // Running mean of first data set thus far
value2 m2 = 0; // Running mean of second data set thus far
OutputType3 nc = 0; // Running covariance times the number of samples
while (first1 != last1)
{
value1 x1 = *first1++;
value1 d1 = x1 - m1;
m1 += d1 / N;
value2 x2 = *first2++;
value2 d2 = x2 - m2;
m2 += d2 / N;
nc += d1*(x2 - m2);
++N;
}
mean1 = m1;
mean2 = m2;
ncovar = nc;
return N-1;
}
/**
* Compute means and the population covariance using Welford's algorithm.
*
* @param[in] first1 Beginning of the first input range.
* @param[in] last1 Exclusive end of first input range.
* @param[in] first2 Beginning of the second input range.
* @param[out] mean1 Mean of the first data set.
* @param[out] mean2 Mean of the second data set.
* @param[out] covar The covariance of the two sets.
*
* @returns the number data values processed within <tt>[first1, last1)</tt>.
*/
template <typename InputIterator1,
typename InputIterator2,
typename OutputType1,
typename OutputType2,
typename OutputType3>
std::size_t welford_covariance_population(InputIterator1 first1,
InputIterator1 last1,
InputIterator2 first2,
OutputType1& mean1,
OutputType2& mean2,
OutputType3& covar)
{
using std::size_t;
size_t N = welford_ncovariance(first1, last1, first2, mean1, mean2, covar);
covar /= N;
return N;
}
/**
* Compute means and the sample covariance using Welford's algorithm.
*
* @param[in] first1 Beginning of the first input range.
* @param[in] last1 Exclusive end of first input range.
* @param[in] first2 Beginning of the second input range.
* @param[out] mean1 Mean of the first data set.
* @param[out] mean2 Mean of the second data set.
* @param[out] covar The covariance of the two sets.
*
* @returns the number data values processed within <tt>[first1, last1)</tt>.
*/
template <typename InputIterator1,
typename InputIterator2,
typename OutputType1,
typename OutputType2,
typename OutputType3>
std::size_t welford_covariance_sample(InputIterator1 first1,
InputIterator1 last1,
InputIterator2 first2,
OutputType1& mean1,
OutputType2& mean2,
OutputType3& covar)
{
using std::size_t;
size_t N = welford_ncovariance(first1, last1, first2, mean1, mean2, covar);
covar /= (N - 1);
return N;
}
/**
* Compute the inner product of <tt>[first, last)</tt> with itself using \ref
* welford_nvariance. Welford's algorithm is combined with the linearity of
* the expectation operator to compute a more expensive but also more
* numerically stable result than can be had using <tt>std::inner_product</tt>.
*
* @param[in] first Beginning of the input data range.
* @param[in] last Exclusive end of the input data range.
* @param[in] init Initial value, often zero, establishing the result type.
*
* @returns The inner product of <tt>[first, last)</tt> with itself.
*/
template <typename InputIterator,
typename ValueType>
ValueType welford_inner_product(InputIterator first,
InputIterator last,
ValueType init)
{
typename std::iterator_traits<InputIterator>::value_type mean;
ValueType nvar;
const std::size_t N = welford_nvariance(first, last, mean, nvar);
return init + (nvar + N*(mean*mean));
}
/**
* Compute the inner product of <tt>[first1, last1)</tt> against <tt>[first2,
* ...)</tt> using \ref welford_ncovariance. Welford's algorithm is combined
* with the linearity of the expectation operator to compute a more expensive
* but also numerically stable result than can be had using
* <tt>std::inner_product</tt>.
*
* @param[in] first1 Beginning of the first input range.
* @param[in] last1 Exclusive end of first input range.
* @param[in] first2 Beginning of the second input range.
* @param[in] init Initial value, often zero, establishing the result type.
*
* @returns The inner product of <tt>[first1, last1)</tt> against
* <tt>[first2, ...)</tt>.
*/
template <typename InputIterator1,
typename InputIterator2,
typename ValueType>
ValueType welford_inner_product(InputIterator1 first1,
InputIterator1 last1,
InputIterator2 first2,
ValueType init)
{
typename std::iterator_traits<InputIterator1>::value_type mean1;
typename std::iterator_traits<InputIterator2>::value_type mean2;
ValueType ncovar;
const std::size_t N = welford_ncovariance(
first1, last1, first2, mean1, mean2, ncovar);
return init + (ncovar + N*(mean1*mean2));
}
/**
* @}
*/
/////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////
/**
* Algorithms for autoregressive parameter estimation and manipulation.
*
* @{
*/
#if _MSC_VER > 1400
# pragma float_control(push)
# pragma float_control(precise, on)
#endif
/**
* Robustly compute negative one half the reflection coefficient assuming
* \f$\vec{a}\f$ and \f$\vec{b}\f$ contain real-valued backward and forward
* prediction error sequences, respectively. Zero is returned whenever the
* reflection coefficient numerator is identically zero, as otherwise
* constant zero signals produce undesired NaN reflection coefficients.
* The constant zero special case does not defeat NaN detection as any data
* introducing NaN into the denominator must introduce NaN into the numerator.
*
* @param[in] a_first Beginning of the first input range \f$\vec{a}\f$.
* @param[in] a_last Exclusive end of first input range \f$\vec{a}\f$.
* @param[in] b_first Beginning of the second input range \f$\vec{b}\f$.
*
* @return \f$\frac{\vec{a}\cdot\vec{b}}
* {\vec{a}\cdot\vec{a} + \vec{b}\cdot\vec{b}}\f$
* when that numerator is nonzero, else zero.
*
* @see Wikipedia's article on <a href="">Kahan summation</a> for
* background on how the accumulation error is reduced in the result.
*/
template <typename ValueType,
typename InputIterator1,
typename InputIterator2>
ValueType
#if (AR_GCC_VERSION > 40305)
__attribute__((__optimize__("no-associative-math")))
#endif
negative_half_reflection_coefficient(InputIterator1 a_first,
InputIterator1 a_last,
InputIterator2 b_first)
#if (AR_GCC_VERSION > 40305) || (_MSC_VER > 1400)
{
ValueType ns = 0, nt, nc = 0, ny; // Kahan numerator accumulation
ValueType ds = 0, dt, dc = 0, dy; // Kahan denominator accumulation
while (a_first != a_last)
{
ValueType xa = *a_first++; // Denominator: \vec{a}\cdot\vec{a}
dy = (xa * xa) - dc;
dt = ds + dy;
dc = (dt - ds) - dy;
ds = dt;
ValueType xb = *b_first++; // Denominator: \vec{b}\cdot\vec{b}
dy = (xb * xb) - dc;
dt = ds + dy;
dc = (dt - ds) - dy;
ds = dt;
ny = (xa * xb) - nc; // Numerator: \vec{a}\cdot\vec{b}
nt = ns + ny;
nc = (nt - ns) - ny;
ns = nt;
}
return ns + nc == 0 // Does special zero case apply?
? 0 // Yes, to avoid NaN from 0 / 0
: (ns + nc) / (ds + dc); // No, correct final sums and form ratio
}
#else
#warning Using Non-Kahan version of ar::negative_half_reflection_coefficient.
{
ValueType ns = 0;
ValueType ds = 0;
while (a_first != a_last)
{
ValueType xa = *a_first++;
ValueType xb = *b_first++;
ns += xa * xb; // Numerator
ds += xa * xa + xb * xb; // Denominator
}
return ns == 0 // Does special zero case apply?
? 0 // Yes, to avoid NaN from 0 / 0
: ns / ds; // No, form ratio
}
#endif
#if _MSC_VER > 1400
# pragma float_control(pop)
#endif
/**
* Fit an autoregressive model to stationary time series data using %Burg's
* method. That is, find coefficients \f$a_i\f$ such that the sum of the
* squared errors in the forward predictions \f$x_n = -a_1 x_{n-1} - \dots -
* a_p x_{n-p}\f$ and backward predictions \f$x_n = -a_1 x_{n+1} - \dots - a_p
* x_{n+p}\f$ are both minimized. Either a single model of given order or a
* hierarchy of models up to and including a maximum order may fit.
*
* The input data \f$\vec{x}\f$ are read from <tt>[data_first, data_last)</tt>
* in a single pass. The mean is computed, returned in \c mean, and \e
* removed from further consideration whenever \c subtract_mean is true.
* The estimated model parameters \f$a_i\f$ are output using \c params_first
* with the behavior determined by the amount of data read, <tt>maxorder</tt>,
* and the \c hierarchy flag:
* <ul>
* <li>If \c hierarchy is \c false, only the \f$a_1, \dots,
* a_\text{maxorder}\f$ parameters for an AR(<tt>maxorder</tt>) process
* are output.</li>
* <li>If \c hierarchy is \c true, the <tt>maxorder*(maxorder+1)/2</tt>
* parameters \f$a_1, \dots, a_m\f$ for models AR(0), AR(1), AR(2),
* ..., AR(maxorder) are output. Notice AR(0) has no parameters.
* </li>
* </ul>
* Note that the latter case is \e always computed; the \c hierarchy flag
* merely controls what is output. In both cases, the maximum order is limited
* by the number of data samples provided and is output to \c maxorder.
*
* One mean squared discrepancy \f$\sigma^2_\epsilon\f$, also called the
* innovation variance, and gain, defined as \f$\sigma^2_x /
* \sigma^2_\epsilon\f$, are output for each model, including the trivial
* zeroth order model when \c maxorder is zero or \c hierarchy is \c true,
* using \c sigma2e_first and \c gain_first. The autocorrelations for lags
* <tt>[0,k]</tt> are output using \c autocor_first. When \c hierarchy is \c
* true, only lags <tt>[0,m]</tt> should be applied for some AR(<tt>m</tt>)
* model. Outputting the lag \c k autocorrelation is technically redundant as
* it may be computed from \f$a_i\f$ and lags <tt>0, ..., k-1</tt>.
* Autocovariances may be computed by multiplying the autocorrelations by the
* gain times \f$\sigma^2_\epsilon\f$.
*
* The software aspects of the implementation differs from many other sources.
* In particular,
* <ul>
* <li>iterators are employed,</li>
* <li>the working precision is selectable using \c mean,</li>
* <li>the mean squared discrepancy calculation has been added,</li>
* <li>some loop index transformations have been performed,</li>
* <li>working storage may be passed into the method to reduce allocations
* across many invocations, and</li>
* <li>and all lower order models may be output during the recursion using
* \c hierarchy.</li>
* </ul>
* Gain and autocorrelation calculations have been added based on sections 5.2
* and 5.3 of Broersen, P. M. T. Automatic autocorrelation and spectral
* analysis. Springer, 2006. http://dx.doi.org/10.1007/1-84628-329-9. The
* classical algorithm, rather than the variant using denominator recursion due
* to Andersen (http://dx.doi.org/10.1109/PROC.1978.11160), has been chosen as
* the latter can be numerically unstable.
*
* @param[in] data_first Beginning of the input data range.
* @param[in] data_last Exclusive end of the input data range.
* @param[out] mean Mean of data.
* @param[in,out] maxorder On input, the maximum model order desired.
* On output, the maximum model order computed.
* @param[out] params_first Model parameters for a single model or
* for an entire hierarchy of models. At most
* <tt>!hierarchy ? maxorder :
* maxorder*(maxorder+1)/2</tt> values will be
* output.
* @param[out] sigma2e_first The mean squared discrepancy for only
* AR(<tt>maxorder</tt>) or for an entire
* hierarchy. Either one or at most
* <tt>maxorder + 1</tt> values will be output.
* @param[out] gain_first The model gain for only AR(<tt>maxorder</tt>)
* or an entire hierarchy. Either one or at most
* <tt>maxorder + 1</tt> values will be output.
* @param[out] autocor_first Lag one through lag maxorder autocorrelations.
* At most <tt>maxorder + 1</tt> values will be
* output.
* @param[in] subtract_mean Should \c mean be subtracted from the data?
* @param[in] hierarchy Should the entire hierarchy of estimated
* models be output?
* @param[in] f Working storage. Reuse across invocations
* may speed execution by avoiding allocations.
* @param[in] b Working storage similar to \c f.
* @param[in] Ak Working storage similar to \c f.
* @param[in] ac Working storage similar to \c f.
*
* @returns the number data values processed within
* <tt>[data_first, data_last)</tt>.
*/
template <class InputIterator,
class Value,
class OutputIterator1,
class OutputIterator2,
class OutputIterator3,
class OutputIterator4,
class Vector>
std::size_t burg_method(InputIterator data_first,
InputIterator data_last,
Value& mean,
std::size_t& maxorder,
OutputIterator1 params_first,
OutputIterator2 sigma2e_first,
OutputIterator3 gain_first,
OutputIterator4 autocor_first,
const bool subtract_mean,
const bool hierarchy,
Vector& f,
Vector& b,
Vector& Ak,
Vector& ac)
{
using std::bind2nd;
using std::copy;
using std::distance;
using std::fill;
using std::inner_product;
using std::min;
using std::minus;
using std::size_t;
// Initialize f from [data_first, data_last) and fix number of samples
f.assign(data_first, data_last);
const size_t N = f.size();
// Stably compute the incoming data's mean and population variance
mean = 0;
Value sigma2e = 0;
welford_variance_population(f.begin(), f.end(), mean, sigma2e);
// When requested, subtract the just-computed mean from the data.
// Adjust, if necessary, to make sigma2e the second moment.
if (subtract_mean)
{
transform(f.begin(), f.end(), f.begin(), bind2nd(minus<Value>(), mean));
}
else
{
sigma2e += mean*mean;
}
// At most maxorder N-1 can be fit from N samples. Beware N is unsigned.
maxorder = (N == 0) ? 0 : min(static_cast<size_t>(maxorder), N-1);
// Output sigma2e and gain for a zeroth order model, if requested.
Value gain = 1;
if (hierarchy || maxorder == 0)
{
*sigma2e_first++ = sigma2e;
*gain_first++ = gain;
}
// Initialize and perform Burg recursion
if (maxorder) b = f; // Copy iff non-trivial work required
Ak.assign(maxorder + 1, Value(0));
Ak[0] = 1;
ac.clear();
ac.reserve(maxorder);
for (size_t kp1 = 1; kp1 <= maxorder; ++kp1)
{
// Compute mu from f, b, and Dk and then update sigma2e and Ak using mu
// Afterwards, Ak[1:kp1] contains AR(k) coefficients by the recurrence
// Must treat mu result of 0 / 0 as 0 to avoid NaNs on constant signals
// By the recurrence, Ak[kp1] will also be the reflection coefficient
Value mu = -2 * negative_half_reflection_coefficient<Value>(
f.begin() + kp1, f.end(), b.begin());
sigma2e *= (1 - mu*mu);
for (size_t n = 0; n <= kp1/2; ++n)
{
Value t1 = Ak[n] + mu*Ak[kp1 - n];
Value t2 = Ak[kp1 - n] + mu*Ak[n];
Ak[n] = t1;
Ak[kp1 - n] = t2;
}
// Update the gain per Broersen 2006 equation (5.25)
gain *= 1 / (1 - Ak[kp1]*Ak[kp1]);
// Compute and output the next autocorrelation coefficient
// See Broersen 2006 equations (5.28) and (5.31) for details
ac.push_back(-inner_product(ac.rbegin(), ac.rend(),
Ak.begin() + 1, Ak[kp1]));
// Output parameters and the input and output variances when requested
if (hierarchy || kp1 == maxorder)
{
params_first = copy(Ak.begin() + 1, Ak.begin() + kp1 + 1,
params_first);
*sigma2e_first++ = sigma2e;
*gain_first++ = gain;
}
// Update f and b for the next iteration if another remains
if (kp1 < maxorder)
{
for (size_t n = 0; n < N - kp1; ++n)
{
Value t1 = f[n + kp1] + mu*b[n];
Value t2 = b[n] + mu*f[n + kp1];
f[n + kp1] = t1;
b[n] = t2;
}
}
}
// Output the lag [0,maxorder] autocorrelation coefficients in single pass
*autocor_first++ = 1;
copy(ac.begin(), ac.end(), autocor_first);
// Return the number of values processed in [data_first, data_last)
return N;
}
/** \copydoc burg_method(InputIterator,InputIterator,Value&,std::size_t&,OutputIterator1,OutputIterator2,OutputIterator3,OutputIterator4,const bool,const bool,Vector&,Vector&,Vector&,Vector&) */
template <class InputIterator,
class Value,
class OutputIterator1,
class OutputIterator2,
class OutputIterator3,
class OutputIterator4>
std::size_t burg_method(InputIterator data_first,
InputIterator data_last,
Value& mean,
std::size_t& maxorder,
OutputIterator1 params_first,
OutputIterator2 sigma2e_first,
OutputIterator3 gain_first,
OutputIterator4 autocor_first,
const bool subtract_mean = false,
const bool hierarchy = false)
{
using std::vector;
vector<Value> f, b, Ak, ac; // Working storage
return burg_method(data_first, data_last, mean, maxorder,
params_first, sigma2e_first, gain_first,
autocor_first, subtract_mean, hierarchy,
f, b, Ak, ac);
}
// Type erasure for NoiseGenerator parameters within predictor.
// Either std::tr1::function or boost::function would better provide the
// desired capability but both add additional, undesired dependencies.
namespace
{
/** Abstract base class for NoiseGenerator-related type erasure. */
template <typename Value>
struct nullary
{
virtual ~nullary() {}
virtual Value operator()() = 0;
virtual nullary* clone() = 0;
};
/** A nullary function always returning zero. */
template<typename Value>
struct nullary_impl0 : public nullary<Value>
{
Value operator()()
{
return 0;
}
nullary_impl0* clone()
{
return new nullary_impl0();
}
};
/** A nullary function always invoking t(). */
template<typename Value, class T>
struct nullary_impl1 : public nullary<Value>
{
nullary_impl1(T t) : t(t) {}
Value operator()()
{
return t();
}
nullary_impl1* clone()
{
return new nullary_impl1(t);
}
T t;
};
}
/**
* Simulate an autoregressive model process with an InputIterator interface.
*/
template <typename Value, typename Index = std::size_t>
class predictor
: public std::iterator<std::input_iterator_tag, Value,
std::ptrdiff_t, const Value*, const Value&>
{
private:
typedef std::iterator<std::input_iterator_tag, Value,
std::ptrdiff_t, const Value*, const Value&> base;
public:
typedef typename base::difference_type difference_type;
typedef typename base::iterator_category iterator_category;
typedef typename base::pointer pointer;
typedef typename base::reference reference;
typedef typename base::value_type value_type;
/** Singular instance marking prediction index \c n. */
explicit predictor(Index n = 0) : n(n), d(), g(0), xn()
{
#ifndef NDEBUG
using std::numeric_limits;
if (numeric_limits<Value>::has_quiet_NaN)
xn = numeric_limits<Value>::quiet_NaN();
#endif
}
/**
* Iterate on the process \f$x_n + a_1 x_{n - 1} + \dots + a_p x_{n - p} =
* 0\f$. Presumably \ref initial_conditions will be used to specify some
* initial state as otherwise the process is identically zero. The process
* order \f$p\f$ is set by <tt>std::distance(params_first,
* params_last)</tt>.
*
* @param params_first Beginning of the process parameter range
* starting with \f$a_1\f$.
* @param params_last End of the process parameter range.
*/
template <class RandomAccessIterator>
predictor(RandomAccessIterator params_first,
RandomAccessIterator params_last)
: n(0),
d(2*std::distance(params_first, params_last), 0),
g(new nullary_impl0<Value>()),
xn((*g)())
{
// Finish preparing d = [ a_p, ..., a_1, 0, ..., 0 ]
using std::vector;
typename vector<Value>::size_type i = d.size() / 2;
while (i --> 0) d[i] = *params_first++;
// Now x_n = 0 because x_{n-p} = ... = x_{n-1} = 0 by construction.
}
/**
* Iterate on the process \f$x_n + a_1 x_{n - 1} + \dots + a_p x_{n - p} =
* \epsilon_n\f$ given zero initial conditions. The process order \f$p\f$
* is set by <tt>std::distance(params_first,params_last)</tt>. The class
* <tt>std::tr1::variate_generator</tt> may be helpful in constructing
* normally distributed input.
*
* @param params_first Beginning of the process parameter range
* starting with \f$a_1\f$.
* @param params_last End of the process parameter range.
* @param generator A nullary callback for generating \f$\epsilon_n\f$.
* For example, a random number generator distributed
* like \f$N\left(0, \sigma^2_\epsilon\right)\f$.
*/
template <class RandomAccessIterator,
class NoiseGenerator>
predictor(RandomAccessIterator params_first,
RandomAccessIterator params_last,
NoiseGenerator generator)
: n(0),
d(2*std::distance(params_first, params_last), 0),
g(new nullary_impl1<Value,NoiseGenerator>(generator)),
xn((*g)())
{
// Finish preparing d = [ a_p, ..., a_1, 0, ..., 0 ]
using std::vector;
typename vector<Value>::size_type i = d.size() / 2;
while (i --> 0) d[i] = *params_first++;
// Here x_0 = \epsilon_0 because x_{0-p} = ... = x_{0-1} = 0.
}
/** Copy constructor */
predictor(const predictor& other)
: n(other.n),
d(other.d),
g(other.g ? other.g->clone() : 0),
xn(other.xn)
{}
/** Assignment operator */
predictor& operator=(const predictor& other)
{
if (this != &other)
{
nullary<Value> *tmp = 0;
try
{
tmp = other.g ? other.g->clone() : 0;
}
catch (...)
{
delete tmp;
throw;
}
base::operator=(other);
n = other.n;
d = other.d;
delete g;
g = tmp;
xn = other.xn;
}
return *this;
}
/** Destructor */
~predictor()
{
delete g;
}
/**
* Specify process initial conditions \f$x_{n-1}, \dots, x_{n-p}\f$ where
* \f$p\f$ is the process order fixed by the constructor. The simulation
* index \f$n\f$ is reset to zero and, optionally, \f$x_0\f$ is additively
* adjusted by \c x0adjust.
*
* @param initial_first Beginning of the initial condition range
* \f$x_{n-1}, \dots, x_{n-p}\f$
* which must contain \f$p\f$ values.
* @param x0adjust An additive adjustment made to \f$\epsilon_0\f$.
*/
template <class InputIterator>
predictor& initial_conditions(InputIterator initial_first,
const Value x0adjust = 0)
{
// Zero the simulation time.
n = 0;
// Set d = [ a_p, ..., a_1, x_{n-p}, ..., x_{n-1} ]
using std::vector;
typename vector<Value>::size_type i = d.size();
typename vector<Value>::size_type p = i / 2;
while (i --> p) d[i] = *initial_first++;
// Make x_n := - a_p*x_{n-p} - ... - a_1*x_{n-1} + x_n + x0adjust.
// By design, x_n was whatever it happened to be.
using std::inner_product;
xn += x0adjust;
xn = -inner_product(d.begin(), d.begin() + p, d.begin() + p, -xn);
return *this;
}
// Concept: InputIterator
/** Prefix increment. */
predictor& operator++()
{
using std::distance;
using std::inner_product;
using std::vector;
if (g)
{
typename vector<Value>::size_type p = d.size() / 2;
if (p)
{
// Make x_n = - a_p*x_{n-p} - ... - a_1*x_{n-1} + \epsilon_n
// by (conceptually) storing previously computed x_n into
// circular buffer, updating ++n, and computing x_{n+1}.
typename vector<Value>::iterator ab = d.begin();
typename vector<Value>::iterator xb = ab + p;
typename vector<Value>::iterator c = xb + n % p;
typename vector<Value>::iterator xe = d.end();
*c++ = xn;
xn = inner_product(c, xe, ab, -(*g)());
xn = -inner_product(xb, c, ab + distance(c, xe), xn );
}
else
{