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.gitignore vendored
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@ -50,3 +50,7 @@ modules.order
Module.symvers Module.symvers
Mkfile.old Mkfile.old
dkms.conf dkms.conf
build/
.DS_Store
*.sh

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CMakeLists.txt Normal file
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cmake_minimum_required(VERSION 3.15.2)
# 设置工程名称
project(LibEEMD VERSION 1.4.1 LANGUAGES C)
# 添加配置配件编写的函数
include(CMakePackageConfigHelpers)
message(STATUS "Platform: " ${CMAKE_HOST_SYSTEM_NAME})
# CMake默认的安装路径 Windows下为C:/Program\ Files/${Project_Name} Linux/Unix下为/usr/local
message(STATUS "Install prefix: " ${CMAKE_INSTALL_PREFIX})
# CMake默认的变异类型为空
message(STATUS "Build type: " ${CMAKE_BUILD_TYPE})
# 添加源文件地址
add_subdirectory(src/)

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COPYING Normal file
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If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:
<program> Copyright (C) <year> <name of author>
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License. Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".
You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
<http://www.gnu.org/licenses/>.
The GNU General Public License does not permit incorporating your program
into proprietary programs. If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library. If this is what you want to do, use the GNU Lesser General
Public License instead of this License. But first, please read
<http://www.gnu.org/philosophy/why-not-lgpl.html>.

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Notable changes from previous release:
In development:
Version 1.4.1:
* Fix bug in CEEMDAN SNR fixing introduced in 1.4; in some cases
(especially N=16) the noise can have zero variance causing an overflow
Version 1.4:
* pyeemd splitted to a separate project
* Easier installation on OSX
* The GSL error handler is disabled and realistic errors are instead
handled by libeemd
* In CEEMDAN the SNR is fixed separately for each mode as done by Torres
et al
Version 1.3.1:
* Minor documentation improvements
* Minor bug fix about possible redefinition of NDEBUG
Version 1.3:
* New default stopping parameters for pyeemd
* pyeemd warns if using num_siftings=0
* Completely reworked S-number criterion handling
* Minor bug fixes
* Minor improvements to unit tests
Version 1.2:
* Rudimentary error reporting in the C API
* Allow setting number of IMFs to compute
Version 1.1:
* Implementation of CEEMDAN
* Allow setting the RNG seed
* Documentation improvements
* Minor optimizations

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@PACKAGE_INIT@
set(@PROJECT_NAME@_Version "@PROJECT_VERSION@")
set_and_check(@PROJECT_NAME@_INSTALL_PREFIX "${PACKAGE_PREFIX_DIR}")
set_and_check(@PROJECT_NAME@_INC_DIR "${PACKAGE_PREFIX_DIR}/include")
set_and_check(@PROJECT_NAME@_INCLUDE_DIR "${PACKAGE_PREFIX_DIR}/include")
set_and_check(@PROJECT_NAME@_LIB_DIR "${PACKAGE_PREFIX_DIR}/lib")
set_and_check(@PROJECT_NAME@_LIBRARY_DIR "${PACKAGE_PREFIX_DIR}/lib")
set(@PROJECT_NAME@_LIB eemd)
set(@PROJECT_NAME@_LIBRARY eemd)
set(@PROJECT_NAME@_FOUND 1)
# include target information
include("${CMAKE_CURRENT_LIST_DIR}/@PROJECT_NAME@Targets.cmake")

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libeemd – a C library for performing the ensemble empirical mode decomposition
==============================================================================
libeemd is a C library for performing the ensemble empirical mode decomposition
(EEMD), its complete variant (CEEMDAN) or the regular empirical mode
decomposition (EMD). The details of what libeemd actually computes are
available as a separate [article][], which you should read if you are unsure
about what EMD, EEMD and CEEMDAN are.
[article]: https://dx.doi.org/10.1007/s00180-015-0603-9
Acquiring libeemd
-----------------
The easiest way to get up-to-date versions of libeemd is to use [Bitbucket][],
which is a site built for distributing software using the amazing version
control system [Git][]. By using libeemd's [Bitbucket site][webpage] you can see
recent changes made to the program, report and track bugs found in the program,
access user-generated documentation and even create your own versions of
libeemd.
[bitbucket]: https://bitbucket.org
[git]: http://git-scm.com
Program license
---------------
libeemd is free software: you can redistribute it and/or modify it under the
terms of the GNU General Public License as published by the Free Software
Foundation, either version 3 of the License, or (at your option) any later
version.
libeemd is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A
PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with
libeemd. If not, see <http://www.gnu.org/licenses/>.
[webpage]: https://bitbucket.org/luukko/libeemd
Installation
------------
### Dependencies
To compile libeemd you need:
* A fairly recent C compiler (something that understands C99)
* GNU [Scientific Library (GSL)][GSL]
If you want to use the easy route and use the `Makefile` distributed with
libeemd, you should have:
* GNU [Make][]
* GNU [Compiler Collection (GCC)][GCC]
[Make]: http://www.gnu.org/software/make/
[GCC]: http://gcc.gnu.org/
[GSL]: http://www.gnu.org/software/gsl/
### Installation
If you have Make and GCC installed, you can simply run
make
in the top-level directory of libeemd (the one with the `Makefile`). This
command compiles libeemd into a static library `libeemd.a`, a dynamic library
`libeemd.so`, and copies the header file `eemd.h` to the top-level directory.
You can then copy these files to wherever you need them.
You can use `make install` to install the library files to your system. By
default this command installs the files under `/usr` (so you'll need root
privileges), but you can specify another installation location like this:
PREFIX=$HOME/usr make install
If you set a `PREFIX` you need to make sure other programs will find libeemd
in this location. For example, if you set `PREFIX` to `$HOME/usr`, the
directory `$HOME/usr/lib` should be in `LIBRARY_PATH` and `LD_LIBRARY_PATH`.
Using the C interface
------------
To use libeemd in your program include `eemd.h` in your header file and link
your program against `libeemd.a` or `libeemd.so` and [GSL][]. The routines
exported by libeemd are documented in the header file `eemd.h`.
To see a short example of libeemd in action, please see the `examples`
subdirectory.
pyeemd, the Python interface to libeemd
-------------------------
The official Python interface to libeemd is called [pyeemd][]. You should
definitely try it, since data analysis with Python is much more fun than with
C. More documentation about [pyeemd][] can be found at [Read the Docs](http://pyeemd.readthedocs.org/).
[pyeemd]: https://bitbucket.org/luukko/pyeemd
Rlibeemd, the R interface to libeemd
----------------------------------------
There is also a R interface to libeemd. It is available separately at
<https://github.com/helske/Rlibeemd>.
Developing libeemd
------------------
Please report any issues using the Bitbucket issue tracker. If submitting pull
requests, please use `develop` as a target branch.

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To all,
This is a CMake redistribution of the libeemd library. It is created for easy installation cross different platforms. Please send all questions and regards to the original authors. THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
Yi Zhang
2021-11-20

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# 设定源文件文件夹
aux_source_directory(lib LIBEEMD_SRC)
# 以下部分为库的编译
# 注意目标名必须唯一 所以不能直接生成相同名称的动态库与静态库
# 注意此处不必为目标名称添加lib前缀和相应后缀，cmake会自行添加
add_library(eemd SHARED ${LIBEEMD_SRC})
# 首先添加静态库的生成命令
add_library(eemd_static STATIC ${LIBEEMD_SRC})
# 设置静态库的输出名称从而获得与动态库名称相同的静态库
set_target_properties(eemd_static PROPERTIES OUTPUT_NAME "eemd")
# 设置输出目标属性以同时输出动态库与静态库
set_target_properties(eemd PROPERTIES CLEAN_DIRECT_OUTPUT 1)
set_target_properties(eemd_static PROPERTIES CLEAN_DIRECT_OUTPUT 1)
# 设置动态库的版本号
set_target_properties(eemd PROPERTIES VERSION ${PROJECT_VERSION} SOVERSION ${PROJECT_VERSION_MAJOR}.${PROJECT_VERSION_MINOR})
# 设置库文件的输出地址
set(LIBRARY_OUTPUT_PATH ${PROJECT_BINARY_DIR}/lib)
# 设置编译选项
set(CMAKE_C_FLAGS "${CMAKE_C_FLAGS} -O3")
# 添加GSL库的编译命令
find_package(GSL REQUIRED)
if(GSL_FOUND)
message(STATUS "GSL Found.")
include_directories(${GSL_INCLUDE_DIRS})
target_link_libraries(eemd PUBLIC GSL::gsl)
target_link_libraries(eemd_static GSL::gsl)
endif()
# 添加openmp的编译命令 设置编译选项
find_package(OpenMP REQUIRED)
if (OpenMP_C_FOUND)
message(STATUS "OpenMP Found.")
set(CMAKE_C_FLAGS "${CMAKE_C_FLAGS} ${OpenMP_C_FLAGS}")
set(CMAKE_EXE_LINKER_FLAGS "${CMAKE_EXE_LINKER_FLAGS} ${OpenMP_EXE_LINKER_FLAGS}")
set(CMAKE_SHARED_LINKER_FLAGS "${CMAKE_SHARED_LINKER_FLAGS} ${OpenMP_SHARED_LINKER_FLAGS}")
target_link_libraries(eemd PUBLIC OpenMP::OpenMP_C)
target_link_libraries(eemd_static OpenMP::OpenMP_C)
endif()
set(CONFIG_FILE_PATH lib/cmake/${PROJECT_NAME})
configure_package_config_file(${PROJECT_SOURCE_DIR}/${PROJECT_NAME}Config.cmake.in
${CMAKE_BINARY_DIR}/${PROJECT_NAME}Config.cmake
INSTALL_DESTINATION ${CONFIG_FILE_PATH}
NO_CHECK_REQUIRED_COMPONENTS_MACRO)
write_basic_package_version_file(${CMAKE_BINARY_DIR}/${PROJECT_NAME}ConfigVersion.cmake
VERSION ${PROJECT_VERSION}
COMPATIBILITY SameMajorVersion)
# 库的安装命令
if(WIN32)
install(TARGETS eemd DESTINATION lib)
install(TARGETS eemd_static DESTINATION lib)
else()
install(TARGETS eemd eemd_static
EXPORT ${PROJECT_NAME}Targets
LIBRARY DESTINATION lib
ARCHIVE DESTINATION lib)
install(EXPORT ${PROJECT_NAME}Targets
DESTINATION ${CONFIG_FILE_PATH})
install(FILES
${CMAKE_BINARY_DIR}/${PROJECT_NAME}Config.cmake
${CMAKE_BINARY_DIR}/${PROJECT_NAME}ConfigVersion.cmake
DESTINATION ${CONFIG_FILE_PATH})
endif()
# 头文件安装命令
install(FILES lib/eemd.h DESTINATION include)
# 以下部分为例子程序的编译
# 设置可执行文件的输出地址
set(EXECUTABLE_OUTPUT_PATH ${PROJECT_BINARY_DIR}/bin)
# 例子的编译方法
macro(add_sample name)
# 添加可执行文件 命令行
add_executable(${name} examples/${name}.c)
# 为安装文件添加动态库的搜索地址 在Windows下并没有什么用 直接忽略
set_target_properties(${name} PROPERTIES INSTALL_RPATH ${CMAKE_INSTALL_PREFIX}/lib)
# 链接动态库
target_link_libraries(${name} PUBLIC eemd)
endmacro()
add_sample(eemd_example)
add_sample(ceemdan_example)

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`eemd_example` is a simple example program for the C interface of libeemd. It
writes its output to `eemd_example.out`. If you have installed `pyeemd` you can
plot the results with the Python script `eemd_example_plot.py`. To compile the
example please run `make`.

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/* Copyright 2013 Perttu Luukko
* This file is part of libeemd.
* libeemd is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
* libeemd is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
* You should have received a copy of the GNU General Public License
* along with libeemd. If not, see <http://www.gnu.org/licenses/>.
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_math.h>
const double pi = M_PI;
#include "../lib/eemd.h"
const size_t ensemble_size = 500;
const unsigned int S_number = 4;
const unsigned int num_siftings = 50;
const double noise_strength = 0.02;
const unsigned long int rng_seed = 0;
const char outfile[] = "ceemdan_example.out";
const size_t N = 512;
int main(void) {
libeemd_error_code err;
// As an example decompose a Dirac signal as in the original CEEMDAN paper
double* inp = malloc(N*sizeof(double));
memset(inp, 0x00, N*sizeof(double));
inp[N/2] = 1.0;
// Allocate memory for output data
size_t M = emd_num_imfs(N);
double* outp = malloc(M*N*sizeof(double));
// Run CEEMDAN
err = ceemdan(inp, N, outp, M, ensemble_size, noise_strength, S_number, num_siftings, rng_seed);
if (err != EMD_SUCCESS) {
emd_report_if_error(err);
exit(1);
}
// Write output to file
FILE* fp = fopen(outfile, "w");
for (size_t j=0; j<N; j++) {
fprintf(fp, "%f ", inp[j]);
}
fprintf(fp, "\n");
for (size_t i=0; i<M; i++) {
for (size_t j=0; j<N; j++) {
fprintf(fp, "%f ", outp[i*N+j]);
}
fprintf(fp, "\n");
}
printf("Done!\n");
// Cleanup
fclose(fp);
free(inp); inp = NULL;
free(outp); outp = NULL;
}

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#!/usr/bin/env python2
# Copyright 2013 Perttu Luukko
# This file is part of libeemd.
#
# libeemd is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# libeemd is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with libeemd. If not, see <http://www.gnu.org/licenses/>.
from numpy import loadtxt
from pylab import plot, show
from pyeemd.utils import plot_imfs
def main():
data = loadtxt("ceemdan_example.out")
orig = data[0]
imfs = data[1:]
plot(orig, label="Original data")
plot_imfs(imfs)
show()
if __name__ == "__main__":
main()

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/* Copyright 2013 Perttu Luukko
* This file is part of libeemd.
* libeemd is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
* libeemd is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
* You should have received a copy of the GNU General Public License
* along with libeemd. If not, see <http://www.gnu.org/licenses/>.
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_math.h>
const double pi = M_PI;
#include "../lib/eemd.h"
const size_t ensemble_size = 250;
const unsigned int S_number = 4;
const unsigned int num_siftings = 50;
const double noise_strength = 0.2;
const unsigned long int rng_seed = 0;
const char outfile[] = "eemd_example.out";
// An example signal to decompose
const size_t N = 1024;
static inline double input_signal(double x) {
const double omega = 2*pi/(N-1);
return sin(17*omega*x)+0.5*(1.0-exp(-0.002*x))*sin(51*omega*x+1);
}
int main(void) {
libeemd_error_code err;
// Define input data
double* inp = malloc(N*sizeof(double));
for (size_t i=0; i<N; i++) {
inp[i] = input_signal((double)i);
}
// Allocate memory for output data
size_t M = emd_num_imfs(N);
double* outp = malloc(M*N*sizeof(double));
// Run eemd
err = eemd(inp, N, outp, M, ensemble_size, noise_strength, S_number, num_siftings, rng_seed);
if (err != EMD_SUCCESS) {
emd_report_if_error(err);
exit(1);
}
// Write output to file
FILE* fp = fopen(outfile, "w");
for (size_t j=0; j<N; j++) {
fprintf(fp, "%f ", inp[j]);
}
fprintf(fp, "\n");
for (size_t i=0; i<M; i++) {
for (size_t j=0; j<N; j++) {
fprintf(fp, "%f ", outp[i*N+j]);
}
fprintf(fp, "\n");
}
printf("Done!\n");
// Cleanup
fclose(fp);
free(inp); inp = NULL;
free(outp); outp = NULL;
}

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#!/usr/bin/env python2
# Copyright 2013 Perttu Luukko
# This file is part of libeemd.
#
# libeemd is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# libeemd is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with libeemd. If not, see <http://www.gnu.org/licenses/>.
from numpy import loadtxt
from pylab import plot, show
from pyeemd.utils import plot_imfs
def main():
data = loadtxt("eemd_example.out")
orig = data[0]
imfs = data[1:]
plot(orig, label="Original data")
plot_imfs(imfs)
show()
if __name__ == "__main__":
main()

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/* Copyright 2013 Perttu Luukko
* This file is part of libeemd.
* libeemd is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
* libeemd is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
* You should have received a copy of the GNU General Public License
* along with libeemd. If not, see <http://www.gnu.org/licenses/>.
*/
#include "eemd.h"
// If we are using OpenMP for parallel computation, we need locks to ensure
// that the same output data is not written by several threads at the same
// time.
#ifdef _OPENMP
typedef omp_lock_t lock;
inline static void init_lock(lock* l) { omp_init_lock(l); }
inline static void destroy_lock(lock* l) { omp_destroy_lock(l); }
inline static void get_lock(lock* l) { omp_set_lock(l); }
inline static void release_lock(lock* l) { omp_unset_lock(l); }
#else
// If we don't use OpenMP, we provide a dummy lock that does nothing. This
// avoids littering the code with too many #ifdefs for _OPENMP.
typedef char lock;
inline static void init_lock(__attribute__((unused)) lock* l) {}
inline static void destroy_lock(__attribute__((unused)) lock* l) {}
inline static void get_lock(__attribute__((unused)) lock* l) {}
inline static void release_lock(__attribute__((unused)) lock* l) {}
#endif
// Helper functions for working with data arrays
inline static void array_copy(double const* restrict src, size_t n, double* restrict dest) {
memcpy(dest, src, n*sizeof(double));
}
inline static void array_add(double const* src, size_t n, double* dest) {
for (size_t i=0; i<n; i++)
dest[i] += src[i];
}
inline static void array_add_to(double const* src1, double const* src2, size_t n, double* dest) {
for (size_t i=0; i<n; i++)
dest[i] = src1[i] + src2[i];
}
inline static void array_addmul_to(double const* src1, double const* src2, double val, size_t n, double* dest) {
for (size_t i=0; i<n; i++)
dest[i] = src1[i] + val*src2[i];
}
inline static void array_sub(double const* src, size_t n, double* dest) {
for (size_t i=0; i<n; i++)
dest[i] -= src[i];
}
inline static void array_mult(double* dest, size_t n, double val) {
for (size_t i=0; i<n; i++)
dest[i] *= val;
}
// Helper function for extrapolating data at the ends. For a line passing
// through (x0, y0), (x1, y1), and (x, y), return y for a given x.
inline static double linear_extrapolate(double x0, double y0,
double x1, double y1, double x) {
assert(x1 != x0);
return y0 + (y1-y0)*(x-x0)/(x1-x0);
}
// In the following part the necessary workspace memory structures for several
// EMD operations are defined
// For sifting we need arrays for storing the found extrema of the signal, and memory required
// to form the spline envelopes
typedef struct {
// Number of samples in the signal
size_t N;
// Found extrema
double* restrict maxx;
double* restrict maxy;
double* restrict minx;
double* restrict miny;
// Upper and lower envelope spline values
double* restrict maxspline;
double* restrict minspline;
// Extra memory required for spline evaluation
double* restrict spline_workspace;
} sifting_workspace;
sifting_workspace* allocate_sifting_workspace(size_t N) {
sifting_workspace* w = malloc(sizeof(sifting_workspace));
w->N = N;
w->maxx = malloc(N*sizeof(double));
w->maxy = malloc(N*sizeof(double));
w->minx = malloc(N*sizeof(double));
w->miny = malloc(N*sizeof(double));
w->maxspline = malloc(N*sizeof(double));
w->minspline = malloc(N*sizeof(double));
// Spline evaluation requires 5*m-10 doubles where m is the number of
// extrema. The worst case scenario is that every point is an extrema, so
// use m=N to be safe.
const size_t spline_workspace_size = (N > 2)? 5*N-10 : 0;
w->spline_workspace = malloc(spline_workspace_size*sizeof(double));
return w;
}
void free_sifting_workspace(sifting_workspace* w) {
free(w->spline_workspace); w->spline_workspace = NULL;
free(w->minspline); w->minspline = NULL;
free(w->maxspline); w->maxspline = NULL;
free(w->miny); w->miny = NULL;
free(w->minx); w->minx = NULL;
free(w->maxy); w->maxy = NULL;
free(w->maxx); w->maxx = NULL;
free(w); w = NULL;
}
// For EMD we need space to do the sifting and somewhere to save the residual from the previous run.
// We also leave room for an array of locks to protect multi-threaded EMD.
typedef struct {
size_t N;
// Previous residual for EMD
double* restrict res;
// What is needed for sifting
sifting_workspace* restrict sift_w;
// A pointer for shared locks. These locks are used to make EMD thread-safe
// even when several threads run EMD with the same output matrix (we'll do
// this in EEMD).
lock** locks;
} emd_workspace;
emd_workspace* allocate_emd_workspace(size_t N) {
emd_workspace* w = malloc(sizeof(emd_workspace));
w->N = N;
w->res = malloc(N*sizeof(double));
w->sift_w = allocate_sifting_workspace(N);
w->locks = NULL; // The locks are assumed to be allocated and freed independently
return w;
}
void free_emd_workspace(emd_workspace* w) {
free_sifting_workspace(w->sift_w);
free(w->res); w->res = NULL;
free(w); w = NULL;
}
// EEMD needs a random number generator in addition to emd_workspace. We also need a place to store
// the member of the ensemble (input signal + realization of noise) to be worked on.
typedef struct {
size_t N;
// The random number generator
gsl_rng* r;
// The ensemble member signal
double* restrict x;
// What is needed for running EMD
emd_workspace* restrict emd_w;
} eemd_workspace;
eemd_workspace* allocate_eemd_workspace(size_t N) {
eemd_workspace* w = malloc(sizeof(eemd_workspace));
w->N = N;
w->r = gsl_rng_alloc(gsl_rng_mt19937);
w->x = malloc(N*sizeof(double));
w->emd_w = allocate_emd_workspace(N);
return w;
}
void set_rng_seed(eemd_workspace* w, unsigned long int rng_seed) {
gsl_rng_set(w->r, rng_seed);
}
void free_eemd_workspace(eemd_workspace* w) {
free_emd_workspace(w->emd_w);
free(w->x); w->x = NULL;
gsl_rng_free(w->r); w->r = NULL;
free(w); w = NULL;
}
// Forward declaration of a helper function used internally for making a single
// EMD run with a preallocated workspace
static libeemd_error_code _emd(double* restrict input, emd_workspace* restrict w,
double* restrict output, size_t M,
unsigned int S_number, unsigned int num_siftings);
// Forward declaration of a helper function for applying the sifting procedure to
// input until it is reduced to an IMF according to the stopping criteria given
// by S_number and num_siftings
static libeemd_error_code _sift(double* restrict input, sifting_workspace*
restrict w, unsigned int S_number, unsigned int num_siftings, unsigned int*
sift_counter);
// Forward declaration of a helper function for parameter validation shared by functions eemd and ceemdan
static inline libeemd_error_code _validate_eemd_parameters(unsigned int ensemble_size, double noise_strength, unsigned int S_number, unsigned int num_siftings);
// Main EEMD decomposition routine definition
libeemd_error_code eemd(double const* restrict input, size_t N,
double* restrict output, size_t M,
unsigned int ensemble_size, double noise_strength, unsigned int
S_number, unsigned int num_siftings, unsigned long int rng_seed) {
gsl_set_error_handler_off();
// Validate parameters
libeemd_error_code validation_result = _validate_eemd_parameters(ensemble_size, noise_strength, S_number, num_siftings);
if (validation_result != EMD_SUCCESS) {
return validation_result;
}
// For empty data we have nothing to do
if (N == 0) {
return EMD_SUCCESS;
}
if (M == 0) {
M = emd_num_imfs(N);
}
// The noise standard deviation is noise_strength times the standard deviation of input data
const double noise_sigma = (noise_strength != 0)? gsl_stats_sd(input, 1, N)*noise_strength : 0;
// Initialize output data to zero
memset(output, 0x00, M*N*sizeof(double));
// Each thread gets a separate workspace if we are using OpenMP
eemd_workspace** ws = NULL;
// The locks are shared among all threads
lock** locks;
// Don't start unnecessary threads if the ensemble is small
#ifdef _OPENMP
if (omp_get_num_threads() > (int)ensemble_size) {
omp_set_num_threads(ensemble_size);
}
#endif
unsigned int ensemble_counter = 0;
// The following section is executed in parallel
libeemd_error_code emd_err = EMD_SUCCESS;
#pragma omp parallel
{
#ifdef _OPENMP
const int num_threads = omp_get_num_threads();
const int thread_id = omp_get_thread_num();
#if EEMD_DEBUG >= 1
#pragma omp single
fprintf(stderr, "Using %d thread(s) with OpenMP.\n", num_threads);
#endif
#else
const int num_threads = 1;
const int thread_id = 0;
#endif
#pragma omp single
{
ws = malloc(num_threads*sizeof(eemd_workspace*));
locks = malloc(M*sizeof(lock*));
for (size_t i=0; i<M; i++) {
locks[i] = malloc(sizeof(lock));
init_lock(locks[i]);
}
}
// Each thread allocates its own workspace
ws[thread_id] = allocate_eemd_workspace(N);
eemd_workspace* w = ws[thread_id];
// All threads share the same array of locks
w->emd_w->locks = locks;
// Loop over all ensemble members, dividing them among the threads
#pragma omp for
for (size_t en_i=0; en_i<ensemble_size; en_i++) {
// Check if an error has occured in other threads
#pragma omp flush(emd_err)
if (emd_err != EMD_SUCCESS) {
continue;
}
// Initialize ensemble member as input data + noise
if (noise_strength == 0.0) {
array_copy(input, N, w->x);
}
else {
// set rng seed based on ensemble member to ensure
// reproducibility even in a multithreaded case
set_rng_seed(w, rng_seed+en_i);
for (size_t i=0; i<N; i++) {
w->x[i] = input[i] + gsl_ran_gaussian(w->r, noise_sigma);
}
}
// Extract IMFs with EMD
emd_err = _emd(w->x, w->emd_w, output, M, S_number, num_siftings);
#pragma omp flush(emd_err)
#pragma omp atomic
ensemble_counter++;
#if EEMD_DEBUG >= 1
fprintf(stderr, "Ensemble iteration %u/%u done.\n", ensemble_counter, ensemble_size);
#endif
}
// Free resources
free_eemd_workspace(w);
#pragma omp single
{
free(ws); ws = NULL;
for (size_t i=0; i<M; i++) {
destroy_lock(locks[i]);
free(locks[i]);
}
free(locks); locks = NULL;
}
} // End of parallel block
if (emd_err != EMD_SUCCESS) {
return emd_err;
}
// Divide output data by the ensemble size to get the average
if (ensemble_size != 1) {
const double one_per_ensemble_size = 1.0/ensemble_size;
array_mult(output, N*M, one_per_ensemble_size);
}
return EMD_SUCCESS;
}
// Main CEEMDAN decomposition routine definition
libeemd_error_code ceemdan(double const* restrict input, size_t N,
double* restrict output, size_t M,
unsigned int ensemble_size, double noise_strength, unsigned int
S_number, unsigned int num_siftings, unsigned long int rng_seed) {
gsl_set_error_handler_off();
// Validate parameters
libeemd_error_code validation_result = _validate_eemd_parameters(ensemble_size, noise_strength, S_number, num_siftings);
if (validation_result != EMD_SUCCESS) {
return validation_result;
}
// For empty data we have nothing to do
if (N == 0) {
return EMD_SUCCESS;
}
// For M == 1 the only "IMF" is the residual
if (M == 1) {
memcpy(output, input, N*sizeof(double));
return EMD_SUCCESS;
}
if (M == 0) {
M = emd_num_imfs(N);
}
const double one_per_ensemble_size = 1.0/ensemble_size;
// Initialize output data to zero
memset(output, 0x00, M*N*sizeof(double));
// Each thread gets a separate workspace if we are using OpenMP
eemd_workspace** ws = NULL;
// All threads need to write to the same row of the output matrix
// so we need only one shared lock
lock* output_lock = malloc(sizeof(lock));
init_lock(output_lock);
// The threads also share the same precomputed noise
double* noises = malloc(ensemble_size*N*sizeof(double));
// Since we need to decompose this noise by EMD, we also need arrays for storing
// the residuals
double* noise_residuals = malloc(ensemble_size*N*sizeof(double));
// Don't start unnecessary threads if the ensemble is small
#ifdef _OPENMP
if (omp_get_num_threads() > (int)ensemble_size) {
omp_set_num_threads(ensemble_size);
}
#endif
int num_threads;
// The following section is executed in parallel
#pragma omp parallel
{
#ifdef _OPENMP
num_threads = omp_get_num_threads();
const int thread_id = omp_get_thread_num();
#if EEMD_DEBUG >= 1
#pragma omp single
fprintf(stderr, "Using %d thread(s) with OpenMP.\n", num_threads);
#endif
#else
num_threads = 1;
const int thread_id = 0;
#endif
#pragma omp single
{
ws = malloc(num_threads*sizeof(eemd_workspace*));
}
// Each thread allocates its own workspace
ws[thread_id] = allocate_eemd_workspace(N);
eemd_workspace* w = ws[thread_id];
// Precompute and store white noise, since for each mode of the data we
// need the same mode of the corresponding realization of noise
#pragma omp for
for (size_t en_i=0; en_i<ensemble_size; en_i++) {
// set rng seed based on ensemble member to ensure
// reproducibility even in a multithreaded case
set_rng_seed(w, rng_seed+en_i);
for (size_t j=0; j<N; j++) {
noises[N*en_i+j] = gsl_ran_gaussian(w->r, 1.0);
}
}
} // Return to sequental mode
// Allocate memory for the residual shared among all threads
double* restrict res = malloc(N*sizeof(double));
// For the first iteration the residual is the input signal
array_copy(input, N, res);
// Each mode is extracted sequentially, but we use parallelization in the inner loop
// to loop over ensemble members
for (size_t imf_i=0; imf_i<M; imf_i++) {
// Provide a pointer to the output vector where this IMF will be stored
double* const imf = &output[imf_i*N];
// Then we go parallel to compute the different ensemble members
libeemd_error_code sift_err = EMD_SUCCESS;
#pragma omp parallel
{
#ifdef _OPENMP
const int thread_id = omp_get_thread_num();
#else
const int thread_id = 0;
#endif
eemd_workspace* w = ws[thread_id];
unsigned int sift_counter = 0;
#pragma omp for
for (size_t en_i=0; en_i<ensemble_size; en_i++) {
// Check if an error has occured in other threads
#pragma omp flush(sift_err)
if (sift_err != EMD_SUCCESS) {
continue;
}
// Provide a pointer to the noise vector and noise residual used by
// this ensemble member
double* const noise = &noises[N*en_i];
double* const noise_residual = &noise_residuals[N*en_i];
// Initialize input signal as data + noise.
// The noise standard deviation is noise_strength times the
// standard deviation of input data divided by the standard
// deviation of the noise. This is used to fix the SNR at each
// stage.
const double noise_sd = gsl_stats_sd(noise, 1, N);
const double noise_sigma = (noise_sd != 0)? noise_strength*gsl_stats_sd(res, 1, N)/noise_sd : 0;
array_addmul_to(res, noise, noise_sigma, N, w->x);
// Sift to extract first EMD mode
sift_err = _sift(w->x, w->emd_w->sift_w, S_number, num_siftings, &sift_counter);
#pragma omp flush(sift_err)
// Sum to output vector
get_lock(output_lock);
array_add(w->x, N, imf);
release_lock(output_lock);
// Extract next EMD mode of the noise. This is used as the noise for
// the next mode extracted from the data
if (imf_i == 0) {
array_copy(noise, N, noise_residual);
}
else {
array_copy(noise_residual, N, noise);
}
sift_err = _sift(noise, w->emd_w->sift_w, S_number, num_siftings, &sift_counter);
#pragma omp flush(sift_err)
array_sub(noise, N, noise_residual);
}
} // Parallel section ends
if (sift_err != EMD_SUCCESS) {
return sift_err;
}
// Divide with ensemble size to get the average
array_mult(imf, N, one_per_ensemble_size);
// Subtract this IMF from the previous residual to form the new one
array_sub(imf, N, res);
}
// Save final residual
get_lock(output_lock);
array_add(res, N, output+N*(M-1));
release_lock(output_lock);
// Free global resources
for (int thread_id=0; thread_id<num_threads; thread_id++) {
free_eemd_workspace(ws[thread_id]);
}
free(ws); ws = NULL;
free(res); res = NULL;
free(noise_residuals); noise_residuals = NULL;
free(noises); noises = NULL;
destroy_lock(output_lock);
free(output_lock); output_lock = NULL;
return EMD_SUCCESS;
}
static inline libeemd_error_code _validate_eemd_parameters(unsigned int ensemble_size, double noise_strength, unsigned int S_number, unsigned int num_siftings) {
if (ensemble_size < 1) {
return EMD_INVALID_ENSEMBLE_SIZE;
}
if (noise_strength < 0) {
return EMD_INVALID_NOISE_STRENGTH;
}
if (ensemble_size == 1 && noise_strength > 0) {
return EMD_NOISE_ADDED_TO_EMD;
}
if (ensemble_size > 1 && noise_strength == 0) {
return EMD_NO_NOISE_ADDED_TO_EEMD;
}
if (S_number == 0 && num_siftings == 0) {
return EMD_NO_CONVERGENCE_POSSIBLE;
}
return EMD_SUCCESS;
}
// Helper function for applying the sifting procedure to input until it is
// reduced to an IMF according to the stopping criteria given by S_number and
// num_siftings. The required number of siftings is saved to sift_counter.
static libeemd_error_code _sift(double* restrict input, sifting_workspace*
restrict w, unsigned int S_number, unsigned int num_siftings,
unsigned int* sift_counter) {
const size_t N = w->N;
// Provide some shorthands to avoid excessive '->' operators
double* const maxx = w->maxx;
double* const maxy = w->maxy;
double* const minx = w->minx;
double* const miny = w->miny;
// Initialize counters that keep track of the number of siftings
// and the S number
*sift_counter = 0;
unsigned int S_counter = 0;
// Numbers of extrema and zero crossings are initialized to dummy values
size_t num_max = (size_t)(-1);
size_t num_min = (size_t)(-1);
size_t num_zc = (size_t)(-1);
size_t prev_num_max = (size_t)(-1);
size_t prev_num_min = (size_t)(-1);
size_t prev_num_zc = (size_t)(-1);
while (num_siftings == 0 || *sift_counter < num_siftings) {
(*sift_counter)++;
#if EEMD_DEBUG >= 1
if (*sift_counter == 10000) {
fprintf(stderr, "Something is probably wrong. Sift counter has reached 10000.\n");
}
#endif
prev_num_max = num_max;
prev_num_min = num_min;
prev_num_zc = num_zc;
// Find extrema and count zero crossings
emd_find_extrema(input, N, maxx, maxy, &num_max, minx, miny, &num_min, &num_zc);
// Check if we are finished based on the S-number criteria
if (S_number != 0) {
const int max_diff = (int)num_max - (int)prev_num_max;
const int min_diff = (int)num_min - (int)prev_num_min;
const int zc_diff = (int)num_zc - (int)prev_num_zc;
if (abs(max_diff)+abs(min_diff)+abs(zc_diff) <= 1) {
S_counter++;
if (S_counter >= S_number) {
const int num_diff = (int)num_min + (int)num_max - 4 - (int)num_zc;
if (abs(num_diff) <= 1) {
// Number of extrema has been stable for S_number steps
// and the number of *interior* extrema and zero
// crossings differ by at most one -- we are converged
// according to the S-number criterion
break;
}
}
}
else {
S_counter = 0;
}
}
// Fit splines, choose order of spline based on the number of extrema
libeemd_error_code max_errcode = emd_evaluate_spline(maxx, maxy, num_max, w->maxspline, w->spline_workspace);
if (max_errcode != EMD_SUCCESS) {
return max_errcode;
}
libeemd_error_code min_errcode = emd_evaluate_spline(minx, miny, num_min, w->minspline, w->spline_workspace);
if (min_errcode != EMD_SUCCESS) {
return min_errcode;
}
// Subtract envelope mean from the data
for (size_t i=0; i<N; i++) {
input[i] -= 0.5*(w->maxspline[i] + w->minspline[i]);
}
}
return EMD_SUCCESS;
}
// Helper function for extracting all IMFs from input using the sifting
// procedure defined by _sift. The contents of the input array are destroyed in
// the process.
static libeemd_error_code _emd(double* restrict input, emd_workspace* restrict w,
double* restrict output, size_t M,
unsigned int S_number, unsigned int num_siftings) {
// Provide some shorthands to avoid excessive '->' operators
const size_t N = w->N;
double* const res = w->res;
lock** locks = w->locks;
if (M == 0) {
M = emd_num_imfs(N);
}
// We need to store a copy of the original signal so that once it is
// reduced to an IMF we have something to subtract the IMF from to form
// the residual for the next iteration
array_copy(input, N, res);
// Loop over all IMFs to be separated from input
unsigned int sift_counter;
for (size_t imf_i=0; imf_i<M-1; imf_i++) {
if (imf_i != 0) {
// Except for the first iteration, restore the previous residual
// and use it as an input
array_copy(res, N, input);
}
// Perform siftings on input until it is an IMF
libeemd_error_code sift_err = _sift(input, w->sift_w, S_number, num_siftings, &sift_counter);
if (sift_err != EMD_SUCCESS) {
return sift_err;
}
// Subtract this IMF from the saved copy to form the residual for
// the next round
array_sub(input, N, res);
// Add the discovered IMF to the output matrix. Use locks to ensure
// other threads are not writing to the same row of the output matrix
// at the same time
get_lock(locks[imf_i]);
array_add(input, N, output+N*imf_i);
release_lock(locks[imf_i]);
#if EEMD_DEBUG >= 2
fprintf(stderr, "IMF %zd saved after %u siftings.\n", imf_i+1, sift_counter);
#endif
}
// Save final residual
get_lock(locks[M-1]);
array_add(res, N, output+N*(M-1));
release_lock(locks[M-1]);
return EMD_SUCCESS;
}
void emd_find_extrema(double const* restrict x, size_t N,
double* restrict maxx, double* restrict maxy, size_t* nmax,
double* restrict minx, double* restrict miny, size_t* nmin,
size_t* nzc) {
// Set the number of extrema and zero crossings to zero initially
*nmax = 0;
*nmin = 0;
*nzc = 0;
// Handle empty array as a special case
if (N == 0) {
return;
}
// Add the ends of the data as both local minima and maxima. These
// might be changed later by linear extrapolation.
maxx[0] = 0;
maxy[0] = x[0];
(*nmax)++;
minx[0] = 0;
miny[0] = x[0];
(*nmin)++;
// If we had only one data point this is it
if (N == 1) {
return;
}
// Now starts the main extrema-finding loop. The loop detects points where
// the slope of the data changes sign. In the case of flat regions at the
// extrema, the center point of the flat region will be considered the
// extremal point. While detecting extrema, the loop also counts the number
// of zero crossings that occur.
enum slope { UP, DOWN, NONE };
enum sign { POS, NEG, ZERO };
enum slope previous_slope = NONE;
enum sign previous_sign = (x[0] < -0)? NEG : ((x[0] > 0)? POS : ZERO);
int flat_counter = 0;
for (size_t i=0; i<N-1; i++) {
if (x[i+1] > x[i]) { // Going up
if (previous_slope == DOWN) {
// Was going down before -> local minimum found
minx[*nmin] = (double)(i)-(double)(flat_counter)/2;
miny[*nmin] = x[i];
(*nmin)++;
}
if (previous_sign == NEG && x[i+1] > 0) { // zero crossing from neg to pos
(*nzc)++;
previous_sign = POS;
}
else if (previous_sign == ZERO && x[i+1] > 0) {
// this needs to be handled as an unfortunate special case
previous_sign = POS;
}
previous_slope = UP;
flat_counter = 0;
}
else if (x[i+1] < x[i]) { // Going down
if (previous_slope == UP) {
// Was going up before -> local maximum found
maxx[*nmax] = (double)(i)-(double)(flat_counter)/2;
maxy[*nmax] = x[i];
(*nmax)++;
}
if (previous_sign == POS && x[i+1] < -0) { // zero crossing from pos to neg
(*nzc)++;
previous_sign = NEG;
}
else if (previous_sign == ZERO && x[i+1] < -0) {
// this needs to be handled as an unfortunate special case
previous_sign = NEG;
}
previous_slope = DOWN;
flat_counter = 0;
}
else { // Staying flat
flat_counter++;
#if EEMD_DEBUG >= 3
fprintf(stderr, "Warning: a flat slope found in data. The results will differ from the reference EEMD implementation.\n");
#endif
}
}
// Add the other end of the data as extrema as well.
maxx[*nmax] = N-1;
maxy[*nmax] = x[N-1];
(*nmax)++;
minx[*nmin] = N-1;
miny[*nmin] = x[N-1];
(*nmin)++;
// If we have at least two interior extrema, test if linear extrapolation provides
// a more extremal value.
if (*nmax >= 4) {
const double max_el = linear_extrapolate(maxx[1], maxy[1],
maxx[2], maxy[2], 0);
if (max_el > maxy[0])
maxy[0] = max_el;
const double max_er = linear_extrapolate(maxx[*nmax-3], maxy[*nmax-3],
maxx[*nmax-2], maxy[*nmax-2], N-1);
if (max_er > maxy[*nmax-1])
maxy[*nmax-1] = max_er;
}
if (*nmin >= 4) {
const double min_el = linear_extrapolate(minx[1], miny[1],
minx[2], miny[2], 0);
if (min_el < miny[0])
miny[0] = min_el;
const double min_er = linear_extrapolate(minx[*nmin-3], miny[*nmin-3],
minx[*nmin-2], miny[*nmin-2], N-1);
if (min_er < miny[*nmin-1])
miny[*nmin-1] = min_er;
}
return;
}
size_t emd_num_imfs(size_t N) {
if (N == 0) {
return 0;
}
if (N <= 3) {
return 1;
}
return (size_t)(log2(N));
}
libeemd_error_code emd_evaluate_spline(double const* restrict x, double const* restrict y,
size_t N, double* restrict spline_y, double* restrict spline_workspace) {
gsl_set_error_handler_off();
const size_t n = N-1;
const size_t max_j = (size_t)x[n];
if (N <= 1) {
return EMD_NOT_ENOUGH_POINTS_FOR_SPLINE;
}
// perform more assertions only if EEMD_DEBUG is on,
// as this function is meant only for internal use
#if EEMD_DEBUG >= 1
if (x[0] != 0) {
return EMD_INVALID_SPLINE_POINTS;
}
for (size_t i=1; i<N; i++) {
if (x[i] <= x[i-1]) {
return EMD_INVALID_SPLINE_POINTS;
}
}
#endif
// Fall back to linear interpolation (for N==2) or polynomial interpolation
// (for N==3)
if (N <= 3) {
int gsl_status = gsl_poly_dd_init(spline_workspace, x, y, N);
if (gsl_status != GSL_SUCCESS) {
fprintf(stderr, "Error reported by gsl_poly_dd_init: %s\n",
gsl_strerror(gsl_status));
return EMD_GSL_ERROR;
}
for (size_t j=0; j<=max_j; j++) {
spline_y[j] = gsl_poly_dd_eval(spline_workspace, x, N, j);
}
return EMD_SUCCESS;
}
// For N >= 4, interpolate by using cubic splines with not-a-node end conditions.
// This algorithm is described in "Numerical Algorithms with C" by
// G. Engeln-Müllges and F. Uhlig, page 257.
//
// Extra homework assignment for anyone reading this: Implement this
// algorithm in GSL, so that next time someone needs these end conditions
// they can just use GSL.
const size_t sys_size = N-2;
double* const c = spline_workspace;
double* const diag = c+N;
double* const supdiag = diag + sys_size;
double* const subdiag = supdiag + (sys_size-1);
double* const g = subdiag + (sys_size-1);
// Define some constants for easier comparison with Engeln-Mullges & Uhlig
// and let the compiler optimize them away.
const double h_0 = x[1]-x[0];
const double h_1 = x[2]-x[1];
const double h_nm1 = x[n]-x[n-1];
const double h_nm2 = x[n-1]-x[n-2];
// Describe the (N-2)x(N-2) linear system Ac=g with the tridiagonal
// matrix A defined by subdiag, diag and supdiag
// first row
diag[0] = h_0 + 2*h_1;
supdiag[0] = h_1 - h_0;
g[0] = 3.0/(h_0 + h_1)*((y[2]-y[1]) - (h_1/h_0)*(y[1]-y[0]));
// rows 2 to n-2
for (size_t i=2; i<=n-2; i++) {
const double h_i = x[i+1] - x[i];
const double h_im1 = x[i] - x[i-1];
subdiag[i-2] = h_im1;
diag[i-1] = 2*(h_im1 + h_i);
supdiag[i-1] = h_i;
g[i-1] = 3.0*((y[i+1]-y[i])/h_i - (y[i]-y[i-1])/h_im1);
}
// final row
subdiag[n-3] = h_nm2 - h_nm1;
diag[n-2] = 2*h_nm2 + h_nm1;
g[n-2] = 3.0/(h_nm1 + h_nm2)*((h_nm2/h_nm1)*(y[n]-y[n-1]) - (y[n-1]-y[n-2]));
// Solve to get c_1 ... c_{n-1}
gsl_vector_view diag_vec = gsl_vector_view_array(diag, n-1);
gsl_vector_view supdiag_vec = gsl_vector_view_array(supdiag, n-2);
gsl_vector_view subdiag_vec = gsl_vector_view_array(subdiag, n-2);
gsl_vector_view g_vec = gsl_vector_view_array(g, n-1);
gsl_vector_view solution_vec = gsl_vector_view_array(c+1, n-1);
int gsl_status = gsl_linalg_solve_tridiag(&diag_vec.vector,
&supdiag_vec.vector,
&subdiag_vec.vector,
&g_vec.vector,
&solution_vec.vector);
if (gsl_status != GSL_SUCCESS) {
fprintf(stderr, "Error reported by gsl_linalg_solve_tridiag: %s\n",
gsl_strerror(gsl_status));
return EMD_GSL_ERROR;
}
// Compute c[0] and c[n]
c[0] = c[1] + (h_0/h_1)*(c[1]-c[2]);
c[n] = c[n-1] + (h_nm1/h_nm2)*(c[n-1]-c[n-2]);
// The coefficients b_i and d_i are computed from the c_i's, so just
// evaluate the spline at the required points. In this case it is easy to
// find the required interval for spline evaluation, since the evaluation
// points j just increase monotonically from 0 to max_j.
size_t i = 0;
for (size_t j=0; j<=max_j; j++) {
if (j > x[i+1]) {
i++;
assert(i < n);
}
const double dx = j-x[i];
if (dx == 0) {
spline_y[j] = y[i];
continue;
}
// Compute coefficients b_i and d_i
const double h_i = x[i+1] - x[i];
const double a_i = y[i];
const double b_i = (y[i+1]-y[i])/h_i - (h_i/3.0)*(c[i+1]+2*c[i]);
const double c_i = c[i];
const double d_i = (c[i+1]-c[i])/(3.0*h_i);
// evaluate spline at x=j using the Horner scheme
spline_y[j] = a_i + dx*(b_i + dx*(c_i + dx*d_i));
}
return EMD_SUCCESS;
}
// Helper functions for printing what error codes mean
void emd_report_to_file_if_error(FILE* file, libeemd_error_code err) {
if (err == EMD_SUCCESS) {
return;
}
fprintf(file, "libeemd error: ");
switch (err) {
case EMD_INVALID_ENSEMBLE_SIZE :
fprintf(file, "Invalid ensemble size (zero or negative)\n");
break;
case EMD_INVALID_NOISE_STRENGTH :
fprintf(file, "Invalid noise strength (negative)\n");
break;
case EMD_NOISE_ADDED_TO_EMD :
fprintf(file, "Positive noise strength but ensemble size is one (regular EMD)\n");
break;
case EMD_NO_NOISE_ADDED_TO_EEMD :
fprintf(file, "Ensemble size is more than one (EEMD) but noise strength is zero\n");
break;
case EMD_NO_CONVERGENCE_POSSIBLE :
fprintf(file, "Stopping criteria invalid: would never converge\n");
break;
case EMD_NOT_ENOUGH_POINTS_FOR_SPLINE :
fprintf(file, "Spline evaluation tried with insufficient points\n");
break;
case EMD_INVALID_SPLINE_POINTS :
fprintf(file, "Spline evaluation points invalid\n");
break;
case EMD_GSL_ERROR :
fprintf(file, "Error reported by GSL library\n");
break;
default :
fprintf(file, "Error code with unknown meaning. Please file a bug!\n");
}
}
void emd_report_if_error(libeemd_error_code err) {
emd_report_to_file_if_error(stderr, err);
}

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/* Copyright 2013 Perttu Luukko
* This file is part of libeemd.
* libeemd is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
* libeemd is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
* You should have received a copy of the GNU General Public License
* along with libeemd. If not, see <http://www.gnu.org/licenses/>.
*/
#ifndef _EEMD_H_
#define _EEMD_H_
#ifndef EEMD_DEBUG
#define EEMD_DEBUG 0
#endif
#if EEMD_DEBUG == 0
#ifndef NDEBUG
#define NDEBUG
#endif
#endif
#include <assert.h>
#include <limits.h>
#include <string.h>
#include <math.h>
#include <stdbool.h>
#include <gsl/gsl_statistics_double.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_poly.h>
#ifdef _OPENMP
#include <omp.h>
#endif
// Possible error codes returned by functions eemd, ceemdan and
// emd_evaluate_spline
typedef enum {
EMD_SUCCESS = 0,
// Errors from invalid parameters
EMD_INVALID_ENSEMBLE_SIZE = 1,
EMD_INVALID_NOISE_STRENGTH = 2,
EMD_NOISE_ADDED_TO_EMD = 3,
EMD_NO_NOISE_ADDED_TO_EEMD = 4,
EMD_NO_CONVERGENCE_POSSIBLE = 5,
EMD_NOT_ENOUGH_POINTS_FOR_SPLINE = 6,
EMD_INVALID_SPLINE_POINTS = 7,
// Other errors
EMD_GSL_ERROR = 8
} libeemd_error_code;
// Helper functions to print an error message if an error occured
void emd_report_if_error(libeemd_error_code err);
void emd_report_to_file_if_error(FILE* file, libeemd_error_code err);
// Main EEMD decomposition routine as described in:
// Z. Wu and N. Huang,
// Ensemble Empirical Mode Decomposition: A Noise-Assisted Data Analysis
// Method, Advances in Adaptive Data Analysis,
// Vol. 1, No. 1 (2009) 1–41
//
// Parameters 'input' and 'N' denote the input data and its length,
// respectively. Output from the routine is written to array 'output', which
// needs to be able to store at least N*M doubles, where M is the number of
// Intrinsic Mode Functions (IMFs) to compute. If M is set to zero, a value of
// M = emd_num_imfs(N) will be used, which corresponds to a maximal number of
// IMFs. Note that the final residual is also counted as an IMF in this
// respect, so you most likely want at least num_imfs=2. The following
// parameters are the ensemble size and the relative noise standard deviation,
// respectively. These are followed by the parameters for the stopping
// criterion. The stopping parameter can be defined by a S-number (see the
// article for details) or a fixed number of siftings. If both are specified,
// the sifting ends when either criterion is fulfilled. The final parameter is
// the seed given to the random number generator. A value of zero denotes a
// RNG-specific default value.
libeemd_error_code eemd(double const* restrict input, size_t N,
double* restrict output, size_t M,
unsigned int ensemble_size, double noise_strength, unsigned int
S_number, unsigned int num_siftings, unsigned long int rng_seed);
// A complete variant of EEMD as described in:
// M. Torres et al,
// A Complete Ensemble Empirical Mode Decomposition with Adaptive Noise
// IEEE Int. Conf. on Acoust., Speech and Signal Proc. ICASSP-11,
// (2011) 4144-4147
//
// Parameters are identical to routine eemd
libeemd_error_code ceemdan(double const* restrict input, size_t N,
double* restrict output, size_t M,
unsigned int ensemble_size, double noise_strength, unsigned int
S_number, unsigned int num_siftings, unsigned long int rng_seed);
// A method for finding the local minima and maxima from input data specified
// with parameters x and N. The memory for storing the coordinates of the
// extrema and their number are passed as the rest of the parameters. The
// arrays for the coordinates must be at least size N. The method also counts
// the number of zero crossings in the data, and saves the results into the
// pointer given as num_zero_crossings_ptr.
void emd_find_extrema(double const* restrict x, size_t N,
double* restrict maxx, double* restrict maxy, size_t* num_max_ptr,
double* restrict minx, double* restrict miny, size_t* num_min_ptr,
size_t* num_zero_crossings_ptr);
// Return the number of IMFs that can be extracted from input data of length N,
// including the final residual.
size_t emd_num_imfs(size_t N);
// This routine evaluates a cubic spline with nodes defined by the arrays x and
// y, each of length N. The spline is evaluated using the not-a-node end point
// conditions (same as Matlab). The y values of the spline curve will be
// evaluated at integer points from 0 to x[N-1], and these y values will be
// written to the array spline_y. The endpoint x[N-1] is assumed to be an
// integer, and the x values are assumed to be in ascending order, with x[0]
// equal to 0. The workspace required is 5*N-10 doubles, except that N==2
// requires no extra memory. For N<=3 the routine falls back to polynomial
// interpolation, same as Matlab.
//
// This routine is mainly exported so that it can be tested separately to
// produce identical results to the Matlab routine 'spline'.
libeemd_error_code emd_evaluate_spline(double const* restrict x, double const* restrict y,
size_t N, double* restrict spline_y, double* spline_workspace);
#endif // _EEMD_H_