Peak detection in Python

readme.txt

Musings about the peakdetect functions by Sixten Bergman
Note that this code should work with both python 2.7 and python3.x.


All the peak detection functions in __all__ of peakdetect.py will work on
consistent waveforms, but only peakdetect.peakdetect can properly handle
offsets.

The most accurate method for pure sine seems to be peakdetect_parabola,
which for a 50Hz sine wave lasting 0.1s with 10k samples has an error in
the order of 1e-10, whilst a naive most extreme sample will have an error
in the order of 7e-5 for the position and 4e-7 for the amplitude

Do note that this accuracy most likely doesn't stay true for any real world
data where you'll have noise and harmonics in the signal which may produce
errors in the functions, which may be smaller or larger then the error of
naively using the highest/lowest point in a local maxima/minima.


The sine fit function seem to perform even worse than a just retrieving the
highest or lowest data point and is as such not recommended. The reason for
this as far as I can tell is that the scipy.optimize.curve_fit can't optimize
the variables. 


For parabola fit to function well, it must be fitted to a small section of the
peak as the curvature will start to mismatch with the function, but this also
means that the parabola should be quite sensitive to noise


FFT interpolation has between 0 to 2 orders of magnitude improvement over a 
raw peak fit. To obtain this improvement the wave needs to be heavily padded
in length

Spline seems to have similar performance to a FFT interpolation of the time
domain. Spline does however seem to be better at estimating amplitude than the
FFT method, but is unknown if this will hold true for wave-shapes that are
noisy.

It should also be noted that the errors as given in "Missmatch data.txt"
generated by the test routine are for pure functions with no noise, so the only
error being reduced by the "non-raw" peakdetect functions are errors stemming
low time resolution and are in no way an indication of how the functions can
handle any kind of noise that real signals will have.


Automatic tests for sine fitted peak detection is disabled due to it's problems



Avoid using the following functions as they're questionable in performance:
    peakdetect_sine
    peakdetect_sine_locked

analytic_wfm.py

#!/usr/bin/python2


# Copyright (C) 2016 Sixten Bergman
# License WTFPL
#
# This program is free software. It comes without any warranty, to the extent
# permitted by applicable law. 
# You can redistribute it and/or modify it under the terms of the Do What The
# Fuck You Want To Public License, Version 2, as published by Sam Hocevar. See
# http://www.wtfpl.net/ for more details.
#

import numpy as np
from math import pi, sqrt

__all__ = [
        'ACV_A1',
        'ACV_A2',
        'ACV_A3',
        'ACV_A4',
        'ACV_A5',
        'ACV_A6',
        'ACV_A7',
        'ACV_A8'
        ]



#Heavyside step function
H_num = lambda t: 1 if t > 0 else 0
H = lambda T: np.asarray([1 if t > 0 else 0 for t in T])

# pure sine
def ACV_A1(T, Hz=50):
    """
    Generate a pure sine wave at a specified frequency
    
    keyword arguments:
    T -- time points to generate the waveform given in seconds
    Hz -- The desired frequency of the signal (default:50)
    """
    ampl = 1000
    T = np.asarray(T, dtype=np.float64)
    return ampl * sqrt(2) * np.sin(2*pi*Hz * T)

    
def ACV_A2(T, Hz=50):
    """
    Generate a pure sine wave with a DC offset at a specified frequency
    
    keyword arguments:
    T -- time points to generate the waveform given in seconds
    Hz -- The desired frequency of the signal (default:50)
    """
    ampl = 1000
    offset = 500
    T = np.asarray(T, dtype=np.float64)
    return ampl * sqrt(2) * np.sin(2*pi*Hz * T) + offset

    
def ACV_A3(T, Hz=50):
    """
    Generate a fundamental with a 3rd overtone
    
    keyword arguments:
    T -- time points to generate the waveform given in seconds
    Hz -- The desired frequency of the signal (default:50)
    """
    ampl = 1000
    T = np.asarray(T, dtype=np.float64)
    main_wave = np.sin(2*pi*Hz * T)
    harmonic_wave = 0.05 * np.sin(2*pi*Hz * T * 4 + pi * 2 / 3)
    return ampl * sqrt(2) * (main_wave + harmonic_wave)

    
def ACV_A4(T, Hz=50):
    """
    Generate a fundamental with a 4th overtone
    
    keyword arguments:
    T -- time points to generate the waveform given in seconds
    Hz -- The desired frequency of the signal (default:50)
    """
    ampl = 1000
    T = np.asarray(T, dtype=np.float64)
    main_wave = np.sin(2*pi*Hz * T)
    harmonic_wave = 0.07 * np.sin(2*pi*Hz * T * 5 + pi * 22 / 18)
    return ampl * sqrt(2) * (main_wave + harmonic_wave)
    
    
def ACV_A5(T, Hz=50):
    """
    Generate a realistic triangle wave
    
    keyword arguments:
    T -- time points to generate the waveform given in seconds
    Hz -- The desired frequency of the signal (default:50)
    """
    ampl = 1000
    T = np.asarray(T, dtype=np.float64)
    wave_1 = np.sin(2*pi*Hz * T)
    wave_2 = 0.05 * np.sin(2*pi*Hz * T * 3 - pi)
    wave_3 = 0.05 * np.sin(2*pi*Hz * T * 5)
    wave_4 = 0.02 * np.sin(2*pi*Hz * T * 7 - pi)
    wave_5 = 0.01 * np.sin(2*pi*Hz * T * 9)
    return ampl * sqrt(2) * (wave_1 + wave_2 + wave_3 + wave_4 + wave_5)
    
    
def ACV_A6(T, Hz=50):
    """
    Generate a realistic triangle wave
    
    keyword arguments:
    T -- time points to generate the waveform given in seconds
    Hz -- The desired frequency of the signal (default:50)
    """
    ampl = 1000
    T = np.asarray(T, dtype=np.float64)
    wave_1 = np.sin(2*pi*Hz * T)
    wave_2 = 0.02 * np.sin(2*pi*Hz * T * 3 - pi)
    wave_3 = 0.02 * np.sin(2*pi*Hz * T * 5)
    wave_4 = 0.0015 * np.sin(2*pi*Hz * T * 7 - pi)
    wave_5 = 0.009 * np.sin(2*pi*Hz * T * 9)
    return ampl * sqrt(2) * (wave_1 + wave_2 + wave_3 + wave_4 + wave_5)
    
    
def ACV_A7(T, Hz=50):
    """
    Generate a growing sine wave, where the wave starts at 0 and reaches 0.9 of
    full amplitude at 250 cycles. Thereafter it will linearly increase to full
    amplitude at 500 cycles and terminate to 0
    
    Frequency locked to 50Hz and = 0 at t>10
    
    keyword arguments:
    T -- time points to generate the waveform given in seconds
    Hz -- The desired frequency of the signal (default:50)
    """
    ampl = 1000
    Hz = 50
    T = np.asarray(T, dtype=np.float64)
    wave_main = np.sin(2*pi*Hz * T)
    step_func = (0.9 * T / 5 * H(5-T) + H(T-5) * H(10-T) * (0.9 + 0.1 * (T-5) / 5))
    return ampl * sqrt(2) * wave_main * step_func
    
def ACV_A8(T, Hz=50):
    """
    Generate a growing sine wave, which reaches 100 times the amplitude at
    500 cycles
    
    frequency not implemented and signal = 0 at t>1000*pi
    signal frequency = 0.15915494309189535 Hz?
    
    keyword arguments:
    T -- time points to generate the waveform given in seconds
    Hz -- The desired frequency of the signal (default:50)
    """
    ampl = 1000
    Hz = 50
    T = np.asarray(T, dtype=np.float64)
    wave_main = np.sin(T)
    step_func = T / (10 * pi) * H(10 - T / (2*pi*Hz))
    return ampl * sqrt(2) * wave_main * step_func
    
 
_ACV_A1_L = lambda T, Hz = 50: 1000 * sqrt(2) * np.sin(2*pi*Hz * T)
# 
_ACV_A2_L = lambda T, Hz = 50: 1000 * sqrt(2) * np.sin(2*pi*Hz * T) + 500
# 
_ACV_A3_L = lambda T, Hz = 50: 1000 * sqrt(2) * (np.sin(2*pi*Hz * T) + 
                            0.05 * np.sin(2*pi*Hz * T * 4 + pi * 2 / 3))
#     
_ACV_A4_L = lambda T, Hz = 50:( 1000 * sqrt(2) * (np.sin(2*pi*Hz * T) + 
                            0.07 * np.sin(2*pi*Hz * T * 5 + pi * 22 / 18)))

# Realistic triangle
_ACV_A5_L = lambda T, Hz = 50:( 1000 * sqrt(2) * (np.sin(2*pi*Hz * T) + 
                            0.05 * np.sin(2*pi*Hz * T * 3 - pi) + 
                            0.05 * np.sin(2*pi*Hz * T * 5) + 
                            0.02 * np.sin(2*pi*Hz * T * 7 - pi) +
                            0.01 * np.sin(2*pi*Hz * T * 9)))
# 
_ACV_A6_L = lambda T, Hz = 50:( 1000 * sqrt(2) * (np.sin(2*pi*Hz * T) + 
                            0.02 * np.sin(2*pi*Hz * T * 3 - pi) + 
                            0.02 * np.sin(2*pi*Hz * T * 5) + 
                            0.0015 * np.sin(2*pi*Hz * T * 7 - pi) + 
                            0.009 * np.sin(2*pi*Hz * T * 9)))

#A7 & A8 convert so that a input of 16*pi corresponds to a input 0.25 in the current version
_ACV_A7_OLD = lambda T: [1000 * sqrt(2) * np.sin(100 * pi * t) *
        (0.9 * t / 5 * H_num(5-t) + H_num(t-5) * H_num(10-t) * (0.9 + 0.1 * (t-5) / 5)) for t in T]
_ACV_A8_OLD = lambda T: [1000 * sqrt(2) * np.sin(t) * 
        t / (10 * pi) * H_num(10 - t / (100 * pi)) for t in T]



if __name__ == "__main__":
    #create 1 period triangle
    x = np.linspace(0, 0.02, 4000)
    y = ACV_A5(x)
    
    

peakdetect.py

#!/usr/bin/python2


# Copyright (C) 2016 Sixten Bergman
# License WTFPL
#
# This program is free software. It comes without any warranty, to the extent
# permitted by applicable law. 
# You can redistribute it and/or modify it under the terms of the Do What The
# Fuck You Want To Public License, Version 2, as published by Sam Hocevar. See
# http://www.wtfpl.net/ for more details.
#
# note that the function peakdetect is derived from code which was released to
# public domain see: http://billauer.co.il/peakdet.html
#

import logging
from math import pi, log
import numpy as np
import pylab
from scipy import fft, ifft
from scipy.optimize import curve_fit
from scipy.signal import cspline1d_eval, cspline1d

__all__ = [
        "peakdetect",
        "peakdetect_fft",
        "peakdetect_parabola",
        "peakdetect_sine",
        "peakdetect_sine_locked",
        "peakdetect_spline",
        "peakdetect_zero_crossing",
        "zero_crossings",
        "zero_crossings_sine_fit"
        ]



def _datacheck_peakdetect(x_axis, y_axis):
    if x_axis is None:
        x_axis = range(len(y_axis))
    
    if len(y_axis) != len(x_axis):
        raise ValueError( 
                "Input vectors y_axis and x_axis must have same length")
    
    #needs to be a numpy array
    y_axis = np.array(y_axis)
    x_axis = np.array(x_axis)
    return x_axis, y_axis
    

def _pad(fft_data, pad_len):
    """
    Pads fft data to interpolate in time domain
    
    keyword arguments:
    fft_data -- the fft
    pad_len --  By how many times the time resolution should be increased by
    
    return: padded list
    """
    l = len(fft_data)
    n = _n(l * pad_len)
    fft_data = list(fft_data)
    
    return fft_data[:l // 2] + [0] * (2**n-l) + fft_data[l // 2:]
    
def _n(x):
    """
    Find the smallest value for n, which fulfils 2**n >= x
    
    keyword arguments:
    x -- the value, which 2**n must surpass
    
    return: the integer n
    """
    return int(log(x)/log(2)) + 1
    
    
def _peakdetect_parabola_fitter(raw_peaks, x_axis, y_axis, points):
    """
    Performs the actual parabola fitting for the peakdetect_parabola function.
        
    keyword arguments:
    raw_peaks -- A list of either the maxima or the minima peaks, as given
        by the peakdetect functions, with index used as x-axis
    
    x_axis -- A numpy array of all the x values
    
    y_axis -- A numpy array of all the y values
    
    points -- How many points around the peak should be used during curve
        fitting, must be odd.
    
    
    return: A list giving all the peaks and the fitted waveform, format:
        [[x, y, [fitted_x, fitted_y]]]
        
    """
    func = lambda x, a, tau, c: a * ((x - tau) ** 2) + c
    fitted_peaks = []
    distance = abs(x_axis[raw_peaks[1][0]] - x_axis[raw_peaks[0][0]]) / 4
    for peak in raw_peaks:
        index = peak[0]
        x_data = x_axis[index - points // 2: index + points // 2 + 1]
        y_data = y_axis[index - points // 2: index + points // 2 + 1]
        # get a first approximation of tau (peak position in time)
        tau = x_axis[index]
        # get a first approximation of peak amplitude
        c = peak[1]
        a = np.sign(c) * (-1) * (np.sqrt(abs(c))/distance)**2
        """Derived from ABC formula to result in a solution where A=(rot(c)/t)**2"""
        
        # build list of approximations
        
        p0 = (a, tau, c)
        popt, pcov = curve_fit(func, x_data, y_data, p0)
        # retrieve tau and c i.e x and y value of peak
        x, y = popt[1:3]
        
        # create a high resolution data set for the fitted waveform
        x2 = np.linspace(x_data[0], x_data[-1], points * 10)
        y2 = func(x2, *popt)
        
        fitted_peaks.append([x, y, [x2, y2]])
        
    return fitted_peaks
    
    
def peakdetect_parabole(*args, **kwargs):
    """
    Misspelling of peakdetect_parabola
    function is deprecated please use peakdetect_parabola
    """
    logging.warn("peakdetect_parabole is deprecated due to misspelling use: peakdetect_parabola")
    
    return peakdetect_parabola(*args, **kwargs)
    
    
def peakdetect(y_axis, x_axis = None, lookahead = 200, delta=0):
    """
    Converted from/based on a MATLAB script at: 
    http://billauer.co.il/peakdet.html
    
    function for detecting local maxima and minima in a signal.
    Discovers peaks by searching for values which are surrounded by lower
    or larger values for maxima and minima respectively
    
    keyword arguments:
    y_axis -- A list containing the signal over which to find peaks
    
    x_axis -- A x-axis whose values correspond to the y_axis list and is used
        in the return to specify the position of the peaks. If omitted an
        index of the y_axis is used.
        (default: None)
    
    lookahead -- distance to look ahead from a peak candidate to determine if
        it is the actual peak
        (default: 200) 
        '(samples / period) / f' where '4 >= f >= 1.25' might be a good value
    
    delta -- this specifies a minimum difference between a peak and
        the following points, before a peak may be considered a peak. Useful
        to hinder the function from picking up false peaks towards to end of
        the signal. To work well delta should be set to delta >= RMSnoise * 5.
        (default: 0)
            When omitted delta function causes a 20% decrease in speed.
            When used Correctly it can double the speed of the function
    
    
    return: two lists [max_peaks, min_peaks] containing the positive and
        negative peaks respectively. Each cell of the lists contains a tuple
        of: (position, peak_value) 
        to get the average peak value do: np.mean(max_peaks, 0)[1] on the
        results to unpack one of the lists into x, y coordinates do: 
        x, y = zip(*max_peaks)
    """
    max_peaks = []
    min_peaks = []
    dump = []   #Used to pop the first hit which almost always is false
       
    # check input data
    x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
    # store data length for later use
    length = len(y_axis)
    
    
    #perform some checks
    if lookahead < 1:
        raise ValueError("Lookahead must be '1' or above in value")
    if not (np.isscalar(delta) and delta >= 0):
        raise ValueError("delta must be a positive number")
    
    #maxima and minima candidates are temporarily stored in
    #mx and mn respectively
    mn, mx = np.Inf, -np.Inf
    
    #Only detect peak if there is 'lookahead' amount of points after it
    for index, (x, y) in enumerate(zip(x_axis[:-lookahead], 
                                        y_axis[:-lookahead])):
        if y > mx:
            mx = y
            mxpos = x
        if y < mn:
            mn = y
            mnpos = x
        
        ####look for max####
        if y < mx-delta and mx != np.Inf:
            #Maxima peak candidate found
            #look ahead in signal to ensure that this is a peak and not jitter
            if y_axis[index:index+lookahead].max() < mx:
                max_peaks.append([mxpos, mx])
                dump.append(True)
                #set algorithm to only find minima now
                mx = np.Inf
                mn = np.Inf
                if index+lookahead >= length:
                    #end is within lookahead no more peaks can be found
                    break
                continue
            #else:  #slows shit down this does
            #    mx = ahead
            #    mxpos = x_axis[np.where(y_axis[index:index+lookahead]==mx)]
        
        ####look for min####
        if y > mn+delta and mn != -np.Inf:
            #Minima peak candidate found 
            #look ahead in signal to ensure that this is a peak and not jitter
            if y_axis[index:index+lookahead].min() > mn:
                min_peaks.append([mnpos, mn])
                dump.append(False)
                #set algorithm to only find maxima now
                mn = -np.Inf
                mx = -np.Inf
                if index+lookahead >= length:
                    #end is within lookahead no more peaks can be found
                    break
            #else:  #slows shit down this does
            #    mn = ahead
            #    mnpos = x_axis[np.where(y_axis[index:index+lookahead]==mn)]
    
    
    #Remove the false hit on the first value of the y_axis
    try:
        if dump[0]:
            max_peaks.pop(0)
        else:
            min_peaks.pop(0)
        del dump
    except IndexError:
        #no peaks were found, should the function return empty lists?
        pass
        
    return [max_peaks, min_peaks]
    
    
def peakdetect_fft(y_axis, x_axis, pad_len = 20):
    """
    Performs a FFT calculation on the data and zero-pads the results to
    increase the time domain resolution after performing the inverse fft and
    send the data to the 'peakdetect' function for peak 
    detection.
    
    Omitting the x_axis is forbidden as it would make the resulting x_axis
    value silly if it was returned as the index 50.234 or similar.
    
    Will find at least 1 less peak then the 'peakdetect_zero_crossing'
    function, but should result in a more precise value of the peak as
    resolution has been increased. Some peaks are lost in an attempt to
    minimize spectral leakage by calculating the fft between two zero
    crossings for n amount of signal periods.
    
    The biggest time eater in this function is the ifft and thereafter it's
    the 'peakdetect' function which takes only half the time of the ifft.
    Speed improvements could include to check if 2**n points could be used for
    fft and ifft or change the 'peakdetect' to the 'peakdetect_zero_crossing',
    which is maybe 10 times faster than 'peakdetct'. The pro of 'peakdetect'
    is that it results in one less lost peak. It should also be noted that the
    time used by the ifft function can change greatly depending on the input.
    
    keyword arguments:
    y_axis -- A list containing the signal over which to find peaks
    
    x_axis -- A x-axis whose values correspond to the y_axis list and is used
        in the return to specify the position of the peaks.
    
    pad_len -- By how many times the time resolution should be
        increased by, e.g. 1 doubles the resolution. The amount is rounded up
        to the nearest 2**n amount
        (default: 20)
    
    
    return: two lists [max_peaks, min_peaks] containing the positive and
        negative peaks respectively. Each cell of the lists contains a tuple
        of: (position, peak_value) 
        to get the average peak value do: np.mean(max_peaks, 0)[1] on the
        results to unpack one of the lists into x, y coordinates do: 
        x, y = zip(*max_peaks)
    """
    # check input data
    x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
    zero_indices = zero_crossings(y_axis, window_len = 11)
    #select a n amount of periods
    last_indice = - 1 - (1 - len(zero_indices) & 1)
    ###
    # Calculate the fft between the first and last zero crossing
    # this method could be ignored if the beginning and the end of the signal
    # are unnecessary as any errors induced from not using whole periods
    # should mainly manifest in the beginning and the end of the signal, but
    # not in the rest of the signal
    # this is also unnecessary if the given data is an amount of whole periods
    ###
    fft_data = fft(y_axis[zero_indices[0]:zero_indices[last_indice]])
    padd = lambda x, c: x[:len(x) // 2] + [0] * c + x[len(x) // 2:]
    n = lambda x: int(log(x)/log(2)) + 1
    # pads to 2**n amount of samples
    fft_padded = padd(list(fft_data), 2 ** 
                n(len(fft_data) * pad_len) - len(fft_data))
    
    # There is amplitude decrease directly proportional to the sample increase
    sf = len(fft_padded) / float(len(fft_data))
    # There might be a leakage giving the result an imaginary component
    # Return only the real component
    y_axis_ifft = ifft(fft_padded).real * sf #(pad_len + 1)
    x_axis_ifft = np.linspace(
                x_axis[zero_indices[0]], x_axis[zero_indices[last_indice]],
                len(y_axis_ifft))
    # get the peaks to the interpolated waveform
    max_peaks, min_peaks = peakdetect(y_axis_ifft, x_axis_ifft, 500,
                                    delta = abs(np.diff(y_axis).max() * 2))
    #max_peaks, min_peaks = peakdetect_zero_crossing(y_axis_ifft, x_axis_ifft)
    
    # store one 20th of a period as waveform data
    data_len = int(np.diff(zero_indices).mean()) / 10
    data_len += 1 - data_len & 1
    
    
    return [max_peaks, min_peaks]
    
    
def peakdetect_parabola(y_axis, x_axis, points = 31):
    """
    Function for detecting local maxima and minima in a signal.
    Discovers peaks by fitting the model function: y = k (x - tau) ** 2 + m
    to the peaks. The amount of points used in the fitting is set by the
    points argument.
    
    Omitting the x_axis is forbidden as it would make the resulting x_axis
    value silly, if it was returned as index 50.234 or similar.
    
    will find the same amount of peaks as the 'peakdetect_zero_crossing'
    function, but might result in a more precise value of the peak.
    
    keyword arguments:
    y_axis -- A list containing the signal over which to find peaks
    
    x_axis -- A x-axis whose values correspond to the y_axis list and is used
        in the return to specify the position of the peaks.
    
    points -- How many points around the peak should be used during curve
        fitting (default: 31)
    
    
    return: two lists [max_peaks, min_peaks] containing the positive and
        negative peaks respectively. Each cell of the lists contains a tuple
        of: (position, peak_value) 
        to get the average peak value do: np.mean(max_peaks, 0)[1] on the
        results to unpack one of the lists into x, y coordinates do: 
        x, y = zip(*max_peaks)
    """
    # check input data
    x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
    # make the points argument odd
    points += 1 - points % 2
    #points += 1 - int(points) & 1 slower when int conversion needed
    
    # get raw peaks
    max_raw, min_raw = peakdetect_zero_crossing(y_axis)
    
    # define output variable
    max_peaks = []
    min_peaks = []
    
    max_ = _peakdetect_parabola_fitter(max_raw, x_axis, y_axis, points)
    min_ = _peakdetect_parabola_fitter(min_raw, x_axis, y_axis, points)
    
    max_peaks = map(lambda x: [x[0], x[1]], max_)
    max_fitted = map(lambda x: x[-1], max_)
    min_peaks = map(lambda x: [x[0], x[1]], min_)
    min_fitted = map(lambda x: x[-1], min_)
    
    return [max_peaks, min_peaks]
    

def peakdetect_sine(y_axis, x_axis, points = 31, lock_frequency = False):
    """
    Function for detecting local maxima and minima in a signal.
    Discovers peaks by fitting the model function:
    y = A * sin(2 * pi * f * (x - tau)) to the peaks. The amount of points used
    in the fitting is set by the points argument.
    
    Omitting the x_axis is forbidden as it would make the resulting x_axis
    value silly if it was returned as index 50.234 or similar.
    
    will find the same amount of peaks as the 'peakdetect_zero_crossing'
    function, but might result in a more precise value of the peak.
    
    The function might have some problems if the sine wave has a
    non-negligible total angle i.e. a k*x component, as this messes with the
    internal offset calculation of the peaks, might be fixed by fitting a 
    y = k * x + m function to the peaks for offset calculation.
    
    keyword arguments:
    y_axis -- A list containing the signal over which to find peaks
    
    x_axis -- A x-axis whose values correspond to the y_axis list and is used
        in the return to specify the position of the peaks.
    
    points -- How many points around the peak should be used during curve
        fitting (default: 31)
    
    lock_frequency -- Specifies if the frequency argument of the model
        function should be locked to the value calculated from the raw peaks
        or if optimization process may tinker with it.
        (default: False)
    
    
    return: two lists [max_peaks, min_peaks] containing the positive and
        negative peaks respectively. Each cell of the lists contains a tuple
        of: (position, peak_value) 
        to get the average peak value do: np.mean(max_peaks, 0)[1] on the
        results to unpack one of the lists into x, y coordinates do: 
        x, y = zip(*max_peaks)
    """
    # check input data
    x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
    # make the points argument odd
    points += 1 - points % 2
    #points += 1 - int(points) & 1 slower when int conversion needed
    
    # get raw peaks
    max_raw, min_raw = peakdetect_zero_crossing(y_axis)
    
    # define output variable
    max_peaks = []
    min_peaks = []
    
    # get global offset
    offset = np.mean([np.mean(max_raw, 0)[1], np.mean(min_raw, 0)[1]])
    # fitting a k * x + m function to the peaks might be better
    #offset_func = lambda x, k, m: k * x + m
    
    # calculate an approximate frequency of the signal
    Hz_h_peak = np.diff(zip(*max_raw)[0]).mean()
    Hz_l_peak = np.diff(zip(*min_raw)[0]).mean()
    Hz = 1 / np.mean([Hz_h_peak, Hz_l_peak])
    
    
    
    # model function
    # if cosine is used then tau could equal the x position of the peak
    # if sine were to be used then tau would be the first zero crossing
    if lock_frequency:
        func = lambda x_ax, A, tau: A * np.sin(
            2 * pi * Hz * (x_ax - tau) + pi / 2)
    else:
        func = lambda x_ax, A, Hz, tau: A * np.sin(
            2 * pi * Hz * (x_ax - tau) + pi / 2)
    #func = lambda x_ax, A, Hz, tau: A * np.cos(2 * pi * Hz * (x_ax - tau))
    
    
    #get peaks
    fitted_peaks = []
    for raw_peaks in [max_raw, min_raw]:
        peak_data = []
        for peak in raw_peaks:
            index = peak[0]
            x_data = x_axis[index - points // 2: index + points // 2 + 1]
            y_data = y_axis[index - points // 2: index + points // 2 + 1]
            # get a first approximation of tau (peak position in time)
            tau = x_axis[index]
            # get a first approximation of peak amplitude
            A = peak[1]
            
            # build list of approximations
            if lock_frequency:
                p0 = (A, tau)
            else:
                p0 = (A, Hz, tau)
            
            # subtract offset from wave-shape
            y_data -= offset
            popt, pcov = curve_fit(func, x_data, y_data, p0)
            # retrieve tau and A i.e x and y value of peak
            x = popt[-1]
            y = popt[0]
            
            # create a high resolution data set for the fitted waveform
            x2 = np.linspace(x_data[0], x_data[-1], points * 10)
            y2 = func(x2, *popt)
            
            # add the offset to the results
            y += offset
            y2 += offset
            y_data += offset
            
            peak_data.append([x, y, [x2, y2]])
       
        fitted_peaks.append(peak_data)
    
    # structure date for output
    max_peaks = map(lambda x: [x[0], x[1]], fitted_peaks[0])
    max_fitted = map(lambda x: x[-1], fitted_peaks[0])
    min_peaks = map(lambda x: [x[0], x[1]], fitted_peaks[1])
    min_fitted = map(lambda x: x[-1], fitted_peaks[1])
    
    
    return [max_peaks, min_peaks]

    
def peakdetect_sine_locked(y_axis, x_axis, points = 31):
    """
    Convenience function for calling the 'peakdetect_sine' function with
    the lock_frequency argument as True.
    
    keyword arguments:
    y_axis -- A list containing the signal over which to find peaks
    x_axis -- A x-axis whose values correspond to the y_axis list and is used
        in the return to specify the position of the peaks.
    points -- How many points around the peak should be used during curve
        fitting (default: 31)
    
    return: see the function 'peakdetect_sine'
    """
    return peakdetect_sine(y_axis, x_axis, points, True)
    
    
def peakdetect_spline(y_axis, x_axis, pad_len=20):
    """
    Performs a b-spline interpolation on the data to increase resolution and
    send the data to the 'peakdetect_zero_crossing' function for peak 
    detection.
    
    Omitting the x_axis is forbidden as it would make the resulting x_axis
    value silly if it was returned as the index 50.234 or similar.
    
    will find the same amount of peaks as the 'peakdetect_zero_crossing'
    function, but might result in a more precise value of the peak.
    
    keyword arguments:
    y_axis -- A list containing the signal over which to find peaks
    
    x_axis -- A x-axis whose values correspond to the y_axis list and is used
        in the return to specify the position of the peaks. 
        x-axis must be equally spaced.
    
    pad_len -- By how many times the time resolution should be increased by,
        e.g. 1 doubles the resolution.
        (default: 20)
    
    
    return: two lists [max_peaks, min_peaks] containing the positive and
        negative peaks respectively. Each cell of the lists contains a tuple
        of: (position, peak_value) 
        to get the average peak value do: np.mean(max_peaks, 0)[1] on the
        results to unpack one of the lists into x, y coordinates do: 
        x, y = zip(*max_peaks)
    """
    # check input data
    x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
    # could perform a check if x_axis is equally spaced
    #if np.std(np.diff(x_axis)) > 1e-15: raise ValueError
    # perform spline interpolations
    dx = x_axis[1] - x_axis[0]
    x_interpolated = np.linspace(x_axis.min(), x_axis.max(), len(x_axis) * (pad_len + 1))
    cj = cspline1d(y_axis)
    y_interpolated = cspline1d_eval(cj, x_interpolated, dx=dx,x0=x_axis[0])
    # get peaks
    max_peaks, min_peaks = peakdetect_zero_crossing(y_interpolated, x_interpolated)
    
    return [max_peaks, min_peaks]
    
def peakdetect_zero_crossing(y_axis, x_axis = None, window = 11):
    """
    Function for detecting local maxima and minima in a signal.
    Discovers peaks by dividing the signal into bins and retrieving the
    maximum and minimum value of each the even and odd bins respectively.
    Division into bins is performed by smoothing the curve and finding the
    zero crossings.
    
    Suitable for repeatable signals, where some noise is tolerated. Executes
    faster than 'peakdetect', although this function will break if the offset
    of the signal is too large. It should also be noted that the first and
    last peak will probably not be found, as this function only can find peaks
    between the first and last zero crossing.
    
    keyword arguments:
    y_axis -- A list containing the signal over which to find peaks
    
    x_axis -- A x-axis whose values correspond to the y_axis list
        and is used in the return to specify the position of the peaks. If
        omitted an index of the y_axis is used.
        (default: None)
    
    window -- the dimension of the smoothing window; should be an odd integer
        (default: 11)
    
    
    return: two lists [max_peaks, min_peaks] containing the positive and
        negative peaks respectively. Each cell of the lists contains a tuple
        of: (position, peak_value) 
        to get the average peak value do: np.mean(max_peaks, 0)[1] on the
        results to unpack one of the lists into x, y coordinates do: 
        x, y = zip(*max_peaks)
    """
    # check input data
    x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
    
    zero_indices = zero_crossings(y_axis, window_len = window)
    period_lengths = np.diff(zero_indices)
            
    bins_y = [y_axis[index:index + diff] for index, diff in 
        zip(zero_indices, period_lengths)]
    bins_x = [x_axis[index:index + diff] for index, diff in 
        zip(zero_indices, period_lengths)]
        
    even_bins_y = bins_y[::2]
    odd_bins_y = bins_y[1::2]
    even_bins_x = bins_x[::2]
    odd_bins_x = bins_x[1::2]
    hi_peaks_x = []
    lo_peaks_x = []
    
    #check if even bin contains maxima
    if abs(even_bins_y[0].max()) > abs(even_bins_y[0].min()):
        hi_peaks = [bin.max() for bin in even_bins_y]
        lo_peaks = [bin.min() for bin in odd_bins_y]
        # get x values for peak
        for bin_x, bin_y, peak in zip(even_bins_x, even_bins_y, hi_peaks):
            hi_peaks_x.append(bin_x[np.where(bin_y==peak)[0][0]])
        for bin_x, bin_y, peak in zip(odd_bins_x, odd_bins_y, lo_peaks):
            lo_peaks_x.append(bin_x[np.where(bin_y==peak)[0][0]])
    else:
        hi_peaks = [bin.max() for bin in odd_bins_y]
        lo_peaks = [bin.min() for bin in even_bins_y]
        # get x values for peak
        for bin_x, bin_y, peak in zip(odd_bins_x, odd_bins_y, hi_peaks):
            hi_peaks_x.append(bin_x[np.where(bin_y==peak)[0][0]])
        for bin_x, bin_y, peak in zip(even_bins_x, even_bins_y, lo_peaks):
            lo_peaks_x.append(bin_x[np.where(bin_y==peak)[0][0]])
    
    max_peaks = [[x, y] for x,y in zip(hi_peaks_x, hi_peaks)]
    min_peaks = [[x, y] for x,y in zip(lo_peaks_x, lo_peaks)]
    
    return [max_peaks, min_peaks]
        
    
def _smooth(x, window_len=11, window="hanning"):
    """
    smooth the data using a window of the requested size.
    
    This method is based on the convolution of a scaled window on the signal.
    The signal is prepared by introducing reflected copies of the signal 
    (with the window size) in both ends so that transient parts are minimized
    in the beginning and end part of the output signal.
    
    keyword arguments:
    x -- the input signal 
    
    window_len -- the dimension of the smoothing window; should be an odd
        integer (default: 11)
    
    window -- the type of window from 'flat', 'hanning', 'hamming', 
        'bartlett', 'blackman', where flat is a moving average
        (default: 'hanning')

    
    return: the smoothed signal
        
    example:

    t = linspace(-2,2,0.1)
    x = sin(t)+randn(len(t))*0.1
    y = _smooth(x)
    
    see also: 
    
    numpy.hanning, numpy.hamming, numpy.bartlett, numpy.blackman, 
    numpy.convolve, scipy.signal.lfilter 
    """
    if x.ndim != 1:
        raise ValueError("smooth only accepts 1 dimension arrays.")

    if x.size < window_len:
        raise ValueError("Input vector needs to be bigger than window size.")
    
    if window_len<3:
        return x
    #declare valid windows in a dictionary
    window_funcs = {
        "flat": lambda _len: np.ones(_len, "d"),
        "hanning": np.hanning,
        "hamming": np.hamming,
        "bartlett": np.bartlett,
        "blackman": np.blackman
        }
    
    s = np.r_[x[window_len-1:0:-1], x, x[-1:-window_len:-1]]
    try:
        w = window_funcs[window](window_len)
    except KeyError:
        raise ValueError(
            "Window is not one of '{0}', '{1}', '{2}', '{3}', '{4}'".format(
            *window_funcs.keys()))
    
    y = np.convolve(w / w.sum(), s, mode = "valid")
    
    return y
    
    
def zero_crossings(y_axis, window_len = 11, 
    window_f="hanning", offset_corrected=False):
    """
    Algorithm to find zero crossings. Smooths the curve and finds the
    zero-crossings by looking for a sign change.
    
    
    keyword arguments:
    y_axis -- A list containing the signal over which to find zero-crossings
    
    window_len -- the dimension of the smoothing window; should be an odd
        integer (default: 11)
    
    window_f -- the type of window from 'flat', 'hanning', 'hamming', 
        'bartlett', 'blackman' (default: 'hanning')
    
    offset_corrected -- Used for recursive calling to remove offset when needed
    
    
    return: the index for each zero-crossing
    """
    # smooth the curve
    length = len(y_axis)
    
    # discard tail of smoothed signal
    y_axis = _smooth(y_axis, window_len, window_f)[:length]
    indices = np.where(np.diff(np.sign(y_axis)))[0]
    
    # check if zero-crossings are valid
    diff = np.diff(indices)
    if diff.std() / diff.mean() > 0.1:
        #Possibly bad zero crossing, see if it's offsets
        if ((diff[::2].std() / diff[::2].mean()) < 0.1 and 
        (diff[1::2].std() / diff[1::2].mean()) < 0.1 and
        not offset_corrected):
            #offset present attempt to correct by subtracting the average
            offset = np.mean([y_axis.max(), y_axis.min()])
            return zero_crossings(y_axis-offset, window_len, window_f, True)
        #Invalid zero crossings and the offset has been removed
        print(diff.std() / diff.mean())
        print(np.diff(indices))
        raise ValueError(
            "False zero-crossings found, indicates problem {0!s} or {1!s}".format(
            "with smoothing window", "unhandled problem with offset"))
    # check if any zero crossings were found
    if len(indices) < 1:
        raise ValueError("No zero crossings found")
    #remove offset from indices due to filter function when returning
    return indices - (window_len // 2 - 1)
    # used this to test the fft function's sensitivity to spectral leakage
    #return indices + np.asarray(30 * np.random.randn(len(indices)), int)
    
############################Frequency calculation#############################
#    diff = np.diff(indices)
#    time_p_period = diff.mean()
#    
#    if diff.std() / time_p_period > 0.1:
#        raise ValueError(
#            "smoothing window too small, false zero-crossing found")
#    
#    #return frequency
#    return 1.0 / time_p_period
##############################################################################


def zero_crossings_sine_fit(y_axis, x_axis, fit_window = None, smooth_window = 11):
    """
    Detects the zero crossings of a signal by fitting a sine model function
    around the zero crossings:
    y = A * sin(2 * pi * Hz * (x - tau)) + k * x + m
    Only tau (the zero crossing) is varied during fitting.
    
    Offset and a linear drift of offset is accounted for by fitting a linear
    function the negative respective positive raw peaks of the wave-shape and
    the amplitude is calculated using data from the offset calculation i.e.
    the 'm' constant from the negative peaks is subtracted from the positive
    one to obtain amplitude.
    
    Frequency is calculated using the mean time between raw peaks.
    
    Algorithm seems to be sensitive to first guess e.g. a large smooth_window
    will give an error in the results.
    
    
    keyword arguments:
    y_axis -- A list containing the signal over which to find peaks
    
    x_axis -- A x-axis whose values correspond to the y_axis list
        and is used in the return to specify the position of the peaks. If
        omitted an index of the y_axis is used. (default: None)
    
    fit_window -- Number of points around the approximate zero crossing that
        should be used when fitting the sine wave. Must be small enough that
        no other zero crossing will be seen. If set to none then the mean
        distance between zero crossings will be used (default: None)
    
    smooth_window -- the dimension of the smoothing window; should be an odd
        integer (default: 11)
    
    
    return: A list containing the positions of all the zero crossings.
    """
    # check input data
    x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
    #get first guess
    zero_indices = zero_crossings(y_axis, window_len = smooth_window)
    #modify fit_window to show distance per direction
    if fit_window == None:
        fit_window = np.diff(zero_indices).mean() // 3
    else:
        fit_window = fit_window // 2
    
    #x_axis is a np array, use the indices to get a subset with zero crossings
    approx_crossings = x_axis[zero_indices]
    
    
    
    #get raw peaks for calculation of offsets and frequency
    raw_peaks = peakdetect_zero_crossing(y_axis, x_axis)
    #Use mean time between peaks for frequency
    ext = lambda x: list(zip(*x)[0])
    _diff = map(np.diff, map(ext, raw_peaks))
    
    
    Hz = 1 / np.mean(map(np.mean, _diff))
    #Hz = 1 / np.diff(approx_crossings).mean() #probably bad precision
    
    
    #offset model function
    offset_func = lambda x, k, m: k * x + m
    k = []
    m = []
    amplitude = []
    
    for peaks in raw_peaks:
        #get peak data as nparray
        x_data, y_data = map(np.asarray, zip(*peaks))
        #x_data = np.asarray(x_data)
        #y_data = np.asarray(y_data)
        #calc first guess
        A = np.mean(y_data)
        p0 = (0, A)
        popt, pcov = curve_fit(offset_func, x_data, y_data, p0)
        #append results
        k.append(popt[0])
        m.append(popt[1])
        amplitude.append(abs(A))
    
    #store offset constants
    p_offset = (np.mean(k), np.mean(m))
    A = m[0] - m[1]
    #define model function to fit to zero crossing
    #y = A * sin(2*pi * Hz * (x - tau)) + k * x + m
    func = lambda x, tau: A * np.sin(2 * pi * Hz * (x - tau)) + offset_func(x, *p_offset)
    
    
    #get true crossings
    true_crossings = []
    for indice, crossing in zip(zero_indices, approx_crossings):
        p0 = (crossing, )
        subset_start = max(indice - fit_window, 0.0)
        subset_end = min(indice + fit_window + 1, len(x_axis) - 1.0)
        x_subset = np.asarray(x_axis[subset_start:subset_end])
        y_subset = np.asarray(y_axis[subset_start:subset_end])
        #fit
        popt, pcov = curve_fit(func, x_subset, y_subset, p0)
        
        true_crossings.append(popt[0])
    
    
    return true_crossings
        
        
    
    
def _test_zero():
    _max, _min = peakdetect_zero_crossing(y,x)
def _test():
    _max, _min = peakdetect(y,x, delta=0.30)
    
    
def _test_graph():
    i = 10000
    x = np.linspace(0,3.7*pi,i)
    y = (0.3*np.sin(x) + np.sin(1.3 * x) + 0.9 * np.sin(4.2 * x) + 0.06 *
    np.random.randn(i))
    y *= -1
    x = range(i)
    
    _max, _min = peakdetect(y,x,750, 0.30)
    xm = [p[0] for p in _max]
    ym = [p[1] for p in _max]
    xn = [p[0] for p in _min]
    yn = [p[1] for p in _min]
    
    plot = pylab.plot(x,y)
    pylab.hold(True)
    pylab.plot(xm, ym, "r+")
    pylab.plot(xn, yn, "g+")
    
    _max, _min = peak_det_bad.peakdetect(y, 0.7, x)
    xm = [p[0] for p in _max]
    ym = [p[1] for p in _max]
    xn = [p[0] for p in _min]
    yn = [p[1] for p in _min]
    pylab.plot(xm, ym, "y*")
    pylab.plot(xn, yn, "k*")
    pylab.show()
    
def _test_graph_cross(window = 11):
    i = 10000
    x = np.linspace(0,8.7*pi,i)
    y = (2*np.sin(x) + 0.006 *
    np.random.randn(i))
    y *= -1
    pylab.plot(x,y)
    #pylab.show()
    
    
    crossings = zero_crossings_sine_fit(y,x, smooth_window = window)
    y_cross = [0] * len(crossings)
    
    
    plot = pylab.plot(x,y)
    pylab.hold(True)
    pylab.plot(crossings, y_cross, "b+")
    pylab.show()
    
    
    
if __name__ == "__main__":
    from math import pi
    import pylab
    
    i = 10000
    x = np.linspace(0,3.7*pi,i)
    y = (0.3*np.sin(x) + np.sin(1.3 * x) + 0.9 * np.sin(4.2 * x) + 0.06 * 
    np.random.randn(i))
    y *= -1
    
    _max, _min = peakdetect(y, x, 750, 0.30)
    xm = [p[0] for p in _max]
    ym = [p[1] for p in _max]
    xn = [p[0] for p in _min]
    yn = [p[1] for p in _min]
    
    plot = pylab.plot(x, y)
    pylab.hold(True)
    pylab.plot(xm, ym, "r+")
    pylab.plot(xn, yn, "g+")
    
    
    pylab.show()

test.py

#!/usr/bin/python2


# Copyright (C) 2016 Sixten Bergman
# License WTFPL
#
# This program is free software. It comes without any warranty, to the extent
# permitted by applicable law. 
# You can redistribute it and/or modify it under the terms of the Do What The
# Fuck You Want To Public License, Version 2, as published by Sam Hocevar. See
# http://www.wtfpl.net/ for more details.
#

import analytic_wfm
import numpy as np
import peakdetect
import unittest
import pdb

#generate time axis for 5 cycles @ 50 Hz
linspace_standard = np.linspace(0, 0.10, 1000)
linspace_peakdetect = np.linspace(0, 0.10, 10000)

def prng():
    """
    A numpy random number generator with a known starting state
    
    return: a random number generator
    """
    return np.random.RandomState(773889874)


def _write_log(file, header, message):
    with open(file, "ab") as f:
        f.write(header)
        f.write("\n")
        f.writelines(message)
        f.write("\n")
        f.write("\n")
        
        
def _calculate_missmatch(received, expected):
    """
    Calculates the mean mismatch between received and expected data
    
    keyword arguments:
    received -- [[time of peak], [ampl of peak]]
    expected -- [[time of peak], [ampl of peak]]
    
    return (time mismatch, ampl mismatch)
    """
    #t_diff = np.abs(np.asarray(received[0]) - expected[0])
    t_diff = np.asarray(received[0]) - expected[0]
    a_diff = np.abs(np.asarray(received[1]) - expected[1])
    
    
    #t_diff /= np.abs(expected[0]) time error in absolute terms
    a_diff /= np.abs(expected[1])
    
    return (t_diff, a_diff)
    
    
def _log_diff(t_max, y_max, 
            t_min, y_min,
            t_max_expected, y_max_expected,
            t_min_expected, y_min_expected,
            file, name
            ):
    """
    keyword arguments:
    t_max -- time of maxima
    y_max -- amplitude of maxima
    t_min -- time of minima
    y_min -- amplitude of maxima
    t_max_expected -- expected time of maxima
    y_max_expected -- expected amplitude of maxima
    t_min_expected -- expected time of minima
    y_min_expected -- expected amplitude of maxima
    file -- log file to write to
    name -- name of the test performed
    """
    t_diff_h, a_diff_h = _calculate_missmatch([t_max, y_max],
            [t_max_expected, y_max_expected])
    
    
    t_diff_l, a_diff_l = _calculate_missmatch([t_min, y_min],
            [t_min_expected, y_min_expected])
            
    #data = ["\t{0:.2e}\t{1:.2e}\t{2:.2e}\t{3:.2e}".format(*d) for d in 
    #        [t_diff_h, t_diff_l, a_diff_h, a_diff_l]
    #    ]
    frt = "val:{0} error:{1:.2e}"
    data = ["\t{0}".format("\t".join(map(frt.format, val, err))) for val, err in
        [(t_max, t_diff_h), 
        (t_min, t_diff_l), 
        (y_max, a_diff_h), 
        (y_min, a_diff_l)]
        ]
    
    _write_log(file, name, "\n".join(data))
    
    
    
def _is_close(max_p, min_p, 
        expected_max, expected_min, 
        atol_time, tol_ampl,
        file, name):
    """
    Determines if the peaks are within the given tolerance
    
    keyword arguments:
    max_p -- location and value of maxima
    min_p -- location and value of minima
    expected_max -- expected location and value of maxima
    expected_min -- expected location and value of minima
    atol_time -- absolute tolerance of location of vertex
    tol_ampl -- relative tolerance of value of vertex
    file -- log file to write to
    name -- name of the test performed
    """
    if len(max_p) == 5:
        t_max_expected, y_max_expected = zip(*expected_max)
    else:
        if abs(max_p[0][0] - expected_max[0][0]) > 0.001:
            t_max_expected, y_max_expected = zip(*expected_max[1:])
        else:
            t_max_expected, y_max_expected = zip(*expected_max[:-1])
        
    if len(min_p) == 5:
        t_min_expected, y_min_expected = zip(*expected_min)
    else:
        t_min_expected, y_min_expected = zip(*expected_min[:-1])
    
    t_max, y_max = zip(*max_p)
    t_min, y_min = zip(*min_p)
    
    t_max_close = np.isclose(t_max, t_max_expected, atol=atol_time, rtol=1e-12)
    y_max_close = np.isclose(y_max, y_max_expected, tol_ampl)
    t_min_close = np.isclose(t_min, t_min_expected, atol=atol_time, rtol=1e-12)
    y_min_close = np.isclose(y_min, y_min_expected, tol_ampl)
    
    
    _log_diff(t_max, y_max, t_min, y_min, 
            t_max_expected, y_max_expected,
            t_min_expected, y_min_expected,
            file, name)

    return(t_max_close, y_max_close, t_min_close, y_min_close)




class Test_analytic_wfm(unittest.TestCase):
    def test_ACV1(self):
        #compare with previous lambda implementation
        old = analytic_wfm._ACV_A1_L(linspace_standard)
        acv = analytic_wfm.ACV_A1(linspace_standard)
        
        self.assertTrue(np.allclose(acv, old, rtol=1e-9))
        
    def test_ACV2(self):
        #compare with previous lambda implementation
        old = analytic_wfm._ACV_A2_L(linspace_standard)
        acv = analytic_wfm.ACV_A2(linspace_standard)
        
        self.assertTrue(np.allclose(acv, old, rtol=1e-9))
        
    def test_ACV3(self):
        #compare with previous lambda implementation
        old = analytic_wfm._ACV_A3_L(linspace_standard)
        acv = analytic_wfm.ACV_A3(linspace_standard)
        
        self.assertTrue(np.allclose(acv, old, rtol=1e-9))
        
    def test_ACV4(self):
        #compare with previous lambda implementation
        old = analytic_wfm._ACV_A4_L(linspace_standard)
        acv = analytic_wfm.ACV_A4(linspace_standard)
        
        self.assertTrue(np.allclose(acv, old, rtol=1e-9))
        
    def test_ACV5(self):
        #compare with previous lambda implementation
        old = analytic_wfm._ACV_A5_L(linspace_standard)
        acv = analytic_wfm.ACV_A5(linspace_standard)
        
        self.assertTrue(np.allclose(acv, old, rtol=1e-9))
        
    def test_ACV6(self):
        #compare with previous lambda implementation
        old = analytic_wfm._ACV_A6_L(linspace_standard)
        acv = analytic_wfm.ACV_A6(linspace_standard)
        
        self.assertTrue(np.allclose(acv, old, rtol=1e-9))
        
        
    def test_ACV7(self):
        num = np.linspace(0, 20, 1000)
        old = analytic_wfm._ACV_A7_OLD(num)
        acv = analytic_wfm.ACV_A7(num)
        
        self.assertTrue(np.allclose(acv, old, rtol=1e-9))
        
        
    def test_ACV8(self):
        num = np.linspace(0, 3150, 10000)
        old = analytic_wfm._ACV_A8_OLD(num)
        acv = analytic_wfm.ACV_A8(num)
        
        self.assertTrue(np.allclose(acv, old, rtol=1e-9))
        
        
        
        
class _Test_peakdetect_template(unittest.TestCase):
    func = None
    file = "Mismatch data.txt"
    name = "template"
    args = []
    kwargs = {}
    msg_t = "Time of {0!s} not within tolerance:\n\t{1}"
    msg_y = "Amplitude of {0!s} not within tolerance:\n\t{1}"
    
    def _test_peak_template(self, waveform, 
            expected_max, expected_min, 
            wav_name,
            atol_time = 1e-5, tol_ampl = 1e-5):
        """
        
        keyword arguments:
        waveform -- a function that given x can generate a test waveform
        expected_max -- position and amplitude where maxima are expected
        expected_min -- position and amplitude where minima are expected
        wav_name -- Name of the test waveform
        atol_time -- absolute tolerance for position of vertex (default: 1e-5)
        tol_ampl -- relative tolerance for position of vertex (default: 1e-5)
        """
        
        y = waveform(linspace_peakdetect)
        max_p, min_p = self.func(y, linspace_peakdetect, 
            *self.args, **self.kwargs
            )
        #check if the correct amount of peaks were discovered
        self.assertIn(len(max_p), [4,5])
        self.assertIn(len(min_p), [4,5])
        
        #
        # check if position and amplitude is within 0.001% which is approx the
        # numeric uncertainty from the amount of samples used
        #
        t_max_close, y_max_close, t_min_close, y_min_close = _is_close(max_p,
            min_p, 
            expected_max, 
            expected_min, 
            atol_time, tol_ampl,
            self.file, "{0}: {1}".format(wav_name, self.name))
        
        #assert if values are outside of tolerance
        self.assertTrue(np.all(t_max_close),
            msg=self.msg_t.format("maxima", t_max_close))
        self.assertTrue(np.all(y_max_close),
            msg=self.msg_y.format("maxima", y_max_close))
        
        self.assertTrue(np.all(t_min_close),
            msg=self.msg_t.format("minima", t_min_close))
        self.assertTrue(np.all(y_min_close),
            msg=self.msg_y.format("minima", y_min_close))
            
            
    def test_peak_ACV1(self):
        peak_pos = 1000*np.sqrt(2) #1414.2135623730951
        peak_neg = -peak_pos
        expected_max = [
                (0.005, peak_pos),
                (0.025, peak_pos),
                (0.045, peak_pos),
                (0.065, peak_pos),
                (0.085, peak_pos)
                ]
        expected_min = [
                (0.015, peak_neg),
                (0.035, peak_neg),
                (0.055, peak_neg),
                (0.075, peak_neg),
                (0.095, peak_neg)
                ]
        atol_time = 1e-5
        tol_ampl = 1e-6
        
        self._test_peak_template(analytic_wfm.ACV_A1,
            expected_max, expected_min,
            "ACV1",
            atol_time, tol_ampl
            )
            
    def test_peak_ACV2(self):
        peak_pos = 1000*np.sqrt(2) + 500 #1414.2135623730951 + 500
        peak_neg = (-1000*np.sqrt(2)) + 500 #-914.2135623730951
        expected_max = [
                (0.005, peak_pos),
                (0.025, peak_pos),
                (0.045, peak_pos),
                (0.065, peak_pos),
                (0.085, peak_pos)
                ]
        expected_min = [
                (0.015, peak_neg),
                (0.035, peak_neg),
                (0.055, peak_neg),
                (0.075, peak_neg),
                (0.095, peak_neg)
                ]
        atol_time = 1e-5
        tol_ampl = 2e-6
        
        self._test_peak_template(analytic_wfm.ACV_A2,
            expected_max, expected_min, 
            "ACV2",
            atol_time, tol_ampl
            )
    
    
    def test_peak_ACV3(self):
        """
        Sine wave with a 3rd overtone
        
        WolframAlpha solution
        
        max{y = sin(100 pi x)+0.05 sin(400 pi x+(2 pi)/3)}~~
        sin(6.28319 n+1.51306)-0.05 sin(25.1327 n+5.00505) 
        at x~~0.00481623+0.02 n for integer n
        
        min{y = sin(100 pi x)+0.05 sin(400 pi x+(2 pi)/3)}~~
        0.05 sin(6.55488-25.1327 n)-sin(1.37692-6.28319 n)
        at x~~-0.00438287+0.02 n for integer n
        
        Derivative for 50 Hz in 2 alternative forms
        y = 100pi*cos(100pi*x) - 25pi*cos(400pi*x)-0.3464*50*pi*sin(400pi*x)
        y = 100pi*cos(100pi*x) + 20pi*cos(400pi*x + 2*pi/3)
        
        root 0 = 1/(50 * pi) * (pi*0 - 0.68846026579266880983)
        The exact solution according to WolframAlpha - I haven't the foggiest
        (tan^(-1)(root of 
            {#1^2-3&, 11 #2^8-8 #1 #2^7-8 #2^6+56 #1 #2^5+70 #2^4-56 #1 #2^3-48 #2^2+8 #1 #2-9&}(x) 
            near x = -0.822751)+pi n) / (50 * pi)
            
            
        root 1 = 1/(50 * pi) * (pi*0 + 0.75653155241276430710)
        
        period = 0.02
        """
        base = 1000*np.sqrt(2)
        
        #def peak_pos(n):
        #    return base * (np.sin(6.28319 * n + 1.51306) 
        #        -0.05*np.sin(25.1327 * n + 5.00505))
        #def peak_neg(n):
        #    return base * (0.05 * np.sin(6.55488 - 25.1327 * n)
        #        - np.sin(1.37692 - 6.28319 * n))
        
        
        def peak_pos(n):
            return base * (np.sin(2*np.pi * n + 1.51306) 
                -0.05*np.sin(8*np.pi * n + 5.00505))
        def peak_neg(n):
            return base * (0.05 * np.sin(6.55488 - 8*np.pi * n)
                - np.sin(1.37692 - 2*np.pi * n))
        t_max = [
                0.75653155241276430710/(50*np.pi)+0.00,#0.004816229446859069
                0.75653155241276430710/(50*np.pi)+0.02,#0.024816229446859069
                0.75653155241276430710/(50*np.pi)+0.04,#0.044816229446859069
                0.75653155241276430710/(50*np.pi)+0.06,#0.064816229446859069
                0.75653155241276430710/(50*np.pi)+0.08 #0.084816229446859069
                ]
        t_min = [
                -0.68846026579266880983/(50*np.pi)+0.02,#0.015617125823069466
                -0.68846026579266880983/(50*np.pi)+0.04,#0.035617125823069466
                -0.68846026579266880983/(50*np.pi)+0.06,#0.055617125823069466
                -0.68846026579266880983/(50*np.pi)+0.08,#0.075617125823069466
                -0.68846026579266880983/(50*np.pi)+0.10 #0.095617125823069466
                ]        
        
        expected_max = [
                (t_max[0], analytic_wfm.ACV_A3(t_max[0])),
                (t_max[1], analytic_wfm.ACV_A3(t_max[1])),
                (t_max[2], analytic_wfm.ACV_A3(t_max[2])),
                (t_max[3], analytic_wfm.ACV_A3(t_max[3])),
                (t_max[4], analytic_wfm.ACV_A3(t_max[4])),
                ]
        expected_min = [
                (t_min[0], analytic_wfm.ACV_A3(t_min[0])),
                (t_min[1], analytic_wfm.ACV_A3(t_min[1])),
                (t_min[2], analytic_wfm.ACV_A3(t_min[2])),
                (t_min[3], analytic_wfm.ACV_A3(t_min[3])),
                (t_min[4], analytic_wfm.ACV_A3(t_min[4])),
                ]
        atol_time = 1e-5
        tol_ampl = 2e-6
        #reduced tolerance since the expected values are only approximated
        
        self._test_peak_template(analytic_wfm.ACV_A3,
            expected_max, expected_min,
            "ACV3",
            atol_time, tol_ampl
            )
            
    def test_peak_ACV4(self):
        """
        Sine wave with a 4th overtone
        
        Expected data is from a numerical solution using 1e8 samples
        The numerical solution used about 2 GB memory and required 64-bit
        python
        
        Test is currently disabled as it pushes time index forward enough to
        change what peaks are discovers by peakdetect_fft, such that the last
        maxima is lost instead of the first one, which is expected from all the
        other functions
        """
        expected_max = [
                (0.0059351920593519207, 1409.2119572886963),
                (0.025935191259351911, 1409.2119572887088),
                (0.045935191459351918, 1409.2119572887223),
                (0.065935191659351911, 1409.2119572887243),
                (0.085935191859351917, 1409.2119572887166)
                ]
        expected_min = [
                (0.015935191159351911, -1409.2119572886984),
                (0.035935191359351915, -1409.2119572887166),
                (0.055935191559351914, -1409.2119572887245),
                (0.075935191759351914, -1409.2119572887223),
                (0.09593519195935192, -1409.2119572887068)
                ]
        atol_time = 1e-5
        tol_ampl = 2.5e-6
        #reduced tolerance since the expected values are only approximated
        
        self._test_peak_template(analytic_wfm.ACV_A4,
            expected_max, expected_min,
            "ACV4",
            atol_time, tol_ampl
            )
     
     
    def test_peak_ACV5(self):
        """
        Realistic triangle wave
        
        Easy enough to solve, but here is the numerical solution from 1e8
        samples. Numerical solution used about 2 GB memory and required
        64-bit python
        
        expected_max = [
                [0.0050000000500000008, 1598.0613254815967]
                [0.025000000250000001, 1598.0613254815778],
                [0.045000000450000008, 1598.0613254815346],
                [0.064999999650000001, 1598.0613254815594],
                [0.084999999849999994, 1598.0613254815908]
                ]
        expected_min = [
                [0.015000000150000001, -1598.0613254815908],
                [0.035000000350000005, -1598.0613254815594],
                [0.054999999549999998, -1598.0613254815346],
                [0.074999999750000004, -1598.0613254815778],
                [0.094999999949999997, -1598.0613254815967]
                ]
        """
        peak_pos = 1130*np.sqrt(2)    #1598.0613254815976
        peak_neg = -1130*np.sqrt(2)   #-1598.0613254815967
        expected_max = [
                (0.005, peak_pos),
                (0.025, peak_pos),
                (0.045, peak_pos),
                (0.065, peak_pos),
                (0.085, peak_pos)
                ]
        expected_min = [
                (0.015, peak_neg),
                (0.035, peak_neg),
                (0.055, peak_neg),
                (0.075, peak_neg),
                (0.095, peak_neg)
                ]
        atol_time = 1e-5
        tol_ampl = 4e-6
        
        self._test_peak_template(analytic_wfm.ACV_A5,
            expected_max, expected_min,
            "ACV5",
            atol_time, tol_ampl
            )
     
     
    def test_peak_ACV6(self):
        """
        Realistic triangle wave
        
        Easy enough to solve, but here is the numerical solution from 1e8
        samples. Numerical solution used about 2 GB memory and required
        64-bit python
        
        expected_max = [
                [0.0050000000500000008, 1485.6313472729362],
                [0.025000000250000001, 1485.6313472729255],
                [0.045000000450000008, 1485.6313472729012],
                [0.064999999650000001, 1485.6313472729153],
                [0.084999999849999994, 1485.6313472729323]
                ]
        expected_min = [
                [0.015000000150000001, -1485.6313472729323],
                [0.035000000350000005, -1485.6313472729153],
                [0.054999999549999998, -1485.6313472729012],
                [0.074999999750000004, -1485.6313472729255],
                [0.094999999949999997, -1485.6313472729362]
                ]
        """
        peak_pos = 1050.5*np.sqrt(2)    #1485.6313472729364
        peak_neg = -1050.5*np.sqrt(2)   #1485.6313472729255
        expected_max = [
                (0.005, peak_pos),
                (0.025, peak_pos),
                (0.045, peak_pos),
                (0.065, peak_pos),
                (0.085, peak_pos)
                ]
        expected_min = [
                (0.015, peak_neg),
                (0.035, peak_neg),
                (0.055, peak_neg),
                (0.075, peak_neg),
                (0.095, peak_neg)
                ]
        atol_time = 1e-5
        tol_ampl = 2.5e-6
        
        self._test_peak_template(analytic_wfm.ACV_A6,
            expected_max, expected_min,
            "ACV6",
            atol_time, tol_ampl
            )
            
    
            
            
            
class Test_peakdetect(_Test_peakdetect_template):
    name = "peakdetect"
    def __init__(self, *args, **kwargs):
        super(Test_peakdetect, self).__init__(*args, **kwargs)
        self.func = peakdetect.peakdetect
 
 
class Test_peakdetect_fft(_Test_peakdetect_template):
    name = "peakdetect_fft"
    def __init__(self, *args, **kwargs):
        super(Test_peakdetect_fft, self).__init__(*args, **kwargs)
        self.func = peakdetect.peakdetect_fft
            
        
class Test_peakdetect_parabola(_Test_peakdetect_template):
    name = "peakdetect_parabola"
    def __init__(self, *args, **kwargs):
        super(Test_peakdetect_parabola, self).__init__(*args, **kwargs)
        self.func = peakdetect.peakdetect_parabola
            
        
class Test_peakdetect_sine(_Test_peakdetect_template):
    name = "peakdetect_sine"
    def __init__(self, *args, **kwargs):
        super(Test_peakdetect_sine, self).__init__(*args, **kwargs)
        self.func = peakdetect.peakdetect_sine
            
        
class Test_peakdetect_sine_locked(_Test_peakdetect_template):
    name = "peakdetect_sine_locked"
    def __init__(self, *args, **kwargs):
        super(Test_peakdetect_sine_locked, self).__init__(*args, **kwargs)
        self.func = peakdetect.peakdetect_sine_locked
            
        
class Test_peakdetect_spline(_Test_peakdetect_template):
    name = "peakdetect_spline"
    def __init__(self, *args, **kwargs):
        super(Test_peakdetect_spline, self).__init__(*args, **kwargs)
        self.func = peakdetect.peakdetect_spline
            
        
class Test_peakdetect_zero_crossing(_Test_peakdetect_template):
    name = "peakdetect_zero_crossing"
    def __init__(self, *args, **kwargs):
        super(Test_peakdetect_zero_crossing, self).__init__(*args, **kwargs)
        self.func = peakdetect.peakdetect_zero_crossing
        
        
        
        
class Test_peakdetect_misc(unittest.TestCase):
    def test__pad(self):
        data = [1,2,3,4,5,6,5,4,3,2,1]
        pad_len = 2
        pad = lambda x, c: x[:len(x) // 2] + [0] * c + x[len(x) // 2:]
        expected = pad(list(data), 2 ** 
                peakdetect._n(len(data) * pad_len) - len(data))
        received = peakdetect._pad(data, pad_len)
        
        self.assertListEqual(received, expected)
    def test__n(self):
        self.assertEqual(2**peakdetect._n(1000), 1024)
        
    def test_zero_crossings(self):
        y = analytic_wfm.ACV_A1(linspace_peakdetect)
        expected_indice = [1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000]
        indice = peakdetect.zero_crossings(y, 50)
        msg = "index:{0:d} should be within 1 of expected:{1:d}"
        for rec, exp in zip(indice, expected_indice):
            self.assertAlmostEqual(rec, exp, delta=1, msg=msg.format(rec, exp))
        
    
        
            
        
#class zero_crossings(unittest.TestCase):
            
        
        
        
if __name__ == "__main__":
    tests_to_run = [
                #Test_analytic_wfm,
                Test_peakdetect,
                Test_peakdetect_parabola,
                Test_peakdetect_fft,
                #Test_peakdetect_sine,  #sine tests disabled pending rework
                #Test_peakdetect_sine_locked,
                Test_peakdetect_spline,
                Test_peakdetect_zero_crossing,
                Test_peakdetect_misc
                ]
    
    suites_list = [unittest.TestLoader().loadTestsFromTestCase(test_class) for test_class in tests_to_run]
    big_suite = unittest.TestSuite(suites_list)
    unittest.TextTestRunner(verbosity=2).run(big_suite)

Thanks that is pretty useful!

Found an embarrassing error in the peakdetect_zero_crossing function, where it flipped the x and y values in the return. Also added some functions for fitting model functions to the waveform and a fft based interpolation, although at this time I only recommend the peakdetect and peakdetect_zero_crossing function. The reason for this is that I don't yet know if the other functions actually increases the accuracy of the peak detection. Being a metrologist I want to know the uncertainty of my peak detection funtion. For raw peak detection function this can easily be calculated as a function of the time resolution and signal frequency.

Oh the new fitting functions look good. I migth give those a try. I found a Savitzky-Golay filter pretty helpfull for my data. (Analytical chemistry)
Thanks for the update

Correct me if I'm wrong, but isn't the if index+lookahead >= length unnecessary in the peakdetect function, considering that the loop is from [0,length-lookahead) ?

This is very useful! Thanks!

great piece of code

great code, just a small issue correct me if I'm wrong, should
line 140 be mn = y
line 158 be mx = y

thanks

very useful, nice!

Thanks for the code.

What is the license associated with this code?

I have a negative signal that has a peak, but this code lists all the values of x and y

Hi Sixtenbe! I implemented the peakdetect.py into the algorithm of my master thesis and it works like a charm:). Therefore I wanted to ask you what the license for this code is and how you want me to reference you? Thanks!

great tool. i am using it for an acoustic experiment designed to behave like a quantum state. lots of data sets with lots of peaks; lots of manual work done by your script.
Thanks

highly appreciate your work.

Great work, @sixtenbe! Would you mind adding a license to the script? I would like to include it in a BSD-licensed library for metabolomics, https://github.com/metabolite-atlas/metatlas.

This works really well for identifying peaks in a saw-tooth pattern from atomic force microscopy. I would also want to ask for the license.

Hi, @sixtenbe, thank you for coding this. Do you release this under the MIT license?

Killer code. Thanks a bunch for sharing!

Just saw that I didn't actually get any kind of notifications for this code and lots of people are asking for the license.

The original piece of code that this was based on is in public domain see http://billauer.co.il/peakdet.html. The contributions I've made to the code can be considered licensed under http://www.wtfpl.net/, which means that you can do whatever you want with the code.

I might do some more with this code as I want to implement some automatic tests of it to verify that everything is returning sane values for a few analytic waveforms that will be based on an IEC standard for high voltage measurement systems.

Refactored a bit of the code and added some automatic testing of the peakdetection functions so that any new function can be verified to function to some standard of bare minimum.

I also changed the name of peakdetect_parabole to peakdetect_parabola, but added a legacy function so that the old name will still work.
I changed the name of the window variable in zero_crossings to window_len and added the variable window_f to enable specification of window type.

Future improvements that I really should do are to make all the function handle offsets without breaking and possibly add some further tests for triangle waveshapes ac defined in ACV_5 or ACV_6, but for this I would need to decide how I define what the maxima and minima are.

please how to use this with PyOpenCl ?

@algerianmaster I'm not sure what you're asking as to use something with PyOpenCl you would also have to be interested in making calls to the OpenCl API and why would even want from that. To use this with PyOpenCl you would have to rewrite all the appropriate math to use OpenCl calls and once again: Why?

Very useful code, and I have been quite happy with the peakdetect function itself. But after trying the peakdetect_parabola, I am wondering if the zero_crossing function has some prerequisites for the data? In particular, my data has a DC offset. So the line

indices = np.where(np.diff(np.sign(y_axis)))[0]
# check if zero-crossings are valid
diff = np.diff(indices)

results in indices and diff being empty arrays. This fails the test in the next line (nan) and finally comes out with the ValueError:

    if len(indices) < 1:
        raise ValueError("No zero crossings found")

So it seems the offset correction in the middle is ineffective.

EDIT: Just noticed your note that this function is ineffective for large offset. So this is not a bug.

@subhacom A quick fix you can do on your end is to simply do a offset = np.mean(np.max(my_data), np.min(my_data))
and subtract that from your data and you should end up with zero crossing that are close enough to the real ones for all peak detect methods to find their peaks.

As you probably found from the code it currently only takes DC offsets less than half the amplitude as it must have a zero crossing or it assumes it's non-periodic data.

And you can also use the peakdetect function, which only finds the raw values and will work with any offsets. It can take a constant DC offset and even a linearly increasing offset as it's searching for local extremes. The downside is that it's a bit slower than the other functions due to how it crawls over all data points.

Any chance to get this published to pypi? Would cool to make this installable!

published: https://pypi.python.org/pypi/analytic-wfm

Thanks. The code saved me a lot of time. @cancan101 Is the pypi version being updated regularly? Not that I'm going to use it, but I might have to suggest the pypi version to my friends.

@Sakthi-G33k it not like this gist is seeing much in terms of updates, so I don't see any pypi getting regular updates either. There isn't much to update, unless adding additional logic like processing the indata for frequency domain padding for time domain interpolation to reduce edge effects, but personally I would prefer to just sample 2**n samples over x amount of whole periods of the signal.

Revised and published on PyPI and GitHub: https://pypi.org/project/peakdetect/

Hi,
The code is not working under python 3.8.6.
Below I put the diff for changes I made to make it work.

diff --git a/peakdetect.py b/peakdetect.py
index e388408..5d070fa 100644
--- a/peakdetect.py
+++ b/peakdetect.py
@@ -333,7 +333,7 @@ def peakdetect_fft(y_axis, x_axis, pad_len = 20):
     #max_peaks, min_peaks = peakdetect_zero_crossing(y_axis_ifft, x_axis_ifft)
     
     # store one 20th of a period as waveform data
-    data_len = int(np.diff(zero_indices).mean()) / 10
+    data_len = int(np.diff(zero_indices).mean()) // 10
     data_len += 1 - data_len & 1
diff --git a/test.py b/test.py
index 680c2a8..c7c6dfa 10644
--- a/test.py
+++ b/test.py
@@ -31,7 +31,7 @@ def prng():
 
 
 def _write_log(file, header, message):
-    with open(file, "ab") as f:
+    with open(file, "a") as f:
         f.write(header)
         f.write("\n")
         f.writelines(message)
@@ -241,6 +241,8 @@ class _Test_peakdetect_template(unittest.TestCase):
             *self.args, **self.kwargs
             )
         #check if the correct amount of peaks were discovered
+        max_p = list(max_p)
+        min_p = list(min_p)
         self.assertIn(len(max_p), [4,5])
         self.assertIn(len(min_p), [4,5])