<jupyter_start><jupyter_text>Detection of peaks in data> <NAME>  > Laboratory of Biomechanics and Motor Control ([http://demotu.org/](http://demotu.org/))  > Federal University of ABC, Brazil One way to detect peaks (local maxima) or valleys (local minima) in data is to use the property that a peak (or valley) must be greater (or smaller) than its immediate neighbors. The function `detect_peaks.py` (code at the end of this notebook) detects peaks (or valleys) based on this feature and other characteristics. The function signature is:```pythonind = detect_peaks(x, mph=None, mpd=1, threshold=0, edge='rising', kpsh=False, valley=False, show=False, ax=None, title=True)```The parameters `mph`, `mpd`, and `threshold` follow the convention of the Matlab function `findpeaks.m`.   Let's see how to use `detect_peaks.py`; first let's import the necessary Python libraries and configure the environment:<jupyter_code>import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
import sys
sys.path.insert(1, r'./../functions')  # add to pythonpath
from detect_peaks import detect_peaks<jupyter_output><empty_output><jupyter_text>Running the function examples:<jupyter_code>>>> from detect_peaks import detect_peaks
    >>> x = np.random.randn(100)
    >>> x[60:81] = np.nan
    >>> # detect all peaks and plot data
    >>> ind = detect_peaks(x, show=True)
    >>> print(ind)

    >>> x = np.sin(2*np.pi*5*np.linspace(0, 1, 200)) + np.random.randn(200)/5
    >>> # set minimum peak height = 0 and minimum peak distance = 20
    >>> detect_peaks(x, mph=0, mpd=20, show=True)

    >>> x = [0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 0]
    >>> # set minimum peak distance = 2
    >>> detect_peaks(x, mpd=2, show=True)

    >>> x = np.sin(2*np.pi*5*np.linspace(0, 1, 200)) + np.random.randn(200)/5
    >>> # detection of valleys instead of peaks
    >>> detect_peaks(x, mph=-1.2, mpd=20, valley=True, show=True)

    >>> x = [0, 1, 1, 0, 1, 1, 0]
    >>> # detect both edges
    >>> detect_peaks(x, edge='both', show=True)

    >>> x = [-2, 1, -2, 2, 1, 1, 3, 0]
    >>> # set threshold = 2
    >>> detect_peaks(x, threshold = 2, show=True)
     
    >>> x = [-2, 1, -2, 2, 1, 1, 3, 0]
    >>> fig, axs = plt.subplots(ncols=2, nrows=1, figsize=(10, 4))
    >>> detect_peaks(x, show=True, ax=axs[0], threshold=0.5, title=False)
    >>> detect_peaks(x, show=True, ax=axs[1], threshold=1.5, title=False)<jupyter_output><empty_output><jupyter_text>Function performanceThe function `detect_peaks.py` is relatively fast but the parameter minimum peak distance (mpd) slows down the function if the data has several peaks (>1000). Try to decrease the number of peaks by tuning the other parameters or smooth the data before calling this function with several peaks in the data.  Here is a simple test of its performance:<jupyter_code>x = np.random.randn(10000)
ind = detect_peaks(x)
print('Data with %d points and %d peaks\n' %(x.size, ind.size))
print('Performance (without the minimum peak distance parameter):')
print('detect_peaks(x)')
%timeit detect_peaks(x)
print('\nPerformance (using the minimum peak distance parameter):')
print('detect_peaks(x, mpd=10)')
%timeit detect_peaks(x, mpd=10)<jupyter_output>Data with 10000 points and 3327 peaks

Performance (without the minimum peak distance parameter):
detect_peaks(x)
91.8 µs ± 1.51 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

Performance (using the minimum peak distance parameter):
detect_peaks(x, mpd=10)
7.52 ms ± 48.8 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)<jupyter_text>Function `detect_peaks.py`<jupyter_code># %load ./../functions/detect_peaks.py
"""Detect peaks in data based on their amplitude and other features."""

from __future__ import division, print_function
import numpy as np

__author__ = "<NAME>, https://github.com/demotu/BMC"
__version__ = "1.0.6"
__license__ = "MIT"


def detect_peaks(x, mph=None, mpd=1, threshold=0, edge='rising',
                 kpsh=False, valley=False, show=False, ax=None, title=True):

    """Detect peaks in data based on their amplitude and other features.

    Parameters
    ----------
    x : 1D array_like
        data.
    mph : {None, number}, optional (default = None)
        detect peaks that are greater than minimum peak height (if parameter
        `valley` is False) or peaks that are smaller than maximum peak height
         (if parameter `valley` is True).
    mpd : positive integer, optional (default = 1)
        detect peaks that are at least separated by minimum peak distance (in
        number of data).
    threshold : positive number, optional (default = 0)
        detect peaks (valleys) that are greater (smaller) than `threshold`
        in relation to their immediate neighbors.
    edge : {None, 'rising', 'falling', 'both'}, optional (default = 'rising')
        for a flat peak, keep only the rising edge ('rising'), only the
        falling edge ('falling'), both edges ('both'), or don't detect a
        flat peak (None).
    kpsh : bool, optional (default = False)
        keep peaks with same height even if they are closer than `mpd`.
    valley : bool, optional (default = False)
        if True (1), detect valleys (local minima) instead of peaks.
    show : bool, optional (default = False)
        if True (1), plot data in matplotlib figure.
    ax : a matplotlib.axes.Axes instance, optional (default = None).
    title : bool or string, optional (default = True)
        if True, show standard title. If False or empty string, doesn't show
        any title. If string, shows string as title.

    Returns
    -------
    ind : 1D array_like
        indeces of the peaks in `x`.

    Notes
    -----
    The detection of valleys instead of peaks is performed internally by simply
    negating the data: `ind_valleys = detect_peaks(-x)`

    The function can handle NaN's

    See this IPython Notebook [1]_.

    References
    ----------
    .. [1] http://nbviewer.ipython.org/github/demotu/BMC/blob/master/notebooks/DetectPeaks.ipynb

    Examples
    --------
    >>> from detect_peaks import detect_peaks
    >>> x = np.random.randn(100)
    >>> x[60:81] = np.nan
    >>> # detect all peaks and plot data
    >>> ind = detect_peaks(x, show=True)
    >>> print(ind)

    >>> x = np.sin(2*np.pi*5*np.linspace(0, 1, 200)) + np.random.randn(200)/5
    >>> # set minimum peak height = 0 and minimum peak distance = 20
    >>> detect_peaks(x, mph=0, mpd=20, show=True)

    >>> x = [0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 0]
    >>> # set minimum peak distance = 2
    >>> detect_peaks(x, mpd=2, show=True)

    >>> x = np.sin(2*np.pi*5*np.linspace(0, 1, 200)) + np.random.randn(200)/5
    >>> # detection of valleys instead of peaks
    >>> detect_peaks(x, mph=-1.2, mpd=20, valley=True, show=True)

    >>> x = [0, 1, 1, 0, 1, 1, 0]
    >>> # detect both edges
    >>> detect_peaks(x, edge='both', show=True)

    >>> x = [-2, 1, -2, 2, 1, 1, 3, 0]
    >>> # set threshold = 2
    >>> detect_peaks(x, threshold = 2, show=True)

    >>> x = [-2, 1, -2, 2, 1, 1, 3, 0]
    >>> fig, axs = plt.subplots(ncols=2, nrows=1, figsize=(10, 4))
    >>> detect_peaks(x, show=True, ax=axs[0], threshold=0.5, title=False)
    >>> detect_peaks(x, show=True, ax=axs[1], threshold=1.5, title=False)

    Version history
    ---------------
    '1.0.6':
        Fix issue of when specifying ax object only the first plot was shown
        Add parameter to choose if a title is shown and input a title
    '1.0.5':
        The sign of `mph` is inverted if parameter `valley` is True

    """

    x = np.atleast_1d(x).astype('float64')
    if x.size < 3:
        return np.array([], dtype=int)
    if valley:
        x = -x
        if mph is not None:
            mph = -mph
    # find indices of all peaks
    dx = x[1:] - x[:-1]
    # handle NaN's
    indnan = np.where(np.isnan(x))[0]
    if indnan.size:
        x[indnan] = np.inf
        dx[np.where(np.isnan(dx))[0]] = np.inf
    ine, ire, ife = np.array([[], [], []], dtype=int)
    if not edge:
        ine = np.where((np.hstack((dx, 0)) < 0) & (np.hstack((0, dx)) > 0))[0]
    else:
        if edge.lower() in ['rising', 'both']:
            ire = np.where((np.hstack((dx, 0)) <= 0) & (np.hstack((0, dx)) > 0))[0]
        if edge.lower() in ['falling', 'both']:
            ife = np.where((np.hstack((dx, 0)) < 0) & (np.hstack((0, dx)) >= 0))[0]
    ind = np.unique(np.hstack((ine, ire, ife)))
    # handle NaN's
    if ind.size and indnan.size:
        # NaN's and values close to NaN's cannot be peaks
        ind = ind[np.in1d(ind, np.unique(np.hstack((indnan, indnan-1, indnan+1))), invert=True)]
    # first and last values of x cannot be peaks
    if ind.size and ind[0] == 0:
        ind = ind[1:]
    if ind.size and ind[-1] == x.size-1:
        ind = ind[:-1]
    # remove peaks < minimum peak height
    if ind.size and mph is not None:
        ind = ind[x[ind] >= mph]
    # remove peaks - neighbors < threshold
    if ind.size and threshold > 0:
        dx = np.min(np.vstack([x[ind]-x[ind-1], x[ind]-x[ind+1]]), axis=0)
        ind = np.delete(ind, np.where(dx < threshold)[0])
    # detect small peaks closer than minimum peak distance
    if ind.size and mpd > 1:
        ind = ind[np.argsort(x[ind])][::-1]  # sort ind by peak height
        idel = np.zeros(ind.size, dtype=bool)
        for i in range(ind.size):
            if not idel[i]:
                # keep peaks with the same height if kpsh is True
                idel = idel | (ind >= ind[i] - mpd) & (ind <= ind[i] + mpd) \
                       & (x[ind[i]] > x[ind] if kpsh else True)
                idel[i] = 0  # Keep current peak
        # remove the small peaks and sort back the indices by their occurrence
        ind = np.sort(ind[~idel])

    if show:
        if indnan.size:
            x[indnan] = np.nan
        if valley:
            x = -x
            if mph is not None:
                mph = -mph
        _plot(x, mph, mpd, threshold, edge, valley, ax, ind, title)

    return ind


def _plot(x, mph, mpd, threshold, edge, valley, ax, ind, title):
    """Plot results of the detect_peaks function, see its help."""
    try:
        import matplotlib.pyplot as plt
    except ImportError:
        print('matplotlib is not available.')
    else:
        if ax is None:
            _, ax = plt.subplots(1, 1, figsize=(8, 4))
            no_ax = True
        else:
            no_ax = False

        ax.plot(x, 'b', lw=1)
        if ind.size:
            label = 'valley' if valley else 'peak'
            label = label + 's' if ind.size > 1 else label
            ax.plot(ind, x[ind], '+', mfc=None, mec='r', mew=2, ms=8,
                    label='%d %s' % (ind.size, label))
            ax.legend(loc='best', framealpha=.5, numpoints=1)
        ax.set_xlim(-.02*x.size, x.size*1.02-1)
        ymin, ymax = x[np.isfinite(x)].min(), x[np.isfinite(x)].max()
        yrange = ymax - ymin if ymax > ymin else 1
        ax.set_ylim(ymin - 0.1*yrange, ymax + 0.1*yrange)
        ax.set_xlabel('Data #', fontsize=14)
        ax.set_ylabel('Amplitude', fontsize=14)
        if title:
            if not isinstance(title, str):
                mode = 'Valley detection' if valley else 'Peak detection'
                title = "%s (mph=%s, mpd=%d, threshold=%s, edge='%s')"% \
                        (mode, str(mph), mpd, str(threshold), edge)
            ax.set_title(title)
        # plt.grid()
        if no_ax:
            plt.show()<jupyter_output><empty_output>
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