package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Return (x, phi, psi) at dyadic points ``K/2**J`` from filter coefficients.

Parameters ---------- hk : array_like Coefficients of low-pass filter. J : int, optional Values will be computed at grid points ``K/2**J``. Default is 7.

Returns ------- x : ndarray The dyadic points ``K/2**J`` for ``K=0...N * (2**J)-1`` where ``len(hk) = len(gk) = N+1``. phi : ndarray The scaling function ``phi(x)`` at `x`: ``phi(x) = sum(hk * phi(2x-k))``, where k is from 0 to N. psi : ndarray, optional The wavelet function ``psi(x)`` at `x`: ``phi(x) = sum(gk * phi(2x-k))``, where k is from 0 to N. `psi` is only returned if `gk` is not None.

Notes ----- The algorithm uses the vector cascade algorithm described by Strang and Nguyen in 'Wavelets and Filter Banks'. It builds a dictionary of values and slices for quick reuse. Then inserts vectors into final vector at the end.

The number of combinations of N things taken k at a time.

This is often expressed as 'N choose k'.

Parameters ---------- N : int, ndarray Number of things. k : int, ndarray Number of elements taken. exact : bool, optional If `exact` is False, then floating point precision is used, otherwise exact long integer is computed. repetition : bool, optional If `repetition` is True, then the number of combinations with repetition is computed.

Returns ------- val : int, float, ndarray The total number of combinations.

See Also -------- binom : Binomial coefficient ufunc

Notes -----

• Array arguments accepted only for exact=False case.
• If N < 0, or k < 0, then 0 is returned.
• If k > N and repetition=False, then 0 is returned.

Examples -------- >>> from scipy.special import comb >>> k = np.array(3, 4) >>> n = np.array(10, 10) >>> comb(n, k, exact=False) array( 120., 210.) >>> comb(10, 3, exact=True) 120 >>> comb(10, 3, exact=True, repetition=True) 220

Convolve two N-dimensional arrays.

Convolve `in1` and `in2`, with the output size determined by the `mode` argument.

Parameters ---------- in1 : array_like First input. in2 : array_like Second input. Should have the same number of dimensions as `in1`. mode : str 'full', 'valid', 'same', optional A string indicating the size of the output:

``full`` The output is the full discrete linear convolution of the inputs. (Default) ``valid`` The output consists only of those elements that do not rely on the zero-padding. In 'valid' mode, either `in1` or `in2` must be at least as large as the other in every dimension. ``same`` The output is the same size as `in1`, centered with respect to the 'full' output. method : str 'auto', 'direct', 'fft', optional A string indicating which method to use to calculate the convolution.

``direct`` The convolution is determined directly from sums, the definition of convolution. ``fft`` The Fourier Transform is used to perform the convolution by calling `fftconvolve`. ``auto`` Automatically chooses direct or Fourier method based on an estimate of which is faster (default). See Notes for more detail.

.. versionadded:: 0.19.0

Returns ------- convolve : array An N-dimensional array containing a subset of the discrete linear convolution of `in1` with `in2`.

See Also -------- numpy.polymul : performs polynomial multiplication (same operation, but also accepts poly1d objects) choose_conv_method : chooses the fastest appropriate convolution method fftconvolve : Always uses the FFT method. oaconvolve : Uses the overlap-add method to do convolution, which is generally faster when the input arrays are large and significantly different in size.

Notes ----- By default, `convolve` and `correlate` use ``method='auto'``, which calls `choose_conv_method` to choose the fastest method using pre-computed values (`choose_conv_method` can also measure real-world timing with a keyword argument). Because `fftconvolve` relies on floating point numbers, there are certain constraints that may force `method=direct` (more detail in `choose_conv_method` docstring).

Examples -------- Smooth a square pulse using a Hann window:

>>> from scipy import signal >>> sig = np.repeat(0., 1., 0., 100) >>> win = signal.hann(50) >>> filtered = signal.convolve(sig, win, mode='same') / sum(win)

>>> import matplotlib.pyplot as plt >>> fig, (ax_orig, ax_win, ax_filt) = plt.subplots(3, 1, sharex=True) >>> ax_orig.plot(sig) >>> ax_orig.set_title('Original pulse') >>> ax_orig.margins(0, 0.1) >>> ax_win.plot(win) >>> ax_win.set_title('Filter impulse response') >>> ax_win.margins(0, 0.1) >>> ax_filt.plot(filtered) >>> ax_filt.set_title('Filtered signal') >>> ax_filt.margins(0, 0.1) >>> fig.tight_layout() >>> fig.show()

Continuous wavelet transform.

Performs a continuous wavelet transform on `data`, using the `wavelet` function. A CWT performs a convolution with `data` using the `wavelet` function, which is characterized by a width parameter and length parameter. The `wavelet` function is allowed to be complex.

Parameters ---------- data : (N,) ndarray data on which to perform the transform. wavelet : function Wavelet function, which should take 2 arguments. The first argument is the number of points that the returned vector will have (len(wavelet(length,width)) == length). The second is a width parameter, defining the size of the wavelet (e.g. standard deviation of a gaussian). See `ricker`, which satisfies these requirements. widths : (M,) sequence Widths to use for transform. dtype : data-type, optional The desired data type of output. Defaults to ``float64`` if the output of `wavelet` is real and ``complex128`` if it is complex.

.. versionadded:: 1.4.0

kwargs Keyword arguments passed to wavelet function.

.. versionadded:: 1.4.0

Returns ------- cwt: (M, N) ndarray Will have shape of (len(widths), len(data)).

Notes -----

.. versionadded:: 1.4.0

For non-symmetric, complex-valued wavelets, the input signal is convolved with the time-reversed complex-conjugate of the wavelet data 1.

::

length = min(10 * widthii, len(data)) cwtii,: = signal.convolve(data, np.conj(wavelet(length, widthii, **kwargs))::-1, mode='same')

References ---------- .. 1 S. Mallat, 'A Wavelet Tour of Signal Processing (3rd Edition)', Academic Press, 2009.

Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> t = np.linspace(-1, 1, 200, endpoint=False) >>> sig = np.cos(2 * np.pi * 7 * t) + signal.gausspulse(t - 0.4, fc=2) >>> widths = np.arange(1, 31) >>> cwtmatr = signal.cwt(sig, signal.ricker, widths) >>> plt.imshow(cwtmatr, extent=-1, 1, 1, 31, cmap='PRGn', aspect='auto', ... vmax=abs(cwtmatr).max(), vmin=-abs(cwtmatr).max()) >>> plt.show()

The coefficients for the FIR low-pass filter producing Daubechies wavelets.

p>=1 gives the order of the zero at f=1/2. There are 2p filter coefficients.

Parameters ---------- p : int Order of the zero at f=1/2, can have values from 1 to 34.

Returns ------- daub : ndarray Return

Solve an ordinary or generalized eigenvalue problem of a square matrix.

Find eigenvalues w and right or left eigenvectors of a general matrix::

a vr:,i = wi b vr:,i a.H vl:,i = wi.conj() b.H vl:,i

where ``.H`` is the Hermitian conjugation.

Parameters ---------- a : (M, M) array_like A complex or real matrix whose eigenvalues and eigenvectors will be computed. b : (M, M) array_like, optional Right-hand side matrix in a generalized eigenvalue problem. Default is None, identity matrix is assumed. left : bool, optional Whether to calculate and return left eigenvectors. Default is False. right : bool, optional Whether to calculate and return right eigenvectors. Default is True. overwrite_a : bool, optional Whether to overwrite `a`; may improve performance. Default is False. overwrite_b : bool, optional Whether to overwrite `b`; may improve performance. Default is False. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. homogeneous_eigvals : bool, optional If True, return the eigenvalues in homogeneous coordinates. In this case ``w`` is a (2, M) array so that::

w1,i a vr:,i = w0,i b vr:,i

Default is False.

Returns ------- w : (M,) or (2, M) double or complex ndarray The eigenvalues, each repeated according to its multiplicity. The shape is (M,) unless ``homogeneous_eigvals=True``. vl : (M, M) double or complex ndarray The normalized left eigenvector corresponding to the eigenvalue ``wi`` is the column vl:,i. Only returned if ``left=True``. vr : (M, M) double or complex ndarray The normalized right eigenvector corresponding to the eigenvalue ``wi`` is the column ``vr:,i``. Only returned if ``right=True``.

Raises ------ LinAlgError If eigenvalue computation does not converge.

See Also -------- eigvals : eigenvalues of general arrays eigh : Eigenvalues and right eigenvectors for symmetric/Hermitian arrays. eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian band matrices eigh_tridiagonal : eigenvalues and right eiegenvectors for symmetric/Hermitian tridiagonal matrices

Examples -------- >>> from scipy import linalg >>> a = np.array([0., -1.], [1., 0.]) >>> linalg.eigvals(a) array(0.+1.j, 0.-1.j)

>>> b = np.array([0., 1.], [1., 1.]) >>> linalg.eigvals(a, b) array( 1.+0.j, -1.+0.j)

>>> a = np.array([3., 0., 0.], [0., 8., 0.], [0., 0., 7.]) >>> linalg.eigvals(a, homogeneous_eigvals=True) array([3.+0.j, 8.+0.j, 7.+0.j], [1.+0.j, 1.+0.j, 1.+0.j])

>>> a = np.array([0., -1.], [1., 0.]) >>> linalg.eigvals(a) == linalg.eig(a)0 array( True, True) >>> linalg.eig(a, left=True, right=False)1 # normalized left eigenvector array([-0.70710678+0.j , -0.70710678-0.j ], [-0. +0.70710678j, -0. -0.70710678j]) >>> linalg.eig(a, left=False, right=True)1 # normalized right eigenvector array([0.70710678+0.j , 0.70710678-0.j ], [0. -0.70710678j, 0. +0.70710678j])

Complex Morlet wavelet.

Parameters ---------- M : int Length of the wavelet. w : float, optional Omega0. Default is 5 s : float, optional Scaling factor, windowed from ``-s*2*pi`` to ``+s*2*pi``. Default is 1. complete : bool, optional Whether to use the complete or the standard version.

Returns ------- morlet : (M,) ndarray

See Also -------- morlet2 : Implementation of Morlet wavelet, compatible with `cwt`. scipy.signal.gausspulse

Notes ----- The standard version::

pi**-0.25 * exp(1j*w*x) * exp(-0.5*(x**2))

This commonly used wavelet is often referred to simply as the Morlet wavelet. Note that this simplified version can cause admissibility problems at low values of `w`.

The complete version::

pi**-0.25 * (exp(1j*w*x) - exp(-0.5*(w**2))) * exp(-0.5*(x**2))

This version has a correction term to improve admissibility. For `w` greater than 5, the correction term is negligible.

Note that the energy of the return wavelet is not normalised according to `s`.

The fundamental frequency of this wavelet in Hz is given by ``f = 2*s*w*r / M`` where `r` is the sampling rate.

Note: This function was created before `cwt` and is not compatible with it.

Complex Morlet wavelet, designed to work with `cwt`.

Returns the complete version of morlet wavelet, normalised according to `s`::

exp(1j*w*x/s) * exp(-0.5*(x/s)**2) * pi**(-0.25) * sqrt(1/s)

Parameters ---------- M : int Length of the wavelet. s : float Width parameter of the wavelet. w : float, optional Omega0. Default is 5

Returns ------- morlet : (M,) ndarray

See Also -------- morlet : Implementation of Morlet wavelet, incompatible with `cwt`

Notes -----

.. versionadded:: 1.4.0

This function was designed to work with `cwt`. Because `morlet2` returns an array of complex numbers, the `dtype` argument of `cwt` should be set to `complex128` for best results.

Note the difference in implementation with `morlet`. The fundamental frequency of this wavelet in Hz is given by::

f = w*fs / (2*s*np.pi)

where ``fs`` is the sampling rate and `s` is the wavelet width parameter. Similarly we can get the wavelet width parameter at ``f``::

s = w*fs / (2*f*np.pi)

Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt

>>> M = 100 >>> s = 4.0 >>> w = 2.0 >>> wavelet = signal.morlet2(M, s, w) >>> plt.plot(abs(wavelet)) >>> plt.show()

This example shows basic use of `morlet2` with `cwt` in time-frequency analysis:

>>> from scipy import signal >>> import matplotlib.pyplot as plt >>> t, dt = np.linspace(0, 1, 200, retstep=True) >>> fs = 1/dt >>> w = 6. >>> sig = np.cos(2*np.pi*(50 + 10*t)*t) + np.sin(40*np.pi*t) >>> freq = np.linspace(1, fs/2, 100) >>> widths = w*fs / (2*freq*np.pi) >>> cwtm = signal.cwt(sig, signal.morlet2, widths, w=w) >>> plt.pcolormesh(t, freq, np.abs(cwtm), cmap='viridis', shading='gouraud') >>> plt.show()

Return high-pass qmf filter from low-pass

Parameters ---------- hk : array_like Coefficients of high-pass filter.

Return a Ricker wavelet, also known as the 'Mexican hat wavelet'.

It models the function:

``A * (1 - (x/a)**2) * exp(-0.5*(x/a)**2)``,

where ``A = 2/(sqrt(3*a)*(pi**0.25))``.

Parameters ---------- points : int Number of points in `vector`. Will be centered around 0. a : scalar Width parameter of the wavelet.

Returns ------- vector : (N,) ndarray Array of length `points` in shape of ricker curve.

Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt

>>> points = 100 >>> a = 4.0 >>> vec2 = signal.ricker(points, a) >>> print(len(vec2)) 100 >>> plt.plot(vec2) >>> plt.show()