API¶

Summary¶

Series and transformed data¶

`Series`(x[, dt])	Series.
`Trans`(W, scales, omega, wavelet[, x_mean, dt])	Continuous wavelet transformed series.

Functions¶

`cwt`(s, wavelet[, dj, scale0, scales])	Continuous wavelet transform.
`icwt`(T)	Inverse continuous wavelet transform.

Wavelet functions¶

The The `Wavelet` abstract class¶

All the wavelet function classes inherit from the abstract class Wavelet. The Wavelet has the following abstract methods:

`Wavelet.time`(t[, scale, dt])	Wavelet function in the time domain psi0(t/s).
`Wavelet.freq`(omega[, scale, dt])	Wavelet function in the frequency domain psi0_hat(s*omega).
`Wavelet.fourier_period`(scale)	Computes the equivalent Fourier period (wavelength) from wavelet scale.
`Wavelet.efolding_time`(scale)	Returns the e-folding time tau_s.

Moreover, the Wavelet exposes the following methods (available to the subclasses):

`Wavelet.wavelet_scale`(lmbd)	Computes the wavelet scale from equivalent Fourier period (wavelength).
`Wavelet.efolding_time`(scale)	Returns the e-folding time tau_s.
`Wavelet.smallest_scale`(dt)	Returns the smallest resolvable scale.
`Wavelet.auto_scales`(dt, dj, N[, scale0])	Computes the equivalent Fourier period (wavelength) from wavelet scale.

Available wavelet classes¶

`Morlet`([omega0])	Morlet wavelet function.
`Paul`([m])	Paul Wavelet function.
`DOG`([m])	Derivative Of Gaussian (DOG) Wavelet function.

Classes and functions¶

class cwave.Series(x, dt=1)¶

Series.

Example

>>> import cwave
>>> import numpy as np
>>> x = np.random.sample(16)
>>> dt=2
>>> s = cwave.Series(x, dt)

dt¶: Get the time step (float).

x¶: Get the series (1d numpy array).

class cwave.Trans(W, scales, omega, wavelet, x_mean=0, dt=1)¶

Continuous wavelet transformed series.

S()¶: Returns the wavelet power spectrum abs(W)^2.

W¶: Get the transformed series (2d numpy array).

dt¶: Get the time step (float).

omega¶: Get the angular frequencies (1d numpy array).

scales¶: Get the wavelet scales (1d numpy array).

var()¶: Returns the variance (eq. 14 in Torrence 1998)

wavelet¶: Get the wavelet (Wavelet).

x_mean¶: Get the mean value of the original (non-transformed) series (float).

cwave.cwt(s, wavelet, dj=0.25, scale0=None, scales=None)¶

Continuous wavelet transform.

Parameters:	s (`cwave.Series` object) – series. wavelet (`cwave.Wavelet` object) – wavelet function. dj (float) – scale resolution (spacing between scales). A smaller dj will give better scale resolution. If scales is not None, this parameter is ignored. scale0 (float) – the smallest scale. If scale0=None it is chosen so that the equivalent Fourier period is 2dt. If scales is not None, this parameter is ignored. scales* (None, float or 1d array_like object) – wavelet scale(s). If scales=None, the scales are automatically computed as fractional powers of two, with a scale resolution dj and the smallest scale scale0. If the parameter scales is provided, dj and scale0 are automatically ignored.
Returns:	T – continuous wavelet transformed series.
Return type:	`cwave.Trans` object

Example

>>> import cwave
>>> import numpy as np
>>> x = np.random.sample(8)
>>> dt=2
>>> T = cwave.cwt(cwave.Series(x, dt), cwave.DOG(), scales=[2, 4, 8])
>>> T.S()
array([[  1.19017195e-01,   1.54253543e-01,   9.07432163e-02,
          2.96227293e-02,   5.89519189e-04,   1.09486971e-01,
          1.15546678e-02,   3.93108223e-02],
       [  4.43627788e-01,   2.27266757e-01,   3.35649411e-05,
          1.86687786e-01,   3.78349904e-01,   2.24894861e-01,
          3.22011576e-03,   1.84538790e-01],
       [  8.01208492e-03,   4.40859411e-03,   1.92688516e-05,
          3.62275119e-03,   8.01206572e-03,   4.40859342e-03,
          1.92697929e-05,   3.62275056e-03]])

cwave.icwt(T)¶

Inverse continuous wavelet transform.

Parameters:	T (`cwave.Trans` object) – continuous wavelet transformed series.
Returns:	s – series.
Return type:	`cwave.Series` object

Example

>>> import cwave
>>> import numpy as np
>>> x = np.random.normal(2, 1, 16)
>>> x
array([ 1.6960901 ,  3.27146653,  2.55896222,  2.39484518,  2.34977766,
        3.48575552,  1.7372688 , -0.21329766,  3.5425618 ,  3.34657898,
        1.54359934,  2.96181542,  2.43294205,  1.65980233,  3.44710306,
        1.69615204])
>>> dt=1
>>> T = cwave.cwt(cwave.Series(x, dt), cwave.DOG())
>>> sr = cwave.icwt(T)
>>> sr.x
array([ 1.64067578,  3.28517018,  2.78434897,  2.38949828,  2.58014315,
        3.52751356,  1.34776275, -0.41078628,  3.3648406 ,  3.56461166,
        1.7286081 ,  2.88596331,  2.40275636,  1.81964648,  3.28932397,
        1.7113465 ])

class cwave.Wavelet¶

The abstract wavelet base class.

auto_scales(dt, dj, N, scale0=None)¶

Computes the equivalent Fourier period (wavelength) from wavelet scale.

Parameters:	dt (float) – time step. dj (float) – scale resolution. For the Morlet wavelet, a dj of about 0.5 is the largest value that still gives adequate sampling in scale, while for the other wavelet functions al larger value can be used. N (integer) – number of data samples. scale0 (float) – the smallest scale. If scale0=None it is chosen so that the equivalent Fourier period is 2*dt.
Returns:	scale – scales.
Return type:	1d numpy array

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.auto_scales(dt=1, dj=0.125, N=64, scale0=1)
array([  1.        ,   1.09050773,   1.18920712,   1.29683955,
         1.41421356,   1.54221083,   1.68179283,   1.83400809,
         2.        ,   2.18101547,   2.37841423,   2.59367911,
         2.82842712,   3.08442165,   3.36358566,   3.66801617,
         4.        ,   4.36203093,   4.75682846,   5.18735822,
         5.65685425,   6.1688433 ,   6.72717132,   7.33603235,
         8.        ,   8.72406186,   9.51365692,  10.37471644,
         11.3137085 ,  12.3376866 ,  13.45434264,  14.67206469,
         16.        ,  17.44812372,  19.02731384,  20.74943287,
         22.627417  ,  24.67537321,  26.90868529,  29.34412938,
         32.        ,  34.89624745,  38.05462768,  41.49886575,
         45.254834  ,  49.35074641,  53.81737058,  58.68825877,  64.        ])

efolding_time(scale)¶

Returns the e-folding time tau_s.

Parameters:	scale (float) – wavelet scale.
Returns:	tau – e-folding time.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.efolding_time(1)
1.4142135623730951

fourier_period(scale)¶

Computes the equivalent Fourier period (wavelength) from wavelet scale.

Parameters:	scale (float) – wavelet scale.
Returns:	lmbd – equivalent Fourier period.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.fourier_period(scale=1)
1.0330436477492537

freq(omega, scale=1, dt=None)¶

Wavelet function in the frequency domain psi0_hat(s*omega).

Parameters:	omega (float or 1d array_like object) – angular frequency (length N). scale (float) – wavelet scale. dt (None or float) – time step. If dt is not None, the method returns the energy-normalized psi_hat (psi_hat0).
Returns:	psi0_hat – the wavelet in the frequency domain
Return type:	float/complex or 1D numpy array

Example

>>> import numpy as np
>>> import cwave
>>> w = cwave.Morlet()
>>> omega = 2 * np.pi * np.fft.fftfreq(32, 1)
>>> w.freq(omega, 10)
array([  0.00000000e+000,   2.17596717e-004,   8.76094852e-002,
         7.46634798e-001,   1.34686366e-001,   5.14277294e-004,
         4.15651521e-008,   7.11082035e-014,   2.57494732e-021,
         1.97367345e-030,   3.20214178e-041,   1.09967442e-053,
         7.99366604e-068,   1.22994640e-083,   4.00575772e-101,
         2.76147731e-120,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000])

smallest_scale(dt)¶

Returns the smallest resolvable scale. It is chosen so that the equivalent Fourier period is 2*dt.

Parameters:	dt (float) – time step.
Returns:	scale0 – smallest scale.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.smallest_scale(dt=1)
1.9360266183887822

time(t, scale=1, dt=None)¶

Wavelet function in the time domain psi0(t/s).

Parameters:	t (float) – time. If df is not None each element of t should be multiple of dt. scale (float) – wavelet scale. dt (None or float) – time step. If dt is not None, the method returns the energy-normalized psi (psi0).
Returns:	psi0 – the wavelet in the time domain.
Return type:	float/complex or 1D numpy array

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> t = range(-8, 8)
>>> w.time(t, 10)
array([ 0.04772449+0.54333716j, -0.28822893+0.51240757j,
       -0.56261977+0.27763414j, -0.65623233-0.09354365j,
       -0.51129130-0.46835014j, -0.16314795-0.69929479j,
        0.26678672-0.68621589j,  0.61683875-0.42200209j,
        0.75112554+0.j        ,  0.61683875+0.42200209j,
        0.26678672+0.68621589j, -0.16314795+0.69929479j,
       -0.51129130+0.46835014j, -0.65623233+0.09354365j,
       -0.56261977-0.27763414j, -0.28822893-0.51240757j])

wavelet_scale(lmbd)¶

Computes the wavelet scale from equivalent Fourier period (wavelength).

Parameters:	lmbd (float) – equivalent Fourier period.
Returns:	scale – wavelet scale.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.wavelet_scale(lmbd=10)
9.6801330919439117

class cwave.Morlet(omega0=6)¶

Morlet wavelet function.

Example

>>> import cwave
>>> w = cwave.Morlet(omega0=6)

efolding_time(scale)¶

Returns the e-folding time tau_s.

Parameters:	scale (float) – wavelet scale.
Returns:	tau – e-folding time.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.efolding_time(1)
1.4142135623730951

fourier_period(scale)¶

Computes the equivalent Fourier period (wavelength) from wavelet scale.

Parameters:	scale (float) – wavelet scale.
Returns:	lmbd – equivalent Fourier period.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.fourier_period(scale=1)
1.0330436477492537

freq(omega, scale=1, dt=None)¶

Wavelet function in the frequency domain psi0_hat(s*omega).

Parameters:	omega (float or 1d array_like object) – angular frequency (length N). scale (float) – wavelet scale. dt (None or float) – time step. If dt is not None, the method returns the energy-normalized psi_hat (psi_hat0).
Returns:	psi0_hat – the wavelet in the frequency domain
Return type:	float/complex or 1D numpy array

Example

>>> import numpy as np
>>> import cwave
>>> w = cwave.Morlet()
>>> omega = 2 * np.pi * np.fft.fftfreq(32, 1)
>>> w.freq(omega, 10)
array([  0.00000000e+000,   2.17596717e-004,   8.76094852e-002,
         7.46634798e-001,   1.34686366e-001,   5.14277294e-004,
         4.15651521e-008,   7.11082035e-014,   2.57494732e-021,
         1.97367345e-030,   3.20214178e-041,   1.09967442e-053,
         7.99366604e-068,   1.22994640e-083,   4.00575772e-101,
         2.76147731e-120,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000])

omega0¶: Get the omega0 parameter

time(t, scale=1, dt=None)¶

Wavelet function in the time domain psi0(t/s).

Parameters:	t (float) – time. If df is not None each element of t should be multiple of dt. scale (float) – wavelet scale. dt (None or float) – time step. If dt is not None, the method returns the energy-normalized psi (psi0).
Returns:	psi0 – the wavelet in the time domain.
Return type:	float/complex or 1D numpy array

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> t = range(-8, 8)
>>> w.time(t, 10)
array([ 0.04772449+0.54333716j, -0.28822893+0.51240757j,
       -0.56261977+0.27763414j, -0.65623233-0.09354365j,
       -0.51129130-0.46835014j, -0.16314795-0.69929479j,
        0.26678672-0.68621589j,  0.61683875-0.42200209j,
        0.75112554+0.j        ,  0.61683875+0.42200209j,
        0.26678672+0.68621589j, -0.16314795+0.69929479j,
       -0.51129130+0.46835014j, -0.65623233+0.09354365j,
       -0.56261977-0.27763414j, -0.28822893-0.51240757j])

class cwave.Paul(m=4)¶

Paul Wavelet function.

Example

>>> import cwave
>>> w = cwave.Paul(m=4)

efolding_time(scale)¶

Returns the e-folding time tau_s.

Parameters:	scale (float) – wavelet scale.
Returns:	tau – e-folding time.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.efolding_time(1)
1.4142135623730951

fourier_period(scale)¶

Computes the equivalent Fourier period (wavelength) from wavelet scale.

Parameters:	scale (float) – wavelet scale.
Returns:	lmbd – equivalent Fourier period.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.fourier_period(scale=1)
1.0330436477492537

freq(omega, scale=1, dt=None)¶

Wavelet function in the frequency domain psi0_hat(s*omega).

Parameters:	omega (float or 1d array_like object) – angular frequency (length N). scale (float) – wavelet scale. dt (None or float) – time step. If dt is not None, the method returns the energy-normalized psi_hat (psi_hat0).
Returns:	psi0_hat – the wavelet in the frequency domain
Return type:	float/complex or 1D numpy array

Example

>>> import numpy as np
>>> import cwave
>>> w = cwave.Morlet()
>>> omega = 2 * np.pi * np.fft.fftfreq(32, 1)
>>> w.freq(omega, 10)
array([  0.00000000e+000,   2.17596717e-004,   8.76094852e-002,
         7.46634798e-001,   1.34686366e-001,   5.14277294e-004,
         4.15651521e-008,   7.11082035e-014,   2.57494732e-021,
         1.97367345e-030,   3.20214178e-041,   1.09967442e-053,
         7.99366604e-068,   1.22994640e-083,   4.00575772e-101,
         2.76147731e-120,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000])

m¶: Get the m parameter (order)

time(t, scale=1, dt=None)¶

Wavelet function in the time domain psi0(t/s).

Parameters:	t (float) – time. If df is not None each element of t should be multiple of dt. scale (float) – wavelet scale. dt (None or float) – time step. If dt is not None, the method returns the energy-normalized psi (psi0).
Returns:	psi0 – the wavelet in the time domain.
Return type:	float/complex or 1D numpy array

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> t = range(-8, 8)
>>> w.time(t, 10)
array([ 0.04772449+0.54333716j, -0.28822893+0.51240757j,
       -0.56261977+0.27763414j, -0.65623233-0.09354365j,
       -0.51129130-0.46835014j, -0.16314795-0.69929479j,
        0.26678672-0.68621589j,  0.61683875-0.42200209j,
        0.75112554+0.j        ,  0.61683875+0.42200209j,
        0.26678672+0.68621589j, -0.16314795+0.69929479j,
       -0.51129130+0.46835014j, -0.65623233+0.09354365j,
       -0.56261977-0.27763414j, -0.28822893-0.51240757j])

class cwave.DOG(m=2)¶

Derivative Of Gaussian (DOG) Wavelet function.

Example

>>> import cwave
>>> w = cwave.DOG(m=2)

efolding_time(scale)¶

Returns the e-folding time tau_s.

Parameters:	scale (float) – wavelet scale.
Returns:	tau – e-folding time.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.efolding_time(1)
1.4142135623730951

fourier_period(scale)¶

Computes the equivalent Fourier period (wavelength) from wavelet scale.

Parameters:	scale (float) – wavelet scale.
Returns:	lmbd – equivalent Fourier period.
Return type:	float

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> w.fourier_period(scale=1)
1.0330436477492537

freq(omega, scale=1, dt=None)¶

Wavelet function in the frequency domain psi0_hat(s*omega).

Parameters:	omega (float or 1d array_like object) – angular frequency (length N). scale (float) – wavelet scale. dt (None or float) – time step. If dt is not None, the method returns the energy-normalized psi_hat (psi_hat0).
Returns:	psi0_hat – the wavelet in the frequency domain
Return type:	float/complex or 1D numpy array

Example

>>> import numpy as np
>>> import cwave
>>> w = cwave.Morlet()
>>> omega = 2 * np.pi * np.fft.fftfreq(32, 1)
>>> w.freq(omega, 10)
array([  0.00000000e+000,   2.17596717e-004,   8.76094852e-002,
         7.46634798e-001,   1.34686366e-001,   5.14277294e-004,
         4.15651521e-008,   7.11082035e-014,   2.57494732e-021,
         1.97367345e-030,   3.20214178e-041,   1.09967442e-053,
         7.99366604e-068,   1.22994640e-083,   4.00575772e-101,
         2.76147731e-120,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000,   0.00000000e+000,
         0.00000000e+000,   0.00000000e+000])

m¶: Get the m parameter (derivative)

time(t, scale=1, dt=None)¶

Wavelet function in the time domain psi0(t/s).

Parameters:	t (float) – time. If df is not None each element of t should be multiple of dt. scale (float) – wavelet scale. dt (None or float) – time step. If dt is not None, the method returns the energy-normalized psi (psi0).
Returns:	psi0 – the wavelet in the time domain.
Return type:	float/complex or 1D numpy array

Example

>>> import cwave
>>> w = cwave.Morlet()
>>> t = range(-8, 8)
>>> w.time(t, 10)
array([ 0.04772449+0.54333716j, -0.28822893+0.51240757j,
       -0.56261977+0.27763414j, -0.65623233-0.09354365j,
       -0.51129130-0.46835014j, -0.16314795-0.69929479j,
        0.26678672-0.68621589j,  0.61683875-0.42200209j,
        0.75112554+0.j        ,  0.61683875+0.42200209j,
        0.26678672+0.68621589j, -0.16314795+0.69929479j,
       -0.51129130+0.46835014j, -0.65623233+0.09354365j,
       -0.56261977-0.27763414j, -0.28822893-0.51240757j])

API¶

Summary¶

Series and transformed data¶

Functions¶

Wavelet functions¶

The The Wavelet abstract class¶

Available wavelet classes¶

Classes and functions¶

The The `Wavelet` abstract class¶