{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%load_ext load_style\n",
"%load_style talk.css\n",
"from IPython.display import YouTubeVideo, Image"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Wavelets in Python"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"[Nicolas Fauchereau](mailto:Nicolas.Fauchereau@gmail.com)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This notebook contains quick comparisons of 3 convenient packages implementing wavelet analysis in Python: \n",
" \n",
"+ waipy \n",
"+ kPywavelets\n",
"+ wavelets"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"we will try and reproduce the example in \n",
"\n",
"[A practical guide to wavelet analysis](http://journals.ametsoc.org/doi/pdf/10.1175/1520-0477%281998%29079%3C0061%3AAPGTWA%3E2.0.CO%3B2)\n",
"\n",
"from Christopher Torrence and Gil P. Compo\n",
"\n",
"using the NINO3.4 seasonal time series\n",
"\n",
"See also the [Interactive Wavelet page](http://ion.exelisvis.com/)\n",
"\n",
"We will also see the example of rectification of the bias that exist in favor of large scales"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### references"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"+ Torrence, C., and G. P. Compo, 1998: [A practical guide to wavelet analysis](http://journals.ametsoc.org/doi/pdf/10.1175/1520-0477%281998%29079%3C0061%3AAPGTWA%3E2.0.CO%3B2). Bull. Amer. Meteor. Soc., 79, 6178. \n",
"\n",
"\n",
"+ Liu, Y., X.S. Liang, and R.H. Weisberg, 2007: Rectification of the bias in the wavelet power spectrum. Journal of Atmospheric and Oceanic Technology, 24(12), 2093-2102. [http://ocgweb.marine.usf.edu/~liu/wavelet.html](http://ocgweb.marine.usf.edu/~liu/wavelet.html)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import numpy as np \n",
"from matplotlib import pyplot as plt"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### loads the data "
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"data = np.loadtxt('../data/sst_nino3.dat')"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"N = data.size"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"t0=1871\n",
"dt=0.25\n",
"units='^{\\circ}C'\n",
"label='NINO3 SST'"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"time = np.arange(0, N) * dt + t0"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1. [Waipy](Waipy)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import waipy"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"data_norm = waipy.normalize(data)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Lag-1 autocorrelation = 0.77\n"
]
}
],
"source": [
"alpha = np.corrcoef(data_norm[0:-1], data_norm[1:])[0,1]; \n",
"print(\"Lag-1 autocorrelation = {:4.2f}\".format(alpha))"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"result = waipy.cwt(data_norm, 1, 1, 0.25, 2, 4/0.25, alpha, 6, mother='Morlet',name='x')"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"waipy.wavelet_plot?"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"collapsed": false
},
"outputs": [
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"waipy.wavelet_plot(label, time, data_norm, 0.03125, result); "
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from waipy import wavelet_plot"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"wavelet_plot??"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 2. [kPyWavelet](https://github.com/duducosmos/kPyWavelet)"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import kPyWavelet as wavelet"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"slevel = 0.95 # Significance level\n",
"\n",
"std = data.std() # Standard deviation\n",
"std2 = std ** 2 # Variance\n",
"var = (data - data.mean()) / std # Calculating anomaly and normalizing\n",
"\n",
"dj = 0.25 # Four sub-octaves per octaves\n",
"s0 = -1 #2 * dt # Starting scale, here 6 months\n",
"J = -1 # 7 / dj # Seven powers of two with dj sub-octaves\n",
"\n",
"mother = wavelet.Morlet(6.) # Morlet mother wavelet with wavenumber=6\n",
"\n",
"# The following routines perform the wavelet transform and siginificance\n",
"# analysis for the chosen data set.\n",
"\n",
"wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(data, dt, dj, s0, J,\n",
" mother)\n",
"iwave = wavelet.icwt(wave, scales, dt, dj, mother)\n",
"power = (abs(wave)) ** 2 # Normalized wavelet power spectrum\n",
"fft_power = std2 * abs(fft) ** 2 # FFT power spectrum\n",
"period = 1. / freqs\n",
"\n",
"signif, fft_theor = wavelet.significance(1.0, dt, scales, 0, alpha,\n",
" significance_level=slevel, wavelet=mother)\n",
"sig95 = (signif * np.ones((N, 1))).transpose()\n",
"sig95 = power / sig95 # Where ratio > 1, power is significant\n",
"\n",
"# Calculates the global wavelet spectrum and determines its significance level.\n",
"glbl_power = std2 * power.mean(axis=1)\n",
"dof = N - scales # Correction for padding at edges\n",
"glbl_signif, tmp = wavelet.significance(std2, dt, scales, 1, alpha,\n",
" significance_level=slevel, dof=dof, wavelet=mother)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Second sub-plot, the normalized wavelet power spectrum and significance level\n",
"# contour lines and cone of influece hatched area.\n",
"f, ax = plt.subplots(figsize=(15,10))\n",
"levels = [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]\n",
"ax.contourf(time, np.log2(period), np.log2(power), np.log2(levels),\n",
" extend='both')\n",
"ax.contour(time, np.log2(period), sig95, [-99, 1], colors='k',\n",
" linewidths=2.)\n",
"#ax.fill(numpy.concatenate([time[:1]-dt, time, time[-1:]+dt, time[-1:]+dt,\n",
"# time[:1]-dt, time[:1]-dt]), numpy.log2(numpy.concatenate([[1e-9], coi,\n",
"# [1e-9], period[-1:], period[-1:], [1e-9]])), 'k', alpha='0.3',\n",
"# hatch='x')\n",
"\n",
"ax.plot(time,np.log2(coi), '0.8', lw=4)\n",
"\n",
"ax.plot(time,np.log2(coi), 'w--', lw=3)\n",
"#ax.plot(time,numpy.log2(coi), 'k:', lw=3)\n",
"\n",
"ax.set_title('b) %s Wavelet Power Spectrum (%s)' % (label, mother.name))\n",
"ax.set_ylabel('Period (years)')\n",
"Yticks = 2 ** np.arange(np.ceil(np.log2(period.min())),\n",
" np.ceil(np.log2(period.max())))\n",
"ax.set_yticks(np.log2(Yticks))\n",
"ax.set_yticklabels(Yticks)\n",
"ax.invert_yaxis()\n",
"ylim = ax.get_ylim()\n",
"ax.set_ylim(ylim[0], -1);"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Third sub-plot, the global wavelet and Fourier power spectra and theoretical\n",
"# noise spectra.\n",
"f, cx = plt.subplots()\n",
"cx.plot(glbl_signif, np.log2(period), 'k--')\n",
"cx.plot(fft_power, np.log2(1./fftfreqs), '-', color=[0.7, 0.7, 0.7],\n",
" linewidth=1.)\n",
"cx.plot(glbl_power, np.log2(period), 'k-', linewidth=1.5)\n",
"cx.set_title('c) Global Wavelet Spectrum')\n",
"cx.set_xlabel(r'Power')\n",
"#cx.set_xlim([0, glbl_power.max() + std2])\n",
"cx.set_ylim(np.log2([period.min(), period.max()]))\n",
"cx.set_yticks(np.log2(Yticks))\n",
"cx.set_yticklabels(Yticks)\n",
"cx.set_ylabel('Period (years)')\n",
"\n",
"cx.invert_yaxis()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 3. wavelet from Torrence and Compo"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The implementation is from **Dr. Toru Miyama**\n",
"\n",
"see: \n",
"\n",
"+ [http://iprc.soest.hawaii.edu/people/miyama.php](http://iprc.soest.hawaii.edu/people/miyama.php)\n",
"+ [https://sites.google.com/site/drtorumiyama/](https://sites.google.com/site/drtorumiyama/)\n",
"\n",
"The following is taken from his IPython notebook available at: \n",
"\n",
"[http://nbviewer.ipython.org/urls/dl.dropbox.com/s/ryeoz3wxrkg1hhz/wavelet_test_ElNino3_Liu.ipynb](http://nbviewer.ipython.org/urls/dl.dropbox.com/s/ryeoz3wxrkg1hhz/wavelet_test_ElNino3_Liu.ipynb)\n",
"\n",
"See also: \n",
"\n",
"[http://nbviewer.ipython.org/urls/dl.dropbox.com/s/wjch3iysnj6165q/wavelet_test_sine.ipynb](http://nbviewer.ipython.org/urls/dl.dropbox.com/s/wjch3iysnj6165q/wavelet_test_sine.ipynb)"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import sys"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"sys.path.append('./')"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from wavelib import *"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(1870, 2000)"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"f, ax = plt.subplots()\n",
"ax.plot(time,data)\n",
"xlim = [1870,2000] # plotting range\n",
"ax.set_xlim(xlim)"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"mean= 3.52451753849e-18\n",
"std= 1.0\n"
]
}
],
"source": [
"variance = np.std(data)**2\n",
"mean=np.mean(data)\n",
"data = (data - np.mean(data))/np.sqrt(variance)\n",
"print \"mean=\",mean\n",
"print \"std=\", np.sqrt(variance)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### set wavelet parameters"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"pad = 1 # pad the time series with zeroes (recommended)\n",
"dj = 0.25 # this will do 4 sub-octaves per octave\n",
"s0 = 2.*dt # this says start at a scale of 6 months\n",
"j1 = 7./dj # this says do 7 powers-of-two with dj sub-octaves each\n",
"lag1 = 0.72 # lag-1 autocorrelation for red noise background\n",
"mother = 'Morlet'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### wavelet transform"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"wave,period,scale,coi = wavelet(data,dt,pad,dj,s0,j1,mother);\n",
"power = (np.abs(wave))**2 # compute wavelet power spectrum"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### significance levels"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"signif,fft_theor = wave_signif(1.0,dt,scale,0,lag1,-1,-1,mother)\n",
"sig95 = np.dot(signif.reshape(len(signif),1),np.ones((1,N))) # expand signif --> (J+1)x(N) array\n",
"sig95 = power / sig95 # where ratio > 1, power is significant"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"global_ws = variance*power.sum(axis=1)/float(N) # time-average over all times\n",
"dof = N - scale # the -scale corrects for padding at edges\n",
"global_signif,fft_theor = wave_signif(variance,dt,scale,1,lag1,-1,dof,mother)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### scale-average between 2 and 8 years"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"avg = (scale >= 2) & (scale < 8)\n",
"Cdelta = 0.776; # this is for the MORLET wavelet\n",
"scale_avg = np.dot(scale.reshape(len(scale),1),np.ones((1,N))) # expand scale --> (J+1)x(N) array\n",
"scale_avg = power / scale_avg # [Eqn(24)]\n",
"scale_avg = variance*dj*dt/Cdelta*sum(scale_avg[avg,:]) # [Eqn(24)]\n",
"scaleavg_signif ,fft_theor= wave_signif(variance,dt,scale,2,lag1,-1,[2,7.9],mother)"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"root square mean error 0.105722742588 deg C\n"
]
}
],
"source": [
"iwave=wavelet_inverse(wave, scale, dt, dj, \"Morlet\")\n",
"print \"root square mean error\",np.sqrt(np.sum((data-iwave)**2)/float(len(data)))*np.sqrt(variance),\"deg C\""
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {
"collapsed": false
},
"outputs": [
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"#figure size\n",
"fig=plt.figure(figsize=(10,10))\n",
"\n",
"# subplot positions\n",
"width= 0.65\n",
"hight = 0.28;\n",
"pos1a = [0.1, 0.75, width, 0.2]\n",
"pos1b = [0.1, 0.37, width, hight]\n",
"pos1c = [0.79, 0.37, 0.18, hight]\n",
"pos1d = [0.1, 0.07, width, 0.2]\n",
"\n",
"#########################################\n",
"#---- a) Original signal\n",
"#########################################\n",
"ax=fig.add_axes(pos1a)\n",
"#original\n",
"ax.plot(time,data*np.sqrt(variance)+mean,\"r-\")\n",
"#reconstruction\n",
"ax.plot(time,iwave*np.sqrt(variance)+mean,\"k--\")\n",
"\n",
"ax.set_ylabel('NINO3 SST (degC)')\n",
"plt.title('a) NINO3 Sea Surface Temperature (seasonal)')\n",
"\n",
"#########################################\n",
"# b) Wavelet spectrum\n",
"#########################################\n",
"\n",
"#--- Contour plot wavelet power spectrum\n",
"bx=fig.add_axes(pos1b,sharex=ax)\n",
"levels = [0.0625,0.125,0.25,0.5,1,2,4,8,16] \n",
"Yticks = 2**(np.arange(np.int(np.log2(np.min(period))),np.int(np.log2(np.max(period)))+1))\n",
"bx.contour(time,np.log2(period),np.log2(power),np.log2(levels))\n",
"bx.contourf(time,np.log2(period),np.log2(powers),np.log2(levels), extend='both', cmap=plt.get_cmap('rainbow'))\n",
"bx.set_xlabel('Time (year)')\n",
"bx.set_ylabel('Period (years)')\n",
"import matplotlib.ticker as ticker\n",
"ymajorLocator=ticker.FixedLocator(np.log2(Yticks))\n",
"bx.yaxis.set_major_locator( ymajorLocator )\n",
"ticks=bx.yaxis.set_ticklabels(Yticks)\n",
"plt.title('b) Wavelet Power Spectrum')\n",
"\n",
"# 95% significance contour, levels at -99 (fake) and 1 (95% signif)\n",
"cs = bx.contour(time,np.log2(period),sig95,[1],color='k',linewidth=1)\n",
"\n",
"# cone-of-influence, anything \"below\" is dubious\n",
"ts = time;\n",
"coi_area = np.concatenate([[np.max(scale)], coi, [np.max(scale)],[np.max(scale)]])\n",
"ts_area = np.concatenate([[ts[0]], ts, [ts[-1]] ,[ts[0]]]);\n",
"L = bx.plot(ts_area,np.log2(coi_area),'k',linewidth=3)\n",
"F=bx.fill(ts_area,np.log2(coi_area),'k',alpha=0.3,hatch=\"x\")\n",
"\n",
"#########################################\n",
"# c) Global Wavelet spectrum\n",
"#########################################\n",
"\n",
"#--- Plot global wavelet spectrum\n",
"cx=fig.add_axes(pos1c,sharey=bx)\n",
"cx.plot(global_ws,np.log2(period),\"r-\")\n",
"cx.plot(global_signif,np.log2(period),'k--')\n",
"ylim=cx.set_ylim(np.log2([period.min(),period.max()]))\n",
"cx.invert_yaxis()\n",
"plt.title('c) Global Wavelet Spectrum')\n",
"xrangec=cx.set_xlim([0,1.25*np.max(global_ws)])\n",
"\n",
"#########################################\n",
"# d) Scale average Spectrum\n",
"#########################################\n",
"#--- Plot Scale-averaged spectrum -----------------\n",
"dx=fig.add_axes(pos1d,sharex=bx)\n",
"dx.plot(time,scale_avg,\"r-\")\n",
"dx.plot([time[0],time[-1]],[scaleavg_signif,scaleavg_signif],\"k--\")\n",
"xrange=dx.set_xlim(xlim)\n",
"dx.set_ylabel('Avg variance (degC$^2$)')\n",
"title=plt.title('d) Scale-average Time Series')\n",
"plt.savefig(\"nino3_TorrenceCompo.png\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Bias rectification"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"divides by scales"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"########################\n",
"# Spectrum\n",
"########################\n",
"powers=np.zeros_like(power)\n",
"for k in range(len(scale)):\n",
" powers[k,:] = power[k,:]/scale[k]\n",
"#significance: sig95 is already normalized = 1\n",
"\n",
"########################\n",
"# Spectrum\n",
"########################\n",
"global_wss = global_ws/scale \n",
"global_signifs=global_signif/scale\n",
"\n",
"########################\n",
"# Scale-average between El Nino periods of 2--8 years\n",
"########################\n",
"# No need to change \n",
"# because in Eqn(24) of Torrence and Compo [1998], division by scale has been done.\n",
"scale_avgs=scale_avg\n",
"scaleavg_signifs=scaleavg_signif"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {
"collapsed": false
},
"outputs": [
{
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"text/plain": [
""
]
},
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],
"source": [
"#figure size\n",
"fig=plt.figure(figsize=(10,10))\n",
"\n",
"# subplot positions\n",
"width= 0.65\n",
"hight = 0.28;\n",
"pos1a = [0.1, 0.75, width, 0.2]\n",
"pos1b = [0.1, 0.37, width, hight]\n",
"pos1c = [0.79, 0.37, 0.18, hight]\n",
"pos1d = [0.1, 0.07, width, 0.2]\n",
"\n",
"#########################################\n",
"#---- a) Original signal\n",
"#########################################\n",
"ax=fig.add_axes(pos1a)\n",
"#original\n",
"ax.plot(time,data*np.sqrt(variance)+mean,\"r-\")\n",
"#reconstruction\n",
"ax.plot(time,iwave*np.sqrt(variance)+mean,\"k--\")\n",
"\n",
"ax.set_ylabel('NINO3 SST (degC)')\n",
"plt.title('a) NINO3 Sea Surface Temperature (seasonal)')\n",
"\n",
"#########################################\n",
"# b) Wavelet spectrum\n",
"#########################################\n",
"\n",
"#--- Contour plot wavelet power spectrum\n",
"bx=fig.add_axes(pos1b,sharex=ax)\n",
"levels = [0.0625,0.125,0.25,0.5,1,2,4,8,16] \n",
"Yticks = 2**(np.arange(np.int(np.log2(np.min(period))),np.int(np.log2(np.max(period)))+1))\n",
"bx.contour(time,np.log2(period),np.log2(powers),np.log2(levels))\n",
"bx.contourf(time,np.log2(period),np.log2(powers),np.log2(levels), extend='both', cmap=plt.get_cmap('rainbow'))\n",
"bx.set_xlabel('Time (year)')\n",
"bx.set_ylabel('Period (years)')\n",
"import matplotlib.ticker as ticker\n",
"ymajorLocator=ticker.FixedLocator(np.log2(Yticks))\n",
"bx.yaxis.set_major_locator( ymajorLocator )\n",
"ticks=bx.yaxis.set_ticklabels(Yticks)\n",
"plt.title('b) Wavelet Power Spectrum')\n",
"\n",
"# 95% significance contour, levels at -99 (fake) and 1 (95% signif)\n",
"cs = bx.contour(time,np.log2(period),sig95,[1],color='k',linewidth=1)\n",
"\n",
"# cone-of-influence, anything \"below\" is dubious\n",
"ts = time;\n",
"coi_area = np.concatenate([[np.max(scale)], coi, [np.max(scale)],[np.max(scale)]])\n",
"ts_area = np.concatenate([[ts[0]], ts, [ts[-1]] ,[ts[0]]]);\n",
"L = bx.plot(ts_area,np.log2(coi_area),'k',linewidth=3)\n",
"F=bx.fill(ts_area,np.log2(coi_area),'k',alpha=0.3,hatch=\"x\")\n",
"\n",
"#########################################\n",
"# c) Global Wavelet spectrum\n",
"#########################################\n",
"\n",
"#--- Plot global wavelet spectrum\n",
"cx=fig.add_axes(pos1c,sharey=bx)\n",
"cx.plot(global_wss,np.log2(period),\"r-\")\n",
"cx.plot(global_signifs,np.log2(period),'k--')\n",
"ylim=cx.set_ylim(np.log2([period.min(),period.max()]))\n",
"cx.invert_yaxis()\n",
"plt.title('c) Global Wavelet Spectrum')\n",
"xrangec=cx.set_xlim([0,1.25*np.max(global_wss)])\n",
"\n",
"#########################################\n",
"# d) Global Wavelet spectrum\n",
"#########################################\n",
"#--- Plot Scale-averaged spectrum -----------------\n",
"dx=fig.add_axes(pos1d,sharex=bx)\n",
"dx.plot(time,scale_avgs,\"r-\")\n",
"dx.plot([time[0],time[-1]],[scaleavg_signifs,scaleavg_signifs],\"k--\")\n",
"xrange=dx.set_xlim(xlim)\n",
"dx.set_ylabel('Avg variance (degC$^2$)')\n",
"title=plt.title('d) Scale-average Time Series')"
]
}
],
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