{ "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, 61–78. \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", 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[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": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAA3sAAAJZCAYAAAAQ3ixqAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", 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"# 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": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAYoAAAEZCAYAAACJjGL9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xl8FPX9x/HXOwk3IiAenOItVbxBvLGexQEUtd62aL3P\n", 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"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": 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"ZdkiZpmxAAAAAElFTkSuQmCC\n" ], "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": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAArsAAAKkCAYAAAAA4/fMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsnXecXFXZx7/PTtves8kmm0YSAgRCCS2ASQDpVQWkSLEh\n", "goJSBAVeogiIiKggllewoSAgVcSCEkAQXkogQEhCem+bbC/TzvvHObN7d3b6zO5MlvP9fPazM3Pv\n", "nPvcO7f8znOe8zyilMJisVgsFovFYhmJFOXbAIvFYrFYLBaLZaiwYtdisVgsFovFMmKxYtdisVgs\n", "FovFMmKxYtdisVgsFovFMmKxYtdisVgsFovFMmKxYtdisVgsFovFMmKxYtcyZIjIKSLyUL7t+Kgj\n", "It8VkW0isjHftlhSR0T2EpHX821HrhGR1SJylHn9FRH5Xr5tslgsIxsrdi1DhlLqaWCGiOwTbx3z\n", "4NsiIqWOz74gIs873odFZDfzer55f6Zjudt8NsHx2WEi8m8RaRORFhF5SkT2dCzfS0TeEJEdZvnL\n", 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"rP8HCD9OZ5Jd/PwAAAAASUVORK5CYII=\n" ], "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": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAArsAAAKkCAYAAAAA4/fMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsnXecXFXZx7/PTtves8kmm0YSAgRCCS2ASQDpVQWkSLEh\n", "goJSBAVeogiIiKggllewoSAgVcSCEkAQXkogQEhCem+bbC/TzvvHObN7d3b6zO5MlvP9fPazM3Pv\n", "nPvcO7f8znOe8zyilMJisVgsFovFYhmJFOXbAIvFYrFYLBaLZaiwYtdisVgsFovFMmKxYtdisVgs\n", "FovFMmKxYtdisVgsFovFMmKxYtdisVgsFovFMmKxYtdisVgsFovFMmKxYtcyZIjIKSLyUL7t+Kgj\n", "It8VkW0isjHftlhSR0T2EpHX821HrhGR1SJylHn9FRH5Xr5tslgsIxsrdi1DhlLqaWCGiOwTbx3z\n", "4NsiIqWOz74gIs873odFZDfzer55f6Zjudt8NsHx2WEi8m8RaRORFhF5SkT2dCzfS0TeEJEdZvnL\n", 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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')" ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.9" } }, "nbformat": 4, "nbformat_minor": 0 }