{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "7dc613c6",
   "metadata": {},
   "source": [
    "# Variational Mode Decomposition:  Python 3.8"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "63861019",
   "metadata": {},
   "source": [
    "Jan 21, 2023 by Teiji Sawa, Department of Anesthesiology, Kyoto Prefectural University of Medicine"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3465306c",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/0j/j_y5y9pj73g9f1r01zqzycxm0000gp/T/ipykernel_5324/2066908370.py:12: DeprecationWarning: `set_matplotlib_formats` is deprecated since IPython 7.23, directly use `matplotlib_inline.backend_inline.set_matplotlib_formats()`\n",
      "  set_matplotlib_formats('svg')\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt \n",
    "import matplotlib.patches as patches\n",
    "from scipy.fftpack import fft\n",
    "from scipy import signal as sig #Signal function import from SciPy package\n",
    "from scipy import ndimage\n",
    "\n",
    "import emd\n",
    "\n",
    "from IPython.display import set_matplotlib_formats\n",
    "set_matplotlib_formats('svg')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bdb54d5",
   "metadata": {},
   "source": [
    "# try EEG waveform"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2ede15c9",
   "metadata": {},
   "outputs": [],
   "source": [
    "df_tsv = pd.read_table('./eeg_bis_20220629_sev_EME.tsv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f0b449ba",
   "metadata": {},
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>.Ch</th>\n",
       "      <th>Time</th>\n",
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       "      <th>ch[1]</th>\n",
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       "      <td>6.70</td>\n",
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       "      <td>1.75</td>\n",
       "      <td>-1.75</td>\n",
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       "      <td>26.30</td>\n",
       "      <td>26.60</td>\n",
       "      <td>25.60</td>\n",
       "      <td>23.00</td>\n",
       "      <td>17.85</td>\n",
       "      <td>8.90</td>\n",
       "      <td>0.20</td>\n",
       "      <td>-0.95</td>\n",
       "      <td>4.10</td>\n",
       "      <td>7.10</td>\n",
       "      <td>7.00</td>\n",
       "      <td>6.20</td>\n",
       "      <td>7.05</td>\n",
       "      <td>10.10</td>\n",
       "      <td>13.05</td>\n",
       "      <td>13.15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>ch1:</td>\n",
       "      <td>12:15:17</td>\n",
       "      <td>10.45</td>\n",
       "      <td>4.45</td>\n",
       "      <td>-2.90</td>\n",
       "      <td>-12.30</td>\n",
       "      <td>-24.95</td>\n",
       "      <td>-35.85</td>\n",
       "      <td>-38.35</td>\n",
       "      <td>-30.90</td>\n",
       "      <td>-17.70</td>\n",
       "      <td>-5.70</td>\n",
       "      <td>4.10</td>\n",
       "      <td>11.35</td>\n",
       "      <td>11.60</td>\n",
       "      <td>6.05</td>\n",
       "      <td>-4.35</td>\n",
       "      <td>-20.85</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>ch1:</td>\n",
       "      <td>12:15:17</td>\n",
       "      <td>-39.00</td>\n",
       "      <td>-52.55</td>\n",
       "      <td>-64.65</td>\n",
       "      <td>-76.25</td>\n",
       "      <td>-83.35</td>\n",
       "      <td>-82.55</td>\n",
       "      <td>-75.25</td>\n",
       "      <td>-64.40</td>\n",
       "      <td>-52.55</td>\n",
       "      <td>-42.90</td>\n",
       "      <td>-36.65</td>\n",
       "      <td>-32.25</td>\n",
       "      <td>-29.05</td>\n",
       "      <td>-24.50</td>\n",
       "      <td>-19.45</td>\n",
       "      <td>-20.40</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>ch1:</td>\n",
       "      <td>12:15:17</td>\n",
       "      <td>-27.60</td>\n",
       "      <td>-34.85</td>\n",
       "      <td>-38.25</td>\n",
       "      <td>-35.30</td>\n",
       "      <td>-26.20</td>\n",
       "      <td>-15.60</td>\n",
       "      <td>-7.65</td>\n",
       "      <td>-3.50</td>\n",
       "      <td>-2.55</td>\n",
       "      <td>-4.50</td>\n",
       "      <td>-7.75</td>\n",
       "      <td>-12.00</td>\n",
       "      <td>-20.40</td>\n",
       "      <td>-28.80</td>\n",
       "      <td>-30.85</td>\n",
       "      <td>-27.15</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    .Ch      Time  ch[0]  ch[1]  ch[2]  ch[3]  ch[4]  ch[5]  ch[6]  ch[7]  \\\n",
       "0  ch1:  12:15:17   6.70   3.80   1.75  -1.75  -1.95   1.10   5.15  10.30   \n",
       "1  ch1:  12:15:17  26.30  26.60  25.60  23.00  17.85   8.90   0.20  -0.95   \n",
       "2  ch1:  12:15:17  10.45   4.45  -2.90 -12.30 -24.95 -35.85 -38.35 -30.90   \n",
       "3  ch1:  12:15:17 -39.00 -52.55 -64.65 -76.25 -83.35 -82.55 -75.25 -64.40   \n",
       "4  ch1:  12:15:17 -27.60 -34.85 -38.25 -35.30 -26.20 -15.60  -7.65  -3.50   \n",
       "\n",
       "   ch[8]  ch[9]  ch[10]  ch[11]  ch[12]  ch[13]  ch[14]  ch[15]  \n",
       "0  15.70  19.65   20.60   17.55   12.90   12.75   18.00   23.80  \n",
       "1   4.10   7.10    7.00    6.20    7.05   10.10   13.05   13.15  \n",
       "2 -17.70  -5.70    4.10   11.35   11.60    6.05   -4.35  -20.85  \n",
       "3 -52.55 -42.90  -36.65  -32.25  -29.05  -24.50  -19.45  -20.40  \n",
       "4  -2.55  -4.50   -7.75  -12.00  -20.40  -28.80  -30.85  -27.15  "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_tsv.head(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c2247f2d",
   "metadata": {},
   "outputs": [
    {
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       "      <td>10.60</td>\n",
       "      <td>21.15</td>\n",
       "      <td>13.30</td>\n",
       "      <td>-5.60</td>\n",
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       "    <tr>\n",
       "      <th>4679</th>\n",
       "      <td>ch1:</td>\n",
       "      <td>12:25:18</td>\n",
       "      <td>6.65</td>\n",
       "      <td>20.25</td>\n",
       "      <td>17.00</td>\n",
       "      <td>10.60</td>\n",
       "      <td>12.55</td>\n",
       "      <td>10.20</td>\n",
       "      <td>20.5</td>\n",
       "      <td>18.85</td>\n",
       "      <td>12.50</td>\n",
       "      <td>22.15</td>\n",
       "      <td>33.80</td>\n",
       "      <td>22.90</td>\n",
       "      <td>14.50</td>\n",
       "      <td>12.85</td>\n",
       "      <td>21.85</td>\n",
       "      <td>22.60</td>\n",
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       "    <tr>\n",
       "      <th>4680</th>\n",
       "      <td>ch1:</td>\n",
       "      <td>12:25:18</td>\n",
       "      <td>17.05</td>\n",
       "      <td>14.30</td>\n",
       "      <td>15.10</td>\n",
       "      <td>16.15</td>\n",
       "      <td>29.15</td>\n",
       "      <td>29.00</td>\n",
       "      <td>31.9</td>\n",
       "      <td>36.95</td>\n",
       "      <td>32.90</td>\n",
       "      <td>28.40</td>\n",
       "      <td>36.15</td>\n",
       "      <td>34.20</td>\n",
       "      <td>36.05</td>\n",
       "      <td>48.30</td>\n",
       "      <td>41.95</td>\n",
       "      <td>39.25</td>\n",
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       "    <tr>\n",
       "      <th>4681</th>\n",
       "      <td>ch1:</td>\n",
       "      <td>12:25:18</td>\n",
       "      <td>53.75</td>\n",
       "      <td>55.75</td>\n",
       "      <td>49.75</td>\n",
       "      <td>58.85</td>\n",
       "      <td>65.15</td>\n",
       "      <td>69.65</td>\n",
       "      <td>70.7</td>\n",
       "      <td>72.40</td>\n",
       "      <td>82.45</td>\n",
       "      <td>87.25</td>\n",
       "      <td>85.80</td>\n",
       "      <td>93.45</td>\n",
       "      <td>95.95</td>\n",
       "      <td>91.35</td>\n",
       "      <td>89.50</td>\n",
       "      <td>94.80</td>\n",
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       "    <tr>\n",
       "      <th>4682</th>\n",
       "      <td>ch1:</td>\n",
       "      <td>12:25:18</td>\n",
       "      <td>105.20</td>\n",
       "      <td>113.00</td>\n",
       "      <td>104.95</td>\n",
       "      <td>99.35</td>\n",
       "      <td>108.80</td>\n",
       "      <td>111.40</td>\n",
       "      <td>109.3</td>\n",
       "      <td>112.30</td>\n",
       "      <td>111.55</td>\n",
       "      <td>108.90</td>\n",
       "      <td>111.80</td>\n",
       "      <td>113.35</td>\n",
       "      <td>107.90</td>\n",
       "      <td>110.50</td>\n",
       "      <td>114.30</td>\n",
       "      <td>111.75</td>\n",
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      "text/plain": [
       "       .Ch      Time   ch[0]   ch[1]   ch[2]  ch[3]   ch[4]   ch[5]  ch[6]  \\\n",
       "4678  ch1:  12:25:18    5.65   -1.30   -5.40  -1.50   10.25   13.10   13.4   \n",
       "4679  ch1:  12:25:18    6.65   20.25   17.00  10.60   12.55   10.20   20.5   \n",
       "4680  ch1:  12:25:18   17.05   14.30   15.10  16.15   29.15   29.00   31.9   \n",
       "4681  ch1:  12:25:18   53.75   55.75   49.75  58.85   65.15   69.65   70.7   \n",
       "4682  ch1:  12:25:18  105.20  113.00  104.95  99.35  108.80  111.40  109.3   \n",
       "\n",
       "       ch[7]   ch[8]   ch[9]  ch[10]  ch[11]  ch[12]  ch[13]  ch[14]  ch[15]  \n",
       "4678    8.70    9.85   17.95   16.30    5.70   10.60   21.15   13.30   -5.60  \n",
       "4679   18.85   12.50   22.15   33.80   22.90   14.50   12.85   21.85   22.60  \n",
       "4680   36.95   32.90   28.40   36.15   34.20   36.05   48.30   41.95   39.25  \n",
       "4681   72.40   82.45   87.25   85.80   93.45   95.95   91.35   89.50   94.80  \n",
       "4682  112.30  111.55  108.90  111.80  113.35  107.90  110.50  114.30  111.75  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_tsv.tail(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "0c42f071",
   "metadata": {},
   "outputs": [],
   "source": [
    "cols = ['Time', 'eeg']\n",
    "df_eeg = pd.DataFrame(data=None, index=[], columns=cols)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "6692f46d",
   "metadata": {},
   "outputs": [],
   "source": [
    "for row in range(0, 4682): \n",
    "    # Import only 1024 lines of data. Since 8 lines / second, 128 seconds = 2 minutes 8 seconds \n",
    "    # Number of data = 16,384\n",
    "    # By playing with the range here, you can change which brain waves you can take out.\n",
    "    \n",
    "    t_tmp = df_tsv.iat[row,1]\n",
    "    for column in range(2, 18):\n",
    "              eeg_tmp = df_tsv.iat[row, column]\n",
    "              data_tmp = pd.Series( [t_tmp, eeg_tmp], index=df_eeg.columns )\n",
    "              df_eeg = df_eeg.append(data_tmp, ignore_index=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "e6948ca1",
   "metadata": {},
   "outputs": [
    {
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       "      <th>74907</th>\n",
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       "      <th>74908</th>\n",
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       "      <th>74909</th>\n",
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       "      <td>12:25:18</td>\n",
       "      <td>94.80</td>\n",
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       "</table>\n",
       "<p>74912 rows × 2 columns</p>\n",
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      ],
      "text/plain": [
       "           Time    eeg\n",
       "0      12:15:17   6.70\n",
       "1      12:15:17   3.80\n",
       "2      12:15:17   1.75\n",
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       "...         ...    ...\n",
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       "\n",
       "[74912 rows x 2 columns]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_eeg"
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  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f548fded",
   "metadata": {},
   "outputs": [],
   "source": [
    "df_eeg.to_csv(\"data_eeg/eeg_linealized_eeg_bis_std2.csv\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "4e62a04b",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Place brain wave data in the data frame\n",
    "time = df_eeg['Time'].values\n",
    "eeg = df_eeg['eeg'].values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "10e74afc",
   "metadata": {},
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   ],
   "source": [
    "# Time-series graph display of EEG data\n",
    "fig, ax = plt.subplots(figsize=(18,6))\n",
    "plt.plot(eeg)\n",
    "plt.savefig('data_eeg/spectral_analysis_3_fig_1.svg')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "b1c5d2b1",
   "metadata": {},
   "outputs": [],
   "source": [
    "eeg_ip = df_eeg['eeg'].values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "b8a93e72",
   "metadata": {},
   "outputs": [],
   "source": [
    "eeg_512 = eeg_ip[5000:5512]\n",
    "time_512 = time[5000:5512]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "3f56a3ba",
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      "text/plain": [
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     },
     "metadata": {
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     "output_type": "display_data"
    }
   ],
   "source": [
    "# Time-series graph display of EEG data that complements and corrects missing values\n",
    "fig, ax = plt.subplots(figsize=(18,6))\n",
    "plt.ylim(-100, 100)\n",
    "plt.plot(eeg_512[0:511], linewidth =0.5)\n",
    "plt.savefig('data_eeg/VMD_fig_eeg.svg')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "9f9a5142",
   "metadata": {},
   "outputs": [],
   "source": [
    "N=len(eeg_512[0:511]) # 総データポイント数\n",
    "f_s = 128 # サンプリング周波数\n",
    "T = 1.0/f_s  # データ幅(秒)\n",
    "s_rate = 1/f_s # サンプリングレート"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "65d9d68d",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.fftpack import fft\n",
    "\n",
    "xfft = np.linspace(0, int(1.0/(2.0*T)), int(N/2)) \n",
    "yfft=fft(eeg_512[0:511])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "459df62c",
   "metadata": {},
   "outputs": [
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    "plt.plot(xfft, 2.0/N*abs(yfft[0:int(N/2)]), linewidth =0.5,  label=\"periodogram\")\n",
    "plt.legend(bbox_to_anchor=(1, 1), loc='upper right', borderaxespad=1, fontsize=12)\n",
    "plt.grid()\n",
    "plt.yscale(\"log\")\n",
    "plt.savefig('data_eeg/eeg_fft_maintenance.svg')"
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    "def VMD(signal, alpha, tau, K, DC, init, tol):\n",
    "    # ---------------------\n",
    "    #  signal  - the time domain signal (1D) to be decomposed \n",
    "    #            \n",
    "    #  alpha   - the balancing parameter of the data-fidelity constraint \n",
    "    #            \n",
    "    #  tau     - time-step of the dual ascent ( pick 0 for noise-slack )　\n",
    "    #            \n",
    "    #  K       - the number of modes to be recovered　\n",
    "    #  DC      - true if the first mode is put and kept at DC (0-freq)　\n",
    "    #            \n",
    "    #  init    - 0 = all omegas start at 0　\n",
    "    #                     1 = all omegas start uniformly distributed\n",
    "    #                     2 = all omegas initialized randomly\n",
    "    #  tol     - tolerance of convergence criterion; typically around 1e-6　\n",
    "    #             \n",
    "    #\n",
    "    #  Output:\n",
    "    #  -------\n",
    "    #  u       - the collection of decomposed modes \n",
    "    #  u_hat   - spectra of the modes \n",
    "    #  omega   - estimated mode center-frequencies \n",
    "    #\n",
    "\n",
    "    import numpy as np\n",
    "    import math\n",
    "    import matplotlib.pyplot as plt\n",
    "    # Period and sampling frequency of input signal\n",
    "    save_T=len(signal)\n",
    "    fs=1/float(save_T)\n",
    "    #print(\"save_T=:\")\n",
    "    #print(save_T)\n",
    "\n",
    "    # extend the signal by mirroring \n",
    "    # size: 2048 data points\n",
    "    T=save_T\n",
    "    f_mirror=np.zeros(2*T)\n",
    "    f_mirror[0:T//2]=signal[T//2-1::-1]\n",
    "    f_mirror[T//2:3*T//2]= signal\n",
    "    f_mirror[3*T//2:2*T]=signal[-1:-T//2-1:-1]\n",
    "      \n",
    "    # signal size =2048 data points to f   \n",
    "    f=f_mirror\n",
    "    T=float(len(f))\n",
    "    t=np.linspace(1/float(T),1,int(T),endpoint=True)\n",
    "\n",
    "    # Spectral Domain discretization \n",
    "    freqs=t-0.5-1/T\n",
    "    N=2000\n",
    "\n",
    "    # For future generalizations: individual alpha for each mode\n",
    "    # penalty index、balance parameter\n",
    "    Alpha=alpha*np.ones(K,dtype=complex)\n",
    "    #print('-----')\n",
    "\n",
    "    # Construct and center f_hat Fourier transform of signal\n",
    "    f_hat=np.fft.fftshift(np.fft.fft(f))\n",
    "    f_hat_plus=f_hat\n",
    "    f_hat_plus[0:int(int(T)/2)]=0\n",
    "    \n",
    "    # matrix keeping track of every iterant\n",
    "    # could be discarded for mem\n",
    "    u_hat_plus=np.zeros((N,len(freqs),K),dtype=complex)\n",
    "    \n",
    "    #print(\"u_hat_plus.shape:=\", u_hat_plus.shape)\n",
    "    #print(\"u_hat_plus:=\", u_hat_plus)\n",
    "    \n",
    "    # Initialization of omega_k　\n",
    "    omega_plus=np.zeros((N,K),dtype=complex)\n",
    "\n",
    "    #print(\"omega_plus.shape:=\", omega_plus.shape)\n",
    "    #print(\"omega_plus:=\", omega_plus)\n",
    "    \n",
    "    if (init==1):\n",
    "        for i in range(1,K+1):\n",
    "            omega_plus[0,i-1]=(0.5/K)*(i-1)\n",
    "    elif (init==2):\n",
    "        omega_plus[0,:]=np.sort(math.exp(math.log(fs))+(math.log(0.5)\n",
    "        -math.log(fs))*np.random.rand(1,K))\n",
    "    else:\n",
    "        omega_plus[0,:]=0\n",
    "\n",
    "    if (DC):\n",
    "        omega_plus[0,0]=0\n",
    "    \n",
    "    # start with empty dual variables: Lagrange multiplier λ \n",
    "    lamda_hat=np.zeros((N,len(freqs)),dtype=complex)\n",
    "\n",
    "    # other inits\n",
    "    uDiff=tol+2.2204e-16 #updata step \n",
    "    n=1 #loop counter\n",
    "    sum_uk=0 #accumulator\n",
    "    T=int(T)\n",
    "\n",
    "    # ----------- Main loop for iterative updates\n",
    "    # renew n、and minimize u_hat& omega_hat\n",
    "    # Algorithm 2 Complete optimization of VMD\n",
    "    \n",
    "    while uDiff > tol and n<N:\n",
    "    # while uDiff > tol and n<4:\n",
    "        # update first mode accumulator\n",
    "        k=1\n",
    "        sum_uk = u_hat_plus[n-1,:,K-1] + sum_uk - u_hat_plus[n-1,:,0]\n",
    "        \n",
    "        #update spectrum of first mode through Wiener filter of residuals\n",
    "        # equation in the paper (27)\n",
    "        freqs_square = np.square(freqs - omega_plus[n-1,k-1])       \n",
    "        u_hat_plus[n,:,k-1]=(f_hat_plus - sum_uk\\\n",
    "        - lamda_hat[n-1,:]/2)/(1 + Alpha[k-1] *  freqs_square)\n",
    "    \n",
    "        # update first omega if not held at 0\n",
    "        # equation(28) The calculation of the integral is substituted by the inner product.\n",
    "        if DC==False:            \n",
    "            omega_plus[n,k-1]= (np.dot(freqs[T//2:T],\n",
    "            (np.square(np.abs(u_hat_plus[n,T//2:T,k-1]))).T))\\\n",
    "            /(np.sum(np.square(np.abs(u_hat_plus[n,T//2:T,k-1]))))\n",
    "            #print(\"n=:\",n, \"k=:\",k, \"omega_plus[n,k-1]:=\", omega_plus[n,k-1])\n",
    "\n",
    "        for k in range(2,K+1):\n",
    "            # accumulator\n",
    "            sum_uk = u_hat_plus[n,:,k-2] + sum_uk - u_hat_plus[n-1,:,k-1]\n",
    "            \n",
    "            # mode spectrum            \n",
    "            u_hat_plus[n,:,k-1]= (f_hat_plus - sum_uk \n",
    "            - lamda_hat[n-1,:]/2)/(1+Alpha[k-1] \n",
    "            * np.square(freqs - omega_plus[n-1,k-1]))\n",
    "\n",
    "            # center frequencies\n",
    "            omega_plus[n,k-1]=np.dot(freqs[T//2:T], \n",
    "            np.square(np.abs(u_hat_plus[n,T//2:T,k-1])).T)\\\n",
    "            /np.sum(np.square(np.abs(u_hat_plus[n,T//2:T:,k-1]))) \n",
    "            \n",
    "            # Dual ascent\n",
    "            # equation in the paper(29)\n",
    "            lamda_hat[n,:]=lamda_hat[n-1,:]\\\n",
    "            +tau*(np.sum(u_hat_plus[n,:,:],axis=1) - f_hat_plus)            \n",
    "\n",
    "        #loop counter : Comparison with convergence criteria\n",
    "        n=n+1\n",
    "        uDiff=2.2204e-16\n",
    "\n",
    "        for i in range(1,K+1):            \n",
    "            uDiff=uDiff+1/float(T)*(np.dot(np.conj(u_hat_plus[n-1,:,i-1] \n",
    "            - u_hat_plus[n-2,:,i-1]).conj().T, \n",
    "              np.conj(u_hat_plus[n-1,:,i-1] \n",
    "            - u_hat_plus[n-2,:,i-1]).conj().T))    \n",
    "        \n",
    "        uDiff=np.abs(uDiff)\n",
    "        #print(\"n=:\", n)\n",
    "                \n",
    "    # ------ Postprocessing and cleanup\n",
    "    # discard empty space if converged early\n",
    "    N=np.minimum(N,n)\n",
    "    omega = omega_plus[0:N,:]\n",
    "        \n",
    "    # Signal reconstruction.　Calculate IMF from frequency components by inverse Fourier transform\n",
    "    u_hat = np.zeros((T,K),dtype=complex)\n",
    "    u_hat[T//2:T,:]= np.squeeze(u_hat_plus[N-1,T//2:T,:])\n",
    "    u_hat[T//2:0:-1,:]=np.squeeze(np.conj(u_hat_plus[N-1,T//2:T,:]))\n",
    "    u_hat[0,:]=np.conj(u_hat[-1,:])\n",
    "    u=np.zeros((K,len(t)),dtype=complex)\n",
    "\n",
    "    for k in range(1,K+1):\n",
    "        u[k-1,:]= np.real(np.fft.ifft(np.fft.ifftshift(u_hat[:,k-1]))) \n",
    "    \n",
    "    # remove mirror part \n",
    "    u=u[:,T//4:3*T//4]\n",
    "    \n",
    "    for k in range(1,K+1):   \n",
    "        filename_ifft = \"data_eeg/{}_imf-%d.txt\" .format(str(n).zfill(3)) %(k) \n",
    "        np.savetxt(filename_ifft, u[k-1,:]) \n",
    "       \n",
    "    # recompute spectrum　Obtain the frequency spectrum from the IMF by Fourier transform.\n",
    "    u_hat = np.zeros((T//2, K),dtype=complex)\n",
    "\n",
    "    for k in range(1,K+1):\n",
    "        # u_hat[:, k-1]= ((np.fft.fftshift(np.fft.fft(u[k-1,:]))).conj()).T\n",
    "        u_hat[:, k-1]= ((np.fft.fft(u[k-1,:])).conj()).T\n",
    "        #print(\"u_hat.shape=:\", u_hat.shape)\n",
    "    \n",
    "    fft_amp = np.zeros((T//2, K), dtype = 'float64')\n",
    "    for k in range(1,K+1): \n",
    "        filename_fft = \"data_eeg/{}_fft-%d.txt\" .format(str(n).zfill(3)) %(k) \n",
    "        np.savetxt(filename_fft, np.abs(u_hat[:,k-1]/ (128 / 2)))      \n",
    "        fft_amp[:, k-1] = np.abs(u_hat[:,k-1] / (128 / 2))\n",
    "        \n",
    "    \n",
    "    fft_axis = np.linspace(0, 128, int(T/2))\n",
    "                       \n",
    "    fig = plt.figure()\n",
    "    \n",
    "    plt.plot(signal)\n",
    "    plt.show()\n",
    "    filename1 = \"data_eeg/eeg.svg\"\n",
    "    fig.savefig(filename1)\n",
    "    \n",
    "    \n",
    "    fig4 = plt.figure()\n",
    "    N=len(signal) \n",
    "    f_sig= fft(signal)\n",
    "    plt.plot(fft_axis[0:int(T/4)], 2.0/N*abs(f_sig[0:int(N/2)]))\n",
    "    plt.show()\n",
    "    filename4 = \"data_eeg/spectrum.svg\"\n",
    "    fig4.savefig(filename4)\n",
    "\n",
    "\n",
    "    for i in range(1,K+1):\n",
    "         fig2 = plt.figure()\n",
    "         plt.plot(u[i-1,:])\n",
    "         plt.show()\n",
    "         filename2 = \"data_eeg/{}_imf-%d.svg\" .format(str(n).zfill(3)) %(i) \n",
    "         fig2.savefig(filename2)\n",
    "    #print(\"FFT Power Spectrum\")\n",
    "    for i in range(1,K+1):\n",
    "         fig3 = plt.figure()\n",
    "         plt.plot(fft_axis[0:int(T/4)], fft_amp[0:int(T/4),i-1]) \n",
    "         plt.show()               \n",
    "         filename3 = \"data_eeg/{}_fft-%d.svg\" .format(str(n).zfill(3)) %(i) \n",
    "         fig3.savefig(filename3)\n",
    "                                       \n",
    "    return (u,u_hat,omega)"
   ]
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   "source": [
    "# signal=eeg8[0:511]\n",
    "alpha = 2000.0     \n",
    "tau = 0         \n",
    "DC = 0         \n",
    "init = 1         \n",
    "tol = 1e-7\n",
    "\n",
    "signal=eeg_512\n",
    "K = 6\n",
    "u,u_hat,omega = VMD(signal, alpha, tau, K, DC, init, tol)\n",
    "#print(u.shape)\n",
    "#print(u_hat.shape)\n",
    "#print(omega.shape)"
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  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "2b658835",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
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     "text": [
      "/var/folders/0j/j_y5y9pj73g9f1r01zqzycxm0000gp/T/ipykernel_5324/4259685197.py:5: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  u_t[j, i] = u[i, j]\n"
     ]
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   ],
   "source": [
    "u_t = np.zeros((len(signal),K), dtype = 'float64')\n",
    "\n",
    "for i in range(len(u)):\n",
    "    for j in range(len(u[i])):\n",
    "        u_t[j, i] = u[i, j]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "f217a9c5",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Get the default configuration for a sift\n",
    "config = emd.sift.get_config('sift')\n",
    "# Adjust the threshold for accepting an IMF\n",
    "config['imf_opts/sd_thresh'] = 0.05\n",
    "imf = emd.sift.sift(signal)"
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  },
  {
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   "execution_count": 21,
   "id": "0f64ac37",
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    "emd.plotting.plot_imfs(imf, cmap=True)\n",
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   "cell_type": "code",
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   "id": "e9b87b2b",
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       "<Figure size 1152x720 with 1 Axes>"
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     "metadata": {
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   ],
   "source": [
    "emd.plotting.plot_imfs(u_t, cmap=True)\n",
    "plt.savefig('data_eeg/vmd.svg')"
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  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "30eb5335",
   "metadata": {},
   "outputs": [],
   "source": [
    "sample_rate = 128\n",
    "seconds = 4\n",
    "num_samples = sample_rate*seconds\n",
    "time_vect = np.linspace(0, seconds, num_samples)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "6a98d846",
   "metadata": {},
   "outputs": [],
   "source": [
    "IP, IF, IA = emd.spectra.frequency_transform(u_t, sample_rate, 'hilbert')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "801bce0d",
   "metadata": {},
   "outputs": [],
   "source": [
    "freq_edges, freq_bins = emd.spectra.define_hist_bins(0.1, 64, 128)\n",
    " #hht = emd.spectra.hilberthuang(IF, IA, freq_edges)\n",
    "\n",
    "freq_range = (0, 64, 128)  # 0 to 64Hz in 128 steps\n",
    "f, hht = emd.spectra.hilberthuang(IF, IA, freq_range, sum_time=False)"
   ]
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  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "51cb82bf",
   "metadata": {},
   "outputs": [],
   "source": [
    "freq_edges, freq_centres = emd.spectra.define_hist_bins(0, 100, 128, 'linear')\n",
    "\n",
    "# Amplitude weighted HHT per IMF\n",
    "f, spec_weighted = emd.spectra.hilberthuang(IF, IA, freq_edges, sum_imfs=False)\n",
    "\n",
    "# Unweighted HHT per IMF - we replace the instantaneous amplitude values with ones\n",
    "f, spec_unweighted = emd.spectra.hilberthuang(IF, np.ones_like(IA), freq_edges, sum_imfs=False)"
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  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "676bdde7",
   "metadata": {},
   "outputs": [
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   "source": [
    "plt.figure(figsize=(10, 4))\n",
    "plt.subplots_adjust(hspace=0.4)\n",
    "plt.subplot(121)\n",
    "plt.plot(freq_centres, spec_unweighted)\n",
    "plt.xticks(np.arange(10)*10)\n",
    "plt.xlim(0, 64)\n",
    "plt.xlabel('Frequency (Hz)')\n",
    "plt.ylabel('Count')\n",
    "plt.title('unweighted\\nHilbert-Huang Transform')\n",
    "\n",
    "plt.subplot(122)\n",
    "plt.plot(freq_centres, spec_weighted)\n",
    "plt.xticks(np.arange(10)*10)\n",
    "plt.xlim(0, 64)\n",
    "plt.xlabel('Frequency (Hz)')\n",
    "plt.ylabel('Power')\n",
    "plt.title('IA-weighted\\nHilbert-Huang Transform')\n",
    "plt.legend(['IMF-1', 'IMF-2', 'IMF-3', 'IMF-4', 'IMF-5', 'IMF-6'], frameon=False)\n",
    "plt.savefig('data_eeg/HHT_freq.svg')"
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  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "1af41cc8",
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   ],
   "source": [
    "fig = plt.figure(figsize=(6, 4))\n",
    "emd.plotting.plot_hilberthuang(hht, time_vect, freq_bins,\n",
    "                               time_lims=(0, 4), freq_lims=(0.1, 32), vmax=400,\n",
    "                               fig=fig, log_y=False)\n",
    "plt.savefig('data_eeg/HHT_spectrum-1.svg')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "c336c254",
   "metadata": {},
   "outputs": [
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   "source": [
    "fig = plt.figure(figsize=(6, 4))\n",
    "emd.plotting.plot_hilberthuang(hht, time_vect, freq_bins,cmap='ocean_r',\n",
    "                               time_lims=(0, 4), freq_lims=(0.1, 32), vmax=200,\n",
    "                               fig=fig, log_y=False)\n",
    "plt.savefig('data_eeg/HHT_spectrum-1_1.svg')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "44b5d4b6",
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   "source": [
    "fig = plt.figure(figsize=(6, 4))\n",
    "emd.plotting.plot_hilberthuang(hht, time_vect, freq_bins, cmap='jet',\n",
    "                               time_lims=(0, 4), freq_lims=(0.1, 32),\n",
    "                               fig=fig, log_y=False)\n",
    "plt.savefig('data_eeg/HHT_spectrum-2.svg')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "0302463b",
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "fig = plt.figure(figsize=(6, 4))\n",
    "emd.plotting.plot_hilberthuang(hht, time_vect, freq_bins, cmap='gist_heat_r',\n",
    "                               time_lims=(0, 4), freq_lims=(0.1, 32),\n",
    "                               fig=fig, log_y=False)\n",
    "plt.savefig('data_eeg/HHT_spectrum-3.svg')"
   ]
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