Exercício sobre série temporal

serie_temporal.ex.ipynb

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   "source": [
    "# Exercício sobre série temporal"
   ]
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   "source": [
    "Neste exercício, vamos testar o que vimos até agora em aula. Para isso usaremos os seguintes dados:"
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       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>t</th>\n",
       "      <th>y0</th>\n",
       "      <th>y1</th>\n",
       "      <th>y2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.2</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-0.01</td>\n",
       "      <td>-0.04</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.4</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-0.01</td>\n",
       "      <td>0.09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.6</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-0.01</td>\n",
       "      <td>-0.03</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.8</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.01</td>\n",
       "      <td>-0.02</td>\n",
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      "text/plain": [
       "     t   y0    y1    y2\n",
       "0  0.0  0.0  0.00  0.06\n",
       "1  0.2  0.0 -0.01 -0.04\n",
       "2  0.4  0.0 -0.01  0.09\n",
       "3  0.6  0.0 -0.01 -0.03\n",
       "4  0.8  0.0  0.01 -0.02"
      ]
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   "source": [
    "import pandas as pd\n",
    "\n",
    "data = pd.read_csv('http://pastebin.com/raw/jmSftLRe')\n",
    "data.columns = ['t', 'y0', 'y1', 'y2']\n",
    "data.head()"
   ]
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   "source": [
    "A coluna `t` é o tempo em segundos que a media foi feita. A coluna `y0` é a \n",
    "medida da intensidade do sinal e `y'`e `y2` é o sinal com ruído."
   ]
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   "source": [
    "Exercício:\n",
    "\n",
    "1. Faça um gráfico para cada série temporal\n",
    "2. Calcule a frequência máxima que é possível obter para este sinal. \n",
    "   A resposta deve ser em Hertz.\n",
    "3. Calcule e plote a transformada de Fourier dos sinais.\n",
    "4. Calcule a densidade espectral (PSD).\n",
    "\n",
    "Para os items 3 e 4 talvez seja necessário escrever uma função para calcular o espectro e outra para o PSD.\n",
    "O pacote `numpy` fornece diversas ferramentas para Transformada de Fourier neste [link][1]. Também é\n",
    "interessante olhar a biblioetca [`thinkdsp`][2] que utilizamos em aula.\n",
    "\n",
    "A equação da PSD é: \n",
    "\n",
    "$PSD = |H(f)|^2 + |H(-f)|^2$\n",
    "\n",
    "[1]: http://docs.scipy.org/doc/numpy/reference/routines.fft.html\n",
    "[2]: https://github.com/AllenDowney/ThinkDSP/blob/master/code/thinkdsp.py"
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