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gangulymadhura/Time Series Frequency Analysis.ipynb

Last active October 21, 2020 06:38
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Time Series Frequency Analysis.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:13.946251Z",
"start_time": "2020-10-21T06:35:12.482Z"
}
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"\n",
"Attaching package: 'zoo'\n",
"\n",
"The following objects are masked from 'package:base':\n",
"\n",
" as.Date, as.Date.numeric\n",
"\n",
"Loading required package: lattice\n",
"Loading required package: survival\n",
"Loading required package: Formula\n",
"Loading required package: ggplot2\n",
"Registered S3 methods overwritten by 'ggplot2':\n",
" method from \n",
" [.quosures rlang\n",
" c.quosures rlang\n",
" print.quosures rlang\n",
"\n",
"Attaching package: 'Hmisc'\n",
"\n",
"The following objects are masked from 'package:base':\n",
"\n",
" format.pval, units\n",
"\n",
"Warning message:\n",
"\"package 'pracma' was built under R version 3.6.3\""
]
}
],
"source": [
"#library(Hmisc)\n",
"suppressWarnings(library(zoo))\n",
"suppressWarnings(library(Hmisc))\n",
"suppressMessages(library(pracma))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Read data"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.072249Z",
"start_time": "2020-10-21T06:35:12.484Z"
}
},
"outputs": [],
"source": [
"df = read.table(\"airlines_passengers_data.csv\",sep=',',header=1)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.097249Z",
"start_time": "2020-10-21T06:35:12.485Z"
}
},
"outputs": [
{
"data": {
"text/html": [
"<table>\n",
"<thead><tr><th scope=col>Month</th><th scope=col>Passengers</th></tr></thead>\n",
"<tbody>\n",
"\t<tr><td>1949-01</td><td>112 </td></tr>\n",
"\t<tr><td>1949-02</td><td>118 </td></tr>\n",
"\t<tr><td>1949-03</td><td>132 </td></tr>\n",
"\t<tr><td>1949-04</td><td>129 </td></tr>\n",
"\t<tr><td>1949-05</td><td>121 </td></tr>\n",
"\t<tr><td>1949-06</td><td>135 </td></tr>\n",
"</tbody>\n",
"</table>\n"
],
"text/latex": [
"\\begin{tabular}{r|ll}\n",
" Month & Passengers\\\\\n",
"\\hline\n",
"\t 1949-01 & 112 \\\\\n",
"\t 1949-02 & 118 \\\\\n",
"\t 1949-03 & 132 \\\\\n",
"\t 1949-04 & 129 \\\\\n",
"\t 1949-05 & 121 \\\\\n",
"\t 1949-06 & 135 \\\\\n",
"\\end{tabular}\n"
],
"text/markdown": [
"\n",
"| Month | Passengers |\n",
"|---|---|\n",
"| 1949-01 | 112 |\n",
"| 1949-02 | 118 |\n",
"| 1949-03 | 132 |\n",
"| 1949-04 | 129 |\n",
"| 1949-05 | 121 |\n",
"| 1949-06 | 135 |\n",
"\n"
],
"text/plain": [
" Month Passengers\n",
"1 1949-01 112 \n",
"2 1949-02 118 \n",
"3 1949-03 132 \n",
"4 1949-04 129 \n",
"5 1949-05 121 \n",
"6 1949-06 135 "
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"head(df)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.122250Z",
"start_time": "2020-10-21T06:35:12.487Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"df \n",
"\n",
" 2 Variables 144 Observations\n",
"--------------------------------------------------------------------------------\n",
"Month \n",
" n missing distinct \n",
" 144 0 144 \n",
"\n",
"lowest : 1949-01 1949-02 1949-03 1949-04 1949-05\n",
"highest: 1960-08 1960-09 1960-10 1960-11 1960-12\n",
"--------------------------------------------------------------------------------\n",
"Passengers \n",
" n missing distinct Info Mean Gmd .05 .10 \n",
" 144 0 118 1 280.3 135.9 121.6 135.3 \n",
" .25 .50 .75 .90 .95 \n",
" 180.0 265.5 360.5 453.2 488.1 \n",
"\n",
"lowest : 104 112 114 115 118, highest: 535 548 559 606 622\n",
"--------------------------------------------------------------------------------"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"describe(df)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.144290Z",
"start_time": "2020-10-21T06:35:12.489Z"
}
},
"outputs": [],
"source": [
"# create date feature from year and month\n",
"yr <- substr(df$Month,1,4)\n",
"mo <- substr(df$Month,6,7)\n",
"dates <- as.Date(paste(yr,mo,'01',sep=\"-\"))"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.163288Z",
"start_time": "2020-10-21T06:35:12.490Z"
}
},
"outputs": [],
"source": [
"# create zoo object\n",
"df_2 = zoo(df$Passengers,order.by=dates)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.226135Z",
"start_time": "2020-10-21T06:35:12.491Z"
}
},
"outputs": [
{
"data": {
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XOBSMNDDKaAuUCk4SEGU8BcIFJbyGmT\n57eNbP7Cl+ObAtICkVpCLvKYy+SBqkmIlDBxRPIyRtrf7gycTqStnPOrbPKzbMMX5JcCEmOZ\nIhVFXKVGilQckK5yqCb0QKSEsSPSu9F6nipOLNJOLogE3iiI5Hsq1ptc/GsxrUhbuV4kyz1P\n7f6Ou+JySnaHnisqRFoOofsqvNugpVxyIl0KL47FAenSG3ffyAe3d4i0HELPReYVqf7+R435\nRcpPWXGFlG/O/XEHyc7Xcup2qaI0awUzgUjeixgfUpLJ9T19Lc4H9VNAfCRwb8URqbOAUZF2\nzgPLV9x3L4VfCrDNOJF8g1MXKeS4zhEpSXoaWFt5x6xWlIdI8pmdX6SN3L3jHtdIl1s5xTVS\nOkgeQ6TvUroiedVhWpHuu63/2KBtrddu4xQQkRaDf0usB3TO+gWlJ1JNDY/Iv0N5HynbHbmP\nlArWROq76pHmxDJFGpQCTON/kdEI6Jr1C+oVqVe0ISJp949FaOWItBjMidSjRhIiMUQoPRYu\nkn9nydQiXXblwNVbfxxDhFLEnEg9algVaVtdHknWbxJDhFIEkfpq5BVyku29EOkk+944bsim\niD2RevrXvrr1ejrj6nFTipTJvTor9ei16xkiJDJRFyBMCSL11cgrpDyt8xSJI1KKfLVL74CO\nWb+gnlM3Z2O0KdLmeUQqfrehD4YIJYh8vfpHBAX+nr+4CoaI1CdILTDCNdJDjFN/IEOE0iNY\nJG8lhkX1n2naFCnfeXVnP2GIUHIgUk+FvEOK+0iy8/gD2eEpwC42RfJYlDmRJgGRlsKaRBrx\nqVbI9xL6FoFISyEBkXyrPqlI9UE//n91jkjJsHyRvKseSyRx3xr6KstvNqTBUkWS7zcG1CEw\nsTtknxU/aHfJ5C/fue8N5X8ZIiUHIgUupSPk8BytcJVtfu+7KXvfyba8I8upXTLEEem3zHCR\nqiLqfQOjT+1qE/2XbWeRc45I6SA/E94RIZEmRPJY6vCQ7H1Eyrz6P25b2d0RKRkQKSCxK6QY\nP5eX10gHzx/SPz4iECkVZhOpK2ypIr3Hz22LA5LHeLvHwWvT/2cSiLQQFDqREamiGiJUHJaK\np1J4sUekVECkgMQKIQZTgAaIFJBYIcRgCtBgqSLpNzBEghEoiOR9EPGKWqxIx43PSIVRKcAu\n1kTyPbyZE+nITxavGkQKyewI8foT83EpwDDBIvmfpA2KWqxIE/1wFiItA2md9IwIiExfpF3A\ng8YGpgC7LFKkopA5kW5ZwIPGhqUAuyBSWObuEJ6PtGoQKSxzdwgirZoZRWoPW6xIE4FIy2Cp\nIk3QvhAJhiMd034R/oFrEMn/QWODU4BVECkssyMk4EFjQ1OAWSKJ5O/MYkUKedDYwBRgF0QK\ny9wdEvKgsYEpwC6IFJa5OyTkQWMDU4BdECksc3dIyIPGBqYAs0jnjF+Ib+QKRAp60NiwFGCW\nhYo0yVDrsb12QQ8aG5YCrIJIoZkdITxobL2YE2n48IrRMLIBXgRv9VlFGnhvN6BYEIgELyKI\nNOx8bA0inTZ5ftvIRvevkhBpDoIvHRBp1DLrIeXPeJfPPVI1CZHmwJpI4v48KZG2ci7vIfn9\ngP6gFBCLcSJ57TQbIk2BwsiGa/GkPkY2LB9zIonz85bsyxZpJxdESoHgO/6I9JV/eMhWrpfi\nGWOc2i0fCd7ucUTqO/OrzS1XpEvRz3As1vmiVqUckWYhWCRxzvqEOANfB0gvkarCyxUpP2Xl\ns8w3ukMbEGkGJhRJnGW6T92ch5m0RJoGRJqB4JYYR6TOxSKSiRTwzZQiObsNEClnZENCzCZS\nt1/uKjW6jxcuEiMb0mHSUzv/q52v+vicML7rvlyRGNmQDsEdyKEi+Z2kNd5djUiMbEiHSUVy\nNnNEYmRDQowVyX0141ICkRjZkBDTiVQteYhIzvr8iDRnsxnf2cDIhkSQ2v8BAd3zzQ/8Di0h\n1ZGvqYkeH+nH6O5vRjakwsQiBSzIsyJJiTQNiDQDoSL9lESkyUMMpoAvpPHiH+B4o/cD1+cr\nFelvN7YmvSlgUhBpFGNFOvDoy0RYnkjvIt+vczBSpI9H9NotnODWqCdSa4F1iZTJOd/K7bZl\nrN3SQaRRjBSpOKM7Po5GV27ILp0ERJoTBZEuxZMouEZaOuNF6gwNE8m/Gz4hkXaPU7ubbPI/\nRFo6VkTq+cmTljATrWWkSOXfI5UPZOYZsgvHhkjS95MnLWEmWsvY7u9jMbeXcpyQHiY2zcow\nIZK8Z9Ym0jSY2DQrw4JINYcQSQMTm2ZlTCdS/xJXLtLtkEl2uGtW5zsFRCNUpJZyiDQo5Fb+\n6IlkN9UKNVJAPORnwrO88y3PBTaU8BYpoOTkjBBpL9t7ft/q9tc1U0A0pGXKM6AvEpHcIZkU\nZ3W34i/NtbGxbVaFBZHqOq1IpNedswlWxMa2WRWGRCpe/SqBSJ4pIBoaInWEIpI7BJFSApHG\ngUhQgkjjGCVSg5lrBeNApHEgEpQsViQjbWWESBNiZOOsiXlF+jq2+P5oKiLNnwKaSOukX0BP\n5HQi+R+7JgeRoCRUpNZC40SqHxQRSQUjG2dNINI4EAlKbInk2wAQafYU0GQ6kbwXh0jqGNk4\nK0I6pr0C3O8OEMm7Acz7TKQ6iAQFiDSSkSKVx2D9QUJWts56QKSRjBAp253+ECkRfEUSd5nB\nIg282klCpGpk0P58Q6Tl4y2SOMuMEWnITk9CpPvfafccZ7c/XeeuFYxCOme+y4mrBCINDBF5\n26RWpRyR4uMvkvMkTFrmphRp3oeL1dG5RnrY5PPLDX/HSrrdoechMFa2znoIEMnV5hFpQEjo\nNdJ9U/ujC/dTYKxsnfUQIpLj86/F9JRuKRyKmZYy5tTueto/tXgcm/p/J/Ig2bm6lLpdMveP\nhZvZPKvBU6S+HYNIA0OKs7q95zVSJp8eiav7R7zMbJ7VMIFI0nv8akauXKTc+xpJfgI1awVj\naLu26S/nLiC1fz41QCTPaySOSHZZrEhmGClSCI9rpEv1O+FcI1ljApHep3WeO3Ph+zyiSNWT\n/Z5snJ0TC9+oy0Ocsx4f/BZ49X377suF7/OYIuV/h/I+UrY7ch/JFvoiyesFkRRDDKaAOog0\nFkSCfEKRgv7YdcnEPbVjiJBRDIi0cCKKxBAhuyDSWOJ2fzNEKBKhG9BTpP7FytfravZlRJG4\nIRuPwA34XXy4SJ9OhoFVWSoRReoZIjTZL/KvkcAt+LszPMt1FlndHuSIlCSIFBuGCCVJ4Mg1\nRBoNQ4RSRAI3ISKNhiFCKYJI0WFkQ4pIHrYNEWk0iJQis4u0vh04g0inTDanaVOsHXn/F1De\n+Ybz7d8i69uBMUW67iQ75UeGCE0OIkUnokjX0qCD7O/5bSfOY9L69oMugSK1lGwP9llkqMSJ\nEFGkfXHv6FDdib3LZooUUDGRSF5LRKQJQ6q4ahPvajPaKaAk9PQKkcYTXaRzdU7HEKEJCe2C\nRqTxRD2127+GM9z3DBGakNlFWuH+iyjSPXufz4n7gLTGHaEJIsUn6n2kw0ufzHk8WuWO0ASR\n4sPIhgRBpPggUoIoiOT9XmuhFe4/REoQfZFCOtRllfsPkdJDvl59y3e/GfikozXuP0RKD/mZ\n8Czf+SYi9YJI6aEu0vOyx3txa9x9iJQeiDQDiJQe2iK9ehoQyQEipcc0Inn/xBciTRdiMEXC\nqIj0uxREcoFI6TGRSJzauUCk9JhKJO/0a9x9iJQcv1c3/gFt7wbvjFX+dDsiJUeoSB2FRogU\nGpACiJQcyiKF74tV7j1EWgJB2wOR5gCRFkDYRQcizQEiLYBJRepa+HCRVgki2Wf4w47647od\nRaQgEMk+I54a1hPoPNRJ4wXcIJJ5JA/aINI581PSvVRECgGRzDORSL0XXogUAiJZR97/BZRv\nnXGUcxRgX3iBSNaZWSR2hR+IZJzg5oxIs4BIxkGkZYBIxplKJI9bTKGZVw0i2Sb8il8ccz4f\nfJVgV/iBSLZBpIWASLYJFkmcsx4ffBVhV/iBSLYJHvGGSPOASLZBpIWASLaZWyT2hCeIZBr5\nmfCO6IlDJF0QyTSItBQQyTSjReoK9FkeIgWASKZBpKWASKYJFumnHCLFAZFMg0hLAZEsIy1T\nvhHuQK/lCXvCG0SyjLROekY44xBJGUSyzOwisSN8QSTLINJiQCTLKIjUEYhIyiCSZRBpMSCS\nYaRj2i/CFei7NHaEL4hkGERaDohkGERaDogUl6A1m0QkCVlaujtCG0SKS8iaSedMyOJHiLTO\nxyoPA5GiEtQyxTHnFdL2loQMoUt2P+iDSFGZQaSfxSDSFCBSVEJE+jmUhMe0vCchQ+iS3Q/6\nIFJMgkaBDhCpv4dOgmqR6n6YAESKiRWRfCuR6n6YAESKSP/ViXzO/Vp72zwSuN+s+uwS3b5z\ngkgR8RDp41KASJ3y/bz5XHSi23dOECkiXiK9/p9QpFS375wgUjw8boU6RHJEitMPRIoBIsXD\nW6T2TomhItXeTnO7mgCR4jGdSO5FI1IEECka7X0ILUXy9h7qPpE6P0ekCCBSNDyuTz4itZRy\nieS8Q+XuiwAVECkaISK1lXKK4rrJikgRQKRYyNerq0zYh303WREpAogUi9EiuTsTBt9mAhUQ\nKRYTizQ2NYwDkSLhdViYVqQEt6odECkSiJQ2iBQJREobRIoEIqUNIsXBb3QBIi0WRIqDtE52\nFwr5uH9reYxOgnEgUhw0ROr6HJEMgEhRkI7p7lIBn3uKlNxGNQUiRQGRUgeRojClSB4bC5Em\nB5GigEipg0hRmFmkskxyG9UUiBQD6ZzpLub9OSJZIKpIf8edFOwOf1OlsImSSO0FEMkCEUW6\nb+TDdpIUVhHHnM8HzgKIZIGIIh0kO1/Lqdslk8MUKaxiQaTUtqkxIoqUyfU9fZVsihRW8RPJ\nd4RCcBgiTU9EkRq/KuD++enUdvrsIuU8DnZiOCLFAJGSJ+410uVWTnGN5FHKuwQiWSBm9/e2\n1mu3uU+SwibinO19v6cIIlkg7n2kQ3kfKdsd13Uf6Xt1Bg8+RSSzMLIhApOK5LeteLbYxCBS\nBCYSSXyjAorBQBgiFIFpROp4RKZnHUAXhghF4Gd1Bo8+/RSR1z1WRDIBQ4QiMIFI0vwP5oYb\nshHwE8mz9632gkh2sDNESOoMTGGT37VBpOTgiDSAQNk9+6zDRSpfjG+rtcAQoQEEHgn0RZLa\nnPFttRYYIjSAoH5nRFoFDBEKp3l65V3e8Zb/uSIiGYWRDeEoiNR2ZzVgWfVecNvbajUgUjjK\nIoWtbPP6DJGsEFOk+15ke3kuxLkU241Dvl59y7e/F7iu3yLZ3lTrIeYQoawaaFctJHWRekwZ\nJVL9cGZ7U62HqN3fp4dNp6wcZrd2kb4G+4QlRyR7RL0hW77css1t0SJ5HUykz5TQ88N6HCLZ\nI6JIr31+324RCZESI6JIG3ndhN1sVyBSjypDRfruqDO9qVZERJFOsn9O3WSbhEiOevaKFNyJ\n3rLkQfEwCTG7vw9vey499/FNtw4fkeonYOoimd46ayXqDdnr7jV126cvUo8q4l6CI73prbNW\noopkKcVgvM6raiJ1lUGkpECkUKRz5vttRFoRiBSKv0iNE7yOQoiUCIgUSqBI3b0qX3eE/Ctg\neeusFkQKJUCk3Nk5OVikATEwNYgUiodIPYMaah8iUiogUiDinHW82VoKkVIBkQKZXySwCCL1\n4zqZa6upb+3pf0sIROrHdQsWkaAEkfoJFMm78oiUEIjUT60P+6diiAQliNSL+2/0ft9CpDWC\nSL182rvXBVFA3U2tJowCkXpBJOgHkXoR50htn248x5IhERCpl+lEgnRYu0gev17/EqlnXF3r\nLKyF1YvUm+1domek9+8MrAhE6kv3KtFVyHW7FlbDykVyXPw0iiASuEGkvnwBIuHRekGkvnxP\nkTqLOMYPwXpApL58z7M/RAIX6xbJNYiu+YmjRh4LgeRBpJ6E/iLh0ZpBpJ6E/VVBJECk3oyI\nBD4gUk9Gj6p49FhA6qxaJJ97qYgEPiCSO6VPTRAJEKknpb9IeLRqEMmd0qsm/SNfIXUQyZ0S\nkcCLNYskjjn3u22FEGnVIJI7JyKBF4jkzOlZEcdfWcA6QCRnTkQCP1YskjhngyqCSKsHkZw5\nfSvCrw+vHURy5fSuByKtnXREEo+fqHMmaYlFJPAkIZFCA/tF8l9akMGQIMmIFD5ytFekMSR3\nagoAAAlMSURBVFbCylivSP3eIBJ4g0idb+AG+INIXW/gEQSQikjyMxGcw2foHUA7iNT+Dh5B\nEOsSya2KuD4EcJGcSM5YcbritwyAFhIRyfNgIk5XEAkGsyqRxH0KiEgwGERqi0QkCCQ9kRzB\nUvvUJRIeQShpiCSdMz/FHCI5PwNwsSaRpP4xIoEmCYrUGS0//3csBpEglCRE8hwlVz8WIRKo\nslSRXCdziATRWaxIjkEKHdGNTjlEAlUWKlLpQme7bw/vF8l1/QTgYsEidXfA9YjUuDHbvVgA\nfxCpc7EA/ixapK5+A+cFUDntvIxCJAhmmSI1ro6co+ba3+v89azuHj0AF8sWqavfoE8kdw85\nIkEwyxepV5q29xAJdFm4SHn7bwV7HqQ6SiESBLNIkRr3Ynt7FrwWWSuGRxDO0kXq7er2W2K9\nHCJBOIj0Uw6RIJzFi+RXwrvOiASDWKJIPhUQ56wzEJEgHET6CUQkCCd5kUI7tBEJhrBAkbzy\nf0TqGPzQGYhHMIDlieR/Q+j5EvZkWQ5IMITFiRTQj/1+QSSYmlWIpJUboIuoIv0dd1KwO/wN\nTRFyQyikOMAoIop038iH7bAUQd1vYeUBRhBRpINk52s5dbtkcpgixc8yEAniEFGkTK7v6atk\nU6T4XgYeQSQiitTohXZ3SSMSLIx0j0iMP4WIxL1GutzKqSjXSAz2gYjE7P7e1nrtNvdJUjQX\ngkgQi7j3kQ7lfaRsdxx8HykARIJ4LG5kQ/SlAHiASAAKLG2IEIBJFjZECMAm6Q4RAohIwjdk\nAeJhZ4iQ1BmYAmAmOCIBKJDwECGAeCQ8RAggHgkPEQKIR8ojGwCigUgACiASgAKIBKAAIgEo\nEHVkg/fgBUSChRFRpBMiQbLEPLW7Zu4/nlBIATAPUa+Rru6BQRopAGYhbmfDqTZudaIUAHNA\nrx2AAogEoAAiAShgVCSAhTGgleuLMzExaxwxV5qpUl0ta9mHQDNYUKpUV8ta9iHQDBaUKtXV\nspZ9CDSDBaVKdbWsZR8CzWBBqVJdLWvZh0AzWFCqVFfLWvYh0AwWlCrV1bKWfQg0gwWlSnW1\nrGUfAs1gQalSXS1r2YdAM1hQqlRXy1r2IdAMFpQq1dWyln0INIMFpUp1taxlB0gERAJQAJEA\nFEAkAAUQCUABRAJQAJEAFEAkAAUQCUABRAJQAJEAFEAkAAUQCUABRAJQAJEAFEAkAAXMi3R6\n1fCQyfZSTtV/6Pzxbna4x8l12ujl6kn14E9r1/Skuu5F9rcoue6ae6slVX1dVBuGB9ZFur5a\n1rbcHcfqrfeuqd7dRMl1KKcylZ3Tk+rBPVPaNT2pLopr1ZPrllW5VKxtSVVfF9WG4YNxka7Z\n62gg23t+3xdP27zK7vXxn2TXosxfhFxX2d+Lz/bTpyrYDXm2yIBU2WML3ne+zwMelWtfZjlM\ntgVr66LaMLywLdJjKz2317bcJrdiI52qr5+CgxTH9PPnjQlz7aoPNdp3X6q8WCcdkfpSncvG\nfZcsQi6ZeAvW1kWzYfhhW6THlmlufdkWG/H0+nwnxWnC13f5RLlexRS2WH+q27uhTJxq7/lE\nbY1cz5NVDWlbU9XWRbNheNYoWqYhXL+/xoqXnVz2jwvJr3cnz1VxL/bY9Km2ctMRqS/VRvJj\nVp60Tp/r+Dy1UzhMtKaqrYtmw/DDtkj5e1tsyq+Yv2rXlGxz9e3lzFVxkkuEVEc56zWCni1Y\nzmic2fXmyk9Fb0P2fYxXS1VbF0T64bktjrK759dttb3ORU9qccowjUjtuUpumdLJgjNVeUqi\nLVLXFiwu0PdaFxPuLXj8dLBNk+q9Loj0w2tblF2nta6se9G3OY1I7bnKiUzhxK4/1abowdUW\nqWsLFtcVN62OYmeuU3Fq92joOoek31S1dUGkH17b4rEDsmN9yxST2TQiteYq2KrdmHCl2pen\nj+oita7VRF9Frbk2Uly+3JWk/U1VWxflhuFTnWiZBtLYFtfaTqhOwIsT5JtW54wz1yPPZqs1\nAsCZSt5MnkqzU783l660v6lq66LcMHyqEy3TQJ7bKyu/zU7Flqkmy410LL+7Lzq3E3tyPdJo\nndf1pJpGJNcWvGmtmjNXdZjQuWfVlqq2LsoNw6c60TIN5Lm9yvvhf5viyvVQnmqXt9yUb2A7\nc6k1tv5U9RITp3pcUZQDA84Rcj0m7883JklVWxdGNvzw3F73apzW7jP5vA3S7J2eMNd+isNE\nx2rVSkyd6hhvCz4HwOke/Gqp6uui2zB8qhMv1TBe7en2aMi76tu6GES8Ob0nM7XjtzPXNBcu\n7atVLzF1qss21hZ8DsmeLlVtXXQbhk91IuYCSBZEAlAAkQAUQCQABRAJQAFEAlAAkQAUQCQA\nBRAJQAFEAlAAkQAUQCQABRAJQAFEAlAAkQAUQCQABRAJQAFEAlAAkQAUQCQABRAJQAFEAlAA\nkQAUQCQABRAJQAFEAlAAkQAUQCQABRAJQAFEAlAAkQAUQCQABRAJQAFEsky2O1XPUb+ddt2P\nMC4fV6f2gD8YBJvfMiLFo4bz6gG2XYU2UhWNVCdohc1vGZFNdSDKNt2iCCIZgM1vGZGDXB+v\n18crIpmGzW8ZkYsUDwQ/yfkpymnzfsj6bSfZ8fm09VKkQzlfPttbtpf5ar1KEMkyInfZPV53\ncqtE2pbabMuPsmLy+BFpV0ycCuvkOQXxQCTLPAwpuxIkq07dzpJd82sm5+Kj7f3hzOZzavea\nz4qzwXMxBfFAJMuU52t/+Z/sK112UpyxXYpDkjzeryR6ifSZ57QuPohkmYcX58fZ2/FxCCp1\neV4o1fVpdjYU/x8eZ3nX61xVXiuIZJmHF7fH4WcrtwCR8mNx9ZTdZqrzSkEkyxReZHJ/XCKF\niPQ4+TtsuEaKCyJZpvBiL4dieEPjGmnXI1JjCmLA5rZMYcP5cZ52zlt67V4FRG55fX5TfEyv\nXWQQyTKFFzcpTfm9j/QqsJH3qd/bPKk68SAaiGSZ0o6s8OR1qnbK3iMbXv//bZoiVSMb8Cgu\niASgACIBKIBIAAogEoACiASgACIBKIBIAAogEoACiASgACIBKIBIAAogEoACiASgACIBKIBI\nAAogEoACiASgACIBKIBIAAogEoACiASgACIBKIBIAAogEoACiASgACIBKIBIAAogEoACiASg\nACIBKIBIAAr8A1RwnsHJX+HwAAAAAElFTkSuQmCC",
"text/plain": [
"Plot with title \"Airlines passengers across years\""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# plot time series\n",
"plot(df_2,xlab=\"Months\",ylab=\"# Passengers\",main=\"Airlines passengers across years\")"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.246174Z",
"start_time": "2020-10-21T06:35:12.492Z"
}
},
"outputs": [],
"source": [
"# take log to remove increasing variance in seasonality\n",
"df_2_log = zoo(log(df$Passengers),order.by=dates)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.310134Z",
"start_time": "2020-10-21T06:35:12.493Z"
}
},
"outputs": [
{
"data": {
"image/png": 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OApQAFE8ss8OyTCFKAAIvllnh0SYQpQ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\nIBKAAv8B/1HA3iYuruYAAAAASUVORK5CYII=",
"text/plain": [
"Plot with title \"Airlines passengers across years\""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# plot time series\n",
"plot(df_2_log,xlab=\"Months\",ylab=\"# Passengers\",main=\"Airlines passengers across years\")"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.339136Z",
"start_time": "2020-10-21T06:35:12.494Z"
}
},
"outputs": [],
"source": [
"# par(mfrow=c(2,2))\n",
"# plot(df_2_log[1:12],xlab=\"Months\",ylab=\"# Passengers\",main=min(yr[1:12]))\n",
"# plot(df_2_log[13:24],xlab=\"Months\",ylab=\"# Passengers\",main=min(yr[13:24]))\n",
"# plot(df_2_log[25:36],xlab=\"Months\",ylab=\"# Passengers\",main=min(yr[25:36]))\n",
"# plot(df_2_log[37:48],xlab=\"Months\",ylab=\"# Passengers\",main=min(yr[37:48]))"
]
},
{
"cell_type": "markdown",
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-20T15:11:47.923619Z",
"start_time": "2020-10-20T15:11:47.905Z"
}
},
"source": [
"### Calculating Periodogram by hand"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.363137Z",
"start_time": "2020-10-21T06:35:12.496Z"
}
},
"outputs": [
{
"data": {
"text/html": [
"<table>\n",
"<thead><tr><th scope=col>Passengers</th></tr></thead>\n",
"<tbody>\n",
"\t<tr><td>-0.105217793</td></tr>\n",
"\t<tr><td>-0.063080421</td></tr>\n",
"\t<tr><td> 0.038988495</td></tr>\n",
"\t<tr><td> 0.005950595</td></tr>\n",
"\t<tr><td>-0.068119645</td></tr>\n",
"\t<tr><td> 0.031316206</td></tr>\n",
"</tbody>\n",
"</table>\n"
],
"text/latex": [
"\\begin{tabular}{r|l}\n",
" Passengers\\\\\n",
"\\hline\n",
"\t -0.105217793\\\\\n",
"\t -0.063080421\\\\\n",
"\t 0.038988495\\\\\n",
"\t 0.005950595\\\\\n",
"\t -0.068119645\\\\\n",
"\t 0.031316206\\\\\n",
"\\end{tabular}\n"
],
"text/markdown": [
"\n",
"| Passengers |\n",
"|---|\n",
"| -0.105217793 |\n",
"| -0.063080421 |\n",
"| 0.038988495 |\n",
"| 0.005950595 |\n",
"| -0.068119645 |\n",
"| 0.031316206 |\n",
"\n"
],
"text/plain": [
" Passengers \n",
"1 -0.105217793\n",
"2 -0.063080421\n",
"3 0.038988495\n",
"4 0.005950595\n",
"5 -0.068119645\n",
"6 0.031316206"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# remove trend from data\n",
"df_detrend = data.frame(\"Passengers\"=detrend(log(df$Passengers), 'linear'))\n",
"head(df_detrend)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.384144Z",
"start_time": "2020-10-21T06:35:12.497Z"
}
},
"outputs": [],
"source": [
"# create zoo object\n",
"df_detrend = zoo(df_detrend$Passengers,order.by=dates)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:14.444138Z",
"start_time": "2020-10-21T06:35:12.498Z"
}
},
"outputs": [
{
"data": {
"image/png": 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QGREGlQgkiqNQxvI1Jg1npECt5FRIpaAyIh0rALkVRrQCTFFi4nUtAa3kWS6IdI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",
"text/plain": [
"Plot with title \"\""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# plot detrended time series\n",
"plot(df_detrend)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Fourier transform"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:37.740809Z",
"start_time": "2020-10-21T06:35:37.716Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
" 1949-01-01 1949-02-01 \n",
"-2.042810e-14+0.000000e+00i -3.413459e+00+1.836112e-01i\n",
" 1949-03-01 1949-04-01 \n",
"-1.385319e+00-5.684846e-01i 1.836019e+00+1.359261e+00i\n",
" 1949-05-01 1949-06-01 \n",
" 8.815098e-02+2.633051e-02i 1.121880e-01+7.153653e-01i\n",
" 1949-07-01 1949-08-01 \n",
" 6.001707e-01-9.846591e-01i 2.247522e-02-5.483008e-01i\n",
" 1949-09-01 1949-10-01 \n",
" 2.532679e-01+4.666718e-01i 1.001089e-01-3.796122e-01i"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# get fourier transform of time series\n",
"fft_df = fft(df_detrend)\n",
"fft_df[1:10]\n",
"n = length(fft_df)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:38.234997Z",
"start_time": "2020-10-21T06:35:38.213Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1] \"2nd Fourier coefficient \" \"-3.41345865984196+0.18361121180606i\"\n",
"[3] \"144th Fourier coefficient \" \"-3.41345865984196-0.18361121180607i\"\n",
"[1] \"3rd Fourier coefficient \" \"-1.38531925117481-0.56848455637024i\"\n",
"[3] \"143th Fourier coefficient \" \"-1.38531925117481+0.56848455637024i\"\n"
]
}
],
"source": [
"# FFT values are symmetric\n",
"print(paste(c(\"2nd Fourier coefficient \" ,as.character(fft_df[2]) ,\"144th Fourier coefficient \" ,as.character(fft_df[n]))))\n",
"print(paste(c(\"3rd Fourier coefficient \" ,as.character(fft_df[3]) ,\"143th Fourier coefficient \" ,as.character(fft_df[n-1]))))\n"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:39.096653Z",
"start_time": "2020-10-21T06:35:39.077Z"
}
},
"outputs": [],
"source": [
"# FFT values are symmetric hence we only consider the fourier coffecicients till (n+1)/2 and multiply by 2\n",
"\n",
"periodogram = (abs(fft_df)^2)/n\n",
"periodogram = 2*periodogram[1:((n+1)/2)]"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:35:41.868362Z",
"start_time": "2020-10-21T06:35:41.811Z"
}
},
"outputs": [
{
"data": {
"image/png": 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fTiz2v9conn0/Wk49cLM5WQmvh6D2idTkf0qZLzl5FDWn1cYnv++8Pp9ldK+29X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\ngg==",
"text/plain": [
"plot without title"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# plot periodogram\n",
"spec_freq =(0:((n+1)/2-1))/n\n",
"spec_perdiodogram = periodogram\n",
"plot(spec_freq, spec_perdiodogram, xlab=\"Cycles per year\", ylab=\"Spectral Density\", type=\"l\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The data shows spikes at 1 cycle per year and 2 cycle per year, which can be interpreted as an annual cycle that repeats once every year and there is a 6 month cycle that repeats twice each year."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Spectral Analysis\n",
"\n",
"**Spectrum** function in R removes any linear trend in data and estimates the strength at each frequency also known as the periodogram.\n",
"\n",
"Documenatation :\n",
"https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/spectrum\n"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:36:06.265229Z",
"start_time": "2020-10-21T06:36:06.197Z"
}
},
"outputs": [
{
"data": {
"image/png": 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JPjLdLHkJeaT0+22k1+fd2FkHgwdR4j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",
"text/plain": [
"Plot with title \"Periodogram\""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"density_spectrum <- spectrum(log(df$Passengers),log='no',main=\"Periodogram\",taper=0)\n",
"n = length(df$Passengers)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The frequency is number of cycles per observations, this is equivalent to cycles per month in this case as data in months.\n",
"\n",
"We can convert this to cycles per quarter or cycles per year as well."
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"ExecuteTime": {
"end_time": "2020-10-21T06:36:10.412441Z",
"start_time": "2020-10-21T06:36:10.356Z"
}
},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
"plot without title"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# frequency as cycles per year\n",
"sampling_freq = 12\n",
"spec_freq = density_spectrum$freq * sampling_freq\n",
"spec_perdiodogram = 2 * density_spectrum$spec\n",
"plot(spec_freq, spec_perdiodogram, xlab=\"Cycles per year\", ylab=\"Periodogram\", type=\"l\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "R",
"language": "R",
"name": "ir"
},
"language_info": {
"codemirror_mode": "r",
"file_extension": ".r",
"mimetype": "text/x-r-source",
"name": "R",
"pygments_lexer": "r",
"version": "3.6.1"
},
"toc": {
"base_numbering": 1,
"nav_menu": {},
"number_sections": true,
"sideBar": true,
"skip_h1_title": false,
"title_cell": "Table of Contents",
"title_sidebar": "Contents",
"toc_cell": false,
"toc_position": {
"height": "calc(100% - 180px)",
"left": "10px",
"top": "150px",
"width": "384px"
},
"toc_section_display": true,
"toc_window_display": true
}
},
"nbformat": 4,
"nbformat_minor": 4
}
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