{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"# https://cran.r-project.org/web/views/TimeSeries.html\n",
"# https://www.datacamp.com/tracks/time-series-with-r\n",
"# https://www.datascience.com/blog/introduction-to-forecasting-with-arima-in-r-learn-data-science-tutorials\n",
"# https://www.otexts.org/fpp/8/7"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"gdata: read.xls support for 'XLS' (Excel 97-2004) files ENABLED.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"gdata: read.xls support for 'XLSX' (Excel 2007+) files ENABLED.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"\n",
"Attaching package: ‘gdata’\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"The following object is masked from ‘package:stats’:\n",
"\n",
" nobs\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"The following object is masked from ‘package:utils’:\n",
"\n",
" object.size\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"The following object is masked from ‘package:base’:\n",
"\n",
" startsWith\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"\n",
"Attaching package: ‘tseries’\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"The following object is masked from ‘package:imputeTS’:\n",
"\n",
" na.remove\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Loading required package: timeDate\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Loading required package: timeSeries\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Loading required package: fBasics\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Loading required package: parallel\n"
]
}
],
"source": [
"library(gdata)\n",
"library(MASS)\n",
"library(imputeTS)\n",
"library(tseries)\n",
"library(forecast)\n",
"library(fUnitRoots)\n",
"library(portes)\n",
"library(nortest)\n",
"library(tsoutliers)\n",
"library(boot)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"T <- read.xls(\"Dados.xlsx\", sheet = 1, header = TRUE, stringsAsFactors = FALSE)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"| Ano | Mes | AnoF | AnoF2 | INMET | AGRITEM | ICEA |
|---|
\n",
"\n",
"\t| 1961 | 1 | 22296 | 1961.04 | 26.38 | NA | NA |
\n",
"\t| 1961 | 2 | 22327 | 1961.13 | 26.54 | NA | NA |
\n",
"\t| 1961 | 3 | 22355 | 1961.21 | 26.70 | NA | NA |
\n",
"\t| 1961 | 4 | 22386 | 1961.29 | 26.65 | NA | NA |
\n",
"\t| 1961 | 5 | 22416 | 1961.38 | 25.50 | NA | NA |
\n",
"\t| 1961 | 6 | 22447 | 1961.46 | 22.78 | NA | NA |
\n",
"\n",
"
\n"
]
},
"execution_count": 4,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"head(T)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\t- Ano
\n",
"\t\t- 'integer'
\n",
"\t- Mes
\n",
"\t\t- 'integer'
\n",
"\t- AnoF
\n",
"\t\t- 'integer'
\n",
"\t- AnoF2
\n",
"\t\t- 'numeric'
\n",
"\t- INMET
\n",
"\t\t- 'numeric'
\n",
"\t- AGRITEM
\n",
"\t\t- 'numeric'
\n",
"\t- ICEA
\n",
"\t\t- 'numeric'
\n",
"
\n"
]
},
"execution_count": 5,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"sapply(T, class) # Verificando os tipos"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
" Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec\n",
"1961 26.38 26.54 26.70 26.65 25.50 22.78 23.34 27.72 28.90 27.69 27.37 26.92\n",
"1962 26.43 26.85 27.18 26.13 24.49 22.36 20.55 25.37 27.94 26.43 28.49 26.34\n",
"1963 26.39 26.24 26.34 26.57 24.75 22.97 23.63 26.74 28.01 29.04 28.10 28.49\n",
"1964 27.64 27.46 26.70 27.59 24.57 23.51 22.70 27.69 27.46 26.44 27.01 26.44\n",
"1965 26.57 26.54 25.36 25.90 25.67 25.10 23.96 26.33 27.44 27.19 27.07 27.09\n",
"1966 27.30 26.33 26.91 26.45 25.49 24.68 24.61 24.32 26.83 28.07 28.12 28.44\n",
"1967 27.49 26.15 26.37 25.64 23.66 22.02 22.37 25.11 28.03 27.83 26.88 26.88\n",
"1968 26.19 25.62 25.98 23.72 20.93 21.78 22.03 24.92 24.49 27.01 28.04 26.12\n",
"1969 26.58 26.61 26.72 26.11 24.55 23.03 21.83 23.83 28.04 26.94 27.43 26.94\n",
"1970 27.19 26.33 26.47 26.30 23.58 22.74 21.36 23.56 27.11 27.52 27.77 27.15\n",
"1971 26.57 25.59 26.06 24.88 22.78 20.92 21.88 23.21 27.09 25.90 26.44 26.64\n",
"1972 26.08 25.88 26.57 23.99 24.95 23.25 22.16 24.63 27.19 27.52 27.29 27.08\n",
"1973 27.67 27.06 27.38 26.89 23.39 23.95 21.24 24.42 27.17 27.63 26.71 26.56\n",
"1974 25.98 26.37 25.57 25.00 23.73 24.41 22.54 26.04 26.16 27.24 27.50 26.26\n",
"1975 26.53 26.38 26.21 25.93 23.52 23.19 22.45 25.38 27.84 27.24 26.13 26.35\n",
"1976 26.88 26.04 25.33 25.24 24.11 21.74 22.97 25.44 25.35 26.87 26.07 26.28\n",
"1977 26.11 25.70 26.82 25.03 22.96 23.62 24.74 25.04 26.11 26.25 26.68 26.50\n",
"1978 26.48 27.14 26.60 26.32 24.31 23.37 25.09 22.79 26.26 27.46 26.83 26.66\n",
"1979 26.40 26.68 26.36 25.35 24.63 21.82 23.73 26.23 26.25 28.34 27.09 26.83\n",
"1980 26.86 26.25 26.78 26.42 25.17 23.35 22.80 25.85 25.48 28.22 26.42 26.36\n",
"1981 26.45 26.18 26.54 26.55 25.73 21.87 20.75 25.28 25.04 27.38 27.20 26.72\n",
"1982 26.33 26.32 26.05 25.79 24.27 24.15 24.47 25.24 25.64 27.46 27.22 26.80\n",
"1983 26.95 27.51 26.80 27.19 26.12 22.08 23.04 22.93 25.46 25.75 25.62 26.22\n",
"1984 26.74 26.20 26.60 25.41 25.17 22.72 23.56 23.40 25.84 27.88 27.08 26.35\n",
"1985 26.06 27.13 26.84 26.27 25.94 21.97 23.07 24.20 26.41 27.72 27.48 28.41\n",
"1986 27.16 26.57 26.62 27.18 25.59 23.20 23.31 25.89 24.79 26.30 28.51 27.21\n",
"1987 27.16 26.61 26.14 27.17 24.59 23.36 24.90 23.86 27.65 28.63 28.00 27.12\n",
"1988 27.15 26.64 27.27 26.81 24.46 22.90 20.86 25.18 27.40 28.58 27.71 27.02\n",
"1989 26.08 26.37 26.22 26.54 23.89 24.44 21.86 25.10 25.16 27.68 27.88 26.73\n",
"1990 NA NA NA NA NA NA NA NA NA NA NA NA\n",
"1991 NA NA NA NA NA NA NA NA NA NA NA NA\n",
"1992 NA NA NA NA NA NA NA NA NA NA NA NA\n",
"1993 NA NA NA NA NA NA NA NA NA NA NA NA\n",
"1994 NA NA NA NA NA NA NA NA NA NA NA NA\n",
"1995 NA 26.28 27.09 25.52 24.72 23.77 24.57 25.01 28.15 28.25 27.41 28.52\n",
"1996 NA NA NA NA NA NA NA NA NA NA NA NA\n",
"1997 NA NA NA NA NA NA NA NA NA NA NA NA\n",
"1998 28.52 27.86 27.97 27.74 23.92 23.62 25.02 26.11 26.30 28.04 27.95 27.23\n",
"1999 27.17 27.44 26.28 26.03 24.68 24.35 23.53 24.88 28.32 28.29 26.97 27.55\n",
"2000 27.76 26.99 26.45 26.59 25.23 24.40 21.90 26.84 26.73 28.22 27.34 27.52\n",
"2001 26.92 27.41 26.58 27.34 24.24 22.96 24.82 26.88 27.92 27.73 27.16 26.38\n",
"2002 27.40 26.46 27.01 27.12 25.91 23.34 24.05 27.20 26.83 29.34 28.65 27.90\n",
"2003 27.01 26.67 23.35 25.97 24.88 24.13 23.18 24.33 27.20 27.50 27.23 28.08\n",
"2004 27.22 27.05 27.70 27.02 23.34 23.74 23.48 25.26 27.54 28.25 27.20 27.83\n",
"2005 NA NA NA NA NA NA 23.85 26.28 25.79 28.26 27.87 27.73\n",
"2006 27.22 27.06 26.94 26.13 22.97 24.81 NA 27.22 26.49 27.46 28.41 27.31\n",
"2007 27.61 26.89 27.43 27.44 23.61 23.89 23.45 24.26 28.50 28.06 27.18 27.16\n",
"2008 26.42 26.92 26.80 NA NA NA NA NA NA NA NA NA\n",
"2009 NA NA NA NA 25.52 22.62 23.90 25.66 27.08 28.72 28.24 NA\n",
"2010 26.91 27.48 27.91 26.58 23.66 24.71 22.82 24.73 29.37 28.71 27.04 27.64\n",
"2011 26.82 26.82 26.75 27.46 24.76 24.01 24.27 26.02 28.96 28.57 28.57 27.38\n",
"2012 26.61 26.71 26.69 27.55 25.20 24.25 23.52 25.82 28.63 29.37 NA NA\n",
"2013 NA NA 27.48 26.71 25.62 26.09 23.98 24.22 28.38 28.26 28.05 27.40\n",
"2014 26.48 26.35 27.14 27.05 25.23 24.56 23.31 26.45 29.47 29.43 28.53 27.44\n",
"2015 27.55 27.34 27.16 27.07 26.32 24.98 24.52 27.61 29.85 NA NA NA\n",
"2016 NA NA NA 27.11 25.47 22.98 23.93 26.19 27.11 28.04 27.07 27.31\n",
"2017 27.21 27.10 27.38 27.14 27.06 24.13 23.33 27.33 29.08 28.46 28.72 27.63"
]
},
"execution_count": 6,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 6,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"INMET <- ts(T$INMET,start=c(1961,1),f=12)\n",
"INMET\n",
"plot(INMET)"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
" Jan Feb Mar Apr May Jun Jul Aug\n",
"1961 NA NA NA NA NA NA NA NA\n",
"1962 NA NA NA NA NA NA NA NA\n",
"1963 NA NA NA NA 25.39000 24.90000 25.71000 28.59000\n",
"1964 28.44000 28.25000 27.63000 28.76000 25.80000 24.73000 23.76000 29.25000\n",
"1965 27.45000 27.47000 26.37000 26.92000 26.84000 26.65000 25.20000 27.76000\n",
"1966 27.89000 27.09000 27.50000 27.82000 26.68000 26.04000 26.30000 26.13000\n",
"1967 28.59000 27.96000 28.08000 27.01000 27.53000 24.56000 26.27000 29.12000\n",
"1968 27.99000 27.06000 27.69000 26.02000 24.07000 26.19000 26.61000 27.75000\n",
"1969 28.37000 28.02000 28.78000 28.54000 27.77000 25.76000 26.32000 27.01000\n",
"1970 28.83000 27.50000 28.95000 28.41000 26.15000 26.52000 25.37000 27.75000\n",
"1971 27.90000 27.16000 27.80000 27.08000 25.06000 24.12000 25.81000 27.12000\n",
"1972 27.83000 27.38000 28.06000 26.61000 28.48000 27.10000 25.48000 26.78000\n",
"1973 29.48000 28.66000 29.40000 29.71000 26.38000 27.23000 24.35000 26.03000\n",
"1974 27.57000 27.64000 26.67000 26.18000 25.63000 26.20000 24.87000 27.36000\n",
"1975 27.41000 27.65000 27.46000 27.13000 24.94000 24.98000 23.88000 27.09000\n",
"1976 27.58000 26.51000 26.11000 26.57000 25.58000 23.42000 24.86000 27.08000\n",
"1977 27.28000 26.96000 28.42000 26.46000 24.61000 24.88000 27.47000 26.89000\n",
"1978 27.74000 27.25000 28.41000 27.46000 25.93000 25.87000 27.83000 24.33000\n",
"1979 27.56000 27.51000 27.63000 26.23000 26.33000 24.14000 26.12000 29.42000\n",
"1980 27.05000 26.63000 27.90000 28.25000 26.97000 24.00000 24.78000 27.27000\n",
"1981 27.49000 27.18000 27.42000 27.94000 27.69000 23.35000 22.28000 27.55000\n",
"1982 27.11000 27.03000 26.90000 26.70000 23.84000 26.36000 27.43000 27.20000\n",
"1983 27.34000 27.99000 27.35000 27.61000 26.97000 22.18000 23.70000 24.23000\n",
"1984 27.25000 26.59000 26.82000 25.39000 25.77000 23.65000 24.81000 24.32000\n",
"1985 26.44000 27.41000 27.36000 26.59000 26.61000 23.16000 24.15000 24.63000\n",
"1986 27.54000 26.93000 26.84000 27.79000 26.14000 24.28000 23.98000 26.29000\n",
"1987 27.77000 27.01000 26.55000 27.70000 25.22000 23.88000 26.45000 24.84000\n",
"1988 27.43000 26.61000 27.48000 26.92000 24.42000 23.44000 21.90000 26.83000\n",
"1989 26.51000 26.80000 26.52000 27.11000 24.37000 25.31000 22.94000 26.08000\n",
"1990 26.48009 26.07057 26.80931 26.39176 24.50474 23.62948 22.11987 25.52453\n",
"1991 26.39176 26.64872 25.75740 25.78149 25.53256 23.99886 23.71781 25.17925\n",
"1992 26.02239 NA NA NA NA 22.23229 24.48868 24.54489\n",
"1993 26.63266 25.22743 NA NA NA 23.79008 23.29223 NA\n",
"1994 26.69690 26.36767 26.25525 25.84573 25.75740 24.50474 23.66160 25.55665\n",
"1995 25.66104 26.53000 27.63000 NA NA 24.86000 26.28000 26.30000\n",
"1996 NA 26.34358 26.50418 26.21510 25.50044 22.14396 24.07113 27.02612\n",
"1997 NA NA NA NA NA NA NA NA\n",
"1998 NA NA NA NA NA NA NA NA\n",
"1999 NA NA NA NA NA NA NA NA\n",
"2000 NA NA NA NA NA NA NA NA\n",
"2001 NA NA NA NA NA NA 26.89000 29.82000\n",
"2002 29.09000 28.36000 29.24000 29.48000 28.36000 25.65000 26.46000 29.65000\n",
"2003 26.62000 26.20000 26.26000 25.79000 25.07000 24.91000 24.09000 25.26000\n",
"2004 26.58000 27.51000 27.20000 28.85000 23.66000 23.98000 23.17000 25.72000\n",
"2005 25.95012 26.22313 25.73331 25.23546 25.53256 25.85376 23.62000 27.00000\n",
"2006 26.75000 26.54000 26.49000 22.99000 25.93000 26.15000 25.08289 27.08000\n",
"2007 29.58000 28.77000 29.56000 29.82000 24.69000 24.39000 23.42000 24.56000\n",
"2008 26.07000 26.62000 26.49000 24.87411 23.31632 22.80241 24.83397 26.87355\n",
"2009 26.24722 25.80558 25.23546 25.62892 25.44000 22.79000 23.20000 25.97000\n",
"2010 26.95000 27.47000 27.36000 26.55000 23.80000 25.08000 23.09000 25.80000\n",
"2011 26.21000 26.26000 25.91000 26.89000 24.48000 24.30000 24.78000 26.05000\n",
"2012 26.04000 25.98000 25.79000 26.16000 24.56000 23.40000 22.97000 26.00000\n",
"2013 25.35591 25.22743 26.57000 25.38000 24.79000 25.26000 23.49000 24.20000\n",
"2014 25.41000 25.24000 26.04000 25.87000 24.16000 24.11000 23.08000 27.09000\n",
"2015 26.70000 26.22000 26.30000 26.41000 25.67000 24.99000 24.61000 28.43000\n",
"2016 26.01436 26.80128 26.07057 26.97000 25.04000 22.86000 25.48000 23.84000\n",
"2017 26.81000 26.58000 26.92000 26.49000 26.71000 24.27000 23.75000 27.72000\n",
" Sep Oct Nov Dec\n",
"1961 NA NA NA NA\n",
"1962 NA NA NA NA\n",
"1963 29.36000 31.36000 29.51000 29.84000\n",
"1964 28.96000 27.48000 27.70000 27.52000\n",
"1965 28.50000 28.15000 28.12000 27.99000\n",
"1966 28.38000 29.77000 29.56000 29.39000\n",
"1967 30.34000 30.06000 29.00000 28.34000\n",
"1968 26.61000 29.22000 30.11000 27.50000\n",
"1969 30.18000 28.50000 28.84000 28.54000\n",
"1970 29.39000 29.60000 29.92000 29.40000\n",
"1971 29.75000 27.35000 28.14000 28.32000\n",
"1972 29.17000 29.27000 28.38000 28.77000\n",
"1973 28.55000 29.55000 27.99000 27.82000\n",
"1974 27.00000 28.30000 28.78000 27.05000\n",
"1975 29.20000 28.64000 27.39000 27.73000\n",
"1976 26.15000 27.82000 27.04000 27.56000\n",
"1977 27.55000 28.08000 28.42000 27.65000\n",
"1978 28.23000 28.01000 27.30000 27.33000\n",
"1979 27.85000 29.74000 28.15000 27.13000\n",
"1980 27.09000 29.76000 27.61000 26.34000\n",
"1981 26.99000 28.08000 28.04000 26.84000\n",
"1982 26.70000 28.75000 27.92000 27.54000\n",
"1983 25.72000 26.07000 25.85000 26.72000\n",
"1984 26.65000 28.57000 27.64000 26.92000\n",
"1985 26.68000 28.00000 27.78000 28.71000\n",
"1986 25.46000 26.03000 28.91000 27.74000\n",
"1987 27.99000 29.06000 28.24000 27.29000\n",
"1988 28.73000 29.14000 28.11000 27.42000\n",
"1989 26.03000 28.52000 28.44000 26.96000\n",
"1990 24.74564 27.50791 27.06627 26.81734\n",
"1991 27.09839 26.67281 26.63266 26.76113\n",
"1992 26.32752 25.93406 26.55236 26.63266\n",
"1993 26.74507 27.34731 27.46776 26.94582\n",
"1994 27.08233 27.56412 27.58018 26.02239\n",
"1995 28.83000 28.46000 27.64000 27.07000\n",
"1996 26.10268 27.01006 26.39979 26.89764\n",
"1997 NA NA NA NA\n",
"1998 NA NA NA NA\n",
"1999 NA NA NA NA\n",
"2000 NA NA NA NA\n",
"2001 30.18000 29.73000 29.17000 27.94000\n",
"2002 29.16000 29.97000 28.51000 27.49000\n",
"2003 27.30000 27.43000 26.69000 27.51000\n",
"2004 28.51000 28.13000 27.50000 27.33000\n",
"2005 25.32000 27.60000 27.41000 26.87000\n",
"2006 28.55000 28.72000 29.98000 29.05000\n",
"2007 29.44000 28.33000 26.77000 26.45000\n",
"2008 26.31949 26.98597 27.01809 26.01436\n",
"2009 27.05000 28.21000 27.95000 25.87785\n",
"2010 30.05000 28.40000 26.72000 27.37000\n",
"2011 23.40000 NA 27.99000 26.88000\n",
"2012 28.79000 28.62000 25.69316 26.09465\n",
"2013 27.54000 26.79000 26.70000 26.18000\n",
"2014 29.80000 29.77000 28.26000 26.49000\n",
"2015 29.72000 27.77290 27.29914 27.20278\n",
"2016 26.06000 27.70000 26.54000 26.61000\n",
"2017 29.42000 27.99000 28.14000 27.07000"
]
},
"execution_count": 49,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 49,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"ICEA <- ts(T$ICEA,start=c(1961,1),f=12)\n",
"ICEA\n",
"plot(ICEA, cex.lab=1.5, cex.axis=2)\n",
"\n",
"# cex.main: Size of main title\n",
"# cex.lab: Size of axis labels (the text describing the axis)\n",
"# cex.axis: Size of axis text (the values that indicate the axis tick labels)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Observe que as séries possuem falhas. Inicialmente as preencheremos com uma regressão entre INMET e ICEA."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"\n",
"Call:\n",
"lm(formula = INMET ~ ICEA, data = T)\n",
"\n",
"Residuals:\n",
" Min 1Q Median 3Q Max \n",
"-3.7194 -0.6334 -0.0078 0.7152 5.7882 \n",
"\n",
"Coefficients:\n",
" Estimate Std. Error t value Pr(>|t|) \n",
"(Intercept) 4.38188 0.81258 5.393 1.08e-07 ***\n",
"ICEA 0.80299 0.03012 26.659 < 2e-16 ***\n",
"---\n",
"Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
"\n",
"Residual standard error: 1.13 on 493 degrees of freedom\n",
" (189 observations deleted due to missingness)\n",
"Multiple R-squared: 0.5904,\tAdjusted R-squared: 0.5896 \n",
"F-statistic: 710.7 on 1 and 493 DF, p-value: < 2.2e-16\n"
]
},
"execution_count": 8,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"mod1 = lm(INMET ~ ICEA, data = T)\n",
"summary(mod1)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"aux <- T[is.na(T$INMET)&!is.na(T$ICEA),]"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"aux <- predict(mod1, newdata=T[is.na(T$INMET)&!is.na(T$ICEA),]) # Estima INMET com base no ICEA"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\t- 349
\n",
"\t\t- 26.4800887468442
\n",
"\t- 350
\n",
"\t\t- 26.0705653435554
\n",
"\t- 351
\n",
"\t\t- 26.8093134436057
\n",
"\t- 352
\n",
"\t\t- 26.3917601696642
\n",
"\t- 353
\n",
"\t\t- 24.5047405662748
\n",
"\t- 354
\n",
"\t\t- 23.6294846651282
\n",
"
\n"
]
},
"execution_count": 11,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"head(aux)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"T[is.na(T$INMET)&!is.na(T$ICEA),] <- aux"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
" Jan Feb Mar Apr May Jun Jul Aug\n",
"1961 26.38000 26.54000 26.70000 26.65000 25.50000 22.78000 23.34000 27.72000\n",
"1962 26.43000 26.85000 27.18000 26.13000 24.49000 22.36000 20.55000 25.37000\n",
"1963 26.39000 26.24000 26.34000 26.57000 24.75000 22.97000 23.63000 26.74000\n",
"1964 27.64000 27.46000 26.70000 27.59000 24.57000 23.51000 22.70000 27.69000\n",
"1965 26.57000 26.54000 25.36000 25.90000 25.67000 25.10000 23.96000 26.33000\n",
"1966 27.30000 26.33000 26.91000 26.45000 25.49000 24.68000 24.61000 24.32000\n",
"1967 27.49000 26.15000 26.37000 25.64000 23.66000 22.02000 22.37000 25.11000\n",
"1968 26.19000 25.62000 25.98000 23.72000 20.93000 21.78000 22.03000 24.92000\n",
"1969 26.58000 26.61000 26.72000 26.11000 24.55000 23.03000 21.83000 23.83000\n",
"1970 27.19000 26.33000 26.47000 26.30000 23.58000 22.74000 21.36000 23.56000\n",
"1971 26.57000 25.59000 26.06000 24.88000 22.78000 20.92000 21.88000 23.21000\n",
"1972 26.08000 25.88000 26.57000 23.99000 24.95000 23.25000 22.16000 24.63000\n",
"1973 27.67000 27.06000 27.38000 26.89000 23.39000 23.95000 21.24000 24.42000\n",
"1974 25.98000 26.37000 25.57000 25.00000 23.73000 24.41000 22.54000 26.04000\n",
"1975 26.53000 26.38000 26.21000 25.93000 23.52000 23.19000 22.45000 25.38000\n",
"1976 26.88000 26.04000 25.33000 25.24000 24.11000 21.74000 22.97000 25.44000\n",
"1977 26.11000 25.70000 26.82000 25.03000 22.96000 23.62000 24.74000 25.04000\n",
"1978 26.48000 27.14000 26.60000 26.32000 24.31000 23.37000 25.09000 22.79000\n",
"1979 26.40000 26.68000 26.36000 25.35000 24.63000 21.82000 23.73000 26.23000\n",
"1980 26.86000 26.25000 26.78000 26.42000 25.17000 23.35000 22.80000 25.85000\n",
"1981 26.45000 26.18000 26.54000 26.55000 25.73000 21.87000 20.75000 25.28000\n",
"1982 26.33000 26.32000 26.05000 25.79000 24.27000 24.15000 24.47000 25.24000\n",
"1983 26.95000 27.51000 26.80000 27.19000 26.12000 22.08000 23.04000 22.93000\n",
"1984 26.74000 26.20000 26.60000 25.41000 25.17000 22.72000 23.56000 23.40000\n",
"1985 26.06000 27.13000 26.84000 26.27000 25.94000 21.97000 23.07000 24.20000\n",
"1986 27.16000 26.57000 26.62000 27.18000 25.59000 23.20000 23.31000 25.89000\n",
"1987 27.16000 26.61000 26.14000 27.17000 24.59000 23.36000 24.90000 23.86000\n",
"1988 27.15000 26.64000 27.27000 26.81000 24.46000 22.90000 20.86000 25.18000\n",
"1989 26.08000 26.37000 26.22000 26.54000 23.89000 24.44000 21.86000 25.10000\n",
"1990 26.48009 26.07057 26.80931 26.39176 24.50474 23.62948 22.11987 25.52453\n",
"1991 26.39176 26.64872 25.75740 25.78149 25.53256 23.99886 23.71781 25.17925\n",
"1992 26.02239 NA NA NA NA 22.23229 24.48868 24.54489\n",
"1993 26.63266 25.22743 NA NA NA 23.79008 23.29223 NA\n",
"1994 26.69690 26.36767 26.25525 25.84573 25.75740 24.50474 23.66160 25.55665\n",
"1995 25.66104 26.28000 27.09000 25.52000 24.72000 23.77000 24.57000 25.01000\n",
"1996 NA 26.34358 26.50418 26.21510 25.50044 22.14396 24.07113 27.02612\n",
"1997 NA NA NA NA NA NA NA NA\n",
"1998 28.52000 27.86000 27.97000 27.74000 23.92000 23.62000 25.02000 26.11000\n",
"1999 27.17000 27.44000 26.28000 26.03000 24.68000 24.35000 23.53000 24.88000\n",
"2000 27.76000 26.99000 26.45000 26.59000 25.23000 24.40000 21.90000 26.84000\n",
"2001 26.92000 27.41000 26.58000 27.34000 24.24000 22.96000 24.82000 26.88000\n",
"2002 27.40000 26.46000 27.01000 27.12000 25.91000 23.34000 24.05000 27.20000\n",
"2003 27.01000 26.67000 23.35000 25.97000 24.88000 24.13000 23.18000 24.33000\n",
"2004 27.22000 27.05000 27.70000 27.02000 23.34000 23.74000 23.48000 25.26000\n",
"2005 25.95012 26.22313 25.73331 25.23546 25.53256 25.85376 23.85000 26.28000\n",
"2006 27.22000 27.06000 26.94000 26.13000 22.97000 24.81000 25.08289 27.22000\n",
"2007 27.61000 26.89000 27.43000 27.44000 23.61000 23.89000 23.45000 24.26000\n",
"2008 26.42000 26.92000 26.80000 24.87411 23.31632 22.80241 24.83397 26.87355\n",
"2009 26.24722 25.80558 25.23546 25.62892 25.52000 22.62000 23.90000 25.66000\n",
"2010 26.91000 27.48000 27.91000 26.58000 23.66000 24.71000 22.82000 24.73000\n",
"2011 26.82000 26.82000 26.75000 27.46000 24.76000 24.01000 24.27000 26.02000\n",
"2012 26.61000 26.71000 26.69000 27.55000 25.20000 24.25000 23.52000 25.82000\n",
"2013 25.35591 25.22743 27.48000 26.71000 25.62000 26.09000 23.98000 24.22000\n",
"2014 26.48000 26.35000 27.14000 27.05000 25.23000 24.56000 23.31000 26.45000\n",
"2015 27.55000 27.34000 27.16000 27.07000 26.32000 24.98000 24.52000 27.61000\n",
"2016 26.01436 26.80128 26.07057 27.11000 25.47000 22.98000 23.93000 26.19000\n",
"2017 27.21000 27.10000 27.38000 27.14000 27.06000 24.13000 23.33000 27.33000\n",
" Sep Oct Nov Dec\n",
"1961 28.90000 27.69000 27.37000 26.92000\n",
"1962 27.94000 26.43000 28.49000 26.34000\n",
"1963 28.01000 29.04000 28.10000 28.49000\n",
"1964 27.46000 26.44000 27.01000 26.44000\n",
"1965 27.44000 27.19000 27.07000 27.09000\n",
"1966 26.83000 28.07000 28.12000 28.44000\n",
"1967 28.03000 27.83000 26.88000 26.88000\n",
"1968 24.49000 27.01000 28.04000 26.12000\n",
"1969 28.04000 26.94000 27.43000 26.94000\n",
"1970 27.11000 27.52000 27.77000 27.15000\n",
"1971 27.09000 25.90000 26.44000 26.64000\n",
"1972 27.19000 27.52000 27.29000 27.08000\n",
"1973 27.17000 27.63000 26.71000 26.56000\n",
"1974 26.16000 27.24000 27.50000 26.26000\n",
"1975 27.84000 27.24000 26.13000 26.35000\n",
"1976 25.35000 26.87000 26.07000 26.28000\n",
"1977 26.11000 26.25000 26.68000 26.50000\n",
"1978 26.26000 27.46000 26.83000 26.66000\n",
"1979 26.25000 28.34000 27.09000 26.83000\n",
"1980 25.48000 28.22000 26.42000 26.36000\n",
"1981 25.04000 27.38000 27.20000 26.72000\n",
"1982 25.64000 27.46000 27.22000 26.80000\n",
"1983 25.46000 25.75000 25.62000 26.22000\n",
"1984 25.84000 27.88000 27.08000 26.35000\n",
"1985 26.41000 27.72000 27.48000 28.41000\n",
"1986 24.79000 26.30000 28.51000 27.21000\n",
"1987 27.65000 28.63000 28.00000 27.12000\n",
"1988 27.40000 28.58000 27.71000 27.02000\n",
"1989 25.16000 27.68000 27.88000 26.73000\n",
"1990 24.74564 27.50791 27.06627 26.81734\n",
"1991 27.09839 26.67281 26.63266 26.76113\n",
"1992 26.32752 25.93406 26.55236 26.63266\n",
"1993 26.74507 27.34731 27.46776 26.94582\n",
"1994 27.08233 27.56412 27.58018 26.02239\n",
"1995 28.15000 28.25000 27.41000 28.52000\n",
"1996 26.10268 27.01006 26.39979 26.89764\n",
"1997 NA NA NA NA\n",
"1998 26.30000 28.04000 27.95000 27.23000\n",
"1999 28.32000 28.29000 26.97000 27.55000\n",
"2000 26.73000 28.22000 27.34000 27.52000\n",
"2001 27.92000 27.73000 27.16000 26.38000\n",
"2002 26.83000 29.34000 28.65000 27.90000\n",
"2003 27.20000 27.50000 27.23000 28.08000\n",
"2004 27.54000 28.25000 27.20000 27.83000\n",
"2005 25.79000 28.26000 27.87000 27.73000\n",
"2006 26.49000 27.46000 28.41000 27.31000\n",
"2007 28.50000 28.06000 27.18000 27.16000\n",
"2008 26.31949 26.98597 27.01809 26.01436\n",
"2009 27.08000 28.72000 28.24000 25.87785\n",
"2010 29.37000 28.71000 27.04000 27.64000\n",
"2011 28.96000 28.57000 28.57000 27.38000\n",
"2012 28.63000 29.37000 25.69316 26.09465\n",
"2013 28.38000 28.26000 28.05000 27.40000\n",
"2014 29.47000 29.43000 28.53000 27.44000\n",
"2015 29.85000 27.77290 27.29914 27.20278\n",
"2016 27.11000 28.04000 27.07000 27.31000\n",
"2017 29.08000 28.46000 28.72000 27.63000"
]
},
"execution_count": 13,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 13,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"INMET <- ts(T$INMET,start=c(1961,1),f=12)\n",
"INMET\n",
"plot(INMET)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 14,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"# As falhas que sobraram foram preenchidas pelo estimador de Kalman.\n",
"INMET <- na.kalman(INMET, model = \"auto.arima\")\n",
"plot(INMET)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 15,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"# Decomposição da série temporal (aditiva, possivelmente baseada na transformáda rápida de Fourier, e apenas para fins de análise preliminar)\n",
"plot(decompose(INMET))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# ARIMA"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Etapas:\n",
"a) Verificar se existe a necessidade de uma transformação na série original, com objetivo de estabilizar a variância;\n",
"b) Tornar a série estacionária por meio de diferenças, de modo que o processo dZt seja reduzido a um ARMA(p,q)\n",
"c) Identificar o processo ARMA(p,q) resultante.\n",
"d) Verificação da estacionariedade e da invertibilidade.\n",
"\n",
"\n",
"FAC : correlação simples entre Zt e Zt – k em função da defasagem k.\n",
"FACP: correlação entre Zt e Zt – k em função da defasagem k, filtrado o efeito de todas as outras defasagens sobre Zt e Zt – k.\n",
"\n",
"FACP -> AR\n",
"\n",
"D -> I\n",
"\n",
"FAC -> MA\n",
"\n",
"1- Número de AR (auto-regressivo) termos (p): termos AR são apenas defasagens da variável dependente. Por exemplo, se o símbolo p representa 5, os preditores de x (t) irá ser X (t-1) … .x (T-5).\n",
"2- Número de MA (média móvel) termos (q): termos MA estão defasados erros de previsão na equação de predição. Por exemplo, se q é 5, os preditores para x (t) será E (t-1) … .e (t-5) onde e (i) é a diferença entre a média móvel ao valor imediato e real.\n",
"3- Número de Diferenças (d): Estes são o número de diferenças não sazonal, ou seja, neste caso, tomamos a primeira diferença de ordem. Assim, ou nós podemos passar essa variável e colocar d = 0, ou passar a variável original e coloca -d = 1. Ambos irão gerar mesmos resultados."
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Notação: arima(p, d, q), sendo p relacionado a autocorrelação parcial, d a diferença entre os valores, q associado a autocorrelação)."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"1"
]
},
"execution_count": 16,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"ndiffs(INMET) # Valor de d para fazer a série estacionária."
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"\n",
"Title:\n",
" KPSS Unit Root Test\n",
"\n",
"Test Results:\n",
" NA\n",
"\n",
"Description:\n",
" Thu Dec 6 14:05:52 2018 by user: \n"
]
},
"execution_count": 17,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 17,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 17,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"urkpssTest(INMET, type = c(\"tau\"), lags = c(\"short\"),use.lag = NULL, doplot = TRUE)\n",
"tsestacionaria = diff(INMET, differences=1)\n",
"plot(tsestacionaria)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"Warning message in adf.test(INMET, alternative = \"stationary\"):\n",
"“p-value smaller than printed p-value”"
]
},
{
"data": {
"text/plain": [
"\n",
"\tAugmented Dickey-Fuller Test\n",
"\n",
"data: INMET\n",
"Dickey-Fuller = -15.038, Lag order = 8, p-value = 0.01\n",
"alternative hypothesis: stationary\n"
]
},
"execution_count": 18,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"# Teste da estacionariedade da série\n",
"adf.test(INMET, alternative=\"stationary\")"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 19,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 19,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"acf(INMET)\n",
"pacf(INMET)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"O teste não rejeitou a hipótese da estacionariedade, mas o ACF e a função sugerem utilizar d=1."
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 20,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 20,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"# Calculando as correlações para uma diferença de ordem 1\n",
"acf(diff(INMET, differences=1))\n",
"pacf(diff(INMET, differences=1))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Como o ACF cai depois do primeiro lag, podemos partir de p = 1. Para o PACF o q poderia ser igual a 0, temos, então, um ARIMA(p, d, q) = ARIMA (1, 1, 0) com os mesmos parâmetros para sazonalidade."
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"Series: INMET \n",
"ARIMA(1,1,1) \n",
"\n",
"Coefficients:\n",
" ar1 ma1\n",
" -0.2652 0.3130\n",
"s.e. 0.4061 0.3994\n",
"\n",
"sigma^2 estimated as 2.154: log likelihood=-1230.11\n",
"AIC=2466.22 AICc=2466.26 BIC=2479.8"
]
},
"execution_count": 21,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"ajuste_0 <- Arima(INMET, c(1, 1, 1))\n",
"ajuste_0"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"Series: INMET \n",
"ARIMA(1,1,1)(1,1,1)[12] \n",
"\n",
"Coefficients:\n",
" ar1 ma1 sar1 sma1\n",
" 0.1792 -0.9649 0.0360 -0.9129\n",
"s.e. 0.0412 0.0143 0.0479 0.0322\n",
"\n",
"sigma^2 estimated as 0.7475: log likelihood=-864.86\n",
"AIC=1739.73 AICc=1739.82 BIC=1762.27"
]
},
"execution_count": 22,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"ajuste_1 <- Arima(INMET, c(1, 1, 1), seasonal=list(order = c(1, 1, 1), period = 12))\n",
"ajuste_1"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Utilizando o \"automático\""
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
" ARIMA(0,0,0)(0,1,0)[12] : 2101.997\n",
" ARIMA(0,0,0)(0,1,0)[12] with drift : 2103.936\n",
" ARIMA(0,0,0)(0,1,1)[12] : 1777.447\n",
" ARIMA(0,0,0)(0,1,1)[12] with drift : 1776.128\n",
" ARIMA(0,0,0)(0,1,2)[12] : 1779.456\n",
" ARIMA(0,0,0)(0,1,2)[12] with drift : 1777.932\n",
" ARIMA(0,0,0)(1,1,0)[12] : 1916.368\n",
" ARIMA(0,0,0)(1,1,0)[12] with drift : 1918.209\n",
" ARIMA(0,0,0)(1,1,1)[12] : 1779.455\n",
" ARIMA(0,0,0)(1,1,1)[12] with drift : 1777.902\n",
" ARIMA(0,0,0)(1,1,2)[12] : Inf\n",
" ARIMA(0,0,0)(1,1,2)[12] with drift : 1780.115\n",
" ARIMA(0,0,0)(2,1,0)[12] : 1841.745\n",
" ARIMA(0,0,0)(2,1,0)[12] with drift : 1843.425\n",
" ARIMA(0,0,0)(2,1,1)[12] : 1780.857\n",
" ARIMA(0,0,0)(2,1,1)[12] with drift : Inf\n",
" ARIMA(0,0,0)(2,1,2)[12] : 1783.475\n",
" ARIMA(0,0,0)(2,1,2)[12] with drift : Inf\n",
" ARIMA(0,0,1)(0,1,0)[12] : 2079.529\n",
" ARIMA(0,0,1)(0,1,0)[12] with drift : 2081.493\n",
" ARIMA(0,0,1)(0,1,1)[12] : 1750.092\n",
" ARIMA(0,0,1)(0,1,1)[12] with drift : 1748.647\n",
" ARIMA(0,0,1)(0,1,2)[12] : 1751.788\n",
" ARIMA(0,0,1)(0,1,2)[12] with drift : 1749.72\n",
" ARIMA(0,0,1)(1,1,0)[12] : 1898.631\n",
" ARIMA(0,0,1)(1,1,0)[12] with drift : 1900.521\n",
" ARIMA(0,0,1)(1,1,1)[12] : 1751.763\n",
" ARIMA(0,0,1)(1,1,1)[12] with drift : 1749.588\n",
" ARIMA(0,0,1)(1,1,2)[12] : 1754.146\n",
" ARIMA(0,0,1)(1,1,2)[12] with drift : Inf\n",
" ARIMA(0,0,1)(2,1,0)[12] : 1828.566\n",
" ARIMA(0,0,1)(2,1,0)[12] with drift : 1830.327\n",
" ARIMA(0,0,1)(2,1,1)[12] : 1753.174\n",
" ARIMA(0,0,1)(2,1,1)[12] with drift : Inf\n",
" ARIMA(0,0,1)(2,1,2)[12] : Inf\n",
" ARIMA(0,0,1)(2,1,2)[12] with drift : Inf\n",
" ARIMA(0,0,2)(0,1,0)[12] : 2081.476\n",
" ARIMA(0,0,2)(0,1,0)[12] with drift : 2083.447\n",
" ARIMA(0,0,2)(0,1,1)[12] : 1748.593\n",
" ARIMA(0,0,2)(0,1,1)[12] with drift : 1747.289\n",
" ARIMA(0,0,2)(0,1,2)[12] : 1750.456\n",
" ARIMA(0,0,2)(0,1,2)[12] with drift : 1748.672\n",
" ARIMA(0,0,2)(1,1,0)[12] : 1898.542\n",
" ARIMA(0,0,2)(1,1,0)[12] with drift : 1900.447\n",
" ARIMA(0,0,2)(1,1,1)[12] : 1750.443\n",
" ARIMA(0,0,2)(1,1,1)[12] with drift : 1748.581\n",
" ARIMA(0,0,2)(1,1,2)[12] : Inf\n",
" ARIMA(0,0,2)(1,1,2)[12] with drift : Inf\n",
" ARIMA(0,0,2)(2,1,0)[12] : 1827.098\n",
" ARIMA(0,0,2)(2,1,0)[12] with drift : 1828.888\n",
" ARIMA(0,0,2)(2,1,1)[12] : 1751.819\n",
" ARIMA(0,0,2)(2,1,1)[12] with drift : Inf\n",
" ARIMA(0,0,3)(0,1,0)[12] : 2074.456\n",
" ARIMA(0,0,3)(0,1,0)[12] with drift : 2076.439\n",
" ARIMA(0,0,3)(0,1,1)[12] : 1742.418\n",
" ARIMA(0,0,3)(0,1,1)[12] with drift : 1741.336\n",
" ARIMA(0,0,3)(0,1,2)[12] : 1744.072\n",
" ARIMA(0,0,3)(0,1,2)[12] with drift : Inf\n",
" ARIMA(0,0,3)(1,1,0)[12] : 1892.367\n",
" ARIMA(0,0,3)(1,1,0)[12] with drift : 1894.294\n",
" ARIMA(0,0,3)(1,1,1)[12] : 1744.034\n",
" ARIMA(0,0,3)(1,1,1)[12] with drift : Inf\n",
" ARIMA(0,0,3)(2,1,0)[12] : 1823.79\n",
" ARIMA(0,0,3)(2,1,0)[12] with drift : 1825.615\n",
" ARIMA(0,0,4)(0,1,0)[12] : 2076.328\n",
" ARIMA(0,0,4)(0,1,0)[12] with drift : 2078.318\n",
" ARIMA(0,0,4)(0,1,1)[12] : 1744.056\n",
" ARIMA(0,0,4)(0,1,1)[12] with drift : 1743.052\n",
" ARIMA(0,0,4)(1,1,0)[12] : 1894.373\n",
" ARIMA(0,0,4)(1,1,0)[12] with drift : 1896.306\n",
" ARIMA(0,0,5)(0,1,0)[12] : 2076.84\n",
" ARIMA(0,0,5)(0,1,0)[12] with drift : 2078.837\n",
" ARIMA(1,0,0)(0,1,0)[12] : 2078.792\n",
" ARIMA(1,0,0)(0,1,0)[12] with drift : 2080.76\n",
" ARIMA(1,0,0)(0,1,1)[12] : 1745.57\n",
" ARIMA(1,0,0)(0,1,1)[12] with drift : 1744.233\n",
" ARIMA(1,0,0)(0,1,2)[12] : 1747.296\n",
" ARIMA(1,0,0)(0,1,2)[12] with drift : 1745.376\n",
" ARIMA(1,0,0)(1,1,0)[12] : 1896.282\n",
" ARIMA(1,0,0)(1,1,0)[12] with drift : 1898.182\n",
" ARIMA(1,0,0)(1,1,1)[12] : 1747.273\n",
" ARIMA(1,0,0)(1,1,1)[12] with drift : Inf\n",
" ARIMA(1,0,0)(1,1,2)[12] : Inf\n",
" ARIMA(1,0,0)(1,1,2)[12] with drift : Inf\n",
" ARIMA(1,0,0)(2,1,0)[12] : 1826.302\n",
" ARIMA(1,0,0)(2,1,0)[12] with drift : 1828.082\n",
" ARIMA(1,0,0)(2,1,1)[12] : 1748.642\n",
" ARIMA(1,0,0)(2,1,1)[12] with drift : Inf\n",
" ARIMA(1,0,0)(2,1,2)[12] : Inf\n",
" ARIMA(1,0,0)(2,1,2)[12] with drift : Inf\n",
" ARIMA(1,0,1)(0,1,0)[12] : 2078.217\n",
" ARIMA(1,0,1)(0,1,0)[12] with drift : 2080.198\n",
" ARIMA(1,0,1)(0,1,1)[12] : 1736.005\n",
" ARIMA(1,0,1)(0,1,1)[12] with drift : 1736.06\n",
" ARIMA(1,0,1)(0,1,2)[12] : 1737.915\n",
" ARIMA(1,0,1)(0,1,2)[12] with drift : 1737.722\n",
" ARIMA(1,0,1)(1,1,0)[12] : 1890.475\n",
" ARIMA(1,0,1)(1,1,0)[12] with drift : 1892.418\n",
" ARIMA(1,0,1)(1,1,1)[12] : 1737.903\n",
" ARIMA(1,0,1)(1,1,1)[12] with drift : 1737.671\n",
" ARIMA(1,0,1)(1,1,2)[12] : Inf\n",
" ARIMA(1,0,1)(1,1,2)[12] with drift : Inf\n",
" ARIMA(1,0,1)(2,1,0)[12] : 1818.362\n",
" ARIMA(1,0,1)(2,1,0)[12] with drift : 1820.24\n",
" ARIMA(1,0,1)(2,1,1)[12] : Inf\n",
" ARIMA(1,0,1)(2,1,1)[12] with drift : Inf\n",
" ARIMA(1,0,2)(0,1,0)[12] : 2078.237\n",
" ARIMA(1,0,2)(0,1,0)[12] with drift : 2080.225\n",
" ARIMA(1,0,2)(0,1,1)[12] : 1733.894\n",
" ARIMA(1,0,2)(0,1,1)[12] with drift : 1734.251\n",
" ARIMA(1,0,2)(0,1,2)[12] : 1735.836\n",
" ARIMA(1,0,2)(0,1,2)[12] with drift : Inf\n",
" ARIMA(1,0,2)(1,1,0)[12] : 1891.939\n",
" ARIMA(1,0,2)(1,1,0)[12] with drift : 1893.89\n",
" ARIMA(1,0,2)(1,1,1)[12] : 1735.828\n",
" ARIMA(1,0,2)(1,1,1)[12] with drift : Inf\n",
" ARIMA(1,0,2)(2,1,0)[12] : 1820.178\n",
" ARIMA(1,0,2)(2,1,0)[12] with drift : 1822.065\n",
" ARIMA(1,0,3)(0,1,0)[12] : 2075.174\n",
" ARIMA(1,0,3)(0,1,0)[12] with drift : 2077.168\n",
" ARIMA(1,0,3)(0,1,1)[12] : 1735.885\n",
" ARIMA(1,0,3)(0,1,1)[12] with drift : 1736.175\n",
" ARIMA(1,0,3)(1,1,0)[12] : 1892.711\n",
" ARIMA(1,0,3)(1,1,0)[12] with drift : 1894.668\n",
" ARIMA(1,0,4)(0,1,0)[12] : 2076.587\n",
" ARIMA(1,0,4)(0,1,0)[12] with drift : 2078.587\n",
" ARIMA(2,0,0)(0,1,0)[12] : 2080.563\n",
" ARIMA(2,0,0)(0,1,0)[12] with drift : 2082.538\n",
" ARIMA(2,0,0)(0,1,1)[12] : 1743.982\n",
" ARIMA(2,0,0)(0,1,1)[12] with drift : 1742.908\n",
" ARIMA(2,0,0)(0,1,2)[12] : 1745.839\n",
" ARIMA(2,0,0)(0,1,2)[12] with drift : 1744.312\n",
" ARIMA(2,0,0)(1,1,0)[12] : 1895.702\n",
" ARIMA(2,0,0)(1,1,0)[12] with drift : 1897.619\n",
" ARIMA(2,0,0)(1,1,1)[12] : 1745.824\n",
" ARIMA(2,0,0)(1,1,1)[12] with drift : Inf\n",
" ARIMA(2,0,0)(1,1,2)[12] : Inf\n",
" ARIMA(2,0,0)(1,1,2)[12] with drift : Inf\n",
" ARIMA(2,0,0)(2,1,0)[12] : 1824.487\n",
" ARIMA(2,0,0)(2,1,0)[12] with drift : 1826.3\n",
" ARIMA(2,0,0)(2,1,1)[12] : 1747.059\n",
" ARIMA(2,0,0)(2,1,1)[12] with drift : Inf\n",
" ARIMA(2,0,1)(0,1,0)[12] : 2078.787\n",
" ARIMA(2,0,1)(0,1,0)[12] with drift : 2080.775\n",
" ARIMA(2,0,1)(0,1,1)[12] : 1733.85\n",
" ARIMA(2,0,1)(0,1,1)[12] with drift : 1734.396\n",
" ARIMA(2,0,1)(0,1,2)[12] : Inf\n",
" ARIMA(2,0,1)(0,1,2)[12] with drift : Inf\n",
" ARIMA(2,0,1)(1,1,0)[12] : 1892.015\n",
" ARIMA(2,0,1)(1,1,0)[12] with drift : 1893.967\n",
" ARIMA(2,0,1)(1,1,1)[12] : Inf\n",
" ARIMA(2,0,1)(1,1,1)[12] with drift : Inf\n",
" ARIMA(2,0,1)(2,1,0)[12] : Inf\n",
" ARIMA(2,0,1)(2,1,0)[12] with drift : Inf\n",
" ARIMA(2,0,2)(0,1,0)[12] : Inf\n",
" ARIMA(2,0,2)(0,1,0)[12] with drift : Inf\n",
" ARIMA(2,0,2)(0,1,1)[12] : 1735.387\n",
" ARIMA(2,0,2)(0,1,1)[12] with drift : 1735.626\n",
" ARIMA(2,0,2)(1,1,0)[12] : 1892.068\n",
" ARIMA(2,0,2)(1,1,0)[12] with drift : 1894.027\n",
" ARIMA(2,0,3)(0,1,0)[12] : 2075.99\n",
" ARIMA(2,0,3)(0,1,0)[12] with drift : 2077.991\n",
" ARIMA(3,0,0)(0,1,0)[12] : 2073.867\n",
" ARIMA(3,0,0)(0,1,0)[12] with drift : 2075.853\n",
" ARIMA(3,0,0)(0,1,1)[12] : 1739.145\n",
" ARIMA(3,0,0)(0,1,1)[12] with drift : 1738.463\n",
" ARIMA(3,0,0)(0,1,2)[12] : 1740.747\n",
" ARIMA(3,0,0)(0,1,2)[12] with drift : Inf\n",
" ARIMA(3,0,0)(1,1,0)[12] : 1891.143\n",
" ARIMA(3,0,0)(1,1,0)[12] with drift : 1893.081\n",
" ARIMA(3,0,0)(1,1,1)[12] : 1740.704\n",
" ARIMA(3,0,0)(1,1,1)[12] with drift : Inf\n",
" ARIMA(3,0,0)(2,1,0)[12] : 1821.908\n",
" ARIMA(3,0,0)(2,1,0)[12] with drift : 1823.756\n",
" ARIMA(3,0,1)(0,1,0)[12] : 2075.092\n",
" ARIMA(3,0,1)(0,1,0)[12] with drift : 2077.084\n",
" ARIMA(3,0,1)(0,1,1)[12] : Inf\n",
" ARIMA(3,0,1)(0,1,1)[12] with drift : 1736.36\n",
" ARIMA(3,0,1)(1,1,0)[12] : 1892.573\n",
" ARIMA(3,0,1)(1,1,0)[12] with drift : 1894.514\n",
" ARIMA(3,0,2)(0,1,0)[12] : 2075.913\n",
" ARIMA(3,0,2)(0,1,0)[12] with drift : 2077.914\n",
" ARIMA(4,0,0)(0,1,0)[12] : 2075.789\n",
" ARIMA(4,0,0)(0,1,0)[12] with drift : 2077.781\n",
" ARIMA(4,0,0)(0,1,1)[12] : 1741.105\n",
" ARIMA(4,0,0)(0,1,1)[12] with drift : 1740.471\n",
" ARIMA(4,0,0)(1,1,0)[12] : 1893.098\n",
" ARIMA(4,0,0)(1,1,0)[12] with drift : 1895.041\n",
" ARIMA(4,0,1)(0,1,0)[12] : 2076.993\n",
" ARIMA(4,0,1)(0,1,0)[12] with drift : 2078.991\n",
" ARIMA(5,0,0)(0,1,0)[12] : 2074.108\n",
" ARIMA(5,0,0)(0,1,0)[12] with drift : 2076.108\n",
"\n",
"\n",
"\n",
" Best model: ARIMA(2,0,1)(0,1,1)[12] \n",
"\n"
]
},
{
"data": {
"text/plain": [
"Series: INMET \n",
"ARIMA(2,0,1)(0,1,1)[12] \n",
"\n",
"Coefficients:\n",
" ar1 ar2 ma1 sma1\n",
" 1.1203 -0.1462 -0.9194 -0.8870\n",
"s.e. 0.1073 0.0667 0.0910 0.0341\n",
"\n",
"sigma^2 estimated as 0.7459: log likelihood=-861.88\n",
"AIC=1733.76 AICc=1733.85 BIC=1756.31"
]
},
"execution_count": 23,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"# Automatizando\n",
"ajuste_auto <- auto.arima(INMET, stepwise=FALSE, approximation=FALSE, trace=TRUE)\n",
"ajuste_auto"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"No caso, obteve um AIC apenas um pouco melhor."
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 24,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 24,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"acf(ajuste_auto$residuals)\n",
"pacf(ajuste_auto$residuals)"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"\n",
"\tLilliefors (Kolmogorov-Smirnov) normality test\n",
"\n",
"data: ajuste_auto$residuals\n",
"D = 0.055364, p-value = 3.59e-05\n"
]
},
"execution_count": 25,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 25,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"qqnorm(ajuste_auto$residuals)\n",
"qqline(ajuste_auto$residuals)\n",
"lillie.test(ajuste_auto$residuals)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"https://stats.stackexchange.com/questions/101274/how-to-interpret-a-qq-plot\n",
"\n",
"Não adere a Normal, mas também não apresenta comportamento anômalo. É o caso de tentar uma transformação de Box-Cox."
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"1.99992424816297"
]
},
"execution_count": 26,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"l <- BoxCox.lambda(INMET)\n",
"l"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
" ARIMA(0,0,0)(0,1,0)[12] : 6459.492\n",
" ARIMA(0,0,0)(0,1,0)[12] with drift : 6461.425\n",
" ARIMA(0,0,0)(0,1,1)[12] : 6130.443\n",
" ARIMA(0,0,0)(0,1,1)[12] with drift : 6129.316\n",
" ARIMA(0,0,0)(0,1,2)[12] : 6132.377\n",
" ARIMA(0,0,0)(0,1,2)[12] with drift : 6131.338\n",
" ARIMA(0,0,0)(1,1,0)[12] : 6269.231\n",
" ARIMA(0,0,0)(1,1,0)[12] with drift : 6271.07\n",
" ARIMA(0,0,0)(1,1,1)[12] : 6132.373\n",
" ARIMA(0,0,0)(1,1,1)[12] with drift : 6131.338\n",
" ARIMA(0,0,0)(1,1,2)[12] : 6134.386\n",
" ARIMA(0,0,0)(1,1,2)[12] with drift : 6133.339\n",
" ARIMA(0,0,0)(2,1,0)[12] : 6190.466\n",
" ARIMA(0,0,0)(2,1,0)[12] with drift : 6192.138\n",
" ARIMA(0,0,0)(2,1,1)[12] : 6134.065\n",
" ARIMA(0,0,0)(2,1,1)[12] with drift : 6132.104\n",
" ARIMA(0,0,0)(2,1,2)[12] : 6136.426\n",
" ARIMA(0,0,0)(2,1,2)[12] with drift : 6135.332\n",
" ARIMA(0,0,1)(0,1,0)[12] : 6435.915\n",
" ARIMA(0,0,1)(0,1,0)[12] with drift : 6437.876\n",
" ARIMA(0,0,1)(0,1,1)[12] : 6100.853\n",
" ARIMA(0,0,1)(0,1,1)[12] with drift : 6099.665\n",
" ARIMA(0,0,1)(0,1,2)[12] : 6102.849\n",
" ARIMA(0,0,1)(0,1,2)[12] with drift : 6101.42\n",
" ARIMA(0,0,1)(1,1,0)[12] : 6249.518\n",
" ARIMA(0,0,1)(1,1,0)[12] with drift : 6251.407\n",
" ARIMA(0,0,1)(1,1,1)[12] : 6102.847\n",
" ARIMA(0,0,1)(1,1,1)[12] with drift : 6101.388\n",
" ARIMA(0,0,1)(1,1,2)[12] : Inf\n",
" ARIMA(0,0,1)(1,1,2)[12] with drift : 6103.716\n",
" ARIMA(0,0,1)(2,1,0)[12] : 6175.527\n",
" ARIMA(0,0,1)(2,1,0)[12] with drift : 6177.286\n",
" ARIMA(0,0,1)(2,1,1)[12] : 6104.446\n",
" ARIMA(0,0,1)(2,1,1)[12] with drift : Inf\n",
" ARIMA(0,0,1)(2,1,2)[12] : Inf\n",
" ARIMA(0,0,1)(2,1,2)[12] with drift : 6104.97\n",
" ARIMA(0,0,2)(0,1,0)[12] : 6437.798\n",
" ARIMA(0,0,2)(0,1,0)[12] with drift : 6439.766\n",
" ARIMA(0,0,2)(0,1,1)[12] : 6099.46\n",
" ARIMA(0,0,2)(0,1,1)[12] with drift : 6098.418\n",
" ARIMA(0,0,2)(0,1,2)[12] : 6101.49\n",
" ARIMA(0,0,2)(0,1,2)[12] with drift : 6100.323\n",
" ARIMA(0,0,2)(1,1,0)[12] : 6249.341\n",
" ARIMA(0,0,2)(1,1,0)[12] with drift : 6251.245\n",
" ARIMA(0,0,2)(1,1,1)[12] : 6101.49\n",
" ARIMA(0,0,2)(1,1,1)[12] with drift : 6100.308\n",
" ARIMA(0,0,2)(1,1,2)[12] : Inf\n",
" ARIMA(0,0,2)(1,1,2)[12] with drift : Inf\n",
" ARIMA(0,0,2)(2,1,0)[12] : 6174.002\n",
" ARIMA(0,0,2)(2,1,0)[12] with drift : 6175.789\n",
" ARIMA(0,0,2)(2,1,1)[12] : 6103.076\n",
" ARIMA(0,0,2)(2,1,1)[12] with drift : Inf\n",
" ARIMA(0,0,3)(0,1,0)[12] : 6429.49\n",
" ARIMA(0,0,3)(0,1,0)[12] with drift : 6431.47\n",
" ARIMA(0,0,3)(0,1,1)[12] : 6092.457\n",
" ARIMA(0,0,3)(0,1,1)[12] with drift : 6091.68\n",
" ARIMA(0,0,3)(0,1,2)[12] : 6094.437\n",
" ARIMA(0,0,3)(0,1,2)[12] with drift : 6093.404\n",
" ARIMA(0,0,3)(1,1,0)[12] : 6241.902\n",
" ARIMA(0,0,3)(1,1,0)[12] with drift : 6243.829\n",
" ARIMA(0,0,3)(1,1,1)[12] : 6094.432\n",
" ARIMA(0,0,3)(1,1,1)[12] with drift : 6093.358\n",
" ARIMA(0,0,3)(2,1,0)[12] : 6169.704\n",
" ARIMA(0,0,3)(2,1,0)[12] with drift : 6171.529\n",
" ARIMA(0,0,4)(0,1,0)[12] : 6431.336\n",
" ARIMA(0,0,4)(0,1,0)[12] with drift : 6433.323\n",
" ARIMA(0,0,4)(0,1,1)[12] : 6094.109\n",
" ARIMA(0,0,4)(0,1,1)[12] with drift : 6093.402\n",
" ARIMA(0,0,4)(1,1,0)[12] : 6243.923\n",
" ARIMA(0,0,4)(1,1,0)[12] with drift : 6245.856\n",
" ARIMA(0,0,5)(0,1,0)[12] : 6431.87\n",
" ARIMA(0,0,5)(0,1,0)[12] with drift : 6433.864\n",
" ARIMA(1,0,0)(0,1,0)[12] : 6434.885\n",
" ARIMA(1,0,0)(0,1,0)[12] with drift : 6436.849\n",
" ARIMA(1,0,0)(0,1,1)[12] : 6096.043\n",
" ARIMA(1,0,0)(0,1,1)[12] with drift : 6094.989\n",
" ARIMA(1,0,0)(0,1,2)[12] : 6098.047\n",
" ARIMA(1,0,0)(0,1,2)[12] with drift : 6096.783\n",
" ARIMA(1,0,0)(1,1,0)[12] : 6246.859\n",
" ARIMA(1,0,0)(1,1,0)[12] with drift : 6248.759\n",
" ARIMA(1,0,0)(1,1,1)[12] : 6098.046\n",
" ARIMA(1,0,0)(1,1,1)[12] with drift : 6096.756\n",
" ARIMA(1,0,0)(1,1,2)[12] : Inf\n",
" ARIMA(1,0,0)(1,1,2)[12] with drift : Inf\n",
" ARIMA(1,0,0)(2,1,0)[12] : 6172.965\n",
" ARIMA(1,0,0)(2,1,0)[12] with drift : 6174.745\n",
" ARIMA(1,0,0)(2,1,1)[12] : 6099.601\n",
" ARIMA(1,0,0)(2,1,1)[12] with drift : Inf\n",
" ARIMA(1,0,0)(2,1,2)[12] : Inf\n",
" ARIMA(1,0,0)(2,1,2)[12] with drift : 6100.293\n",
" ARIMA(1,0,1)(0,1,0)[12] : 6433.408\n",
" ARIMA(1,0,1)(0,1,0)[12] with drift : 6435.386\n",
" ARIMA(1,0,1)(0,1,1)[12] : 6086.428\n",
" ARIMA(1,0,1)(0,1,1)[12] with drift : 6086.654\n",
" ARIMA(1,0,1)(0,1,2)[12] : 6088.457\n",
" ARIMA(1,0,1)(0,1,2)[12] with drift : 6088.653\n",
" ARIMA(1,0,1)(1,1,0)[12] : 6240.708\n",
" ARIMA(1,0,1)(1,1,0)[12] with drift : 6242.651\n",
" ARIMA(1,0,1)(1,1,1)[12] : 6088.458\n",
" ARIMA(1,0,1)(1,1,1)[12] with drift : 6088.648\n",
" ARIMA(1,0,1)(1,1,2)[12] : Inf\n",
" ARIMA(1,0,1)(1,1,2)[12] with drift : Inf\n",
" ARIMA(1,0,1)(2,1,0)[12] : 6164.485\n",
" ARIMA(1,0,1)(2,1,0)[12] with drift : 6166.362\n",
" ARIMA(1,0,1)(2,1,1)[12] : 6089.714\n",
" ARIMA(1,0,1)(2,1,1)[12] with drift : Inf\n",
" ARIMA(1,0,2)(0,1,0)[12] : 6433.686\n",
" ARIMA(1,0,2)(0,1,0)[12] with drift : 6435.671\n",
" ARIMA(1,0,2)(0,1,1)[12] : 6084.258\n",
" ARIMA(1,0,2)(0,1,1)[12] with drift : 6084.714\n",
" ARIMA(1,0,2)(0,1,2)[12] : 6086.287\n",
" ARIMA(1,0,2)(0,1,2)[12] with drift : 6086.739\n",
" ARIMA(1,0,2)(1,1,0)[12] : 6242.11\n",
" ARIMA(1,0,2)(1,1,0)[12] with drift : 6244.061\n",
" ARIMA(1,0,2)(1,1,1)[12] : 6086.287\n",
" ARIMA(1,0,2)(1,1,1)[12] with drift : 6086.738\n",
" ARIMA(1,0,2)(2,1,0)[12] : 6166.235\n",
" ARIMA(1,0,2)(2,1,0)[12] with drift : 6168.122\n",
" ARIMA(1,0,3)(0,1,0)[12] : 6430.348\n",
" ARIMA(1,0,3)(0,1,0)[12] with drift : 6432.339\n",
" ARIMA(1,0,3)(0,1,1)[12] : 6086.092\n",
" ARIMA(1,0,3)(0,1,1)[12] with drift : 6086.483\n",
" ARIMA(1,0,3)(1,1,0)[12] : 6242.756\n",
" ARIMA(1,0,3)(1,1,0)[12] with drift : 6244.712\n",
" ARIMA(1,0,4)(0,1,0)[12] : 6431.563\n",
" ARIMA(1,0,4)(0,1,0)[12] with drift : 6433.56\n",
" ARIMA(2,0,0)(0,1,0)[12] : 6436.474\n",
" ARIMA(2,0,0)(0,1,0)[12] with drift : 6438.446\n",
" ARIMA(2,0,0)(0,1,1)[12] : 6094.514\n",
" ARIMA(2,0,0)(0,1,1)[12] with drift : 6093.724\n",
" ARIMA(2,0,0)(0,1,2)[12] : 6096.544\n",
" ARIMA(2,0,0)(0,1,2)[12] with drift : 6095.637\n",
" ARIMA(2,0,0)(1,1,0)[12] : 6246.088\n",
" ARIMA(2,0,0)(1,1,0)[12] with drift : 6248.005\n",
" ARIMA(2,0,0)(1,1,1)[12] : 6096.544\n",
" ARIMA(2,0,0)(1,1,1)[12] with drift : 6095.622\n",
" ARIMA(2,0,0)(1,1,2)[12] : Inf\n",
" ARIMA(2,0,0)(1,1,2)[12] with drift : Inf\n",
" ARIMA(2,0,0)(2,1,0)[12] : 6170.995\n",
" ARIMA(2,0,0)(2,1,0)[12] with drift : 6172.807\n",
" ARIMA(2,0,0)(2,1,1)[12] : 6097.998\n",
" ARIMA(2,0,0)(2,1,1)[12] with drift : Inf\n",
" ARIMA(2,0,1)(0,1,0)[12] : 6434.174\n",
" ARIMA(2,0,1)(0,1,0)[12] with drift : 6436.159\n",
" ARIMA(2,0,1)(0,1,1)[12] : 6084.44\n",
" ARIMA(2,0,1)(0,1,1)[12] with drift : 6084.961\n",
" ARIMA(2,0,1)(0,1,2)[12] : 6086.462\n",
" ARIMA(2,0,1)(0,1,2)[12] with drift : 6086.998\n",
" ARIMA(2,0,1)(1,1,0)[12] : 6242.201\n",
" ARIMA(2,0,1)(1,1,0)[12] with drift : 6244.151\n",
" ARIMA(2,0,1)(1,1,1)[12] : 6086.461\n",
" ARIMA(2,0,1)(1,1,1)[12] with drift : 6086.995\n",
" ARIMA(2,0,1)(2,1,0)[12] : Inf\n",
" ARIMA(2,0,1)(2,1,0)[12] with drift : Inf\n",
" ARIMA(2,0,2)(0,1,0)[12] : Inf\n",
" ARIMA(2,0,2)(0,1,0)[12] with drift : Inf\n",
" ARIMA(2,0,2)(0,1,1)[12] : 6085.4\n",
" ARIMA(2,0,2)(0,1,1)[12] with drift : 6085.776\n",
" ARIMA(2,0,2)(1,1,0)[12] : 6241.891\n",
" ARIMA(2,0,2)(1,1,0)[12] with drift : 6243.85\n",
" ARIMA(2,0,3)(0,1,0)[12] : 6430.935\n",
" ARIMA(2,0,3)(0,1,0)[12] with drift : 6432.932\n",
" ARIMA(3,0,0)(0,1,0)[12] : 6428.772\n",
" ARIMA(3,0,0)(0,1,0)[12] with drift : 6430.756\n",
" ARIMA(3,0,0)(0,1,1)[12] : 6089.022\n",
" ARIMA(3,0,0)(0,1,1)[12] with drift : 6088.626\n",
" ARIMA(3,0,0)(0,1,2)[12] : 6090.979\n",
" ARIMA(3,0,0)(0,1,2)[12] with drift : 6090.353\n",
" ARIMA(3,0,0)(1,1,0)[12] : 6240.788\n",
" ARIMA(3,0,0)(1,1,0)[12] with drift : 6242.726\n",
" ARIMA(3,0,0)(1,1,1)[12] : 6090.972\n",
" ARIMA(3,0,0)(1,1,1)[12] with drift : 6090.311\n",
" ARIMA(3,0,0)(2,1,0)[12] : 6167.684\n",
" ARIMA(3,0,0)(2,1,0)[12] with drift : 6169.533\n",
" ARIMA(3,0,1)(0,1,0)[12] : 6429.903\n",
" ARIMA(3,0,1)(0,1,0)[12] with drift : 6431.891\n",
" ARIMA(3,0,1)(0,1,1)[12] : 6086.378\n",
" ARIMA(3,0,1)(0,1,1)[12] with drift : 6086.774\n",
" ARIMA(3,0,1)(1,1,0)[12] : 6241.918\n",
" ARIMA(3,0,1)(1,1,0)[12] with drift : 6243.856\n",
" ARIMA(3,0,2)(0,1,0)[12] : 6430.82\n",
" ARIMA(3,0,2)(0,1,0)[12] with drift : 6432.816\n",
" ARIMA(4,0,0)(0,1,0)[12] : 6430.646\n",
" ARIMA(4,0,0)(0,1,0)[12] with drift : 6432.636\n",
" ARIMA(4,0,0)(0,1,1)[12] : 6091.012\n",
" ARIMA(4,0,0)(0,1,1)[12] with drift : 6090.649\n",
" ARIMA(4,0,0)(1,1,0)[12] : 6242.63\n",
" ARIMA(4,0,0)(1,1,0)[12] with drift : 6244.572\n",
" ARIMA(4,0,1)(0,1,0)[12] : 6431.798\n",
" ARIMA(4,0,1)(0,1,0)[12] with drift : 6433.794\n",
" ARIMA(5,0,0)(0,1,0)[12] : 6429.072\n",
" ARIMA(5,0,0)(0,1,0)[12] with drift : 6431.069\n",
"\n",
"\n",
"\n",
" Best model: ARIMA(1,0,2)(0,1,1)[12] \n",
"\n"
]
},
{
"data": {
"text/plain": [
"Series: INMET \n",
"ARIMA(1,0,2)(0,1,1)[12] \n",
"Box Cox transformation: lambda= 1.999924 \n",
"\n",
"Coefficients:\n",
" ar1 ma1 ma2 sma1\n",
" 0.9225 -0.7169 -0.1025 -0.8690\n",
"s.e. 0.0642 0.0740 0.0542 0.0299\n",
"\n",
"sigma^2 estimated as 484.2: log likelihood=-3037.08\n",
"AIC=6084.17 AICc=6084.26 BIC=6106.72"
]
},
"execution_count": 27,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"ajuste_auto_bc <- auto.arima(INMET, stepwise=FALSE, approximation=FALSE, trace=TRUE, lambda=l)\n",
"ajuste_auto_bc"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"\n",
"\tLilliefors (Kolmogorov-Smirnov) normality test\n",
"\n",
"data: ajuste_auto_bc$residuals\n",
"D = 0.050203, p-value = 0.0003183\n"
]
},
"execution_count": 28,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 28,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 28,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"qqnorm(ajuste_auto_bc$residuals)\n",
"qqline(ajuste_auto_bc$residuals)\n",
"lillie.test(ajuste_auto_bc$residuals)\n",
"hist(ajuste_auto_bc$residuals)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Melhorou, mas ainda não adere a Normal. Tem outliers... Por hora, segue abaixo."
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"# A função arima não possue o parâmetro lambda\n",
"# outliers <- tso(BoxCox(INMET,l),tsmethod = \"auto.arima\", args.tsmethod = list(lambda=l), types = c(\"AO\",\"LS\",\"TC\"))"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"outliers <- tso(BoxCox(INMET,l),tsmethod = \"arima\", args.tsmethod = list(order=c(1, 0, 2), seasonal=list(order=c(0, 1, 1), period=12)), types = c(\"AO\",\"LS\",\"TC\"))"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"\n",
"Call:\n",
"structure(list(method = NULL), .Names = \"method\")\n",
"\n",
"Coefficients:\n",
" ar1 ma1 ma2 sma1 AO507\n",
" 0.9210 -0.7180 -0.0957 -0.8664 -85.6234\n",
"s.e. 0.0591 0.0695 0.0524 0.0297 20.4189\n",
"\n",
"sigma^2 estimated as 469.3: log likelihood = -3028.46, aic = 6068.92\n",
"\n",
"Outliers:\n",
" type ind time coefhat tstat\n",
"1 AO 507 2003:03 -85.62 -4.193"
]
},
"execution_count": 31,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"outliers"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"INMET[outliers$outliers$ind] <- NA"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"# Substituindo os outliers por estimativas...\n",
"INMET <- na.kalman(INMET, model = \"auto.arima\")"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
" Jan Feb Mar Apr May Jun Jul Aug\n",
"1961 26.38000 26.54000 26.70000 26.65000 25.50000 22.78000 23.34000 27.72000\n",
"1962 26.43000 26.85000 27.18000 26.13000 24.49000 22.36000 20.55000 25.37000\n",
"1963 26.39000 26.24000 26.34000 26.57000 24.75000 22.97000 23.63000 26.74000\n",
"1964 27.64000 27.46000 26.70000 27.59000 24.57000 23.51000 22.70000 27.69000\n",
"1965 26.57000 26.54000 25.36000 25.90000 25.67000 25.10000 23.96000 26.33000\n",
"1966 27.30000 26.33000 26.91000 26.45000 25.49000 24.68000 24.61000 24.32000\n",
"1967 27.49000 26.15000 26.37000 25.64000 23.66000 22.02000 22.37000 25.11000\n",
"1968 26.19000 25.62000 25.98000 23.72000 20.93000 21.78000 22.03000 24.92000\n",
"1969 26.58000 26.61000 26.72000 26.11000 24.55000 23.03000 21.83000 23.83000\n",
"1970 27.19000 26.33000 26.47000 26.30000 23.58000 22.74000 21.36000 23.56000\n",
"1971 26.57000 25.59000 26.06000 24.88000 22.78000 20.92000 21.88000 23.21000\n",
"1972 26.08000 25.88000 26.57000 23.99000 24.95000 23.25000 22.16000 24.63000\n",
"1973 27.67000 27.06000 27.38000 26.89000 23.39000 23.95000 21.24000 24.42000\n",
"1974 25.98000 26.37000 25.57000 25.00000 23.73000 24.41000 22.54000 26.04000\n",
"1975 26.53000 26.38000 26.21000 25.93000 23.52000 23.19000 22.45000 25.38000\n",
"1976 26.88000 26.04000 25.33000 25.24000 24.11000 21.74000 22.97000 25.44000\n",
"1977 26.11000 25.70000 26.82000 25.03000 22.96000 23.62000 24.74000 25.04000\n",
"1978 26.48000 27.14000 26.60000 26.32000 24.31000 23.37000 25.09000 22.79000\n",
"1979 26.40000 26.68000 26.36000 25.35000 24.63000 21.82000 23.73000 26.23000\n",
"1980 26.86000 26.25000 26.78000 26.42000 25.17000 23.35000 22.80000 25.85000\n",
"1981 26.45000 26.18000 26.54000 26.55000 25.73000 21.87000 20.75000 25.28000\n",
"1982 26.33000 26.32000 26.05000 25.79000 24.27000 24.15000 24.47000 25.24000\n",
"1983 26.95000 27.51000 26.80000 27.19000 26.12000 22.08000 23.04000 22.93000\n",
"1984 26.74000 26.20000 26.60000 25.41000 25.17000 22.72000 23.56000 23.40000\n",
"1985 26.06000 27.13000 26.84000 26.27000 25.94000 21.97000 23.07000 24.20000\n",
"1986 27.16000 26.57000 26.62000 27.18000 25.59000 23.20000 23.31000 25.89000\n",
"1987 27.16000 26.61000 26.14000 27.17000 24.59000 23.36000 24.90000 23.86000\n",
"1988 27.15000 26.64000 27.27000 26.81000 24.46000 22.90000 20.86000 25.18000\n",
"1989 26.08000 26.37000 26.22000 26.54000 23.89000 24.44000 21.86000 25.10000\n",
"1990 26.48009 26.07057 26.80931 26.39176 24.50474 23.62948 22.11987 25.52453\n",
"1991 26.39176 26.64872 25.75740 25.78149 25.53256 23.99886 23.71781 25.17925\n",
"1992 26.02239 26.21318 26.20898 26.00565 24.51632 22.23229 24.48868 24.54489\n",
"1993 26.63266 25.22743 26.20072 26.06485 24.65986 23.79008 23.29223 25.29206\n",
"1994 26.69690 26.36767 26.25525 25.84573 25.75740 24.50474 23.66160 25.55665\n",
"1995 25.66104 26.28000 27.09000 25.52000 24.72000 23.77000 24.57000 25.01000\n",
"1996 26.99359 26.34358 26.50418 26.21510 25.50044 22.14396 24.07113 27.02612\n",
"1997 26.74070 26.57819 26.54203 26.42550 24.82866 23.70641 23.84208 25.87136\n",
"1998 28.52000 27.86000 27.97000 27.74000 23.92000 23.62000 25.02000 26.11000\n",
"1999 27.17000 27.44000 26.28000 26.03000 24.68000 24.35000 23.53000 24.88000\n",
"2000 27.76000 26.99000 26.45000 26.59000 25.23000 24.40000 21.90000 26.84000\n",
"2001 26.92000 27.41000 26.58000 27.34000 24.24000 22.96000 24.82000 26.88000\n",
"2002 27.40000 26.46000 27.01000 27.12000 25.91000 23.34000 24.05000 27.20000\n",
"2003 27.01000 26.67000 26.83418 25.97000 24.88000 24.13000 23.18000 24.33000\n",
"2004 27.22000 27.05000 27.70000 27.02000 23.34000 23.74000 23.48000 25.26000\n",
"2005 25.95012 26.22313 25.73331 25.23546 25.53256 25.85376 23.85000 26.28000\n",
"2006 27.22000 27.06000 26.94000 26.13000 22.97000 24.81000 25.08289 27.22000\n",
"2007 27.61000 26.89000 27.43000 27.44000 23.61000 23.89000 23.45000 24.26000\n",
"2008 26.42000 26.92000 26.80000 24.87411 23.31632 22.80241 24.83397 26.87355\n",
"2009 26.24722 25.80558 25.23546 25.62892 25.52000 22.62000 23.90000 25.66000\n",
"2010 26.91000 27.48000 27.91000 26.58000 23.66000 24.71000 22.82000 24.73000\n",
"2011 26.82000 26.82000 26.75000 27.46000 24.76000 24.01000 24.27000 26.02000\n",
"2012 26.61000 26.71000 26.69000 27.55000 25.20000 24.25000 23.52000 25.82000\n",
"2013 25.35591 25.22743 27.48000 26.71000 25.62000 26.09000 23.98000 24.22000\n",
"2014 26.48000 26.35000 27.14000 27.05000 25.23000 24.56000 23.31000 26.45000\n",
"2015 27.55000 27.34000 27.16000 27.07000 26.32000 24.98000 24.52000 27.61000\n",
"2016 26.01436 26.80128 26.07057 27.11000 25.47000 22.98000 23.93000 26.19000\n",
"2017 27.21000 27.10000 27.38000 27.14000 27.06000 24.13000 23.33000 27.33000\n",
" Sep Oct Nov Dec\n",
"1961 28.90000 27.69000 27.37000 26.92000\n",
"1962 27.94000 26.43000 28.49000 26.34000\n",
"1963 28.01000 29.04000 28.10000 28.49000\n",
"1964 27.46000 26.44000 27.01000 26.44000\n",
"1965 27.44000 27.19000 27.07000 27.09000\n",
"1966 26.83000 28.07000 28.12000 28.44000\n",
"1967 28.03000 27.83000 26.88000 26.88000\n",
"1968 24.49000 27.01000 28.04000 26.12000\n",
"1969 28.04000 26.94000 27.43000 26.94000\n",
"1970 27.11000 27.52000 27.77000 27.15000\n",
"1971 27.09000 25.90000 26.44000 26.64000\n",
"1972 27.19000 27.52000 27.29000 27.08000\n",
"1973 27.17000 27.63000 26.71000 26.56000\n",
"1974 26.16000 27.24000 27.50000 26.26000\n",
"1975 27.84000 27.24000 26.13000 26.35000\n",
"1976 25.35000 26.87000 26.07000 26.28000\n",
"1977 26.11000 26.25000 26.68000 26.50000\n",
"1978 26.26000 27.46000 26.83000 26.66000\n",
"1979 26.25000 28.34000 27.09000 26.83000\n",
"1980 25.48000 28.22000 26.42000 26.36000\n",
"1981 25.04000 27.38000 27.20000 26.72000\n",
"1982 25.64000 27.46000 27.22000 26.80000\n",
"1983 25.46000 25.75000 25.62000 26.22000\n",
"1984 25.84000 27.88000 27.08000 26.35000\n",
"1985 26.41000 27.72000 27.48000 28.41000\n",
"1986 24.79000 26.30000 28.51000 27.21000\n",
"1987 27.65000 28.63000 28.00000 27.12000\n",
"1988 27.40000 28.58000 27.71000 27.02000\n",
"1989 25.16000 27.68000 27.88000 26.73000\n",
"1990 24.74564 27.50791 27.06627 26.81734\n",
"1991 27.09839 26.67281 26.63266 26.76113\n",
"1992 26.32752 25.93406 26.55236 26.63266\n",
"1993 26.74507 27.34731 27.46776 26.94582\n",
"1994 27.08233 27.56412 27.58018 26.02239\n",
"1995 28.15000 28.25000 27.41000 28.52000\n",
"1996 26.10268 27.01006 26.39979 26.89764\n",
"1997 27.21277 28.12264 27.78926 27.66938\n",
"1998 26.30000 28.04000 27.95000 27.23000\n",
"1999 28.32000 28.29000 26.97000 27.55000\n",
"2000 26.73000 28.22000 27.34000 27.52000\n",
"2001 27.92000 27.73000 27.16000 26.38000\n",
"2002 26.83000 29.34000 28.65000 27.90000\n",
"2003 27.20000 27.50000 27.23000 28.08000\n",
"2004 27.54000 28.25000 27.20000 27.83000\n",
"2005 25.79000 28.26000 27.87000 27.73000\n",
"2006 26.49000 27.46000 28.41000 27.31000\n",
"2007 28.50000 28.06000 27.18000 27.16000\n",
"2008 26.31949 26.98597 27.01809 26.01436\n",
"2009 27.08000 28.72000 28.24000 25.87785\n",
"2010 29.37000 28.71000 27.04000 27.64000\n",
"2011 28.96000 28.57000 28.57000 27.38000\n",
"2012 28.63000 29.37000 25.69316 26.09465\n",
"2013 28.38000 28.26000 28.05000 27.40000\n",
"2014 29.47000 29.43000 28.53000 27.44000\n",
"2015 29.85000 27.77290 27.29914 27.20278\n",
"2016 27.11000 28.04000 27.07000 27.31000\n",
"2017 29.08000 28.46000 28.72000 27.63000"
]
},
"execution_count": 34,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"INMET"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"1.99992424816297"
]
},
"execution_count": 35,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"l <- BoxCox.lambda(INMET)\n",
"l"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"\n",
"Call:\n",
"structure(list(method = NULL), .Names = \"method\")\n",
"\n",
"Coefficients:\n",
" ar1 ma1 ma2 sma1\n",
" 0.9200 -0.7173 -0.0948 -0.8660\n",
"s.e. 0.0589 0.0693 0.0521 0.0297\n",
"\n",
"sigma^2 estimated as 469.3: log likelihood = -3028.46, aic = 6066.93\n",
"\n",
"No outliers were detected."
]
},
"execution_count": 36,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"outliers2 <- tso(BoxCox(INMET,l),tsmethod = \"arima\", args.tsmethod = list(order=c(1, 0, 2), seasonal=list(order=c(0, 1, 1), period=12)), types = c(\"AO\",\"LS\",\"TC\"))\n",
"outliers2"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
" ARIMA(0,0,0)(0,1,0)[12] : 6435.896\n",
" ARIMA(0,0,0)(0,1,0)[12] with drift : 6437.827\n",
" ARIMA(0,0,0)(0,1,1)[12] : 6114.469\n",
" ARIMA(0,0,0)(0,1,1)[12] with drift : 6113.411\n",
" ARIMA(0,0,0)(0,1,2)[12] : 6116.485\n",
" ARIMA(0,0,0)(0,1,2)[12] with drift : 6115.348\n",
" ARIMA(0,0,0)(1,1,0)[12] : 6254.366\n",
" ARIMA(0,0,0)(1,1,0)[12] with drift : 6256.202\n",
" ARIMA(0,0,0)(1,1,1)[12] : 6116.485\n",
" ARIMA(0,0,0)(1,1,1)[12] with drift : 6115.341\n",
" ARIMA(0,0,0)(1,1,2)[12] : 6118.508\n",
" ARIMA(0,0,0)(1,1,2)[12] with drift : 6117.428\n",
" ARIMA(0,0,0)(2,1,0)[12] : 6174.55\n",
" ARIMA(0,0,0)(2,1,0)[12] with drift : 6176.217\n",
" ARIMA(0,0,0)(2,1,1)[12] : 6118.397\n",
" ARIMA(0,0,0)(2,1,1)[12] with drift : 6116.586\n",
" ARIMA(0,0,0)(2,1,2)[12] : 6120.492\n",
" ARIMA(0,0,0)(2,1,2)[12] with drift : 6119.407\n",
" ARIMA(0,0,1)(0,1,0)[12] : 6413.01\n",
" ARIMA(0,0,1)(0,1,0)[12] with drift : 6414.968\n",
" ARIMA(0,0,1)(0,1,1)[12] : 6085.25\n",
" ARIMA(0,0,1)(0,1,1)[12] with drift : 6084.092\n",
" ARIMA(0,0,1)(0,1,2)[12] : 6087.093\n",
" ARIMA(0,0,1)(0,1,2)[12] with drift : 6085.503\n",
" ARIMA(0,0,1)(1,1,0)[12] : 6235.142\n",
" ARIMA(0,0,1)(1,1,0)[12] with drift : 6237.029\n",
" ARIMA(0,0,1)(1,1,1)[12] : 6087.086\n",
" ARIMA(0,0,1)(1,1,1)[12] with drift : 6085.447\n",
" ARIMA(0,0,1)(1,1,2)[12] : 6089.296\n",
" ARIMA(0,0,1)(1,1,2)[12] with drift : Inf\n",
" ARIMA(0,0,1)(2,1,0)[12] : 6160.08\n",
" ARIMA(0,0,1)(2,1,0)[12] with drift : 6161.834\n",
" ARIMA(0,0,1)(2,1,1)[12] : 6088.951\n",
" ARIMA(0,0,1)(2,1,1)[12] with drift : Inf\n",
" ARIMA(0,0,1)(2,1,2)[12] : 6091.12\n",
" ARIMA(0,0,1)(2,1,2)[12] with drift : 6089.31\n",
" ARIMA(0,0,2)(0,1,0)[12] : 6414.942\n",
" ARIMA(0,0,2)(0,1,0)[12] with drift : 6416.908\n",
" ARIMA(0,0,2)(0,1,1)[12] : 6083.756\n",
" ARIMA(0,0,2)(0,1,1)[12] with drift : 6082.758\n",
" ARIMA(0,0,2)(0,1,2)[12] : 6085.73\n",
" ARIMA(0,0,2)(0,1,2)[12] with drift : 6084.445\n",
" ARIMA(0,0,2)(1,1,0)[12] : 6234.712\n",
" ARIMA(0,0,2)(1,1,0)[12] with drift : 6236.614\n",
" ARIMA(0,0,2)(1,1,1)[12] : 6085.728\n",
" ARIMA(0,0,2)(1,1,1)[12] with drift : 6084.413\n",
" ARIMA(0,0,2)(1,1,2)[12] : 6087.821\n",
" ARIMA(0,0,2)(1,1,2)[12] with drift : 6086.826\n",
" ARIMA(0,0,2)(2,1,0)[12] : 6158.322\n",
" ARIMA(0,0,2)(2,1,0)[12] with drift : 6160.104\n",
" ARIMA(0,0,2)(2,1,1)[12] : 6087.575\n",
" ARIMA(0,0,2)(2,1,1)[12] with drift : Inf\n",
" ARIMA(0,0,3)(0,1,0)[12] : 6404.489\n",
" ARIMA(0,0,3)(0,1,0)[12] with drift : 6406.468\n",
" ARIMA(0,0,3)(0,1,1)[12] : 6075.612\n",
" ARIMA(0,0,3)(0,1,1)[12] with drift : 6074.871\n",
" ARIMA(0,0,3)(0,1,2)[12] : 6077.386\n",
" ARIMA(0,0,3)(0,1,2)[12] with drift : 6076.196\n",
" ARIMA(0,0,3)(1,1,0)[12] : 6226.586\n",
" ARIMA(0,0,3)(1,1,0)[12] with drift : 6228.511\n",
" ARIMA(0,0,3)(1,1,1)[12] : 6077.369\n",
" ARIMA(0,0,3)(1,1,1)[12] with drift : 6076.112\n",
" ARIMA(0,0,3)(2,1,0)[12] : 6153.76\n",
" ARIMA(0,0,3)(2,1,0)[12] with drift : 6155.58\n",
" ARIMA(0,0,4)(0,1,0)[12] : 6406.022\n",
" ARIMA(0,0,4)(0,1,0)[12] with drift : 6408.008\n",
" ARIMA(0,0,4)(0,1,1)[12] : 6077.086\n",
" ARIMA(0,0,4)(0,1,1)[12] with drift : 6076.428\n",
" ARIMA(0,0,4)(1,1,0)[12] : 6228.529\n",
" ARIMA(0,0,4)(1,1,0)[12] with drift : 6230.462\n",
" ARIMA(0,0,5)(0,1,0)[12] : 6406.42\n",
" ARIMA(0,0,5)(0,1,0)[12] with drift : 6408.413\n",
" ARIMA(1,0,0)(0,1,0)[12] : 6412.02\n",
" ARIMA(1,0,0)(0,1,0)[12] with drift : 6413.982\n",
" ARIMA(1,0,0)(0,1,1)[12] : 6080.297\n",
" ARIMA(1,0,0)(0,1,1)[12] with drift : 6079.273\n",
" ARIMA(1,0,0)(0,1,2)[12] : 6082.174\n",
" ARIMA(1,0,0)(0,1,2)[12] with drift : 6080.766\n",
" ARIMA(1,0,0)(1,1,0)[12] : 6232.366\n",
" ARIMA(1,0,0)(1,1,0)[12] with drift : 6234.264\n",
" ARIMA(1,0,0)(1,1,1)[12] : 6082.168\n",
" ARIMA(1,0,0)(1,1,1)[12] with drift : 6080.717\n",
" ARIMA(1,0,0)(1,1,2)[12] : 6084.351\n",
" ARIMA(1,0,0)(1,1,2)[12] with drift : Inf\n",
" ARIMA(1,0,0)(2,1,0)[12] : 6157.48\n",
" ARIMA(1,0,0)(2,1,0)[12] with drift : 6159.254\n",
" ARIMA(1,0,0)(2,1,1)[12] : 6084.002\n",
" ARIMA(1,0,0)(2,1,1)[12] with drift : Inf\n",
" ARIMA(1,0,0)(2,1,2)[12] : 6086.184\n",
" ARIMA(1,0,0)(2,1,2)[12] with drift : 6084.544\n",
" ARIMA(1,0,1)(0,1,0)[12] : 6409.066\n",
" ARIMA(1,0,1)(0,1,0)[12] with drift : 6411.044\n",
" ARIMA(1,0,1)(0,1,1)[12] : 6068.718\n",
" ARIMA(1,0,1)(0,1,1)[12] with drift : 6069.036\n",
" ARIMA(1,0,1)(0,1,2)[12] : 6070.702\n",
" ARIMA(1,0,1)(0,1,2)[12] with drift : 6070.884\n",
" ARIMA(1,0,1)(1,1,0)[12] : 6225.001\n",
" ARIMA(1,0,1)(1,1,0)[12] with drift : 6226.945\n",
" ARIMA(1,0,1)(1,1,1)[12] : 6070.699\n",
" ARIMA(1,0,1)(1,1,1)[12] with drift : 6070.867\n",
" ARIMA(1,0,1)(1,1,2)[12] : 6072.779\n",
" ARIMA(1,0,1)(1,1,2)[12] with drift : 6073.114\n",
" ARIMA(1,0,1)(2,1,0)[12] : 6147.981\n",
" ARIMA(1,0,1)(2,1,0)[12] with drift : 6149.858\n",
" ARIMA(1,0,1)(2,1,1)[12] : 6072.342\n",
" ARIMA(1,0,1)(2,1,1)[12] with drift : Inf\n",
" ARIMA(1,0,2)(0,1,0)[12] : 6409.74\n",
" ARIMA(1,0,2)(0,1,0)[12] with drift : 6411.725\n",
" ARIMA(1,0,2)(0,1,1)[12] : 6067.018\n",
" ARIMA(1,0,2)(0,1,1)[12] with drift : 6067.505\n",
" ARIMA(1,0,2)(0,1,2)[12] : 6069.024\n",
" ARIMA(1,0,2)(0,1,2)[12] with drift : 6069.4\n",
" ARIMA(1,0,2)(1,1,0)[12] : 6226.595\n",
" ARIMA(1,0,2)(1,1,0)[12] with drift : 6228.546\n",
" ARIMA(1,0,2)(1,1,1)[12] : 6069.023\n",
" ARIMA(1,0,2)(1,1,1)[12] with drift : 6069.389\n",
" ARIMA(1,0,2)(2,1,0)[12] : 6149.848\n",
" ARIMA(1,0,2)(2,1,0)[12] with drift : 6151.733\n",
" ARIMA(1,0,3)(0,1,0)[12] : 6404.849\n",
" ARIMA(1,0,3)(0,1,0)[12] with drift : 6406.839\n",
" ARIMA(1,0,3)(0,1,1)[12] : 6068.717\n",
" ARIMA(1,0,3)(0,1,1)[12] with drift : 6069.141\n",
" ARIMA(1,0,3)(1,1,0)[12] : 6227.219\n",
" ARIMA(1,0,3)(1,1,0)[12] with drift : 6229.175\n",
" ARIMA(1,0,4)(0,1,0)[12] : 6405.895\n",
" ARIMA(1,0,4)(0,1,0)[12] with drift : 6407.892\n",
" ARIMA(2,0,0)(0,1,0)[12] : 6413.518\n",
" ARIMA(2,0,0)(0,1,0)[12] with drift : 6415.488\n",
" ARIMA(2,0,0)(0,1,1)[12] : 6078.312\n",
" ARIMA(2,0,0)(0,1,1)[12] with drift : 6077.576\n",
" ARIMA(2,0,0)(0,1,2)[12] : 6080.283\n",
" ARIMA(2,0,0)(0,1,2)[12] with drift : 6079.286\n",
" ARIMA(2,0,0)(1,1,0)[12] : 6231.139\n",
" ARIMA(2,0,0)(1,1,0)[12] with drift : 6233.055\n",
" ARIMA(2,0,0)(1,1,1)[12] : 6080.28\n",
" ARIMA(2,0,0)(1,1,1)[12] with drift : 6079.254\n",
" ARIMA(2,0,0)(1,1,2)[12] : 6082.373\n",
" ARIMA(2,0,0)(1,1,2)[12] with drift : 6081.65\n",
" ARIMA(2,0,0)(2,1,0)[12] : 6155.098\n",
" ARIMA(2,0,0)(2,1,0)[12] with drift : 6156.908\n",
" ARIMA(2,0,0)(2,1,1)[12] : 6082.041\n",
" ARIMA(2,0,0)(2,1,1)[12] with drift : Inf\n",
" ARIMA(2,0,1)(0,1,0)[12] : 6410.142\n",
" ARIMA(2,0,1)(0,1,0)[12] with drift : 6412.126\n",
" ARIMA(2,0,1)(0,1,1)[12] : 6067.252\n",
" ARIMA(2,0,1)(0,1,1)[12] with drift : 6067.771\n",
" ARIMA(2,0,1)(0,1,2)[12] : 6069.278\n",
" ARIMA(2,0,1)(0,1,2)[12] with drift : 6069.693\n",
" ARIMA(2,0,1)(1,1,0)[12] : 6226.658\n",
" ARIMA(2,0,1)(1,1,0)[12] with drift : 6228.609\n",
" ARIMA(2,0,1)(1,1,1)[12] : 6069.27\n",
" ARIMA(2,0,1)(1,1,1)[12] with drift : 6069.684\n",
" ARIMA(2,0,1)(2,1,0)[12] : Inf\n",
" ARIMA(2,0,1)(2,1,0)[12] with drift : Inf\n",
" ARIMA(2,0,2)(0,1,0)[12] : Inf\n",
" ARIMA(2,0,2)(0,1,0)[12] with drift : Inf\n",
" ARIMA(2,0,2)(0,1,1)[12] : 6067.849\n",
" ARIMA(2,0,2)(0,1,1)[12] with drift : 6068.268\n",
" ARIMA(2,0,2)(1,1,0)[12] : 6226.584\n",
" ARIMA(2,0,2)(1,1,0)[12] with drift : 6228.543\n",
" ARIMA(2,0,3)(0,1,0)[12] : 6405.307\n",
" ARIMA(2,0,3)(0,1,0)[12] with drift : 6407.304\n",
" ARIMA(3,0,0)(0,1,0)[12] : 6403.278\n",
" ARIMA(3,0,0)(0,1,0)[12] with drift : 6405.261\n",
" ARIMA(3,0,0)(0,1,1)[12] : 6071.603\n",
" ARIMA(3,0,0)(0,1,1)[12] with drift : 6071.266\n",
" ARIMA(3,0,0)(0,1,2)[12] : 6073.304\n",
" ARIMA(3,0,0)(0,1,2)[12] with drift : 6072.566\n",
" ARIMA(3,0,0)(1,1,0)[12] : 6225.121\n",
" ARIMA(3,0,0)(1,1,0)[12] with drift : 6227.059\n",
" ARIMA(3,0,0)(1,1,1)[12] : 6073.283\n",
" ARIMA(3,0,0)(1,1,1)[12] with drift : 6072.487\n",
" ARIMA(3,0,0)(2,1,0)[12] : 6151.402\n",
" ARIMA(3,0,0)(2,1,0)[12] with drift : 6153.25\n",
" ARIMA(3,0,1)(0,1,0)[12] : 6404.729\n",
" ARIMA(3,0,1)(0,1,0)[12] with drift : 6406.716\n",
" ARIMA(3,0,1)(0,1,1)[12] : 6069.052\n",
" ARIMA(3,0,1)(0,1,1)[12] with drift : 6069.488\n",
" ARIMA(3,0,1)(1,1,0)[12] : 6226.582\n",
" ARIMA(3,0,1)(1,1,0)[12] with drift : 6228.523\n",
" ARIMA(3,0,2)(0,1,0)[12] : 6405.177\n",
" ARIMA(3,0,2)(0,1,0)[12] with drift : 6407.174\n",
" ARIMA(4,0,0)(0,1,0)[12] : 6405.238\n",
" ARIMA(4,0,0)(0,1,0)[12] with drift : 6407.227\n",
" ARIMA(4,0,0)(0,1,1)[12] : 6073.562\n",
" ARIMA(4,0,0)(0,1,1)[12] with drift : 6073.267\n",
" ARIMA(4,0,0)(1,1,0)[12] : 6227.043\n",
" ARIMA(4,0,0)(1,1,0)[12] with drift : 6228.986\n",
" ARIMA(4,0,1)(0,1,0)[12] : 6406.43\n",
" ARIMA(4,0,1)(0,1,0)[12] with drift : 6408.424\n",
" ARIMA(5,0,0)(0,1,0)[12] : 6403.707\n",
" ARIMA(5,0,0)(0,1,0)[12] with drift : 6405.703\n",
"\n",
"\n",
"\n",
" Best model: ARIMA(1,0,2)(0,1,1)[12] \n",
"\n"
]
},
{
"data": {
"text/plain": [
"Series: INMET \n",
"ARIMA(1,0,2)(0,1,1)[12] \n",
"Box Cox transformation: lambda= 1.999924 \n",
"\n",
"Coefficients:\n",
" ar1 ma1 ma2 sma1\n",
" 0.9200 -0.7173 -0.0948 -0.8660\n",
"s.e. 0.0589 0.0693 0.0521 0.0297\n",
"\n",
"sigma^2 estimated as 472.1: log likelihood=-3028.46\n",
"AIC=6066.93 AICc=6067.02 BIC=6089.48"
]
},
"execution_count": 37,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"# Reajustando\n",
"ajuste_auto_bc_outliers <- auto.arima(INMET, stepwise=FALSE, approximation=FALSE, trace=TRUE, lambda=l)\n",
"ajuste_auto_bc_outliers"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"\n",
"\tLilliefors (Kolmogorov-Smirnov) normality test\n",
"\n",
"data: ajuste_auto_bc_outliers$residuals\n",
"D = 0.048307, p-value = 0.000666\n"
]
},
"execution_count": 38,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 38,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 38,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"qqnorm(ajuste_auto_bc_outliers$residuals)\n",
"qqline(ajuste_auto_bc_outliers$residuals)\n",
"lillie.test(ajuste_auto_bc_outliers$residuals)\n",
"hist(ajuste_auto_bc_outliers$residuals)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Ainda tem 9 indivíduos que poderiam ser considerados outliers, mas paramos, por hora, por aqui. Verificar a possibilidade de bootstrap.\n",
"\n",
"https://cran.r-project.org/web/packages/TSA/TSA.pdf\n",
"\n",
"https://www.rdocumentation.org/packages/boot/versions/1.3-20/topics/tsboot\n"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"\n",
" | lags | statistic | p-value |
|---|
\n",
"\n",
"\t | 5 | 2.042474 | 0.7622378 |
|---|
\n",
"\t | 10 | 6.002297 | 0.6383616 |
|---|
\n",
"\t | 15 | 10.464827 | 0.5104895 |
|---|
\n",
"\t | 20 | 15.461162 | 0.4055944 |
|---|
\n",
"\t | 25 | 20.134118 | 0.3546454 |
|---|
\n",
"\t | 30 | 24.229868 | 0.3326673 |
|---|
\n",
"\n",
"
\n"
]
},
"execution_count": 39,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"portest(ajuste_auto_bc_outliers$residuals) # Testes de portmanteau (ruído branco)"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"\n",
"\tBox-Ljung test\n",
"\n",
"data: ajuste_auto_bc$residuals\n",
"X-squared = 18.186, df = 20, p-value = 0.5751\n"
]
},
"execution_count": 40,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"# Testando diretamente\n",
"Box.test(ajuste_auto_bc$residuals, lag=20, type=\"Ljung-Box\") # Testa se existe autocorrelação entre os resíduos"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
" Point Forecast Lo 80 Hi 80 Lo 95 Hi 95\n",
"Jan 2018 27.02914 25.96459 28.05332 25.38299 28.58063\n",
"Feb 2018 27.00269 25.91430 28.04888 25.31921 28.58721\n",
"Mar 2018 27.08752 25.99865 28.13429 25.40335 28.67295"
]
},
"execution_count": 41,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"\n",
" | ME | RMSE | MAE | MPE | MAPE | MASE | ACF1 |
|---|
\n",
"\n",
"\t| Training set | 0.01058867 | 0.8544344 | 0.6447901 | -0.05951616 | 2.527899 | 0.7231742 | -0.005222841 |
|---|
\n",
"\n",
"
\n"
]
},
"execution_count": 41,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"image/svg+xml": "\n\n"
},
"execution_count": 41,
"metadata": {
"image/svg+xml": {
"isolated": true
}
},
"output_type": "execute_result"
}
],
"source": [
"previsoes_auto_bc <- forecast(ajuste_auto_bc, h=3)\n",
"previsoes_auto_bc\n",
"accuracy(previsoes_auto_bc)\n",
"plot(previsoes_auto_bc)"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"library(boot)\n"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"AIC_Boot <- function(ts) {\n",
" aux <- Arima(ts, lambda=l, order=c(1, 0, 2), seasonal=list(order=c(0, 1, 1), period=12))\n",
" return(aux$aic)\n",
"}"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"aux <- Arima(INMET, lambda=l, order=c(1, 0, 2), seasonal=list(order=c(0, 1, 1), period=12))"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"6066.92790473731"
]
},
"execution_count": 45,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"AIC_Boot(INMET)"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"# Bootstrap estacionário com bloco de 20\n",
"boot_ajuste_auto_bc_outliers <- tsboot(INMET, AIC_Boot, R = 99, l = 20, sim = \"geom\")"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"\n",
"STATIONARY BOOTSTRAP FOR TIME SERIES\n",
"\n",
"Average Block Length of 20 \n",
"\n",
"Call:\n",
"tsboot(tseries = INMET, statistic = AIC_Boot, R = 99, l = 20, \n",
" sim = \"geom\")\n",
"\n",
"\n",
"Bootstrap Statistics :\n",
" original bias std. error\n",
"t1* 6066.928 577.0108 51.53266"
]
},
"execution_count": 47,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"boot_ajuste_auto_bc_outliers"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"Warning message in boot.ci(boot_ajuste_auto_bc_outliers):\n",
"“bootstrap variances needed for studentized intervals”"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Warning message in boot.ci(boot_ajuste_auto_bc_outliers):\n",
"“BCa intervals not defined for time series bootstraps”"
]
},
{
"data": {
"text/plain": [
"BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS\n",
"Based on 99 bootstrap replicates\n",
"\n",
"CALL : \n",
"boot.ci(boot.out = boot_ajuste_auto_bc_outliers)\n",
"\n",
"Intervals : \n",
"Level Normal Basic Percentile \n",
"95% (5389, 5591 ) (5390, 5602 ) (6532, 6743 ) \n",
"Calculations and Intervals on Original Scale\n",
"Some basic intervals may be unstable\n",
"Some percentile intervals may be unstable"
]
},
"execution_count": 48,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"boot.ci(boot_ajuste_auto_bc_outliers)"
]
}
],
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