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Project: Pesquisa
Views: 261
Kernel: R (R-Project)
# https://cran.r-project.org/web/views/TimeSeries.html
# https://www.datacamp.com/tracks/time-series-with-r
# https://www.datascience.com/blog/introduction-to-forecasting-with-arima-in-r-learn-data-science-tutorials
# https://www.otexts.org/fpp/8/7

library(gdata)
library(MASS)
library(imputeTS)
library(tseries)
library(forecast)
library(fUnitRoots)
library(portes)
library(nortest)
library(tsoutliers)
library(boot)
gdata: read.xls support for 'XLS' (Excel 97-2004) files ENABLED. gdata: read.xls support for 'XLSX' (Excel 2007+) files ENABLED. Attaching package: ‘gdata’ The following object is masked from ‘package:stats’: nobs The following object is masked from ‘package:utils’: object.size The following object is masked from ‘package:base’: startsWith Attaching package: ‘tseries’ The following object is masked from ‘package:imputeTS’: na.remove Loading required package: timeDate Loading required package: timeSeries Loading required package: fBasics Loading required package: parallel 
T <- read.xls("Dados.xlsx", sheet = 1, header = TRUE, stringsAsFactors = FALSE)

head(T)
AnoMesAnoFAnoF2INMETAGRITEMICEA
1961 1 22296 1961.0426.38 NA NA
1961 2 22327 1961.1326.54 NA NA
1961 3 22355 1961.2126.70 NA NA
1961 4 22386 1961.2926.65 NA NA
1961 5 22416 1961.3825.50 NA NA
1961 6 22447 1961.4622.78 NA NA 
sapply(T, class) # Verificando os tipos
Ano
'integer'
Mes
'integer'
AnoF
'integer'
AnoF2
'numeric'
INMET
'numeric'
AGRITEM
'numeric'
ICEA
'numeric'
INMET <- ts(T$INMET,start=c(1961,1),f=12)
INMET
plot(INMET)
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1990 NA NA NA NA NA NA NA NA NA NA NA NA 1991 NA NA NA NA NA NA NA NA NA NA NA NA 1992 NA NA NA NA NA NA NA NA NA NA NA NA 1993 NA NA NA NA NA NA NA NA NA NA NA NA 1994 NA NA NA NA NA NA NA NA NA NA NA NA 1995 NA 26.28 27.09 25.52 24.72 23.77 24.57 25.01 28.15 28.25 27.41 28.52 1996 NA NA NA NA NA NA NA NA NA NA NA NA 1997 NA NA NA NA NA NA NA NA NA NA NA NA 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 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 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 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 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 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 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 2005 NA NA NA NA NA NA 23.85 26.28 25.79 28.26 27.87 27.73 2006 27.22 27.06 26.94 26.13 22.97 24.81 NA 27.22 26.49 27.46 28.41 27.31 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 2008 26.42 26.92 26.80 NA NA NA NA NA NA NA NA NA 2009 NA NA NA NA 25.52 22.62 23.90 25.66 27.08 28.72 28.24 NA 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 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 2012 26.61 26.71 26.69 27.55 25.20 24.25 23.52 25.82 28.63 29.37 NA NA 2013 NA NA 27.48 26.71 25.62 26.09 23.98 24.22 28.38 28.26 28.05 27.40 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 2015 27.55 27.34 27.16 27.07 26.32 24.98 24.52 27.61 29.85 NA NA NA 2016 NA NA NA 27.11 25.47 22.98 23.93 26.19 27.11 28.04 27.07 27.31 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
Image in a Jupyter notebook
ICEA <- ts(T$ICEA,start=c(1961,1),f=12)
ICEA
plot(ICEA, cex.lab=1.5, cex.axis=2)

# cex.main: Size of main title
# cex.lab: Size of axis labels (the text describing the axis)
# cex.axis: Size of axis text (the values that indicate the axis tick labels)
Jan Feb Mar Apr May Jun Jul Aug 1961 NA NA NA NA NA NA NA NA 1962 NA NA NA NA NA NA NA NA 1963 NA NA NA NA 25.39000 24.90000 25.71000 28.59000 1964 28.44000 28.25000 27.63000 28.76000 25.80000 24.73000 23.76000 29.25000 1965 27.45000 27.47000 26.37000 26.92000 26.84000 26.65000 25.20000 27.76000 1966 27.89000 27.09000 27.50000 27.82000 26.68000 26.04000 26.30000 26.13000 1967 28.59000 27.96000 28.08000 27.01000 27.53000 24.56000 26.27000 29.12000 1968 27.99000 27.06000 27.69000 26.02000 24.07000 26.19000 26.61000 27.75000 1969 28.37000 28.02000 28.78000 28.54000 27.77000 25.76000 26.32000 27.01000 1970 28.83000 27.50000 28.95000 28.41000 26.15000 26.52000 25.37000 27.75000 1971 27.90000 27.16000 27.80000 27.08000 25.06000 24.12000 25.81000 27.12000 1972 27.83000 27.38000 28.06000 26.61000 28.48000 27.10000 25.48000 26.78000 1973 29.48000 28.66000 29.40000 29.71000 26.38000 27.23000 24.35000 26.03000 1974 27.57000 27.64000 26.67000 26.18000 25.63000 26.20000 24.87000 27.36000 1975 27.41000 27.65000 27.46000 27.13000 24.94000 24.98000 23.88000 27.09000 1976 27.58000 26.51000 26.11000 26.57000 25.58000 23.42000 24.86000 27.08000 1977 27.28000 26.96000 28.42000 26.46000 24.61000 24.88000 27.47000 26.89000 1978 27.74000 27.25000 28.41000 27.46000 25.93000 25.87000 27.83000 24.33000 1979 27.56000 27.51000 27.63000 26.23000 26.33000 24.14000 26.12000 29.42000 1980 27.05000 26.63000 27.90000 28.25000 26.97000 24.00000 24.78000 27.27000 1981 27.49000 27.18000 27.42000 27.94000 27.69000 23.35000 22.28000 27.55000 1982 27.11000 27.03000 26.90000 26.70000 23.84000 26.36000 27.43000 27.20000 1983 27.34000 27.99000 27.35000 27.61000 26.97000 22.18000 23.70000 24.23000 1984 27.25000 26.59000 26.82000 25.39000 25.77000 23.65000 24.81000 24.32000 1985 26.44000 27.41000 27.36000 26.59000 26.61000 23.16000 24.15000 24.63000 1986 27.54000 26.93000 26.84000 27.79000 26.14000 24.28000 23.98000 26.29000 1987 27.77000 27.01000 26.55000 27.70000 25.22000 23.88000 26.45000 24.84000 1988 27.43000 26.61000 27.48000 26.92000 24.42000 23.44000 21.90000 26.83000 1989 26.51000 26.80000 26.52000 27.11000 24.37000 25.31000 22.94000 26.08000 1990 26.48009 26.07057 26.80931 26.39176 24.50474 23.62948 22.11987 25.52453 1991 26.39176 26.64872 25.75740 25.78149 25.53256 23.99886 23.71781 25.17925 1992 26.02239 NA NA NA NA 22.23229 24.48868 24.54489 1993 26.63266 25.22743 NA NA NA 23.79008 23.29223 NA 1994 26.69690 26.36767 26.25525 25.84573 25.75740 24.50474 23.66160 25.55665 1995 25.66104 26.53000 27.63000 NA NA 24.86000 26.28000 26.30000 1996 NA 26.34358 26.50418 26.21510 25.50044 22.14396 24.07113 27.02612 1997 NA NA NA NA NA NA NA NA 1998 NA NA NA NA NA NA NA NA 1999 NA NA NA NA NA NA NA NA 2000 NA NA NA NA NA NA NA NA 2001 NA NA NA NA NA NA 26.89000 29.82000 2002 29.09000 28.36000 29.24000 29.48000 28.36000 25.65000 26.46000 29.65000 2003 26.62000 26.20000 26.26000 25.79000 25.07000 24.91000 24.09000 25.26000 2004 26.58000 27.51000 27.20000 28.85000 23.66000 23.98000 23.17000 25.72000 2005 25.95012 26.22313 25.73331 25.23546 25.53256 25.85376 23.62000 27.00000 2006 26.75000 26.54000 26.49000 22.99000 25.93000 26.15000 25.08289 27.08000 2007 29.58000 28.77000 29.56000 29.82000 24.69000 24.39000 23.42000 24.56000 2008 26.07000 26.62000 26.49000 24.87411 23.31632 22.80241 24.83397 26.87355 2009 26.24722 25.80558 25.23546 25.62892 25.44000 22.79000 23.20000 25.97000 2010 26.95000 27.47000 27.36000 26.55000 23.80000 25.08000 23.09000 25.80000 2011 26.21000 26.26000 25.91000 26.89000 24.48000 24.30000 24.78000 26.05000 2012 26.04000 25.98000 25.79000 26.16000 24.56000 23.40000 22.97000 26.00000 2013 25.35591 25.22743 26.57000 25.38000 24.79000 25.26000 23.49000 24.20000 2014 25.41000 25.24000 26.04000 25.87000 24.16000 24.11000 23.08000 27.09000 2015 26.70000 26.22000 26.30000 26.41000 25.67000 24.99000 24.61000 28.43000 2016 26.01436 26.80128 26.07057 26.97000 25.04000 22.86000 25.48000 23.84000 2017 26.81000 26.58000 26.92000 26.49000 26.71000 24.27000 23.75000 27.72000 Sep Oct Nov Dec 1961 NA NA NA NA 1962 NA NA NA NA 1963 29.36000 31.36000 29.51000 29.84000 1964 28.96000 27.48000 27.70000 27.52000 1965 28.50000 28.15000 28.12000 27.99000 1966 28.38000 29.77000 29.56000 29.39000 1967 30.34000 30.06000 29.00000 28.34000 1968 26.61000 29.22000 30.11000 27.50000 1969 30.18000 28.50000 28.84000 28.54000 1970 29.39000 29.60000 29.92000 29.40000 1971 29.75000 27.35000 28.14000 28.32000 1972 29.17000 29.27000 28.38000 28.77000 1973 28.55000 29.55000 27.99000 27.82000 1974 27.00000 28.30000 28.78000 27.05000 1975 29.20000 28.64000 27.39000 27.73000 1976 26.15000 27.82000 27.04000 27.56000 1977 27.55000 28.08000 28.42000 27.65000 1978 28.23000 28.01000 27.30000 27.33000 1979 27.85000 29.74000 28.15000 27.13000 1980 27.09000 29.76000 27.61000 26.34000 1981 26.99000 28.08000 28.04000 26.84000 1982 26.70000 28.75000 27.92000 27.54000 1983 25.72000 26.07000 25.85000 26.72000 1984 26.65000 28.57000 27.64000 26.92000 1985 26.68000 28.00000 27.78000 28.71000 1986 25.46000 26.03000 28.91000 27.74000 1987 27.99000 29.06000 28.24000 27.29000 1988 28.73000 29.14000 28.11000 27.42000 1989 26.03000 28.52000 28.44000 26.96000 1990 24.74564 27.50791 27.06627 26.81734 1991 27.09839 26.67281 26.63266 26.76113 1992 26.32752 25.93406 26.55236 26.63266 1993 26.74507 27.34731 27.46776 26.94582 1994 27.08233 27.56412 27.58018 26.02239 1995 28.83000 28.46000 27.64000 27.07000 1996 26.10268 27.01006 26.39979 26.89764 1997 NA NA NA NA 1998 NA NA NA NA 1999 NA NA NA NA 2000 NA NA NA NA 2001 30.18000 29.73000 29.17000 27.94000 2002 29.16000 29.97000 28.51000 27.49000 2003 27.30000 27.43000 26.69000 27.51000 2004 28.51000 28.13000 27.50000 27.33000 2005 25.32000 27.60000 27.41000 26.87000 2006 28.55000 28.72000 29.98000 29.05000 2007 29.44000 28.33000 26.77000 26.45000 2008 26.31949 26.98597 27.01809 26.01436 2009 27.05000 28.21000 27.95000 25.87785 2010 30.05000 28.40000 26.72000 27.37000 2011 23.40000 NA 27.99000 26.88000 2012 28.79000 28.62000 25.69316 26.09465 2013 27.54000 26.79000 26.70000 26.18000 2014 29.80000 29.77000 28.26000 26.49000 2015 29.72000 27.77290 27.29914 27.20278 2016 26.06000 27.70000 26.54000 26.61000 2017 29.42000 27.99000 28.14000 27.07000
Image in a Jupyter notebook

Observe que as séries possuem falhas. Inicialmente as preencheremos com uma regressão entre INMET e ICEA.

mod1 = lm(INMET ~ ICEA, data = T)
summary(mod1)
Call: lm(formula = INMET ~ ICEA, data = T) Residuals: Min 1Q Median 3Q Max -3.7194 -0.6334 -0.0078 0.7152 5.7882 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.38188 0.81258 5.393 1.08e-07 *** ICEA 0.80299 0.03012 26.659 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.13 on 493 degrees of freedom (189 observations deleted due to missingness) Multiple R-squared: 0.5904, Adjusted R-squared: 0.5896 F-statistic: 710.7 on 1 and 493 DF, p-value: < 2.2e-16 
aux <- T[is.na(T$INMET)&!is.na(T$ICEA),]

aux <- predict(mod1, newdata=T[is.na(T$INMET)&!is.na(T$ICEA),]) # Estima INMET com base no ICEA

head(aux)
349
26.4800887468442
350
26.0705653435554
351
26.8093134436057
352
26.3917601696642
353
24.5047405662748
354
23.6294846651282
T[is.na(T$INMET)&!is.na(T$ICEA),] <- aux

INMET <- ts(T$INMET,start=c(1961,1),f=12)
INMET
plot(INMET)
Jan Feb Mar Apr May Jun Jul Aug 1961 26.38000 26.54000 26.70000 26.65000 25.50000 22.78000 23.34000 27.72000 1962 26.43000 26.85000 27.18000 26.13000 24.49000 22.36000 20.55000 25.37000 1963 26.39000 26.24000 26.34000 26.57000 24.75000 22.97000 23.63000 26.74000 1964 27.64000 27.46000 26.70000 27.59000 24.57000 23.51000 22.70000 27.69000 1965 26.57000 26.54000 25.36000 25.90000 25.67000 25.10000 23.96000 26.33000 1966 27.30000 26.33000 26.91000 26.45000 25.49000 24.68000 24.61000 24.32000 1967 27.49000 26.15000 26.37000 25.64000 23.66000 22.02000 22.37000 25.11000 1968 26.19000 25.62000 25.98000 23.72000 20.93000 21.78000 22.03000 24.92000 1969 26.58000 26.61000 26.72000 26.11000 24.55000 23.03000 21.83000 23.83000 1970 27.19000 26.33000 26.47000 26.30000 23.58000 22.74000 21.36000 23.56000 1971 26.57000 25.59000 26.06000 24.88000 22.78000 20.92000 21.88000 23.21000 1972 26.08000 25.88000 26.57000 23.99000 24.95000 23.25000 22.16000 24.63000 1973 27.67000 27.06000 27.38000 26.89000 23.39000 23.95000 21.24000 24.42000 1974 25.98000 26.37000 25.57000 25.00000 23.73000 24.41000 22.54000 26.04000 1975 26.53000 26.38000 26.21000 25.93000 23.52000 23.19000 22.45000 25.38000 1976 26.88000 26.04000 25.33000 25.24000 24.11000 21.74000 22.97000 25.44000 1977 26.11000 25.70000 26.82000 25.03000 22.96000 23.62000 24.74000 25.04000 1978 26.48000 27.14000 26.60000 26.32000 24.31000 23.37000 25.09000 22.79000 1979 26.40000 26.68000 26.36000 25.35000 24.63000 21.82000 23.73000 26.23000 1980 26.86000 26.25000 26.78000 26.42000 25.17000 23.35000 22.80000 25.85000 1981 26.45000 26.18000 26.54000 26.55000 25.73000 21.87000 20.75000 25.28000 1982 26.33000 26.32000 26.05000 25.79000 24.27000 24.15000 24.47000 25.24000 1983 26.95000 27.51000 26.80000 27.19000 26.12000 22.08000 23.04000 22.93000 1984 26.74000 26.20000 26.60000 25.41000 25.17000 22.72000 23.56000 23.40000 1985 26.06000 27.13000 26.84000 26.27000 25.94000 21.97000 23.07000 24.20000 1986 27.16000 26.57000 26.62000 27.18000 25.59000 23.20000 23.31000 25.89000 1987 27.16000 26.61000 26.14000 27.17000 24.59000 23.36000 24.90000 23.86000 1988 27.15000 26.64000 27.27000 26.81000 24.46000 22.90000 20.86000 25.18000 1989 26.08000 26.37000 26.22000 26.54000 23.89000 24.44000 21.86000 25.10000 1990 26.48009 26.07057 26.80931 26.39176 24.50474 23.62948 22.11987 25.52453 1991 26.39176 26.64872 25.75740 25.78149 25.53256 23.99886 23.71781 25.17925 1992 26.02239 NA NA NA NA 22.23229 24.48868 24.54489 1993 26.63266 25.22743 NA NA NA 23.79008 23.29223 NA 1994 26.69690 26.36767 26.25525 25.84573 25.75740 24.50474 23.66160 25.55665 1995 25.66104 26.28000 27.09000 25.52000 24.72000 23.77000 24.57000 25.01000 1996 NA 26.34358 26.50418 26.21510 25.50044 22.14396 24.07113 27.02612 1997 NA NA NA NA NA NA NA NA 1998 28.52000 27.86000 27.97000 27.74000 23.92000 23.62000 25.02000 26.11000 1999 27.17000 27.44000 26.28000 26.03000 24.68000 24.35000 23.53000 24.88000 2000 27.76000 26.99000 26.45000 26.59000 25.23000 24.40000 21.90000 26.84000 2001 26.92000 27.41000 26.58000 27.34000 24.24000 22.96000 24.82000 26.88000 2002 27.40000 26.46000 27.01000 27.12000 25.91000 23.34000 24.05000 27.20000 2003 27.01000 26.67000 23.35000 25.97000 24.88000 24.13000 23.18000 24.33000 2004 27.22000 27.05000 27.70000 27.02000 23.34000 23.74000 23.48000 25.26000 2005 25.95012 26.22313 25.73331 25.23546 25.53256 25.85376 23.85000 26.28000 2006 27.22000 27.06000 26.94000 26.13000 22.97000 24.81000 25.08289 27.22000 2007 27.61000 26.89000 27.43000 27.44000 23.61000 23.89000 23.45000 24.26000 2008 26.42000 26.92000 26.80000 24.87411 23.31632 22.80241 24.83397 26.87355 2009 26.24722 25.80558 25.23546 25.62892 25.52000 22.62000 23.90000 25.66000 2010 26.91000 27.48000 27.91000 26.58000 23.66000 24.71000 22.82000 24.73000 2011 26.82000 26.82000 26.75000 27.46000 24.76000 24.01000 24.27000 26.02000 2012 26.61000 26.71000 26.69000 27.55000 25.20000 24.25000 23.52000 25.82000 2013 25.35591 25.22743 27.48000 26.71000 25.62000 26.09000 23.98000 24.22000 2014 26.48000 26.35000 27.14000 27.05000 25.23000 24.56000 23.31000 26.45000 2015 27.55000 27.34000 27.16000 27.07000 26.32000 24.98000 24.52000 27.61000 2016 26.01436 26.80128 26.07057 27.11000 25.47000 22.98000 23.93000 26.19000 2017 27.21000 27.10000 27.38000 27.14000 27.06000 24.13000 23.33000 27.33000 Sep Oct Nov Dec 1961 28.90000 27.69000 27.37000 26.92000 1962 27.94000 26.43000 28.49000 26.34000 1963 28.01000 29.04000 28.10000 28.49000 1964 27.46000 26.44000 27.01000 26.44000 1965 27.44000 27.19000 27.07000 27.09000 1966 26.83000 28.07000 28.12000 28.44000 1967 28.03000 27.83000 26.88000 26.88000 1968 24.49000 27.01000 28.04000 26.12000 1969 28.04000 26.94000 27.43000 26.94000 1970 27.11000 27.52000 27.77000 27.15000 1971 27.09000 25.90000 26.44000 26.64000 1972 27.19000 27.52000 27.29000 27.08000 1973 27.17000 27.63000 26.71000 26.56000 1974 26.16000 27.24000 27.50000 26.26000 1975 27.84000 27.24000 26.13000 26.35000 1976 25.35000 26.87000 26.07000 26.28000 1977 26.11000 26.25000 26.68000 26.50000 1978 26.26000 27.46000 26.83000 26.66000 1979 26.25000 28.34000 27.09000 26.83000 1980 25.48000 28.22000 26.42000 26.36000 1981 25.04000 27.38000 27.20000 26.72000 1982 25.64000 27.46000 27.22000 26.80000 1983 25.46000 25.75000 25.62000 26.22000 1984 25.84000 27.88000 27.08000 26.35000 1985 26.41000 27.72000 27.48000 28.41000 1986 24.79000 26.30000 28.51000 27.21000 1987 27.65000 28.63000 28.00000 27.12000 1988 27.40000 28.58000 27.71000 27.02000 1989 25.16000 27.68000 27.88000 26.73000 1990 24.74564 27.50791 27.06627 26.81734 1991 27.09839 26.67281 26.63266 26.76113 1992 26.32752 25.93406 26.55236 26.63266 1993 26.74507 27.34731 27.46776 26.94582 1994 27.08233 27.56412 27.58018 26.02239 1995 28.15000 28.25000 27.41000 28.52000 1996 26.10268 27.01006 26.39979 26.89764 1997 NA NA NA NA 1998 26.30000 28.04000 27.95000 27.23000 1999 28.32000 28.29000 26.97000 27.55000 2000 26.73000 28.22000 27.34000 27.52000 2001 27.92000 27.73000 27.16000 26.38000 2002 26.83000 29.34000 28.65000 27.90000 2003 27.20000 27.50000 27.23000 28.08000 2004 27.54000 28.25000 27.20000 27.83000 2005 25.79000 28.26000 27.87000 27.73000 2006 26.49000 27.46000 28.41000 27.31000 2007 28.50000 28.06000 27.18000 27.16000 2008 26.31949 26.98597 27.01809 26.01436 2009 27.08000 28.72000 28.24000 25.87785 2010 29.37000 28.71000 27.04000 27.64000 2011 28.96000 28.57000 28.57000 27.38000 2012 28.63000 29.37000 25.69316 26.09465 2013 28.38000 28.26000 28.05000 27.40000 2014 29.47000 29.43000 28.53000 27.44000 2015 29.85000 27.77290 27.29914 27.20278 2016 27.11000 28.04000 27.07000 27.31000 2017 29.08000 28.46000 28.72000 27.63000
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# As falhas que sobraram foram preenchidas pelo estimador de Kalman.
INMET <- na.kalman(INMET, model = "auto.arima")
plot(INMET)
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# Decomposição da série temporal (aditiva, possivelmente baseada na transformáda rápida de Fourier, e apenas para fins de análise preliminar)
plot(decompose(INMET))
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ARIMA

Etapas: a) Verificar se existe a necessidade de uma transformação na série original, com objetivo de estabilizar a variância; b) Tornar a série estacionária por meio de diferenças, de modo que o processo dZt seja reduzido a um ARMA(p,q) c) Identificar o processo ARMA(p,q) resultante. d) Verificação da estacionariedade e da invertibilidade.

FAC : correlação simples entre Zt e Zt – k em função da defasagem k. 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.

FACP -> AR

D -> I

FAC -> MA

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). 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. 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.

Notação: arima(p, d, q), sendo p relacionado a autocorrelação parcial, d a diferença entre os valores, q associado a autocorrelação).

ndiffs(INMET) # Valor de d para fazer a série estacionária.
1
urkpssTest(INMET, type = c("tau"), lags = c("short"),use.lag = NULL, doplot = TRUE)
tsestacionaria = diff(INMET, differences=1)
plot(tsestacionaria)
Title: KPSS Unit Root Test Test Results: NA Description: Thu Dec 6 14:05:52 2018 by user:
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# Teste da estacionariedade da série
adf.test(INMET, alternative="stationary")
Warning message in adf.test(INMET, alternative = "stationary"): “p-value smaller than printed p-value”
Augmented Dickey-Fuller Test data: INMET Dickey-Fuller = -15.038, Lag order = 8, p-value = 0.01 alternative hypothesis: stationary 
acf(INMET)
pacf(INMET)
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O teste não rejeitou a hipótese da estacionariedade, mas o ACF e a função sugerem utilizar d=1.

# Calculando as correlações para uma diferença de ordem 1
acf(diff(INMET, differences=1))
pacf(diff(INMET, differences=1))
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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.

ajuste_0 <- Arima(INMET, c(1, 1, 1))
ajuste_0
Series: INMET ARIMA(1,1,1) Coefficients: ar1 ma1 -0.2652 0.3130 s.e. 0.4061 0.3994 sigma^2 estimated as 2.154: log likelihood=-1230.11 AIC=2466.22 AICc=2466.26 BIC=2479.8
ajuste_1 <- Arima(INMET, c(1, 1, 1), seasonal=list(order = c(1, 1, 1), period = 12))
ajuste_1
Series: INMET ARIMA(1,1,1)(1,1,1)[12] Coefficients: ar1 ma1 sar1 sma1 0.1792 -0.9649 0.0360 -0.9129 s.e. 0.0412 0.0143 0.0479 0.0322 sigma^2 estimated as 0.7475: log likelihood=-864.86 AIC=1739.73 AICc=1739.82 BIC=1762.27

Utilizando o "automático"

# Automatizando
ajuste_auto <- auto.arima(INMET, stepwise=FALSE, approximation=FALSE, trace=TRUE)
ajuste_auto
ARIMA(0,0,0)(0,1,0)[12] : 2101.997 ARIMA(0,0,0)(0,1,0)[12] with drift : 2103.936 ARIMA(0,0,0)(0,1,1)[12] : 1777.447 ARIMA(0,0,0)(0,1,1)[12] with drift : 1776.128 ARIMA(0,0,0)(0,1,2)[12] : 1779.456 ARIMA(0,0,0)(0,1,2)[12] with drift : 1777.932 ARIMA(0,0,0)(1,1,0)[12] : 1916.368 ARIMA(0,0,0)(1,1,0)[12] with drift : 1918.209 ARIMA(0,0,0)(1,1,1)[12] : 1779.455 ARIMA(0,0,0)(1,1,1)[12] with drift : 1777.902 ARIMA(0,0,0)(1,1,2)[12] : Inf ARIMA(0,0,0)(1,1,2)[12] with drift : 1780.115 ARIMA(0,0,0)(2,1,0)[12] : 1841.745 ARIMA(0,0,0)(2,1,0)[12] with drift : 1843.425 ARIMA(0,0,0)(2,1,1)[12] : 1780.857 ARIMA(0,0,0)(2,1,1)[12] with drift : Inf ARIMA(0,0,0)(2,1,2)[12] : 1783.475 ARIMA(0,0,0)(2,1,2)[12] with drift : Inf ARIMA(0,0,1)(0,1,0)[12] : 2079.529 ARIMA(0,0,1)(0,1,0)[12] with drift : 2081.493 ARIMA(0,0,1)(0,1,1)[12] : 1750.092 ARIMA(0,0,1)(0,1,1)[12] with drift : 1748.647 ARIMA(0,0,1)(0,1,2)[12] : 1751.788 ARIMA(0,0,1)(0,1,2)[12] with drift : 1749.72 ARIMA(0,0,1)(1,1,0)[12] : 1898.631 ARIMA(0,0,1)(1,1,0)[12] with drift : 1900.521 ARIMA(0,0,1)(1,1,1)[12] : 1751.763 ARIMA(0,0,1)(1,1,1)[12] with drift : 1749.588 ARIMA(0,0,1)(1,1,2)[12] : 1754.146 ARIMA(0,0,1)(1,1,2)[12] with drift : Inf ARIMA(0,0,1)(2,1,0)[12] : 1828.566 ARIMA(0,0,1)(2,1,0)[12] with drift : 1830.327 ARIMA(0,0,1)(2,1,1)[12] : 1753.174 ARIMA(0,0,1)(2,1,1)[12] with drift : Inf ARIMA(0,0,1)(2,1,2)[12] : Inf ARIMA(0,0,1)(2,1,2)[12] with drift : Inf ARIMA(0,0,2)(0,1,0)[12] : 2081.476 ARIMA(0,0,2)(0,1,0)[12] with drift : 2083.447 ARIMA(0,0,2)(0,1,1)[12] : 1748.593 ARIMA(0,0,2)(0,1,1)[12] with drift : 1747.289 ARIMA(0,0,2)(0,1,2)[12] : 1750.456 ARIMA(0,0,2)(0,1,2)[12] with drift : 1748.672 ARIMA(0,0,2)(1,1,0)[12] : 1898.542 ARIMA(0,0,2)(1,1,0)[12] with drift : 1900.447 ARIMA(0,0,2)(1,1,1)[12] : 1750.443 ARIMA(0,0,2)(1,1,1)[12] with drift : 1748.581 ARIMA(0,0,2)(1,1,2)[12] : Inf ARIMA(0,0,2)(1,1,2)[12] with drift : Inf ARIMA(0,0,2)(2,1,0)[12] : 1827.098 ARIMA(0,0,2)(2,1,0)[12] with drift : 1828.888 ARIMA(0,0,2)(2,1,1)[12] : 1751.819 ARIMA(0,0,2)(2,1,1)[12] with drift : Inf ARIMA(0,0,3)(0,1,0)[12] : 2074.456 ARIMA(0,0,3)(0,1,0)[12] with drift : 2076.439 ARIMA(0,0,3)(0,1,1)[12] : 1742.418 ARIMA(0,0,3)(0,1,1)[12] with drift : 1741.336 ARIMA(0,0,3)(0,1,2)[12] : 1744.072 ARIMA(0,0,3)(0,1,2)[12] with drift : Inf ARIMA(0,0,3)(1,1,0)[12] : 1892.367 ARIMA(0,0,3)(1,1,0)[12] with drift : 1894.294 ARIMA(0,0,3)(1,1,1)[12] : 1744.034 ARIMA(0,0,3)(1,1,1)[12] with drift : Inf ARIMA(0,0,3)(2,1,0)[12] : 1823.79 ARIMA(0,0,3)(2,1,0)[12] with drift : 1825.615 ARIMA(0,0,4)(0,1,0)[12] : 2076.328 ARIMA(0,0,4)(0,1,0)[12] with drift : 2078.318 ARIMA(0,0,4)(0,1,1)[12] : 1744.056 ARIMA(0,0,4)(0,1,1)[12] with drift : 1743.052 ARIMA(0,0,4)(1,1,0)[12] : 1894.373 ARIMA(0,0,4)(1,1,0)[12] with drift : 1896.306 ARIMA(0,0,5)(0,1,0)[12] : 2076.84 ARIMA(0,0,5)(0,1,0)[12] with drift : 2078.837 ARIMA(1,0,0)(0,1,0)[12] : 2078.792 ARIMA(1,0,0)(0,1,0)[12] with drift : 2080.76 ARIMA(1,0,0)(0,1,1)[12] : 1745.57 ARIMA(1,0,0)(0,1,1)[12] with drift : 1744.233 ARIMA(1,0,0)(0,1,2)[12] : 1747.296 ARIMA(1,0,0)(0,1,2)[12] with drift : 1745.376 ARIMA(1,0,0)(1,1,0)[12] : 1896.282 ARIMA(1,0,0)(1,1,0)[12] with drift : 1898.182 ARIMA(1,0,0)(1,1,1)[12] : 1747.273 ARIMA(1,0,0)(1,1,1)[12] with drift : Inf ARIMA(1,0,0)(1,1,2)[12] : Inf ARIMA(1,0,0)(1,1,2)[12] with drift : Inf ARIMA(1,0,0)(2,1,0)[12] : 1826.302 ARIMA(1,0,0)(2,1,0)[12] with drift : 1828.082 ARIMA(1,0,0)(2,1,1)[12] : 1748.642 ARIMA(1,0,0)(2,1,1)[12] with drift : Inf ARIMA(1,0,0)(2,1,2)[12] : Inf ARIMA(1,0,0)(2,1,2)[12] with drift : Inf ARIMA(1,0,1)(0,1,0)[12] : 2078.217 ARIMA(1,0,1)(0,1,0)[12] with drift : 2080.198 ARIMA(1,0,1)(0,1,1)[12] : 1736.005 ARIMA(1,0,1)(0,1,1)[12] with drift : 1736.06 ARIMA(1,0,1)(0,1,2)[12] : 1737.915 ARIMA(1,0,1)(0,1,2)[12] with drift : 1737.722 ARIMA(1,0,1)(1,1,0)[12] : 1890.475 ARIMA(1,0,1)(1,1,0)[12] with drift : 1892.418 ARIMA(1,0,1)(1,1,1)[12] : 1737.903 ARIMA(1,0,1)(1,1,1)[12] with drift : 1737.671 ARIMA(1,0,1)(1,1,2)[12] : Inf ARIMA(1,0,1)(1,1,2)[12] with drift : Inf ARIMA(1,0,1)(2,1,0)[12] : 1818.362 ARIMA(1,0,1)(2,1,0)[12] with drift : 1820.24 ARIMA(1,0,1)(2,1,1)[12] : Inf ARIMA(1,0,1)(2,1,1)[12] with drift : Inf ARIMA(1,0,2)(0,1,0)[12] : 2078.237 ARIMA(1,0,2)(0,1,0)[12] with drift : 2080.225 ARIMA(1,0,2)(0,1,1)[12] : 1733.894 ARIMA(1,0,2)(0,1,1)[12] with drift : 1734.251 ARIMA(1,0,2)(0,1,2)[12] : 1735.836 ARIMA(1,0,2)(0,1,2)[12] with drift : Inf ARIMA(1,0,2)(1,1,0)[12] : 1891.939 ARIMA(1,0,2)(1,1,0)[12] with drift : 1893.89 ARIMA(1,0,2)(1,1,1)[12] : 1735.828 ARIMA(1,0,2)(1,1,1)[12] with drift : Inf ARIMA(1,0,2)(2,1,0)[12] : 1820.178 ARIMA(1,0,2)(2,1,0)[12] with drift : 1822.065 ARIMA(1,0,3)(0,1,0)[12] : 2075.174 ARIMA(1,0,3)(0,1,0)[12] with drift : 2077.168 ARIMA(1,0,3)(0,1,1)[12] : 1735.885 ARIMA(1,0,3)(0,1,1)[12] with drift : 1736.175 ARIMA(1,0,3)(1,1,0)[12] : 1892.711 ARIMA(1,0,3)(1,1,0)[12] with drift : 1894.668 ARIMA(1,0,4)(0,1,0)[12] : 2076.587 ARIMA(1,0,4)(0,1,0)[12] with drift : 2078.587 ARIMA(2,0,0)(0,1,0)[12] : 2080.563 ARIMA(2,0,0)(0,1,0)[12] with drift : 2082.538 ARIMA(2,0,0)(0,1,1)[12] : 1743.982 ARIMA(2,0,0)(0,1,1)[12] with drift : 1742.908 ARIMA(2,0,0)(0,1,2)[12] : 1745.839 ARIMA(2,0,0)(0,1,2)[12] with drift : 1744.312 ARIMA(2,0,0)(1,1,0)[12] : 1895.702 ARIMA(2,0,0)(1,1,0)[12] with drift : 1897.619 ARIMA(2,0,0)(1,1,1)[12] : 1745.824 ARIMA(2,0,0)(1,1,1)[12] with drift : Inf ARIMA(2,0,0)(1,1,2)[12] : Inf ARIMA(2,0,0)(1,1,2)[12] with drift : Inf ARIMA(2,0,0)(2,1,0)[12] : 1824.487 ARIMA(2,0,0)(2,1,0)[12] with drift : 1826.3 ARIMA(2,0,0)(2,1,1)[12] : 1747.059 ARIMA(2,0,0)(2,1,1)[12] with drift : Inf ARIMA(2,0,1)(0,1,0)[12] : 2078.787 ARIMA(2,0,1)(0,1,0)[12] with drift : 2080.775 ARIMA(2,0,1)(0,1,1)[12] : 1733.85 ARIMA(2,0,1)(0,1,1)[12] with drift : 1734.396 ARIMA(2,0,1)(0,1,2)[12] : Inf ARIMA(2,0,1)(0,1,2)[12] with drift : Inf ARIMA(2,0,1)(1,1,0)[12] : 1892.015 ARIMA(2,0,1)(1,1,0)[12] with drift : 1893.967 ARIMA(2,0,1)(1,1,1)[12] : Inf ARIMA(2,0,1)(1,1,1)[12] with drift : Inf ARIMA(2,0,1)(2,1,0)[12] : Inf ARIMA(2,0,1)(2,1,0)[12] with drift : Inf ARIMA(2,0,2)(0,1,0)[12] : Inf ARIMA(2,0,2)(0,1,0)[12] with drift : Inf ARIMA(2,0,2)(0,1,1)[12] : 1735.387 ARIMA(2,0,2)(0,1,1)[12] with drift : 1735.626 ARIMA(2,0,2)(1,1,0)[12] : 1892.068 ARIMA(2,0,2)(1,1,0)[12] with drift : 1894.027 ARIMA(2,0,3)(0,1,0)[12] : 2075.99 ARIMA(2,0,3)(0,1,0)[12] with drift : 2077.991 ARIMA(3,0,0)(0,1,0)[12] : 2073.867 ARIMA(3,0,0)(0,1,0)[12] with drift : 2075.853 ARIMA(3,0,0)(0,1,1)[12] : 1739.145 ARIMA(3,0,0)(0,1,1)[12] with drift : 1738.463 ARIMA(3,0,0)(0,1,2)[12] : 1740.747 ARIMA(3,0,0)(0,1,2)[12] with drift : Inf ARIMA(3,0,0)(1,1,0)[12] : 1891.143 ARIMA(3,0,0)(1,1,0)[12] with drift : 1893.081 ARIMA(3,0,0)(1,1,1)[12] : 1740.704 ARIMA(3,0,0)(1,1,1)[12] with drift : Inf ARIMA(3,0,0)(2,1,0)[12] : 1821.908 ARIMA(3,0,0)(2,1,0)[12] with drift : 1823.756 ARIMA(3,0,1)(0,1,0)[12] : 2075.092 ARIMA(3,0,1)(0,1,0)[12] with drift : 2077.084 ARIMA(3,0,1)(0,1,1)[12] : Inf ARIMA(3,0,1)(0,1,1)[12] with drift : 1736.36 ARIMA(3,0,1)(1,1,0)[12] : 1892.573 ARIMA(3,0,1)(1,1,0)[12] with drift : 1894.514 ARIMA(3,0,2)(0,1,0)[12] : 2075.913 ARIMA(3,0,2)(0,1,0)[12] with drift : 2077.914 ARIMA(4,0,0)(0,1,0)[12] : 2075.789 ARIMA(4,0,0)(0,1,0)[12] with drift : 2077.781 ARIMA(4,0,0)(0,1,1)[12] : 1741.105 ARIMA(4,0,0)(0,1,1)[12] with drift : 1740.471 ARIMA(4,0,0)(1,1,0)[12] : 1893.098 ARIMA(4,0,0)(1,1,0)[12] with drift : 1895.041 ARIMA(4,0,1)(0,1,0)[12] : 2076.993 ARIMA(4,0,1)(0,1,0)[12] with drift : 2078.991 ARIMA(5,0,0)(0,1,0)[12] : 2074.108 ARIMA(5,0,0)(0,1,0)[12] with drift : 2076.108 Best model: ARIMA(2,0,1)(0,1,1)[12]
Series: INMET ARIMA(2,0,1)(0,1,1)[12] Coefficients: ar1 ar2 ma1 sma1 1.1203 -0.1462 -0.9194 -0.8870 s.e. 0.1073 0.0667 0.0910 0.0341 sigma^2 estimated as 0.7459: log likelihood=-861.88 AIC=1733.76 AICc=1733.85 BIC=1756.31

No caso, obteve um AIC apenas um pouco melhor.

acf(ajuste_auto$residuals)
pacf(ajuste_auto$residuals)
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qqnorm(ajuste_auto$residuals)
qqline(ajuste_auto$residuals)
lillie.test(ajuste_auto$residuals)
Lilliefors (Kolmogorov-Smirnov) normality test data: ajuste_auto$residuals D = 0.055364, p-value = 3.59e-05
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https://stats.stackexchange.com/questions/101274/how-to-interpret-a-qq-plot

Não adere a Normal, mas também não apresenta comportamento anômalo. É o caso de tentar uma transformação de Box-Cox.

l <- BoxCox.lambda(INMET)
l
1.99992424816297
ajuste_auto_bc <- auto.arima(INMET, stepwise=FALSE, approximation=FALSE, trace=TRUE, lambda=l)
ajuste_auto_bc
ARIMA(0,0,0)(0,1,0)[12] : 6459.492 ARIMA(0,0,0)(0,1,0)[12] with drift : 6461.425 ARIMA(0,0,0)(0,1,1)[12] : 6130.443 ARIMA(0,0,0)(0,1,1)[12] with drift : 6129.316 ARIMA(0,0,0)(0,1,2)[12] : 6132.377 ARIMA(0,0,0)(0,1,2)[12] with drift : 6131.338 ARIMA(0,0,0)(1,1,0)[12] : 6269.231 ARIMA(0,0,0)(1,1,0)[12] with drift : 6271.07 ARIMA(0,0,0)(1,1,1)[12] : 6132.373 ARIMA(0,0,0)(1,1,1)[12] with drift : 6131.338 ARIMA(0,0,0)(1,1,2)[12] : 6134.386 ARIMA(0,0,0)(1,1,2)[12] with drift : 6133.339 ARIMA(0,0,0)(2,1,0)[12] : 6190.466 ARIMA(0,0,0)(2,1,0)[12] with drift : 6192.138 ARIMA(0,0,0)(2,1,1)[12] : 6134.065 ARIMA(0,0,0)(2,1,1)[12] with drift : 6132.104 ARIMA(0,0,0)(2,1,2)[12] : 6136.426 ARIMA(0,0,0)(2,1,2)[12] with drift : 6135.332 ARIMA(0,0,1)(0,1,0)[12] : 6435.915 ARIMA(0,0,1)(0,1,0)[12] with drift : 6437.876 ARIMA(0,0,1)(0,1,1)[12] : 6100.853 ARIMA(0,0,1)(0,1,1)[12] with drift : 6099.665 ARIMA(0,0,1)(0,1,2)[12] : 6102.849 ARIMA(0,0,1)(0,1,2)[12] with drift : 6101.42 ARIMA(0,0,1)(1,1,0)[12] : 6249.518 ARIMA(0,0,1)(1,1,0)[12] with drift : 6251.407 ARIMA(0,0,1)(1,1,1)[12] : 6102.847 ARIMA(0,0,1)(1,1,1)[12] with drift : 6101.388 ARIMA(0,0,1)(1,1,2)[12] : Inf ARIMA(0,0,1)(1,1,2)[12] with drift : 6103.716 ARIMA(0,0,1)(2,1,0)[12] : 6175.527 ARIMA(0,0,1)(2,1,0)[12] with drift : 6177.286 ARIMA(0,0,1)(2,1,1)[12] : 6104.446 ARIMA(0,0,1)(2,1,1)[12] with drift : Inf ARIMA(0,0,1)(2,1,2)[12] : Inf ARIMA(0,0,1)(2,1,2)[12] with drift : 6104.97 ARIMA(0,0,2)(0,1,0)[12] : 6437.798 ARIMA(0,0,2)(0,1,0)[12] with drift : 6439.766 ARIMA(0,0,2)(0,1,1)[12] : 6099.46 ARIMA(0,0,2)(0,1,1)[12] with drift : 6098.418 ARIMA(0,0,2)(0,1,2)[12] : 6101.49 ARIMA(0,0,2)(0,1,2)[12] with drift : 6100.323 ARIMA(0,0,2)(1,1,0)[12] : 6249.341 ARIMA(0,0,2)(1,1,0)[12] with drift : 6251.245 ARIMA(0,0,2)(1,1,1)[12] : 6101.49 ARIMA(0,0,2)(1,1,1)[12] with drift : 6100.308 ARIMA(0,0,2)(1,1,2)[12] : Inf ARIMA(0,0,2)(1,1,2)[12] with drift : Inf ARIMA(0,0,2)(2,1,0)[12] : 6174.002 ARIMA(0,0,2)(2,1,0)[12] with drift : 6175.789 ARIMA(0,0,2)(2,1,1)[12] : 6103.076 ARIMA(0,0,2)(2,1,1)[12] with drift : Inf ARIMA(0,0,3)(0,1,0)[12] : 6429.49 ARIMA(0,0,3)(0,1,0)[12] with drift : 6431.47 ARIMA(0,0,3)(0,1,1)[12] : 6092.457 ARIMA(0,0,3)(0,1,1)[12] with drift : 6091.68 ARIMA(0,0,3)(0,1,2)[12] : 6094.437 ARIMA(0,0,3)(0,1,2)[12] with drift : 6093.404 ARIMA(0,0,3)(1,1,0)[12] : 6241.902 ARIMA(0,0,3)(1,1,0)[12] with drift : 6243.829 ARIMA(0,0,3)(1,1,1)[12] : 6094.432 ARIMA(0,0,3)(1,1,1)[12] with drift : 6093.358 ARIMA(0,0,3)(2,1,0)[12] : 6169.704 ARIMA(0,0,3)(2,1,0)[12] with drift : 6171.529 ARIMA(0,0,4)(0,1,0)[12] : 6431.336 ARIMA(0,0,4)(0,1,0)[12] with drift : 6433.323 ARIMA(0,0,4)(0,1,1)[12] : 6094.109 ARIMA(0,0,4)(0,1,1)[12] with drift : 6093.402 ARIMA(0,0,4)(1,1,0)[12] : 6243.923 ARIMA(0,0,4)(1,1,0)[12] with drift : 6245.856 ARIMA(0,0,5)(0,1,0)[12] : 6431.87 ARIMA(0,0,5)(0,1,0)[12] with drift : 6433.864 ARIMA(1,0,0)(0,1,0)[12] : 6434.885 ARIMA(1,0,0)(0,1,0)[12] with drift : 6436.849 ARIMA(1,0,0)(0,1,1)[12] : 6096.043 ARIMA(1,0,0)(0,1,1)[12] with drift : 6094.989 ARIMA(1,0,0)(0,1,2)[12] : 6098.047 ARIMA(1,0,0)(0,1,2)[12] with drift : 6096.783 ARIMA(1,0,0)(1,1,0)[12] : 6246.859 ARIMA(1,0,0)(1,1,0)[12] with drift : 6248.759 ARIMA(1,0,0)(1,1,1)[12] : 6098.046 ARIMA(1,0,0)(1,1,1)[12] with drift : 6096.756 ARIMA(1,0,0)(1,1,2)[12] : Inf ARIMA(1,0,0)(1,1,2)[12] with drift : Inf ARIMA(1,0,0)(2,1,0)[12] : 6172.965 ARIMA(1,0,0)(2,1,0)[12] with drift : 6174.745 ARIMA(1,0,0)(2,1,1)[12] : 6099.601 ARIMA(1,0,0)(2,1,1)[12] with drift : Inf ARIMA(1,0,0)(2,1,2)[12] : Inf ARIMA(1,0,0)(2,1,2)[12] with drift : 6100.293 ARIMA(1,0,1)(0,1,0)[12] : 6433.408 ARIMA(1,0,1)(0,1,0)[12] with drift : 6435.386 ARIMA(1,0,1)(0,1,1)[12] : 6086.428 ARIMA(1,0,1)(0,1,1)[12] with drift : 6086.654 ARIMA(1,0,1)(0,1,2)[12] : 6088.457 ARIMA(1,0,1)(0,1,2)[12] with drift : 6088.653 ARIMA(1,0,1)(1,1,0)[12] : 6240.708 ARIMA(1,0,1)(1,1,0)[12] with drift : 6242.651 ARIMA(1,0,1)(1,1,1)[12] : 6088.458 ARIMA(1,0,1)(1,1,1)[12] with drift : 6088.648 ARIMA(1,0,1)(1,1,2)[12] : Inf ARIMA(1,0,1)(1,1,2)[12] with drift : Inf ARIMA(1,0,1)(2,1,0)[12] : 6164.485 ARIMA(1,0,1)(2,1,0)[12] with drift : 6166.362 ARIMA(1,0,1)(2,1,1)[12] : 6089.714 ARIMA(1,0,1)(2,1,1)[12] with drift : Inf ARIMA(1,0,2)(0,1,0)[12] : 6433.686 ARIMA(1,0,2)(0,1,0)[12] with drift : 6435.671 ARIMA(1,0,2)(0,1,1)[12] : 6084.258 ARIMA(1,0,2)(0,1,1)[12] with drift : 6084.714 ARIMA(1,0,2)(0,1,2)[12] : 6086.287 ARIMA(1,0,2)(0,1,2)[12] with drift : 6086.739 ARIMA(1,0,2)(1,1,0)[12] : 6242.11 ARIMA(1,0,2)(1,1,0)[12] with drift : 6244.061 ARIMA(1,0,2)(1,1,1)[12] : 6086.287 ARIMA(1,0,2)(1,1,1)[12] with drift : 6086.738 ARIMA(1,0,2)(2,1,0)[12] : 6166.235 ARIMA(1,0,2)(2,1,0)[12] with drift : 6168.122 ARIMA(1,0,3)(0,1,0)[12] : 6430.348 ARIMA(1,0,3)(0,1,0)[12] with drift : 6432.339 ARIMA(1,0,3)(0,1,1)[12] : 6086.092 ARIMA(1,0,3)(0,1,1)[12] with drift : 6086.483 ARIMA(1,0,3)(1,1,0)[12] : 6242.756 ARIMA(1,0,3)(1,1,0)[12] with drift : 6244.712 ARIMA(1,0,4)(0,1,0)[12] : 6431.563 ARIMA(1,0,4)(0,1,0)[12] with drift : 6433.56 ARIMA(2,0,0)(0,1,0)[12] : 6436.474 ARIMA(2,0,0)(0,1,0)[12] with drift : 6438.446 ARIMA(2,0,0)(0,1,1)[12] : 6094.514 ARIMA(2,0,0)(0,1,1)[12] with drift : 6093.724 ARIMA(2,0,0)(0,1,2)[12] : 6096.544 ARIMA(2,0,0)(0,1,2)[12] with drift : 6095.637 ARIMA(2,0,0)(1,1,0)[12] : 6246.088 ARIMA(2,0,0)(1,1,0)[12] with drift : 6248.005 ARIMA(2,0,0)(1,1,1)[12] : 6096.544 ARIMA(2,0,0)(1,1,1)[12] with drift : 6095.622 ARIMA(2,0,0)(1,1,2)[12] : Inf ARIMA(2,0,0)(1,1,2)[12] with drift : Inf ARIMA(2,0,0)(2,1,0)[12] : 6170.995 ARIMA(2,0,0)(2,1,0)[12] with drift : 6172.807 ARIMA(2,0,0)(2,1,1)[12] : 6097.998 ARIMA(2,0,0)(2,1,1)[12] with drift : Inf ARIMA(2,0,1)(0,1,0)[12] : 6434.174 ARIMA(2,0,1)(0,1,0)[12] with drift : 6436.159 ARIMA(2,0,1)(0,1,1)[12] : 6084.44 ARIMA(2,0,1)(0,1,1)[12] with drift : 6084.961 ARIMA(2,0,1)(0,1,2)[12] : 6086.462 ARIMA(2,0,1)(0,1,2)[12] with drift : 6086.998 ARIMA(2,0,1)(1,1,0)[12] : 6242.201 ARIMA(2,0,1)(1,1,0)[12] with drift : 6244.151 ARIMA(2,0,1)(1,1,1)[12] : 6086.461 ARIMA(2,0,1)(1,1,1)[12] with drift : 6086.995 ARIMA(2,0,1)(2,1,0)[12] : Inf ARIMA(2,0,1)(2,1,0)[12] with drift : Inf ARIMA(2,0,2)(0,1,0)[12] : Inf ARIMA(2,0,2)(0,1,0)[12] with drift : Inf ARIMA(2,0,2)(0,1,1)[12] : 6085.4 ARIMA(2,0,2)(0,1,1)[12] with drift : 6085.776 ARIMA(2,0,2)(1,1,0)[12] : 6241.891 ARIMA(2,0,2)(1,1,0)[12] with drift : 6243.85 ARIMA(2,0,3)(0,1,0)[12] : 6430.935 ARIMA(2,0,3)(0,1,0)[12] with drift : 6432.932 ARIMA(3,0,0)(0,1,0)[12] : 6428.772 ARIMA(3,0,0)(0,1,0)[12] with drift : 6430.756 ARIMA(3,0,0)(0,1,1)[12] : 6089.022 ARIMA(3,0,0)(0,1,1)[12] with drift : 6088.626 ARIMA(3,0,0)(0,1,2)[12] : 6090.979 ARIMA(3,0,0)(0,1,2)[12] with drift : 6090.353 ARIMA(3,0,0)(1,1,0)[12] : 6240.788 ARIMA(3,0,0)(1,1,0)[12] with drift : 6242.726 ARIMA(3,0,0)(1,1,1)[12] : 6090.972 ARIMA(3,0,0)(1,1,1)[12] with drift : 6090.311 ARIMA(3,0,0)(2,1,0)[12] : 6167.684 ARIMA(3,0,0)(2,1,0)[12] with drift : 6169.533 ARIMA(3,0,1)(0,1,0)[12] : 6429.903 ARIMA(3,0,1)(0,1,0)[12] with drift : 6431.891 ARIMA(3,0,1)(0,1,1)[12] : 6086.378 ARIMA(3,0,1)(0,1,1)[12] with drift : 6086.774 ARIMA(3,0,1)(1,1,0)[12] : 6241.918 ARIMA(3,0,1)(1,1,0)[12] with drift : 6243.856 ARIMA(3,0,2)(0,1,0)[12] : 6430.82 ARIMA(3,0,2)(0,1,0)[12] with drift : 6432.816 ARIMA(4,0,0)(0,1,0)[12] : 6430.646 ARIMA(4,0,0)(0,1,0)[12] with drift : 6432.636 ARIMA(4,0,0)(0,1,1)[12] : 6091.012 ARIMA(4,0,0)(0,1,1)[12] with drift : 6090.649 ARIMA(4,0,0)(1,1,0)[12] : 6242.63 ARIMA(4,0,0)(1,1,0)[12] with drift : 6244.572 ARIMA(4,0,1)(0,1,0)[12] : 6431.798 ARIMA(4,0,1)(0,1,0)[12] with drift : 6433.794 ARIMA(5,0,0)(0,1,0)[12] : 6429.072 ARIMA(5,0,0)(0,1,0)[12] with drift : 6431.069 Best model: ARIMA(1,0,2)(0,1,1)[12]
Series: INMET ARIMA(1,0,2)(0,1,1)[12] Box Cox transformation: lambda= 1.999924 Coefficients: ar1 ma1 ma2 sma1 0.9225 -0.7169 -0.1025 -0.8690 s.e. 0.0642 0.0740 0.0542 0.0299 sigma^2 estimated as 484.2: log likelihood=-3037.08 AIC=6084.17 AICc=6084.26 BIC=6106.72
qqnorm(ajuste_auto_bc$residuals)
qqline(ajuste_auto_bc$residuals)
lillie.test(ajuste_auto_bc$residuals)
hist(ajuste_auto_bc$residuals)
Lilliefors (Kolmogorov-Smirnov) normality test data: ajuste_auto_bc$residuals D = 0.050203, p-value = 0.0003183
Image in a Jupyter notebookImage in a Jupyter notebook

Melhorou, mas ainda não adere a Normal. Tem outliers... Por hora, segue abaixo.

# A função arima não possue o parâmetro lambda
# outliers <- tso(BoxCox(INMET,l),tsmethod = "auto.arima", args.tsmethod = list(lambda=l), types = c("AO","LS","TC"))

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"))

outliers
Call: structure(list(method = NULL), .Names = "method") Coefficients: ar1 ma1 ma2 sma1 AO507 0.9210 -0.7180 -0.0957 -0.8664 -85.6234 s.e. 0.0591 0.0695 0.0524 0.0297 20.4189 sigma^2 estimated as 469.3: log likelihood = -3028.46, aic = 6068.92 Outliers: type ind time coefhat tstat 1 AO 507 2003:03 -85.62 -4.193
INMET[outliers$outliers$ind] <- NA

# Substituindo os outliers por estimativas...
INMET <- na.kalman(INMET, model = "auto.arima")

INMET
Jan Feb Mar Apr May Jun Jul Aug 1961 26.38000 26.54000 26.70000 26.65000 25.50000 22.78000 23.34000 27.72000 1962 26.43000 26.85000 27.18000 26.13000 24.49000 22.36000 20.55000 25.37000 1963 26.39000 26.24000 26.34000 26.57000 24.75000 22.97000 23.63000 26.74000 1964 27.64000 27.46000 26.70000 27.59000 24.57000 23.51000 22.70000 27.69000 1965 26.57000 26.54000 25.36000 25.90000 25.67000 25.10000 23.96000 26.33000 1966 27.30000 26.33000 26.91000 26.45000 25.49000 24.68000 24.61000 24.32000 1967 27.49000 26.15000 26.37000 25.64000 23.66000 22.02000 22.37000 25.11000 1968 26.19000 25.62000 25.98000 23.72000 20.93000 21.78000 22.03000 24.92000 1969 26.58000 26.61000 26.72000 26.11000 24.55000 23.03000 21.83000 23.83000 1970 27.19000 26.33000 26.47000 26.30000 23.58000 22.74000 21.36000 23.56000 1971 26.57000 25.59000 26.06000 24.88000 22.78000 20.92000 21.88000 23.21000 1972 26.08000 25.88000 26.57000 23.99000 24.95000 23.25000 22.16000 24.63000 1973 27.67000 27.06000 27.38000 26.89000 23.39000 23.95000 21.24000 24.42000 1974 25.98000 26.37000 25.57000 25.00000 23.73000 24.41000 22.54000 26.04000 1975 26.53000 26.38000 26.21000 25.93000 23.52000 23.19000 22.45000 25.38000 1976 26.88000 26.04000 25.33000 25.24000 24.11000 21.74000 22.97000 25.44000 1977 26.11000 25.70000 26.82000 25.03000 22.96000 23.62000 24.74000 25.04000 1978 26.48000 27.14000 26.60000 26.32000 24.31000 23.37000 25.09000 22.79000 1979 26.40000 26.68000 26.36000 25.35000 24.63000 21.82000 23.73000 26.23000 1980 26.86000 26.25000 26.78000 26.42000 25.17000 23.35000 22.80000 25.85000 1981 26.45000 26.18000 26.54000 26.55000 25.73000 21.87000 20.75000 25.28000 1982 26.33000 26.32000 26.05000 25.79000 24.27000 24.15000 24.47000 25.24000 1983 26.95000 27.51000 26.80000 27.19000 26.12000 22.08000 23.04000 22.93000 1984 26.74000 26.20000 26.60000 25.41000 25.17000 22.72000 23.56000 23.40000 1985 26.06000 27.13000 26.84000 26.27000 25.94000 21.97000 23.07000 24.20000 1986 27.16000 26.57000 26.62000 27.18000 25.59000 23.20000 23.31000 25.89000 1987 27.16000 26.61000 26.14000 27.17000 24.59000 23.36000 24.90000 23.86000 1988 27.15000 26.64000 27.27000 26.81000 24.46000 22.90000 20.86000 25.18000 1989 26.08000 26.37000 26.22000 26.54000 23.89000 24.44000 21.86000 25.10000 1990 26.48009 26.07057 26.80931 26.39176 24.50474 23.62948 22.11987 25.52453 1991 26.39176 26.64872 25.75740 25.78149 25.53256 23.99886 23.71781 25.17925 1992 26.02239 26.21318 26.20898 26.00565 24.51632 22.23229 24.48868 24.54489 1993 26.63266 25.22743 26.20072 26.06485 24.65986 23.79008 23.29223 25.29206 1994 26.69690 26.36767 26.25525 25.84573 25.75740 24.50474 23.66160 25.55665 1995 25.66104 26.28000 27.09000 25.52000 24.72000 23.77000 24.57000 25.01000 1996 26.99359 26.34358 26.50418 26.21510 25.50044 22.14396 24.07113 27.02612 1997 26.74070 26.57819 26.54203 26.42550 24.82866 23.70641 23.84208 25.87136 1998 28.52000 27.86000 27.97000 27.74000 23.92000 23.62000 25.02000 26.11000 1999 27.17000 27.44000 26.28000 26.03000 24.68000 24.35000 23.53000 24.88000 2000 27.76000 26.99000 26.45000 26.59000 25.23000 24.40000 21.90000 26.84000 2001 26.92000 27.41000 26.58000 27.34000 24.24000 22.96000 24.82000 26.88000 2002 27.40000 26.46000 27.01000 27.12000 25.91000 23.34000 24.05000 27.20000 2003 27.01000 26.67000 26.83418 25.97000 24.88000 24.13000 23.18000 24.33000 2004 27.22000 27.05000 27.70000 27.02000 23.34000 23.74000 23.48000 25.26000 2005 25.95012 26.22313 25.73331 25.23546 25.53256 25.85376 23.85000 26.28000 2006 27.22000 27.06000 26.94000 26.13000 22.97000 24.81000 25.08289 27.22000 2007 27.61000 26.89000 27.43000 27.44000 23.61000 23.89000 23.45000 24.26000 2008 26.42000 26.92000 26.80000 24.87411 23.31632 22.80241 24.83397 26.87355 2009 26.24722 25.80558 25.23546 25.62892 25.52000 22.62000 23.90000 25.66000 2010 26.91000 27.48000 27.91000 26.58000 23.66000 24.71000 22.82000 24.73000 2011 26.82000 26.82000 26.75000 27.46000 24.76000 24.01000 24.27000 26.02000 2012 26.61000 26.71000 26.69000 27.55000 25.20000 24.25000 23.52000 25.82000 2013 25.35591 25.22743 27.48000 26.71000 25.62000 26.09000 23.98000 24.22000 2014 26.48000 26.35000 27.14000 27.05000 25.23000 24.56000 23.31000 26.45000 2015 27.55000 27.34000 27.16000 27.07000 26.32000 24.98000 24.52000 27.61000 2016 26.01436 26.80128 26.07057 27.11000 25.47000 22.98000 23.93000 26.19000 2017 27.21000 27.10000 27.38000 27.14000 27.06000 24.13000 23.33000 27.33000 Sep Oct Nov Dec 1961 28.90000 27.69000 27.37000 26.92000 1962 27.94000 26.43000 28.49000 26.34000 1963 28.01000 29.04000 28.10000 28.49000 1964 27.46000 26.44000 27.01000 26.44000 1965 27.44000 27.19000 27.07000 27.09000 1966 26.83000 28.07000 28.12000 28.44000 1967 28.03000 27.83000 26.88000 26.88000 1968 24.49000 27.01000 28.04000 26.12000 1969 28.04000 26.94000 27.43000 26.94000 1970 27.11000 27.52000 27.77000 27.15000 1971 27.09000 25.90000 26.44000 26.64000 1972 27.19000 27.52000 27.29000 27.08000 1973 27.17000 27.63000 26.71000 26.56000 1974 26.16000 27.24000 27.50000 26.26000 1975 27.84000 27.24000 26.13000 26.35000 1976 25.35000 26.87000 26.07000 26.28000 1977 26.11000 26.25000 26.68000 26.50000 1978 26.26000 27.46000 26.83000 26.66000 1979 26.25000 28.34000 27.09000 26.83000 1980 25.48000 28.22000 26.42000 26.36000 1981 25.04000 27.38000 27.20000 26.72000 1982 25.64000 27.46000 27.22000 26.80000 1983 25.46000 25.75000 25.62000 26.22000 1984 25.84000 27.88000 27.08000 26.35000 1985 26.41000 27.72000 27.48000 28.41000 1986 24.79000 26.30000 28.51000 27.21000 1987 27.65000 28.63000 28.00000 27.12000 1988 27.40000 28.58000 27.71000 27.02000 1989 25.16000 27.68000 27.88000 26.73000 1990 24.74564 27.50791 27.06627 26.81734 1991 27.09839 26.67281 26.63266 26.76113 1992 26.32752 25.93406 26.55236 26.63266 1993 26.74507 27.34731 27.46776 26.94582 1994 27.08233 27.56412 27.58018 26.02239 1995 28.15000 28.25000 27.41000 28.52000 1996 26.10268 27.01006 26.39979 26.89764 1997 27.21277 28.12264 27.78926 27.66938 1998 26.30000 28.04000 27.95000 27.23000 1999 28.32000 28.29000 26.97000 27.55000 2000 26.73000 28.22000 27.34000 27.52000 2001 27.92000 27.73000 27.16000 26.38000 2002 26.83000 29.34000 28.65000 27.90000 2003 27.20000 27.50000 27.23000 28.08000 2004 27.54000 28.25000 27.20000 27.83000 2005 25.79000 28.26000 27.87000 27.73000 2006 26.49000 27.46000 28.41000 27.31000 2007 28.50000 28.06000 27.18000 27.16000 2008 26.31949 26.98597 27.01809 26.01436 2009 27.08000 28.72000 28.24000 25.87785 2010 29.37000 28.71000 27.04000 27.64000 2011 28.96000 28.57000 28.57000 27.38000 2012 28.63000 29.37000 25.69316 26.09465 2013 28.38000 28.26000 28.05000 27.40000 2014 29.47000 29.43000 28.53000 27.44000 2015 29.85000 27.77290 27.29914 27.20278 2016 27.11000 28.04000 27.07000 27.31000 2017 29.08000 28.46000 28.72000 27.63000
l <- BoxCox.lambda(INMET)
l
1.99992424816297
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"))
outliers2
Call: structure(list(method = NULL), .Names = "method") Coefficients: ar1 ma1 ma2 sma1 0.9200 -0.7173 -0.0948 -0.8660 s.e. 0.0589 0.0693 0.0521 0.0297 sigma^2 estimated as 469.3: log likelihood = -3028.46, aic = 6066.93 No outliers were detected.
# Reajustando
ajuste_auto_bc_outliers <- auto.arima(INMET, stepwise=FALSE, approximation=FALSE, trace=TRUE, lambda=l)
ajuste_auto_bc_outliers
ARIMA(0,0,0)(0,1,0)[12] : 6435.896 ARIMA(0,0,0)(0,1,0)[12] with drift : 6437.827 ARIMA(0,0,0)(0,1,1)[12] : 6114.469 ARIMA(0,0,0)(0,1,1)[12] with drift : 6113.411 ARIMA(0,0,0)(0,1,2)[12] : 6116.485 ARIMA(0,0,0)(0,1,2)[12] with drift : 6115.348 ARIMA(0,0,0)(1,1,0)[12] : 6254.366 ARIMA(0,0,0)(1,1,0)[12] with drift : 6256.202 ARIMA(0,0,0)(1,1,1)[12] : 6116.485 ARIMA(0,0,0)(1,1,1)[12] with drift : 6115.341 ARIMA(0,0,0)(1,1,2)[12] : 6118.508 ARIMA(0,0,0)(1,1,2)[12] with drift : 6117.428 ARIMA(0,0,0)(2,1,0)[12] : 6174.55 ARIMA(0,0,0)(2,1,0)[12] with drift : 6176.217 ARIMA(0,0,0)(2,1,1)[12] : 6118.397 ARIMA(0,0,0)(2,1,1)[12] with drift : 6116.586 ARIMA(0,0,0)(2,1,2)[12] : 6120.492 ARIMA(0,0,0)(2,1,2)[12] with drift : 6119.407 ARIMA(0,0,1)(0,1,0)[12] : 6413.01 ARIMA(0,0,1)(0,1,0)[12] with drift : 6414.968 ARIMA(0,0,1)(0,1,1)[12] : 6085.25 ARIMA(0,0,1)(0,1,1)[12] with drift : 6084.092 ARIMA(0,0,1)(0,1,2)[12] : 6087.093 ARIMA(0,0,1)(0,1,2)[12] with drift : 6085.503 ARIMA(0,0,1)(1,1,0)[12] : 6235.142 ARIMA(0,0,1)(1,1,0)[12] with drift : 6237.029 ARIMA(0,0,1)(1,1,1)[12] : 6087.086 ARIMA(0,0,1)(1,1,1)[12] with drift : 6085.447 ARIMA(0,0,1)(1,1,2)[12] : 6089.296 ARIMA(0,0,1)(1,1,2)[12] with drift : Inf ARIMA(0,0,1)(2,1,0)[12] : 6160.08 ARIMA(0,0,1)(2,1,0)[12] with drift : 6161.834 ARIMA(0,0,1)(2,1,1)[12] : 6088.951 ARIMA(0,0,1)(2,1,1)[12] with drift : Inf ARIMA(0,0,1)(2,1,2)[12] : 6091.12 ARIMA(0,0,1)(2,1,2)[12] with drift : 6089.31 ARIMA(0,0,2)(0,1,0)[12] : 6414.942 ARIMA(0,0,2)(0,1,0)[12] with drift : 6416.908 ARIMA(0,0,2)(0,1,1)[12] : 6083.756 ARIMA(0,0,2)(0,1,1)[12] with drift : 6082.758 ARIMA(0,0,2)(0,1,2)[12] : 6085.73 ARIMA(0,0,2)(0,1,2)[12] with drift : 6084.445 ARIMA(0,0,2)(1,1,0)[12] : 6234.712 ARIMA(0,0,2)(1,1,0)[12] with drift : 6236.614 ARIMA(0,0,2)(1,1,1)[12] : 6085.728 ARIMA(0,0,2)(1,1,1)[12] with drift : 6084.413 ARIMA(0,0,2)(1,1,2)[12] : 6087.821 ARIMA(0,0,2)(1,1,2)[12] with drift : 6086.826 ARIMA(0,0,2)(2,1,0)[12] : 6158.322 ARIMA(0,0,2)(2,1,0)[12] with drift : 6160.104 ARIMA(0,0,2)(2,1,1)[12] : 6087.575 ARIMA(0,0,2)(2,1,1)[12] with drift : Inf ARIMA(0,0,3)(0,1,0)[12] : 6404.489 ARIMA(0,0,3)(0,1,0)[12] with drift : 6406.468 ARIMA(0,0,3)(0,1,1)[12] : 6075.612 ARIMA(0,0,3)(0,1,1)[12] with drift : 6074.871 ARIMA(0,0,3)(0,1,2)[12] : 6077.386 ARIMA(0,0,3)(0,1,2)[12] with drift : 6076.196 ARIMA(0,0,3)(1,1,0)[12] : 6226.586 ARIMA(0,0,3)(1,1,0)[12] with drift : 6228.511 ARIMA(0,0,3)(1,1,1)[12] : 6077.369 ARIMA(0,0,3)(1,1,1)[12] with drift : 6076.112 ARIMA(0,0,3)(2,1,0)[12] : 6153.76 ARIMA(0,0,3)(2,1,0)[12] with drift : 6155.58 ARIMA(0,0,4)(0,1,0)[12] : 6406.022 ARIMA(0,0,4)(0,1,0)[12] with drift : 6408.008 ARIMA(0,0,4)(0,1,1)[12] : 6077.086 ARIMA(0,0,4)(0,1,1)[12] with drift : 6076.428 ARIMA(0,0,4)(1,1,0)[12] : 6228.529 ARIMA(0,0,4)(1,1,0)[12] with drift : 6230.462 ARIMA(0,0,5)(0,1,0)[12] : 6406.42 ARIMA(0,0,5)(0,1,0)[12] with drift : 6408.413 ARIMA(1,0,0)(0,1,0)[12] : 6412.02 ARIMA(1,0,0)(0,1,0)[12] with drift : 6413.982 ARIMA(1,0,0)(0,1,1)[12] : 6080.297 ARIMA(1,0,0)(0,1,1)[12] with drift : 6079.273 ARIMA(1,0,0)(0,1,2)[12] : 6082.174 ARIMA(1,0,0)(0,1,2)[12] with drift : 6080.766 ARIMA(1,0,0)(1,1,0)[12] : 6232.366 ARIMA(1,0,0)(1,1,0)[12] with drift : 6234.264 ARIMA(1,0,0)(1,1,1)[12] : 6082.168 ARIMA(1,0,0)(1,1,1)[12] with drift : 6080.717 ARIMA(1,0,0)(1,1,2)[12] : 6084.351 ARIMA(1,0,0)(1,1,2)[12] with drift : Inf ARIMA(1,0,0)(2,1,0)[12] : 6157.48 ARIMA(1,0,0)(2,1,0)[12] with drift : 6159.254 ARIMA(1,0,0)(2,1,1)[12] : 6084.002 ARIMA(1,0,0)(2,1,1)[12] with drift : Inf ARIMA(1,0,0)(2,1,2)[12] : 6086.184 ARIMA(1,0,0)(2,1,2)[12] with drift : 6084.544 ARIMA(1,0,1)(0,1,0)[12] : 6409.066 ARIMA(1,0,1)(0,1,0)[12] with drift : 6411.044 ARIMA(1,0,1)(0,1,1)[12] : 6068.718 ARIMA(1,0,1)(0,1,1)[12] with drift : 6069.036 ARIMA(1,0,1)(0,1,2)[12] : 6070.702 ARIMA(1,0,1)(0,1,2)[12] with drift : 6070.884 ARIMA(1,0,1)(1,1,0)[12] : 6225.001 ARIMA(1,0,1)(1,1,0)[12] with drift : 6226.945 ARIMA(1,0,1)(1,1,1)[12] : 6070.699 ARIMA(1,0,1)(1,1,1)[12] with drift : 6070.867 ARIMA(1,0,1)(1,1,2)[12] : 6072.779 ARIMA(1,0,1)(1,1,2)[12] with drift : 6073.114 ARIMA(1,0,1)(2,1,0)[12] : 6147.981 ARIMA(1,0,1)(2,1,0)[12] with drift : 6149.858 ARIMA(1,0,1)(2,1,1)[12] : 6072.342 ARIMA(1,0,1)(2,1,1)[12] with drift : Inf ARIMA(1,0,2)(0,1,0)[12] : 6409.74 ARIMA(1,0,2)(0,1,0)[12] with drift : 6411.725 ARIMA(1,0,2)(0,1,1)[12] : 6067.018 ARIMA(1,0,2)(0,1,1)[12] with drift : 6067.505 ARIMA(1,0,2)(0,1,2)[12] : 6069.024 ARIMA(1,0,2)(0,1,2)[12] with drift : 6069.4 ARIMA(1,0,2)(1,1,0)[12] : 6226.595 ARIMA(1,0,2)(1,1,0)[12] with drift : 6228.546 ARIMA(1,0,2)(1,1,1)[12] : 6069.023 ARIMA(1,0,2)(1,1,1)[12] with drift : 6069.389 ARIMA(1,0,2)(2,1,0)[12] : 6149.848 ARIMA(1,0,2)(2,1,0)[12] with drift : 6151.733 ARIMA(1,0,3)(0,1,0)[12] : 6404.849 ARIMA(1,0,3)(0,1,0)[12] with drift : 6406.839 ARIMA(1,0,3)(0,1,1)[12] : 6068.717 ARIMA(1,0,3)(0,1,1)[12] with drift : 6069.141 ARIMA(1,0,3)(1,1,0)[12] : 6227.219 ARIMA(1,0,3)(1,1,0)[12] with drift : 6229.175 ARIMA(1,0,4)(0,1,0)[12] : 6405.895 ARIMA(1,0,4)(0,1,0)[12] with drift : 6407.892 ARIMA(2,0,0)(0,1,0)[12] : 6413.518 ARIMA(2,0,0)(0,1,0)[12] with drift : 6415.488 ARIMA(2,0,0)(0,1,1)[12] : 6078.312 ARIMA(2,0,0)(0,1,1)[12] with drift : 6077.576 ARIMA(2,0,0)(0,1,2)[12] : 6080.283 ARIMA(2,0,0)(0,1,2)[12] with drift : 6079.286 ARIMA(2,0,0)(1,1,0)[12] : 6231.139 ARIMA(2,0,0)(1,1,0)[12] with drift : 6233.055 ARIMA(2,0,0)(1,1,1)[12] : 6080.28 ARIMA(2,0,0)(1,1,1)[12] with drift : 6079.254 ARIMA(2,0,0)(1,1,2)[12] : 6082.373 ARIMA(2,0,0)(1,1,2)[12] with drift : 6081.65 ARIMA(2,0,0)(2,1,0)[12] : 6155.098 ARIMA(2,0,0)(2,1,0)[12] with drift : 6156.908 ARIMA(2,0,0)(2,1,1)[12] : 6082.041 ARIMA(2,0,0)(2,1,1)[12] with drift : Inf ARIMA(2,0,1)(0,1,0)[12] : 6410.142 ARIMA(2,0,1)(0,1,0)[12] with drift : 6412.126 ARIMA(2,0,1)(0,1,1)[12] : 6067.252 ARIMA(2,0,1)(0,1,1)[12] with drift : 6067.771 ARIMA(2,0,1)(0,1,2)[12] : 6069.278 ARIMA(2,0,1)(0,1,2)[12] with drift : 6069.693 ARIMA(2,0,1)(1,1,0)[12] : 6226.658 ARIMA(2,0,1)(1,1,0)[12] with drift : 6228.609 ARIMA(2,0,1)(1,1,1)[12] : 6069.27 ARIMA(2,0,1)(1,1,1)[12] with drift : 6069.684 ARIMA(2,0,1)(2,1,0)[12] : Inf ARIMA(2,0,1)(2,1,0)[12] with drift : Inf ARIMA(2,0,2)(0,1,0)[12] : Inf ARIMA(2,0,2)(0,1,0)[12] with drift : Inf ARIMA(2,0,2)(0,1,1)[12] : 6067.849 ARIMA(2,0,2)(0,1,1)[12] with drift : 6068.268 ARIMA(2,0,2)(1,1,0)[12] : 6226.584 ARIMA(2,0,2)(1,1,0)[12] with drift : 6228.543 ARIMA(2,0,3)(0,1,0)[12] : 6405.307 ARIMA(2,0,3)(0,1,0)[12] with drift : 6407.304 ARIMA(3,0,0)(0,1,0)[12] : 6403.278 ARIMA(3,0,0)(0,1,0)[12] with drift : 6405.261 ARIMA(3,0,0)(0,1,1)[12] : 6071.603 ARIMA(3,0,0)(0,1,1)[12] with drift : 6071.266 ARIMA(3,0,0)(0,1,2)[12] : 6073.304 ARIMA(3,0,0)(0,1,2)[12] with drift : 6072.566 ARIMA(3,0,0)(1,1,0)[12] : 6225.121 ARIMA(3,0,0)(1,1,0)[12] with drift : 6227.059 ARIMA(3,0,0)(1,1,1)[12] : 6073.283 ARIMA(3,0,0)(1,1,1)[12] with drift : 6072.487 ARIMA(3,0,0)(2,1,0)[12] : 6151.402 ARIMA(3,0,0)(2,1,0)[12] with drift : 6153.25 ARIMA(3,0,1)(0,1,0)[12] : 6404.729 ARIMA(3,0,1)(0,1,0)[12] with drift : 6406.716 ARIMA(3,0,1)(0,1,1)[12] : 6069.052 ARIMA(3,0,1)(0,1,1)[12] with drift : 6069.488 ARIMA(3,0,1)(1,1,0)[12] : 6226.582 ARIMA(3,0,1)(1,1,0)[12] with drift : 6228.523 ARIMA(3,0,2)(0,1,0)[12] : 6405.177 ARIMA(3,0,2)(0,1,0)[12] with drift : 6407.174 ARIMA(4,0,0)(0,1,0)[12] : 6405.238 ARIMA(4,0,0)(0,1,0)[12] with drift : 6407.227 ARIMA(4,0,0)(0,1,1)[12] : 6073.562 ARIMA(4,0,0)(0,1,1)[12] with drift : 6073.267 ARIMA(4,0,0)(1,1,0)[12] : 6227.043 ARIMA(4,0,0)(1,1,0)[12] with drift : 6228.986 ARIMA(4,0,1)(0,1,0)[12] : 6406.43 ARIMA(4,0,1)(0,1,0)[12] with drift : 6408.424 ARIMA(5,0,0)(0,1,0)[12] : 6403.707 ARIMA(5,0,0)(0,1,0)[12] with drift : 6405.703 Best model: ARIMA(1,0,2)(0,1,1)[12]
Series: INMET ARIMA(1,0,2)(0,1,1)[12] Box Cox transformation: lambda= 1.999924 Coefficients: ar1 ma1 ma2 sma1 0.9200 -0.7173 -0.0948 -0.8660 s.e. 0.0589 0.0693 0.0521 0.0297 sigma^2 estimated as 472.1: log likelihood=-3028.46 AIC=6066.93 AICc=6067.02 BIC=6089.48
qqnorm(ajuste_auto_bc_outliers$residuals)
qqline(ajuste_auto_bc_outliers$residuals)
lillie.test(ajuste_auto_bc_outliers$residuals)
hist(ajuste_auto_bc_outliers$residuals)
Lilliefors (Kolmogorov-Smirnov) normality test data: ajuste_auto_bc_outliers$residuals D = 0.048307, p-value = 0.000666
Image in a Jupyter notebookImage in a Jupyter notebook

Ainda tem 9 indivíduos que poderiam ser considerados outliers, mas paramos, por hora, por aqui. Verificar a possibilidade de bootstrap.

https://cran.r-project.org/web/packages/TSA/TSA.pdf

https://www.rdocumentation.org/packages/boot/versions/1.3-20/topics/tsboot

portest(ajuste_auto_bc_outliers$residuals) # Testes de portmanteau (ruído branco)
lagsstatisticp-value
5 2.0424740.7622378
10 6.0022970.6383616
15 10.4648270.5104895
20 15.4611620.4055944
25 20.1341180.3546454
30 24.2298680.3326673
 
# Testando diretamente
Box.test(ajuste_auto_bc$residuals, lag=20, type="Ljung-Box") # Testa se existe autocorrelação entre os resíduos
Box-Ljung test data: ajuste_auto_bc$residuals X-squared = 18.186, df = 20, p-value = 0.5751 
previsoes_auto_bc <- forecast(ajuste_auto_bc, h=3)
previsoes_auto_bc
accuracy(previsoes_auto_bc)
plot(previsoes_auto_bc)
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 Jan 2018 27.02914 25.96459 28.05332 25.38299 28.58063 Feb 2018 27.00269 25.91430 28.04888 25.31921 28.58721 Mar 2018 27.08752 25.99865 28.13429 25.40335 28.67295
MERMSEMAEMPEMAPEMASEACF1
Training set0.01058867 0.8544344 0.6447901 -0.05951616 2.527899 0.7231742 -0.005222841
Image in a Jupyter notebook


library(boot)


AIC_Boot <- function(ts) {
    aux <- Arima(ts, lambda=l, order=c(1, 0, 2), seasonal=list(order=c(0, 1, 1), period=12))
    return(aux$aic)
}

aux <- Arima(INMET, lambda=l, order=c(1, 0, 2), seasonal=list(order=c(0, 1, 1), period=12))

AIC_Boot(INMET)
6066.92790473731
# Bootstrap estacionário com bloco de 20
boot_ajuste_auto_bc_outliers <- tsboot(INMET, AIC_Boot, R = 99, l = 20, sim = "geom")

boot_ajuste_auto_bc_outliers
STATIONARY BOOTSTRAP FOR TIME SERIES Average Block Length of 20 Call: tsboot(tseries = INMET, statistic = AIC_Boot, R = 99, l = 20, sim = "geom") Bootstrap Statistics : original bias std. error t1* 6066.928 577.0108 51.53266
boot.ci(boot_ajuste_auto_bc_outliers)
Warning message in boot.ci(boot_ajuste_auto_bc_outliers): “bootstrap variances needed for studentized intervals”Warning message in boot.ci(boot_ajuste_auto_bc_outliers): “BCa intervals not defined for time series bootstraps”
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS Based on 99 bootstrap replicates CALL : boot.ci(boot.out = boot_ajuste_auto_bc_outliers) Intervals : Level Normal Basic Percentile 95% (5389, 5591 ) (5390, 5602 ) (6532, 6743 ) Calculations and Intervals on Original Scale Some basic intervals may be unstable Some percentile intervals may be unstable