By Salerno | December 18, 2019
Introduction
The main idea here is breaking the ice in terms of exponential smoothing models
First of all it is importan to show some behaviours patterns usually found in time series
Trends: it is usually observed when the time series follow one specific direction. It can be linear or not.
Seasonality: it is a pattern that repeat in a certain times (specific period)
Cycle: Like seasonality but it appears in non specific time
library(fpp)
## Carregando pacotes exigidos: forecast
## Registered S3 method overwritten by 'xts':
## method from
## as.zoo.xts zoo
## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
## Registered S3 methods overwritten by 'forecast':
## method from
## fitted.fracdiff fracdiff
## residuals.fracdiff fracdiff
## Carregando pacotes exigidos: fma
## Carregando pacotes exigidos: expsmooth
## Carregando pacotes exigidos: lmtest
## Carregando pacotes exigidos: zoo
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
## Carregando pacotes exigidos: tseries
data(elecequip)
plot(elecequip, xlab = "Time", ylab = "New Orders Index")
decomp <- decompose(elecequip, type = "additive")
plot(decomp)
seasonality_adjust <- elecequip - decomp$seasonal
plot(seasonality_adjust)
library(mFilter)
hpfilter(elecequip, type = "lambda")
##
## Title:
## Hodrick-Prescott Filter
##
## Call:
## hpfilter(x = elecequip, type = "lambda")
##
## Method:
## hpfilter
##
## Filter Type:
## lambda
##
## Series:
## elecequip
##
## elecequip Trend Cycle
## Jan 1996 79.43 77.77 1.6588
## Feb 1996 75.86 78.17 -2.3142
## Mar 1996 86.40 78.58 7.8227
## Apr 1996 72.67 78.98 -6.3103
## May 1996 74.93 79.38 -4.4535
## Jun 1996 83.88 79.79 4.0933
## Jul 1996 79.88 80.19 -0.3099
## Aug 1996 62.47 80.59 -18.1232
## Sep 1996 85.50 81.00 4.5035
## Oct 1996 83.19 81.40 1.7903
## Nov 1996 84.29 81.80 2.4873
## Dec 1996 89.79 82.21 7.5845
## Jan 1997 78.72 82.61 -3.8879
## Feb 1997 77.49 83.01 -5.5199
## Mar 1997 89.94 83.41 6.5285
## Apr 1997 81.35 83.81 -2.4627
## May 1997 78.76 84.21 -5.4533
## Jun 1997 89.59 84.61 4.9767
## Jul 1997 83.75 85.01 -1.2626
## Aug 1997 69.87 85.41 -15.5412
## Sep 1997 91.18 85.81 5.3710
## Oct 1997 89.52 86.21 3.3142
## Nov 1997 91.12 86.60 4.5185
## Dec 1997 92.97 87.00 5.9741
## Jan 1998 81.97 87.39 -5.4190
## Feb 1998 85.26 87.78 -2.5207
## Mar 1998 93.09 88.17 4.9190
## Apr 1998 81.19 88.56 -7.3697
## May 1998 85.74 88.95 -3.2068
## Jun 1998 91.24 89.33 1.9079
## Jul 1998 83.56 89.72 -6.1556
## Aug 1998 66.45 90.10 -23.6470
## Sep 1998 93.45 90.48 2.9737
## Oct 1998 86.03 90.85 -4.8230
## Nov 1998 86.91 91.23 -4.3169
## Dec 1998 93.42 91.60 1.8226
## Jan 1999 81.68 91.96 -10.2844
## Feb 1999 81.68 92.33 -10.6472
## Mar 1999 91.35 92.69 -1.3356
## Apr 1999 79.55 93.04 -13.4889
## May 1999 87.08 93.39 -6.3065
## Jun 1999 96.71 93.73 2.9822
## Jul 1999 98.10 94.06 4.0379
## Aug 1999 79.22 94.39 -15.1686
## Sep 1999 103.68 94.71 8.9732
## Oct 1999 101.00 95.02 5.9842
## Nov 1999 99.52 95.31 4.2050
## Dec 1999 111.94 95.60 16.3364
## Jan 2000 95.42 95.88 -0.4610
## Feb 2000 98.49 96.15 2.3432
## Mar 2000 116.37 96.40 19.9697
## Apr 2000 101.09 96.64 4.4488
## May 2000 104.20 96.87 7.3309
## Jun 2000 114.79 97.08 17.7064
## Jul 2000 107.75 97.28 10.4654
## Aug 2000 96.23 97.47 -1.2419
## Sep 2000 123.65 97.65 26.0045
## Oct 2000 116.24 97.81 18.4346
## Nov 2000 117.00 97.95 19.0483
## Dec 2000 128.75 98.08 30.6653
## Jan 2001 100.69 98.20 2.4851
## Feb 2001 102.99 98.31 4.6771
## Mar 2001 119.21 98.41 20.8005
## Apr 2001 92.56 98.50 -5.9355
## May 2001 98.86 98.57 0.2884
## Jun 2001 111.26 98.64 12.6211
## Jul 2001 96.25 98.70 -2.4480
## Aug 2001 79.81 98.75 -18.9401
## Sep 2001 102.18 98.80 3.3840
## Oct 2001 96.28 98.84 -2.5565
## Nov 2001 101.38 98.87 2.5076
## Dec 2001 109.97 98.90 11.0654
## Jan 2002 89.66 98.93 -9.2739
## Feb 2002 89.23 98.96 -9.7311
## Mar 2002 104.36 98.99 5.3729
## Apr 2002 87.17 99.01 -11.8426
## May 2002 89.43 99.04 -9.6085
## Jun 2002 102.25 99.07 3.1845
## Jul 2002 88.26 99.09 -10.8342
## Aug 2002 75.73 99.13 -23.3952
## Sep 2002 99.60 99.16 0.4409
## Oct 2002 96.57 99.20 -2.6263
## Nov 2002 96.22 99.24 -3.0173
## Dec 2002 101.12 99.28 1.8377
## Jan 2003 89.45 99.33 -9.8818
## Feb 2003 86.87 99.39 -12.5161
## Mar 2003 98.94 99.45 -0.5055
## Apr 2003 85.62 99.51 -13.8902
## May 2003 85.31 99.58 -14.2703
## Jun 2003 101.22 99.66 1.5639
## Jul 2003 91.93 99.74 -7.8073
## Aug 2003 77.01 99.82 -22.8141
## Sep 2003 104.50 99.92 4.5836
## Oct 2003 99.83 100.01 -0.1838
## Nov 2003 101.10 100.12 0.9837
## Dec 2003 109.16 100.22 8.9365
## Jan 2004 89.93 100.34 -10.4052
## Feb 2004 92.73 100.45 -7.7214
## Mar 2004 105.22 100.57 4.6482
## Apr 2004 91.56 100.70 -9.1361
## May 2004 92.60 100.82 -8.2241
## Jun 2004 104.46 100.96 3.5046
## Jul 2004 96.28 101.09 -4.8097
## Aug 2004 79.61 101.23 -21.6166
## Sep 2004 105.55 101.37 4.1842
## Oct 2004 99.15 101.51 -2.3567
## Nov 2004 99.81 101.65 -1.8387
## Dec 2004 113.72 101.79 11.9287
## Jan 2005 91.73 101.93 -10.2041
## Feb 2005 90.45 102.08 -11.6264
## Mar 2005 105.56 102.22 3.3423
## Apr 2005 92.15 102.36 -10.2076
## May 2005 91.23 102.50 -11.2652
## Jun 2005 108.95 102.63 6.3201
## Jul 2005 99.33 102.76 -3.4311
## Aug 2005 83.30 102.89 -19.5878
## Sep 2005 110.85 103.01 7.8405
## Oct 2005 104.99 103.13 1.8648
## Nov 2005 107.10 103.23 3.8660
## Dec 2005 114.38 103.34 11.0449
## Jan 2006 99.09 103.43 -4.3378
## Feb 2006 99.73 103.51 -3.7813
## Mar 2006 116.05 103.58 12.4651
## Apr 2006 103.51 103.65 -0.1378
## May 2006 102.99 103.70 -0.7093
## Jun 2006 119.45 103.74 15.7112
## Jul 2006 107.98 103.77 4.2145
## Aug 2006 90.50 103.78 -13.2789
## Sep 2006 121.85 103.78 18.0715
## Oct 2006 117.12 103.76 13.3563
## Nov 2006 113.66 103.73 9.9261
## Dec 2006 120.35 103.69 16.6612
## Jan 2007 103.92 103.63 0.2920
## Feb 2007 103.97 103.55 0.4187
## Mar 2007 125.63 103.46 22.1713
## Apr 2007 104.69 103.35 1.3402
## May 2007 108.36 103.22 5.1352
## Jun 2007 123.09 103.08 20.0065
## Jul 2007 108.88 102.93 5.9539
## Aug 2007 93.98 102.75 -8.7727
## Sep 2007 121.94 102.56 19.3764
## Oct 2007 116.79 102.36 14.4311
## Nov 2007 115.78 102.14 13.6409
## Dec 2007 127.28 101.90 25.3756
## Jan 2008 109.35 101.66 7.6945
## Feb 2008 105.64 101.39 4.2470
## Mar 2008 121.30 101.12 20.1821
## Apr 2008 108.62 100.83 7.7892
## May 2008 103.13 100.53 2.5971
## Jun 2008 117.84 100.23 17.6149
## Jul 2008 103.62 99.91 3.7116
## Aug 2008 89.22 99.58 -10.3642
## Sep 2008 109.41 99.25 10.1563
## Oct 2008 103.93 98.92 5.0121
## Nov 2008 100.07 98.58 1.4919
## Dec 2008 101.15 98.24 2.9144
## Jan 2009 77.33 97.89 -20.5617
## Feb 2009 75.01 97.55 -22.5376
## Mar 2009 86.31 97.20 -10.8945
## Apr 2009 74.09 96.86 -22.7734
## May 2009 74.09 96.53 -22.4351
## Jun 2009 85.58 96.19 -10.6104
## Jul 2009 79.84 95.86 -16.0198
## Aug 2009 65.24 95.53 -30.2937
## Sep 2009 87.92 95.21 -7.2925
## Oct 2009 84.45 94.90 -10.4462
## Nov 2009 87.93 94.58 -6.6550
## Dec 2009 102.42 94.28 8.1413
## Jan 2010 79.16 93.98 -14.8173
## Feb 2010 78.40 93.68 -15.2807
## Mar 2010 94.32 93.39 0.9311
## Apr 2010 84.45 93.10 -8.6514
## May 2010 84.92 92.82 -7.8982
## Jun 2010 103.18 92.54 10.6412
## Jul 2010 89.42 92.26 -2.8429
## Aug 2010 77.66 91.99 -14.3301
## Sep 2010 95.68 91.72 3.9598
## Oct 2010 94.03 91.45 2.5772
## Nov 2010 100.99 91.19 9.8027
## Dec 2010 101.26 90.92 10.3365
## Jan 2011 91.47 90.66 0.8090
## Feb 2011 87.66 90.40 -2.7396
## Mar 2011 103.33 90.14 13.1909
## Apr 2011 88.56 89.88 -1.3193
## May 2011 92.32 89.62 2.7001
## Jun 2011 102.21 89.36 12.8491
## Jul 2011 92.80 89.10 3.6979
## Aug 2011 76.44 88.84 -12.4034
## Sep 2011 94.00 88.58 5.4150
## Oct 2011 91.67 88.33 3.3434
## Nov 2011 91.98 88.07 3.9117
par(mfrow= c(2,1))
plot(elecequip, xlab = "Time", ylab = "New Orders Index")
lines(hpfilter(elecequip, type = "lambda")$trend, col = "red", lwd = 2)
legend(1996, 200, c("Original Serie", "Trend - HP Filter"), col = c("black", "red"), lwd = c(1,2), bty = "n")
plot(hpfilter(elecequip, type = "lambda")$cycle, xlab = "Time", ylab = "New Orders Index")
data("cafe")
hpfilter(cafe, type = "lambda")
##
## Title:
## Hodrick-Prescott Filter
##
## Call:
## hpfilter(x = cafe, type = "lambda")
##
## Method:
## hpfilter
##
## Filter Type:
## lambda
##
## Series:
## cafe
##
## cafe Trend Cycle
## 1982 Q2 1013 953.8 59.3606
## 1982 Q3 1012 987.7 24.1833
## 1982 Q4 1166 1021.6 144.5689
## 1983 Q1 1082 1055.6 26.8652
## 1983 Q2 1059 1089.9 -31.1704
## 1983 Q3 1118 1124.5 -6.3973
## 1983 Q4 1224 1159.7 64.0448
## 1984 Q1 1164 1195.5 -31.7802
## 1984 Q2 1179 1232.1 -53.3481
## 1984 Q3 1197 1269.8 -73.1150
## 1984 Q4 1349 1308.6 40.4965
## 1985 Q1 1234 1348.6 -115.0907
## 1985 Q2 1273 1389.9 -117.0789
## 1985 Q3 1370 1432.5 -62.7984
## 1985 Q4 1563 1476.4 86.6934
## 1986 Q1 1438 1521.5 -84.0213
## 1986 Q2 1513 1567.8 -55.0147
## 1986 Q3 1602 1615.2 -13.1065
## 1986 Q4 1797 1663.6 133.8182
## 1987 Q1 1640 1712.8 -72.5178
## 1987 Q2 1666 1762.9 -97.3750
## 1987 Q3 1775 1813.7 -38.6690
## 1987 Q4 1985 1865.1 119.9458
## 1988 Q1 1816 1916.9 -100.4610
## 1988 Q2 1878 1969.0 -91.0946
## 1988 Q3 1975 2021.3 -46.1976
## 1988 Q4 2115 2073.6 41.3445
## 1989 Q1 2107 2125.5 -18.8252
## 1989 Q2 2126 2177.0 -51.3891
## 1989 Q3 2254 2227.7 25.9821
## 1989 Q4 2544 2277.5 266.4495
## 1990 Q1 2462 2325.9 136.2583
## 1990 Q2 2412 2373.1 39.3871
## 1990 Q3 2455 2419.0 36.4293
## 1990 Q4 2630 2463.5 166.6536
## 1991 Q1 2446 2506.9 -60.7940
## 1991 Q2 2419 2549.2 -129.8715
## 1991 Q3 2565 2590.5 -25.5991
## 1991 Q4 2941 2630.9 310.0842
## 1992 Q1 2736 2670.4 65.4553
## 1992 Q2 2730 2709.3 20.5977
## 1992 Q3 2681 2747.7 -66.8466
## 1992 Q4 2913 2786.0 127.2520
## 1993 Q1 2626 2824.4 -198.7352
## 1993 Q2 2542 2863.2 -321.4167
## 1993 Q3 2654 2902.6 -248.4765
## 1993 Q4 2994 2942.5 51.2021
## 1994 Q1 2901 2982.8 -81.5088
## 1994 Q2 2815 3023.4 -208.0690
## 1994 Q3 3072 3064.0 7.5123
## 1994 Q4 3320 3104.3 215.9564
## 1995 Q1 3157 3144.1 12.6794
## 1995 Q2 3196 3183.1 12.9629
## 1995 Q3 3330 3221.2 108.9803
## 1995 Q4 3676 3258.2 417.2970
## 1996 Q1 3521 3294.0 227.1101
## 1996 Q2 3424 3328.7 95.2563
## 1996 Q3 3389 3362.8 25.9299
## 1996 Q4 3502 3396.4 105.9659
## 1997 Q1 3388 3430.1 -42.5169
## 1997 Q2 3425 3464.3 -39.1660
## 1997 Q3 3492 3499.3 -7.3023
## 1997 Q4 3695 3535.6 159.5777
## 1998 Q1 3377 3573.6 -196.6176
## 1998 Q2 3339 3613.8 -274.6801
## 1998 Q3 3456 3656.5 -199.9783
## 1998 Q4 3770 3701.9 67.6907
## 1999 Q1 3614 3750.1 -136.3451
## 1999 Q2 3715 3801.3 -85.9001
## 1999 Q3 3714 3855.4 -141.3035
## 1999 Q4 4088 3912.4 175.5694
## 2000 Q1 3829 3972.3 -143.5686
## 2000 Q2 3809 4034.9 -225.6144
## 2000 Q3 4079 4100.3 -21.1751
## 2000 Q4 4416 4168.1 247.5831
## 2001 Q1 4330 4238.2 91.8075
## 2001 Q2 4285 4310.4 -25.4097
## 2001 Q3 4419 4384.7 34.4665
## 2001 Q4 4583 4461.1 121.6871
## 2002 Q1 4291 4539.5 -248.7187
## 2002 Q2 4367 4620.0 -252.7976
## 2002 Q3 4574 4702.4 -128.4408
## 2002 Q4 4862 4786.6 75.9184
## 2003 Q1 4616 4872.1 -255.6731
## 2003 Q2 4801 4958.6 -157.9157
## 2003 Q3 5147 5045.7 100.8500
## 2003 Q4 5765 5132.9 632.1820
## 2004 Q1 5534 5219.6 314.1754
## 2004 Q2 5478 5305.8 172.6299
## 2004 Q3 5650 5391.5 258.3492
## 2004 Q4 5796 5476.9 319.4289
## 2005 Q1 5240 5562.4 -322.1968
## 2005 Q2 5367 5648.6 -281.7934
## 2005 Q3 5528 5735.8 -207.6250
## 2005 Q4 6096 5824.3 271.6204
## 2006 Q1 5648 5914.0 -266.3153
## 2006 Q2 5915 6005.3 -90.0601
## 2006 Q3 6101 6098.1 3.0244
## 2006 Q4 6520 6192.5 327.5330
## 2007 Q1 6228 6288.4 -60.4414
## 2007 Q2 6413 6386.2 27.0894
## 2007 Q3 6616 6485.9 129.8512
## 2007 Q4 7003 6587.8 415.1532
## 2008 Q1 6410 6692.2 -282.5769
## 2008 Q2 6415 6799.5 -384.6707
## 2008 Q3 6501 6910.1 -409.2831
## 2008 Q4 7025 7024.1 0.4713
## 2009 Q1 6691 7141.5 -450.4664
## 2009 Q2 6992 7262.0 -270.3553
## 2009 Q3 7292 7385.2 -93.2727
## 2009 Q4 8068 7510.5 557.4726
## 2010 Q1 7451 7637.4 -186.6696
## 2010 Q2 7608 7765.4 -157.0979
## 2010 Q3 8317 7894.2 422.6056
## 2010 Q4 8426 8023.2 403.2570
par(mfrow= c(2,1))
plot(cafe, xlab = "Time", ylab = "Expenditures Quarters")
lines(hpfilter(cafe, type = "lambda")$trend, col = "red", lwd = 2)
legend(1985, 8000, c("Original Serie", "Trend - HP Filter"), col = c("black", "red"), lwd = c(1,2), bty = "n")
plot(hpfilter(cafe, type = "lambda")$cycle, xlab = "Time", ylab = "Cycle Component")
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