Daily Climate Forecasting
Dimas Abdullah Lubis
2025-05-17
1 Introduction
Kondisi cuaca kadang menjadi salah satu faktor dalam melaksanakan suatu kegiatan, dengan mengetahui cuaca yang akan terjadi dapat memberikan informasi kapan akan melakukan suatu kegiatan. kali ini kita akan mencoba membuat model untuk melakukan perkiraan cuaca dengan menggunakan dataset Daily Climate di negara India dari 1 Januari 2013 sampai 1 Januari 2017 yang didapat dari Kaggle.com.
2 Data Preparation
2.1 Import Library
# load library
library(dplyr) # data wrangling
library(lubridate) # date manipulation
library(padr) # complete data frame
library(zoo) # Missing value imputation
library(forecast) # time series library
library(TTR) # for Simple moving average function
library(MLmetrics) # calculate error
library(tseries) # adf.test
library(TSstudio) # visualisasi timeseries
library(ggplot2)
library(tidyr)2.2 Read Data
climate <- read.csv("DelhiClimate_with_WeatherIndex_and_Categories.csv")
rmarkdown::paged_table(climate)#> Rows: 1,462
#> Columns: 11
#> $ date <chr> "2013-01-01", "2013-01-02", "2013-01-03", "2013-01-0…
#> $ meantemp <dbl> 10.000000, 7.400000, 7.166667, 8.666667, 6.000000, 7…
#> $ humidity <dbl> 84.50000, 92.00000, 87.00000, 71.33333, 86.83333, 82…
#> $ wind_speed <dbl> 0.0000000, 2.9800000, 4.6333333, 1.2333333, 3.700000…
#> $ meanpressure <dbl> 1015.667, 1017.800, 1018.667, 1017.167, 1016.500, 10…
#> $ meantemp_norm <dbl> 0.12227074, 0.04279476, 0.03566230, 0.08151383, 0.00…
#> $ humidity_norm <dbl> 0.8209571, 0.9075908, 0.8498350, 0.6688669, 0.847909…
#> $ wind_speed_norm <dbl> 0.00000000, 0.07058266, 0.10974262, 0.02921206, 0.08…
#> $ meanpressure_norm <dbl> 0.1326033, 0.1328810, 0.1329938, 0.1327986, 0.132711…
#> $ weather_index <dbl> 0.3084558, 0.3167998, 0.3044633, 0.2523879, 0.285171…
#> $ weather_category <chr> "Cukup", "Cukup", "Cukup", "Bagus", "Bagus", "Bagus"…Dari hasil pembacaan dataset di atas, terdapat 1.462 data observasi dengan 11 kolom. Adapun untuk penjelasan masing-masing kolom adalah sebagai berikut.
- date : tanggal observasi cuaca dengan format YYYY-MM-DD.
- meantemp : Suhu rata-rata dari beberapa interval 3 jam dalam sehari.
- humidity : Nilai kelembapan harian (satuannya gram uap air per meter kubik volume udara).
- wind_speed : Kecepatan angin diukur dalam kmph.
- meanpressure : Pembacaan tekanan cuaca (diukur dalam atm).
- (col_name)_norm : normalisasi terhadap nilai dari masing-masing kolom dengan rumus
\[ X_{\text{norm}} = \frac{X - X_{\text{min}}}{X_{\text{max}} - X_{\text{min}}} \]
- weather_index : perhitungan indeks cuaca dari beberapa variabel di atas dengan pembobotan masing-masing variabel seperti rumus di bawah ini.
\[\text{Weather Index} = (0.4 \times T_{\text{norm}}) + (0.3 \times RH_{\text{norm}}) + (0.2 \times Wind_{\text{norm}}) + (0.1 \times P_{\text{norm}})\]
- weather_category : indeks cuaca dikategorikan ke dalam 4 kategori dengan range masing-masing. Bagus (0.0 - 0.3), Cukup (0.3 - 0.6), Buruk (0.6 - 0.8) dan Ekstrem ( > 0.8).
3 Data Preprocessing
3.1 Data Type
Masih terdapat tipe variabel yang belum sesuai dengan nilainya, maka perlu melakukan penyesuaian tipe data.
3.2 Select Variabel
Karena pada Time Series Forecasting dataset yang akan digunakan
adalah yang kolom yang berisi data yang menunjukan waktu yaitu
**date** dengan kolom yang berisi data yang ingin kita
amati yaitu **weather_index** maka kita perlu melakukan
Data Aggregation terhadap dataframe Climate.
3.3 Data Date Checking
Perlu melakukan pengecekan terhadap karakteristik Data Time Series kita apakkah sudah sesuai dan bisa dilakukan analisis lanjutan.
3.3.1 Data Arrange
# Memastikan apakah data sudah terurut
climate_daily <- arrange(climate_daily, date)
climate_daily %>% tail()# Mendefinisikan deret waktu yang lengkap
complete_day <- seq.Date(from = min(climate_daily$date), # data terlampau
to = max(climate_daily$date), # data terbaru
by = "day") # interval
complete_day %>% head()#> [1] "2013-01-01" "2013-01-02" "2013-01-03" "2013-01-04" "2013-01-05"
#> [6] "2013-01-06"4 Exploratory Data Analysis
4.1 Time-Series Object
Pembuatan object time-series menggunakan fungsi
ts(data, start, frequency) dengan parameter sebagai
berikut.
data= data (kolom) yang akan diprediksi (nilai numerik)start= waktu awal mula data dalam bentuk vektorc(year, month, day)frequency= pola berulang yang ingin dianalisis
4.2 Decomposition
Decomposition adalah tahapan dalam time series analisis yang digunakan untuk menguraikan beberapa komponen dalam time series data. Komponen/unsur dalam time series:
- Trend \((T_t)\) : pola general data, biasa digunakan untuk melihat kenaikan atau penurunan data.
- Seasonality \((S_t)\) : pola musiman yang membentuk pola berulang pada periode waktu yang tetap.
- Error/Random \((E_t)\) : pola yang tidak dapat ditangkap dalam trend dan seasonality.
#> $x
#> Time Series:
#> Start = c(2013, 1)
#> End = c(2017, 2)
#> Frequency = 365
#> [1] 0.3084558 0.3167998 0.3044633 0.2523879 0.2851714 0.2729247 0.2812291
#> [8] 0.2563283 0.3013723 0.2777670 0.3133098 0.3836416 0.3693553 0.3587201
#> [15] 0.3247922 0.3628475 0.4000974 0.4242756 0.3398262 0.3324356 0.3077239
#> [22] 0.3002789 0.2921309 0.3020104 0.3016801 0.3028635 0.3041531 0.2863463
#> [29] 0.3083997 0.3223716 0.3328259 0.3524874 0.3676145 0.3494708 0.4165245
#> [36] 0.4628272 0.3852785 0.3408617 0.3524728 0.3382532 0.3440788 0.3570108
#> [43] 0.3597317 0.3643921 0.3470117 0.4023054 0.4311674 0.3723087 0.4136830
#> [50] 0.3989515 0.3964282 0.4024024 0.4229518 0.4097389 0.4078401 0.3742341
#> [57] 0.4375849 0.3774472 0.3754882 0.3902710 0.3873047 0.3883960 0.3809393
#> [64] 0.3827491 0.4071739 0.4191890 0.3778364 0.4015133 0.3957689 0.4210437
#> [71] 0.4205990 0.4267563 0.4353687 0.3842403 0.3940026 0.4161989 0.4639261
#> [78] 0.4733527 0.4080258 0.4371419 0.4221186 0.4488093 0.4439345 0.4139229
#> [85] 0.4043057 0.3736601 0.4199686 0.4232762 0.4229765 0.4094785 0.3820484
#> [92] 0.4187062 0.4102055 0.3915748 0.3873984 0.3692332 0.3791090 0.3867624
#> [99] 0.4087477 0.3859849 0.3807708 0.3890143 0.4110853 0.4058452 0.4116101
#> [106] 0.4206724 0.3656289 0.3909877 0.3662732 0.4043320 0.4312907 0.4214845
#> [113] 0.4181053 0.4181466 0.4242462 0.4493227 0.4468304 0.4255056 0.4147206
#> [120] 0.4224580 0.3977597 0.3889252 0.3945366 0.4176091 0.4142051 0.4471592
#> [127] 0.4443618 0.4385296 0.4450496 0.4500590 0.4445683 0.4249925 0.4176135
#> [134] 0.4293542 0.4263437 0.4510426 0.4389658 0.4847383 0.4771802 0.4571198
#> [141] 0.4530257 0.5024708 0.4967623 0.5119665 0.4930801 0.4796610 0.4823301
#> [148] 0.4651556 0.4334785 0.4109269 0.4937158 0.5351928 0.5527348 0.5492912
#> [155] 0.5029309 0.5207973 0.5370920 0.5146358 0.5733899 0.5280046 0.5274264
#> [162] 0.5531572 0.5333464 0.5426094 0.5354494 0.5565897 0.5713622 0.5639737
#> [169] 0.5546025 0.5307182 0.5018918 0.5108256 0.5158047 0.5463064 0.5516101
#> [176] 0.5636564 0.5465943 0.5294462 0.5426542 0.5628996 0.5250931 0.5355634
#> [183] 0.5467511 0.5518425 0.5667008 0.5711160 0.5795389 0.5758588 0.5490918
#> [190] 0.5588637 0.5487918 0.5511970 0.5727518 0.5819360 0.5401104 0.5993128
#> [197] 0.5917199 0.5862733 0.5709610 0.5646243 0.5588027 0.5534775 0.5527270
#> [204] 0.5465154 0.5668381 0.5703276 0.5422450 0.5554764 0.5503869 0.5630809
#> [211] 0.5662264 0.5644322 0.5409393 0.5620255 0.5577962 0.5311947 0.5528360
#> [218] 0.5585751 0.5509433 0.5619386 0.5461652 0.5586179 0.5584531 0.5758515
#> [225] 0.5713372 0.5602237 0.5654449 0.5583492 0.6115423 0.5359492 0.5572519
#> [232] 0.5597706 0.5697141 0.5398331 0.5416955 0.6190906 0.5665902 0.5572115
#> [239] 0.5655543 0.5273159 0.5198157 0.5174831 0.5184528 0.5403356 0.5316454
#> [246] 0.5213977 0.5161879 0.5040420 0.5076463 0.4819687 0.4952641 0.6715471
#> [253] 0.5033605 0.5012186 0.4966036 0.5217349 0.5114981 0.4938696 0.4974804
#> [260] 0.4630921 0.4788271 0.4850367 0.5316792 0.5342746 0.5302934 0.5204385
#> [267] 0.5244401 0.5338084 0.5304453 0.5298373 0.4992167 0.5403717 0.5347054
#> [274] 0.5074121 0.4982415 0.4824755 0.5212734 0.4906822 0.4961923 0.4947981
#> [281] 0.4908125 0.5236453 0.5016419 0.5164849 0.5128617 0.5205934 0.5178457
#> [288] 0.4918142 0.4763215 0.4925287 0.4688235 0.4448513 0.4332321 0.4244917
#> [295] 0.4189577 0.4384313 0.4435084 0.4435640 0.4490233 0.4168166 0.4127005
#> [302] 0.4106668 0.4177674 0.4011844 0.4244069 0.4049743 0.3724580 0.3515846
#> [309] 0.3812544 0.3911732 0.4144337 0.4926288 0.3936260 0.3603798 0.3658010
#> [316] 0.3482643 0.3498934 0.3449377 0.3377944 0.3379435 0.3287476 0.3457208
#> [323] 0.3659582 0.3325712 0.3513570 0.3738412 0.3887747 0.3553131 0.3709887
#> [330] 0.3773564 0.3777750 0.3789550 0.3643223 0.4091565 0.3462160 0.3668832
#> [337] 0.3686268 0.3720286 0.3746924 0.3588940 0.3734452 0.3848779 0.3760798
#> [344] 0.3782296 0.3763627 0.4065604 0.3926616 0.3682549 0.3938416 0.3954810
#> [351] 0.3999200 0.4073673 0.4147330 0.4189073 0.4016634 0.4121767 0.4162310
#> [358] 0.3796071 0.4427960 0.2968504 0.3208997 0.2941786 0.2884768 0.3508730
#> [365] 0.4034682 0.4037902 0.3378877 0.3309216 0.3616849 0.3552648 0.4616369
#> [372] 0.3899411 0.3137506 0.3437609 0.3405397 0.3598492 0.3465084 0.3209019
#> [379] 0.3544283 0.3663421 0.3650055 0.3774274 0.3816339 0.3800344 0.3825173
#> [386] 0.4289812 0.4447851 0.4285830 0.4094729 0.4079538 0.4080715 0.3946170
#> [393] 0.3739072 0.3809138 0.3635760 0.3739208 0.3517369 0.3524050 0.3981648
#> [400] 0.4141983 0.3609710 0.3745305 0.4375738 0.3941849 0.3142327 0.2792012
#> [407] 0.2983474 0.3313993 0.3264077 0.4084907 0.3734377 0.3738226 0.3677504
#> [414] 0.3298887 0.3482743 0.3717986 0.3947319 0.4229877 0.4097498 0.3593127
#> [421] 0.3584903 0.3789639 0.4205802 0.4181033 0.4133122 0.3917221 0.3568704
#> [428] 0.3702305 0.3815585 0.3617954 0.3692645 0.4049360 0.3948077 0.4108120
#> [435] 0.4630858 0.4073199 0.3829059 0.3730385 0.3880049 0.4084401 0.4266954
#> [442] 0.4288288 0.4063540 0.3818299 0.4180266 0.4193859 0.4286341 0.4652380
#> [449] 0.4163202 0.4176706 0.4229842 0.4474696 0.4419959 0.4161165 0.4036829
#> [456] 0.4019361 0.4046064 0.4146920 0.4317255 0.4243435 0.4502022 0.4529140
#> [463] 0.4135324 0.3906321 0.3763650 0.3855123 0.4117054 0.4352847 0.4179858
#> [470] 0.4267126 0.4315985 0.4315254 0.4808169 0.4313422 0.4117319 0.3921828
#> [477] 0.4134908 0.4061488 0.4140300 0.4252441 0.4277159 0.4112213 0.4232410
#> [484] 0.4255027 0.4134089 0.4436958 0.4299886 0.4277447 0.4295328 0.4843308
#> [491] 0.4646666 0.4586903 0.4763371 0.4341369 0.4277288 0.4315906 0.4721188
#> [498] 0.4770418 0.4539275 0.4447941 0.4581959 0.4575600 0.4668403 0.4461566
#> [505] 0.4278145 0.4553132 0.4679999 0.5221483 0.4852228 0.5061259 0.4689319
#> [512] 0.4674397 0.4772304 0.4840458 0.5072745 0.4823973 0.4890673 0.4589853
#> [519] 0.4668327 0.4729767 0.4776748 0.4956564 0.5051306 0.4759358 0.4960355
#> [526] 0.4779113 0.5034503 0.5463299 0.5329564 0.5285278 0.5258825 0.5328160
#> [533] 0.5655497 0.5576509 0.5476679 0.5673002 0.5291274 0.4977943 0.4901040
#> [540] 0.5051256 0.5345108 0.5151089 0.5310837 0.5167883 0.5427562 0.5345659
#> [547] 0.5320827 0.5286277 0.5306279 0.5365740 0.5415169 0.4970257 0.5041428
#> [554] 0.5040156 0.5266007 0.5228106 0.5164017 0.5350699 0.5500634 0.5560407
#> [561] 0.5569512 0.5598512 0.5582503 0.5290159 0.5340261 0.5389697 0.5349097
#> [568] 0.5554980 0.5585509 0.5435737 0.5355280 0.5376711 0.5372961 0.6079679
#> [575] 0.5325755 0.5305156 0.5535538 0.5366627 0.5352750 0.5352471 0.5319017
#> [582] 0.5342102 0.5375110 0.5331578 0.5358696 0.5610169 0.5779666 0.5379340
#> [589] 0.5246907 0.5845812 0.5514753 0.5329756 0.5122247 0.4975694 0.5055426
#> [596] 0.5110174 0.5046721 0.4911405 0.5005557 0.5006404 0.5122568 0.5062587
#> [603] 0.4970996 0.5005279 0.5220694 0.5221047 0.5336418 0.5503039 0.5547012
#> [610] 0.5462494 0.5842758 0.5577571 0.5538276 0.5179490 0.5132169 0.5217698
#> [617] 0.5240006 0.5363877 0.5995910 0.5023657 0.5086097 0.5081061 0.5059894
#> [624] 0.5229303 0.5171139 0.5140842 0.4882468 0.4878007 0.4826432 0.5045940
#> [631] 0.5804867 0.4775015 0.4880713 0.4694239 0.4703300 0.4665110 0.4702532
#> [638] 0.4732370 0.4635391 0.4628430 0.4573167 0.4577053 0.4622411 0.4644037
#> [645] 0.4835694 0.4451997 0.4192345 0.4390932 0.4385605 0.4165191 0.4798016
#> [652] 0.4524332 0.4439950 0.4016206 0.4826815 0.4822312 0.4035586 0.4095533
#> [659] 0.4110497 0.4352005 0.4351536 0.4477772 0.4390472 0.4374270 0.4241933
#> [666] 0.4370407 0.4314815 0.4067187 0.3985874 0.3699769 0.3773254 0.3816581
#> [673] 0.3970593 0.3883211 0.3990659 0.4268772 0.4225481 0.4034114 0.3517878
#> [680] 0.3307634 0.3244748 0.3099113 0.3021685 0.2893019 0.2936203 0.3028769
#> [687] 0.3024462 0.3142780 0.3460843 0.3470904 0.3480405 0.3081353 0.3005412
#> [694] 0.2856446 0.3362677 0.3362770 0.3636993 0.3752627 0.3366336 0.3272766
#> [701] 0.3570207 0.4834878 0.4148295 0.3299800 0.3089195 0.2949386 0.2871944
#> [708] 0.3144367 0.4061924 0.3929694 0.3369245 0.3797606 0.4185771 0.3701580
#> [715] 0.3963934 0.3315165 0.3277375 0.3225984 0.3252648 0.2881282 0.3091382
#> [722] 0.2912382 0.3182634 0.3260184 0.3278426 0.2725120 0.3047958 0.2876535
#> [729] 0.2923091 0.2905050 0.3276232 0.4243793 0.4015137 0.3840798 0.3762483
#> [736] 0.3377237 0.3277334 0.3367169 0.3104253 0.3037371 0.3307669 0.3032027
#> [743] 0.3361211 0.3492959 0.3232989 0.3450603 0.3328304 0.3204374 0.3405879
#> [750] 0.3487397 0.3272384 0.3893067 0.3723019 0.3897530 0.3445821 0.3713058
#> [757] 0.3380059 0.4003792 0.2865909 0.3008350 0.3039571 0.3270695 0.3831384
#> [764] 0.3791242 0.3972110 0.3251690 0.3096289 0.3305931 0.3389120 0.3460886
#> [771] 0.3296840 0.3286577 0.3527046 0.3405531 0.3686523 0.3736898 0.4418102
#> [778] 0.4271349 0.4298339 0.4624232 0.4381651 0.4345667 0.4382043 0.4282917
#> [785] 0.4523966 0.4099681 0.2887483 0.3056179 0.3437499 0.4973286 0.4506008
#> [792] 0.4042694 0.3699180 0.3619314 0.3781649 0.4163149 0.4260200 0.3611320
#> [799] 0.3883764 0.3872214 0.3888770 0.4005962 0.4174909 0.4725441 0.4604049
#> [806] 0.4550095 0.4402966 0.4056470 0.4390609 0.4345708 0.4706213 0.4599356
#> [813] 0.4416928 0.4583524 0.4741057 0.4423026 0.4268313 0.4442159 0.4803780
#> [820] 0.4443665 0.4551295 0.4494418 0.4722392 0.4251333 0.4994961 0.4204364
#> [827] 0.4450978 0.4129988 0.4372416 0.4356005 0.4325610 0.4348972 0.4744876
#> [834] 0.4460069 0.4638375 0.4423596 0.4455697 0.4420888 0.4478611 0.4604971
#> [841] 0.4328454 0.4277752 0.4219208 0.4227447 0.4770684 0.4965076 0.5285647
#> [848] 0.4987330 0.4818103 0.5184871 0.4513586 0.4219047 0.4225702 0.4254236
#> [855] 0.4301022 0.4404954 0.4456087 0.4461354 0.4557163 0.4422688 0.4735590
#> [862] 0.4862507 0.4757242 0.4483510 0.4563353 0.4586305 0.4791922 0.4759283
#> [869] 0.5009832 0.4769603 0.4677946 0.4632135 0.4857465 0.4588204 0.4833105
#> [876] 0.4975707 0.4086767 0.4597815 0.4555760 0.4799036 0.4676954 0.4563230
#> [883] 0.4565215 0.4500764 0.4736196 0.4587756 0.4762487 0.4807046 0.4833961
#> [890] 0.5247439 0.5383567 0.5357992 0.5121124 0.4887410 0.5110031 0.4969833
#> [897] 0.4828355 0.4951219 0.4932486 0.5343823 0.5276865 0.5234446 0.5129284
#> [904] 0.5202808 0.6006977 0.5350603 0.5164328 0.5260853 0.5193557 0.5247720
#> [911] 0.5210536 0.4887571 0.4977333 0.5344271 0.5688979 0.5600294 0.5663880
#> [918] 0.5442379 0.5378879 0.5351461 0.5664749 0.5598813 0.5684463 0.5348889
#> [925] 0.5573686 0.5655687 0.5576409 0.5520161 0.5469385 0.5073592 0.5425926
#> [932] 0.5552479 0.5470170 0.5514118 0.5406176 0.6108002 0.5881600 0.5482058
#> [939] 0.5201368 0.5291975 0.5297003 0.5135582 0.4916304 0.5083547 0.5067175
#> [946] 0.5451427 0.5699922 0.5423572 0.5493795 0.5480612 0.5544636 0.5313884
#> [953] 0.5425181 0.5264845 0.5274826 0.5388284 0.5379538 0.5382846 0.5241465
#> [960] 0.5249368 0.6454023 0.5414772 0.5424812 0.5170618 0.5252128 0.5306258
#> [967] 0.5306628 0.5188606 0.5271764 0.5154590 0.5160248 0.5318141 0.5351997
#> [974] 0.5233614 0.5013652 0.4984064 0.4860046 0.4706801 0.4563677 0.4840356
#> [981] 0.4987403 0.4864344 0.4771024 0.4701059 0.4898877 0.4840636 0.4834523
#> [988] 0.4894800 0.4862840 0.4846372 0.5417231 0.5348513 0.5154570 0.5125224
#> [995] 0.5344016 0.5202745 0.4749637 0.4534018 0.4658724 0.4596325 0.4600664
#> [1002] 0.4821287 0.4498261 0.4412103 0.4943917 0.5304277 0.4506081 0.4586641
#> [1009] 0.4740073 0.5017973 0.4867833 0.4772817 0.4627883 0.4539488 0.4538888
#> [1016] 0.4895305 0.4807942 0.4604833 0.4555639 0.4741791 0.5228702 0.4834964
#> [1023] 0.4707100 0.4425047 0.4108415 0.3886963 0.3886207 0.4055281 0.4137122
#> [1030] 0.4004164 0.3843018 0.3927161 0.4076956 0.4128653 0.4217539 0.4287038
#> [1037] 0.4242551 0.4272025 0.4487709 0.4417685 0.4257916 0.4130966 0.4177455
#> [1044] 0.4264877 0.3889700 0.3978961 0.3979137 0.3899790 0.3832762 0.3485624
#> [1051] 0.3281434 0.3401071 0.3483510 0.3504738 0.3635355 0.3480907 0.3559292
#> [1058] 0.3455830 0.3530257 0.3484367 0.3611654 0.4127155 0.4020756 0.4216450
#> [1065] 0.4093163 0.3748117 0.3863582 0.3647359 0.3625260 0.3723278 0.3862391
#> [1072] 0.3707288 0.4052219 0.4032134 0.3982474 0.3915433 0.2994373 0.3109265
#> [1079] 0.3004285 0.2848115 0.3026786 0.3185745 0.3103091 0.2860157 0.3133792
#> [1086] 0.3202664 0.3447794 0.2973409 0.2895531 0.3035762 0.3379660 0.3558180
#> [1093] 0.3539748 0.3414933 0.3340600 0.3288500 0.3374335 0.3524173 0.3532300
#> [1100] 0.3977354 0.3996329 0.4042690 0.4087574 0.3531191 0.3553157 0.3531291
#> [1107] 0.3835030 0.4034315 0.3827663 0.3951136 0.3903412 0.3436706 0.3414194
#> [1114] 0.3734081 0.3620629 0.3334234 0.3428773 0.3482743 0.3604672 0.3190154
#> [1121] 0.3596406 0.3670897 0.3636364 0.3893677 0.4173505 0.4137480 0.3494829
#> [1128] 0.3157888 0.3204267 0.3100110 0.3034029 0.3391835 0.3644487 0.3631107
#> [1135] 0.3720876 0.3497760 0.3523246 0.3321474 0.3448485 0.3619430 0.3430416
#> [1142] 0.3364319 0.3452361 0.4193422 0.4394680 0.4882818 0.4431211 0.4003711
#> [1149] 0.3771816 0.3712853 0.3937761 0.3765974 0.3924227 0.4107484 0.4244752
#> [1156] 0.4050342 0.4041882 0.4243131 0.4097215 0.4360536 0.4276433 0.4300512
#> [1163] 0.4203786 0.4306503 0.4427343 0.4657771 0.4705161 0.4481282 0.4253645
#> [1170] 0.4328022 0.4238195 0.4217898 0.4575337 0.4520986 0.4420714 0.4272480
#> [1177] 0.4356621 0.4139834 0.4149523 0.4484900 0.4565963 0.4437066 0.5111741
#> [1184] 0.4249874 0.4260088 0.4183084 0.4564858 0.4478596 0.4660039 0.4471336
#> [1191] 0.4338226 0.4081242 0.4388662 0.3980784 0.4413437 0.4128435 0.4549185
#> [1198] 0.4797498 0.4669755 0.4303501 0.4388892 0.4642792 0.4442377 0.4549650
#> [1205] 0.4470246 0.4660183 0.4690118 0.4233214 0.4521404 0.4009893 0.3934173
#> [1212] 0.4075281 0.4225634 0.4422669 0.4337375 0.4107546 0.4389407 0.4668757
#> [1219] 0.4632980 0.4654020 0.4662260 0.4463356 0.4514412 0.4832712 0.5312268
#> [1226] 0.4967187 0.5004998 0.4884944 0.4848434 0.4704581 0.4733931 0.4723558
#> [1233] 0.4866453 0.4904615 0.5349443 0.5434350 0.5437460 0.5293609 0.5239488
#> [1240] 0.5043711 0.5089262 0.5507567 0.5240999 0.5314206 0.5314049 0.5252417
#> [1247] 0.5048795 0.5228726 0.5313543 0.5066394 0.5212237 0.5550057 0.5368914
#> [1254] 0.6225834 0.5290977 0.5246839 0.5277798 0.5167854 0.4992596 0.5442748
#> [1261] 0.5392483 0.5369386 0.5344451 0.5274022 0.5582772 0.5577659 0.5638716
#> [1268] 0.5528737 0.5440812 0.5136903 0.5367882 0.5659405 0.5549095 0.5628056
#> [1275] 0.5554960 0.5697582 0.5675796 0.5798986 0.5423957 0.5346558 0.5492805
#> [1282] 0.5629620 0.5639443 0.5490907 0.5583618 0.5604832 0.5411464 0.5656890
#> [1289] 0.5703004 0.5719281 0.5687652 0.5771397 0.5708343 0.5682970 0.5839773
#> [1296] 0.5924780 0.5789768 0.5619451 0.5460456 0.5484326 0.5387374 0.5370247
#> [1303] 0.5787150 0.5655553 0.5607104 0.5597327 0.5559286 0.5613638 0.5692249
#> [1310] 0.5395388 0.5373671 0.5569472 0.5405863 0.5569757 0.5611573 0.5674402
#> [1317] 0.5732481 0.5794808 0.5839239 0.5609944 0.5462939 0.5623886 0.5751717
#> [1324] 0.5628520 0.5402916 0.5433563 0.5310164 0.5784083 0.5628362 0.5529199
#> [1331] 0.5497817 0.5768226 0.5593783 0.5515857 0.5593201 0.5656170 0.5676441
#> [1338] 0.5688061 0.5687197 0.5570263 0.5801766 0.5509779 0.5389143 0.5483053
#> [1345] 0.5188542 0.5165243 0.5356642 0.5177722 0.5335775 0.5063009 0.5238491
#> [1352] 0.5291775 0.5344189 0.5441849 0.4948114 0.5020456 0.5762588 0.5050045
#> [1359] 0.5016889 0.5343000 0.5333043 0.5182702 0.5361350 0.5046734 0.5032754
#> [1366] 0.4955612 0.4933761 0.5185422 0.5050432 0.5207295 0.5282442 0.5254061
#> [1373] 0.5361016 0.5105976 0.5085912 0.5545499 0.5097526 0.4915294 0.4572253
#> [1380] 0.4754931 0.4725416 0.4737899 0.4652619 0.4516895 0.4469115 0.4473339
#> [1387] 0.4460038 0.4479919 0.4574080 0.4659833 0.4865252 0.4746810 0.4729381
#> [1394] 0.4838772 0.4522243 0.4524185 0.4434985 0.4346382 0.4526261 0.4327163
#> [1401] 0.4079106 0.4186709 0.4245352 0.4182805 0.4340419 0.4063167 0.4099949
#> [1408] 0.4250236 0.3803240 0.3699640 0.3871412 0.3954774 0.4047078 0.3963598
#> [1415] 0.3777552 0.3778402 0.3974012 0.3817047 0.3696988 0.3848261 0.3790641
#> [1422] 0.3811128 0.3819518 0.3729128 0.3592122 0.3721711 0.3579213 0.3276867
#> [1429] 0.3496301 0.4340765 0.4387627 0.4146141 0.3612933 0.3428114 0.3408164
#> [1436] 0.3742329 0.3785605 0.3623786 0.3730371 0.4068249 0.4018569 0.3736297
#> [1443] 0.3585942 0.3614012 0.3585252 0.3248321 0.3589063 0.3441221 0.3554496
#> [1450] 0.3790841 0.3941518 0.3344979 0.3480454 0.4023647 0.4343438 0.4040045
#> [1457] 0.3729772 0.3564808 0.4125747 0.4061477 0.4136029 0.3621730
#>
#> $seasonal
#> Time Series:
#> Start = c(2013, 1)
#> End = c(2017, 2)
#> Frequency = 365
#> [1] -0.0944314775 -0.0812771463 -0.0861971883 -0.0814632051 -0.0713634752
#> [6] -0.0481042666 -0.0737875891 -0.0946933236 -0.1120105645 -0.1145723044
#> [11] -0.0998482712 -0.1033480338 -0.0942969906 -0.0855903675 -0.0861557196
#> [16] -0.0809241484 -0.0964145714 -0.0999115056 -0.0830956339 -0.0833557623
#> [21] -0.0845898215 -0.0554807363 -0.0647338574 -0.0611937293 -0.0905695655
#> [26] -0.0681019171 -0.0812175317 -0.0684786232 -0.0954895052 -0.0871853697
#> [31] -0.0839158788 -0.1050346732 -0.0973370358 -0.0818814924 -0.0739837221
#> [36] -0.1179264331 -0.1066652422 -0.0702403216 -0.0823957468 -0.1036753070
#> [41] -0.1282770797 -0.1213949729 -0.1090783577 -0.1105463006 -0.0681179474
#> [46] -0.0844388930 -0.0637625674 -0.0677476756 -0.0547538203 -0.0310273357
#> [51] -0.0150086213 -0.0236382193 -0.0272673153 -0.0426506952 -0.0534028697
#> [56] -0.0603249135 -0.0996204393 -0.0748792494 -0.0569274900 -0.0028018294
#> [61] -0.0321008027 -0.0594697397 -0.0597816945 -0.0635854837 -0.0560164043
#> [66] -0.0436453280 -0.0277542736 -0.0560175919 -0.0381968683 -0.0170051826
#> [71] -0.0273741861 -0.0300556547 -0.0351847192 -0.0194346853 -0.0142062406
#> [76] -0.0129509083 -0.0178632841 -0.0250190760 -0.0238887328 -0.0167455699
#> [81] -0.0091924370 -0.0068370949 -0.0079466402 -0.0183881198 -0.0015020913
#> [86] -0.0076319160 -0.0089002561 0.0175834860 -0.0077145087 -0.0234794700
#> [91] -0.0230261278 -0.0113039413 -0.0032389032 -0.0072567595 0.0087763124
#> [96] -0.0134357993 -0.0128906768 -0.0264828167 -0.0396895042 -0.0305577683
#> [101] -0.0380126340 -0.0144237554 0.0149440058 -0.0045300761 -0.0078458217
#> [106] -0.0105058831 -0.0009743947 0.0076569776 -0.0033151123 -0.0082877579
#> [111] -0.0177035522 -0.0113231921 -0.0309950941 -0.0185309601 -0.0137638529
#> [116] -0.0090110140 0.0008482470 -0.0001096345 0.0015349128 0.0068561352
#> [121] -0.0131007805 -0.0181394408 -0.0093614937 -0.0090095900 0.0114750670
#> [126] 0.0085928241 0.0016411999 0.0093617371 0.0091060466 0.0185481772
#> [131] 0.0187341435 0.0377162604 0.0318428090 0.0137611422 0.0085414167
#> [136] 0.0146974792 0.0209401735 0.0276724448 0.0303567541 0.0310154825
#> [141] 0.0399423362 0.0427139422 0.0634488309 0.0403561370 0.0489683167
#> [146] 0.0428177813 0.0266307691 0.0380602123 0.0413755398 0.0572416547
#> [151] 0.0428588782 0.0345583882 0.0305146688 0.0337437031 0.0353591402
#> [156] 0.0368483948 0.0599564159 0.0585781813 0.0783070599 0.0676334182
#> [161] 0.0646708618 0.0733680655 0.0760730682 0.0579871135 0.0789540668
#> [166] 0.0717323338 0.0685872926 0.0827966029 0.0772601716 0.0979796517
#> [171] 0.1021858619 0.0901408127 0.0725457434 0.0695186817 0.0912758932
#> [176] 0.0869199537 0.0839978235 0.0887377435 0.0842855204 0.0922384415
#> [181] 0.0929608119 0.0805850243 0.0896069315 0.0911055616 0.1029582759
#> [186] 0.1136562030 0.1036608814 0.0973203883 0.0861232845 0.0958826563
#> [191] 0.1015657300 0.0979860104 0.1141778654 0.1110149235 0.1065588610
#> [196] 0.1292987236 0.1250698858 0.1207815670 0.1042150295 0.0906313082
#> [201] 0.1021587394 0.1032181017 0.1070600483 0.1074486058 0.1055934602
#> [206] 0.1274176757 0.1111710001 0.1021226112 0.1145751818 0.0966384926
#> [211] 0.0970973632 0.0987240293 0.0778667019 0.0899337346 0.0879716681
#> [216] 0.0908451089 0.1071380611 0.1010369655 0.0995283134 0.1037000199
#> [221] 0.1089372403 0.1110365948 0.1013164616 0.0973517625 0.1161474674
#> [226] 0.1052148457 0.1005149400 0.0913293887 0.0995167603 0.0773267073
#> [231] 0.1264324731 0.0905100528 0.0896116593 0.0742327469 0.0775615414
#> [236] 0.1090570401 0.0895996141 0.0795196435 0.0861974363 0.0767816881
#> [241] 0.0744683326 0.0827674040 0.0897419745 0.0945397028 0.0814792247
#> [246] 0.0897128875 0.0749781637 0.0644851559 0.0489485555 0.0479869169
#> [251] 0.0601540249 0.1155248024 0.0604611420 0.0784415504 0.0510487250
#> [256] 0.0595541753 0.0557326745 0.0511247825 0.0569030004 0.0429910120
#> [261] 0.0662691808 0.0573957235 0.0663252757 0.0644883730 0.0777833184
#> [266] 0.0951142045 0.0470151523 0.0464335247 0.0431778402 0.0411755417
#> [271] 0.0297600317 0.0520780115 0.0404012274 0.0251657772 0.0395695188
#> [276] 0.0444497023 0.0308373767 0.0247830114 0.0323959459 0.0475566670
#> [281] 0.0283859593 0.0275177015 0.0219211287 0.0237154672 0.0150803750
#> [286] 0.0505696469 0.0375630372 0.0192857463 -0.0016597557 0.0368779251
#> [291] 0.0450077902 -0.0024127541 -0.0085886721 -0.0204368300 -0.0248290452
#> [296] -0.0257399832 -0.0198959680 -0.0171303860 -0.0130715863 -0.0326144648
#> [301] -0.0350728425 -0.0348242065 -0.0357287763 -0.0422555088 -0.0411343280
#> [306] -0.0429117619 -0.0538309415 -0.0547192750 -0.0405700087 -0.0360180866
#> [311] -0.0243315623 -0.0039761463 -0.0418804648 -0.0673039641 -0.0850798543
#> [316] -0.0901110324 -0.0944673215 -0.1013853120 -0.1103099982 -0.1204225974
#> [321] -0.1272103913 -0.1177204683 -0.1043538566 -0.1042556748 -0.0933422198
#> [326] -0.0907087942 -0.0964308214 -0.1135750912 -0.1108612380 -0.0934564743
#> [331] -0.0891247785 -0.0624967308 -0.0671764871 -0.0586137893 -0.0867948680
#> [336] -0.0814653235 -0.0348632070 -0.0638262031 -0.0919584582 -0.1009938550
#> [341] -0.0961869898 -0.1001725044 -0.0825277308 -0.0518929259 -0.0585550011
#> [346] -0.0693903487 -0.0904070134 -0.0817820570 -0.0928763549 -0.0887589889
#> [351] -0.1029222851 -0.0963760365 -0.0984141345 -0.1042827359 -0.1133368485
#> [356] -0.1005671851 -0.0970093759 -0.1159871652 -0.0949105963 -0.1382943398
#> [361] -0.1372814536 -0.1294949375 -0.1377172619 -0.1195679937 -0.1051445925
#> [366] -0.0944314775 -0.0812771463 -0.0861971883 -0.0814632051 -0.0713634752
#> [371] -0.0481042666 -0.0737875891 -0.0946933236 -0.1120105645 -0.1145723044
#> [376] -0.0998482712 -0.1033480338 -0.0942969906 -0.0855903675 -0.0861557196
#> [381] -0.0809241484 -0.0964145714 -0.0999115056 -0.0830956339 -0.0833557623
#> [386] -0.0845898215 -0.0554807363 -0.0647338574 -0.0611937293 -0.0905695655
#> [391] -0.0681019171 -0.0812175317 -0.0684786232 -0.0954895052 -0.0871853697
#> [396] -0.0839158788 -0.1050346732 -0.0973370358 -0.0818814924 -0.0739837221
#> [401] -0.1179264331 -0.1066652422 -0.0702403216 -0.0823957468 -0.1036753070
#> [406] -0.1282770797 -0.1213949729 -0.1090783577 -0.1105463006 -0.0681179474
#> [411] -0.0844388930 -0.0637625674 -0.0677476756 -0.0547538203 -0.0310273357
#> [416] -0.0150086213 -0.0236382193 -0.0272673153 -0.0426506952 -0.0534028697
#> [421] -0.0603249135 -0.0996204393 -0.0748792494 -0.0569274900 -0.0028018294
#> [426] -0.0321008027 -0.0594697397 -0.0597816945 -0.0635854837 -0.0560164043
#> [431] -0.0436453280 -0.0277542736 -0.0560175919 -0.0381968683 -0.0170051826
#> [436] -0.0273741861 -0.0300556547 -0.0351847192 -0.0194346853 -0.0142062406
#> [441] -0.0129509083 -0.0178632841 -0.0250190760 -0.0238887328 -0.0167455699
#> [446] -0.0091924370 -0.0068370949 -0.0079466402 -0.0183881198 -0.0015020913
#> [451] -0.0076319160 -0.0089002561 0.0175834860 -0.0077145087 -0.0234794700
#> [456] -0.0230261278 -0.0113039413 -0.0032389032 -0.0072567595 0.0087763124
#> [461] -0.0134357993 -0.0128906768 -0.0264828167 -0.0396895042 -0.0305577683
#> [466] -0.0380126340 -0.0144237554 0.0149440058 -0.0045300761 -0.0078458217
#> [471] -0.0105058831 -0.0009743947 0.0076569776 -0.0033151123 -0.0082877579
#> [476] -0.0177035522 -0.0113231921 -0.0309950941 -0.0185309601 -0.0137638529
#> [481] -0.0090110140 0.0008482470 -0.0001096345 0.0015349128 0.0068561352
#> [486] -0.0131007805 -0.0181394408 -0.0093614937 -0.0090095900 0.0114750670
#> [491] 0.0085928241 0.0016411999 0.0093617371 0.0091060466 0.0185481772
#> [496] 0.0187341435 0.0377162604 0.0318428090 0.0137611422 0.0085414167
#> [501] 0.0146974792 0.0209401735 0.0276724448 0.0303567541 0.0310154825
#> [506] 0.0399423362 0.0427139422 0.0634488309 0.0403561370 0.0489683167
#> [511] 0.0428177813 0.0266307691 0.0380602123 0.0413755398 0.0572416547
#> [516] 0.0428588782 0.0345583882 0.0305146688 0.0337437031 0.0353591402
#> [521] 0.0368483948 0.0599564159 0.0585781813 0.0783070599 0.0676334182
#> [526] 0.0646708618 0.0733680655 0.0760730682 0.0579871135 0.0789540668
#> [531] 0.0717323338 0.0685872926 0.0827966029 0.0772601716 0.0979796517
#> [536] 0.1021858619 0.0901408127 0.0725457434 0.0695186817 0.0912758932
#> [541] 0.0869199537 0.0839978235 0.0887377435 0.0842855204 0.0922384415
#> [546] 0.0929608119 0.0805850243 0.0896069315 0.0911055616 0.1029582759
#> [551] 0.1136562030 0.1036608814 0.0973203883 0.0861232845 0.0958826563
#> [556] 0.1015657300 0.0979860104 0.1141778654 0.1110149235 0.1065588610
#> [561] 0.1292987236 0.1250698858 0.1207815670 0.1042150295 0.0906313082
#> [566] 0.1021587394 0.1032181017 0.1070600483 0.1074486058 0.1055934602
#> [571] 0.1274176757 0.1111710001 0.1021226112 0.1145751818 0.0966384926
#> [576] 0.0970973632 0.0987240293 0.0778667019 0.0899337346 0.0879716681
#> [581] 0.0908451089 0.1071380611 0.1010369655 0.0995283134 0.1037000199
#> [586] 0.1089372403 0.1110365948 0.1013164616 0.0973517625 0.1161474674
#> [591] 0.1052148457 0.1005149400 0.0913293887 0.0995167603 0.0773267073
#> [596] 0.1264324731 0.0905100528 0.0896116593 0.0742327469 0.0775615414
#> [601] 0.1090570401 0.0895996141 0.0795196435 0.0861974363 0.0767816881
#> [606] 0.0744683326 0.0827674040 0.0897419745 0.0945397028 0.0814792247
#> [611] 0.0897128875 0.0749781637 0.0644851559 0.0489485555 0.0479869169
#> [616] 0.0601540249 0.1155248024 0.0604611420 0.0784415504 0.0510487250
#> [621] 0.0595541753 0.0557326745 0.0511247825 0.0569030004 0.0429910120
#> [626] 0.0662691808 0.0573957235 0.0663252757 0.0644883730 0.0777833184
#> [631] 0.0951142045 0.0470151523 0.0464335247 0.0431778402 0.0411755417
#> [636] 0.0297600317 0.0520780115 0.0404012274 0.0251657772 0.0395695188
#> [641] 0.0444497023 0.0308373767 0.0247830114 0.0323959459 0.0475566670
#> [646] 0.0283859593 0.0275177015 0.0219211287 0.0237154672 0.0150803750
#> [651] 0.0505696469 0.0375630372 0.0192857463 -0.0016597557 0.0368779251
#> [656] 0.0450077902 -0.0024127541 -0.0085886721 -0.0204368300 -0.0248290452
#> [661] -0.0257399832 -0.0198959680 -0.0171303860 -0.0130715863 -0.0326144648
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#> [1256] 0.0646708618 0.0733680655 0.0760730682 0.0579871135 0.0789540668
#> [1261] 0.0717323338 0.0685872926 0.0827966029 0.0772601716 0.0979796517
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#> [1301] 0.1274176757 0.1111710001 0.1021226112 0.1145751818 0.0966384926
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#> [1406] -0.0243315623 -0.0039761463 -0.0418804648 -0.0673039641 -0.0850798543
#> [1411] -0.0901110324 -0.0944673215 -0.1013853120 -0.1103099982 -0.1204225974
#> [1416] -0.1272103913 -0.1177204683 -0.1043538566 -0.1042556748 -0.0933422198
#> [1421] -0.0907087942 -0.0964308214 -0.1135750912 -0.1108612380 -0.0934564743
#> [1426] -0.0891247785 -0.0624967308 -0.0671764871 -0.0586137893 -0.0867948680
#> [1431] -0.0814653235 -0.0348632070 -0.0638262031 -0.0919584582 -0.1009938550
#> [1436] -0.0961869898 -0.1001725044 -0.0825277308 -0.0518929259 -0.0585550011
#> [1441] -0.0693903487 -0.0904070134 -0.0817820570 -0.0928763549 -0.0887589889
#> [1446] -0.1029222851 -0.0963760365 -0.0984141345 -0.1042827359 -0.1133368485
#> [1451] -0.1005671851 -0.0970093759 -0.1159871652 -0.0949105963 -0.1382943398
#> [1456] -0.1372814536 -0.1294949375 -0.1377172619 -0.1195679937 -0.1051445925
#> [1461] -0.0944314775 -0.0812771463
#>
#> $trend
#> Time Series:
#> Start = c(2013, 1)
#> End = c(2017, 2)
#> Frequency = 365
#> [1] NA NA NA NA NA NA NA
#> [8] NA NA NA NA NA NA NA
#> [15] NA NA NA NA NA NA NA
#> [22] NA NA NA NA NA NA NA
#> [29] NA NA NA NA NA NA NA
#> [36] NA NA NA NA NA NA NA
#> [43] NA NA NA NA NA NA NA
#> [50] NA NA NA NA NA NA NA
#> [57] NA NA NA NA NA NA NA
#> [64] NA NA NA NA NA NA NA
#> [71] NA NA NA NA NA NA NA
#> [78] NA NA NA NA NA NA NA
#> [85] NA NA NA NA NA NA NA
#> [92] NA NA NA NA NA NA NA
#> [99] NA NA NA NA NA NA NA
#> [106] NA NA NA NA NA NA NA
#> [113] NA NA NA NA NA NA NA
#> [120] NA NA NA NA NA NA NA
#> [127] NA NA NA NA NA NA NA
#> [134] NA NA NA NA NA NA NA
#> [141] NA NA NA NA NA NA NA
#> [148] NA NA NA NA NA NA NA
#> [155] NA NA NA NA NA NA NA
#> [162] NA NA NA NA NA NA NA
#> [169] NA NA NA NA NA NA NA
#> [176] NA NA NA NA NA NA NA
#> [183] 0.4472048 0.4474660 0.4475238 0.4475962 0.4478957 0.4480877 0.4486047
#> [190] 0.4489026 0.4490599 0.4491760 0.4493480 0.4494755 0.4493738 0.4492410
#> [197] 0.4492293 0.4493431 0.4493490 0.4492869 0.4491701 0.4492802 0.4494175
#> [204] 0.4497497 0.4501456 0.4505194 0.4508138 0.4511050 0.4513932 0.4516411
#> [211] 0.4518810 0.4520796 0.4521925 0.4523051 0.4523031 0.4522614 0.4523948
#> [218] 0.4523884 0.4521094 0.4520799 0.4523449 0.4524592 0.4523934 0.4522156
#> [225] 0.4520549 0.4519773 0.4518732 0.4520416 0.4519626 0.4518054 0.4517930
#> [232] 0.4515634 0.4514245 0.4513571 0.4513360 0.4513361 0.4513362 0.4512032
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#> [274] 0.4508097 0.4507711 0.4507833 0.4508934 0.4509946 0.4512164 0.4514186
#> [281] 0.4514920 0.4514423 0.4514160 0.4514290 0.4514911 0.4515574 0.4515907
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#> [295] 0.4521581 0.4521254 0.4521141 0.4521168 0.4520576 0.4519601 0.4519539
#> [302] 0.4519834 0.4519586 0.4520845 0.4521970 0.4522880 0.4523206 0.4525128
#> [309] 0.4525607 0.4526000 0.4527036 0.4526737 0.4526125 0.4525769 0.4527060
#> [316] 0.4528689 0.4529362 0.4529867 0.4530063 0.4530573 0.4530082 0.4529232
#> [323] 0.4528429 0.4528492 0.4527548 0.4528243 0.4527511 0.4527868 0.4527574
#> [330] 0.4527166 0.4527497 0.4528882 0.4531522 0.4531212 0.4529948 0.4527380
#> [337] 0.4525121 0.4524300 0.4523118 0.4521983 0.4521723 0.4519053 0.4518177
#> [344] 0.4516820 0.4515459 0.4515814 0.4515550 0.4515360 0.4514519 0.4513463
#> [351] 0.4513506 0.4513589 0.4514054 0.4515846 0.4516347 0.4515854 0.4514314
#> [358] 0.4513041 0.4512242 0.4511379 0.4511424 0.4510716 0.4510164 0.4510423
#> [365] 0.4510328 0.4509831 0.4509250 0.4508425 0.4507614 0.4505353 0.4503388
#> [372] 0.4502153 0.4501269 0.4500558 0.4499604 0.4498572 0.4497699 0.4498135
#> [379] 0.4496975 0.4496102 0.4495334 0.4494185 0.4493346 0.4492803 0.4492294
#> [386] 0.4492370 0.4492700 0.4492062 0.4491109 0.4490984 0.4490486 0.4492063
#> [393] 0.4491227 0.4490249 0.4489951 0.4489834 0.4489101 0.4488483 0.4488503
#> [400] 0.4487992 0.4487415 0.4486928 0.4486214 0.4486621 0.4487151 0.4486588
#> [407] 0.4485187 0.4485550 0.4485310 0.4484420 0.4483157 0.4480034 0.4479201
#> [414] 0.4477934 0.4476425 0.4474272 0.4473196 0.4472071 0.4469144 0.4467491
#> [421] 0.4465845 0.4464063 0.4463919 0.4463982 0.4464425 0.4465297 0.4465691
#> [428] 0.4466091 0.4467814 0.4468953 0.4470317 0.4470599 0.4471455 0.4472181
#> [435] 0.4468139 0.4469044 0.4471739 0.4471897 0.4471537 0.4471444 0.4471776
#> [442] 0.4472473 0.4473953 0.4474919 0.4475007 0.4473805 0.4472391 0.4471687
#> [449] 0.4473332 0.4472046 0.4470793 0.4469121 0.4467490 0.4466594 0.4464673
#> [456] 0.4462989 0.4461787 0.4460817 0.4460128 0.4458387 0.4457607 0.4456736
#> [463] 0.4456429 0.4455179 0.4452319 0.4450605 0.4448470 0.4445830 0.4444713
#> [470] 0.4442921 0.4441611 0.4439564 0.4439294 0.4439662 0.4438530 0.4437882
#> [477] 0.4437513 0.4437958 0.4437869 0.4437985 0.4437862 0.4437544 0.4437746
#> [484] 0.4438413 0.4438983 0.4438681 0.4438609 0.4437118 0.4436361 0.4436613
#> [491] 0.4437859 0.4438052 0.4438268 0.4438609 0.4436689 0.4436957 0.4436722
#> [498] 0.4435762 0.4435110 0.4434015 0.4432843 0.4431515 0.4430300 0.4429591
#> [505] 0.4428406 0.4426990 0.4427360 0.4427243 0.4426536 0.4424327 0.4422827
#> [512] 0.4420488 0.4419363 0.4418226 0.4417808 0.4418107 0.4416121 0.4415602
#> [519] 0.4415331 0.4418478 0.4419651 0.4418426 0.4417057 0.4414906 0.4412230
#> [526] 0.4410541 0.4411307 0.4411762 0.4409854 0.4409501 0.4410879 0.4410230
#> [533] 0.4410255 0.4408381 0.4406200 0.4403675 0.4401110 0.4397999 0.4395176
#> [540] 0.4391752 0.4390071 0.4386872 0.4387721 0.4386395 0.4386686 0.4386664
#> [547] 0.4385059 0.4381964 0.4379878 0.4382247 0.4384181 0.4384795 0.4385370
#> [554] 0.4381975 0.4380270 0.4380900 0.4379986 0.4378978 0.4378181 0.4376995
#> [561] 0.4377412 0.4377271 0.4376092 0.4375545 0.4374324 0.4372647 0.4371566
#> [568] 0.4370641 0.4367853 0.4366333 0.4364791 0.4364251 0.4362515 0.4361508
#> [575] 0.4359957 0.4360682 0.4358098 0.4356379 0.4354462 0.4353786 0.4354628
#> [582] 0.4354107 0.4353641 0.4352660 0.4350882 0.4347951 0.4346437 0.4347310
#> [589] 0.4348693 0.4349523 0.4350107 0.4350494 0.4349403 0.4349410 0.4351273
#> [596] 0.4352899 0.4355638 0.4358765 0.4360583 0.4361675 0.4362092 0.4362600
#> [603] 0.4365150 0.4366560 0.4364089 0.4360939 0.4358902 0.4361204 0.4362817
#> [610] 0.4364115 0.4364107 0.4363569 0.4364018 0.4365307 0.4365884 0.4364962
#> [617] 0.4364347 0.4362268 0.4361763 0.4362248 0.4363466 0.4365782 0.4367206
#> [624] 0.4367981 0.4368295 0.4368276 0.4369844 0.4370297 0.4371701 0.4372559
#> [631] 0.4371914 0.4373065 0.4374611 0.4375141 0.4374575 0.4374636 0.4376396
#> [638] 0.4377511 0.4378968 0.4380197 0.4381773 0.4381593 0.4383652 0.4382836
#> [645] 0.4382622 0.4382608 0.4383885 0.4385507 0.4386796 0.4387432 0.4388506
#> [652] 0.4389274 0.4390291 0.4390586 0.4390970 0.4389909 0.4390362 0.4391698
#> [659] 0.4392812 0.4393203 0.4393635 0.4393874 0.4395294 0.4397179 0.4400394
#> [666] 0.4402462 0.4404005 0.4406883 0.4407093 0.4406872 0.4406730 0.4406618
#> [673] 0.4405132 0.4404470 0.4404111 0.4403284 0.4403875 0.4404273 0.4405423
#> [680] 0.4405810 0.4405774 0.4405621 0.4405938 0.4405950 0.4406542 0.4406791
#> [687] 0.4408293 0.4409640 0.4409982 0.4409851 0.4408853 0.4408130 0.4407505
#> [694] 0.4408289 0.4406679 0.4406201 0.4405421 0.4404672 0.4404269 0.4403372
#> [701] 0.4403304 0.4402845 0.4402863 0.4402345 0.4401813 0.4401144 0.4401348
#> [708] 0.4402135 0.4403791 0.4404677 0.4403740 0.4402528 0.4402048 0.4401256
#> [715] 0.4399887 0.4397958 0.4396193 0.4395829 0.4394744 0.4394588 0.4395003
#> [722] 0.4395830 0.4398448 0.4398463 0.4398499 0.4398362 0.4398433 0.4397940
#> [729] 0.4397570 0.4396383 0.4395536 0.4395640 0.4396526 0.4397033 0.4398934
#> [736] 0.4400032 0.4400960 0.4401194 0.4402391 0.4403582 0.4404496 0.4404080
#> [743] 0.4404117 0.4404353 0.4404292 0.4404122 0.4404613 0.4403882 0.4403981
#> [750] 0.4404538 0.4404306 0.4404110 0.4404029 0.4406092 0.4407475 0.4407774
#> [757] 0.4405368 0.4405275 0.4405253 0.4404157 0.4402923 0.4402186 0.4401404
#> [764] 0.4401767 0.4402747 0.4402880 0.4403324 0.4403658 0.4403479 0.4402203
#> [771] 0.4402328 0.4402377 0.4400813 0.4400467 0.4400603 0.4401317 0.4402045
#> [778] 0.4402576 0.4406258 0.4407267 0.4408673 0.4409125 0.4409799 0.4410302
#> [785] 0.4410970 0.4411567 0.4412297 0.4412116 0.4411949 0.4411899 0.4411485
#> [792] 0.4410626 0.4409397 0.4407044 0.4405078 0.4402800 0.4401113 0.4400314
#> [799] 0.4399683 0.4398654 0.4397029 0.4393482 0.4393140 0.4392467 0.4391792
#> [806] 0.4391340 0.4390336 0.4389446 0.4390203 0.4391480 0.4392238 0.4393056
#> [813] 0.4393873 0.4392223 0.4392154 0.4391204 0.4391107 0.4390813 0.4390637
#> [820] 0.4390962 0.4390321 0.4389709 0.4390573 0.4392577 0.4392382 0.4392284
#> [827] 0.4392547 0.4393047 0.4394186 0.4395776 0.4396425 0.4396847 0.4397871
#> [834] 0.4398137 0.4398914 0.4399366 0.4400844 0.4400611 0.4401724 0.4403915
#> [841] 0.4405590 0.4406452 0.4405784 0.4404512 0.4402891 0.4401973 0.4401323
#> [848] 0.4400671 0.4399227 0.4398165 0.4398191 0.4398582 0.4400001 0.4401409
#> [855] 0.4402576 0.4403402 0.4405058 0.4406228 0.4406198 0.4405939 0.4406332
#> [862] 0.4408378 0.4409973 0.4411984 0.4414395 0.4416801 0.4419376 0.4420881
#> [869] 0.4421573 0.4422605 0.4423539 0.4423659 0.4424109 0.4424111 0.4425420
#> [876] 0.4426654 0.4428500 0.4428834 0.4429516 0.4430859 0.4431593 0.4433922
#> [883] 0.4436170 0.4436657 0.4433996 0.4432624 0.4433515 0.4435253 0.4437754
#> [890] 0.4440043 0.4442530 0.4442448 0.4442593 0.4444089 0.4441889 0.4438939
#> [897] 0.4437029 0.4433972 0.4433182 0.4432931 0.4432594 0.4431519 0.4432211
#> [904] 0.4432515 0.4433982 0.4433409 0.4432410 0.4431745 0.4433539 0.4434936
#> [911] 0.4436753 0.4438101 0.4439294 0.4439328 0.4436946 0.4435601 0.4434755
#> [918] 0.4435344 0.4437040 0.4439137 0.4441111 0.4442281 0.4443694 0.4444306
#> [925] 0.4446506 0.4448350 0.4449267 0.4451235 0.4452476 0.4452773 0.4453347
#> [932] 0.4454247 0.4454612 0.4454781 0.4453509 0.4452851 0.4452048 0.4451348
#> [939] 0.4451028 0.4451825 0.4450818 0.4453634 0.4456826 0.4459834 0.4460449
#> [946] 0.4458603 0.4456995 0.4454606 0.4454010 0.4454820 0.4455747 0.4456410
#> [953] 0.4457122 0.4457673 0.4458321 0.4457758 0.4457876 0.4457692 0.4456852
#> [960] 0.4453965 0.4451721 0.4451434 0.4450805 0.4452178 0.4452412 0.4451376
#> [967] 0.4449976 0.4447753 0.4447310 0.4449717 0.4452095 0.4453930 0.4451934
#> [974] 0.4450686 0.4450684 0.4452174 0.4453483 0.4455069 0.4455380 0.4455490
#> [981] 0.4457113 0.4458272 0.4459792 0.4461899 0.4463815 0.4464654 0.4463362
#> [988] 0.4462605 0.4461751 0.4461244 0.4462665 0.4463023 0.4463228 0.4462040
#> [995] 0.4461375 0.4460616 0.4459427 0.4458725 0.4459116 0.4459579 0.4461413
#> [1002] 0.4459896 0.4459393 0.4458384 0.4458577 0.4457909 0.4459029 0.4457594
#> [1009] 0.4457961 0.4456948 0.4457657 0.4456584 0.4456741 0.4456201 0.4456749
#> [1016] 0.4456893 0.4457468 0.4456550 0.4456455 0.4456968 0.4457027 0.4457222
#> [1023] 0.4456852 0.4457761 0.4458891 0.4458929 0.4459735 0.4457650 0.4454826
#> [1030] 0.4451510 0.4449423 0.4448340 0.4446018 0.4444905 0.4445372 0.4446586
#> [1037] 0.4447624 0.4448591 0.4449296 0.4449316 0.4449461 0.4450216 0.4452653
#> [1044] 0.4453288 0.4453678 0.4454028 0.4455028 0.4455415 0.4455819 0.4455632
#> [1051] 0.4455925 0.4455637 0.4457226 0.4459298 0.4461504 0.4462699 0.4464484
#> [1058] 0.4465061 0.4465372 0.4469264 0.4471026 0.4473104 0.4474515 0.4476092
#> [1065] 0.4477422 0.4479240 0.4481467 0.4482372 0.4484083 0.4486240 0.4487780
#> [1072] 0.4491593 0.4491712 0.4491338 0.4491118 0.4491246 0.4491534 0.4492446
#> [1079] 0.4493604 0.4495086 0.4496163 0.4497099 0.4497754 0.4498578 0.4499685
#> [1086] 0.4500780 0.4501432 0.4499048 0.4499095 0.4500452 0.4501241 0.4502432
#> [1093] 0.4503274 0.4504608 0.4506767 0.4509019 0.4509237 0.4508299 0.4508004
#> [1100] 0.4507910 0.4508450 0.4508757 0.4509393 0.4509229 0.4508716 0.4508640
#> [1107] 0.4509610 0.4510009 0.4510097 0.4510631 0.4511147 0.4511732 0.4513831
#> [1114] 0.4515198 0.4515848 0.4516257 0.4516110 0.4516324 0.4514350 0.4512949
#> [1121] 0.4513785 0.4515029 0.4515892 0.4516715 0.4517876 0.4519786 0.4521454
#> [1128] 0.4522353 0.4522140 0.4521783 0.4521734 0.4521942 0.4522301 0.4522657
#> [1135] 0.4523804 0.4524816 0.4526390 0.4527308 0.4527513 0.4528182 0.4529193
#> [1142] 0.4530253 0.4530674 0.4527878 0.4527591 0.4528576 0.4529830 0.4530589
#> [1149] 0.4531114 0.4532378 0.4533488 0.4534157 0.4535359 0.4536718 0.4537699
#> [1156] 0.4538620 0.4539863 0.4541388 0.4543628 0.4545408 0.4547277 0.4549796
#> [1163] 0.4550750 0.4551237 0.4552586 0.4553700 0.4555439 0.4555889 0.4556979
#> [1170] 0.4558232 0.4559463 0.4561049 0.4561328 0.4560241 0.4561375 0.4561089
#> [1177] 0.4560792 0.4560789 0.4561146 0.4562333 0.4564600 0.4565663 0.4566858
#> [1184] 0.4567831 0.4568139 0.4570022 0.4571770 0.4572492 0.4572432 0.4574481
#> [1191] 0.4576603 0.4577605 0.4577792 0.4579648 0.4580538 0.4581325 0.4581415
#> [1198] 0.4582007 0.4581542 0.4581350 0.4581481 0.4581374 0.4580627 0.4578558
#> [1205] 0.4577531 0.4576908 0.4577317 0.4578827 0.4581508 0.4583865 0.4585712
#> [1212] 0.4587635 0.4589054 0.4590920 0.4592311 0.4593050 0.4594139 0.4594439
#> [1219] 0.4593870 0.4593717 0.4593643 0.4592808 0.4592596 0.4592063 0.4591978
#> [1226] 0.4592177 0.4590913 0.4590392 0.4590097 0.4590030 0.4590434 0.4590792
#> [1233] 0.4591592 0.4592954 0.4594523 0.4595437 0.4595964 0.4596547 0.4597396
#> [1240] 0.4598086 0.4599082 0.4599627 0.4599922 0.4600224 0.4598723 0.4596685
#> [1247] 0.4594712 0.4595390 0.4597142 0.4597916 0.4597822 0.4597282 0.4596418
#> [1254] 0.4596089 0.4596304 0.4595130 0.4594303 0.4594538 0.4594821 0.4596854
#> [1261] 0.4598160 0.4599830 0.4601850 0.4602457 0.4603562 0.4604488 0.4606390
#> [1268] 0.4608190 0.4610215 0.4609933 0.4611322 0.4614413 0.4617996 0.4619805
#> [1275] 0.4620275 0.4620344 0.4622291 0.4624266 0.4626588 0.4627266 NA
#> [1282] NA NA NA NA NA NA NA
#> [1289] NA NA NA NA NA NA NA
#> [1296] NA NA NA NA NA NA NA
#> [1303] NA NA NA NA NA NA NA
#> [1310] NA NA NA NA NA NA NA
#> [1317] NA NA NA NA NA NA NA
#> [1324] NA NA NA NA NA NA NA
#> [1331] NA NA NA NA NA NA NA
#> [1338] NA NA NA NA NA NA NA
#> [1345] NA NA NA NA NA NA NA
#> [1352] NA NA NA NA NA NA NA
#> [1359] NA NA NA NA NA NA NA
#> [1366] NA NA NA NA NA NA NA
#> [1373] NA NA NA NA NA NA NA
#> [1380] NA NA NA NA NA NA NA
#> [1387] NA NA NA NA NA NA NA
#> [1394] NA NA NA NA NA NA NA
#> [1401] NA NA NA NA NA NA NA
#> [1408] NA NA NA NA NA NA NA
#> [1415] NA NA NA NA NA NA NA
#> [1422] NA NA NA NA NA NA NA
#> [1429] NA NA NA NA NA NA NA
#> [1436] NA NA NA NA NA NA NA
#> [1443] NA NA NA NA NA NA NA
#> [1450] NA NA NA NA NA NA NA
#> [1457] NA NA NA NA NA NA
#>
#> $random
#> Time Series:
#> Start = c(2013, 1)
#> End = c(2017, 2)
#> Frequency = 365
#> [1] NA NA NA NA
#> [5] NA NA NA NA
#> [9] NA NA NA NA
#> [13] NA NA NA NA
#> [17] NA NA NA NA
#> [21] NA NA NA NA
#> [25] NA NA NA NA
#> [29] NA NA NA NA
#> [33] NA NA NA NA
#> [37] NA NA NA NA
#> [41] NA NA NA NA
#> [45] NA NA NA NA
#> [49] NA NA NA NA
#> [53] NA NA NA NA
#> [57] NA NA NA NA
#> [61] NA NA NA NA
#> [65] NA NA NA NA
#> [69] NA NA NA NA
#> [73] NA NA NA NA
#> [77] NA NA NA NA
#> [81] NA NA NA NA
#> [85] NA NA NA NA
#> [89] NA NA NA NA
#> [93] NA NA NA NA
#> [97] NA NA NA NA
#> [101] NA NA NA NA
#> [105] NA NA NA NA
#> [109] NA NA NA NA
#> [113] NA NA NA NA
#> [117] NA NA NA NA
#> [121] NA NA NA NA
#> [125] NA NA NA NA
#> [129] NA NA NA NA
#> [133] NA NA NA NA
#> [137] NA NA NA NA
#> [141] NA NA NA NA
#> [145] NA NA NA NA
#> [149] NA NA NA NA
#> [153] NA NA NA NA
#> [157] NA NA NA NA
#> [161] NA NA NA NA
#> [165] NA NA NA NA
#> [169] NA NA NA NA
#> [173] NA NA NA NA
#> [177] NA NA NA NA
#> [181] NA NA 0.00993938634 0.01327097751
#> [185] 0.01621873984 0.00986351683 0.02798231992 0.03045065764
#> [189] 0.01436380748 0.01407849889 -0.00183381782 0.00403495571
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#> [201] 0.00747385384 0.00097916326 -0.00375050167 -0.01068283336
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#> [265] 0.00208353156 -0.02516065489 0.02693344023 0.03684677860
#> [269] 0.03660426590 0.03792317793 0.01866679397 0.03752259057
#> [273] 0.04354895564 0.03143663351 0.00790095753 -0.01275755931
#> [277] 0.03954270544 0.01490461971 0.01257991353 -0.00417716166
#> [281] 0.01093457373 0.04468525114 0.02830485368 0.04134048513
#> [285] 0.04629016620 0.01846631069 0.02869196291 0.02089643581
#> [289] 0.02631930999 0.00380823731 -0.02827297512 -0.00500287709
#> [293] -0.01046639347 -0.00725150778 -0.00837136926 0.01204587316
#> [297] 0.01129028977 0.00857758545 0.01003729000 -0.00252905535
#> [301] -0.00418050795 -0.00649240363 0.00153750802 -0.00864460721
#> [305] 0.01334421313 -0.00440188620 -0.02603170765 -0.04620888100
#> [309] -0.03073628966 -0.02540865368 -0.01393829471 0.04393133156
#> [313] -0.01710600053 -0.02489319727 -0.00182515993 -0.01449353262
#> [317] -0.00857548559 -0.00666369512 -0.00490195922 0.00530879438
#> [321] 0.00294976921 0.01051798266 0.01746905794 -0.01602230987
#> [325] -0.00805556968 0.01172565900 0.03245445588 0.01610136406
#> [329] 0.02909258311 0.01809626375 0.01415006434 -0.01143648927
#> [333] -0.02165336019 0.01464914414 -0.01998398166 -0.00438941582
#> [337] -0.04902202839 -0.01657517810 0.01433904502 0.00768955170
#> [341] 0.01745995557 0.03314508979 0.00678987427 -0.02155951436
#> [345] -0.01662812938 0.02436933925 0.03151359756 -0.00149905349
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#> [1261] 0.00769995433 0.00836826082 -0.00853652522 -0.01010361423
#> [1265] -0.00005860709 -0.00486879723 0.01309171573 0.01950893498
#> [1269] 0.01354100335 -0.03857885885 -0.01126395080 0.02050137367
#> [1273] 0.00437223493 0.01653962530 0.00123009021 0.01476303261
#> [1277] 0.02476546783 0.02786502501 -0.01136868084 -0.03102911323
#> [1281] NA NA NA NA
#> [1285] NA NA NA NA
#> [1289] NA NA NA NA
#> [1293] NA NA NA NA
#> [1297] NA NA NA NA
#> [1301] NA NA NA NA
#> [1305] NA NA NA NA
#> [1309] NA NA NA NA
#> [1313] NA NA NA NA
#> [1317] NA NA NA NA
#> [1321] NA NA NA NA
#> [1325] NA NA NA NA
#> [1329] NA NA NA NA
#> [1333] NA NA NA NA
#> [1337] NA NA NA NA
#> [1341] NA NA NA NA
#> [1345] NA NA NA NA
#> [1349] NA NA NA NA
#> [1353] NA NA NA NA
#> [1357] NA NA NA NA
#> [1361] NA NA NA NA
#> [1365] NA NA NA NA
#> [1369] NA NA NA NA
#> [1373] NA NA NA NA
#> [1377] NA NA NA NA
#> [1381] NA NA NA NA
#> [1385] NA NA NA NA
#> [1389] NA NA NA NA
#> [1393] NA NA NA NA
#> [1397] NA NA NA NA
#> [1401] NA NA NA NA
#> [1405] NA NA NA NA
#> [1409] NA NA NA NA
#> [1413] NA NA NA NA
#> [1417] NA NA NA NA
#> [1421] NA NA NA NA
#> [1425] NA NA NA NA
#> [1429] NA NA NA NA
#> [1433] NA NA NA NA
#> [1437] NA NA NA NA
#> [1441] NA NA NA NA
#> [1445] NA NA NA NA
#> [1449] NA NA NA NA
#> [1453] NA NA NA NA
#> [1457] NA NA NA NA
#> [1461] NA NA
#>
#> $figure
#> [1] -0.0944314775 -0.0812771463 -0.0861971883 -0.0814632051 -0.0713634752
#> [6] -0.0481042666 -0.0737875891 -0.0946933236 -0.1120105645 -0.1145723044
#> [11] -0.0998482712 -0.1033480338 -0.0942969906 -0.0855903675 -0.0861557196
#> [16] -0.0809241484 -0.0964145714 -0.0999115056 -0.0830956339 -0.0833557623
#> [21] -0.0845898215 -0.0554807363 -0.0647338574 -0.0611937293 -0.0905695655
#> [26] -0.0681019171 -0.0812175317 -0.0684786232 -0.0954895052 -0.0871853697
#> [31] -0.0839158788 -0.1050346732 -0.0973370358 -0.0818814924 -0.0739837221
#> [36] -0.1179264331 -0.1066652422 -0.0702403216 -0.0823957468 -0.1036753070
#> [41] -0.1282770797 -0.1213949729 -0.1090783577 -0.1105463006 -0.0681179474
#> [46] -0.0844388930 -0.0637625674 -0.0677476756 -0.0547538203 -0.0310273357
#> [51] -0.0150086213 -0.0236382193 -0.0272673153 -0.0426506952 -0.0534028697
#> [56] -0.0603249135 -0.0996204393 -0.0748792494 -0.0569274900 -0.0028018294
#> [61] -0.0321008027 -0.0594697397 -0.0597816945 -0.0635854837 -0.0560164043
#> [66] -0.0436453280 -0.0277542736 -0.0560175919 -0.0381968683 -0.0170051826
#> [71] -0.0273741861 -0.0300556547 -0.0351847192 -0.0194346853 -0.0142062406
#> [76] -0.0129509083 -0.0178632841 -0.0250190760 -0.0238887328 -0.0167455699
#> [81] -0.0091924370 -0.0068370949 -0.0079466402 -0.0183881198 -0.0015020913
#> [86] -0.0076319160 -0.0089002561 0.0175834860 -0.0077145087 -0.0234794700
#> [91] -0.0230261278 -0.0113039413 -0.0032389032 -0.0072567595 0.0087763124
#> [96] -0.0134357993 -0.0128906768 -0.0264828167 -0.0396895042 -0.0305577683
#> [101] -0.0380126340 -0.0144237554 0.0149440058 -0.0045300761 -0.0078458217
#> [106] -0.0105058831 -0.0009743947 0.0076569776 -0.0033151123 -0.0082877579
#> [111] -0.0177035522 -0.0113231921 -0.0309950941 -0.0185309601 -0.0137638529
#> [116] -0.0090110140 0.0008482470 -0.0001096345 0.0015349128 0.0068561352
#> [121] -0.0131007805 -0.0181394408 -0.0093614937 -0.0090095900 0.0114750670
#> [126] 0.0085928241 0.0016411999 0.0093617371 0.0091060466 0.0185481772
#> [131] 0.0187341435 0.0377162604 0.0318428090 0.0137611422 0.0085414167
#> [136] 0.0146974792 0.0209401735 0.0276724448 0.0303567541 0.0310154825
#> [141] 0.0399423362 0.0427139422 0.0634488309 0.0403561370 0.0489683167
#> [146] 0.0428177813 0.0266307691 0.0380602123 0.0413755398 0.0572416547
#> [151] 0.0428588782 0.0345583882 0.0305146688 0.0337437031 0.0353591402
#> [156] 0.0368483948 0.0599564159 0.0585781813 0.0783070599 0.0676334182
#> [161] 0.0646708618 0.0733680655 0.0760730682 0.0579871135 0.0789540668
#> [166] 0.0717323338 0.0685872926 0.0827966029 0.0772601716 0.0979796517
#> [171] 0.1021858619 0.0901408127 0.0725457434 0.0695186817 0.0912758932
#> [176] 0.0869199537 0.0839978235 0.0887377435 0.0842855204 0.0922384415
#> [181] 0.0929608119 0.0805850243 0.0896069315 0.0911055616 0.1029582759
#> [186] 0.1136562030 0.1036608814 0.0973203883 0.0861232845 0.0958826563
#> [191] 0.1015657300 0.0979860104 0.1141778654 0.1110149235 0.1065588610
#> [196] 0.1292987236 0.1250698858 0.1207815670 0.1042150295 0.0906313082
#> [201] 0.1021587394 0.1032181017 0.1070600483 0.1074486058 0.1055934602
#> [206] 0.1274176757 0.1111710001 0.1021226112 0.1145751818 0.0966384926
#> [211] 0.0970973632 0.0987240293 0.0778667019 0.0899337346 0.0879716681
#> [216] 0.0908451089 0.1071380611 0.1010369655 0.0995283134 0.1037000199
#> [221] 0.1089372403 0.1110365948 0.1013164616 0.0973517625 0.1161474674
#> [226] 0.1052148457 0.1005149400 0.0913293887 0.0995167603 0.0773267073
#> [231] 0.1264324731 0.0905100528 0.0896116593 0.0742327469 0.0775615414
#> [236] 0.1090570401 0.0895996141 0.0795196435 0.0861974363 0.0767816881
#> [241] 0.0744683326 0.0827674040 0.0897419745 0.0945397028 0.0814792247
#> [246] 0.0897128875 0.0749781637 0.0644851559 0.0489485555 0.0479869169
#> [251] 0.0601540249 0.1155248024 0.0604611420 0.0784415504 0.0510487250
#> [256] 0.0595541753 0.0557326745 0.0511247825 0.0569030004 0.0429910120
#> [261] 0.0662691808 0.0573957235 0.0663252757 0.0644883730 0.0777833184
#> [266] 0.0951142045 0.0470151523 0.0464335247 0.0431778402 0.0411755417
#> [271] 0.0297600317 0.0520780115 0.0404012274 0.0251657772 0.0395695188
#> [276] 0.0444497023 0.0308373767 0.0247830114 0.0323959459 0.0475566670
#> [281] 0.0283859593 0.0275177015 0.0219211287 0.0237154672 0.0150803750
#> [286] 0.0505696469 0.0375630372 0.0192857463 -0.0016597557 0.0368779251
#> [291] 0.0450077902 -0.0024127541 -0.0085886721 -0.0204368300 -0.0248290452
#> [296] -0.0257399832 -0.0198959680 -0.0171303860 -0.0130715863 -0.0326144648
#> [301] -0.0350728425 -0.0348242065 -0.0357287763 -0.0422555088 -0.0411343280
#> [306] -0.0429117619 -0.0538309415 -0.0547192750 -0.0405700087 -0.0360180866
#> [311] -0.0243315623 -0.0039761463 -0.0418804648 -0.0673039641 -0.0850798543
#> [316] -0.0901110324 -0.0944673215 -0.1013853120 -0.1103099982 -0.1204225974
#> [321] -0.1272103913 -0.1177204683 -0.1043538566 -0.1042556748 -0.0933422198
#> [326] -0.0907087942 -0.0964308214 -0.1135750912 -0.1108612380 -0.0934564743
#> [331] -0.0891247785 -0.0624967308 -0.0671764871 -0.0586137893 -0.0867948680
#> [336] -0.0814653235 -0.0348632070 -0.0638262031 -0.0919584582 -0.1009938550
#> [341] -0.0961869898 -0.1001725044 -0.0825277308 -0.0518929259 -0.0585550011
#> [346] -0.0693903487 -0.0904070134 -0.0817820570 -0.0928763549 -0.0887589889
#> [351] -0.1029222851 -0.0963760365 -0.0984141345 -0.1042827359 -0.1133368485
#> [356] -0.1005671851 -0.0970093759 -0.1159871652 -0.0949105963 -0.1382943398
#> [361] -0.1372814536 -0.1294949375 -0.1377172619 -0.1195679937 -0.1051445925
#>
#> $type
#> [1] "additive"
#>
#> attr(,"class")
#> [1] "decomposed.ts"
4.2.1 Trend Analysis
Komponen trend time series menunjukkan pergerakan jangka panjang, ini digunakan untuk melihat pergerakan data secara jangka panjang apakah meningkat, menurun, atau tetap.
Untuk mendapatkan pola trend, decompose() menggunakan
Moving Average (MA), metode yang menggunakan rata-rata
pada suatu periode waktu tertentu secara beruntun untuk
merepresentasikan pola general.
climate_trend <- climate_daily %>%
mutate(date = ymd(date),
trend = (climate_ts %>% decompose())$trend) # ekstraksi trend
ggplot(climate_daily, aes(date, weather_index)) +
geom_line() +
geom_line(color='red',data = climate_trend, aes(x=date, y=trend))
4.2.2 Seasonality Analysis
Analisis seasonality bertujuan untuk mengetahui pola berulang yang terjadi pada data, menunjukkan pada waktu kapan saja datanya tinggi/rendah.
Untuk mendapatkan pola seasonality, decompose()
menggunakan Rata-Rata seluruh musim untuk masing-masing
interval waktu (Seasonality untuk januari adalah rata-rata indeks cuaca
januari, dst.)
climate_seasonality <- climate_daily %>%
mutate(date = ymd(date),
month = month(date, label = T)) %>% # ekstraksi bulan
group_by(month) %>%
summarise(mean_index = mean(weather_index)) # rata-rata sales untuk setiap bulan
ggplot(climate_seasonality, aes(month, mean_index, group = 1)) +
geom_line()
5 Cross Validation
Tahapan cross validation akan selalu dilakukan sebelum pembuatan model. Data akan dibagi menjadi data train dan data test. Khusus untuk data deret waktu/time series pembagian data tidak boleh diambil secara acak melainkan dibagi dengan cara dipisah secara berurutan.
- Train data menggunakan data awal
- Test data menggunakan data akhir
Data test akan diibaratkan sebagai data masa depan yang ingin kita lakukan forecasting, sehingga dapat dibandingkan untuk melakukan evaluasi.
Mari kita coba ambil data climate_ts 1 tahun terakhir (1
Januari 2016 - 1 Januari 2017) untuk dijadikan sebagai data test,
sedangkan data diawalnya akan dijadikan sebagai data train.
#> Time Series:
#> Start = c(2016, 1)
#> End = c(2016, 6)
#> Frequency = 365
#> [1] 0.3288500 0.3374335 0.3524173 0.3532300 0.3977354 0.3996329#> Time Series:
#> Start = c(2013, 1)
#> End = c(2013, 6)
#> Frequency = 365
#> [1] 0.3084558 0.3167998 0.3044633 0.2523879 0.2851714 0.2729247Di bawah ini adalah visualisasi terhadap data_train dan
data_test kita.
6 Modelling
Untuk pemodelannya, di sini saya akan menggunakan 2 metode yaitu Triple Exponential Smoothing (Holt-Winters Exponential) dan Seasonal ARIMA (SARIMA) karena data kita memiliki trend dan seasonality dan nantinya akan kita bandingkan.
6.1 Triple Exponential Smoothing (Holt-Winters Exponential)
6.1.1 Model Fitting
Triple Exponential Smoothing (Holt-Winters Exponential) merupakan metode forecasting yang tepat digunakan untuk data yang memiliki efek trend dan seasonal.
Mari kita bandingkan secara visualisasi.
climate_train %>%
autoplot() +
autolayer(climate_test, series = "Data Test") +
autolayer(climate_triple$fitted[,1], series = "Data Fitted")
6.1.2 Forecasting
#> Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
#> 2016.0000 0.3924882 0.349462370 0.4355140 0.3266858880 0.4582905
#> 2016.0027 0.3936101 0.347088684 0.4401315 0.3224617418 0.4647584
#> 2016.0055 0.3545075 0.304735403 0.4042796 0.2783876440 0.4306274
#> 2016.0082 0.3534166 0.300593432 0.4062397 0.2726305546 0.4342026
#> 2016.0110 0.3470831 0.291375808 0.4027904 0.2618861386 0.4322801
#> 2016.0137 0.3960092 0.337559885 0.4544586 0.3066186706 0.4853998
#> 2016.0164 0.3734057 0.312337361 0.4344741 0.2800097121 0.4668018
#> 2016.0192 0.3355635 0.271983920 0.3991432 0.2383269002 0.4328002
#> 2016.0219 0.3314227 0.265427303 0.3974181 0.2304914604 0.4323539
#> 2016.0247 0.3293169 0.260991111 0.3976427 0.2248216319 0.4338121
#> 2016.0274 0.3538009 0.283221647 0.4243802 0.2458592419 0.4617426
#> 2016.0301 0.3347283 0.261965272 0.4074913 0.2234468680 0.4460097
#> 2016.0329 0.3349068 0.260023713 0.4097899 0.2203830077 0.4494306
#> 2016.0356 0.3520106 0.275065817 0.4289553 0.2343337208 0.4696874
#> 2016.0384 0.3472308 0.268278178 0.4261834 0.2264831805 0.4679784
#> 2016.0411 0.3605569 0.279646250 0.4414676 0.2368147207 0.4842991
#> 2016.0438 0.3624905 0.279668022 0.4453129 0.2358244584 0.4891565
#> 2016.0466 0.3637727 0.279081615 0.4484638 0.2342488574 0.4932966
#> 2016.0493 0.3758472 0.289327779 0.4623665 0.2435271870 0.5081671
#> 2016.0521 0.3799413 0.291631478 0.4682511 0.2448830840 0.5149995
#> 2016.0548 0.3978050 0.307740364 0.4878697 0.2600630060 0.5355471
#> 2016.0575 0.4425719 0.350785872 0.5343578 0.3021973070 0.5829464
#> 2016.0603 0.4245556 0.331079960 0.5180312 0.2815969660 0.5675142
#> 2016.0630 0.4193868 0.324251612 0.5145220 0.2738900704 0.5648836
#> 2016.0658 0.3938921 0.297125764 0.4906585 0.2459007415 0.5418835
#> 2016.0685 0.4080856 0.309715149 0.5064561 0.2576409603 0.5585303
#> 2016.0712 0.3842180 0.284269191 0.4841669 0.2313594636 0.5370766
#> 2016.0740 0.3986887 0.297186054 0.5001914 0.2434537790 0.5539237
#> 2016.0767 0.3446439 0.241610772 0.4476769 0.1870683536 0.5022194
#> 2016.0795 0.3524819 0.247940822 0.4570230 0.1926001181 0.5123637
#> 2016.0822 0.3620783 0.256050675 0.4681059 0.1999230378 0.5242336
#> 2016.0849 0.3605377 0.253044098 0.4680314 0.1961404090 0.5249350
#> 2016.0877 0.3774244 0.268484534 0.4863643 0.2108152358 0.5440336
#> 2016.0904 0.3905326 0.280165339 0.5008998 0.2217404642 0.5593246
#> 2016.0932 0.4081404 0.296364117 0.5199167 0.2371933122 0.5790875
#> 2016.0959 0.3474786 0.234310742 0.4606465 0.1744032947 0.5205539
#> 2016.0986 0.3525418 0.237999290 0.4670843 0.1773641483 0.5277194
#> 2016.1014 0.4053914 0.289490521 0.5212922 0.2281363153 0.5826464
#> 2016.1041 0.3935976 0.276354154 0.5108411 0.2142892154 0.5729060
#> 2016.1068 0.3464303 0.227859444 0.4650012 0.1650918187 0.5277688
#> 2016.1096 0.3032328 0.183349218 0.4231164 0.1198866873 0.4865789
#> 2016.1123 0.3020430 0.180860927 0.4232251 0.1167110173 0.4873750
#> 2016.1151 0.3275746 0.205107784 0.4500414 0.1402777826 0.5148714
#> 2016.1178 0.3188963 0.195158112 0.4426345 0.1296550798 0.5081375
#> 2016.1205 0.3800462 0.255049559 0.5050428 0.1888803418 0.5712120
#> 2016.1233 0.3704385 0.244195969 0.4966810 0.1773672072 0.5635098
#> 2016.1260 0.3969962 0.269519973 0.5244725 0.2020381121 0.5919544
#> 2016.1288 0.3757303 0.247032104 0.5044285 0.1789034044 0.5725572
#> 2016.1315 0.3486879 0.218779277 0.4785965 0.1500098232 0.5473659
#> 2016.1342 0.3629200 0.231812120 0.4940278 0.1624078268 0.5634321
#> 2016.1370 0.3592048 0.226908632 0.4915010 0.1568752536 0.5615344
#> 2016.1397 0.3746568 0.241182820 0.5081308 0.1705259578 0.5787877
#> 2016.1425 0.3993130 0.264671485 0.5339544 0.1933965926 0.6052293
#> 2016.1452 0.3952655 0.259466590 0.5310644 0.1875789809 0.6029521
#> 2016.1479 0.3788594 0.241912821 0.5158060 0.1694176727 0.5883012
#> 2016.1507 0.3515396 0.213454915 0.4896244 0.1403572773 0.5627220
#> 2016.1534 0.3163237 0.177110175 0.4555373 0.1034149738 0.5292325
#> 2016.1562 0.3720647 0.231731361 0.5123980 0.1574434016 0.5866859
#> 2016.1589 0.4041181 0.262673909 0.5455623 0.1877978854 0.6204383
#> 2016.1616 0.4668764 0.324329984 0.6094228 0.2488704781 0.6848823
#> 2016.1644 0.4114071 0.267766899 0.5550472 0.1917283875 0.6310857
#> 2016.1671 0.3620621 0.217336447 0.5067878 0.1407233061 0.5834009
#> 2016.1699 0.3523266 0.206523494 0.4981297 0.1293400022 0.5753132
#> 2016.1726 0.3613714 0.214498783 0.5082440 0.1367491237 0.5859936
#> 2016.1753 0.3623985 0.214464097 0.5103329 0.1361523627 0.5886446
#> 2016.1781 0.3799105 0.230921910 0.5288991 0.1520521072 0.6077689
#> 2016.1808 0.3987441 0.248708717 0.5487795 0.1692847660 0.6282035
#> 2016.1836 0.3671011 0.216026118 0.5181761 0.1360518597 0.5981503
#> 2016.1863 0.3968704 0.244762998 0.5489778 0.1642421919 0.6294986
#> 2016.1890 0.4314355 0.278302630 0.5845685 0.1972389620 0.6656321
#> 2016.1918 0.4096542 0.255502599 0.5638058 0.1738996797 0.6454087
#> 2016.1945 0.3966105 0.241446982 0.5517741 0.1593083520 0.6339127
#> 2016.1973 0.3896665 0.233497503 0.5458355 0.1508266340 0.6285064
#> 2016.2000 0.4116939 0.254525927 0.5688619 0.1713262226 0.6520616
#> 2016.2027 0.4071254 0.248964768 0.5652861 0.1652395699 0.6490113
#> 2016.2055 0.4142124 0.255065263 0.5733596 0.1708178475 0.6576070
#> 2016.2082 0.4124620 0.252334441 0.5725896 0.1675680259 0.6573560
#> 2016.2110 0.3895085 0.228406506 0.5506105 0.1431242494 0.6358928
#> 2016.2137 0.3927802 0.230709562 0.5548508 0.1449145660 0.6406458
#> 2016.2164 0.4057603 0.242726859 0.5687937 0.1564221692 0.6550984
#> 2016.2192 0.4237120 0.259721396 0.5877026 0.1729100050 0.6745140
#> 2016.2219 0.4208502 0.255907957 0.5857925 0.1685928052 0.6731076
#> 2016.2247 0.4363306 0.270442216 0.6022190 0.1826261937 0.6900351
#> 2016.2274 0.4232149 0.256385695 0.5900441 0.1680716419 0.6783582
#> 2016.2301 0.4251574 0.257392666 0.5929222 0.1685833755 0.6817314
#> 2016.2329 0.4101819 0.241486792 0.5788770 0.1521850107 0.6681787
#> 2016.2356 0.4216301 0.252009772 0.5912504 0.1622182004 0.6810420
#> 2016.2384 0.4337266 0.263186038 0.6042671 0.1729073342 0.6945458
#> 2016.2411 0.4339354 0.262479587 0.6053912 0.1717163646 0.6961544
#> 2016.2438 0.4025807 0.230214486 0.5749469 0.1389693180 0.6661921
#> 2016.2466 0.4042225 0.230950651 0.5774944 0.1392260702 0.6692189
#> 2016.2493 0.4005551 0.226382351 0.5747279 0.1341808495 0.6669294
#> 2016.2521 0.4153921 0.240323047 0.5904612 0.1476470790 0.6831372
#> 2016.2548 0.4044150 0.228454232 0.5803758 0.1353062146 0.6735238
#> 2016.2575 0.4391183 0.262270316 0.6159664 0.1686526300 0.7095840
#> 2016.2603 0.4148976 0.237166812 0.5926284 0.1430818011 0.6867135
#> 2016.2630 0.4390416 0.260432349 0.6176509 0.1658823232 0.7122009
#> 2016.2658 0.4064002 0.226916773 0.5858835 0.1319040074 0.6808963
#> 2016.2685 0.4034967 0.223143447 0.5838500 0.1276701849 0.6793233
#> 2016.2712 0.3872506 0.206031582 0.5684696 0.1101000340 0.6644011
#> 2016.2740 0.3853304 0.203249832 0.5674111 0.1068621775 0.6637987
#> 2016.2767 0.4003988 0.217460685 0.5833370 0.1206190721 0.6801786
#> 2016.2795 0.4335115 0.249719765 0.6173032 0.1524263113 0.7145966
#> 2016.2822 0.4109540 0.226312700 0.5955953 0.1285694939 0.6933385
#> 2016.2849 0.4247869 0.239299862 0.6102739 0.1411089647 0.7084648
#> 2016.2877 0.4184953 0.232166383 0.6048242 0.1335298252 0.7034607
#> 2016.2904 0.4238045 0.236637495 0.6109715 0.1375572816 0.7100517
#> 2016.2932 0.4537836 0.265782279 0.6417849 0.1662603880 0.7413068
#> 2016.2959 0.4362549 0.247422900 0.6250869 0.1474612830 0.7250485
#> 2016.2986 0.4262639 0.236604845 0.6159229 0.1362054273 0.7163223
#> 2016.3014 0.3962056 0.205723173 0.5866881 0.1048878565 0.6875234
#> 2016.3041 0.4039997 0.212697360 0.5953020 0.1114280201 0.6965714
#> 2016.3068 0.3998548 0.207736092 0.5919735 0.1060345810 0.6936751
#> 2016.3096 0.4061859 0.213254274 0.5991176 0.1111224205 0.7012495
#> 2016.3123 0.4365958 0.242854562 0.6303369 0.1402941718 0.7328973
#> 2016.3151 0.4398761 0.245328754 0.6344234 0.1423416107 0.7374106
#> 2016.3178 0.4365264 0.241176190 0.6318765 0.1377640542 0.7352887
#> 2016.3205 0.4200013 0.223851612 0.6161510 0.1200162230 0.7199864
#> 2016.3233 0.4145138 0.217567745 0.6114598 0.1133108220 0.7157167
#> 2016.3260 0.4250851 0.227345974 0.6228242 0.1226692134 0.7275009
#> 2016.3288 0.4100729 0.211543829 0.6086019 0.1064489083 0.7136968
#> 2016.3315 0.4024417 0.203125904 0.6017575 0.0976144809 0.7072690
#> 2016.3342 0.4112994 0.211199898 0.6113989 0.1052736103 0.7173252
#> 2016.3370 0.4188968 0.218016606 0.6197769 0.1116770711 0.7261165
#> 2016.3397 0.4565642 0.254906394 0.6582220 0.1481552128 0.7649731
#> 2016.3425 0.4589236 0.256491206 0.6613560 0.1493299591 0.7685173
#> 2016.3452 0.4582901 0.255085979 0.6614942 0.1475162296 0.7690639
#> 2016.3479 0.4677442 0.263771382 0.6717171 0.1557946757 0.7796938
#> 2016.3507 0.4483333 0.243594547 0.6530720 0.1352124116 0.7614541
#> 2016.3534 0.4318031 0.226301329 0.6373048 0.1175152762 0.7460909
#> 2016.3562 0.4435049 0.237242991 0.6497669 0.1280545138 0.7589554
#> 2016.3589 0.4664806 0.259461254 0.6734999 0.1498718313 0.7830894
#> 2016.3616 0.4660610 0.258287008 0.6738350 0.1482981012 0.7838239
#> 2016.3644 0.4443013 0.235775395 0.6528272 0.1253884486 0.7632141
#> 2016.3671 0.4445701 0.235295018 0.6538453 0.1245114638 0.7646288
#> 2016.3699 0.4522178 0.242196118 0.6622394 0.1310173703 0.7734182
#> 2016.3726 0.4611914 0.250425844 0.6719569 0.1388533016 0.7835295
#> 2016.3753 0.4626299 0.251123104 0.6741368 0.1391581532 0.7861017
#> 2016.3781 0.4607275 0.248481963 0.6729730 0.1361259729 0.7853290
#> 2016.3808 0.4328538 0.219872149 0.6458354 0.1071264773 0.7585811
#> 2016.3836 0.4423545 0.228639314 0.6560698 0.1155053022 0.7692038
#> 2016.3863 0.4522043 0.237757945 0.6666506 0.1242369211 0.7801716
#> 2016.3890 0.4989271 0.283752152 0.7141020 0.1698454319 0.8280087
#> 2016.3918 0.4743393 0.258438223 0.6902403 0.1441471075 0.8045314
#> 2016.3945 0.5000028 0.283378071 0.7166276 0.1687038495 0.8313018
#> 2016.3973 0.4848551 0.267509051 0.7022012 0.1524529983 0.8172572
#> 2016.4000 0.4407197 0.222654761 0.6587847 0.1072181402 0.7742213
#> 2016.4027 0.4770695 0.258287954 0.6958510 0.1424720156 0.8116669
#> 2016.4055 0.4785671 0.259071422 0.6980629 0.1428774051 0.8142569
#> 2016.4082 0.5035375 0.283329850 0.7237451 0.1667589799 0.8403159
#> 2016.4110 0.4843120 0.263394789 0.7052292 0.1464482804 0.8221757
#> 2016.4137 0.4820557 0.260431134 0.7036802 0.1431101896 0.8210012
#> 2016.4164 0.4661087 0.243779060 0.6884383 0.1260848715 0.8061325
#> 2016.4192 0.4644022 0.241369731 0.6874347 0.1233034773 0.8055009
#> 2016.4219 0.4775137 0.253780646 0.7012468 0.1353434965 0.8196840
#> 2016.4247 0.4714493 0.247017754 0.6958809 0.1282108668 0.8146877
#> 2016.4274 0.4907561 0.265628253 0.7158839 0.1464527756 0.8350594
#> 2016.4301 0.4992296 0.273407604 0.7250516 0.1538646718 0.8445945
#> 2016.4329 0.4843118 0.257797769 0.7108257 0.1378885088 0.8307350
#> 2016.4356 0.5086341 0.281430239 0.7358380 0.1611557658 0.8561125
#> 2016.4384 0.4972560 0.269364342 0.7251477 0.1487257614 0.8457863
#> 2016.4411 0.5027349 0.274157468 0.7313124 0.1531558771 0.8523140
#> 2016.4438 0.5183551 0.289093955 0.7476162 0.1677304384 0.8689798
#> 2016.4466 0.5116848 0.281741988 0.7416276 0.1600176221 0.8633520
#> 2016.4493 0.5259195 0.295297101 0.7565420 0.1732129517 0.8786261
#> 2016.4521 0.5189372 0.287637154 0.7502373 0.1651942788 0.8726802
#> 2016.4548 0.5194405 0.287464767 0.7514163 0.1646642142 0.8742169
#> 2016.4575 0.5486272 0.315977743 0.7812767 0.1928205516 0.9044339
#> 2016.4603 0.5493152 0.315993949 0.7826364 0.1924811483 0.9061492
#> 2016.4630 0.5625399 0.328548866 0.7965310 0.2046814771 0.9203984
#> 2016.4658 0.5648509 0.330191905 0.7995099 0.2059709395 0.9237308
#> 2016.4685 0.5407371 0.305412130 0.7760622 0.1808385925 0.9006357
#> 2016.4712 0.5125292 0.276540079 0.7485184 0.1516149637 0.8734435
#> 2016.4740 0.5044614 0.267809976 0.7411129 0.1425342697 0.8663886
#> 2016.4767 0.5395827 0.302270860 0.7768946 0.1766455418 0.9025199
#> 2016.4795 0.5165126 0.278542168 0.7544831 0.1525682073 0.8804571
#> 2016.4822 0.5047733 0.266146066 0.7434006 0.1398244258 0.8697222
#> 2016.4849 0.5223820 0.283099719 0.7616642 0.1564313524 0.8883326
#> 2016.4877 0.5146741 0.274738620 0.7546095 0.1477244746 0.8816236
#> 2016.4904 0.5332253 0.292638423 0.7738121 0.1652794372 0.9011711
#> 2016.4932 0.5307034 0.289466842 0.7719399 0.1617639467 0.8996428
#> 2016.4959 0.5178537 0.275969239 0.7597381 0.1479233583 0.8877840
#> 2016.4986 0.5269744 0.284443765 0.7695050 0.1560558155 0.8978929
#> 2016.5014 0.5478862 0.304711136 0.7910613 0.1759820263 0.9197904
#> 2016.5041 0.5656912 0.321873325 0.8095090 0.1928039568 0.9385784
#> 2016.5068 0.5605666 0.316107692 0.8050255 0.1866989599 0.9344343
#> 2016.5096 0.5553574 0.310259059 0.8004557 0.1805118509 0.9302029
#> 2016.5123 0.5490193 0.303283234 0.7947553 0.1731984300 0.9248401
#> 2016.5151 0.5421744 0.295802265 0.7885465 0.1653807393 0.9189680
#> 2016.5178 0.5494166 0.302410017 0.7964232 0.1716526367 0.9271805
#> 2016.5205 0.5583235 0.310684081 0.8059629 0.1795917071 0.9370552
#> 2016.5233 0.5478466 0.299575988 0.7961172 0.1681494733 0.9275437
#> 2016.5260 0.5604691 0.311568853 0.8093693 0.1798090460 0.9411291
#> 2016.5288 0.5542350 0.304706752 0.8037632 0.1726144926 0.9358554
#> 2016.5315 0.5568054 0.306650696 0.8069600 0.1742268199 0.9393839
#> 2016.5342 0.5734618 0.322682312 0.8242414 0.1899276472 0.9569960
#> 2016.5370 0.5724189 0.321016045 0.8238218 0.1879314131 0.9569064
#> 2016.5397 0.5707123 0.318687700 0.8227370 0.1852739174 0.9561508
#> 2016.5425 0.5579244 0.305279532 0.8105693 0.1715374087 0.9443114
#> 2016.5452 0.5407860 0.287522339 0.7940496 0.1534526795 0.9281192
#> 2016.5479 0.5594228 0.305542005 0.8133037 0.1711456068 0.9477001
#> 2016.5507 0.5575328 0.303036218 0.8120293 0.1683138748 0.9467517
#> 2016.5534 0.5530966 0.297985834 0.8082074 0.1629383316 0.9432549
#> 2016.5562 0.5501140 0.294390453 0.8058376 0.1590185720 0.9412095
#> 2016.5589 0.5446340 0.288299083 0.8009688 0.1526035995 0.9366643
#> 2016.5616 0.5747111 0.317766399 0.8316558 0.1817480825 0.9676741
#> 2016.5644 0.5474135 0.289860357 0.8049666 0.1535199728 0.9413070
#> 2016.5671 0.5295708 0.271410718 0.7877309 0.1347490247 0.9243926
#> 2016.5699 0.5382956 0.279529994 0.7970613 0.1425477448 0.9340435
#> 2016.5726 0.5325288 0.273159069 0.7918986 0.1358570125 0.9292006
#> 2016.5753 0.5371722 0.277199742 0.7971447 0.1395786217 0.9347658
#> 2016.5781 0.5395583 0.278984477 0.8001321 0.1410450300 0.9380716
#> 2016.5808 0.5229669 0.261793103 0.7841406 0.1235360627 0.9223977
#> 2016.5836 0.5389552 0.277182908 0.8007276 0.1386090023 0.9393015
#> 2016.5863 0.5391334 0.276763841 0.8015029 0.1378737931 0.9403930
#> 2016.5890 0.5454670 0.282501648 0.8084324 0.1432961755 0.9476379
#> 2016.5918 0.5530922 0.289532301 0.8166521 0.1500121171 0.9561723
#> 2016.5945 0.5381273 0.273974267 0.8022804 0.1341400805 0.9421146
#> 2016.5973 0.5390709 0.274326041 0.8038158 0.1341785553 0.9439633
#> 2016.6000 0.5419155 0.276580081 0.8072509 0.1361199942 0.9477110
#> 2016.6027 0.5463693 0.280444664 0.8122939 0.1396726706 0.9530659
#> 2016.6055 0.5417341 0.275221566 0.8082466 0.1341383556 0.9493298
#> 2016.6082 0.5389650 0.271865889 0.8060642 0.1304721465 0.9474579
#> 2016.6110 0.5358156 0.268131121 0.8035000 0.1264275274 0.9452036
#> 2016.6137 0.5563467 0.288078181 0.8246152 0.1460654122 0.9666280
#> 2016.6164 0.5515330 0.282681753 0.8203843 0.1403604805 0.9627056
#> 2016.6192 0.5461413 0.276708545 0.8155741 0.1340794362 0.9582033
#> 2016.6219 0.5389949 0.268981827 0.8090080 0.1260455441 0.9519442
#> 2016.6247 0.5489134 0.278321334 0.8195055 0.1350785364 0.9627483
#> 2016.6274 0.5381664 0.266996528 0.8093363 0.1234478702 0.9528849
#> 2016.6301 0.5907192 0.318972786 0.8624656 0.1751189184 1.0063195
#> 2016.6329 0.5289335 0.256611703 0.8012552 0.1124532716 0.9454137
#> 2016.6356 0.5359540 0.263058116 0.8088499 0.1185957625 0.9533122
#> 2016.6384 0.5234453 0.249976540 0.7969141 0.1052109032 0.9416798
#> 2016.6411 0.5297460 0.255705516 0.8037865 0.1106372293 0.9488548
#> 2016.6438 0.5622253 0.287614278 0.8368364 0.1422439718 0.9822067
#> 2016.6466 0.5555172 0.280336776 0.8306976 0.1346650772 0.9763693
#> 2016.6493 0.5446489 0.268900335 0.8203975 0.1229278649 0.9663699
#> 2016.6521 0.5515361 0.275220519 0.8278516 0.1289478958 0.9741243
#> 2016.6548 0.5381588 0.261277356 0.8150402 0.1147051954 0.9616123
#> 2016.6575 0.5292634 0.251817272 0.8067095 0.1049461846 0.9535805
#> 2016.6603 0.5305013 0.252491665 0.8085109 0.1053222574 0.9556803
#> 2016.6630 0.5274392 0.248867168 0.8060112 0.1014000438 0.9534783
#> 2016.6658 0.5244430 0.245309675 0.8035763 0.0975454347 0.9513405
#> 2016.6685 0.5113595 0.231666040 0.7910529 0.0836052790 0.9391137
#> 2016.6712 0.5181938 0.237941366 0.7984463 0.0895846774 0.9468030
#> 2016.6740 0.5036104 0.222799992 0.7844207 0.0741479645 0.9330727
#> 2016.6767 0.4919047 0.210537578 0.7732719 0.0615907982 0.9222187
#> 2016.6795 0.4787520 0.196829180 0.7606749 0.0475882289 0.9099159
#> 2016.6822 0.4833053 0.200827841 0.7657828 0.0512932987 0.9153173
#> 2016.6849 0.4896475 0.206616490 0.7726785 0.0567889306 0.9225060
#> 2016.6877 0.5445464 0.260962994 0.8281299 0.1108429898 0.9782499
#> 2016.6904 0.5179796 0.233844830 0.8021144 0.0834329492 0.9525263
#> 2016.6932 0.5226119 0.237926815 0.8072970 0.0872236233 0.9580002
#> 2016.6959 0.4980324 0.212798066 0.7832668 0.0618041256 0.9342607
#> 2016.6986 0.4978230 0.212040459 0.7836055 0.0607563280 0.9348896
#> 2016.7014 0.4953018 0.208972167 0.7816315 0.0573984020 0.9332052
#> 2016.7041 0.4908107 0.203934922 0.7776864 0.0520720742 0.9295493
#> 2016.7068 0.4907169 0.203296121 0.7781377 0.0511447405 0.9302891
#> 2016.7096 0.4743316 0.186366815 0.7622965 0.0339274475 0.9147358
#> 2016.7123 0.4941829 0.205675079 0.7826907 0.0529482679 0.9354175
#> 2016.7151 0.4754851 0.186435288 0.7645349 0.0334215731 0.9175486
#> 2016.7178 0.4803740 0.190783207 0.7699647 0.0374831252 0.9232648
#> 2016.7205 0.4897852 0.199654516 0.7799159 0.0460686012 0.9335018
#> 2016.7233 0.5117522 0.221082558 0.8024219 0.0672113414 0.9562931
#> 2016.7260 0.5223845 0.231176928 0.8135921 0.0770209378 0.9677481
#> 2016.7288 0.4797704 0.188025800 0.7715149 0.0335855604 0.9259552
#> 2016.7315 0.4856329 0.193352386 0.7779134 0.0386284201 0.9326374
#> 2016.7342 0.4965698 0.203754275 0.7893853 0.0487471019 0.9443925
#> 2016.7370 0.4996339 0.206284396 0.7929835 0.0509945318 0.9482733
#> 2016.7397 0.4929655 0.199082951 0.7868481 0.0435109104 0.9424202
#> 2016.7425 0.5149530 0.220538340 0.8093677 0.0646846328 0.9652214
#> 2016.7452 0.5031236 0.208177831 0.7980694 0.0520429652 0.9542043
#> 2016.7479 0.4938307 0.198354750 0.7893066 0.0419392319 0.9457222
#> 2016.7507 0.5119680 0.215962795 0.8079731 0.0592671270 0.9646688
#> 2016.7534 0.5086619 0.212128479 0.8051953 0.0551531602 0.9621707
#> 2016.7562 0.4749344 0.177873606 0.7719951 0.0206191353 0.9292496
#> 2016.7589 0.4792661 0.181678902 0.7768532 0.0241457728 0.9343864
#> 2016.7616 0.4874970 0.189384370 0.7856096 0.0315730743 0.9434209
#> 2016.7644 0.5003761 0.201738971 0.7990133 0.0436499991 0.9571023
#> 2016.7671 0.4765613 0.177400449 0.7757221 0.0190342878 0.9340882
#> 2016.7699 0.4763277 0.176644236 0.7760113 0.0180013691 0.9346541
#> 2016.7726 0.4756589 0.175453562 0.7758642 0.0165344716 0.9347833
#> 2016.7753 0.4814619 0.180735666 0.7821881 0.0215408313 0.9413829
#> 2016.7781 0.4816479 0.180401690 0.7828941 0.0209315873 0.9423642
#> 2016.7808 0.5183137 0.216548428 0.8200790 0.0568035326 0.9798239
#> 2016.7836 0.5041375 0.201854011 0.8064210 0.0418347937 0.9664402
#> 2016.7863 0.4834817 0.180680905 0.7862825 0.0203878360 0.9465756
#> 2016.7890 0.4635929 0.160275639 0.7669101 -0.0002908146 0.9274766
#> 2016.7918 0.4936587 0.189825859 0.7974915 0.0289864854 0.9583309
#> 2016.7945 0.5008487 0.196501176 0.8051962 0.0353893444 0.9663080
#> 2016.7973 0.4410879 0.136226618 0.7459492 -0.0251572117 0.9073331
#> 2016.8000 0.4263199 0.120945675 0.7316942 -0.0407096948 0.8933496
#> 2016.8027 0.4083925 0.102506181 0.7142789 -0.0594202734 0.8762053
#> 2016.8055 0.3999852 0.093587646 0.7063828 -0.0686094407 0.8685799
#> 2016.8082 0.4013418 0.094433801 0.7082497 -0.0680334662 0.8707170
#> 2016.8110 0.4145657 0.107148188 0.7219832 -0.0555888119 0.8847202
#> 2016.8137 0.4247413 0.116815110 0.7326675 -0.0461911752 0.8956738
#> 2016.8164 0.4300857 0.121651603 0.7385197 -0.0416235243 0.9017948
#> 2016.8192 0.4102113 0.101270265 0.7191524 -0.0622732630 0.8826959
#> 2016.8219 0.4019689 0.092521631 0.7114162 -0.0712898565 0.8752277
#> 2016.8247 0.4023210 0.092368399 0.7122737 -0.0717106117 0.8763527
#> 2016.8274 0.4020748 0.091617617 0.7125320 -0.0727284808 0.8768781
#> 2016.8301 0.3939464 0.082985497 0.7049073 -0.0816272543 0.8695200
#> 2016.8329 0.3943861 0.082922256 0.7058498 -0.0819567178 0.8707288
#> 2016.8356 0.3934564 0.081490497 0.7054223 -0.0836542699 0.8705670
#> 2016.8384 0.3781869 0.065719719 0.6906541 -0.0996904142 0.8560642
#> 2016.8411 0.3681413 0.055173648 0.6811090 -0.1105014261 0.8467841
#> 2016.8438 0.3749025 0.061435146 0.6883699 -0.1045044466 0.8543094
#> 2016.8466 0.3731239 0.059157673 0.6870902 -0.1070460166 0.8532939
#> 2016.8493 0.3815891 0.067124800 0.6960535 -0.0993425680 0.8625209
#> 2016.8521 0.4097967 0.094835058 0.7247584 -0.0718955719 0.8914890
#> 2016.8548 0.3928692 0.077410966 0.7083273 -0.0895825097 0.8753208
#> 2016.8575 0.3670888 0.051134871 0.6830427 -0.1161210383 0.8502986
#> 2016.8603 0.3387440 0.022295137 0.6551929 -0.1452227943 0.8227109
#> 2016.8630 0.3363848 0.019441746 0.6533280 -0.1483377976 0.8211075
#> 2016.8658 0.3305123 0.013075784 0.6479488 -0.1549649652 0.8159896
#> 2016.8685 0.3228155 0.004886309 0.6407447 -0.1634152399 0.8090462
#> 2016.8712 0.3150220 -0.003399114 0.6334431 -0.1719610601 0.8020050
#> 2016.8740 0.3029630 -0.015949203 0.6218753 -0.1847711437 0.7906972
#> 2016.8767 0.3001476 -0.019255010 0.6195502 -0.1883365459 0.7886318
#> 2016.8795 0.3160993 -0.003792951 0.6359916 -0.1731336841 0.8053323
#> 2016.8822 0.3335197 0.013138513 0.6539008 -0.1564610218 0.8235003
#> 2016.8849 0.3348555 0.013986182 0.6557248 -0.1558717591 0.8255827
#> 2016.8877 0.3423068 0.020950126 0.6636635 -0.1491658293 0.8337795
#> 2016.8904 0.3415188 0.019675412 0.6633621 -0.1506981666 0.8337357
#> 2016.8932 0.3452739 0.022944593 0.6676032 -0.1476862204 0.8382340
#> 2016.8959 0.3334108 0.010596279 0.6562252 -0.1602913822 0.8271129
#> 2016.8986 0.3390677 0.015768734 0.6623666 -0.1553753894 0.8335107
#> 2016.9014 0.3546871 0.030904404 0.6784698 -0.1404957966 0.8498700
#> 2016.9041 0.3630629 0.038797182 0.6873286 -0.1328587149 0.8589845
#> 2016.9068 0.3898966 0.065148604 0.7146446 -0.1067626095 0.8865558
#> 2016.9096 0.3730343 0.047804727 0.6982639 -0.1243614233 0.8704301
#> 2016.9123 0.3792875 0.053577031 0.7049980 -0.1188436796 0.8774187
#> 2016.9151 0.3502361 0.024045441 0.6764267 -0.1486294549 0.8491016
#> 2016.9178 0.3428819 0.016211846 0.6695521 -0.1567168606 0.8424808
#> 2016.9205 0.3820829 0.054934069 0.7092318 -0.1182480775 0.8824139
#> 2016.9233 0.3539959 0.026369016 0.6816229 -0.1470661995 0.8550581
#> 2016.9260 0.3340296 0.005925289 0.6621339 -0.1677626265 0.8358218
#> 2016.9288 0.3308978 0.002316898 0.6594788 -0.1716233503 0.8334190
#> 2016.9315 0.3386882 0.009631243 0.6677451 -0.1645609735 0.8419373
#> 2016.9342 0.3375023 0.007970032 0.6670345 -0.1664737880 0.8414783
#> 2016.9370 0.3631027 0.033095844 0.6931095 -0.1415992177 0.8678046
#> 2016.9397 0.3832563 0.052775563 0.7137371 -0.1221703790 0.8886830
#> 2016.9425 0.3740277 0.043073746 0.7049817 -0.1321227168 0.8801782
#> 2016.9452 0.3650712 0.033644662 0.6964978 -0.1418019649 0.8719444
#> 2016.9479 0.3390329 0.007134403 0.6709313 -0.1685620315 0.8466278
#> 2016.9507 0.3602230 0.027853261 0.6925926 -0.1480926253 0.8685385
#> 2016.9534 0.3544549 0.021614624 0.6872951 -0.1545803618 0.8634901
#> 2016.9562 0.3619491 0.028638936 0.6952592 -0.1478047982 0.8717030
#> 2016.9589 0.3620046 0.028225210 0.6957840 -0.1484669218 0.8724761
#> 2016.9616 0.3740358 0.039787848 0.7082838 -0.1371523323 0.8852239
#> 2016.9644 0.3751359 0.040420061 0.7098518 -0.1367678210 0.8870397
#> 2016.9671 0.3735597 0.038376520 0.7087428 -0.1390587183 0.8861780
#> 2016.9699 0.3788328 0.043183041 0.7144826 -0.1344992086 0.8921648
#> 2016.9726 0.3899491 0.053833383 0.7260648 -0.1240955353 0.9039938
#> 2016.9753 0.3979151 0.061334082 0.7344962 -0.1168411631 0.9126714
#> 2016.9781 0.3704696 0.033423866 0.7075153 -0.1449973669 0.8859366
#> 2016.9808 0.3913131 0.053803293 0.7288228 -0.1248635879 0.9074897
#> 2016.9836 0.3569635 0.018990324 0.6949367 -0.1599218685 0.8738489
#> 2016.9863 0.3426506 0.004214668 0.6810866 -0.1749424993 0.8602437
#> 2016.9890 0.3315065 -0.007391559 0.6704046 -0.1867933670 0.8498064
#> 2016.9918 0.3056883 -0.033671254 0.6450479 -0.2133173691 0.8246940
#> 2016.9945 0.3111831 -0.028637339 0.6510036 -0.2085274304 0.8308937
#> 2016.9973 0.3325933 -0.007687414 0.6728740 -0.1878211507 0.8530078
#> 2017.0000 0.3814185 0.039347484 0.7234896 -0.1417339983 0.9045711
#> 2017.0027 0.3825405 0.040012166 0.7250688 -0.1413113610 0.9063923# Tambahkan nilai forecast pada visualisasi sebelumnya
climate_train %>%
autoplot() +
autolayer(climate_test, series = "Data Test") + # Visualisasi data test
autolayer(climate_triple$fitted[,1], series = "Model") + # Visualisasi hasil smoothing model
autolayer(forecast_triple$mean, series = "Forecast")
Dari hasil visualisasi di atas hasil prediksi memiliki seasonality (membentuk fluktuasi) dan mendekati data test. Namun kita akan melakukan evaluasi model agar lebih jelas apakah model triple menghasilkan prediksi yang bagus.
6.1.3 Model Evaluation
Untuk membandingkan performa model mana yang lebih baik, kita dapat menghitung besar error dari hasil forecast dibandingkan dengan nilai sebenarnya. Mirip seperti regresi, kita dapat menggunakan metrics error seperti MAE (Mean Absolute Error), RMSE (Root Mean Squared Error), dan MAPE (Mean Absolute Percentage Error).
#> ME RMSE MAE MPE MAPE ACF1 Theil's U
#> Test set 0.02118966 0.04205517 0.03390281 4.216693 7.617656 0.6041524 1.79428Untuk mengetahui apakah model sudah memprediksi dengan baik kita bisa
melihat nilai nilai pada MAE dan MAPE. Nilai MAE adalah 0.0339 dimana
rata-rata prediksi meleset sebesar 0.0339 satuan, sedangkan Nilai MAPE
adalah 7.617 dimana berarti prediksi melenceng sebesar 7.6% dari data
asli/aktual. Dari hasil Evaluasi Model di atas, dapat disimpulkan model
**climate_triple** sudah bagus dalam melakukan
prediksi.
6.2 Seasonal Arima (SARIMA)
6.2.1 Stationer
Sebelum melakukan pemodelan dengan menggunakan SARIMA, data harus bersifat stationer. Stasioner berarti data yang digunakan datar dan hanya berfluktuasi di sekitar rata-ratanya.
Data sudah stationer ketika p-value < alpha (0.05).
#>
#> Augmented Dickey-Fuller Test
#>
#> data: climate_train
#> Dickey-Fuller = -2.2804, Lag order = 10, p-value = 0.4596
#> alternative hypothesis: stationaryDari hasil visualisasi dan pengujian p-value dapat disimpulkan jika data train kita masih bersifat stationer sehingga kita perlu melakukan Differencing Seasonality.
#>
#> Augmented Dickey-Fuller Test
#>
#> data: .
#> Dickey-Fuller = -6.9161, Lag order = 8, p-value = 0.01
#> alternative hypothesis: stationarySetelah dilakukan 1x differencing seasonility,
climate_train sudah bersifat stationer dengan p-value <
alpha yaitu 0.01 < 0.05.
Dengan begitu, Nilai parameter Integrated untuk SARIMA:
- D (seasonal) = 1 (karena 1x melakukan diff(lag = frequency))
6.2.2 Model Fitting
#> Series: climate_train
#> ARIMA(4,1,1)(0,1,0)[365]
#>
#> Coefficients:
#> ar1 ar2 ar3 ar4 ma1
#> 0.5476 -0.0139 -0.0226 0.0487 -0.9685
#> s.e. 0.0403 0.0427 0.0428 0.0394 0.0158
#>
#> sigma^2 = 0.001403: log likelihood = 1358.45
#> AIC=-2704.89 AICc=-2704.78 BIC=-2677.346.2.3 Forecasting
Kemudian, kita bisa memvisualisasikan hasil fitted model dan juga forecast hasil prediksi model:
climate_train %>%
autoplot() +
autolayer(climate_test, series = "Data Test") +
autolayer(forecast_auto$fitted, series = "Model") +
autolayer(forecast_auto$mean, series = "Forecast")
6.2.4 Model Evaluation
#> ME RMSE MAE MPE MAPE ACF1
#> Test set 0.002041005 0.04352209 0.03381135 -0.1207572 7.721811 0.5764321
#> Theil's U
#> Test set 1.9006Untuk mengetahui apakah model sudah memprediksi dengan baik kita bisa
melihat nilai nilai pada MAE dan MAPE. Nilai MAE adalah 0.0338 dimana
rata-rata prediksi meleset sebesar 0.0338 satuan, sedangkan Nilai MAPE
adalah 7.721 dimana berarti prediksi melenceng sebesar 7.7% dari data
asli/aktual. Dari hasil Evaluasi Model di atas, dapat disimpulkan model
SARIMA **auto_climate** sudah bagus dalam melakukan
prediksi.
6.2.5 Assumption
Asumsi pada time series diujikan untuk mengukur apakah model SARIMA sudah cukup baik untuk menggambarkan informasi pada data berdasarkan residual model yang terbentuk.
6.2.5.1 No-autocorrelation residual
Pada pengujian ini, kondisi yang diinginkan adalah :
- \(H_0\): residual tidak berkorelasi
- \(H_1\): residual berkorelasi
dan ketika p-value > 0.05 (alpha), maka tidak ada autocorrelation.
#>
#> Box-Ljung test
#>
#> data: auto_climate$residuals
#> X-squared = 0.010763, df = 1, p-value = 0.9174Lulus uji asumsi no-autocorrelation residual. Sehingga tidak ada residual yang memiliki autokorelasi (karena p-value 0.9174 > 0.05)
6.2.5.2 Normality of Residual
Untuk mengecek normality residual pada hasil forecasting time series.
\(H_0\): residual menyebar normal \(H_1\): residual tidak menyebar normal
Kondisi yang diinginkan adalah : p-value > 0.05 (alpha), sehingga residual menyebar normal.
#>
#> Shapiro-Wilk normality test
#>
#> data: auto_climate$residuals
#> W = 0.90456, p-value < 0.00000000000000022Tidak lulus uji asumsi Normality of Residual yaitu p-value 0.00000000000000022 < 0.05. Namun, asumsi ini tidak wajib jika jika tujuan kita hanya untuk melakukan forecasting bukan untuk analisis statistik lanjutan. Cukup asumsi No-autocorrelation residual yang wajib terpenuhi.
Sehingga dari hasil pengujian asumsi di atas, model SARIMA
**auto_climate** bisa digunakan dalam melakukan
forecasting.
7 Conclusion
Model Evaluation menggunakan Exponential Smoothing
#> ME RMSE MAE MPE MAPE ACF1
#> Test set 0.002041005 0.04352209 0.03381135 -0.1207572 7.721811 0.5764321
#> Theil's U
#> Test set 1.9006Model Evaluation menggunakan SARIMA
#> ME RMSE MAE MPE MAPE ACF1 Theil's U
#> Test set 0.02118966 0.04205517 0.03390281 4.216693 7.617656 0.6041524 1.79428Dari hasil Model Evaluation terhadap 2 model tersebut dapat disimpulkan bahwa keduanya sama-sama memiliki hasil prediksi yang sangat bagus dilihat dari nilai MAPE yaitu di ankga 7%, sehingga dalam melakukan forecasting terhadap Daily Climate bisa menggunakan salah satu dari 2 model di atas.