House Hold Energy Data

1. Introduction

The objective of the analysis is to make a forcasting and time series to understand energy and cost used in a building in a unit in an apartment located in San Jose form 2016-10-22 to 2018-10-24 from 00.00.00 to 23.45:00 which the data can be accessed on https://www.kaggle.com/datasets/jaganadhg/house-hold-energy-data

2. Data Pre-Prosessing and Library & Setup

2.1. Library & Setup

library(dplyr)
library(tidyverse)
library(lubridate)
library(forecast)
library(TTR)
library(fpp)
library(tseries)
library(TSstudio)
library(padr)
library(ggplot2)
library(MLmetrics)
library(readxl)
library(nortest)
library(rsconnect)

2.2. Data Import

e.usage <- read_xlsx("Household.xlsx")

head(e.usage)

Column description:

TYPE: This is an information column. The value is ‘Electric usage’ for all the observations.
USAGE TIME: Usage time when electricity is used.
USAGE - Consumption in kWh
UNITS - This column denotes measurement unit. It is kWh for all the observations.
COST - Cost of consumption in $.
NOTES - Usage notes

2.3. Data Manipulation & Aggregation

#change in data type on data
e.usage <- e.usage %>% 
  mutate(USAGE_TIME = as.POSIXct(dmy_hms(USAGE_TIME)))

#change in data into floor_date
e.usage <- e.usage %>% 
  mutate(USAGE_TIME = floor_date(e.usage$USAGE_TIME, unit = "hour"))

#Data aggregation based on cost and hourly
e.usage.agg <- e.usage %>%
  mutate(DATETIME = USAGE_TIME) %>% 
  group_by(DATETIME) %>% 
  summarise(USAGE = sum(unique(USAGE)))

head(e.usage.agg)

2.4. Data Pre-Processing

Data Padding

# time range check

range(e.usage.agg$DATETIME)

## [1] "2016-10-22 00:00:00 UTC" "2018-10-24 23:00:00 UTC"

Time series started at 2016-10-22 00:00:00 UTC and finished at 2018-10-24 23:00:00 UTC

set.seed(192)

e.usage.agg.pad <- pad(e.usage.agg)

tail(e.usage.agg.pad)

Replace NA Value and Missing Value Check

NA value is replaced to hour with no transaction with 0 value

#Replace NA Value

e.usage.agg.pad <- e.usage.agg.pad %>% 
  replace(is.na(.),0)

#Missing Value Check
colSums(is.na(e.usage.agg.pad))

## DATETIME    USAGE 
##        0        0

There is no missing value within the data

Determine Certain Time Filtration

Since hours of electicity usage is 0 o’clock to 23 o’clock, therefore, that duration is determined as the certain time to do the time filtration.

e.usage.dpp <- e.usage.agg.pad %>% 
  filter(hour(DATETIME) >= 0 & hour(DATETIME) <= 23)

head(e.usage.dpp)

3. Time Series Object & Exploratory Data Analysis

#Hourly - Decomposition

usage.ts.hour <- e.usage.dpp$USAGE %>% 
  ts(freq = 24)

usage.ts.hour.dc <- usage.ts.hour %>% 
  head(24*7) %>% 
  decompose()

autoplot(usage.ts.hour.dc)

#plot - visualization

usage.ts.hour <- e.usage.dpp %>% 
  mutate(HOUR =  hour(DATETIME)) %>% 
  group_by(HOUR) %>% 
  summarise(USAGEMEAN = round(mean(USAGE),2))

ggplot(usage.ts.hour, mapping = aes(x = HOUR, y = USAGEMEAN))+
  geom_point()+
  geom_line()+
  geom_text(aes(label = USAGEMEAN, vjust = -0.5))+
  scale_x_continuous(breaks = seq(0,23,1))+
  theme_classic()

In hourly period in a day, the hour when the most electricity being used in the building is in the morning around 7.00 to 8.00 and in the evening around 18.00 to 21.00. While, the hour when the lowest electricity being used in the building is in the dawn around 3.00 to 5.00.

4. Cross Validation

Cross validation assumptions: * test: one week (247) train: the rest of data minus test

msts.fb <- e.usage.dpp$USAGE %>% 
  msts(seasonal.periods = c(24,24*7))

msts.fb %>% 
  head(24*7*4) %>% 
  mstl()

## Multi-Seasonal Time Series:
## Start: 1 1
## Seasonal Periods: 24 168
## Data:
##        Data      Trend    Seasonal24   Seasonal168     Remainder
##   [1,] 0.01 0.11214393 -0.0549618966 -0.0308859767 -1.629606e-02
##   [2,] 0.01 0.11186420 -0.0587112738 -0.0285239775 -1.462895e-02
##   [3,] 0.02 0.11158448 -0.0614405685 -0.0263033053 -3.840603e-03
##   [4,] 0.01 0.11130475 -0.0587424728 -0.0229545783 -1.960770e-02
##   [5,] 0.01 0.11102502  0.0488762810 -0.0366785294 -1.132228e-01
##   [6,] 0.01 0.11074530  0.0421902888 -0.0411990114 -1.017366e-01
##   [7,] 0.01 0.11046557  0.0545045785 -0.0616748194 -9.329533e-02
##   [8,] 0.01 0.11018585  0.0787082752 -0.1155276509 -6.336647e-02
##   [9,] 0.02 0.10990612  0.1053438633 -0.1193531995 -7.589678e-02
##  [10,] 0.01 0.10962639  0.0627545298 -0.0142960703 -1.480849e-01
##  [11,] 0.02 0.10934667  0.0004865384  0.0048743344 -9.470754e-02
##  [12,] 0.07 0.10906694 -0.0282211816  0.0027315103 -1.357727e-02
##  [13,] 0.13 0.10878722 -0.0389344174  0.0659048447 -5.757642e-03
##  [14,] 0.01 0.10850749 -0.0555794807  0.0463631364 -8.929114e-02
##  [15,] 0.06 0.10822776 -0.0590713881  0.1482312567 -1.373876e-01
##  [16,] 0.03 0.10794804 -0.0654189480  0.0859736282 -9.850272e-02
##  [17,] 0.02 0.10766831 -0.0800294634  0.1326763545 -1.403152e-01
##  [18,] 0.02 0.10738858 -0.0492469546 -0.0462621540  8.120524e-03
##  [19,] 0.01 0.10710886  0.0385006046 -0.0651614030 -7.044806e-02
##  [20,] 0.01 0.10682913  0.0695284861  0.3113769407 -4.777346e-01
##  [21,] 0.22 0.10654941  0.1319354682  0.1632732714 -1.817581e-01
##  [22,] 0.06 0.10626968  0.0178464212 -0.0907794451  2.666334e-02
##  [23,] 0.05 0.10598995 -0.0070259107 -0.0173513716 -3.161267e-02
##  [24,] 0.30 0.10571023 -0.0236915763  0.0912253775  1.267560e-01
##  [25,] 0.03 0.10543050 -0.0541557666  0.0395554094 -6.083014e-02
##  [26,] 0.03 0.10515077 -0.0579464787  0.0570070786 -7.421137e-02
##  [27,] 0.02 0.10487105 -0.0606976215  0.0484453226 -7.261875e-02
##  [28,] 0.03 0.10459132 -0.0581432884  0.0097093407 -2.615737e-02
##  [29,] 0.81 0.10431160  0.0426950410  0.1708477312  4.921456e-01
##  [30,] 0.73 0.10403187  0.0361371618  0.1462960813  4.435349e-01
##  [31,] 0.68 0.10377337  0.0472587549  0.1039312616  4.250366e-01
##  [32,] 0.57 0.10351488  0.0721486162  0.0235437929  3.707927e-01
##  [33,] 0.75 0.10325638  0.1009786715  0.0542913694  4.914736e-01
##  [34,] 0.25 0.10299789  0.0613147365  0.0289600627  5.672731e-02
##  [35,] 0.15 0.10273939  0.0042332115  0.1670806326 -1.240532e-01
##  [36,] 0.12 0.10248090 -0.0264410235  0.0739722362 -3.001211e-02
##  [37,] 0.05 0.10222240 -0.0368435205  0.0049459638 -2.032485e-02
##  [38,] 0.03 0.10196391 -0.0523461294  0.0122778049 -3.189558e-02
##  [39,] 0.03 0.10170541 -0.0563640732 -0.0084952632 -6.846077e-03
##  [40,] 0.03 0.10144692 -0.0625017215 -0.0276253153  1.868012e-02
##  [41,] 0.02 0.10118842 -0.0769491071  0.0397630794 -4.400240e-02
##  [42,] 0.01 0.10092993 -0.0458486631  0.0920093450 -1.370906e-01
##  [43,] 0.58 0.10067143  0.0368134083  0.1225449873  3.199702e-01
##  [44,] 0.37 0.10041294  0.0744398008 -0.0508835034  2.460308e-01
##  [45,] 0.35 0.10015444  0.1358438962 -0.0797459289  1.937476e-01
##  [46,] 0.06 0.09989595  0.0182557364 -0.0733572426  1.520556e-02
##  [47,] 0.08 0.09963745 -0.0060073901 -0.0237483978  1.011834e-02
##  [48,] 0.05 0.09937896 -0.0253610897 -0.0326677812  8.649914e-03
##  [49,] 0.02 0.09912046 -0.0533988498 -0.0415027645  1.578115e-02
##  [50,] 0.02 0.09886197 -0.0572099866 -0.0318489925  1.019701e-02
##  [51,] 0.02 0.09860347 -0.0599620674 -0.0279803022  9.338899e-03
##  [52,] 0.02 0.09834498 -0.0575305868 -0.0217856173  9.712285e-04
##  [53,] 0.02 0.09808648  0.0365630527 -0.0309877880 -8.366175e-02
##  [54,] 0.05 0.09782799  0.0301690209 -0.0338403736 -4.415663e-02
##  [55,] 0.01 0.09756949  0.0401336518 -0.0619420539 -6.576109e-02
##  [56,] 0.03 0.09731099  0.0657571364 -0.0169786823 -1.160894e-01
##  [57,] 0.06 0.09705250  0.0968291173  0.0848833285 -2.187649e-01
##  [58,] 0.54 0.09679400  0.0601380394  0.1451503475  2.379176e-01
##  [59,] 0.04 0.09653551  0.0082956688 -0.0642999223 -5.312557e-04
##  [60,] 0.04 0.09631856 -0.0242923934 -0.0338109109  1.784747e-03
##  [61,] 0.03 0.09610161 -0.0343314638 -0.0334068061  1.636664e-03
##  [62,] 0.05 0.09588465 -0.0486436856 -0.0071255911  9.884623e-03
##  [63,] 0.02 0.09566770 -0.0531397329 -0.0410126561  1.848469e-02
##  [64,] 0.02 0.09545075 -0.0590195369 -0.0166458764  2.146626e-04
##  [65,] 0.03 0.09523380 -0.0732708944 -0.0366037717  4.464087e-02
##  [66,] 0.07 0.09501685 -0.0418196169 -0.0181758759  3.497865e-02
##  [67,] 0.07 0.09479990  0.0357898649 -0.0056818055 -5.490795e-02
##  [68,] 0.22 0.09458294  0.0800279326 -0.0435195666  8.890869e-02
##  [69,] 0.09 0.09436599  0.1404423054 -0.0985388027 -4.626949e-02
##  [70,] 0.12 0.09414904  0.0193681969 -0.0470609427  5.354371e-02
##  [71,] 0.18 0.09393209 -0.0042971846 -0.0079728335  9.833793e-02
##  [72,] 0.02 0.09371514 -0.0263503783 -0.0457143586 -1.650400e-03
##  [73,] 0.02 0.09349819 -0.0528283132 -0.0374569325  1.678706e-02
##  [74,] 0.02 0.09328123 -0.0566605297 -0.0330860764  1.646537e-02
##  [75,] 0.01 0.09306428 -0.0595751625 -0.0371051020  1.361598e-02
##  [76,] 0.02 0.09284733 -0.0572033368 -0.0308445179  1.520052e-02
##  [77,] 0.02 0.09263038  0.0208154056 -0.0404644763 -5.298131e-02
##  [78,] 0.02 0.09241343  0.0153881863 -0.0496025767 -3.819904e-02
##  [79,] 0.22 0.09219648  0.0247711141 -0.0354056840  1.384381e-01
##  [80,] 0.39 0.09197952  0.0517182799 -0.0004777659  2.467800e-01
##  [81,] 0.21 0.09176257  0.0848335608 -0.0267021869  6.010605e-02
##  [82,] 0.05 0.09154562  0.0598292030 -0.0654344891 -3.594033e-02
##  [83,] 0.26 0.09132867  0.0187756095  0.0033909192  1.465048e-01
##  [84,] 0.09 0.09111172 -0.0207797596 -0.0023096317  2.197767e-02
##  [85,] 0.03 0.09089476 -0.0304475978 -0.0232291066 -7.218061e-03
##  [86,] 0.04 0.09067781 -0.0406784592 -0.0337156541  2.371630e-02
##  [87,] 0.05 0.09046086 -0.0455409649 -0.0504812204  5.556132e-02
##  [88,] 0.01 0.09024391 -0.0516253030 -0.0419224042  1.330380e-02
##  [89,] 0.02 0.09004777 -0.0645791280 -0.0312221446  2.575350e-02
##  [90,] 0.03 0.08985164 -0.0346410097 -0.0293132364  4.102610e-03
##  [91,] 0.04 0.08965550  0.0310688843 -0.0149505844 -6.577380e-02
##  [92,] 0.27 0.08945936  0.1001425752 -0.0452359171  1.256340e-01
##  [93,] 0.15 0.08926323  0.1508836403 -0.0042179408 -8.592892e-02
##  [94,] 0.08 0.08906709  0.0208074043 -0.0143969700 -1.547752e-02
##  [95,] 0.08 0.08887095 -0.0011034909  0.0444092740 -5.217673e-02
##  [96,] 0.04 0.08867481 -0.0324542239  0.0326443106 -4.886490e-02
##  [97,] 0.03 0.08847868 -0.0532144743  0.0692896462 -7.455385e-02
##  [98,] 0.03 0.08828254 -0.0570417964  0.0262159909 -2.745673e-02
##  [99,] 0.02 0.08808640 -0.0600930069  0.0251524851 -3.314588e-02
## [100,] 0.04 0.08789027 -0.0577548621  0.0559710564 -4.610646e-02
## [101,] 0.02 0.08769413  0.0042509237  0.0252962300 -9.724128e-02
## [102,] 0.02 0.08749799 -0.0001475428  0.0093294603 -7.667991e-02
## [103,] 0.02 0.08730186  0.0087156225  0.0405670473 -1.165845e-01
## [104,] 0.02 0.08710572  0.0370780797  0.0736821026 -1.778659e-01
## [105,] 0.06 0.08690958  0.0723282709 -0.0360209076 -6.321694e-02
## [106,] 0.06 0.08671344  0.0591022436 -0.0723843874 -1.343130e-02
## [107,] 0.04 0.08651731  0.0289517240 -0.0758920147  4.229834e-04
## [108,] 0.05 0.08632117 -0.0174566554 -0.0021837740 -1.668074e-02
## [109,] 0.05 0.08612503 -0.0266389647  0.0266527358 -3.613880e-02
## [110,] 0.07 0.08592890 -0.0326656367  0.0045643400  1.217240e-02
## [111,] 0.03 0.08573276 -0.0377717717 -0.0620158179  4.405483e-02
## [112,] 0.03 0.08553662 -0.0439378148 -0.0354353402  2.383653e-02
## [113,] 0.03 0.08534049 -0.0554799659 -0.0439461468  4.408563e-02
## [114,] 0.02 0.08514435 -0.0269408652 -0.0153050986 -2.289838e-02
## [115,] 0.04 0.08494821  0.0269835826  0.0016509483 -7.358274e-02
## [116,] 0.04 0.08475207  0.1209944228 -0.0682639104 -9.748259e-02
## [117,] 0.24 0.08455594  0.1621637065 -0.0474882205  4.076858e-02
## [118,] 0.21 0.08429454  0.0231868692  0.0622439753  4.027462e-02
## [119,] 0.06 0.08403314  0.0031177494  0.0356529726 -6.280386e-02
## [120,] 0.03 0.08377174 -0.0374432338  0.0118964977 -2.822500e-02
## [121,] 0.02 0.08351034 -0.0453441883 -0.0505490741  3.238292e-02
## [122,] 0.02 0.08324894 -0.0498174129 -0.0502274955  3.679597e-02
## [123,] 0.02 0.08298754 -0.0529609471 -0.0485329063  3.850631e-02
## [124,] 0.02 0.08272614 -0.0508489475 -0.0391661745  2.728898e-02
## [125,] 0.02 0.08246474 -0.0270717244 -0.0264089726 -8.984044e-03
## [126,] 0.02 0.08220334 -0.0299445174  0.0155868946 -4.784572e-02
## [127,] 0.02 0.08194194 -0.0203070509  0.2603603169 -3.019952e-01
## [128,] 0.04 0.08168054  0.0067359837  0.0910180433 -1.394346e-01
## [129,] 0.08 0.08141914  0.0583547189  0.0161961894 -7.597005e-02
## [130,] 0.08 0.08115774  0.0396912245 -0.0082334732 -3.261550e-02
## [131,] 0.05 0.08089634  0.0471035717 -0.0576543705 -2.034555e-02
## [132,] 0.03 0.08063494 -0.0079180224 -0.0396014916 -3.115431e-03
## [133,] 0.03 0.08037355 -0.0137280859 -0.0391039326  2.458473e-03
## [134,] 0.03 0.08011215 -0.0191778474 -0.0621510078  3.121671e-02
## [135,] 0.03 0.07985075 -0.0304364015 -0.0437072358  2.429289e-02
## [136,] 0.02 0.07958935 -0.0335215768 -0.0022040144 -2.386376e-02
## [137,] 0.03 0.07932795 -0.0482106891 -0.0523163089  5.119905e-02
## [138,] 0.14 0.07906655 -0.0124242365  0.0021066260  7.125106e-02
## [139,] 0.12 0.07880515  0.0176548506  0.0496972496 -2.615725e-02
## [140,] 0.08 0.07854375  0.1242833228  0.1318493255 -2.546764e-01
## [141,] 0.25 0.07828235  0.1903757692  0.0705030397 -8.916116e-02
## [142,] 0.07 0.07802095  0.0296651664  0.1916155276 -2.293016e-01
## [143,] 0.08 0.07775955  0.0058687153  0.0239107567 -2.753902e-02
## [144,] 0.03 0.07749815 -0.0375845577  0.0265662526 -3.647985e-02
## [145,] 0.05 0.07723675 -0.0372071421  0.0097789540  1.914369e-04
## [146,] 0.02 0.07697535 -0.0424431316  0.0178426737 -3.237489e-02
## [147,] 0.02 0.07716068 -0.0457958517  0.0258686222 -3.723345e-02
## [148,] 0.02 0.07734600 -0.0440268597  0.0306662576 -4.398540e-02
## [149,] 0.02 0.07753133 -0.0586194246  0.0561610900 -5.507299e-02
## [150,] 0.01 0.07771665 -0.0601077696  0.0598980826 -6.750696e-02
## [151,] 0.02 0.07790197 -0.0498372273  0.0341418751 -4.220662e-02
## [152,] 0.03 0.07808730 -0.0242667703  0.1084969210 -1.323174e-01
## [153,] 0.13 0.07827262  0.0435673536  0.0224400978 -1.428008e-02
## [154,] 0.08 0.07845795  0.0193132371 -0.0788968034  6.112562e-02
## [155,] 0.06 0.07864327  0.0641434903 -0.0825093456 -2.774175e-04
## [156,] 0.02 0.07882860  0.0003637207 -0.0493183779 -9.873940e-03
## [157,] 0.03 0.07901392 -0.0022190579 -0.0437514279 -3.043436e-03
## [158,] 0.03 0.07919925 -0.0072093229 -0.0535197122  1.152979e-02
## [159,] 0.08 0.07938457 -0.0247377103 -0.0690192777  9.437242e-02
## [160,] 0.05 0.07956990 -0.0248594319 -0.0374603413  3.274988e-02
## [161,] 0.02 0.07975522 -0.0427784863 -0.0677934926  5.081676e-02
## [162,] 0.02 0.07994054  0.0001723371 -0.0856860122  2.557313e-02
## [163,] 0.04 0.08012587  0.0063230825 -0.0515410705  5.092119e-03
## [164,] 0.03 0.08031119  0.1255152619 -0.1562021454 -1.962431e-02
## [165,] 0.15 0.08049652  0.2164769463 -0.0205580734 -1.264154e-01
## [166,] 0.11 0.08068184  0.0339786530 -0.0533279853  4.866749e-02
## [167,] 0.07 0.08086717  0.0064395207 -0.0235261173  6.219429e-03
## [168,] 0.05 0.08105249 -0.0399213920  0.0165451427 -7.676243e-03
## [169,] 0.02 0.08123782 -0.0328111419 -0.0312725413  2.845866e-03
## [170,] 0.02 0.08142314 -0.0404728514 -0.0286890504  7.738760e-03
## [171,] 0.03 0.08160847 -0.0440507573 -0.0267320955  1.917439e-02
## [172,] 0.03 0.08179379 -0.0388397468 -0.0227218869  9.767843e-03
## [173,] 0.02 0.08197911 -0.0572890079 -0.0345535087  2.986340e-02
## [174,] 0.02 0.08216444 -0.0590076899 -0.0396649646  3.650822e-02
## [175,] 0.03 0.08234976 -0.0367964496 -0.0603546059  4.480129e-02
## [176,] 0.03 0.08281450 -0.0183132362 -0.1153062679  8.080500e-02
## [177,] 0.02 0.08327924  0.0460545084 -0.1180501873  8.716435e-03
## [178,] 0.03 0.08374398  0.0032478840 -0.0114816577 -4.551021e-02
## [179,] 0.07 0.08420872  0.0448568673  0.0071345802 -6.620017e-02
## [180,] 0.12 0.08467346 -0.0117035343  0.0026608211  4.436925e-02
## [181,] 0.18 0.08513820 -0.0085661193  0.0646765996  3.875132e-02
## [182,] 0.10 0.08560294 -0.0109774284  0.0468472423 -2.147276e-02
## [183,] 0.03 0.08606768 -0.0375870329  0.1544765314 -1.729572e-01
## [184,] 0.12 0.08653242 -0.0357243074  0.0873470605 -1.815517e-02
## [185,] 0.03 0.08699716 -0.0585750252  0.1336257347 -1.320479e-01
## [186,] 0.03 0.08746190 -0.0110801035 -0.0467866017  4.048044e-04
## [187,] 0.04 0.08792664  0.0038214393 -0.0637803923  1.203231e-02
## [188,] 0.85 0.08839138  0.1241772997  0.3131281470  3.243032e-01
## [189,] 0.62 0.08885612  0.2588567537  0.1641087805  1.081783e-01
## [190,] 0.08 0.08932086  0.0572193661 -0.0906242513  2.408403e-02
## [191,] 0.09 0.08978560  0.0075478625 -0.0165566130  9.223151e-03
## [192,] 0.05 0.09025034 -0.0378763287  0.0911052010 -9.347921e-02
## [193,] 0.04 0.09071508 -0.0276369610  0.0437476094 -6.682573e-02
## [194,] 0.04 0.09117982 -0.0377112777  0.0615337677 -7.500231e-02
## [195,] 0.03 0.09164456 -0.0415012566  0.0530333525 -7.317665e-02
## [196,] 0.02 0.09210930 -0.0328351150  0.0119641906 -5.123837e-02
## [197,] 0.02 0.09257404 -0.0551532940  0.1626754069 -1.800962e-01
## [198,] 0.03 0.09303878 -0.0571145347  0.1386148247 -1.445391e-01
## [199,] 0.03 0.09350352 -0.0229748181  0.0964295308 -1.369582e-01
## [200,] 0.02 0.09396826 -0.0116168444  0.0164854227 -7.883683e-02
## [201,] 0.03 0.09443300  0.0492465245  0.0454847273 -1.591642e-01
## [202,] 0.07 0.09489774 -0.0121506038  0.0290509631 -4.179810e-02
## [203,] 0.45 0.09536248  0.0261828864  0.1683711160  1.600835e-01
## [204,] 0.13 0.09582722 -0.0232123705  0.0742206716 -1.683552e-02
## [205,] 0.07 0.09632315 -0.0144089850  0.0057412704 -1.765544e-02
## [206,] 0.04 0.09681909 -0.0142938450  0.0158582140 -5.838346e-02
## [207,] 0.03 0.09731503 -0.0500371736 -0.0072784849 -9.999376e-03
## [208,] 0.04 0.09781097 -0.0462425078 -0.0276568443  1.608838e-02
## [209,] 0.04 0.09830691 -0.0740593730  0.0422300362 -2.647758e-02
## [210,] 0.13 0.09880285 -0.0220548371  0.0978304962 -4.457851e-02
## [211,] 0.17 0.09929879  0.0015630190  0.1160655072 -4.692732e-02
## [212,] 0.04 0.09979473  0.1230612315 -0.0535760031 -1.292800e-01
## [213,] 0.19 0.10029067  0.3014371262 -0.0816658683 -1.300619e-01
## [214,] 0.09 0.10078661  0.0806393152 -0.0729988654 -1.842706e-02
## [215,] 0.07 0.10128255  0.0088060961 -0.0242762311 -1.581242e-02
## [216,] 0.04 0.10177849 -0.0357107178 -0.0337549538  7.687181e-03
## [217,] 0.04 0.10227443 -0.0300265503 -0.0427472938  1.049941e-02
## [218,] 0.04 0.10277037 -0.0398617557 -0.0328056018  9.896988e-03
## [219,] 0.03 0.10326631 -0.0440092886 -0.0287882673 -4.687530e-04
## [220,] 0.04 0.10376225 -0.0337472682 -0.0221276352 -7.887345e-03
## [221,] 0.04 0.10425819 -0.0543669871 -0.0297153170  1.982412e-02
## [222,] 0.03 0.10475413 -0.0588562608 -0.0333504971  1.745263e-02
## [223,] 0.12 0.10525007 -0.0166830803 -0.0625347089  9.396772e-02
## [224,] 0.05 0.10574601  0.0144466746 -0.0138595988 -5.633308e-02
## [225,] 0.43 0.10624195  0.0485450414  0.0882630214  1.869500e-01
## [226,] 0.16 0.10673789 -0.0137686852  0.1425161519 -7.548535e-02
## [227,] 0.03 0.10723383  0.0083381675 -0.0641342158 -2.143778e-02
## [228,] 0.03 0.10772976 -0.0386669046 -0.0336189506 -5.443910e-03
## [229,] 0.07 0.10822570 -0.0278912523 -0.0335059029  2.317145e-02
## [230,] 0.12 0.10872164 -0.0161356368 -0.0084262133  3.584021e-02
## [231,] 0.06 0.10921758 -0.0541075160 -0.0423917412  4.728167e-02
## [232,] 0.01 0.10971352 -0.0549544435 -0.0167333840 -2.802570e-02
## [233,] 0.02 0.11020946 -0.0732371149 -0.0373793763  2.040703e-02
## [234,] 0.06 0.11061793 -0.0343897417 -0.0194816208  3.253437e-03
## [235,] 0.12 0.11102639 -0.0006915023 -0.0039533113  1.361843e-02
## [236,] 0.12 0.11143485  0.1157680858 -0.0453246948 -6.187824e-02
## [237,] 0.41 0.11184331  0.3365687447 -0.0970205738  5.860851e-02
## [238,] 0.08 0.11225178  0.1293377399 -0.0462668773 -1.153226e-01
## [239,] 0.05 0.11266024 -0.0036407954 -0.0099986135 -4.902083e-02
## [240,] 0.05 0.11306870 -0.0420722373 -0.0470679226  2.607146e-02
## [241,] 0.06 0.11347717 -0.0328979464 -0.0389869579  1.840774e-02
## [242,] 0.04 0.11388563 -0.0424186670 -0.0340932430  2.626281e-03
## [243,] 0.03 0.11429409 -0.0468483804 -0.0380100343  5.643230e-04
## [244,] 0.04 0.11470255 -0.0349151076 -0.0314678752 -8.319572e-03
## [245,] 0.04 0.11511102 -0.0537379499 -0.0397809007  1.840783e-02
## [246,] 0.03 0.11551948 -0.0606568401 -0.0492452741  2.438263e-02
## [247,] 0.06 0.11592794 -0.0103517790 -0.0393370896 -6.239075e-03
## [248,] 0.03 0.11633641  0.0406576849 -0.0043968702 -1.225972e-01
## [249,] 0.16 0.11674487  0.0480989774 -0.0283379120  2.349407e-02
## [250,] 0.05 0.11715333 -0.0150234197 -0.0654165484  1.328664e-02
## [251,] 0.06 0.11756179 -0.0090489335  0.0005713176 -4.908418e-02
## [252,] 0.09 0.11797026 -0.0535695497 -0.0031756886  2.877498e-02
## [253,] 0.05 0.11837872 -0.0407273596 -0.0231931670 -4.458194e-03
## [254,] 0.02 0.11878718 -0.0172700398 -0.0344458156 -4.707133e-02
## [255,] 0.03 0.11919565 -0.0574092408 -0.0523569797  2.057057e-02
## [256,] 0.03 0.11960411 -0.0628365329 -0.0422667375  1.549916e-02
## [257,] 0.02 0.12001257 -0.0715607641 -0.0316084250  3.156617e-03
## [258,] 0.10 0.12042103 -0.0458463072 -0.0306959158  5.612119e-02
## [259,] 0.10 0.12082950 -0.0020434382 -0.0133124593 -5.473600e-03
## [260,] 0.12 0.12123796  0.1093696174 -0.0480880602 -6.251952e-02
## [261,] 0.75 0.12164642  0.3725871326 -0.0056634456  2.614299e-01
## [262,] 0.36 0.12205489  0.1789150259 -0.0150301750  7.406026e-02
## [263,] 0.35 0.12225395 -0.0152468488  0.0423885279  2.006044e-01
## [264,] 0.24 0.12245301 -0.0476309422  0.0317495029  1.334284e-01
## [265,] 0.24 0.12265207 -0.0368450475  0.0707484374  8.344454e-02
## [266,] 0.25 0.12285114 -0.0478386021  0.0242550874  1.507324e-01
## [267,] 0.25 0.12305020 -0.0548463099  0.0231060129  1.586901e-01
## [268,] 0.33 0.12324926 -0.0433694707  0.0537931714  1.963270e-01
## [269,] 0.25 0.12344832 -0.0591381710  0.0248162237  1.608736e-01
## [270,] 0.24 0.12364739 -0.0675776373  0.0083507380  1.755795e-01
## [271,] 0.33 0.12384645 -0.0235812248  0.0400098232  1.897250e-01
## [272,] 0.56 0.12404551  0.0411489934  0.0721262131  3.226793e-01
## [273,] 0.10 0.12424457  0.0368829739 -0.0353097752 -2.581777e-02
## [274,] 0.12 0.12444364  0.0000558512 -0.0734013735  6.890189e-02
## [275,] 0.03 0.12464270 -0.0205257380 -0.0762170234  2.100063e-03
## [276,] 0.03 0.12484176 -0.0586589612 -0.0024314913 -3.375131e-02
## [277,] 0.11 0.12504082 -0.0458543619  0.0266919708  4.121567e-03
## [278,] 0.27 0.12523989 -0.0094234971  0.0021459609  1.520377e-01
## [279,] 0.04 0.12543895 -0.0506490460 -0.0636808575  2.889095e-02
## [280,] 0.06 0.12563801 -0.0549862510 -0.0363106276  2.565887e-02
## [281,] 0.02 0.12583707 -0.0294328319 -0.0447291069 -3.167513e-02
## [282,] 0.03 0.12603614 -0.0524103290 -0.0138372441 -2.978856e-02
## [283,] 0.15 0.12623520 -0.0100985571  0.0019149482  3.194841e-02
## [284,] 0.32 0.12643426  0.1279823606 -0.0679386863  1.335221e-01
## [285,] 0.34 0.12663332  0.3812586824 -0.0473469277 -1.205451e-01
## [286,] 0.36 0.12683239  0.1894468484  0.0602713314 -1.655057e-02
## [287,] 0.07 0.12703145 -0.0232342730  0.0391666294 -7.296380e-02
## [288,] 0.05 0.12723051 -0.0580190436  0.0133174794 -3.252895e-02
## [289,] 0.07 0.12742957 -0.0412897316 -0.0524269374  3.628710e-02
## [290,] 0.02 0.12762864 -0.0537114622 -0.0514548563 -2.462317e-03
## [291,] 0.02 0.12782770 -0.0632525063 -0.0499913232  5.416131e-03
## [292,] 0.02 0.12779043 -0.0521874429 -0.0400189991 -1.558398e-02
## [293,] 0.04 0.12775316 -0.0648609886 -0.0267182047  3.826038e-03
## [294,] 0.01 0.12771588 -0.0747800185  0.0154293289 -5.836519e-02
## [295,] 0.22 0.12767861 -0.0370512420  0.2718631173 -1.424905e-01
## [296,] 0.35 0.12764134  0.0414348562  0.0918511656  8.907264e-02
## [297,] 0.22 0.12760407  0.0254966505  0.0156270987  5.127218e-02
## [298,] 0.14 0.12756680  0.0149999280 -0.0091813919  6.614665e-03
## [299,] 0.04 0.12752953 -0.0321189800 -0.0573671115  1.956564e-03
## [300,] 0.05 0.12749226 -0.0638460537 -0.0398188211  2.617262e-02
## [301,] 0.04 0.12745498 -0.0510602887 -0.0392988492  2.904154e-03
## [302,] 0.03 0.12741771 -0.0016587800 -0.0621318405 -3.362709e-02
## [303,] 0.07 0.12738044 -0.0439735781 -0.0452639992  3.185714e-02
## [304,] 0.03 0.12734317 -0.0472235974 -0.0020436183 -4.807595e-02
## [305,] 0.04 0.12730590  0.0125920279 -0.0531385808 -4.675935e-02
## [306,] 0.13 0.12726863 -0.0590928673 -0.0008962834  6.272052e-02
## [307,] 0.22 0.12723136 -0.0182876365  0.0499582910  6.109799e-02
## [308,] 0.21 0.12719408  0.1464497854  0.1453124957 -2.089564e-01
## [309,] 0.58 0.12715681  0.3897735562  0.0733348469 -1.026522e-02
## [310,] 0.89 0.12711954  0.1998106374  0.1929183150  3.701515e-01
## [311,] 0.06 0.12708227 -0.0313782181  0.0262748897 -6.197894e-02
## [312,] 0.04 0.12704500 -0.0685521533  0.0290438528 -4.753670e-02
## [313,] 0.04 0.12700773 -0.0447506989  0.0110871365 -5.334417e-02
## [314,] 0.04 0.12697046 -0.0586493201  0.0204858041 -4.880694e-02
## [315,] 0.04 0.12693318 -0.0688857322  0.0287173059 -4.676476e-02
## [316,] 0.03 0.12689591 -0.0597684342  0.0339943363 -7.112182e-02
## [317,] 0.04 0.12685864 -0.0743365274  0.0597335801 -7.225569e-02
## [318,] 0.04 0.12682137 -0.0808543880  0.0643279764 -7.029496e-02
## [319,] 0.04 0.12678410 -0.0491001116  0.0375829087 -7.526690e-02
## [320,] 0.11 0.12674683  0.0414108382  0.1160952481 -1.742529e-01
## [321,] 0.17 0.12646940  0.0121164263  0.0216851283  9.729048e-03
## [322,] 0.10 0.12619197  0.0174975493 -0.0810081653  3.731865e-02
## [323,] 0.02 0.12591454 -0.0248472875 -0.0829644091  1.897160e-03
## [324,] 0.03 0.12563711 -0.0499200082 -0.0494265334  3.709436e-03
## [325,] 0.03 0.12535967 -0.0420419165 -0.0438676930 -9.450065e-03
## [326,] 0.02 0.12508224  0.0064582329 -0.0539159133 -5.762456e-02
## [327,] 0.02 0.12480481 -0.0388580591 -0.0710841010  5.137346e-03
## [328,] 0.07 0.12452738 -0.0309873228 -0.0384804957  1.494044e-02
## [329,] 0.03 0.12424995  0.0369846251 -0.0683269850 -6.290759e-02
## [330,] 0.01 0.12397252 -0.0575736290 -0.0873134434  3.091455e-02
## [331,] 0.06 0.12369509 -0.0229264681 -0.0523273555  1.155873e-02
## [332,] 0.09 0.12341766  0.1681666570 -0.1567414426 -4.484288e-02
## [333,] 0.63 0.12314023  0.3437854851 -0.0199969860  1.830713e-01
## [334,] 0.26 0.12286280  0.1814887778 -0.0550342710  1.068269e-02
## [335,] 0.08 0.12258537 -0.0347577566 -0.0245603107  1.673270e-02
## [336,] 0.07 0.12230794 -0.0665027322  0.0174534785 -3.258685e-03
## [337,] 0.10 0.12203051 -0.0464704567 -0.0316782077  5.611816e-02
## [338,] 0.06 0.12175308 -0.0619246358 -0.0288731811  2.904474e-02
## [339,] 0.03 0.12147565 -0.0729350830 -0.0271798996  8.639335e-03
## [340,] 0.04 0.12119822 -0.0658442176 -0.0225081406  7.154141e-03
## [341,] 0.04 0.12092079 -0.0824493219 -0.0324473642  3.397590e-02
## [342,] 0.04 0.12064336 -0.0857084770 -0.0381497147  4.321484e-02
## [343,] 0.03 0.12036593 -0.0600711642 -0.0590531099  2.875835e-02
## [344,] 0.04 0.12008849  0.0422747793 -0.1151035231 -7.259751e-03
## [345,] 0.05 0.11981106 -0.0005656967 -0.1167657341  4.752037e-02
## [346,] 0.30 0.11953363  0.0205034140 -0.0086857248  1.686487e-01
## [347,] 0.24 0.11925620 -0.0172882231  0.0093764254  1.286556e-01
## [348,] 0.05 0.11897877 -0.0359274625  0.0025718107 -3.562312e-02
## [349,] 0.19 0.11870134 -0.0331779160  0.0634301126  4.104646e-02
## [350,] 0.36 0.11836787  0.0141983874  0.0473131857  1.801206e-01
## [351,] 0.33 0.11803441 -0.0343418853  0.1607037228  8.560376e-02
## [352,] 0.32 0.11770094 -0.0155728804  0.0887024888  1.291695e-01
## [353,] 0.73 0.11736747  0.0603670826  0.1345571900  4.177083e-01
## [354,] 0.03 0.11703400 -0.0572528379 -0.0473288950  1.754773e-02
## [355,] 0.05 0.11670053 -0.0289520545 -0.0624171494  2.466867e-02
## [356,] 1.00 0.11636706  0.1883682287  0.3148616633  3.804030e-01
## [357,] 0.73 0.11603360  0.2961535691  0.1649266944  1.528861e-01
## [358,] 0.08 0.11570013  0.1613945282 -0.0904865576 -1.066081e-01
## [359,] 0.05 0.11536666 -0.0399592499 -0.0157792442 -9.628164e-03
## [360,] 0.02 0.11503319 -0.0663248307  0.0909677452 -1.196761e-01
## [361,] 0.03 0.11469972 -0.0490529562  0.0479226403 -8.356941e-02
## [362,] 0.05 0.11436625 -0.0624464114  0.0660433981 -6.796324e-02
## [363,] 0.02 0.11403279 -0.0730497438  0.0576044341 -7.858748e-02
## [364,] 0.02 0.11369932 -0.0701104714  0.0142022026 -3.779105e-02
## [365,] 0.02 0.11336585 -0.0837339174  0.1544863550 -1.641183e-01
## [366,] 0.02 0.11303238 -0.0720922079  0.1309169509 -1.518571e-01
## [367,] 0.01 0.11269891 -0.0538155490  0.0889112933 -1.377947e-01
## [368,] 0.02 0.11236544  0.0422499332  0.0094106562 -1.440260e-01
## [369,] 0.02 0.11203197  0.0192335507  0.0366617993 -1.479273e-01
## [370,] 0.10 0.11169851  0.0262362031  0.0291256878 -6.706040e-02
## [371,] 0.19 0.11136504 -0.0114523893  0.1696455341 -7.955818e-02
## [372,] 0.23 0.11103157 -0.0204365165  0.0744531503  6.495180e-02
## [373,] 0.10 0.11069810 -0.0231029253  0.0065207291  5.884095e-03
## [374,] 0.03 0.11036463  0.0130504538  0.0194228908 -1.128380e-01
## [375,] 0.01 0.11003116 -0.0292726604 -0.0060773229 -6.468118e-02
## [376,] 0.01 0.10969770 -0.0062469215 -0.0277038694 -6.574691e-02
## [377,] 0.16 0.10936423  0.0600468124  0.0446816172 -5.409266e-02
## [378,] 0.08 0.10903076 -0.0545017563  0.1036363920 -7.816540e-02
## [379,] 0.06 0.10877392 -0.0294651081  0.1095708921 -1.288797e-01
## [380,] 0.10 0.10851709  0.1762380451 -0.0562835176 -1.284716e-01
## [381,] 0.19 0.10826025  0.2401238974 -0.0836007022 -7.478345e-02
## [382,] 0.11 0.10800342  0.1105350481 -0.0726552623 -3.588320e-02
## [383,] 0.08 0.10774658 -0.0384852329 -0.0248187183  3.555737e-02
## [384,] 0.06 0.10748975 -0.0569337566 -0.0348566600  4.430067e-02
## [385,] 0.05 0.10723291 -0.0498646606 -0.0440062363  3.663799e-02
## [386,] 0.04 0.10697607 -0.0612494339 -0.0337765040  2.804986e-02
## [387,] 0.04 0.10671924 -0.0714976936 -0.0296104049  3.438886e-02
## [388,] 0.04 0.10646240 -0.0727620562 -0.0224837053  2.878336e-02
## [389,] 0.04 0.10620557 -0.0835331557 -0.0284567796  4.578437e-02
## [390,] 0.04 0.10594873 -0.0571198936 -0.0328744354  2.404560e-02
## [391,] 0.03 0.10569190 -0.0463332005 -0.0631410595  3.378236e-02
## [392,] 0.23 0.10543506  0.0432687874 -0.0107540916  9.205024e-02
## [393,] 0.16 0.10517822  0.0398934655  0.0916292428 -7.670093e-02
## [394,] 0.08 0.10492139  0.0326466267  0.1398685901 -1.974366e-01
## [395,] 0.05 0.10466455 -0.0051480584 -0.0639817705  1.446528e-02
## [396,] 0.05 0.10440772 -0.0046862107 -0.0334401466 -1.628136e-02
## [397,] 0.02 0.10415088 -0.0129777124 -0.0336180510 -3.755512e-02
## [398,] 0.12 0.10389405  0.0117519913 -0.0097397817  1.409374e-02
## [399,] 0.02 0.10363721 -0.0245547153 -0.0437836674 -1.529883e-02
## [400,] 0.13 0.10338038  0.0025270066 -0.0168336278  4.092625e-02
## [401,] 0.05 0.10312354  0.0590131870 -0.0381676120 -7.396911e-02
## [402,] 0.04 0.10286670 -0.0526253543 -0.0207998918  1.055854e-02
## [403,] 0.03 0.10260987 -0.0310141655 -0.0022372382 -3.935846e-02
## [404,] 0.25 0.10235303  0.1629688816 -0.0471421390  3.182022e-02
## [405,] 0.08 0.10209620  0.1828522695 -0.0955145560 -1.094339e-01
## [406,] 0.11 0.10183936  0.0583306355 -0.0454849203 -4.685076e-03
## [407,] 0.06 0.10158253 -0.0383934530 -0.0120363991  8.847327e-03
## [408,] 0.06 0.10143181 -0.0489622244 -0.0484333894  5.596380e-02
## [409,] 0.06 0.10128110 -0.0425439845 -0.0405287833  4.179167e-02
## [410,] 0.04 0.10113039 -0.0537782269 -0.0351121389  2.775998e-02
## [411,] 0.03 0.10097967 -0.0613011994 -0.0389266254  2.924815e-02
## [412,] 0.02 0.10082896 -0.0672957329 -0.0321028210  1.856959e-02
## [413,] 0.01 0.10067825 -0.0767943929 -0.0391088431  2.522499e-02
## [414,] 0.02 0.10052754 -0.0408672818 -0.0488994189  9.239164e-03
## [415,] 0.02 0.10037682 -0.0371345850 -0.0432798723  3.763293e-05
## [416,] 0.10 0.10022611  0.0610875223 -0.0083272811 -5.298635e-02
## [417,] 0.09 0.10007540  0.0689128572 -0.0299848733 -4.900338e-02
## [418,] 0.12 0.09992469  0.0398005482 -0.0654097734  4.568454e-02
## [419,] 0.06 0.09977397 -0.0032237790 -0.0022593792 -3.429082e-02
## [420,] 0.06 0.09962326 -0.0017490890 -0.0040527702 -3.382140e-02
## [421,] 0.07 0.09947255 -0.0170795704 -0.0231681816  1.077520e-02
## [422,] 0.14 0.09932184  0.0099407144 -0.0351868609  6.592431e-02
## [423,] 0.02 0.09917112 -0.0069514763 -0.0542435771 -1.797607e-02
## [424,] 0.02 0.09902041 -0.0021205759 -0.0426218633 -3.427797e-02
## [425,] 0.05 0.09886970  0.0219201187 -0.0320054520 -3.878436e-02
## [426,] 0.03 0.09871899 -0.0366329654 -0.0320892959  3.276190e-06
## [427,] 0.08 0.09856827 -0.0238369445 -0.0116850308  1.695370e-02
## [428,] 0.19 0.09841756  0.1113302219 -0.0509508960  3.120311e-02
## [429,] 0.10 0.09826685  0.1236113598 -0.0071196390 -1.147586e-01
## [430,] 0.08 0.09811613  0.0385588429 -0.0156740647 -4.100091e-02
## [431,] 0.05 0.09796542 -0.0360709112  0.0403571012 -5.225161e-02
## [432,] 0.05 0.09781471 -0.0375651518  0.0308440185 -4.109358e-02
## [433,] 0.05 0.09766400 -0.0349392011  0.0721965561 -8.492135e-02
## [434,] 0.04 0.09751328 -0.0460160642  0.0222835152 -3.378073e-02
## [435,] 0.04 0.09736257 -0.0508069011  0.0210488761 -2.760455e-02
## [436,] 0.04 0.09721186 -0.0615247571  0.0516046258 -4.729173e-02
## [437,] 0.04 0.09719937 -0.0697350421  0.0243255608 -1.178989e-02
## [438,] 0.04 0.09718688 -0.0242781467  0.0073613632 -4.027010e-02
## [439,] 0.10 0.09717440 -0.0275835108  0.0394419505 -9.032835e-03
## [440,] 0.24 0.09716191  0.0792908675  0.0705596570 -7.012432e-03
## [441,] 0.27 0.09714942  0.0983490108 -0.0346093275  1.091109e-01
## [442,] 0.06 0.09713693  0.0474033833 -0.0744290553 -1.011126e-02
## [443,] 0.03 0.09712444 -0.0008002133 -0.0765527388  1.022851e-02
## [444,] 0.19 0.09711196  0.0017376920 -0.0026899679  9.384032e-02
## [445,] 0.16 0.09709947 -0.0205813960  0.0267203943  5.676153e-02
## [446,] 0.02 0.09708698  0.0087892209 -0.0002832820 -8.559292e-02
## [447,] 0.02 0.09707449  0.0113712972 -0.0653568134 -2.308898e-02
## [448,] 0.02 0.09706201 -0.0059888729 -0.0371968837 -3.387625e-02
## [449,] 0.04 0.09704952 -0.0143430638 -0.0455230879  2.816633e-03
## [450,] 0.04 0.09703703 -0.0197600906 -0.0123804630 -2.489648e-02
## [451,] 0.16 0.09702454 -0.0157286373  0.0021678223  7.653627e-02
## [452,] 0.06 0.09701206  0.0606524876 -0.0676246404 -3.003990e-02
## [453,] 0.19 0.09699957  0.0653612147 -0.0472168653  7.485608e-02
## [454,] 0.25 0.09698708  0.0198076541  0.0582874047  7.491786e-02
## [455,] 0.08 0.09697459 -0.0327336292  0.0426689509 -2.690991e-02
## [456,] 0.08 0.09696210 -0.0251592028  0.0147270734 -6.529976e-03
## [457,] 0.03 0.09694962 -0.0179370637 -0.0543162651  5.303712e-03
## [458,] 0.02 0.09693713 -0.0306769111 -0.0526937585  6.433540e-03
## [459,] 0.02 0.09692464 -0.0334030243 -0.0514613249  7.939707e-03
## [460,] 0.02 0.09691215 -0.0544559950 -0.0408834518  1.842729e-02
## [461,] 0.02 0.09689967 -0.0751570460 -0.0270391336  2.529651e-02
## [462,] 0.27 0.09688718 -0.0304316048  0.0152599977  1.882844e-01
## [463,] 0.34 0.09687469 -0.0318247361  0.2833540832 -8.404038e-03
## [464,] 0.35 0.09686220  0.0741036774  0.0926723847  8.636173e-02
## [465,] 0.33 0.09684972  0.1051247808  0.0150460360  1.129795e-01
## [466,] 0.27 0.09694198  0.0662275066 -0.0101413513  1.169719e-01
## [467,] 0.05 0.09703425  0.0021012103 -0.0570919618  7.956503e-03
## [468,] 0.02 0.09712651 -0.0097299938 -0.0400483287 -2.734819e-02
## [469,] 0.02 0.09721878 -0.0318616603 -0.0395060126 -5.851108e-03
## [470,] 0.02 0.09731105  0.0117327466 -0.0621249888 -2.691881e-02
## [471,] 0.09 0.09740331  0.0346123024 -0.0468331470  4.817531e-03
## [472,] 0.18 0.09749558 -0.0148403334 -0.0018956754  9.924043e-02
## [473,] 0.03 0.09758785 -0.0260786154 -0.0539733747  1.246414e-02
## [474,] 0.07 0.09768011  0.0098995999 -0.0039118065 -3.366791e-02
## [475,] 0.07 0.09777238 -0.0112547362  0.0502066267 -6.672427e-02
## [476,] 0.13 0.09786465  0.0249425385  0.1587629138 -1.515701e-01
## [477,] 0.21 0.09795691  0.0524919138  0.0761538551 -1.660268e-02
## [478,] 0.13 0.09804918  0.0190762352  0.1942082516 -1.813337e-01
## [479,] 0.06 0.09814145 -0.0330535768  0.0286261201 -3.371399e-02
## [480,] 0.06 0.09823371 -0.0247523092  0.0315084986 -4.498990e-02
## [481,] 0.08 0.09832598 -0.0033750490  0.0123823128 -2.733324e-02
## [482,] 0.04 0.09841824 -0.0177025840  0.0231158765 -6.383154e-02
## [483,] 0.04 0.09851051 -0.0182886771  0.0315528798 -7.177471e-02
## [484,] 0.03 0.09860278 -0.0496014658  0.0373092534 -5.631056e-02
## [485,] 0.03 0.09869504 -0.0826416616  0.0632928568 -4.934624e-02
## [486,] 0.04 0.09878731 -0.0384960531  0.0687446051 -8.903586e-02
## [487,] 0.04 0.09887958 -0.0378253301  0.0410106253 -6.206487e-02
## [488,] 0.26 0.09897184  0.0673734956  0.1236802064 -3.002554e-02
## [489,] 0.29 0.09906411  0.1105739363  0.0209167384  5.944522e-02
## [490,] 0.06 0.09915638  0.0839413927 -0.0831329997 -3.996477e-02
## [491,] 0.04 0.09924864  0.0041621324 -0.0834330146  2.002224e-02
## [492,] 0.04 0.09934091 -0.0217684452 -0.0495483007  1.197584e-02
## [493,] 0.03 0.09943317 -0.0434429543 -0.0439976242  1.800740e-02
## [494,] 0.14 0.09952544  0.0146660402 -0.0543258348  8.013435e-02
## [495,] 0.02 0.09959242  0.0581338731 -0.0731626834 -6.456361e-02
## [496,] 0.01 0.09965940 -0.0231204308 -0.0395144479 -2.702452e-02
## [497,] 0.01 0.09972638 -0.0369774934 -0.0688743136  1.612543e-02
## [498,] 0.06 0.09979336  0.0406612747 -0.0889547495  8.500118e-03
## [499,] 0.06 0.09986034 -0.0054135402 -0.0531275540  1.868076e-02
## [500,] 0.07 0.09992732 -0.0091995046 -0.1572946920  1.365669e-01
## [501,] 0.06 0.09999429  0.0413911301 -0.0194498893 -6.193554e-02
## [502,] 0.06 0.10006127  0.0203139446 -0.0567545860 -3.620632e-03
## [503,] 0.07 0.10012825 -0.0312876304 -0.0256085720  2.676795e-02
## [504,] 0.04 0.10019523 -0.0221427560  0.0183477078 -5.640018e-02
## [505,] 0.03 0.10026221  0.0096650401 -0.0316043201 -4.832293e-02
## [506,] 0.03 0.10032919 -0.0146740957 -0.0287815269 -2.687357e-02
## [507,] 0.03 0.10039617 -0.0146622736 -0.0274203272 -2.831357e-02
## [508,] 0.03 0.10046315 -0.0470067653 -0.0222171146 -1.239268e-03
## [509,] 0.03 0.10053013 -0.0833916583 -0.0301776019  4.303913e-02
## [510,] 0.03 0.10059711 -0.0492716362 -0.0363041167  1.497865e-02
## [511,] 0.03 0.10066409 -0.0261540258 -0.0574658672  1.295581e-02
## [512,] 0.03 0.10073106  0.0608154134 -0.1145891508 -1.695733e-02
## [513,] 0.09 0.10079804  0.0907173242 -0.1152268732  1.371151e-02
## [514,] 0.19 0.10086502  0.0783641978 -0.0051583283  1.592911e-02
## [515,] 0.14 0.10093200  0.0009868924  0.0120127715  2.606833e-02
## [516,] 0.07 0.10099898 -0.0344433039  0.0024393145  1.005009e-03
## [517,] 0.03 0.10106596 -0.0529352931  0.0628154798 -8.094615e-02
## [518,] 0.08 0.10113294  0.0097763812  0.0489711008 -7.988042e-02
## [519,] 0.55 0.10119992  0.0577202670  0.1660461791  2.250336e-01
## [520,] 0.14 0.10126690 -0.0309915777  0.0907855127 -2.106083e-02
## [521,] 0.04 0.10133388 -0.0377770550  0.1378744279 -1.614312e-01
## [522,] 0.08 0.10140086  0.0565921941 -0.0476781297 -3.031492e-02
## [523,] 0.07 0.10146783  0.0011229195 -0.0609639942  2.837324e-02
## [524,] 0.16 0.10185850  0.0007859918  0.3208054402 -2.634499e-01
## [525,] 0.22 0.10224916  0.0455455383  0.1673162690 -9.511097e-02
## [526,] 0.09 0.10263983  0.0225893558 -0.0910467308  5.581755e-02
## [527,] 0.10 0.10303049 -0.0160987094 -0.0151170624  2.818528e-02
## [528,] 0.28 0.10342115 -0.0054283481  0.0895001532  9.250704e-02
## [529,] 0.39 0.10381182  0.0216336198  0.0506853301  2.138692e-01
## [530,] 0.38 0.10420248 -0.0128006093  0.0691807878  2.194173e-01
## [531,] 0.38 0.10459315 -0.0122743649  0.0607256529  2.269556e-01
## [532,] 0.19 0.10498381 -0.0457340521  0.0156607134  1.150895e-01
## [533,] 0.03 0.10537448 -0.0855040812  0.1451148266 -1.349852e-01
## [534,] 0.03 0.10576514 -0.0614500845  0.1222444076 -1.365595e-01
## [535,] 0.03 0.10615580 -0.0159260256  0.0805399224 -1.407697e-01
## [536,] 0.02 0.10654647  0.0528060954  0.0016659825 -1.410185e-01
## [537,] 0.03 0.10693713  0.0694015449  0.0269483803 -1.732871e-01
## [538,] 0.26 0.10732780  0.0713199040  0.0284877959  5.286450e-02
## [539,] 0.31 0.10771846 -0.0036322633  0.1709503197  3.496348e-02
## [540,] 0.11 0.10810912 -0.0485388949  0.0750627347 -2.463296e-02
## [541,] 0.08 0.10849979 -0.0638251811  0.0071705351  2.815486e-02
## [542,] 0.34 0.10889045  0.0035420192  0.0214638561  2.061037e-01
## [543,] 0.24 0.10928112  0.0560148041 -0.0055439534  8.024803e-02
## [544,] 0.07 0.10967178 -0.0401017353 -0.0281623416  2.859230e-02
## [545,] 0.24 0.11006245 -0.0397442294  0.0463260251  1.233558e-01
## [546,] 0.55 0.11045311  0.0714268985  0.1079944171  2.601256e-01
## [547,] 0.08 0.11084377  0.0066345620  0.1026077016 -1.400860e-01
## [548,] 0.08 0.11123444  0.0098144150 -0.0602741186  1.922527e-02
## [549,] 0.09 0.11162510  0.0488106172 -0.0865068568  1.607114e-02
## [550,] 0.10 0.11201577  0.0240431817 -0.0727093030  3.665036e-02
## [551,] 0.05 0.11240643 -0.0016755368 -0.0251132226 -3.561767e-02
## [552,] 0.02 0.11279709  0.0105761486 -0.0354697907 -6.790345e-02
## [553,] 0.02 0.11331477  0.0215119162 -0.0448029182 -7.002377e-02
## [554,] 0.01 0.11383245 -0.0145126042 -0.0343843852 -5.493546e-02
## [555,] 0.02 0.11435013 -0.0143355632 -0.0300885088 -4.992605e-02
## [556,] 0.02 0.11486780 -0.0441931046 -0.0226252538 -2.804944e-02
## [557,] 0.02 0.11538548 -0.0780004072 -0.0269032260  9.518153e-03
## [558,] 0.02 0.11590316 -0.0589532932 -0.0321902408 -4.759624e-03
## [559,] 0.02 0.11642083  0.0400120335 -0.0629782109 -7.345466e-02
## [560,] 0.25 0.11693851  0.0676639233 -0.0075792415  7.297681e-02
## [561,] 0.35 0.11745619  0.0406270172  0.0949425916  9.697420e-02
## [562,] 0.34 0.11797387  0.0442319184  0.1356736405  4.212057e-02
## [563,] 0.04 0.11849154 -0.0167231545 -0.0638437202  2.075331e-03
## [564,] 0.04 0.11900922 -0.0606438865 -0.0334317836  1.506645e-02
## [565,] 0.02 0.11952690 -0.0730694811 -0.0338898321  7.432415e-03
## [566,] 0.03 0.12004458 -0.0114208488 -0.0106319231 -6.799180e-02
## [567,] 0.05 0.12056225  0.0322650988 -0.0449102983 -5.791705e-02
## [568,] 0.03 0.12107993 -0.0546198401 -0.0167537321 -1.970636e-02
## [569,] 0.04 0.12159761 -0.0474809798 -0.0393257621  5.209135e-03
## [570,] 0.10 0.12211528  0.0552554675 -0.0218791751 -5.549158e-02
## [571,] 0.20 0.12263296  0.0030253425 -0.0009436972  7.528539e-02
## [572,] 0.07 0.12315064  0.0591203255 -0.0488189305 -6.345203e-02
## [573,] 0.18 0.12366832  0.0584067376 -0.0947156311  9.264058e-02
## [574,] 0.16 0.12418599  0.0155514088 -0.0453816968  6.564429e-02
## [575,] 0.06 0.12470367  0.0118168595 -0.0140190115 -6.250152e-02
## [576,] 0.02 0.12522135  0.0344656135 -0.0491052633 -9.058170e-02
## [577,] 0.02 0.12573903  0.0219747438 -0.0415002252 -8.621354e-02
## [578,] 0.02 0.12625670 -0.0156487998 -0.0357996429 -5.480826e-02
## [579,] 0.02 0.12677438 -0.0158296943 -0.0395219456 -5.142274e-02
## [580,] 0.02 0.12729206 -0.0420938218 -0.0325703344 -3.262790e-02
## [581,] 0.02 0.12780973 -0.0699526438 -0.0382365932  3.795030e-04
## [582,] 0.02 0.12831518 -0.0559266585 -0.0484078061 -3.980717e-03
## [583,] 0.04 0.12882063  0.0964656900 -0.0468151379 -1.384712e-01
## [584,] 0.13 0.12932608  0.0830080546 -0.0128072592 -6.952687e-02
## [585,] 0.07 0.12983152  0.0123094991 -0.0317037667 -4.043726e-02
## [586,] 0.05 0.13033697  0.0175716483 -0.0649997738 -3.290884e-02
## [587,] 0.03 0.13084242 -0.0294327050 -0.0052686975 -6.614102e-02
## [588,] 0.03 0.13134786 -0.0724139121 -0.0049482830 -2.398567e-02
## [589,] 0.02 0.13185331 -0.0820251900 -0.0230881041 -6.740018e-03
## [590,] 0.02 0.13235876 -0.0261512650 -0.0355796480 -5.062785e-02
## [591,] 0.02 0.13286421  0.0086917060 -0.0559567686 -6.559914e-02
## [592,] 0.02 0.13336965 -0.0690177719 -0.0431203634 -1.231519e-03
## [593,] 0.05 0.13387510 -0.0551505536 -0.0326341086  3.909561e-03
## [594,] 0.07 0.13438055  0.0390982166 -0.0330585664 -7.042020e-02
## [595,] 0.17 0.13488600 -0.0006226933 -0.0101389609  4.587566e-02
## [596,] 0.09 0.13539144  0.1083467403 -0.0535623611 -1.001758e-01
## [597,] 0.12 0.13589689  0.0678826827 -0.0080159727 -7.576360e-02
## [598,] 0.10 0.13640234  0.0068987811 -0.0161970497 -2.710407e-02
## [599,] 0.09 0.13690778  0.0251157154  0.0391252867 -1.111488e-01
## [600,] 0.17 0.13741323  0.0581288524  0.0303735964 -5.591568e-02
## [601,] 0.30 0.13791868  0.0212073540  0.0732497657  6.762420e-02
## [602,] 0.04 0.13842413 -0.0163400562  0.0210040139 -1.030881e-01
## [603,] 0.03 0.13892957 -0.0164681342  0.0197779890 -1.122394e-01
## [604,] 0.03 0.13943502 -0.0415002867  0.0502507760 -1.181855e-01
## [605,] 0.03 0.13994047 -0.0672592645  0.0245758877 -6.725709e-02
## [606,] 0.02 0.14044592 -0.0540099417  0.0070263590 -7.346233e-02
## [607,] 0.21 0.14095136  0.1103984217  0.0397833513 -8.113314e-02
## [608,] 0.14 0.14145681  0.0891034929  0.0706678491 -1.612282e-01
## [609,] 0.08 0.14196226  0.0036936562 -0.0332925182 -3.236340e-02
## [610,] 0.02 0.14246770  0.0092786230 -0.0750843328 -5.666199e-02
## [611,] 0.01 0.14298891 -0.0343118180 -0.0767617907 -2.191530e-02
## [612,] 0.01 0.14351012 -0.0764352736 -0.0023769497 -5.469790e-02
## [613,] 0.05 0.14403133 -0.0849509308  0.0271932823 -3.627368e-02
## [614,] 0.02 0.14455253 -0.0316150957 -0.0023480799 -9.058936e-02
## [615,] 0.02 0.14507374  0.0007014293 -0.0668761852 -5.889898e-02
## [616,] 0.01 0.14559495 -0.0733083514 -0.0381104482 -2.417615e-02
## [617,] 0.02 0.14611615 -0.0569946152 -0.0463554224 -2.276612e-02
## [618,] 0.24 0.14663736  0.0338969075 -0.0114106647  7.087640e-02
## [619,] 0.10 0.14715857 -0.0012172768  0.0031235124 -4.906480e-02
## [620,] 0.18 0.14767977  0.1195373131 -0.0669234042 -2.029368e-02
## [621,] 0.17 0.14820098  0.0708862156 -0.0471151343 -1.972062e-03
## [622,] 0.10 0.14872219  0.0045983160  0.0570110946 -1.103316e-01
## [623,] 0.38 0.14924340  0.0279396105  0.0452353436  1.575817e-01
## [624,] 0.29 0.14976460  0.0649368902  0.0157808611  5.951765e-02
## [625,] 0.03 0.15028581  0.0207321011 -0.0557744768 -8.524343e-02
## [626,] 0.03 0.15080702 -0.0166893705 -0.0537514647 -5.036618e-02
## [627,] 0.02 0.15132822 -0.0167148270 -0.0526749956 -6.193840e-02
## [628,] 0.03 0.15184943 -0.0404651992 -0.0415812255 -3.980301e-02
## [629,] 0.03 0.15237064 -0.0640920473 -0.0271089505 -3.116964e-02
## [630,] 0.02 0.15289184 -0.0515871014  0.0162455308 -9.755027e-02
## [631,] 1.02 0.15341305  0.1248695625  0.2928260080  4.488914e-01
## [632,] 0.29 0.15393426  0.0957551691  0.0944848931 -5.417432e-02
## [633,] 0.06 0.15445546 -0.0043481199  0.0156302093 -1.057376e-01
## [634,] 0.04 0.15497667  0.0015774932 -0.0100783288 -1.064758e-01
## [635,] 0.06 0.15549788 -0.0385987464 -0.0567689524 -1.301798e-04
## [636,] 0.03 0.15601909 -0.0798641614 -0.0403211978 -5.833726e-03
## [637,] 0.02 0.15654029 -0.0872839087 -0.0397097521 -9.546632e-03
## [638,] 0.08 0.15706150 -0.0365066744 -0.0624714075  2.191658e-02
## [639,] 0.03 0.15758271 -0.0067371061 -0.0480276611 -7.281794e-02
## [640,] 0.04 0.15815131 -0.0770677006 -0.0012418669 -3.984174e-02
## [641,] 0.02 0.15871991 -0.0583449108 -0.0548353467 -2.553965e-02
## [642,] 0.07 0.15928851  0.0291518999 -0.0065303042 -1.119101e-01
## [643,] 0.23 0.15985711 -0.0013930234  0.0502336980  2.130222e-02
## [644,] 1.08 0.16042571  0.1311045086  0.1685895126  6.198803e-01
## [645,] 0.42 0.16099431  0.0742241570  0.0784978092  1.062837e-01
## [646,] 0.38 0.16156291  0.0025900451  0.1957896130  2.005743e-02
## [647,] 0.34 0.16213151  0.0310202697  0.0301784380  1.166698e-01
## [648,] 0.39 0.16270011  0.0719662621  0.0331469446  1.221867e-01
## [649,] 0.27 0.16326871  0.0204208246  0.0130859941  7.322447e-02
## [650,] 0.31 0.16383732 -0.0175554587  0.0247548543  1.389633e-01
## [651,] 0.33 0.16440592 -0.0176488227  0.0333251555  1.499178e-01
## [652,] 0.33 0.16497452 -0.0404156221  0.0395063988  1.659347e-01
## [653,] 0.34 0.16554312 -0.0621592229  0.0657540800  1.708620e-01
## [654,] 0.41 0.16611172 -0.0503713616  0.0716805133  2.225791e-01
## [655,] 0.52 0.16668032  0.1356976836  0.0432459058  1.743761e-01
## [656,] 0.73 0.16724892  0.0991741121  0.1293739802  3.342030e-01
## [657,] 0.11 0.16781752 -0.0093600587  0.0207412054 -6.919867e-02
## [658,] 0.01 0.16838612 -0.0037588352 -0.0851246560 -6.950263e-02
## [659,] 0.01 0.16895472 -0.0413736602 -0.0836741275 -3.390694e-02
## [660,] 0.02 0.16952332 -0.0821106762 -0.0495324065 -1.788024e-02
## [661,] 0.02 0.17009193 -0.0888224462 -0.0440068351 -1.726264e-02
## [662,] 0.03 0.17066053 -0.0400217073 -0.0543503614 -4.628846e-02
## [663,] 0.04 0.17122913 -0.0122600608 -0.0754505657 -4.351850e-02
## [664,] 0.02 0.17179773 -0.0796902775 -0.0405458999 -3.156155e-02
## [665,] 0.03 0.17236633 -0.0596007210 -0.0695244352 -1.324117e-02
## [666,] 0.03 0.17293493  0.0255041018 -0.0901829357 -7.825610e-02
## [667,] 0.07 0.17350353 -0.0016103907 -0.0536249350 -4.826820e-02
## [668,] 0.07 0.17407213  0.1406070516 -0.1569616118 -8.771757e-02
## [669,] 0.22 0.17466967  0.0763364716 -0.0185283633 -1.247778e-02
## [670,] 0.05 0.17526721  0.0006052139 -0.0582399226 -6.763250e-02
## [671,] 0.12 0.17586475  0.0337797690 -0.0262400772 -6.340444e-02
## [672,] 0.33 0.17646228  0.0767232761  0.0187243949  5.809005e-02

summary(msts.fb)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.0000  0.0300  0.0500  0.1219  0.1200  2.3600

msts.fb.train <- head(msts.fb, length(msts.fb) - 24*7)
msts.fb.test <- tail(msts.fb, 24*7)

5. Model Building & Evaluation

5.1. Holt Winters Model

Holt Winters Fitting - Forecast

#modeling 
model.fb.hw <- HoltWinters(x = msts.fb.train)

#forcasting
model.fb.hw.for <- forecast(model.fb.hw, 24*7)

Holt Winters Model Evaluation

for.fb.hw <- as.vector(model.fb.hw.for$mean)
acc.fb.hw <- accuracy(for.fb.hw, msts.fb.test)
acc.fb.hw

##                  ME       RMSE        MAE  MPE MAPE      ACF1 Theil's U
## Test set -0.0388817 0.07961334 0.05745913 -Inf  Inf 0.6575339         0

5.2. ARIMA Model

ARIMA Fitting - Forecast

#ARIMA
fb.ar <-stlm(msts.fb.train, method = "arima")

#Forecast
fb.ar.for <- forecast(fb.ar, h = 24*7)

ARIMA Model Evaluation

for.fb.ar <- as.vector(fb.ar.for$mean)
acc.fb.ar <- accuracy(for.fb.ar, msts.fb.test)
acc.fb.ar

##                   ME       RMSE        MAE  MPE MAPE     ACF1 Theil's U
## Test set -0.04103338 0.07777016 0.05650731 -Inf  Inf 0.449382         0

5.3. TBATS Model

TBATS Fitting - Forecast

#TBATS
fb.tb <- tbats(msts.fb.train)

#Forecast
fb.tb.for <- forecast(fb.tb, h = 24*7)

TBATS Model Evaluation

for.fb.tb <- as.vector(fb.tb.for$mean)
acc.fb.tb <- accuracy(for.fb.tb, msts.fb.test)
acc.fb.tb

##                   ME       RMSE        MAE  MPE MAPE      ACF1 Theil's U
## Test set -0.03896522 0.08088287 0.05905175 -Inf  Inf 0.7174149         0

5.4. Model Summary

model.summary <- rbind(acc.fb.tb, acc.fb.hw, acc.fb.ar)
rownames(model.summary) <- c("TBATS", "Holt Winters", "Arima")

model.summary

##                       ME       RMSE        MAE  MPE MAPE      ACF1 Theil's U
## TBATS        -0.03896522 0.08088287 0.05905175 -Inf  Inf 0.7174149         0
## Holt Winters -0.03888170 0.07961334 0.05745913 -Inf  Inf 0.6575339         0
## Arima        -0.04103338 0.07777016 0.05650731 -Inf  Inf 0.4493820         0

Based on model summary from 3 models above, ARIMA model has the lowest RMSE with 0.077 and MAE with 0.056 score compared to other models. Therefore, the model used in this analysis is ARIMA.

6.Assumption Check

6.1. Normality Check

#Anderson Darling Test
ad.test(fb.ar.for$residuals)

## 
##  Anderson-Darling normality test
## 
## data:  fb.ar.for$residuals
## A = 1105, p-value < 2.2e-16

hist(fb.ar.for$residuals, breaks = 20)

According to Anderson Darling Test, p-value result shown is less than 0.05, so that, residuals are not normally distributed.This phenomenon happened due to the data used in this case is not sufficient to build this model. To improve it, more data should be added to make residuals normally distributed and rebuild the model.

6.2. Autocorrelation Check

Box.test(fb.ar.for$residuals, type = "Ljung-Box")

## 
##  Box-Ljung test
## 
## data:  fb.ar.for$residuals
## X-squared = 0.0010572, df = 1, p-value = 0.9741

According to Box-Pierce Test, p-value result shown is 0.9741 (more than 0.05), so that, there is no autocorrelation on residuals.

7. Conclusion

Compared to other models, ARIMA model results the lowest MAE and RMSE.
The peak of electricity usage in a day is in the evening around 19.00 to 21.00
In weekly period, in the evening in weekend around 19.00 to 20.00 is the most consuming electricity time.

House Hold Energy Data - Time Series

Dwi Puji Laksono Gumilang

2022-09-25