Time Series Analysis Assignment
Dina Victoria
2024-11-18
This R Markdown document is intended to analyze a time series dataset.
Load Library
Load the library needed for analysing seasonality, modeling, and forecasting from a dataset given. Execute these R code below.
library(readr)
library(dplyr)
library(tseries)
library(ggplot2)
library(lubridate)
library(forecast)In the process, the ‘seastest’ package is being used to analyze the seasonality in the data.
library(seastests)Load Data
Here is the dataset used for the analysis:
## Rows: 3683 Columns: 2
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (1): rate
## dttm (1): ts
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.## # A tibble: 3,683 × 2
## ts rate
## <dttm> <dbl>
## 1 2020-02-01 00:00:00 100
## 2 2020-02-01 00:10:00 97
## 3 2020-02-01 00:20:00 96
## 4 2020-02-01 00:30:00 100
## 5 2020-02-01 00:40:00 89
## 6 2020-02-01 00:50:00 91
## 7 2020-02-01 01:00:00 90
## 8 2020-02-01 01:10:00 92
## 9 2020-02-01 01:20:00 87
## 10 2020-02-01 01:30:00 83
## # ℹ 3,673 more rowsPre-processing Data
To ensure that the data type of each variable used is appropriate, the check was performed using the following syntax.
glimpse(data)## Rows: 3,683
## Columns: 2
## $ ts <dttm> 2020-02-01 00:00:00, 2020-02-01 00:10:00, 2020-02-01 00:20:00, 2…
## $ rate <dbl> 100, 97, 96, 100, 89, 91, 90, 92, 87, 83, 89, 90, 100, 71, 96, 89…Based in the output, the ‘ts’ variable is already a datetime data type and the ‘rate’ variable is a double. We also need to check the existence of missing value in the data.
anyNA(data)## [1] FALSEThere are no missing value in the dataset based on the output.
Next, the seasonality check is conducted as follow.
hasil_seasonal <- isSeasonal(data$rate, freq = 144)
hasil_seasonal## [1] TRUEThe seasonality of the data is fulfilled.
Plotting the data
plot(data, main = "Time Series - Data", ylab = "Rate", xlab = "Timestamp", type = "l", col = "black")
## Split the Data
In modeling process, first the data must be split into train and test set. The train set is used to create the model, while the test set is being used for forecasting. The data is split based on the date where the test data is the data in the 26th February.
test_date <- as.Date("2020-02-26")
# Split the data
train_data <- data %>% filter(as.Date(ts) != test_date)
test_data <- data %>% filter(as.Date(ts) == test_date)Stationarity Test for the Train Dataset
First, make sure the train set is a time series data.
ts_train <- ts(train_data$rate, frequency = 144)BOX COX Transformation
lambda <- BoxCox.lambda(ts_train)
data_transformed <- BoxCox(ts_train, lambda)
lambda## [1] -0.1506857data_transformed## Time Series:
## Start = c(1, 1)
## End = c(25, 144)
## Frequency = 144
## [1] 3.320773 3.305520 3.300315 3.320773 3.262037 3.273318 3.267714 3.278852
## [9] 3.250461 3.226362 3.262037 3.267714 3.320773 3.145170 3.300315 3.262037
## [17] 3.273318 3.244558 3.238575 3.244558 3.250461 3.278852 3.315748 3.232510
## [25] 3.187587 3.244558 3.273318 3.320773 3.194287 3.046401 3.130106 3.187587
## [33] 3.098411 3.145170 3.180788 3.166873 3.137700 3.098411 2.987915 3.018060
## [41] 3.064389 3.180788 3.122384 3.159753 3.200888 3.173884 3.238575 3.187587
## [49] 3.310664 3.295047 3.335508 3.267714 3.359001 3.256287 3.267714 3.310664
## [57] 3.349757 3.250461 3.278852 3.273318 3.354404 3.335508 3.310664 3.335508
## [65] 3.381275 3.414653 3.449294 3.422602 3.463815 3.510967 3.456618 3.488064
## [73] 3.498036 3.507772 3.529626 3.553244 3.544548 3.538645 3.564555 3.570093
## [81] 3.570093 3.570093 3.535661 3.520403 3.538645 3.553244 3.580946 3.575557
## [89] 3.601806 3.558939 3.575557 3.575557 3.575557 3.604338 3.575557 3.601806
## [97] 3.578260 3.601806 3.631174 3.556102 3.583614 3.588897 3.580946 3.553244
## [105] 3.591512 3.570093 3.529626 3.601806 3.596692 3.556102 3.544548 3.572834
## [113] 3.526575 3.510967 3.494739 3.523501 3.532655 3.523501 3.501307 3.491415
## [121] 3.494739 3.467367 3.460232 3.514136 3.484686 3.514136 3.470890 3.449294
## [129] 3.477845 3.501307 3.491415 3.474382 3.449294 3.488064 3.445584 3.441840
## [137] 3.491415 3.474382 3.517282 3.481280 3.414653 3.402442 3.463815 3.381275
## [145] 3.422602 3.422602 3.363549 3.368050 3.363549 3.430403 3.345059 3.398292
## [153] 3.402442 3.372504 3.359001 3.456618 3.368050 3.449294 3.410622 3.273318
## [161] 3.250461 3.295047 3.310664 3.315748 3.278852 3.273318 3.300315 3.368050
## [169] 3.295047 3.289715 3.410622 3.325740 3.256287 3.213806 3.256287 3.220128
## [177] 3.207393 3.200888 3.226362 3.250461 3.262037 3.273318 3.232510 3.278852
## [185] 3.194287 3.173884 3.187587 3.152520 3.166873 3.226362 3.081711 3.152520
## [193] 3.262037 3.262037 3.300315 3.354404 3.305520 3.315748 3.330651 3.250461
## [201] 3.256287 3.256287 3.430403 3.372504 3.320773 3.410622 3.422602 3.325740
## [209] 3.368050 3.345059 3.398292 3.422602 3.452972 3.481280 3.501307 3.463815
## [217] 3.517282 3.510967 3.535661 3.556102 3.567334 3.538645 3.567334 3.544548
## [225] 3.556102 3.570093 3.580946 3.580946 3.594111 3.591512 3.596692 3.572834
## [233] 3.583614 3.580946 3.591512 3.588897 3.575557 3.614308 3.596692 3.619201
## [241] 3.635862 3.626428 3.609354 3.609354 3.621625 3.621625 3.606854 3.619201
## [249] 3.638186 3.619201 3.626428 3.616762 3.614308 3.628808 3.633525 3.614308
## [257] 3.606854 3.604338 3.596692 3.599257 3.606854 3.604338 3.583614 3.594111
## [265] 3.567334 3.541607 3.550366 3.541607 3.517282 3.532655 3.547467 3.514136
## [273] 3.501307 3.504552 3.474382 3.488064 3.510967 3.556102 3.556102 3.541607
## [281] 3.517282 3.474382 3.467367 3.463815 3.481280 3.514136 3.477845 3.460232
## [289] 3.445584 3.452972 3.406552 3.445584 3.470890 3.477845 3.414653 3.491415
## [297] 3.422602 3.381275 3.349757 3.376912 3.426521 3.389869 3.406552 3.394101
## [305] 3.363549 3.354404 3.330651 3.363549 3.284317 3.310664 3.256287 3.289715
## [313] 3.330651 3.330651 3.310664 3.325740 3.256287 3.226362 3.166873 3.289715
## [321] 3.300315 3.262037 3.232510 3.250461 3.226362 3.349757 3.289715 3.305520
## [329] 3.213806 3.244558 3.262037 3.200888 3.389869 3.335508 3.363549 3.481280
## [337] 3.368050 3.381275 3.345059 3.463815 3.449294 3.398292 3.389869 3.456618
## [345] 3.418646 3.463815 3.526575 3.550366 3.564555 3.606854 3.604338 3.583614
## [353] 3.604338 3.575557 3.609354 3.619201 3.633525 3.631174 3.649602 3.649602
## [361] 3.624034 3.635862 3.635862 3.642793 3.635862 3.626428 3.616762 3.628808
## [369] 3.638186 3.633525 3.635862 3.626428 3.616762 3.631174 3.619201 3.621625
## [377] 3.621625 3.626428 3.624034 3.635862 3.651846 3.624034 3.631174 3.642793
## [385] 3.633525 3.642793 3.638186 3.638186 3.651846 3.642793 3.642793 3.631174
## [393] 3.631174 3.609354 3.621625 3.601806 3.578260 3.604338 3.604338 3.583614
## [401] 3.599257 3.604338 3.599257 3.614308 3.619201 3.614308 3.594111 3.596692
## [409] 3.601806 3.572834 3.561757 3.553244 3.561757 3.580946 3.567334 3.575557
## [417] 3.553244 3.547467 3.544548 3.556102 3.529626 3.529626 3.558939 3.544548
## [425] 3.567334 3.529626 3.514136 3.523501 3.529626 3.547467 3.481280 3.470890
## [433] 3.452972 3.470890 3.410622 3.456618 3.406552 3.414653 3.385593 3.438062
## [441] 3.368050 3.372504 3.394101 3.340310 3.422602 3.467367 3.335508 3.385593
## [449] 3.438062 3.394101 3.349757 3.340310 3.368050 3.414653 3.368050 3.244558
## [457] 3.187587 3.200888 3.256287 3.200888 3.166873 3.130106 3.114531 3.187587
## [465] 3.187587 3.200888 3.267714 3.152520 3.137700 3.145170 3.152520 3.106541
## [473] 3.073130 3.130106 3.145170 3.159753 3.187587 3.330651 3.226362 3.295047
## [481] 3.289715 3.300315 3.414653 3.372504 3.406552 3.430403 3.491415 3.504552
## [489] 3.501307 3.526575 3.541607 3.491415 3.578260 3.614308 3.594111 3.591512
## [497] 3.621625 3.624034 3.621625 3.631174 3.635862 3.631174 3.631174 3.633525
## [505] 3.621625 3.599257 3.611839 3.614308 3.609354 3.609354 3.628808 3.649602
## [513] 3.649602 3.638186 3.656296 3.645076 3.665045 3.656296 3.662876 3.656296
## [521] 3.645076 3.640496 3.628808 3.651846 3.656296 3.635862 3.624034 3.642793
## [529] 3.647346 3.642793 3.665045 3.645076 3.645076 3.640496 3.649602 3.649602
## [537] 3.647346 3.633525 3.624034 3.606854 3.619201 3.621625 3.616762 3.638186
## [545] 3.631174 3.628808 3.601806 3.621625 3.631174 3.628808 3.631174 3.642793
## [553] 3.616762 3.609354 3.594111 3.594111 3.616762 3.604338 3.586264 3.578260
## [561] 3.638186 3.609354 3.596692 3.609354 3.609354 3.628808 3.591512 3.611839
## [569] 3.601806 3.567334 3.567334 3.570093 3.594111 3.575557 3.558939 3.567334
## [577] 3.561757 3.570093 3.507772 3.520403 3.541607 3.514136 3.463815 3.438062
## [585] 3.456618 3.484686 3.456618 3.426521 3.474382 3.463815 3.430403 3.368050
## [593] 3.389869 3.335508 3.305520 3.325740 3.310664 3.267714 3.200888 3.194287
## [601] 3.289715 3.349757 3.220128 3.122384 3.114531 3.220128 3.159753 3.207393
## [609] 3.226362 3.207393 3.122384 3.187587 3.200888 3.180788 3.106541 3.305520
## [617] 3.159753 3.055481 3.180788 3.106541 3.250461 3.250461 3.250461 3.320773
## [625] 3.398292 3.354404 3.330651 3.389869 3.463815 3.406552 3.463815 3.514136
## [633] 3.494739 3.529626 3.501307 3.547467 3.606854 3.619201 3.604338 3.621625
## [641] 3.616762 3.586264 3.614308 3.614308 3.616762 3.601806 3.616762 3.626428
## [649] 3.616762 3.616762 3.628808 3.619201 3.628808 3.609354 3.631174 3.631174
## [657] 3.624034 3.640496 3.628808 3.649602 3.642793 3.642793 3.621625 3.614308
## [665] 3.624034 3.647346 3.638186 3.640496 3.649602 3.660695 3.654077 3.649602
## [673] 3.638186 3.606854 3.599257 3.619201 3.626428 3.619201 3.624034 3.631174
## [681] 3.596692 3.604338 3.624034 3.635862 3.599257 3.575557 3.558939 3.594111
## [689] 3.564555 3.586264 3.567334 3.575557 3.575557 3.578260 3.553244 3.544548
## [697] 3.535661 3.498036 3.526575 3.541607 3.514136 3.507772 3.501307 3.550366
## [705] 3.491415 3.501307 3.498036 3.550366 3.529626 3.538645 3.498036 3.520403
## [713] 3.510967 3.507772 3.481280 3.474382 3.414653 3.406552 3.398292 3.368050
## [721] 3.359001 3.340310 3.354404 3.422602 3.406552 3.320773 3.422602 3.267714
## [729] 3.325740 3.262037 3.250461 3.267714 3.305520 3.335508 3.244558 3.213806
## [737] 3.284317 3.232510 3.345059 3.159753 3.226362 3.238575 3.213806 3.289715
## [745] 3.159753 3.220128 3.114531 3.073130 3.213806 3.180788 2.966706 3.180788
## [753] 2.998177 2.998177 3.046401 2.998177 3.106541 3.130106 3.130106 3.145170
## [761] 3.187587 3.159753 3.008222 3.046401 3.122384 3.152520 3.345059 3.262037
## [769] 3.289715 3.434250 3.305520 3.200888 3.340310 3.368050 3.398292 3.463815
## [777] 3.394101 3.456618 3.510967 3.501307 3.547467 3.550366 3.564555 3.572834
## [785] 3.556102 3.561757 3.575557 3.558939 3.578260 3.575557 3.580946 3.599257
## [793] 3.599257 3.631174 3.642793 3.626428 3.651846 3.658501 3.647346 3.649602
## [801] 3.665045 3.660695 3.651846 3.645076 3.638186 3.642793 3.654077 3.665045
## [809] 3.654077 3.651846 3.649602 3.651846 3.640496 3.642793 3.656296 3.660695
## [817] 3.649602 3.633525 3.621625 3.616762 3.599257 3.616762 3.619201 3.642793
## [825] 3.640496 3.616762 3.628808 3.606854 3.633525 3.647346 3.628808 3.624034
## [833] 3.606854 3.599257 3.586264 3.621625 3.614308 3.596692 3.594111 3.580946
## [841] 3.556102 3.561757 3.556102 3.567334 3.572834 3.578260 3.564555 3.583614
## [849] 3.544548 3.544548 3.572834 3.578260 3.578260 3.580946 3.564555 3.553244
## [857] 3.544548 3.572834 3.541607 3.529626 3.498036 3.488064 3.538645 3.507772
## [865] 3.410622 3.410622 3.414653 3.430403 3.449294 3.463815 3.470890 3.385593
## [873] 3.349757 3.354404 3.340310 3.406552 3.359001 3.320773 3.359001 3.363549
## [881] 3.320773 3.385593 3.359001 3.359001 3.359001 3.310664 3.463815 3.330651
## [889] 3.267714 3.244558 3.330651 3.315748 3.226362 3.238575 3.262037 3.320773
## [897] 3.256287 3.200888 3.173884 3.262037 3.262037 3.180788 3.166873 3.173884
## [905] 3.262037 3.325740 3.295047 3.226362 3.220128 3.325740 3.381275 3.295047
## [913] 3.310664 3.310664 3.267714 3.340310 3.410622 3.467367 3.517282 3.498036
## [921] 3.467367 3.470890 3.570093 3.553244 3.575557 3.635862 3.624034 3.604338
## [929] 3.599257 3.591512 3.611839 3.609354 3.606854 3.621625 3.616762 3.633525
## [937] 3.621625 3.604338 3.619201 3.614308 3.628808 3.599257 3.624034 3.624034
## [945] 3.638186 3.611839 3.588897 3.640496 3.624034 3.635862 3.633525 3.640496
## [953] 3.626428 3.621625 3.635862 3.628808 3.619201 3.633525 3.647346 3.631174
## [961] 3.628808 3.626428 3.609354 3.599257 3.614308 3.606854 3.580946 3.591512
## [969] 3.614308 3.611839 3.570093 3.611839 3.586264 3.572834 3.561757 3.591512
## [977] 3.570093 3.564555 3.564555 3.567334 3.544548 3.504552 3.488064 3.481280
## [985] 3.529626 3.504552 3.532655 3.494739 3.452972 3.456618 3.477845 3.507772
## [993] 3.538645 3.510967 3.494739 3.491415 3.494739 3.491415 3.434250 3.491415
## [1001] 3.418646 3.434250 3.470890 3.363549 3.406552 3.398292 3.445584 3.354404
## [1009] 3.381275 3.406552 3.368050 3.389869 3.463815 3.359001 3.376912 3.330651
## [1017] 3.300315 3.320773 3.359001 3.278852 3.354404 3.284317 3.305520 3.256287
## [1025] 3.389869 3.368050 3.295047 3.220128 3.284317 3.256287 3.273318 3.267714
## [1033] 3.256287 3.295047 3.349757 3.335508 3.238575 3.098411 3.064389 3.200888
## [1041] 3.226362 3.220128 3.200888 3.106541 3.114531 3.114531 3.106541 3.130106
## [1049] 3.180788 3.305520 3.305520 3.300315 3.081711 3.130106 3.220128 3.273318
## [1057] 3.372504 3.256287 3.159753 3.325740 3.256287 3.194287 3.194287 3.173884
## [1065] 3.173884 3.194287 3.315748 3.335508 3.345059 3.305520 3.385593 3.345059
## [1073] 3.300315 3.325740 3.325740 3.481280 3.452972 3.477845 3.501307 3.494739
## [1081] 3.538645 3.570093 3.550366 3.538645 3.588897 3.567334 3.567334 3.580946
## [1089] 3.523501 3.580946 3.580946 3.583614 3.580946 3.567334 3.606854 3.609354
## [1097] 3.588897 3.606854 3.611839 3.609354 3.609354 3.628808 3.616762 3.606854
## [1105] 3.624034 3.638186 3.642793 3.640496 3.624034 3.611839 3.591512 3.599257
## [1113] 3.596692 3.601806 3.604338 3.601806 3.572834 3.621625 3.596692 3.553244
## [1121] 3.535661 3.535661 3.526575 3.578260 3.599257 3.564555 3.604338 3.586264
## [1129] 3.586264 3.594111 3.575557 3.550366 3.586264 3.588897 3.599257 3.586264
## [1137] 3.583614 3.594111 3.586264 3.583614 3.578260 3.567334 3.558939 3.550366
## [1145] 3.556102 3.558939 3.558939 3.494739 3.498036 3.467367 3.426521 3.463815
## [1153] 3.445584 3.410622 3.394101 3.385593 3.422602 3.449294 3.470890 3.491415
## [1161] 3.426521 3.381275 3.359001 3.381275 3.376912 3.398292 3.474382 3.325740
## [1169] 3.345059 3.372504 3.315748 3.330651 3.320773 3.310664 3.250461 3.340310
## [1177] 3.278852 3.330651 3.430403 3.289715 3.363549 3.200888 3.180788 3.187587
## [1185] 3.207393 3.213806 3.226362 3.213806 3.267714 3.220128 3.200888 3.278852
## [1193] 3.220128 3.315748 3.152520 3.152520 3.159753 3.166873 3.244558 3.315748
## [1201] 3.200888 3.262037 3.256287 3.295047 3.166873 3.187587 3.273318 3.278852
## [1209] 3.434250 3.315748 3.410622 3.426521 3.430403 3.363549 3.422602 3.481280
## [1217] 3.456618 3.418646 3.434250 3.463815 3.463815 3.488064 3.510967 3.523501
## [1225] 3.520403 3.514136 3.553244 3.567334 3.564555 3.611839 3.588897 3.588897
## [1233] 3.561757 3.561757 3.601806 3.596692 3.591512 3.619201 3.599257 3.580946
## [1241] 3.561757 3.580946 3.575557 3.599257 3.606854 3.596692 3.596692 3.614308
## [1249] 3.616762 3.628808 3.628808 3.633525 3.616762 3.599257 3.604338 3.611839
## [1257] 3.614308 3.604338 3.609354 3.619201 3.614308 3.616762 3.604338 3.591512
## [1265] 3.611839 3.604338 3.614308 3.621625 3.601806 3.561757 3.572834 3.591512
## [1273] 3.580946 3.594111 3.561757 3.564555 3.544548 3.541607 3.580946 3.572834
## [1281] 3.547467 3.529626 3.523501 3.494739 3.561757 3.538645 3.561757 3.544548
## [1289] 3.561757 3.564555 3.498036 3.544548 3.491415 3.410622 3.406552 3.410622
## [1297] 3.372504 3.345059 3.340310 3.300315 3.394101 3.430403 3.441840 3.320773
## [1305] 3.300315 3.310664 3.438062 3.340310 3.345059 3.330651 3.320773 3.325740
## [1313] 3.376912 3.389869 3.315748 3.359001 3.345059 3.310664 3.295047 3.278852
## [1321] 3.438062 3.295047 3.220128 3.305520 3.250461 3.213806 3.145170 3.159753
## [1329] 3.194287 3.152520 3.159753 3.114531 3.098411 3.200888 3.152520 3.064389
## [1337] 3.073130 3.145170 3.278852 3.220128 3.238575 3.363549 3.345059 3.295047
## [1345] 3.376912 3.477845 3.463815 3.372504 3.398292 3.430403 3.422602 3.467367
## [1353] 3.523501 3.504552 3.529626 3.532655 3.547467 3.594111 3.575557 3.567334
## [1361] 3.594111 3.601806 3.609354 3.616762 3.654077 3.656296 3.647346 3.649602
## [1369] 3.638186 3.640496 3.631174 3.624034 3.611839 3.614308 3.614308 3.631174
## [1377] 3.614308 3.647346 3.647346 3.642793 3.642793 3.640496 3.651846 3.642793
## [1385] 3.645076 3.626428 3.631174 3.631174 3.635862 3.635862 3.633525 3.626428
## [1393] 3.611839 3.640496 3.649602 3.635862 3.631174 3.640496 3.626428 3.619201
## [1401] 3.621625 3.601806 3.594111 3.591512 3.609354 3.588897 3.586264 3.586264
## [1409] 3.564555 3.561757 3.580946 3.578260 3.601806 3.578260 3.572834 3.556102
## [1417] 3.591512 3.564555 3.570093 3.572834 3.564555 3.588897 3.570093 3.572834
## [1425] 3.550366 3.556102 3.564555 3.550366 3.535661 3.541607 3.558939 3.517282
## [1433] 3.488064 3.498036 3.498036 3.517282 3.481280 3.434250 3.426521 3.426521
## [1441] 3.426521 3.406552 3.402442 3.426521 3.456618 3.434250 3.394101 3.394101
## [1449] 3.363549 3.320773 3.410622 3.368050 3.310664 3.315748 3.207393 3.200888
## [1457] 3.213806 3.262037 3.194287 3.300315 3.200888 3.200888 3.262037 3.238575
## [1465] 3.137700 3.137700 3.114531 3.073130 3.137700 3.090136 3.046401 3.055481
## [1473] 3.008222 3.250461 3.027697 3.106541 3.064389 3.106541 2.955740 3.055481
## [1481] 2.977428 3.145170 3.200888 3.359001 3.232510 3.098411 3.200888 3.381275
## [1489] 3.394101 3.345059 3.330651 3.349757 3.368050 3.389869 3.474382 3.368050
## [1497] 3.494739 3.463815 3.535661 3.535661 3.567334 3.606854 3.580946 3.594111
## [1505] 3.609354 3.614308 3.633525 3.649602 3.656296 3.654077 3.645076 3.658501
## [1513] 3.649602 3.647346 3.660695 3.654077 3.651846 3.647346 3.640496 3.633525
## [1521] 3.647346 3.640496 3.631174 3.654077 3.645076 3.656296 3.628808 3.626428
## [1529] 3.635862 3.642793 3.645076 3.642793 3.645076 3.626428 3.638186 3.642793
## [1537] 3.624034 3.631174 3.628808 3.626428 3.628808 3.638186 3.619201 3.635862
## [1545] 3.609354 3.651846 3.642793 3.611839 3.628808 3.619201 3.614308 3.586264
## [1553] 3.609354 3.614308 3.616762 3.621625 3.614308 3.619201 3.614308 3.599257
## [1561] 3.596692 3.609354 3.594111 3.570093 3.556102 3.604338 3.578260 3.588897
## [1569] 3.594111 3.601806 3.586264 3.561757 3.514136 3.544548 3.541607 3.547467
## [1577] 3.541607 3.538645 3.529626 3.510967 3.526575 3.484686 3.484686 3.477845
## [1585] 3.463815 3.494739 3.385593 3.430403 3.422602 3.414653 3.430403 3.372504
## [1593] 3.363549 3.325740 3.300315 3.300315 3.310664 3.325740 3.325740 3.438062
## [1601] 3.376912 3.305520 3.394101 3.345059 3.310664 3.381275 3.452972 3.300315
## [1609] 3.213806 3.106541 3.187587 3.289715 3.098411 3.187587 2.966706 3.159753
## [1617] 3.008222 3.037142 3.106541 3.090136 3.081711 3.073130 3.018060 3.073130
## [1625] 2.944518 3.166873 3.152520 3.152520 3.159753 3.187587 3.180788 3.445584
## [1633] 3.376912 3.278852 3.381275 3.381275 3.441840 3.463815 3.463815 3.484686
## [1641] 3.422602 3.507772 3.535661 3.564555 3.586264 3.601806 3.588897 3.619201
## [1649] 3.624034 3.614308 3.614308 3.609354 3.611839 3.580946 3.578260 3.578260
## [1657] 3.570093 3.583614 3.580946 3.583614 3.604338 3.611839 3.631174 3.624034
## [1665] 3.628808 3.624034 3.631174 3.633525 3.604338 3.594111 3.611839 3.606854
## [1673] 3.606854 3.626428 3.624034 3.631174 3.635862 3.624034 3.628808 3.638186
## [1681] 3.624034 3.594111 3.626428 3.594111 3.628808 3.601806 3.619201 3.624034
## [1689] 3.604338 3.621625 3.628808 3.628808 3.621625 3.624034 3.611839 3.586264
## [1697] 3.596692 3.601806 3.616762 3.601806 3.588897 3.601806 3.604338 3.624034
## [1705] 3.578260 3.611839 3.586264 3.601806 3.558939 3.586264 3.564555 3.532655
## [1713] 3.514136 3.535661 3.558939 3.570093 3.547467 3.556102 3.544548 3.538645
## [1721] 3.498036 3.481280 3.532655 3.504552 3.504552 3.507772 3.529626 3.523501
## [1729] 3.456618 3.510967 3.467367 3.460232 3.430403 3.422602 3.414653 3.418646
## [1737] 3.467367 3.474382 3.406552 3.406552 3.389869 3.418646 3.325740 3.398292
## [1745] 3.376912 3.310664 3.381275 3.359001 3.376912 3.278852 3.256287 3.305520
## [1753] 3.398292 3.320773 3.145170 3.173884 3.187587 3.055481 3.055481 3.137700
## [1761] 3.114531 3.130106 3.055481 2.966706 3.064389 3.130106 3.207393 3.090136
## [1769] 3.315748 3.244558 3.194287 3.130106 3.278852 3.320773 3.363549 3.289715
## [1777] 3.220128 3.250461 3.213806 3.315748 3.389869 3.402442 3.477845 3.376912
## [1785] 3.345059 3.430403 3.514136 3.510967 3.558939 3.572834 3.606854 3.609354
## [1793] 3.594111 3.624034 3.619201 3.624034 3.621625 3.635862 3.616762 3.606854
## [1801] 3.624034 3.628808 3.638186 3.649602 3.614308 3.631174 3.642793 3.651846
## [1809] 3.647346 3.649602 3.645076 3.647346 3.635862 3.631174 3.635862 3.638186
## [1817] 3.645076 3.626428 3.628808 3.638186 3.645076 3.635862 3.651846 3.628808
## [1825] 3.635862 3.645076 3.633525 3.631174 3.642793 3.642793 3.640496 3.631174
## [1833] 3.633525 3.624034 3.601806 3.606854 3.609354 3.616762 3.599257 3.594111
## [1841] 3.588897 3.588897 3.596692 3.601806 3.572834 3.591512 3.588897 3.572834
## [1849] 3.570093 3.583614 3.591512 3.588897 3.604338 3.588897 3.570093 3.570093
## [1857] 3.556102 3.575557 3.547467 3.520403 3.564555 3.547467 3.561757 3.556102
## [1865] 3.578260 3.599257 3.567334 3.556102 3.580946 3.556102 3.575557 3.567334
## [1873] 3.544548 3.526575 3.556102 3.544548 3.520403 3.488064 3.491415 3.488064
## [1881] 3.445584 3.434250 3.474382 3.463815 3.452972 3.481280 3.470890 3.434250
## [1889] 3.354404 3.372504 3.363549 3.359001 3.368050 3.372504 3.410622 3.402442
## [1897] 3.354404 3.372504 3.449294 3.452972 3.430403 3.232510 3.262037 3.345059
## [1905] 3.289715 3.194287 3.173884 3.166873 3.166873 3.130106 3.166873 3.106541
## [1913] 3.106541 3.122384 3.122384 3.166873 3.284317 3.207393 3.152520 3.289715
## [1921] 3.422602 3.422602 3.359001 3.418646 3.460232 3.510967 3.460232 3.456618
## [1929] 3.445584 3.494739 3.532655 3.532655 3.570093 3.558939 3.614308 3.614308
## [1937] 3.588897 3.611839 3.616762 3.606854 3.601806 3.631174 3.635862 3.624034
## [1945] 3.640496 3.651846 3.651846 3.642793 3.635862 3.635862 3.609354 3.628808
## [1953] 3.635862 3.656296 3.645076 3.628808 3.645076 3.642793 3.669347 3.647346
## [1961] 3.640496 3.640496 3.640496 3.628808 3.616762 3.616762 3.606854 3.619201
## [1969] 3.638186 3.638186 3.633525 3.619201 3.626428 3.628808 3.635862 3.611839
## [1977] 3.619201 3.619201 3.619201 3.561757 3.575557 3.514136 3.572834 3.514136
## [1985] 3.558939 3.507772 3.541607 3.498036 3.514136 3.504552 3.452972 3.456618
## [1993] 3.507772 3.441840 3.426521 3.484686 3.438062 3.456618 3.470890 3.449294
## [2001] 3.410622 3.402442 3.385593 3.406552 3.402442 3.430403 3.381275 3.368050
## [2009] 3.376912 3.418646 3.418646 3.402442 3.295047 3.295047 3.363549 3.320773
## [2017] 3.305520 3.289715 3.349757 3.441840 3.310664 3.402442 3.385593 3.284317
## [2025] 3.349757 3.305520 3.310664 3.310664 3.289715 3.278852 3.359001 3.410622
## [2033] 3.256287 3.232510 3.238575 3.244558 3.250461 3.273318 3.278852 3.305520
## [2041] 3.226362 3.238575 3.372504 3.244558 3.207393 3.081711 3.159753 3.244558
## [2049] 3.232510 3.289715 3.438062 3.226362 3.359001 3.145170 3.244558 3.300315
## [2057] 3.130106 3.130106 3.090136 3.180788 3.046401 2.987915 3.122384 3.194287
## [2065] 3.173884 3.200888 3.213806 3.295047 3.226362 3.238575 3.232510 3.398292
## [2073] 3.325740 3.320773 3.335508 3.330651 3.315748 3.325740 3.345059 3.410622
## [2081] 3.456618 3.445584 3.381275 3.456618 3.430403 3.426521 3.449294 3.445584
## [2089] 3.481280 3.498036 3.481280 3.526575 3.532655 3.484686 3.470890 3.535661
## [2097] 3.484686 3.501307 3.504552 3.538645 3.510967 3.526575 3.541607 3.572834
## [2105] 3.580946 3.567334 3.564555 3.575557 3.564555 3.580946 3.561757 3.564555
## [2113] 3.570093 3.596692 3.611839 3.591512 3.567334 3.586264 3.588897 3.599257
## [2121] 3.591512 3.572834 3.561757 3.547467 3.556102 3.556102 3.558939 3.561757
## [2129] 3.535661 3.535661 3.523501 3.504552 3.535661 3.494739 3.474382 3.529626
## [2137] 3.541607 3.523501 3.477845 3.481280 3.501307 3.494739 3.498036 3.463815
## [2145] 3.501307 3.514136 3.426521 3.449294 3.430403 3.445584 3.422602 3.422602
## [2153] 3.418646 3.398292 3.441840 3.394101 3.402442 3.434250 3.398292 3.441840
## [2161] 3.368050 3.359001 3.273318 3.267714 3.300315 3.398292 3.368050 3.340310
## [2169] 3.354404 3.381275 3.381275 3.438062 3.368050 3.354404 3.389869 3.363549
## [2177] 3.310664 3.278852 3.315748 3.376912 3.340310 3.340310 3.325740 3.330651
## [2185] 3.325740 3.278852 3.267714 3.340310 3.273318 3.320773 3.320773 3.262037
## [2193] 3.166873 3.122384 3.159753 3.207393 3.187587 3.081711 3.166873 3.090136
## [2201] 3.090136 3.055481 3.037142 3.073130 3.166873 3.137700 3.152520 3.267714
## [2209] 3.180788 3.187587 3.207393 3.359001 3.194287 3.152520 3.278852 3.398292
## [2217] 3.345059 3.414653 3.320773 3.418646 3.385593 3.398292 3.438062 3.410622
## [2225] 3.430403 3.385593 3.406552 3.398292 3.410622 3.449294 3.520403 3.517282
## [2233] 3.523501 3.510967 3.529626 3.591512 3.561757 3.567334 3.572834 3.578260
## [2241] 3.580946 3.588897 3.578260 3.588897 3.588897 3.606854 3.606854 3.614308
## [2249] 3.611839 3.621625 3.621625 3.619201 3.599257 3.624034 3.626428 3.638186
## [2257] 3.642793 3.628808 3.633525 3.614308 3.609354 3.626428 3.611839 3.628808
## [2265] 3.631174 3.638186 3.611839 3.614308 3.616762 3.604338 3.596692 3.601806
## [2273] 3.616762 3.614308 3.609354 3.614308 3.594111 3.621625 3.609354 3.586264
## [2281] 3.588897 3.604338 3.606854 3.619201 3.624034 3.606854 3.614308 3.606854
## [2289] 3.594111 3.591512 3.604338 3.596692 3.572834 3.580946 3.538645 3.567334
## [2297] 3.532655 3.529626 3.501307 3.481280 3.517282 3.514136 3.491415 3.441840
## [2305] 3.491415 3.441840 3.460232 3.467367 3.449294 3.438062 3.441840 3.449294
## [2313] 3.452972 3.430403 3.449294 3.441840 3.488064 3.422602 3.460232 3.359001
## [2321] 3.376912 3.320773 3.325740 3.310664 3.278852 3.315748 3.349757 3.330651
## [2329] 3.278852 3.256287 3.213806 3.194287 3.180788 3.159753 3.098411 3.173884
## [2337] 3.145170 3.226362 3.137700 3.055481 3.122384 3.114531 3.159753 3.145170
## [2345] 3.137700 3.152520 3.278852 3.213806 3.130106 3.335508 3.295047 3.315748
## [2353] 3.359001 3.368050 3.456618 3.438062 3.422602 3.491415 3.523501 3.517282
## [2361] 3.507772 3.517282 3.578260 3.547467 3.575557 3.601806 3.606854 3.594111
## [2369] 3.624034 3.626428 3.633525 3.642793 3.635862 3.640496 3.626428 3.642793
## [2377] 3.649602 3.660695 3.656296 3.671480 3.675712 3.656296 3.656296 3.638186
## [2385] 3.671480 3.665045 3.642793 3.635862 3.649602 3.671480 3.667202 3.658501
## [2393] 3.658501 3.656296 3.645076 3.649602 3.651846 3.651846 3.654077 3.651846
## [2401] 3.649602 3.640496 3.626428 3.642793 3.649602 3.633525 3.635862 3.638186
## [2409] 3.642793 3.645076 3.635862 3.633525 3.614308 3.635862 3.609354 3.614308
## [2417] 3.614308 3.583614 3.572834 3.599257 3.616762 3.591512 3.596692 3.619201
## [2425] 3.591512 3.580946 3.580946 3.586264 3.601806 3.591512 3.570093 3.561757
## [2433] 3.556102 3.558939 3.547467 3.494739 3.526575 3.570093 3.588897 3.535661
## [2441] 3.526575 3.535661 3.541607 3.567334 3.532655 3.544548 3.553244 3.529626
## [2449] 3.520403 3.494739 3.488064 3.491415 3.481280 3.434250 3.523501 3.463815
## [2457] 3.438062 3.430403 3.410622 3.368050 3.460232 3.406552 3.372504 3.376912
## [2465] 3.372504 3.418646 3.398292 3.394101 3.376912 3.381275 3.463815 3.310664
## [2473] 3.325740 3.310664 3.273318 3.200888 3.200888 3.145170 3.046401 3.145170
## [2481] 3.152520 3.130106 3.166873 3.152520 3.130106 3.295047 3.244558 3.250461
## [2489] 3.098411 3.180788 3.200888 3.145170 3.114531 3.173884 3.226362 3.262037
## [2497] 3.284317 3.315748 3.325740 3.438062 3.494739 3.474382 3.504552 3.517282
## [2505] 3.484686 3.529626 3.544548 3.523501 3.572834 3.633525 3.640496 3.606854
## [2513] 3.638186 3.633525 3.647346 3.640496 3.651846 3.647346 3.649602 3.640496
## [2521] 3.642793 3.651846 3.651846 3.647346 3.656296 3.656296 3.649602 3.656296
## [2529] 3.651846 3.635862 3.658501 3.651846 3.654077 3.662876 3.660695 3.662876
## [2537] 3.667202 3.658501 3.660695 3.669347 3.675712 3.686093 3.673602 3.660695
## [2545] 3.640496 3.647346 3.645076 3.667202 3.654077 3.640496 3.656296 3.658501
## [2553] 3.660695 3.651846 3.647346 3.651846 3.642793 3.640496 3.635862 3.626428
## [2561] 3.619201 3.638186 3.640496 3.616762 3.596692 3.619201 3.604338 3.588897
## [2569] 3.604338 3.580946 3.609354 3.624034 3.611839 3.580946 3.614308 3.624034
## [2577] 3.609354 3.614308 3.596692 3.578260 3.580946 3.586264 3.564555 3.532655
## [2585] 3.520403 3.532655 3.561757 3.514136 3.501307 3.507772 3.514136 3.529626
## [2593] 3.514136 3.385593 3.402442 3.389869 3.394101 3.354404 3.368050 3.359001
## [2601] 3.330651 3.325740 3.354404 3.385593 3.325740 3.330651 3.430403 3.335508
## [2609] 3.430403 3.340310 3.320773 3.320773 3.300315 3.305520 3.349757 3.289715
## [2617] 3.300315 3.207393 3.226362 3.402442 3.244558 3.238575 3.256287 3.300315
## [2625] 3.232510 3.200888 3.200888 3.194287 3.273318 3.267714 3.220128 3.262037
## [2633] 3.300315 3.289715 3.256287 3.310664 3.295047 3.256287 3.445584 3.441840
## [2641] 3.310664 3.385593 3.284317 3.481280 3.410622 3.418646 3.452972 3.449294
## [2649] 3.477845 3.514136 3.538645 3.561757 3.578260 3.614308 3.611839 3.599257
## [2657] 3.596692 3.596692 3.599257 3.624034 3.640496 3.642793 3.642793 3.635862
## [2665] 3.633525 3.645076 3.642793 3.651846 3.621625 3.642793 3.638186 3.640496
## [2673] 3.660695 3.660695 3.660695 3.656296 3.649602 3.642793 3.645076 3.647346
## [2681] 3.635862 3.642793 3.631174 3.633525 3.642793 3.660695 3.642793 3.633525
## [2689] 3.640496 3.638186 3.624034 3.633525 3.635862 3.640496 3.621625 3.633525
## [2697] 3.631174 3.631174 3.642793 3.631174 3.616762 3.624034 3.619201 3.640496
## [2705] 3.596692 3.606854 3.611839 3.633525 3.654077 3.619201 3.619201 3.628808
## [2713] 3.631174 3.604338 3.616762 3.601806 3.578260 3.586264 3.556102 3.556102
## [2721] 3.538645 3.507772 3.553244 3.567334 3.532655 3.588897 3.561757 3.541607
## [2729] 3.529626 3.547467 3.523501 3.494739 3.445584 3.430403 3.481280 3.474382
## [2737] 3.445584 3.422602 3.410622 3.445584 3.438062 3.381275 3.389869 3.381275
## [2745] 3.349757 3.363549 3.368050 3.349757 3.359001 3.372504 3.335508 3.315748
## [2753] 3.389869 3.335508 3.300315 3.256287 3.310664 3.300315 3.335508 3.273318
## [2761] 3.372504 3.402442 3.267714 3.238575 3.220128 3.166873 3.173884 3.200888
## [2769] 3.152520 3.226362 3.200888 3.187587 3.180788 3.187587 3.159753 3.166873
## [2777] 3.226362 3.114531 3.207393 3.207393 3.385593 3.305520 3.273318 3.363549
## [2785] 3.414653 3.385593 3.376912 3.460232 3.438062 3.345059 3.363549 3.289715
## [2793] 3.422602 3.345059 3.398292 3.467367 3.553244 3.547467 3.553244 3.564555
## [2801] 3.541607 3.558939 3.591512 3.572834 3.609354 3.594111 3.624034 3.591512
## [2809] 3.588897 3.645076 3.624034 3.628808 3.647346 3.642793 3.649602 3.662876
## [2817] 3.662876 3.656296 3.654077 3.656296 3.658501 3.649602 3.631174 3.642793
## [2825] 3.658501 3.662876 3.628808 3.654077 3.662876 3.671480 3.671480 3.660695
## [2833] 3.654077 3.660695 3.651846 3.649602 3.649602 3.651846 3.645076 3.667202
## [2841] 3.660695 3.675712 3.673602 3.667202 3.654077 3.640496 3.628808 3.604338
## [2849] 3.570093 3.586264 3.619201 3.588897 3.611839 3.619201 3.609354 3.606854
## [2857] 3.580946 3.567334 3.609354 3.635862 3.642793 3.626428 3.604338 3.624034
## [2865] 3.619201 3.626428 3.626428 3.609354 3.604338 3.578260 3.606854 3.583614
## [2873] 3.599257 3.596692 3.561757 3.591512 3.580946 3.558939 3.578260 3.572834
## [2881] 3.544548 3.556102 3.538645 3.501307 3.520403 3.445584 3.434250 3.426521
## [2889] 3.406552 3.426521 3.385593 3.385593 3.368050 3.273318 3.289715 3.267714
## [2897] 3.238575 3.284317 3.267714 3.250461 3.273318 3.250461 3.244558 3.295047
## [2905] 3.315748 3.207393 3.289715 3.273318 3.238575 3.284317 3.232510 3.305520
## [2913] 3.273318 3.256287 3.207393 3.220128 3.180788 3.152520 3.232510 3.137700
## [2921] 3.152520 3.200888 3.385593 3.278852 3.305520 3.349757 3.238575 3.284317
## [2929] 3.335508 3.452972 3.330651 3.385593 3.474382 3.494739 3.504552 3.514136
## [2937] 3.538645 3.553244 3.561757 3.578260 3.580946 3.588897 3.594111 3.601806
## [2945] 3.596692 3.619201 3.621625 3.647346 3.619201 3.635862 3.628808 3.635862
## [2953] 3.633525 3.642793 3.638186 3.638186 3.616762 3.633525 3.635862 3.642793
## [2961] 3.640496 3.640496 3.656296 3.651846 3.642793 3.628808 3.638186 3.645076
## [2969] 3.640496 3.631174 3.642793 3.654077 3.647346 3.656296 3.651846 3.642793
## [2977] 3.631174 3.638186 3.614308 3.626428 3.638186 3.626428 3.599257 3.609354
## [2985] 3.631174 3.626428 3.638186 3.631174 3.588897 3.591512 3.583614 3.583614
## [2993] 3.583614 3.599257 3.601806 3.583614 3.572834 3.561757 3.550366 3.517282
## [3001] 3.558939 3.588897 3.535661 3.550366 3.494739 3.523501 3.501307 3.463815
## [3009] 3.470890 3.456618 3.463815 3.494739 3.474382 3.460232 3.445584 3.398292
## [3017] 3.410622 3.441840 3.414653 3.434250 3.422602 3.422602 3.278852 3.320773
## [3025] 3.273318 3.315748 3.349757 3.295047 3.320773 3.368050 3.320773 3.402442
## [3033] 3.300315 3.389869 3.335508 3.300315 3.295047 3.262037 3.250461 3.273318
## [3041] 3.278852 3.226362 3.300315 3.349757 3.226362 3.273318 3.267714 3.284317
## [3049] 3.262037 3.278852 3.267714 3.213806 3.244558 3.187587 3.207393 3.194287
## [3057] 3.226362 3.232510 3.232510 3.207393 3.166873 3.226362 3.376912 3.180788
## [3065] 3.159753 3.137700 3.180788 3.194287 3.180788 3.180788 3.300315 3.359001
## [3073] 3.335508 3.244558 3.349757 3.207393 3.310664 3.330651 3.335508 3.372504
## [3081] 3.345059 3.394101 3.418646 3.368050 3.354404 3.381275 3.376912 3.463815
## [3089] 3.452972 3.474382 3.484686 3.507772 3.470890 3.422602 3.418646 3.481280
## [3097] 3.467367 3.488064 3.507772 3.510967 3.547467 3.553244 3.570093 3.583614
## [3105] 3.588897 3.572834 3.596692 3.572834 3.572834 3.580946 3.580946 3.611839
## [3113] 3.591512 3.609354 3.599257 3.583614 3.583614 3.614308 3.586264 3.616762
## [3121] 3.601806 3.614308 3.611839 3.614308 3.606854 3.621625 3.609354 3.631174
## [3129] 3.614308 3.567334 3.583614 3.606854 3.591512 3.586264 3.556102 3.572834
## [3137] 3.586264 3.547467 3.570093 3.558939 3.520403 3.547467 3.526575 3.541607
## [3145] 3.561757 3.575557 3.558939 3.567334 3.553244 3.514136 3.532655 3.572834
## [3153] 3.564555 3.535661 3.532655 3.535661 3.541607 3.514136 3.538645 3.517282
## [3161] 3.491415 3.484686 3.541607 3.463815 3.410622 3.449294 3.449294 3.449294
## [3169] 3.498036 3.467367 3.470890 3.460232 3.414653 3.422602 3.441840 3.426521
## [3177] 3.460232 3.445584 3.430403 3.414653 3.438062 3.406552 3.398292 3.418646
## [3185] 3.474382 3.426521 3.363549 3.376912 3.406552 3.335508 3.345059 3.300315
## [3193] 3.289715 3.300315 3.305520 3.220128 3.207393 3.238575 3.244558 3.256287
## [3201] 3.244558 3.238575 3.273318 3.273318 3.320773 3.152520 3.098411 3.122384
## [3209] 3.106541 3.180788 3.159753 3.166873 3.137700 3.090136 3.106541 3.278852
## [3217] 3.250461 3.220128 3.295047 3.262037 3.267714 3.289715 3.267714 3.200888
## [3225] 3.207393 3.273318 3.262037 3.372504 3.256287 3.244558 3.289715 3.363549
## [3233] 3.402442 3.381275 3.438062 3.456618 3.456618 3.526575 3.504552 3.526575
## [3241] 3.541607 3.544548 3.541607 3.550366 3.580946 3.604338 3.575557 3.567334
## [3249] 3.578260 3.583614 3.580946 3.588897 3.614308 3.621625 3.596692 3.601806
## [3257] 3.626428 3.614308 3.611839 3.642793 3.647346 3.638186 3.633525 3.606854
## [3265] 3.638186 3.635862 3.654077 3.660695 3.651846 3.633525 3.645076 3.638186
## [3273] 3.642793 3.656296 3.633525 3.621625 3.631174 3.638186 3.640496 3.635862
## [3281] 3.640496 3.616762 3.614308 3.626428 3.594111 3.606854 3.586264 3.594111
## [3289] 3.558939 3.614308 3.609354 3.611839 3.604338 3.564555 3.556102 3.580946
## [3297] 3.553244 3.567334 3.547467 3.532655 3.547467 3.556102 3.544548 3.553244
## [3305] 3.558939 3.538645 3.547467 3.517282 3.449294 3.501307 3.488064 3.507772
## [3313] 3.514136 3.523501 3.449294 3.381275 3.349757 3.372504 3.345059 3.335508
## [3321] 3.449294 3.368050 3.354404 3.381275 3.426521 3.410622 3.289715 3.267714
## [3329] 3.267714 3.289715 3.325740 3.363549 3.310664 3.310664 3.305520 3.300315
## [3337] 3.300315 3.295047 3.330651 3.262037 3.220128 3.122384 3.320773 3.114531
## [3345] 3.173884 3.194287 3.200888 3.220128 3.289715 3.315748 3.194287 3.213806
## [3353] 3.159753 3.200888 3.273318 3.213806 3.226362 3.122384 3.114531 3.220128
## [3361] 3.345059 3.335508 3.330651 3.430403 3.501307 3.491415 3.449294 3.558939
## [3369] 3.550366 3.547467 3.556102 3.575557 3.591512 3.604338 3.604338 3.621625
## [3377] 3.606854 3.633525 3.606854 3.631174 3.626428 3.654077 3.649602 3.642793
## [3385] 3.635862 3.656296 3.654077 3.662876 3.647346 3.645076 3.649602 3.651846
## [3393] 3.651846 3.658501 3.658501 3.654077 3.649602 3.651846 3.635862 3.638186
## [3401] 3.647346 3.662876 3.649602 3.651846 3.677811 3.675712 3.671480 3.651846
## [3409] 3.647346 3.647346 3.658501 3.662876 3.681974 3.662876 3.649602 3.654077
## [3417] 3.647346 3.642793 3.649602 3.649602 3.640496 3.645076 3.645076 3.619201
## [3425] 3.619201 3.635862 3.616762 3.626428 3.631174 3.604338 3.631174 3.621625
## [3433] 3.599257 3.606854 3.596692 3.614308 3.601806 3.633525 3.616762 3.614308
## [3441] 3.580946 3.606854 3.591512 3.626428 3.599257 3.601806 3.588897 3.606854
## [3449] 3.529626 3.561757 3.523501 3.588897 3.514136 3.532655 3.572834 3.488064
## [3457] 3.488064 3.467367 3.507772 3.481280 3.504552 3.422602 3.449294 3.372504
## [3465] 3.340310 3.354404 3.325740 3.340310 3.354404 3.330651 3.325740 3.406552
## [3473] 3.345059 3.273318 3.289715 3.305520 3.250461 3.220128 3.220128 3.256287
## [3481] 3.305520 3.289715 3.300315 3.345059 3.130106 3.064389 3.122384 3.027697
## [3489] 3.159753 3.081711 3.130106 3.073130 3.106541 3.180788 3.278852 3.289715
## [3497] 3.090136 3.090136 3.081711 3.130106 3.232510 3.267714 3.187587 3.232510
## [3505] 3.345059 3.434250 3.463815 3.426521 3.514136 3.484686 3.441840 3.434250
## [3513] 3.426521 3.474382 3.538645 3.556102 3.556102 3.583614 3.588897 3.611839
## [3521] 3.609354 3.601806 3.619201 3.628808 3.651846 3.658501 3.645076 3.645076
## [3529] 3.654077 3.651846 3.662876 3.667202 3.658501 3.651846 3.633525 3.651846
## [3537] 3.651846 3.658501 3.658501 3.651846 3.662876 3.660695 3.656296 3.660695
## [3545] 3.658501 3.665045 3.656296 3.671480 3.681974 3.662876 3.690169 3.660695
## [3553] 3.660695 3.662876 3.669347 3.658501 3.654077 3.675712 3.673602 3.662876
## [3561] 3.667202 3.665045 3.645076 3.624034 3.633525 3.621625 3.606854 3.594111
## [3569] 3.611839 3.594111 3.601806 3.611839 3.621625 3.614308 3.601806 3.599257
## [3577] 3.633525 3.611839 3.591512 3.611839 3.604338 3.601806 3.604338 3.619201
## [3585] 3.609354 3.596692 3.580946 3.586264 3.570093 3.586264 3.588897 3.564555
## [3593] 3.567334 3.529626 3.510967 3.481280 3.520403 3.526575 3.510967 3.510967
## attr(,"lambda")
## [1] -0.1506857lambda2 <- BoxCox.lambda(data_transformed)
lambda2## [1] -0.9999242Analyze ACF and PACF
par(mfrow = c(1, 2))
ACF <- acf(data_transformed, main = "ACF of Transformed Data", lag.max = 144)
ACF##
## Autocorrelations of series 'data_transformed', by lag
##
## 0.00000 0.00694 0.01389 0.02083 0.02778 0.03472 0.04167 0.04861 0.05556 0.06250
## 1.000 0.961 0.944 0.931 0.920 0.906 0.887 0.868 0.849 0.826
## 0.06944 0.07639 0.08333 0.09028 0.09722 0.10417 0.11111 0.11806 0.12500 0.13194
## 0.802 0.775 0.750 0.722 0.691 0.662 0.630 0.595 0.560 0.528
## 0.13889 0.14583 0.15278 0.15972 0.16667 0.17361 0.18056 0.18750 0.19444 0.20139
## 0.496 0.462 0.426 0.392 0.358 0.324 0.287 0.251 0.218 0.184
## 0.20833 0.21528 0.22222 0.22917 0.23611 0.24306 0.25000 0.25694 0.26389 0.27083
## 0.149 0.116 0.083 0.050 0.016 -0.015 -0.047 -0.079 -0.110 -0.140
## 0.27778 0.28472 0.29167 0.29861 0.30556 0.31250 0.31944 0.32639 0.33333 0.34028
## -0.168 -0.196 -0.223 -0.251 -0.277 -0.301 -0.327 -0.349 -0.372 -0.392
## 0.34722 0.35417 0.36111 0.36806 0.37500 0.38194 0.38889 0.39583 0.40278 0.40972
## -0.413 -0.433 -0.452 -0.468 -0.484 -0.500 -0.514 -0.526 -0.540 -0.553
## 0.41667 0.42361 0.43056 0.43750 0.44444 0.45139 0.45833 0.46528 0.47222 0.47917
## -0.564 -0.574 -0.582 -0.590 -0.597 -0.603 -0.609 -0.614 -0.618 -0.620
## 0.48611 0.49306 0.50000 0.50694 0.51389 0.52083 0.52778 0.53472 0.54167 0.54861
## -0.621 -0.622 -0.622 -0.622 -0.621 -0.619 -0.615 -0.612 -0.608 -0.603
## 0.55556 0.56250 0.56944 0.57639 0.58333 0.59028 0.59722 0.60417 0.61111 0.61806
## -0.598 -0.591 -0.583 -0.575 -0.565 -0.555 -0.543 -0.531 -0.516 -0.502
## 0.62500 0.63194 0.63889 0.64583 0.65278 0.65972 0.66667 0.67361 0.68056 0.68750
## -0.486 -0.471 -0.454 -0.435 -0.416 -0.396 -0.376 -0.354 -0.332 -0.309
## 0.69444 0.70139 0.70833 0.71528 0.72222 0.72917 0.73611 0.74306 0.75000 0.75694
## -0.285 -0.259 -0.232 -0.206 -0.180 -0.152 -0.124 -0.096 -0.067 -0.036
## 0.76389 0.77083 0.77778 0.78472 0.79167 0.79861 0.80556 0.81250 0.81944 0.82639
## -0.008 0.022 0.052 0.081 0.113 0.146 0.178 0.212 0.245 0.276
## 0.83333 0.84028 0.84722 0.85417 0.86111 0.86806 0.87500 0.88194 0.88889 0.89583
## 0.308 0.342 0.374 0.406 0.436 0.467 0.498 0.530 0.558 0.588
## 0.90278 0.90972 0.91667 0.92361 0.93056 0.93750 0.94444 0.95139 0.95833 0.96528
## 0.618 0.642 0.665 0.689 0.710 0.729 0.746 0.764 0.779 0.790
## 0.97222 0.97917 0.98611 0.99306 1.00000
## 0.800 0.807 0.816 0.819 0.821
par(mfrow = c(1, 2))
PACF <- pacf(data_transformed, main = "PACF of Transformed Data", lag.max = 144)
PACF##
## Partial autocorrelations of series 'data_transformed', by lag
##
## 0.00694 0.01389 0.02083 0.02778 0.03472 0.04167 0.04861 0.05556 0.06250 0.06944
## 0.961 0.266 0.141 0.075 0.002 -0.075 -0.069 -0.053 -0.087 -0.076
## 0.07639 0.08333 0.09028 0.09722 0.10417 0.11111 0.11806 0.12500 0.13194 0.13889
## -0.081 -0.040 -0.058 -0.079 -0.036 -0.056 -0.085 -0.045 -0.001 0.008
## 0.14583 0.15278 0.15972 0.16667 0.17361 0.18056 0.18750 0.19444 0.20139 0.20833
## -0.007 -0.038 -0.002 -0.001 -0.008 -0.050 -0.038 0.008 -0.022 -0.018
## 0.21528 0.22222 0.22917 0.23611 0.24306 0.25000 0.25694 0.26389 0.27083 0.27778
## 0.003 -0.013 -0.023 -0.035 -0.012 -0.027 -0.027 -0.034 -0.011 0.000
## 0.28472 0.29167 0.29861 0.30556 0.31250 0.31944 0.32639 0.33333 0.34028 0.34722
## -0.006 0.001 -0.028 -0.025 -0.017 -0.035 0.001 -0.017 -0.002 -0.007
## 0.35417 0.36111 0.36806 0.37500 0.38194 0.38889 0.39583 0.40278 0.40972 0.41667
## -0.022 -0.027 -0.002 -0.017 -0.017 -0.015 0.002 -0.040 -0.034 -0.029
## 0.42361 0.43056 0.43750 0.44444 0.45139 0.45833 0.46528 0.47222 0.47917 0.48611
## -0.021 -0.011 -0.014 -0.012 -0.033 -0.028 -0.015 -0.029 -0.002 -0.005
## 0.49306 0.50000 0.50694 0.51389 0.52083 0.52778 0.53472 0.54167 0.54861 0.55556
## -0.018 -0.023 -0.020 -0.024 -0.027 -0.011 -0.033 -0.026 -0.020 -0.015
## 0.56250 0.56944 0.57639 0.58333 0.59028 0.59722 0.60417 0.61111 0.61806 0.62500
## -0.018 0.000 -0.017 -0.012 -0.001 0.008 -0.009 0.019 0.007 0.007
## 0.63194 0.63889 0.64583 0.65278 0.65972 0.66667 0.67361 0.68056 0.68750 0.69444
## -0.019 0.002 0.013 0.002 0.006 0.007 0.006 -0.003 0.009 0.011
## 0.70139 0.70833 0.71528 0.72222 0.72917 0.73611 0.74306 0.75000 0.75694 0.76389
## 0.025 0.032 -0.004 -0.001 0.022 0.011 0.007 0.022 0.034 -0.013
## 0.77083 0.77778 0.78472 0.79167 0.79861 0.80556 0.81250 0.81944 0.82639 0.83333
## 0.029 0.012 -0.008 0.052 0.056 0.051 0.058 0.047 0.015 0.037
## 0.84028 0.84722 0.85417 0.86111 0.86806 0.87500 0.88194 0.88889 0.89583 0.90278
## 0.053 0.039 0.048 0.010 0.028 0.047 0.070 0.016 0.056 0.067
## 0.90972 0.91667 0.92361 0.93056 0.93750 0.94444 0.95139 0.95833 0.96528 0.97222
## -0.011 -0.005 0.033 -0.003 0.001 0.012 0.034 0.020 -0.021 0.013
## 0.97917 0.98611 0.99306 1.00000
## -0.022 0.035 -0.010 0.011
Based on the ACF and PACF plot, we get that the data isn’t stationary,
so a differencing must be conducted.
diff_data <- diff(data_transformed, differences = 1)Then, do the ADF test to check its stationarity, for stronger result, plot the ACF and PACF for the data that have been differenced.
adf.test(diff_data)## Warning in adf.test(diff_data): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: diff_data
## Dickey-Fuller = -10.448, Lag order = 15, p-value = 0.01
## alternative hypothesis: stationarypar(mfrow = c(1, 2))
ACF2 <- acf(diff_data , main = "ACF of Differenced Data", lag.max = 144)
PACF2 <- pacf(diff_data , main = "PACF of Differenced Data", lag.max = 144)
Then also do the differencing for seasonal.
diff_seasonal <- diff(diff_data, lag = 144)
data_fix <- na.omit(diff_seasonal)Plot again the ACF and PACF for the new data formed
par(mfrow = c(1, 2))
ACF3 <- acf(data_fix, main = "ACF of Seasonal Differenced Data", lag.max = 144)
PACF3 <- pacf(data_fix, main = "PACF of Seasonal Differenced Data", lag.max = 144)
## The Model
Find the best SARIMA model by trial and error method.
Model 1
sarima1 <- Arima(ts_train, order = c(3, 0, 3), seasonal = list(order = c(0, 0, 0), period = 144), include.drift = FALSE)
summary(sarima1)## Series: ts_train
## ARIMA(3,0,3) with non-zero mean
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 ma3 mean
## 2.0156 -1.0444 0.0269 -1.4112 0.3317 0.0998 143.4824
## s.e. 0.1253 0.2492 0.1241 0.1243 0.1794 0.0586 1.6211
##
## sigma^2 = 84.64: log likelihood = -13095.97
## AIC=26207.93 AICc=26207.97 BIC=26257.44
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.01404469 9.191027 6.89471 -0.7199718 5.791041 0.4187866
## ACF1
## Training set 7.405669e-05Model 2
sarima2 <- Arima(ts_train, order = c(3, 0, 3), seasonal = list(order = c(0, 0, 1), period = 144), include.drift = FALSE)
summary(sarima2)## Series: ts_train
## ARIMA(3,0,3)(0,0,1)[144] with non-zero mean
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 ma3 sma1 mean
## 1.9815 -0.9777 -0.0058 -1.3858 0.2879 0.1182 0.0785 142.7219
## s.e. 0.1279 0.2541 0.1265 0.1263 0.1828 0.0600 0.0174 1.7165
##
## sigma^2 = 84.18: log likelihood = -13086.1
## AIC=26190.2 AICc=26190.25 BIC=26245.9
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.05656364 9.164858 6.875816 -0.6702826 5.770544 0.4176389
## ACF1
## Training set 0.0005633429Model 3
sarima3 <- Arima(ts_train, order = c(3, 0, 3), seasonal = list(order = c(0, 1, 1), period = 144), include.drift = FALSE)
summary(sarima3)## Series: ts_train
## ARIMA(3,0,3)(0,1,1)[144]
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 ma3 sma1
## 0.9250 -0.2131 0.2470 -0.4039 0.1228 -0.1385 -1.0000
## s.e. 0.3407 0.3730 0.1659 0.3403 0.2326 0.0951 0.0218
##
## sigma^2 = 77.69: log likelihood = -12654.47
## AIC=25324.93 AICc=25324.98 BIC=25374.12
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.3020245 8.627579 6.355078 -0.3470499 5.273393 0.3860092
## ACF1
## Training set -0.0003707135Forecasting
forecast <- forecast(sarima3, h = 150)
forecast_mean <- forecast$mean
forecast_mean## Time Series:
## Start = c(26, 1)
## End = c(27, 6)
## Frequency = 144
## [1] 141.05163 137.97606 132.81421 134.01687 133.50162 130.84864 129.94063
## [8] 125.01863 121.98112 119.62839 119.40554 118.55193 120.10593 116.70745
## [15] 114.91658 111.29302 111.55647 107.42671 106.10355 105.98678 103.19618
## [22] 102.21156 105.03274 100.53953 99.05174 96.60921 98.01177 95.17925
## [29] 87.39150 80.12837 78.42970 84.29536 80.72520 81.07909 80.99691
## [36] 77.79852 80.48380 79.85264 78.98492 79.04053 75.09935 79.40130
## [43] 81.50626 80.33413 83.96483 86.03826 89.79433 99.75296 102.67405
## [50] 104.91754 104.20333 112.01136 116.82154 116.83382 121.68811 124.74435
## [57] 125.32247 129.74241 140.88411 143.18751 149.93255 156.43917 158.66732
## [64] 160.29695 162.04799 162.24041 166.07415 170.42917 172.26541 175.10284
## [71] 175.78141 176.78107 177.70179 180.90352 182.58623 184.94988 185.43442
## [78] 184.79983 185.28606 187.85309 187.90087 188.98938 188.99858 189.36845
## [85] 188.97894 190.35003 190.56169 190.81389 190.82660 191.03979 189.93344
## [92] 192.42752 193.24200 194.77685 193.55205 193.12758 192.22341 192.87950
## [99] 193.05585 190.95242 191.80919 191.02614 189.04324 190.62047 190.63780
## [106] 188.81522 186.99269 185.73020 183.46772 182.80524 179.86271 177.80013
## [113] 175.97747 174.67470 174.85180 176.46876 176.08553 173.22211 170.43847
## [120] 170.17457 169.87041 168.04594 166.14116 168.31602 165.09051 165.74460
## [127] 163.63827 163.93147 160.78420 160.67642 158.16809 158.13920 156.38970
## [134] 159.23959 156.08885 153.41733 150.58501 150.11226 148.63898 144.52355
## [141] 139.44643 138.13375 138.18250 134.36892 130.13755 127.70779 123.04403
## [148] 124.47131 124.21723 121.88101check the residual
residuals <- residuals(forecast)
kolmogorov_smirnov_test <- ks.test(residuals, "pnorm", mean(residuals), sd(residuals))
ljung_box_test <- Box.test(residuals, lag = 144, type = "Ljung-Box")kolmogorov_smirnov_test##
## Asymptotic one-sample Kolmogorov-Smirnov test
##
## data: residuals
## D = 0.045061, p-value = 8.951e-07
## alternative hypothesis: two-sidedljung_box_test##
## Box-Ljung test
##
## data: residuals
## X-squared = 192.39, df = 144, p-value = 0.004378Check the Accuracy of the Forecast
accuracy(forecast)## ME RMSE MAE MPE MAPE MASE
## Training set 0.3020245 8.627579 6.355078 -0.3470499 5.273393 0.3860092
## ACF1
## Training set -0.0003707135Also, we can plot the data using time series plot consist of the train set, test set, and the forecast.
df_forecast <- data.frame(
ts = seq(from = max(train_data$ts), by = "10 min", length.out = 150),
rate = forecast$mean,
set = "Forecast"
)
train_data$set <- "Train"
test_data$set <- "Test"
plot_data <- bind_rows(train_data, test_data, df_forecast)
ggplot(plot_data, aes(x = ts, y = rate, color = set)) +
geom_line() +
labs(x = "Date", y = "Rate", title = "Time Series Plot with SARIMA Modeling") +
theme_minimal() +
scale_x_datetime(labels = scales::date_format("%Y-%m-%d %H:%M")) +
scale_color_manual(values = c("Train" = "navy", "Test" = "green", "Forecast" = "red"))