Time Series Individual Assignment
veeya
2024-11-18
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# Set CRAN mirror
options(repos = c(CRAN = "https://cran.r-project.org"))install.packages("seastests")##
## The downloaded binary packages are in
## /var/folders/3m/4k67fkwj4514qw_8n59fky_m0000gn/T//Rtmp9B0Ol9/downloaded_packagesimport libraries
library(tseries)## Warning: package 'tseries' was built under R version 4.3.3## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(forecast)## Warning: package 'forecast' was built under R version 4.3.3library(lubridate)##
## Attaching package: 'lubridate'## The following objects are masked from 'package:base':
##
## date, intersect, setdiff, unionlibrary(ggplot2)
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(tidyr)
library(seastests)load the dataset
data = read.csv("data.csv")
head(data)## ts rate
## 1 2020-02-01T00:00:00Z 100
## 2 2020-02-01T00:10:00Z 97
## 3 2020-02-01T00:20:00Z 96
## 4 2020-02-01T00:30:00Z 100
## 5 2020-02-01T00:40:00Z 89
## 6 2020-02-01T00:50:00Z 91convert ts
data$ts <- as.POSIXct(data$ts, format = "%Y-%m-%dT%H:%M:%S", tz = "UTC")handle missing values
data <- data %>% drop_na()rate_ts
Rate_ts <- ts(data$rate, frequency = 144)seasonalitytest of rate_ts
seasonalitytest <- isSeasonal(Rate_ts)seasonalitytest## [1] TRUEit is true that the rate_ts is seasonal
define the point date
day_26_date <- ymd_hms("2020-02-26 00:00:00", tz = "UTC")split training and testing data
train_data <- data %>% filter(ts < day_26_date)
test_data <- data %>% filter(ts >= day_26_date)display head train and test
head(train_data)## ts rate
## 1 2020-02-01 00:00:00 100
## 2 2020-02-01 00:10:00 97
## 3 2020-02-01 00:20:00 96
## 4 2020-02-01 00:30:00 100
## 5 2020-02-01 00:40:00 89
## 6 2020-02-01 00:50:00 91head(test_data)## ts rate
## 1 2020-02-26 00:00:00 141
## 2 2020-02-26 00:10:00 128
## 3 2020-02-26 00:20:00 130
## 4 2020-02-26 00:30:00 134
## 5 2020-02-26 00:40:00 142
## 6 2020-02-26 00:50:00 142plot the training and testing data
# Plot the training data
plot(train_data, main="Training Data", col="blue", type="l", xlab="Time", ylab="Rate")
# Plot the testing data
plot(test_data, main="Test Data", col="blue", type="l", xlab="Time", ylab="Rate")
plot time series of rate
train_data$set <- "Train"
test_data$set <- "Test"
plot_data <- bind_rows(train_data, test_data)
ggplot(plot_data, aes(x = ts, y = rate, color = set)) +
geom_line() +
labs(x = "Date", y = "Rate", title = "Time Series Plot of Rate") +
theme_minimal() +
scale_x_datetime(labels = scales::date_format("%Y-%m-%d %H:%M")) +
scale_color_manual(values = c("Train" = "blue", "Test" = "green"))
Box-Cox Transformation
#1
lambda <- BoxCox.lambda(train_data$rate)
data_transformed <- BoxCox(train_data$rate, lambda)
print(lambda)## [1] 1.984576print(data_transformed)## [1] 4692.866 4417.563 4327.630 4692.866 3723.802 3891.733 3807.309
## [8] 3977.077 3559.546 3242.064 3723.802 3807.309 4692.866 2377.955
## [15] 4327.630 3723.802 3891.733 3478.797 3398.967 3478.797 3559.546
## [22] 3977.077 4600.181 3320.056 2793.435 3478.797 3891.733 4692.866
## [29] 2865.906 1646.608 2246.835 2793.435 1995.664 2377.955 2721.886
## [36] 2581.549 2311.934 1995.664 1330.837 1484.559 1759.264 2721.886
## [43] 2182.658 2512.763 2939.297 2651.257 3398.967 2793.435 4508.413
## [50] 4238.615 4976.423 3807.309 5467.349 3641.215 3807.309 4508.413
## [57] 5268.229 3559.546 3977.077 3891.733 5367.331 4976.423 4508.413
## [64] 4976.423 5981.174 6850.886 7899.259 7077.460 8388.955 10217.879
## [71] 8142.281 9281.067 9677.087 10081.313 11056.411 12225.466 11780.241
## [78] 11487.977 12831.843 13140.491 13140.491 13140.491 11343.211 10633.044
## [85] 11487.977 12225.466 13768.704 13452.779 15068.778 12526.834 13452.779
## [92] 13452.779 13452.779 15235.378 13452.779 15068.778 13610.287 15068.778
## [99] 17127.932 12375.695 13928.031 14249.413 13768.704 12225.466 14411.468
## [106] 13140.491 11056.411 15068.778 14738.305 12375.695 11780.241 13296.180
## [113] 10914.378 10217.879 9544.168 10773.255 11199.355 10773.255 9810.917
## [120] 9412.162 9544.168 8513.662 8265.162 10355.356 9150.885 10355.356
## [127] 8639.281 7899.259 8893.258 9810.917 9412.162 8765.813 7899.259
## [134] 9281.067 7779.118 7659.891 9412.162 8765.813 10493.744 9021.616
## [141] 6850.886 6517.883 8388.955 5981.174 7077.460 7077.460 5568.282
## [148] 5670.132 5568.282 7307.691 5170.044 6408.711 6517.883 5772.897
## [155] 5467.349 8142.281 5670.132 7899.259 6738.970 3891.733 3559.546
## [162] 4238.615 4508.413 4600.181 3977.077 3891.733 4327.630 5670.132
## [169] 4238.615 4150.517 6738.970 4786.468 3641.215 3088.841 3641.215
## [176] 3164.992 3013.609 2939.297 3242.064 3559.546 3723.802 3891.733
## [183] 3320.056 3977.077 2865.906 2651.257 2793.435 2444.898 2581.549
## [190] 3242.064 1875.617 2444.898 3723.802 3723.802 4327.630 5367.331
## [197] 4417.563 4600.181 4880.987 3559.546 3641.215 3641.215 7307.691
## [204] 5772.897 4692.866 6738.970 7077.460 4786.468 5670.132 5170.044
## [211] 6408.711 7077.460 8020.313 9021.616 9810.917 8388.955 10493.744
## [218] 10217.879 11343.211 12375.695 12985.712 11487.977 12985.712 11780.241
## [225] 12375.695 13140.491 13768.704 13768.704 14574.432 14411.468 14738.305
## [232] 13296.180 13928.031 13768.704 14411.468 14249.413 13452.779 15910.862
## [239] 14738.305 16254.055 17483.838 16775.657 15571.303 15571.303 16427.014
## [246] 16427.014 15402.886 16254.055 17663.152 16254.055 16775.657 16082.005
## [253] 15910.862 16951.340 17305.431 15910.862 15402.886 15235.378 14738.305
## [260] 14903.087 15402.886 15235.378 13928.031 14574.432 12985.712 11633.654
## [267] 12076.147 11633.654 10493.744 11199.355 11927.739 10355.356 9810.917
## [274] 9945.659 8765.813 9281.067 10217.879 12375.695 12375.695 11633.654
## [281] 10493.744 8765.813 8513.662 8388.955 9021.616 10355.356 8893.258
## [288] 8265.162 7779.118 8020.313 6627.969 7779.118 8639.281 8893.258
## [295] 6850.886 9412.162 7077.460 5981.174 5268.229 5876.578 7192.118
## [302] 6193.112 6627.969 6300.454 5568.282 5367.331 4880.987 5568.282
## [309] 4063.338 4508.413 3641.215 4150.517 4880.987 4880.987 4508.413
## [316] 4786.468 3641.215 3242.064 2581.549 4150.517 4327.630 3723.802
## [323] 3320.056 3559.546 3242.064 5268.229 4150.517 4417.563 3088.841
## [330] 3478.797 3723.802 2939.297 6193.112 4976.423 5568.282 9021.616
## [337] 5670.132 5981.174 5170.044 8388.955 7899.259 6408.711 6193.112
## [344] 8142.281 6963.716 8388.955 10914.378 12076.147 12831.843 15402.886
## [351] 15235.378 13928.031 15235.378 13452.779 15571.303 16254.055 17305.431
## [358] 17127.932 18573.340 18573.340 16600.882 17483.838 17483.838 18024.505
## [365] 17483.838 16775.657 16082.005 16951.340 17663.152 17305.431 17483.838
## [372] 16775.657 16082.005 17127.932 16254.055 16427.014 16427.014 16775.657
## [379] 16600.882 17483.838 18758.099 16600.882 17127.932 18024.505 17305.431
## [386] 18024.505 17663.152 17663.152 18758.099 18024.505 18024.505 17127.932
## [393] 17127.932 15571.303 16427.014 15068.778 13610.287 15235.378 15235.378
## [400] 13928.031 14903.087 15235.378 14903.087 15910.862 16254.055 15910.862
## [407] 14574.432 14738.305 15068.778 13296.180 12678.884 12225.466 12678.884
## [414] 13768.704 12985.712 13452.779 12225.466 11927.739 11780.241 12375.695
## [421] 11056.411 11056.411 12526.834 11780.241 12985.712 11056.411 10355.356
## [428] 10773.255 11056.411 11927.739 9021.616 8639.281 8020.313 8639.281
## [435] 6738.970 8142.281 6627.969 6850.886 6086.685 7541.577 5670.132
## [442] 5772.897 6300.454 5072.775 7077.460 8513.662 4976.423 6086.685
## [449] 7541.577 6300.454 5268.229 5072.775 5670.132 6850.886 5670.132
## [456] 3478.797 2793.435 2939.297 3641.215 2939.297 2581.549 2246.835
## [463] 2119.404 2793.435 2793.435 2939.297 3807.309 2444.898 2311.934
## [470] 2377.955 2444.898 2057.072 1816.979 2246.835 2377.955 2512.763
## [477] 2793.435 4880.987 3242.064 4238.615 4150.517 4327.630 6850.886
## [484] 5772.897 6627.969 7307.691 9412.162 9945.659 9810.917 10914.378
## [491] 11633.654 9412.162 13610.287 15910.862 14574.432 14411.468 16427.014
## [498] 16600.882 16427.014 17127.932 17483.838 17127.932 17127.932 17305.431
## [505] 16427.014 14903.087 15740.628 15910.862 15571.303 15571.303 16951.340
## [512] 18573.340 18573.340 17663.152 19130.341 18206.542 19885.707 19130.341
## [519] 19695.505 19130.341 18206.542 17843.375 16951.340 18758.099 19130.341
## [526] 17483.838 16600.882 18024.505 18389.487 18024.505 19885.707 18206.542
## [533] 18206.542 17843.375 18573.340 18573.340 18389.487 17305.431 16600.882
## [540] 15402.886 16254.055 16427.014 16082.005 17663.152 17127.932 16951.340
## [547] 15068.778 16427.014 17127.932 16951.340 17127.932 18024.505 16082.005
## [554] 15571.303 14574.432 14574.432 16082.005 15235.378 14088.268 13610.287
## [561] 17663.152 15571.303 14738.305 15571.303 15571.303 16951.340 14411.468
## [568] 15740.628 15068.778 12985.712 12985.712 13140.491 14574.432 13452.779
## [575] 12526.834 12985.712 12678.884 13140.491 10081.313 10633.044 11633.654
## [582] 10355.356 8388.955 7541.577 8142.281 9150.885 8142.281 7192.118
## [589] 8765.813 8388.955 7307.691 5670.132 6193.112 4976.423 4417.563
## [596] 4786.468 4508.413 3807.309 2939.297 2865.906 4150.517 5268.229
## [603] 3164.992 2182.658 2119.404 3164.992 2512.763 3013.609 3242.064
## [610] 3013.609 2182.658 2793.435 2939.297 2721.886 2057.072 4417.563
## [617] 2512.763 1702.474 2721.886 2057.072 3559.546 3559.546 3559.546
## [624] 4692.866 6408.711 5367.331 4880.987 6193.112 8388.955 6627.969
## [631] 8388.955 10355.356 9544.168 11056.411 9810.917 11927.739 15402.886
## [638] 16254.055 15235.378 16427.014 16082.005 14088.268 15910.862 15910.862
## [645] 16082.005 15068.778 16082.005 16775.657 16082.005 16082.005 16951.340
## [652] 16254.055 16951.340 15571.303 17127.932 17127.932 16600.882 17843.375
## [659] 16951.340 18573.340 18024.505 18024.505 16427.014 15910.862 16600.882
## [666] 18389.487 17663.152 17843.375 18573.340 19506.210 18943.766 18573.340
## [673] 17663.152 15402.886 14903.087 16254.055 16775.657 16254.055 16600.882
## [680] 17127.932 14738.305 15235.378 16600.882 17483.838 14903.087 13452.779
## [687] 12526.834 14574.432 12831.843 14088.268 12985.712 13452.779 13452.779
## [694] 13610.287 12225.466 11780.241 11343.211 9677.087 10914.378 11633.654
## [701] 10355.356 10081.313 9810.917 12076.147 9412.162 9810.917 9677.087
## [708] 12076.147 11056.411 11487.977 9677.087 10633.044 10217.879 10081.313
## [715] 9021.616 8765.813 6850.886 6627.969 6408.711 5670.132 5467.349
## [722] 5072.775 5367.331 7077.460 6627.969 4692.866 7077.460 3807.309
## [729] 4786.468 3723.802 3559.546 3807.309 4417.563 4976.423 3478.797
## [736] 3088.841 4063.338 3320.056 5170.044 2512.763 3242.064 3398.967
## [743] 3088.841 4150.517 2512.763 3164.992 2119.404 1816.979 3088.841
## [750] 2721.886 1232.986 2721.886 1381.152 1381.152 1646.608 1381.152
## [757] 2057.072 2246.835 2246.835 2377.955 2793.435 2512.763 1432.392
## [764] 1646.608 2182.658 2444.898 5170.044 3723.802 4150.517 7424.177
## [771] 4417.563 2939.297 5072.775 5670.132 6408.711 8388.955 6300.454
## [778] 8142.281 10217.879 9810.917 11927.739 12076.147 12831.843 13296.180
## [785] 12375.695 12678.884 13452.779 12526.834 13610.287 13452.779 13768.704
## [792] 14903.087 14903.087 17127.932 18024.505 16775.657 18758.099 19317.822
## [799] 18389.487 18573.340 19885.707 19506.210 18758.099 18206.542 17663.152
## [806] 18024.505 18943.766 19885.707 18943.766 18758.099 18573.340 18758.099
## [813] 17843.375 18024.505 19130.341 19506.210 18573.340 17305.431 16427.014
## [820] 16082.005 14903.087 16082.005 16254.055 18024.505 17843.375 16082.005
## [827] 16951.340 15402.886 17305.431 18389.487 16951.340 16600.882 15402.886
## [834] 14903.087 14088.268 16427.014 15910.862 14738.305 14574.432 13768.704
## [841] 12375.695 12678.884 12375.695 12985.712 13296.180 13610.287 12831.843
## [848] 13928.031 11780.241 11780.241 13296.180 13610.287 13610.287 13768.704
## [855] 12831.843 12225.466 11780.241 13296.180 11633.654 11056.411 9677.087
## [862] 9281.067 11487.977 10081.313 6738.970 6738.970 6850.886 7307.691
## [869] 7899.259 8388.955 8639.281 6086.685 5268.229 5367.331 5072.775
## [876] 6627.969 5467.349 4692.866 5467.349 5568.282 4692.866 6086.685
## [883] 5467.349 5467.349 5467.349 4508.413 8388.955 4880.987 3807.309
## [890] 3478.797 4880.987 4600.181 3242.064 3398.967 3723.802 4692.866
## [897] 3641.215 2939.297 2651.257 3723.802 3723.802 2721.886 2581.549
## [904] 2651.257 3723.802 4786.468 4238.615 3242.064 3164.992 4786.468
## [911] 5981.174 4238.615 4508.413 4508.413 3807.309 5072.775 6738.970
## [918] 8513.662 10493.744 9677.087 8513.662 8639.281 13140.491 12225.466
## [925] 13452.779 17483.838 16600.882 15235.378 14903.087 14411.468 15740.628
## [932] 15571.303 15402.886 16427.014 16082.005 17305.431 16427.014 15235.378
## [939] 16254.055 15910.862 16951.340 14903.087 16600.882 16600.882 17663.152
## [946] 15740.628 14249.413 17843.375 16600.882 17483.838 17305.431 17843.375
## [953] 16775.657 16427.014 17483.838 16951.340 16254.055 17305.431 18389.487
## [960] 17127.932 16951.340 16775.657 15571.303 14903.087 15910.862 15402.886
## [967] 13768.704 14411.468 15910.862 15740.628 13140.491 15740.628 14088.268
## [974] 13296.180 12678.884 14411.468 13140.491 12831.843 12831.843 12985.712
## [981] 11780.241 9945.659 9281.067 9021.616 11056.411 9945.659 11199.355
## [988] 9544.168 8020.313 8142.281 8893.258 10081.313 11487.977 10217.879
## [995] 9544.168 9412.162 9544.168 9412.162 7424.177 9412.162 6963.716
## [1002] 7424.177 8639.281 5568.282 6627.969 6408.711 7779.118 5367.331
## [1009] 5981.174 6627.969 5670.132 6193.112 8388.955 5467.349 5876.578
## [1016] 4880.987 4327.630 4692.866 5467.349 3977.077 5367.331 4063.338
## [1023] 4417.563 3641.215 6193.112 5670.132 4238.615 3164.992 4063.338
## [1030] 3641.215 3891.733 3807.309 3641.215 4238.615 5268.229 4976.423
## [1037] 3398.967 1995.664 1759.264 2939.297 3242.064 3164.992 2939.297
## [1044] 2057.072 2119.404 2119.404 2057.072 2246.835 2721.886 4417.563
## [1051] 4417.563 4327.630 1875.617 2246.835 3164.992 3891.733 5772.897
## [1058] 3641.215 2512.763 4786.468 3641.215 2865.906 2865.906 2651.257
## [1065] 2651.257 2865.906 4600.181 4976.423 5170.044 4417.563 6086.685
## [1072] 5170.044 4327.630 4786.468 4786.468 9021.616 8020.313 8893.258
## [1079] 9810.917 9544.168 11487.977 13140.491 12076.147 11487.977 14249.413
## [1086] 12985.712 12985.712 13768.704 10773.255 13768.704 13768.704 13928.031
## [1093] 13768.704 12985.712 15402.886 15571.303 14249.413 15402.886 15740.628
## [1100] 15571.303 15571.303 16951.340 16082.005 15402.886 16600.882 17663.152
## [1107] 18024.505 17843.375 16600.882 15740.628 14411.468 14903.087 14738.305
## [1114] 15068.778 15235.378 15068.778 13296.180 16427.014 14738.305 12225.466
## [1121] 11343.211 11343.211 10914.378 13610.287 14903.087 12831.843 15235.378
## [1128] 14088.268 14088.268 14574.432 13452.779 12076.147 14088.268 14249.413
## [1135] 14903.087 14088.268 13928.031 14574.432 14088.268 13928.031 13610.287
## [1142] 12985.712 12526.834 12076.147 12375.695 12526.834 12526.834 9544.168
## [1149] 9677.087 8513.662 7192.118 8388.955 7779.118 6738.970 6300.454
## [1156] 6086.685 7077.460 7899.259 8639.281 9412.162 7192.118 5981.174
## [1163] 5467.349 5981.174 5876.578 6408.711 8765.813 4786.468 5170.044
## [1170] 5772.897 4600.181 4880.987 4692.866 4508.413 3559.546 5072.775
## [1177] 3977.077 4880.987 7307.691 4150.517 5568.282 2939.297 2721.886
## [1184] 2793.435 3013.609 3088.841 3242.064 3088.841 3807.309 3164.992
## [1191] 2939.297 3977.077 3164.992 4600.181 2444.898 2444.898 2512.763
## [1198] 2581.549 3478.797 4600.181 2939.297 3723.802 3641.215 4238.615
## [1205] 2581.549 2793.435 3891.733 3977.077 7424.177 4600.181 6738.970
## [1212] 7192.118 7307.691 5568.282 7077.460 9021.616 8142.281 6963.716
## [1219] 7424.177 8388.955 8388.955 9281.067 10217.879 10773.255 10633.044
## [1226] 10355.356 12225.466 12985.712 12831.843 15740.628 14249.413 14249.413
## [1233] 12678.884 12678.884 15068.778 14738.305 14411.468 16254.055 14903.087
## [1240] 13768.704 12678.884 13768.704 13452.779 14903.087 15402.886 14738.305
## [1247] 14738.305 15910.862 16082.005 16951.340 16951.340 17305.431 16082.005
## [1254] 14903.087 15235.378 15740.628 15910.862 15235.378 15571.303 16254.055
## [1261] 15910.862 16082.005 15235.378 14411.468 15740.628 15235.378 15910.862
## [1268] 16427.014 15068.778 12678.884 13296.180 14411.468 13768.704 14574.432
## [1275] 12678.884 12831.843 11780.241 11633.654 13768.704 13296.180 11927.739
## [1282] 11056.411 10773.255 9544.168 12678.884 11487.977 12678.884 11780.241
## [1289] 12678.884 12831.843 9677.087 11780.241 9412.162 6738.970 6627.969
## [1296] 6738.970 5772.897 5170.044 5072.775 4327.630 6300.454 7307.691
## [1303] 7659.891 4692.866 4327.630 4508.413 7541.577 5072.775 5170.044
## [1310] 4880.987 4692.866 4786.468 5876.578 6193.112 4600.181 5467.349
## [1317] 5170.044 4508.413 4238.615 3977.077 7541.577 4238.615 3164.992
## [1324] 4417.563 3559.546 3088.841 2377.955 2512.763 2865.906 2444.898
## [1331] 2512.763 2119.404 1995.664 2939.297 2444.898 1759.264 1816.979
## [1338] 2377.955 3977.077 3164.992 3398.967 5568.282 5170.044 4238.615
## [1345] 5876.578 8893.258 8388.955 5772.897 6408.711 7307.691 7077.460
## [1352] 8513.662 10773.255 9945.659 11056.411 11199.355 11927.739 14574.432
## [1359] 13452.779 12985.712 14574.432 15068.778 15571.303 16082.005 18943.766
## [1366] 19130.341 18389.487 18573.340 17663.152 17843.375 17127.932 16600.882
## [1373] 15740.628 15910.862 15910.862 17127.932 15910.862 18389.487 18389.487
## [1380] 18024.505 18024.505 17843.375 18758.099 18024.505 18206.542 16775.657
## [1387] 17127.932 17127.932 17483.838 17483.838 17305.431 16775.657 15740.628
## [1394] 17843.375 18573.340 17483.838 17127.932 17843.375 16775.657 16254.055
## [1401] 16427.014 15068.778 14574.432 14411.468 15571.303 14249.413 14088.268
## [1408] 14088.268 12831.843 12678.884 13768.704 13610.287 15068.778 13610.287
## [1415] 13296.180 12375.695 14411.468 12831.843 13140.491 13296.180 12831.843
## [1422] 14249.413 13140.491 13296.180 12076.147 12375.695 12831.843 12076.147
## [1429] 11343.211 11633.654 12526.834 10493.744 9281.067 9677.087 9677.087
## [1436] 10493.744 9021.616 7424.177 7192.118 7192.118 7192.118 6627.969
## [1443] 6517.883 7192.118 8142.281 7424.177 6300.454 6300.454 5568.282
## [1450] 4692.866 6738.970 5670.132 4508.413 4600.181 3013.609 2939.297
## [1457] 3088.841 3723.802 2865.906 4327.630 2939.297 2939.297 3723.802
## [1464] 3398.967 2311.934 2311.934 2119.404 1816.979 2311.934 1935.179
## [1471] 1646.608 1702.474 1432.392 3559.546 1537.650 2057.072 1759.264
## [1478] 2057.072 1185.450 1702.474 1281.448 2377.955 2939.297 5467.349
## [1485] 3320.056 1995.664 2939.297 5981.174 6300.454 5170.044 4880.987
## [1492] 5268.229 5670.132 6193.112 8765.813 5670.132 9544.168 8388.955
## [1499] 11343.211 11343.211 12985.712 15402.886 13768.704 14574.432 15571.303
## [1506] 15910.862 17305.431 18573.340 19130.341 18943.766 18206.542 19317.822
## [1513] 18573.340 18389.487 19506.210 18943.766 18758.099 18389.487 17843.375
## [1520] 17305.431 18389.487 17843.375 17127.932 18943.766 18206.542 19130.341
## [1527] 16951.340 16775.657 17483.838 18024.505 18206.542 18024.505 18206.542
## [1534] 16775.657 17663.152 18024.505 16600.882 17127.932 16951.340 16775.657
## [1541] 16951.340 17663.152 16254.055 17483.838 15571.303 18758.099 18024.505
## [1548] 15740.628 16951.340 16254.055 15910.862 14088.268 15571.303 15910.862
## [1555] 16082.005 16427.014 15910.862 16254.055 15910.862 14903.087 14738.305
## [1562] 15571.303 14574.432 13140.491 12375.695 15235.378 13610.287 14249.413
## [1569] 14574.432 15068.778 14088.268 12678.884 10355.356 11780.241 11633.654
## [1576] 11927.739 11633.654 11487.977 11056.411 10217.879 10914.378 9150.885
## [1583] 9150.885 8893.258 8388.955 9544.168 6086.685 7307.691 7077.460
## [1590] 6850.886 7307.691 5772.897 5568.282 4786.468 4327.630 4327.630
## [1597] 4508.413 4786.468 4786.468 7541.577 5876.578 4417.563 6300.454
## [1604] 5170.044 4508.413 5981.174 8020.313 4327.630 3088.841 2057.072
## [1611] 2793.435 4150.517 1995.664 2793.435 1232.986 2512.763 1432.392
## [1618] 1591.667 2057.072 1935.179 1875.617 1816.979 1484.559 1816.979
## [1625] 1138.841 2581.549 2444.898 2444.898 2512.763 2793.435 2721.886
## [1632] 7779.118 5876.578 3977.077 5981.174 5981.174 7659.891 8388.955
## [1639] 8388.955 9150.885 7077.460 10081.313 11343.211 12831.843 14088.268
## [1646] 15068.778 14249.413 16254.055 16600.882 15910.862 15910.862 15571.303
## [1653] 15740.628 13768.704 13610.287 13610.287 13140.491 13928.031 13768.704
## [1660] 13928.031 15235.378 15740.628 17127.932 16600.882 16951.340 16600.882
## [1667] 17127.932 17305.431 15235.378 14574.432 15740.628 15402.886 15402.886
## [1674] 16775.657 16600.882 17127.932 17483.838 16600.882 16951.340 17663.152
## [1681] 16600.882 14574.432 16775.657 14574.432 16951.340 15068.778 16254.055
## [1688] 16600.882 15235.378 16427.014 16951.340 16951.340 16427.014 16600.882
## [1695] 15740.628 14088.268 14738.305 15068.778 16082.005 15068.778 14249.413
## [1702] 15068.778 15235.378 16600.882 13610.287 15740.628 14088.268 15068.778
## [1709] 12526.834 14088.268 12831.843 11199.355 10355.356 11343.211 12526.834
## [1716] 13140.491 11927.739 12375.695 11780.241 11487.977 9677.087 9021.616
## [1723] 11199.355 9945.659 9945.659 10081.313 11056.411 10773.255 8142.281
## [1730] 10217.879 8513.662 8265.162 7307.691 7077.460 6850.886 6963.716
## [1737] 8513.662 8765.813 6627.969 6627.969 6193.112 6963.716 4786.468
## [1744] 6408.711 5876.578 4508.413 5981.174 5467.349 5876.578 3977.077
## [1751] 3641.215 4417.563 6408.711 4692.866 2377.955 2651.257 2793.435
## [1758] 1702.474 1702.474 2311.934 2119.404 2246.835 1702.474 1232.986
## [1765] 1759.264 2246.835 3013.609 1935.179 4600.181 3478.797 2865.906
## [1772] 2246.835 3977.077 4692.866 5568.282 4150.517 3164.992 3559.546
## [1779] 3088.841 4600.181 6193.112 6517.883 8893.258 5876.578 5170.044
## [1786] 7307.691 10355.356 10217.879 12526.834 13296.180 15402.886 15571.303
## [1793] 14574.432 16600.882 16254.055 16600.882 16427.014 17483.838 16082.005
## [1800] 15402.886 16600.882 16951.340 17663.152 18573.340 15910.862 17127.932
## [1807] 18024.505 18758.099 18389.487 18573.340 18206.542 18389.487 17483.838
## [1814] 17127.932 17483.838 17663.152 18206.542 16775.657 16951.340 17663.152
## [1821] 18206.542 17483.838 18758.099 16951.340 17483.838 18206.542 17305.431
## [1828] 17127.932 18024.505 18024.505 17843.375 17127.932 17305.431 16600.882
## [1835] 15068.778 15402.886 15571.303 16082.005 14903.087 14574.432 14249.413
## [1842] 14249.413 14738.305 15068.778 13296.180 14411.468 14249.413 13296.180
## [1849] 13140.491 13928.031 14411.468 14249.413 15235.378 14249.413 13140.491
## [1856] 13140.491 12375.695 13452.779 11927.739 10633.044 12831.843 11927.739
## [1863] 12678.884 12375.695 13610.287 14903.087 12985.712 12375.695 13768.704
## [1870] 12375.695 13452.779 12985.712 11780.241 10914.378 12375.695 11780.241
## [1877] 10633.044 9281.067 9412.162 9281.067 7779.118 7424.177 8765.813
## [1884] 8388.955 8020.313 9021.616 8639.281 7424.177 5367.331 5772.897
## [1891] 5568.282 5467.349 5670.132 5772.897 6738.970 6517.883 5367.331
## [1898] 5772.897 7899.259 8020.313 7307.691 3320.056 3723.802 5170.044
## [1905] 4150.517 2865.906 2651.257 2581.549 2581.549 2246.835 2581.549
## [1912] 2057.072 2057.072 2182.658 2182.658 2581.549 4063.338 3013.609
## [1919] 2444.898 4150.517 7077.460 7077.460 5467.349 6963.716 8265.162
## [1926] 10217.879 8265.162 8142.281 7779.118 9544.168 11199.355 11199.355
## [1933] 13140.491 12526.834 15910.862 15910.862 14249.413 15740.628 16082.005
## [1940] 15402.886 15068.778 17127.932 17483.838 16600.882 17843.375 18758.099
## [1947] 18758.099 18024.505 17483.838 17483.838 15571.303 16951.340 17483.838
## [1954] 19130.341 18206.542 16951.340 18206.542 18024.505 20268.832 18389.487
## [1961] 17843.375 17843.375 17843.375 16951.340 16082.005 16082.005 15402.886
## [1968] 16254.055 17663.152 17663.152 17305.431 16254.055 16775.657 16951.340
## [1975] 17483.838 15740.628 16254.055 16254.055 16254.055 12678.884 13452.779
## [1982] 10355.356 13296.180 10355.356 12526.834 10081.313 11633.654 9677.087
## [1989] 10355.356 9945.659 8020.313 8142.281 10081.313 7659.891 7192.118
## [1996] 9150.885 7541.577 8142.281 8639.281 7899.259 6738.970 6517.883
## [2003] 6086.685 6627.969 6517.883 7307.691 5981.174 5670.132 5876.578
## [2010] 6963.716 6963.716 6517.883 4238.615 4238.615 5568.282 4692.866
## [2017] 4417.563 4150.517 5268.229 7659.891 4508.413 6517.883 6086.685
## [2024] 4063.338 5268.229 4417.563 4508.413 4508.413 4150.517 3977.077
## [2031] 5467.349 6738.970 3641.215 3320.056 3398.967 3478.797 3559.546
## [2038] 3891.733 3977.077 4417.563 3242.064 3398.967 5772.897 3478.797
## [2045] 3013.609 1875.617 2512.763 3478.797 3320.056 4150.517 7541.577
## [2052] 3242.064 5467.349 2377.955 3478.797 4327.630 2246.835 2246.835
## [2059] 1935.179 2721.886 1646.608 1330.837 2182.658 2865.906 2651.257
## [2066] 2939.297 3088.841 4238.615 3242.064 3398.967 3320.056 6408.711
## [2073] 4786.468 4692.866 4976.423 4880.987 4600.181 4786.468 5170.044
## [2080] 6738.970 8142.281 7779.118 5981.174 8142.281 7307.691 7192.118
## [2087] 7899.259 7779.118 9021.616 9677.087 9021.616 10914.378 11199.355
## [2094] 9150.885 8639.281 11343.211 9150.885 9810.917 9945.659 11487.977
## [2101] 10217.879 10914.378 11633.654 13296.180 13768.704 12985.712 12831.843
## [2108] 13452.779 12831.843 13768.704 12678.884 12831.843 13140.491 14738.305
## [2115] 15740.628 14411.468 12985.712 14088.268 14249.413 14903.087 14411.468
## [2122] 13296.180 12678.884 11927.739 12375.695 12375.695 12526.834 12678.884
## [2129] 11343.211 11343.211 10773.255 9945.659 11343.211 9544.168 8765.813
## [2136] 11056.411 11633.654 10773.255 8893.258 9021.616 9810.917 9544.168
## [2143] 9677.087 8388.955 9810.917 10355.356 7192.118 7899.259 7307.691
## [2150] 7779.118 7077.460 7077.460 6963.716 6408.711 7659.891 6300.454
## [2157] 6517.883 7424.177 6408.711 7659.891 5670.132 5467.349 3891.733
## [2164] 3807.309 4327.630 6408.711 5670.132 5072.775 5367.331 5981.174
## [2171] 5981.174 7541.577 5670.132 5367.331 6193.112 5568.282 4508.413
## [2178] 3977.077 4600.181 5876.578 5072.775 5072.775 4786.468 4880.987
## [2185] 4786.468 3977.077 3807.309 5072.775 3891.733 4692.866 4692.866
## [2192] 3723.802 2581.549 2182.658 2512.763 3013.609 2793.435 1875.617
## [2199] 2581.549 1935.179 1935.179 1702.474 1591.667 1816.979 2581.549
## [2206] 2311.934 2444.898 3807.309 2721.886 2793.435 3013.609 5467.349
## [2213] 2865.906 2444.898 3977.077 6408.711 5170.044 6850.886 4692.866
## [2220] 6963.716 6086.685 6408.711 7541.577 6738.970 7307.691 6086.685
## [2227] 6627.969 6408.711 6738.970 7899.259 10633.044 10493.744 10773.255
## [2234] 10217.879 11056.411 14411.468 12678.884 12985.712 13296.180 13610.287
## [2241] 13768.704 14249.413 13610.287 14249.413 14249.413 15402.886 15402.886
## [2248] 15910.862 15740.628 16427.014 16427.014 16254.055 14903.087 16600.882
## [2255] 16775.657 17663.152 18024.505 16951.340 17305.431 15910.862 15571.303
## [2262] 16775.657 15740.628 16951.340 17127.932 17663.152 15740.628 15910.862
## [2269] 16082.005 15235.378 14738.305 15068.778 16082.005 15910.862 15571.303
## [2276] 15910.862 14574.432 16427.014 15571.303 14088.268 14249.413 15235.378
## [2283] 15402.886 16254.055 16600.882 15402.886 15910.862 15402.886 14574.432
## [2290] 14411.468 15235.378 14738.305 13296.180 13768.704 11487.977 12985.712
## [2297] 11199.355 11056.411 9810.917 9021.616 10493.744 10355.356 9412.162
## [2304] 7659.891 9412.162 7659.891 8265.162 8513.662 7899.259 7541.577
## [2311] 7659.891 7899.259 8020.313 7307.691 7899.259 7659.891 9281.067
## [2318] 7077.460 8265.162 5467.349 5876.578 4692.866 4786.468 4508.413
## [2325] 3977.077 4600.181 5268.229 4880.987 3977.077 3641.215 3088.841
## [2332] 2865.906 2721.886 2512.763 1995.664 2651.257 2377.955 3242.064
## [2339] 2311.934 1702.474 2182.658 2119.404 2512.763 2377.955 2311.934
## [2346] 2444.898 3977.077 3088.841 2246.835 4976.423 4238.615 4600.181
## [2353] 5467.349 5670.132 8142.281 7541.577 7077.460 9412.162 10773.255
## [2360] 10493.744 10081.313 10493.744 13610.287 11927.739 13452.779 15068.778
## [2367] 15402.886 14574.432 16600.882 16775.657 17305.431 18024.505 17483.838
## [2374] 17843.375 16775.657 18024.505 18573.340 19506.210 19130.341 20461.755
## [2381] 20850.319 19130.341 19130.341 17663.152 20461.755 19885.707 18024.505
## [2388] 17483.838 18573.340 20461.755 20076.816 19317.822 19317.822 19130.341
## [2395] 18206.542 18573.340 18758.099 18758.099 18943.766 18758.099 18573.340
## [2402] 17843.375 16775.657 18024.505 18573.340 17305.431 17483.838 17663.152
## [2409] 18024.505 18206.542 17483.838 17305.431 15910.862 17483.838 15571.303
## [2416] 15910.862 15910.862 13928.031 13296.180 14903.087 16082.005 14411.468
## [2423] 14738.305 16254.055 14411.468 13768.704 13768.704 14088.268 15068.778
## [2430] 14411.468 13140.491 12678.884 12375.695 12526.834 11927.739 9544.168
## [2437] 10914.378 13140.491 14249.413 11343.211 10914.378 11343.211 11633.654
## [2444] 12985.712 11199.355 11780.241 12225.466 11056.411 10633.044 9544.168
## [2451] 9281.067 9412.162 9021.616 7424.177 10773.255 8388.955 7541.577
## [2458] 7307.691 6738.970 5670.132 8265.162 6627.969 5772.897 5876.578
## [2465] 5772.897 6963.716 6408.711 6300.454 5876.578 5981.174 8388.955
## [2472] 4508.413 4786.468 4508.413 3891.733 2939.297 2939.297 2377.955
## [2479] 1646.608 2377.955 2444.898 2246.835 2581.549 2444.898 2246.835
## [2486] 4238.615 3478.797 3559.546 1995.664 2721.886 2939.297 2377.955
## [2493] 2119.404 2651.257 3242.064 3723.802 4063.338 4600.181 4786.468
## [2500] 7541.577 9544.168 8765.813 9945.659 10493.744 9150.885 11056.411
## [2507] 11780.241 10773.255 13296.180 17305.431 17843.375 15402.886 17663.152
## [2514] 17305.431 18389.487 17843.375 18758.099 18389.487 18573.340 17843.375
## [2521] 18024.505 18758.099 18758.099 18389.487 19130.341 19130.341 18573.340
## [2528] 19130.341 18758.099 17483.838 19317.822 18758.099 18943.766 19695.505
## [2535] 19506.210 19695.505 20076.816 19317.822 19506.210 20268.832 20850.319
## [2542] 21837.594 20655.584 19506.210 17843.375 18389.487 18206.542 20076.816
## [2549] 18943.766 17843.375 19130.341 19317.822 19506.210 18758.099 18389.487
## [2556] 18758.099 18024.505 17843.375 17483.838 16775.657 16254.055 17663.152
## [2563] 17843.375 16082.005 14738.305 16254.055 15235.378 14249.413 15235.378
## [2570] 13768.704 15571.303 16600.882 15740.628 13768.704 15910.862 16600.882
## [2577] 15571.303 15910.862 14738.305 13610.287 13768.704 14088.268 12831.843
## [2584] 11199.355 10633.044 11199.355 12678.884 10355.356 9810.917 10081.313
## [2591] 10355.356 11056.411 10355.356 6086.685 6517.883 6193.112 6300.454
## [2598] 5367.331 5670.132 5467.349 4880.987 4786.468 5367.331 6086.685
## [2605] 4786.468 4880.987 7307.691 4976.423 7307.691 5072.775 4692.866
## [2612] 4692.866 4327.630 4417.563 5268.229 4150.517 4327.630 3013.609
## [2619] 3242.064 6517.883 3478.797 3398.967 3641.215 4327.630 3320.056
## [2626] 2939.297 2939.297 2865.906 3891.733 3807.309 3164.992 3723.802
## [2633] 4327.630 4150.517 3641.215 4508.413 4238.615 3641.215 7779.118
## [2640] 7659.891 4508.413 6086.685 4063.338 9021.616 6738.970 6963.716
## [2647] 8020.313 7899.259 8893.258 10355.356 11487.977 12678.884 13610.287
## [2654] 15910.862 15740.628 14903.087 14738.305 14738.305 14903.087 16600.882
## [2661] 17843.375 18024.505 18024.505 17483.838 17305.431 18206.542 18024.505
## [2668] 18758.099 16427.014 18024.505 17663.152 17843.375 19506.210 19506.210
## [2675] 19506.210 19130.341 18573.340 18024.505 18206.542 18389.487 17483.838
## [2682] 18024.505 17127.932 17305.431 18024.505 19506.210 18024.505 17305.431
## [2689] 17843.375 17663.152 16600.882 17305.431 17483.838 17843.375 16427.014
## [2696] 17305.431 17127.932 17127.932 18024.505 17127.932 16082.005 16600.882
## [2703] 16254.055 17843.375 14738.305 15402.886 15740.628 17305.431 18943.766
## [2710] 16254.055 16254.055 16951.340 17127.932 15235.378 16082.005 15068.778
## [2717] 13610.287 14088.268 12375.695 12375.695 11487.977 10081.313 12225.466
## [2724] 12985.712 11199.355 14249.413 12678.884 11633.654 11056.411 11927.739
## [2731] 10773.255 9544.168 7779.118 7307.691 9021.616 8765.813 7779.118
## [2738] 7077.460 6738.970 7779.118 7541.577 5981.174 6193.112 5981.174
## [2745] 5268.229 5568.282 5670.132 5268.229 5467.349 5772.897 4976.423
## [2752] 4600.181 6193.112 4976.423 4327.630 3641.215 4508.413 4327.630
## [2759] 4976.423 3891.733 5772.897 6517.883 3807.309 3398.967 3164.992
## [2766] 2581.549 2651.257 2939.297 2444.898 3242.064 2939.297 2793.435
## [2773] 2721.886 2793.435 2512.763 2581.549 3242.064 2119.404 3013.609
## [2780] 3013.609 6086.685 4417.563 3891.733 5568.282 6850.886 6086.685
## [2787] 5876.578 8265.162 7541.577 5170.044 5568.282 4150.517 7077.460
## [2794] 5170.044 6408.711 8513.662 12225.466 11927.739 12225.466 12831.843
## [2801] 11633.654 12526.834 14411.468 13296.180 15571.303 14574.432 16600.882
## [2808] 14411.468 14249.413 18206.542 16600.882 16951.340 18389.487 18024.505
## [2815] 18573.340 19695.505 19695.505 19130.341 18943.766 19130.341 19317.822
## [2822] 18573.340 17127.932 18024.505 19317.822 19695.505 16951.340 18943.766
## [2829] 19695.505 20461.755 20461.755 19506.210 18943.766 19506.210 18758.099
## [2836] 18573.340 18573.340 18758.099 18206.542 20076.816 19506.210 20850.319
## [2843] 20655.584 20076.816 18943.766 17843.375 16951.340 15235.378 13140.491
## [2850] 14088.268 16254.055 14249.413 15740.628 16254.055 15571.303 15402.886
## [2857] 13768.704 12985.712 15571.303 17483.838 18024.505 16775.657 15235.378
## [2864] 16600.882 16254.055 16775.657 16775.657 15571.303 15235.378 13610.287
## [2871] 15402.886 13928.031 14903.087 14738.305 12678.884 14411.468 13768.704
## [2878] 12526.834 13610.287 13296.180 11780.241 12375.695 11487.977 9810.917
## [2885] 10633.044 7779.118 7424.177 7192.118 6627.969 7192.118 6086.685
## [2892] 6086.685 5670.132 3891.733 4150.517 3807.309 3398.967 4063.338
## [2899] 3807.309 3559.546 3891.733 3559.546 3478.797 4238.615 4600.181
## [2906] 3013.609 4150.517 3891.733 3398.967 4063.338 3320.056 4417.563
## [2913] 3891.733 3641.215 3013.609 3164.992 2721.886 2444.898 3320.056
## [2920] 2311.934 2444.898 2939.297 6086.685 3977.077 4417.563 5268.229
## [2927] 3398.967 4063.338 4976.423 8020.313 4880.987 6086.685 8765.813
## [2934] 9544.168 9945.659 10355.356 11487.977 12225.466 12678.884 13610.287
## [2941] 13768.704 14249.413 14574.432 15068.778 14738.305 16254.055 16427.014
## [2948] 18389.487 16254.055 17483.838 16951.340 17483.838 17305.431 18024.505
## [2955] 17663.152 17663.152 16082.005 17305.431 17483.838 18024.505 17843.375
## [2962] 17843.375 19130.341 18758.099 18024.505 16951.340 17663.152 18206.542
## [2969] 17843.375 17127.932 18024.505 18943.766 18389.487 19130.341 18758.099
## [2976] 18024.505 17127.932 17663.152 15910.862 16775.657 17663.152 16775.657
## [2983] 14903.087 15571.303 17127.932 16775.657 17663.152 17127.932 14249.413
## [2990] 14411.468 13928.031 13928.031 13928.031 14903.087 15068.778 13928.031
## [2997] 13296.180 12678.884 12076.147 10493.744 12526.834 14249.413 11343.211
## [3004] 12076.147 9544.168 10773.255 9810.917 8388.955 8639.281 8142.281
## [3011] 8388.955 9544.168 8765.813 8265.162 7779.118 6408.711 6738.970
## [3018] 7659.891 6850.886 7424.177 7077.460 7077.460 3977.077 4692.866
## [3025] 3891.733 4600.181 5268.229 4238.615 4692.866 5670.132 4692.866
## [3032] 6517.883 4327.630 6193.112 4976.423 4327.630 4238.615 3723.802
## [3039] 3559.546 3891.733 3977.077 3242.064 4327.630 5268.229 3242.064
## [3046] 3891.733 3807.309 4063.338 3723.802 3977.077 3807.309 3088.841
## [3053] 3478.797 2793.435 3013.609 2865.906 3242.064 3320.056 3320.056
## [3060] 3013.609 2581.549 3242.064 5876.578 2721.886 2512.763 2311.934
## [3067] 2721.886 2865.906 2721.886 2721.886 4327.630 5467.349 4976.423
## [3074] 3478.797 5268.229 3013.609 4508.413 4880.987 4976.423 5772.897
## [3081] 5170.044 6300.454 6963.716 5670.132 5367.331 5981.174 5876.578
## [3088] 8388.955 8020.313 8765.813 9150.885 10081.313 8639.281 7077.460
## [3095] 6963.716 9021.616 8513.662 9281.067 10081.313 10217.879 11927.739
## [3102] 12225.466 13140.491 13928.031 14249.413 13296.180 14738.305 13296.180
## [3109] 13296.180 13768.704 13768.704 15740.628 14411.468 15571.303 14903.087
## [3116] 13928.031 13928.031 15910.862 14088.268 16082.005 15068.778 15910.862
## [3123] 15740.628 15910.862 15402.886 16427.014 15571.303 17127.932 15910.862
## [3130] 12985.712 13928.031 15402.886 14411.468 14088.268 12375.695 13296.180
## [3137] 14088.268 11927.739 13140.491 12526.834 10633.044 11927.739 10914.378
## [3144] 11633.654 12678.884 13452.779 12526.834 12985.712 12225.466 10355.356
## [3151] 11199.355 13296.180 12831.843 11343.211 11199.355 11343.211 11633.654
## [3158] 10355.356 11487.977 10493.744 9412.162 9150.885 11633.654 8388.955
## [3165] 6738.970 7899.259 7899.259 7899.259 9677.087 8513.662 8639.281
## [3172] 8265.162 6850.886 7077.460 7659.891 7192.118 8265.162 7779.118
## [3179] 7307.691 6850.886 7541.577 6627.969 6408.711 6963.716 8765.813
## [3186] 7192.118 5568.282 5876.578 6627.969 4976.423 5170.044 4327.630
## [3193] 4150.517 4327.630 4417.563 3164.992 3013.609 3398.967 3478.797
## [3200] 3641.215 3478.797 3398.967 3891.733 3891.733 4692.866 2444.898
## [3207] 1995.664 2182.658 2057.072 2721.886 2512.763 2581.549 2311.934
## [3214] 1935.179 2057.072 3977.077 3559.546 3164.992 4238.615 3723.802
## [3221] 3807.309 4150.517 3807.309 2939.297 3013.609 3891.733 3723.802
## [3228] 5772.897 3641.215 3478.797 4150.517 5568.282 6517.883 5981.174
## [3235] 7541.577 8142.281 8142.281 10914.378 9945.659 10914.378 11633.654
## [3242] 11780.241 11633.654 12076.147 13768.704 15235.378 13452.779 12985.712
## [3249] 13610.287 13928.031 13768.704 14249.413 15910.862 16427.014 14738.305
## [3256] 15068.778 16775.657 15910.862 15740.628 18024.505 18389.487 17663.152
## [3263] 17305.431 15402.886 17663.152 17483.838 18943.766 19506.210 18758.099
## [3270] 17305.431 18206.542 17663.152 18024.505 19130.341 17305.431 16427.014
## [3277] 17127.932 17663.152 17843.375 17483.838 17843.375 16082.005 15910.862
## [3284] 16775.657 14574.432 15402.886 14088.268 14574.432 12526.834 15910.862
## [3291] 15571.303 15740.628 15235.378 12831.843 12375.695 13768.704 12225.466
## [3298] 12985.712 11927.739 11199.355 11927.739 12375.695 11780.241 12225.466
## [3305] 12526.834 11487.977 11927.739 10493.744 7899.259 9810.917 9281.067
## [3312] 10081.313 10355.356 10773.255 7899.259 5981.174 5268.229 5772.897
## [3319] 5170.044 4976.423 7899.259 5670.132 5367.331 5981.174 7192.118
## [3326] 6738.970 4150.517 3807.309 3807.309 4150.517 4786.468 5568.282
## [3333] 4508.413 4508.413 4417.563 4327.630 4327.630 4238.615 4880.987
## [3340] 3723.802 3164.992 2182.658 4692.866 2119.404 2651.257 2865.906
## [3347] 2939.297 3164.992 4150.517 4600.181 2865.906 3088.841 2512.763
## [3354] 2939.297 3891.733 3088.841 3242.064 2182.658 2119.404 3164.992
## [3361] 5170.044 4976.423 4880.987 7307.691 9810.917 9412.162 7899.259
## [3368] 12526.834 12076.147 11927.739 12375.695 13452.779 14411.468 15235.378
## [3375] 15235.378 16427.014 15402.886 17305.431 15402.886 17127.932 16775.657
## [3382] 18943.766 18573.340 18024.505 17483.838 19130.341 18943.766 19695.505
## [3389] 18389.487 18206.542 18573.340 18758.099 18758.099 19317.822 19317.822
## [3396] 18943.766 18573.340 18758.099 17483.838 17663.152 18389.487 19695.505
## [3403] 18573.340 18758.099 21045.962 20850.319 20461.755 18758.099 18389.487
## [3410] 18389.487 19317.822 19695.505 21439.965 19695.505 18573.340 18943.766
## [3417] 18389.487 18024.505 18573.340 18573.340 17843.375 18206.542 18206.542
## [3424] 16254.055 16254.055 17483.838 16082.005 16775.657 17127.932 15235.378
## [3431] 17127.932 16427.014 14903.087 15402.886 14738.305 15910.862 15068.778
## [3438] 17305.431 16082.005 15910.862 13768.704 15402.886 14411.468 16775.657
## [3445] 14903.087 15068.778 14249.413 15402.886 11056.411 12678.884 10773.255
## [3452] 14249.413 10355.356 11199.355 13296.180 9281.067 9281.067 8513.662
## [3459] 10081.313 9021.616 9945.659 7077.460 7899.259 5772.897 5072.775
## [3466] 5367.331 4786.468 5072.775 5367.331 4880.987 4786.468 6627.969
## [3473] 5170.044 3891.733 4150.517 4417.563 3559.546 3164.992 3164.992
## [3480] 3641.215 4417.563 4150.517 4327.630 5170.044 2246.835 1759.264
## [3487] 2182.658 1537.650 2512.763 1875.617 2246.835 1816.979 2057.072
## [3494] 2721.886 3977.077 4150.517 1935.179 1935.179 1875.617 2246.835
## [3501] 3320.056 3807.309 2793.435 3320.056 5170.044 7424.177 8388.955
## [3508] 7192.118 10355.356 9150.885 7659.891 7424.177 7192.118 8765.813
## [3515] 11487.977 12375.695 12375.695 13928.031 14249.413 15740.628 15571.303
## [3522] 15068.778 16254.055 16951.340 18758.099 19317.822 18206.542 18206.542
## [3529] 18943.766 18758.099 19695.505 20076.816 19317.822 18758.099 17305.431
## [3536] 18758.099 18758.099 19317.822 19317.822 18758.099 19695.505 19506.210
## [3543] 19130.341 19506.210 19317.822 19885.707 19130.341 20461.755 21439.965
## [3550] 19695.505 22238.847 19506.210 19506.210 19695.505 20268.832 19317.822
## [3557] 18943.766 20850.319 20655.584 19695.505 20076.816 19885.707 18206.542
## [3564] 16600.882 17305.431 16427.014 15402.886 14574.432 15740.628 14574.432
## [3571] 15068.778 15740.628 16427.014 15910.862 15068.778 14903.087 17305.431
## [3578] 15740.628 14411.468 15740.628 15235.378 15068.778 15235.378 16254.055
## [3585] 15571.303 14738.305 13768.704 14088.268 13140.491 14088.268 14249.413
## [3592] 12831.843 12985.712 11056.411 10217.879 9021.616 10633.044 10914.378
## [3599] 10217.879 10217.879
## attr(,"lambda")
## [1] 1.984576#2
lambda2 <- BoxCox.lambda(data_transformed)
print(lambda2)## [1] 0.9867136Plot ACF and PACF
# Plot ACF and PACF for training data
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 1 2 3 4 5 6 7 8 9 10
## 1.000 0.977 0.965 0.956 0.946 0.933 0.920 0.905 0.888 0.869 0.849
## 11 12 13 14 15 16 17 18 19 20 21
## 0.827 0.805 0.780 0.755 0.728 0.700 0.670 0.639 0.609 0.578 0.546
## 22 23 24 25 26 27 28 29 30 31 32
## 0.514 0.480 0.447 0.411 0.375 0.340 0.305 0.270 0.234 0.198 0.163
## 33 34 35 36 37 38 39 40 41 42 43
## 0.127 0.090 0.054 0.018 -0.018 -0.053 -0.087 -0.121 -0.154 -0.187 -0.220
## 44 45 46 47 48 49 50 51 52 53 54
## -0.252 -0.282 -0.314 -0.342 -0.371 -0.398 -0.424 -0.449 -0.474 -0.496 -0.517
## 55 56 57 58 59 60 61 62 63 64 65
## -0.538 -0.557 -0.574 -0.593 -0.610 -0.625 -0.638 -0.649 -0.660 -0.669 -0.678
## 66 67 68 69 70 71 72 73 74 75 76
## -0.686 -0.692 -0.697 -0.700 -0.701 -0.702 -0.701 -0.700 -0.698 -0.695 -0.690
## 77 78 79 80 81 82 83 84 85 86 87
## -0.684 -0.678 -0.670 -0.661 -0.651 -0.638 -0.626 -0.612 -0.597 -0.580 -0.562
## 88 89 90 91 92 93 94 95 96 97 98
## -0.543 -0.522 -0.501 -0.480 -0.457 -0.432 -0.407 -0.382 -0.355 -0.328 -0.300
## 99 100 101 102 103 104 105 106 107 108 109
## -0.271 -0.242 -0.212 -0.180 -0.149 -0.118 -0.086 -0.054 -0.022 0.010 0.044
## 110 111 112 113 114 115 116 117 118 119 120
## 0.075 0.108 0.140 0.172 0.205 0.237 0.270 0.302 0.333 0.365 0.396
## 121 122 123 124 125 126 127 128 129 130 131
## 0.427 0.456 0.486 0.513 0.541 0.568 0.595 0.618 0.642 0.666 0.687
## 132 133 134 135 136 137 138 139 140 141 142
## 0.707 0.725 0.742 0.757 0.771 0.784 0.794 0.802 0.810 0.816 0.821
## 143 144
## 0.824 0.824PACF <- pacf(data_transformed, main = "ACF of Transformed Data", lag.max = 144)
PACF##
## Partial autocorrelations of series 'data_transformed', by lag
##
## 1 2 3 4 5 6 7 8 9 10 11
## 0.977 0.251 0.116 0.019 -0.054 -0.070 -0.067 -0.088 -0.094 -0.081 -0.087
## 12 13 14 15 16 17 18 19 20 21 22
## -0.057 -0.072 -0.057 -0.071 -0.059 -0.075 -0.059 -0.016 -0.010 -0.025 -0.023
## 23 24 25 26 27 28 29 30 31 32 33
## -0.035 -0.013 -0.045 -0.053 -0.033 0.006 -0.005 -0.026 -0.010 -0.017 -0.036
## 34 35 36 37 38 39 40 41 42 43 44
## -0.050 -0.025 -0.041 -0.011 -0.031 0.008 0.004 -0.013 -0.010 -0.041 -0.014
## 45 46 47 48 49 50 51 52 53 54 55
## -0.002 -0.041 0.015 -0.004 -0.004 0.000 -0.005 -0.020 0.005 0.007 -0.020
## 56 57 58 59 60 61 62 63 64 65 66
## -0.001 0.021 -0.050 -0.033 -0.019 0.000 0.010 -0.010 -0.008 -0.029 -0.020
## 67 68 69 70 71 72 73 74 75 76 77
## 0.001 -0.013 0.009 0.010 -0.009 -0.012 -0.023 -0.024 -0.017 -0.002 -0.033
## 78 79 80 81 82 83 84 85 86 87 88
## -0.015 -0.002 -0.004 -0.004 0.026 -0.006 0.008 0.018 0.035 0.018 0.048
## 89 90 91 92 93 94 95 96 97 98 99
## 0.037 0.021 -0.017 0.016 0.028 0.013 0.017 0.001 0.015 0.017 0.030
## 100 101 102 103 104 105 106 107 108 109 110
## 0.015 0.037 0.044 0.006 0.012 0.025 0.013 0.012 0.026 0.030 -0.012
## 111 112 113 114 115 116 117 118 119 120 121
## 0.026 0.016 0.004 0.036 0.040 0.045 0.033 0.033 0.025 0.041 0.045
## 122 123 124 125 126 127 128 129 130 131 132
## 0.024 0.027 -0.008 0.028 0.033 0.046 -0.016 0.021 0.044 -0.007 0.011
## 133 134 135 136 137 138 139 140 141 142 143
## 0.002 -0.015 -0.011 0.011 0.015 0.005 -0.036 0.020 -0.011 0.035 -0.017
## 144
## -0.015Do differencing because based on the ACF and PACF plot above, the data isn’t stationary.
#1
diff_data_transformed <- diff(data_transformed, differences = 1)
diff## function (x, ...)
## UseMethod("diff")
## <bytecode: 0x106398d48>
## <environment: namespace:base>ADF Test to check for stationary
# ADF Test
adf.test(diff_data_transformed)## Warning in adf.test(diff_data_transformed): p-value smaller than printed
## p-value##
## Augmented Dickey-Fuller Test
##
## data: diff_data_transformed
## Dickey-Fuller = -9.3262, Lag order = 15, p-value = 0.01
## alternative hypothesis: stationary#2
# ACF and PACF for data differencing
par(mfrow = c(1, 2))
ACF2 <- acf(diff_data_transformed , main = "ACF of Differenced Data", lag.max = 144)
PACF2 <- pacf(diff_data_transformed , main = "PACF of Differenced Data", lag.max = 144)
# Seasonal Differencing
seasonal_diff <- diff(diff_data_transformed, lag = 144)
cleaned_data <- na.omit(seasonal_diff)#3
# ACF and PACF
par(mfrow = c(1, 2))
ACF3 <- acf(cleaned_data, main = "ACF of Seasonal Differenced Data", lag.max = 144)
PACF3 <- pacf(cleaned_data, main = "PACF of Seasonal Differenced Data", lag.max = 144)
ARIMA MODEL
trying out several ARIMA’S Model
arima_model1 <- arima(train_data$rate, order = c(0, 1,1))
summary(arima_model1)##
## Call:
## arima(x = train_data$rate, order = c(0, 1, 1))
##
## Coefficients:
## ma1
## -0.3192
## s.e. 0.0157
##
## sigma^2 estimated as 92.51: log likelihood = -13253.71, aic = 26511.42
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.01983814 9.616851 7.181188 -0.4937123 5.982145 0.9652337
## ACF1
## Training set 0.009202211arima_model2 <- arima(train_data$rate, order = c(1, 1,1))
summary(arima_model2)##
## Call:
## arima(x = train_data$rate, order = c(1, 1, 1))
##
## Coefficients:
## ar1 ma1
## 0.0562 -0.3664
## s.e. 0.0405 0.0363
##
## sigma^2 estimated as 92.46: log likelihood = -13252.76, aic = 26511.53
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.02013964 9.61432 7.180488 -0.4960092 5.980696 0.9651395
## ACF1
## Training set 0.00160246arima_model3 <- arima(train_data$rate, order = c(3, 0,3))
summary(arima_model3)##
## Call:
## arima(x = train_data$rate, order = c(3, 0, 3))
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 ma3 intercept
## 2.0151 -1.0434 0.0264 -1.4107 0.3310 0.1000 143.4839
## s.e. 0.1253 0.2491 0.1241 0.1242 0.1794 0.0586 1.6222
##
## sigma^2 estimated as 84.47: log likelihood = -13095.97, aic = 26207.93
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.01417266 9.191027 6.894706 -0.7198976 5.791019 0.9267272
## ACF1
## Training set 4.43331e-05arima_model4 <- arima(train_data$rate, order = c(2, 1,1))
summary(arima_model4)##
## Call:
## arima(x = train_data$rate, order = c(2, 1, 1))
##
## Coefficients:
## ar1 ar2 ma1
## -0.1335 -0.0936 -0.1758
## s.e. 0.0778 0.0277 0.0771
##
## sigma^2 estimated as 92.2: log likelihood = -13247.59, aic = 26503.18
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.02018231 9.600501 7.177805 -0.4953447 5.979797 0.9647789
## ACF1
## Training set -0.001121417arima_model5 <- arima(train_data$rate, order = c(3, 0,1))
summary(arima_model5)##
## Call:
## arima(x = train_data$rate, order = c(3, 0, 1))
##
## Coefficients:
## ar1 ar2 ar3 ma1 intercept
## 0.8392 0.0530 0.0933 -0.1544 143.5809
## s.e. 0.0814 0.0601 0.0277 0.0807 9.1443
##
## sigma^2 estimated as 91.66: log likelihood = -13242.35, aic = 26496.7
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.01533096 9.574079 7.185471 -0.6970144 6.012371 0.9658094
## ACF1
## Training set -0.001096063comparing three of the models, which is ARIMA(0,1,1), ARIMA(1,1,1), ARIMA(3,0,3), ARIMA(2,1,1), and ARIMA(3,0,1), the ARIMA(3,0,3) suits best in this model.
ARIMA (3,0,3) Time Series Plot of Rate with Forecast
arima_model3$aic## [1] 26207.93 forecast <- forecast(arima_model3, h = 150)
dataframe_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, dataframe_forecast)
ggplot(plot_data, aes(x = ts, y = rate, color = set)) +
geom_line() +
labs(x = "Date", y = "Rate", title = "Time Series Plot of Rate with Forecast") +
theme_minimal() +
scale_x_datetime(labels = scales::date_format("%Y-%m-%d %H:%M")) +
scale_color_manual(values = c("Train" = "blue", "Test" = "green", "Forecast" = "red"))
# Normality Testing
residuals <- residuals(arima_model3)
kolmogorov_smirnov_test <- ks.test(residuals, "pnorm", mean(residuals), sd(residuals))
print(kolmogorov_smirnov_test)##
## Asymptotic one-sample Kolmogorov-Smirnov test
##
## data: residuals
## D = 0.040442, p-value = 1.537e-05
## alternative hypothesis: two-sidedljung_box_test <- Box.test(residuals, lag = 144, type = "Ljung-Box")
print(ljung_box_test)##
## Box-Ljung test
##
## data: residuals
## X-squared = 273.18, df = 144, p-value = 4.749e-10by observing the kolmogorov smirnov and the Box-Ljung test’s results, we can conclude that the data are not normally distributed and there is a significant autocorrelation in the residuals, which means that the residuals are not white noise.
SARIMA MODEL
MODEL 1; SARIMA(3,0,3)(0,0,0)
sarima_model1 <- Arima(train_data$rate, order = c(3, 0, 3),
seasonal = list(order = c(0, 0, 0), period = 144),
include.drift = FALSE)
summary(sarima_model1)## Series: train_data$rate
## ARIMA(3,0,3) with non-zero mean
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 ma3 mean
## 2.0151 -1.0434 0.0264 -1.4107 0.3310 0.1000 143.4839
## s.e. 0.1253 0.2491 0.1241 0.1242 0.1794 0.0586 1.6222
##
## 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 -0.01417266 9.191027 6.894706 -0.7198976 5.791019 0.9267272
## ACF1
## Training set 4.43331e-05MODEL 2; SARIMA(3,0,3)(0,0,1)
sarima_model2 <- Arima(train_data$rate, order = c(3, 0, 3),
seasonal = list(order = c(0, 0, 1), period = 144),
include.drift = FALSE)
summary(sarima_model2)## Series: train_data$rate
## ARIMA(3,0,3)(0,0,1)[144] with non-zero mean
##
## Coefficients:
## ar1 ar2 ar3 ma1 ma2 ma3 sma1 mean
## 1.9819 -0.9784 -0.0055 -1.3861 0.2883 0.1182 0.0784 142.7501
## s.e. 0.1278 0.2538 0.1263 0.1262 0.1826 0.0599 0.0174 1.7164
##
## sigma^2 = 84.18: log likelihood = -13086.09
## AIC=26190.19 AICc=26190.24 BIC=26245.88
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.05403157 9.16483 6.875898 -0.6720062 5.77069 0.9241992
## ACF1
## Training set 0.0005266543MODEL 3; SARIMA(3,0,3)(0,1,1)
sarima_model4 <- Arima(train_data$rate, order = c(3, 0, 3),
seasonal = list(order = c(0, 1, 1), period = 144),
include.drift = FALSE)
summary(sarima_model4)## Series: train_data$rate
## 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 0.3729 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 0.8541951
## ACF1
## Training set -0.0003707099# Forecast
forecast <- forecast(sarima_model4, h = 150)
# View the forecasted mean values
forecast_mean <- forecast$mean
# Print the forecasted values
print(forecast_mean)## Time Series:
## Start = 3601
## End = 3750
## Frequency = 1
## [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 182.80523 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 121.88101residuals <- residuals(forecast)
kolmogorov_smirnov_test <- ks.test(residuals, "pnorm", mean(residuals), sd(residuals))
print(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.test(residuals, lag = 144, type = "Ljung-Box")
print(ljung_box_test)##
## Box-Ljung test
##
## data: residuals
## X-squared = 192.39, df = 144, p-value = 0.004378 dataframe_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_data1 <- bind_rows(train_data, test_data, dataframe_forecast)
ggplot(plot_data1, aes(x = ts, y = rate, color = set)) +
geom_line() +
labs(x = "Date", y = "Rate", title = "Time Series Plot of Rate with Forecast SARIMA") +
theme_minimal() +
scale_x_datetime(labels = scales::date_format("%Y-%m-%d %H:%M")) +
scale_color_manual(values = c("Train" = "blue", "Test" = "green", "Forecast" = "red"))
accuracy(forecast)## ME RMSE MAE MPE MAPE MASE
## Training set 0.3020245 8.627579 6.355078 -0.3470499 5.273393 0.8541951
## ACF1
## Training set -0.0003707099write.csv(forecast_mean, "forecast_predict_lastt.csv", row.names = FALSE)