Time Series: Chicago Crime
Ika Sulistiorini
20 June, 2021
Data
This dataset reflects reported incidents of crime (with the exception of murders where data exists for each victim) that occurred in the City of Chicago from 2001 to present, minus the most recent seven days. Data is extracted from the Chicago Police Department’s CLEAR (Citizen Law Enforcement Analysis and Reporting) system. This data has many variables, namely:
- ID : Unique identifier for the record.
- Case Number : The Chicago Police Department RD Number (Records Division Number), which is unique to the incident.
- Date : Date when the incident occurred. this is sometimes a best estimate.
- Block : The partially redacted address where the incident occurred, placing it on the same block as the actual address.
- IUCR : The Illinois Unifrom Crime Reporting code. This is directly linked to the Primary Type and Description.
- Primary Type : The primary description of the IUCR code.
- Description : The secondary description of the IUCR code, a subcategory of the primary description.
- Location description : Description of the location where the incident occurred.
- Arrest : Indicates whether an arrest was made.
- Domestic : Indicates whether the incident was domestic-related as defined by the Illinois Domestic Violence Act.
- Beat : Indicates the beat where the incident occurred. A beat is the smallest police geographic area – each beat has a dedicated police beat car. Three to five beats make up a police sector, and three sectors make up a police district. The Chicago Police Department has 22 police districts.
- District : Indicates the police district where the incident occurred.
- Ward : The ward (City Council district) where the incident occurred.
- Community Area : Indicates the community area where the incident occurred. Chicago has 77 community areas.
- FBI Code : Indicates the crime classification as outlined in the FBI’s National Incident-Based Reporting System (NIBRS).
- X coordinate : The x coordinate of the location where the incident occurred in State Plane Illinois East NAD 1983 projection. This location is shifted from the actual location for partial redaction but falls on the same block.
- Y Coordinate : The y coordinate of the location where the incident occurred in State Plane Illinois East NAD 1983 projection. This location is shifted from the actual location for partial redaction but falls on the same block.
- Year : Year the incident occurred.
- Update On : Date and time the record was last updated.
- Latitude : The latitude of the location where the incident occurred. This location is shifted from the actual location for partial redaction but falls on the same block.
- Longitude : The longitude of the location where the incident occurred. This location is shifted from the actual location for partial redaction but falls on the same block.
- Location : The location where the incident occurred in a format that allows for creation of maps and other geographic operations on this data portal. This location is shifted from the actual location for partial redaction but falls on the same block.
Cases
Prediction of narcotics crime frequency in Chicago in 2015 - 2020 based on available data from 2001 to present.
Solution
Doing the time series forecasting with the data to predict the frequency of narcotics crime in order to determine preventive measures or other solutions to overcome narcotics crimes.
Analysis
Library
library(readr)
library(dplyr)
library(lubridate)
library(padr)
library(forecast)
library(tseries)
library(MLmetrics) Read Data
crime <- read_csv("Crimes_-_2001_to_Present.csv")##
## -- Column specification --------------------------------------------------------
## cols(
## .default = col_character(),
## ID = col_double(),
## Arrest = col_logical(),
## Domestic = col_logical(),
## Ward = col_double(),
## `Community Area` = col_double(),
## `X Coordinate` = col_double(),
## `Y Coordinate` = col_double(),
## Year = col_double(),
## Latitude = col_double(),
## Longitude = col_double()
## )
## i Use `spec()` for the full column specifications.## Warning: 11 parsing failures.
## row col expected actual file
## 1287470 ID delimiter or quote 'Crimes_-_2001_to_Present.csv'
## 1287470 ID delimiter or quote m 'Crimes_-_2001_to_Present.csv'
## 1287470 ID delimiter or quote 'Crimes_-_2001_to_Present.csv'
## 1287470 ID delimiter or quote I 'Crimes_-_2001_to_Present.csv'
## 1287469 -- 22 columns 8 columns 'Crimes_-_2001_to_Present.csv'
## ....... ... .................. ......... ..............................
## See problems(...) for more details.head(crime)Data Wrangling
glimpse(crime)## Rows: 1,287,471
## Columns: 22
## $ ID <dbl> 10224738, 10224739, 11646166, 10224740, 1022474~
## $ `Case Number` <chr> "HY411648", "HY411615", "JC213529", "HY411595",~
## $ Date <chr> "09/05/2015 01:30:00 PM", "09/04/2015 11:30:00 ~
## $ Block <chr> "043XX S WOOD ST", "008XX N CENTRAL AVE", "082X~
## $ IUCR <chr> "0486", "0870", "0810", "2023", "0560", "0610",~
## $ `Primary Type` <chr> "BATTERY", "THEFT", "THEFT", "NARCOTICS", "ASSA~
## $ Description <chr> "DOMESTIC BATTERY SIMPLE", "POCKET-PICKING", "O~
## $ `Location Description` <chr> "RESIDENCE", "CTA BUS", "RESIDENCE", "SIDEWALK"~
## $ Arrest <lgl> FALSE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE,~
## $ Domestic <lgl> TRUE, FALSE, TRUE, FALSE, TRUE, FALSE, FALSE, F~
## $ Beat <chr> "0924", "1511", "0631", "1412", "1522", "0614",~
## $ District <chr> "009", "015", "006", "014", "015", "006", "014"~
## $ Ward <dbl> 12, 29, 8, 35, 28, 21, 32, 25, 27, 15, 13, 45, ~
## $ `Community Area` <dbl> 61, 25, 44, 21, 25, 71, 24, 31, 27, 63, 65, 11,~
## $ `FBI Code` <chr> "08B", "06", "06", "18", "08A", "05", "05", "06~
## $ `X Coordinate` <dbl> 1165074, 1138875, NA, 1152037, 1141706, 1168430~
## $ `Y Coordinate` <dbl> 1875917, 1904869, NA, 1920384, 1900086, 1850165~
## $ Year <dbl> 2015, 2015, 2018, 2015, 2015, 2015, 2015, 2015,~
## $ `Updated On` <chr> "02/10/2018 03:50:01 PM", "02/10/2018 03:50:01 ~
## $ Latitude <dbl> 41.81512, 41.89508, NA, 41.93741, 41.88190, 41.~
## $ Longitude <dbl> -87.67000, -87.76540, NA, -87.71665, -87.75512,~
## $ Location <chr> "(41.815117282, -87.669999562)", "(41.895080471~From the summary of the data structure above, it can be seen that there are variables whose data types are still wrong, as well as the name of the variable we will change to make it easier to carry out the analysis.
# Changing the name of variables
names(crime) <- c("id",
"case_number",
"date",
"block",
"iucr",
"primary_type",
"description",
"location_description",
"arrest",
"domestic",
"beat",
"district",
"ward",
"community_area",
"fbi_code",
"x",
"y",
"year",
"update_on",
"latitude",
"longitude",
"location")crime <- crime %>%
mutate(date = mdy_hms(date),
primary_type = as.factor(primary_type),
location_description = as.factor(location_description),
beat = as.factor(beat),
district = as.factor(district),
fbi_code = as.factor(fbi_code),
update_on = mdy_hms(update_on),
hour = hour(date),
tgl = date(date),
tanggal = paste0(tgl," ",hour,":00:00"),
tanggal = ymd_hms(tanggal),
year = lubridate::year(date))
glimpse(crime)## Rows: 1,287,471
## Columns: 25
## $ id <dbl> 10224738, 10224739, 11646166, 10224740, 10224741,~
## $ case_number <chr> "HY411648", "HY411615", "JC213529", "HY411595", "~
## $ date <dttm> 2015-09-05 13:30:00, 2015-09-04 11:30:00, 2018-0~
## $ block <chr> "043XX S WOOD ST", "008XX N CENTRAL AVE", "082XX ~
## $ iucr <chr> "0486", "0870", "0810", "2023", "0560", "0610", "~
## $ primary_type <fct> BATTERY, THEFT, THEFT, NARCOTICS, ASSAULT, BURGLA~
## $ description <chr> "DOMESTIC BATTERY SIMPLE", "POCKET-PICKING", "OVE~
## $ location_description <fct> "RESIDENCE", "CTA BUS", "RESIDENCE", "SIDEWALK", ~
## $ arrest <lgl> FALSE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, T~
## $ domestic <lgl> TRUE, FALSE, TRUE, FALSE, TRUE, FALSE, FALSE, FAL~
## $ beat <fct> 0924, 1511, 0631, 1412, 1522, 0614, 1434, 1034, 1~
## $ district <fct> 009, 015, 006, 014, 015, 006, 014, 010, 012, 008,~
## $ ward <dbl> 12, 29, 8, 35, 28, 21, 32, 25, 27, 15, 13, 45, 34~
## $ community_area <dbl> 61, 25, 44, 21, 25, 71, 24, 31, 27, 63, 65, 11, 4~
## $ fbi_code <fct> 08B, 06, 06, 18, 08A, 05, 05, 06, 03, 11, 06, 08B~
## $ x <dbl> 1165074, 1138875, NA, 1152037, 1141706, 1168430, ~
## $ y <dbl> 1875917, 1904869, NA, 1920384, 1900086, 1850165, ~
## $ year <dbl> 2015, 2015, 2018, 2015, 2015, 2015, 2015, 2015, 2~
## $ update_on <dttm> 2018-02-10 15:50:01, 2018-02-10 15:50:01, 2019-0~
## $ latitude <dbl> 41.81512, 41.89508, NA, 41.93741, 41.88190, 41.74~
## $ longitude <dbl> -87.67000, -87.76540, NA, -87.71665, -87.75512, -~
## $ location <chr> "(41.815117282, -87.669999562)", "(41.895080471, ~
## $ hour <int> 13, 11, 0, 12, 13, 10, 18, 13, 11, 0, 14, 11, 3, ~
## $ tgl <date> 2015-09-05, 2015-09-04, 2018-09-01, 2015-09-05, ~
## $ tanggal <dttm> 2015-09-05 13:00:00, 2015-09-04 11:00:00, 2018-0~Data Aggregation
Narcotics type crimes will be analyzed. By extracting crime data where the primary type is narcotics. In this process, data aggregation is carried out to obtain the frequency of narcotics crime rates in 2015 - 2020.
narcotics <- crime %>%
group_by(tanggal, primary_type, year) %>%
summarise(jumlah = sum(n())) %>%
ungroup() %>%
filter(primary_type == "NARCOTICS", year == 2015:2020) %>%
select(-primary_type, -year)head(narcotics)Pre-processing Data
There are 3 mandatory requirements for time series data, namely the data must be sorted by time, there should be no time holes or skipping, there should be no NA or missing values.
Impute missing value
We can perform padding using pad() function from padr package to add missing value.
narcotics <- narcotics %>%
pad() %>%
replace(., is.na(.), 0)
narcoticsData wrangling and aggregation
In this analysis I will try to make an aggregation of daily crime data. This is chosen because in order to obtain daily prediction results that can be used to determine preventive actions or other solutions for drug cases in the future.
daily_crime <- narcotics %>%
mutate(year = year(tanggal),
month = month(tanggal),
day = day(tanggal),
date = paste0(year,"-",month,"-",day),
date = ymd(date)) %>%
select(-tanggal, -year, -month, -day) %>%
group_by(date) %>%
summarise(number_of_narcotics = sum(jumlah))
daily_crimeThis is the range of our data.
range(daily_crime$date)## [1] "2015-01-06" "2020-12-13"TS object
After the data meets the requirements as time series data, the data is ready to be used as a TS object for further analysis.
# making initial ts object
crime_ts <- ts(data = daily_crime$number_of_narcotics,
start = range(daily_crime$date)[[1]],
frequency = 30) # weekly seasonalityMake the simple plot of our ts object.
crime_ts %>% autoplot()
In this section, I tried to explore whether our timeseries object has trend and seasonal properties (one-seasonal/multiseasonal).
# decompose ts object
crime_decompose <- decompose(crime_ts)
autoplot(crime_decompose)
# making 2nd ts object
daily_crime$number_of_narcotics %>%
msts(seasonal.periods = c(30,30*3)) %>%
mstl() %>%
autoplot() 
# making 3rd ts object
daily_crime$number_of_narcotics %>%
msts(seasonal.periods = c(30*3,30*3*4)) %>%
mstl() %>%
autoplot() 
# making 4th ts object
daily_crime$number_of_narcotics %>%
msts(seasonal.periods = c(30*3*4,30*3*4*2)) %>%
mstl() %>%
autoplot() 
From the plot it can be seen that this decomposition is the best. It can be seen that the data has a trend and seasonality, therefore this decomposition will be used to build a time series model.
# assign final ts object
crime_msts <- daily_crime$number_of_narcotics %>%
msts(seasonal.periods = c(30*3*4,30*3*4*2))
# check for stationary
adf.test(crime_msts)## Warning in adf.test(crime_msts): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: crime_msts
## Dickey-Fuller = -6.1797, Lag order = 12, p-value = 0.01
## alternative hypothesis: stationaryCross Validation
The cross-validation scheme for time series should not be sampled randomly, but splitted sequentially.
train <- crime_msts %>% head(length(crime_msts) - 30*3*4)
test <- crime_msts %>% tail(30*3*4)Model Building
Because our data contain both seasonal and trend so I use ETS Holt-Winters to build the model, and using the ARIMA model to compare with the ETS model. Is it the arima or ets model that produces good performance.
# ets Holt-Winters
crime_ets <- stlm(y = train, method = "ets", s.window = 360)
# SARIMA
crime_arima <- stlm(y=train, method = "arima", s.window = 360)Forecasting and Evaluation
# forecast
cforecast_ets <- forecast(crime_ets, h = length(test))
cforecast_arima <- forecast(crime_arima, h = length(test))
cforecast_ets## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 3.512500 -2.137924802 -6.687060132 2.411211 -9.09522691 4.819377
## 3.513889 -4.139763791 -8.699165613 0.419638 -11.11276714 2.833240
## 3.515278 -1.641574726 -6.211219974 2.928071 -8.63024405 5.347095
## 3.516667 -2.143375086 -6.723240850 2.436491 -9.14767534 4.860925
## 3.518056 -0.645236978 -5.235300500 3.944827 -7.66513335 6.374659
## 3.519444 3.352946334 -1.247292338 7.953185 -3.68251159 10.388404
## 3.520833 -1.148862775 -5.759254143 3.461529 -8.19984791 5.902122
## 3.522222 -3.150676152 -7.771197907 1.469846 -10.21715437 3.915802
## 3.523611 -0.652476853 -5.283106832 3.978153 -7.73441427 6.429461
## 3.525000 0.845688094 -3.795028093 5.486404 -6.25167485 7.943051
## 3.526389 5.343864420 0.693083900 9.994645 -1.76889059 12.456619
## 3.527778 5.342058092 0.681234972 10.002881 -1.78605575 12.470172
## 3.529167 -0.159753526 -4.830597656 4.511091 -7.30319318 6.983686
## 3.530556 4.338413921 -0.342429764 9.019258 -2.82031873 11.497147
## 3.531944 -1.663401965 -6.354223889 3.027420 -8.83739502 5.510591
## 3.533333 0.834755248 -3.866023734 5.535534 -6.35446581 8.023976
## 3.534722 5.332948853 0.622233859 10.043664 -1.87146803 12.537366
## 3.536111 -2.668874772 -7.389504866 2.051755 -9.88845549 4.550706
## 3.537500 -0.170632600 -4.901157010 4.559892 -7.40534537 7.064080
## 3.538889 3.327485240 -1.412912836 8.067883 -3.92232799 10.577298
## 3.540278 8.325722931 3.575471713 13.075974 1.06084062 15.590605
## 3.541667 1.323872896 -3.436211069 6.083957 -5.95604731 8.603793
## 3.543056 2.322030715 -2.447865727 7.091927 -4.97289638 9.616958
## 3.544444 -0.679751848 -5.459440623 4.099937 -7.98965502 6.630151
## 3.545833 1.818404335 -2.971056753 6.607865 -5.50644430 9.143253
## 3.547222 0.316594065 -4.482619436 5.115808 -7.02316960 7.656358
## 3.548611 3.814756152 -0.994189985 8.623702 -3.53989230 11.169405
## 3.550000 -0.687067197 -5.505726312 4.131592 -8.05657036 6.682436
## 3.551389 2.311139666 -2.517212888 7.139492 -5.07318834 9.695468
## 3.552778 -0.690687270 -5.528713842 4.147339 -8.08981040 6.708436
## 3.554167 1.307489658 -3.540191625 6.155171 -6.10639908 8.721378
## 3.555556 3.805663805 -1.051653000 8.662981 -3.62296120 11.234289
## 3.556944 6.303862875 1.436929624 11.170796 -1.13946921 13.747195
## 3.558333 8.302020354 3.425489622 13.178551 0.84401019 15.760031
## 3.559722 3.300233585 -1.585875777 8.186343 -4.17242583 10.772893
## 3.561111 2.298404484 -2.597264768 7.194074 -5.18887552 9.785684
## 3.562500 2.796557016 -2.108653493 7.701768 -4.70531508 10.298429
## 3.563889 0.294750760 -4.619982483 5.209484 -7.22168511 7.811187
## 3.565278 -1.207059145 -6.131296708 3.717178 -8.73803061 6.323912
## 3.566667 0.791140226 -4.142583347 5.724864 -6.75433884 8.336619
## 3.568056 3.789307750 -1.153883630 8.732499 -3.77065108 11.349267
## 3.569444 8.787485042 3.834843955 13.740126 1.21307413 16.361896
## 3.570833 1.285663882 -3.676408916 6.247737 -6.30317158 8.874499
## 3.572222 0.783863191 -4.187623425 5.755350 -6.81936947 8.387096
## 3.573611 5.282062966 0.301180324 10.262946 -2.33553967 12.899666
## 3.575000 -0.219768214 -5.210029190 4.770493 -7.85171378 7.412177
## 3.576389 16.278380477 11.278758759 21.278002 8.63211889 23.924642
## 3.577778 1.776557488 -3.232407480 6.785522 -5.88399336 9.437108
## 3.579167 2.274738828 -2.743551993 7.293030 -5.40007469 9.949552
## 3.580556 -2.227056858 -7.254656234 2.800543 -9.91610658 5.461993
## 3.581944 -0.228890824 -5.265781552 4.808000 -7.93215044 7.474369
## 3.583333 7.269317355 2.223152383 12.315482 -0.44812599 14.986761
## 3.584722 2.267480543 -2.787941659 7.322903 -5.46412052 9.999082
## 3.586111 -2.234350516 -7.299013028 2.830312 -9.98008341 5.511382
## 3.587500 19.763844056 14.689958062 24.837730 12.00400506 27.523683
## 3.588889 3.762023420 -1.321069320 8.845116 -4.01189608 11.535943
## 3.590278 -0.239796788 -5.332079627 4.852486 -8.02777133 7.548178
## 3.591667 8.258367018 3.156910634 13.359823 0.45636275 16.060371
## 3.593056 2.256591649 -2.854021813 7.367205 -5.55941716 10.072600
## 3.594444 1.754757620 -3.364996542 6.874512 -6.07523068 9.584746
## 3.595833 -2.747019297 -7.875897868 2.381859 -10.59096218 5.096924
## 3.597222 0.751136439 -4.386850337 5.889123 -7.10673624 8.609009
## 3.598611 -1.250703771 -6.397782636 3.896375 -9.12248160 6.621074
## 3.600000 1.747443493 -3.408711426 6.903598 -6.13821496 9.633102
## 3.601389 6.245636893 1.080421866 11.410852 -1.65387780 14.145152
## 3.602778 11.243821230 6.069561960 16.418080 3.33047456 19.157168
## 3.604167 4.742056033 -0.441231699 9.925344 -3.18509848 12.669211
## 3.605556 2.240219621 -2.952080874 7.432520 -5.70071872 10.181158
## 3.606944 -2.261027731 -7.462325372 2.940270 -10.21572602 5.693671
## 3.608333 2.737721794 -2.472557456 7.948001 -5.23071268 10.706156
## 3.609722 4.736445501 -0.482799903 9.955691 -3.24570153 12.718593
## 3.611111 4.735202201 -0.492993980 9.963398 -3.26063386 12.731038
## 3.612500 7.233930389 1.996798729 12.471062 -0.77557131 15.243432
## 3.613889 5.732716494 0.486664574 10.978768 -2.29042756 13.755861
## 3.615278 2.731447249 -2.523509788 7.986404 -5.30531601 10.768211
## 3.616667 6.230170892 0.966323803 11.494018 -1.82018853 14.280530
## 3.618056 -1.771043405 -7.043765558 3.501679 -9.83497606 6.292889
## 3.619444 2.227730918 -3.053851385 7.509313 -5.84975217 10.305214
## 3.620833 4.226419125 -1.064008489 9.516847 -3.86459170 12.317430
## 3.622222 0.725184636 -4.574073525 6.024443 -7.37933134 8.829701
## 3.623611 5.223947642 -0.084126376 10.532022 -2.89405103 13.341946
## 3.625000 8.722692559 3.405817302 14.039568 0.59123356 16.854152
## 3.626389 2.221391868 -3.104270083 7.547054 -5.92350522 10.366289
## 3.627778 6.220178930 0.885744759 11.554613 -1.93813411 14.378492
## 3.629167 4.718866520 -0.624325471 10.062059 -3.45284045 12.890573
## 3.630556 9.217704942 3.865769463 14.569640 1.03262596 17.402784
## 3.631944 -0.283564025 -5.644228731 5.077101 -8.48199321 7.914865
## 3.633333 3.215194311 -2.154185431 8.584574 -4.99656337 11.426952
## 3.634722 6.713922284 1.335841629 12.092003 -1.51114230 14.938987
## 3.636111 8.212688600 2.825921086 13.599456 -0.02566138 16.451039
## 3.637500 2.211426405 -3.184013982 7.606867 -6.04018759 10.463040
## 3.638889 3.210201068 -2.193898273 8.614300 -5.05465566 11.475058
## 3.640278 0.208937860 -5.203806583 5.621682 -8.06914041 8.487016
## 3.641667 1.707701213 -3.713674546 7.129077 -6.58357752 9.998980
## 3.643056 -2.293578258 -7.723571614 3.136415 -10.59803646 6.010880
## 3.644444 1.205231848 -4.233365449 6.643829 -7.11238495 9.522849
## 3.645833 2.703931719 -2.743255930 8.151119 -5.62682289 11.034686
## 3.647222 3.702688086 -1.753076389 9.158453 -4.64118364 12.046560
## 3.648611 7.201432205 1.737104367 12.665760 -1.15553606 15.558400
## 3.650000 -1.799864891 -7.272742693 3.673013 -10.16990920 6.570179
## 3.651389 3.698952730 -1.782461700 9.180367 -4.68414722 12.082053
## 3.652778 2.697709651 -2.792228133 8.187647 -5.69842565 11.093845
## 3.654167 8.196375163 2.697927237 13.694823 -0.21277527 16.605526
## 3.655556 4.195159381 -1.311785535 9.702104 -4.22698608 12.617305
## 3.656944 0.693916144 -4.821512673 6.209345 -7.74120432 9.129037
## 3.658333 4.692679089 -0.831220598 10.216579 -3.75539646 13.140755
## 3.659722 0.691394260 -4.840963327 6.223752 -7.76961653 9.152405
## 3.661111 2.690188033 -2.850614544 8.230991 -5.78373825 11.164114
## 3.662500 2.688905370 -2.860329344 8.238140 -5.79791676 11.175727
## 3.663889 0.687691413 -4.869962645 6.245345 -7.81200699 9.187390
## 3.665278 5.186404822 -0.379655845 10.752465 -3.32615038 13.698960
## 3.666667 7.686696854 2.112242256 13.261151 -0.83869576 16.212089
## 3.668056 7.186974321 1.604138413 12.769810 -1.35123640 15.725185
## 3.669444 -0.312735597 -5.903940253 5.278469 -8.86374521 8.238274
## 3.670833 2.187535161 -3.412025735 7.787096 -6.37625422 10.751325
## 3.672222 0.187785262 -5.420119422 5.795690 -8.38876484 8.764335
## 3.673611 4.688032921 -0.928203155 10.304269 -3.90125895 13.277325
## 3.675000 -0.811704391 -6.436259518 4.812851 -9.41371915 7.790310
## 3.676389 1.188632166 -4.444229727 6.821494 -7.42608669 9.803351
## 3.677778 -3.311097881 -8.952254307 2.330059 -11.93850213 5.316306
## 3.679167 0.689169629 -4.960269152 6.338608 -7.95090139 9.329241
## 3.680556 0.189458351 -5.468250661 5.847167 -8.46326090 8.842178
## 3.681944 -0.310233015 -5.976200185 5.355734 -8.97558203 8.355116
## 3.683333 1.690023454 -3.984189857 7.364237 -6.98793694 10.367984
## 3.684722 1.690268329 -3.992179155 7.372716 -7.00028515 10.380822
## 3.686111 0.690576221 -5.000093523 6.381246 -8.01255212 9.393705
## 3.687500 3.690824961 -2.008055180 9.389705 -5.02486009 12.406510
## 3.688889 9.191111985 3.484033261 14.898191 0.46288828 17.919336
## 3.690278 1.191411347 -4.523854202 6.906677 -7.54933303 9.932156
## 3.691667 -3.308282519 -9.031723181 2.415158 -12.06152965 5.444965
## 3.693056 -2.308036960 -8.039641074 3.423567 -11.07376901 6.457695
## 3.694444 1.192224927 -4.547531031 6.931981 -7.58597430 9.970424
## 3.695833 3.192513714 -2.555382524 8.940410 -5.59813499 11.983162
## 3.697222 3.192752224 -2.563272784 8.948777 -5.61032836 11.995833
## 3.698611 0.193036681 -5.571105632 5.957179 -8.62245825 9.008532
## 3.700000 -2.306676294 -8.078924498 3.465572 -11.13456812 6.521216
## 3.701389 5.193621985 -0.586720742 10.973965 -3.64664935 14.033893
## 3.702778 6.193874934 0.405449003 11.982301 -2.65875859 15.046508
## 3.704167 6.694145695 0.897647831 12.490644 -2.17083278 15.559124
## 3.705556 8.194438749 2.389880179 13.998997 -0.68286752 17.071745
## 3.706944 0.194758365 -5.617849734 6.007366 -8.69485859 9.084375
## 3.708333 -1.804994716 -7.625641212 4.015652 -10.70690534 7.096916
## 3.709722 0.195269613 -5.633404195 6.023943 -8.71891773 9.109457
## 3.711111 -0.804466059 -6.641156137 5.032224 -9.73091323 8.121981
## 3.712500 3.195821029 -2.648874326 9.040516 -5.74286915 12.134511
## 3.713889 8.696101434 2.843411752 14.548791 -0.25481502 17.647018
## 3.715278 -2.803620970 -8.664294074 3.057052 -11.76674701 6.159505
## 3.716667 3.196645521 -2.672000145 9.065291 -5.77867350 12.171965
## 3.718056 3.696932204 -2.179675208 9.573540 -5.29056326 12.684428
## 3.719444 0.197136975 -5.687421410 6.081695 -8.80251845 9.196792
## 3.720833 7.197435655 1.304937024 13.089934 -1.81436333 16.209235
## 3.722222 3.697741787 -2.202686404 9.598170 -5.32618441 12.721668
## 3.723611 3.698030572 -2.210316536 9.606378 -5.33800657 12.734068
## 3.725000 3.698310142 -2.217945284 9.614566 -5.34982172 12.746442
## 3.726389 4.198588290 -1.725564897 10.122741 -4.86162216 13.258799
## 3.727778 2.198842257 -3.733198177 8.130883 -6.87343070 11.271115
## 3.729167 1.199144586 -4.740772621 7.139062 -7.88517485 10.283464
## 3.730556 10.699394280 4.751610731 16.647178 1.60304431 19.795744
## 3.731944 -1.800321888 -7.755961389 4.155318 -10.90868650 7.308043
## 3.733333 1.699944875 -4.263540229 7.663430 -7.42041855 10.820308
## 3.734722 -0.799751378 -6.771071777 5.171569 -9.93209785 8.332595
## 3.736111 0.700523919 -5.278621506 6.679669 -8.44378990 9.844838
## 3.737500 1.200807752 -4.786152473 7.187768 -7.95545777 10.357073
## 3.738889 6.701060290 0.706295453 12.695825 -2.46714136 15.869262
## 3.740278 -1.298651610 -7.301210912 4.703908 -10.47877386 7.881471
## 3.741667 6.201598549 0.191254890 12.211942 -2.99042885 15.393626
## 3.743056 2.201850596 -3.816267351 8.219969 -7.00206655 11.405768
## 3.744444 0.702092900 -5.323789304 6.727975 -8.51369865 9.917884
## 3.745833 2.702439006 -3.331197464 8.736075 -6.52521168 11.930090
## 3.747222 3.202716344 -2.838664440 9.244097 -6.03677824 12.442211
## 3.748611 -1.796998402 -7.846113584 4.252117 -11.04832173 7.454325
## 3.750000 0.203272629 -5.853567076 6.260112 -9.05986434 9.466410
## 3.751389 11.203555039 5.139000650 17.268109 1.92861948 20.478491
## 3.752778 -2.296163974 -8.368423244 3.776095 -11.58288313 6.990555
## 3.754167 -0.795877297 -6.875831685 5.284077 -10.09436512 8.502611
## 3.755556 5.704383777 -0.383256003 11.792024 -3.60585784 15.014625
## 3.756944 3.704654807 -2.390660674 9.799970 -5.61732578 13.026635
## 3.758333 -1.292629295 -7.395610822 4.810352 -10.62633409 8.041076
## 3.759722 2.210070957 -3.900567000 8.320709 -7.13534334 11.555485
## 3.761111 1.712759829 -4.405524976 7.831045 -7.64434931 11.069869
## 3.762500 7.215454392 1.089532284 13.341377 -2.15333499 16.584244
## 3.763889 4.218113859 -1.915436043 10.351664 -5.16234123 13.598569
## 3.765278 2.220825558 -3.920342663 8.361994 -7.17128074 11.612932
## 3.766667 5.723535370 -0.425241731 11.872312 -3.68020771 15.127278
## 3.768056 3.726206095 -2.430170482 9.882583 -5.68915938 13.141572
## 3.769444 -0.271086876 -6.435053559 5.892880 -9.69806042 9.155887
## 3.770833 6.231569008 0.060021553 12.403116 -3.20699832 15.670136
## 3.772222 2.234304021 -3.944814906 8.413423 -7.21584288 11.684451
## 3.773611 1.736972090 -4.449709043 7.923653 -7.72474020 11.198684
## 3.775000 0.239703049 -5.954531056 6.433937 -9.23356052 9.712967
## 3.776389 1.242407574 -4.959370306 7.444185 -8.24239320 10.727208
## 3.777778 7.745080061 1.535767572 13.954393 -1.75124391 17.241404
## 3.779167 -0.752218876 -6.969056844 5.464619 -10.26005207 8.755614
## 3.780556 6.750489450 0.526135103 12.974844 -2.76883905 16.269818
## 3.781944 -0.746855561 -6.978717221 5.485006 -10.27766551 8.783954
## 3.783333 -2.744136598 -8.983496538 3.495223 -12.28641418 6.798141
## 3.784722 2.258553362 -3.988295859 8.505403 -7.29517808 11.812285
## 3.786111 1.261255041 -4.993074491 7.515585 -8.30391655 10.826427
## 3.787500 0.263883810 -5.997917099 6.525685 -9.31271427 9.840482
## 3.788889 6.266639166 -0.002624216 12.535903 -3.32137178 15.854650
## 3.790278 5.769337258 -0.507379724 12.046054 -3.83007298 15.368747
## 3.791667 0.272026940 -6.012134801 6.556189 -9.33876908 9.882823
## 3.793056 0.774752340 -5.516845351 7.066350 -8.84741598 10.396921
## 3.794444 -1.722543875 -8.021568737 4.576481 -11.35607108 7.910983
## 3.795833 -1.219843614 -7.526286901 5.086600 -10.86471632 8.425029
## 3.797222 0.782826901 -5.531026095 7.096680 -8.87337798 10.439032
## 3.798611 -0.214484620 -6.535738640 6.106769 -9.88200838 9.453039
## 3.800000 4.288219971 -2.040426417 10.616866 -5.39060945 13.967049
## 3.801389 2.790908852 -3.545121279 9.126939 -6.89921302 12.481031
## 3.802778 3.293589538 -3.049815741 9.636995 -6.40781165 12.994991
## 3.804167 -1.703701264 -8.054473126 4.647071 -11.41636867 8.008966
## 3.805556 1.299023393 -5.059106518 7.657153 -8.42489718 11.022944
## 3.806944 -3.198300804 -9.563780259 3.167179 -12.93346154 6.536860
## 3.808333 0.304392747 -6.068427775 6.677213 -9.44199518 10.050781
## 3.809722 -2.192937140 -8.573090283 4.187216 -11.95053934 7.564665
## 3.811111 3.309768006 -3.077709341 9.697245 -6.45903560 13.078572
## 3.812500 -0.687561819 -7.082354980 5.707231 -10.46755399 9.092430
## 3.813889 1.315186277 -5.086914338 7.717287 -8.47598169 11.106354
## 3.815278 -0.682125583 -7.091525322 5.727274 -10.48445659 9.120205
## 3.816667 3.820573998 -2.596116561 10.237265 -5.99290736 13.634055
## 3.818056 4.324540695 -2.099432410 10.748514 -5.50007835 14.149160
## 3.819444 2.828496566 -3.602750837 9.259744 -7.00724756 12.664241
## 3.820833 4.332503646 -2.106009838 10.771017 -5.51435299 14.179360
## 3.822222 -3.663539615 -10.109310988 2.782232 -13.52149624 6.194417
## 3.823611 -2.159593852 -8.612614952 4.293427 -12.02863798 7.709450
## 3.825000 0.844368917 -5.615893774 7.304632 -9.03575026 10.724488
## 3.826389 -1.151651710 -7.619147883 5.315844 -11.04283355 8.739530
## 3.827778 7.852361802 1.377640227 14.327083 -2.04987033 17.754594
## 3.829167 1.356319159 -5.125619764 7.838258 -8.55695096 11.269589
## 3.830556 1.860257746 -4.628890496 8.349406 -8.06403807 11.784554
## 3.831944 -1.135739782 -7.632089344 5.360610 -11.07104907 8.799570
## 3.833333 -2.631850666 -9.135393574 3.871692 -12.57816123 7.314460
## 3.834722 -2.627775748 -9.138504054 3.882953 -12.58507543 7.329524
## 3.836111 4.376156998 -2.141748785 10.894063 -5.59211968 14.344434
## 3.837500 3.380131701 -3.144943663 9.905207 -6.59910991 13.359373
## 3.838889 4.884137296 -1.648099781 11.416374 -5.10605721 14.874332
## 3.840278 4.888091530 -1.651299416 11.427482 -5.11304387 14.889227
## 3.841667 2.892080245 -3.654456753 9.438617 -7.11998410 12.904145
## 3.843056 6.395989026 -0.157686232 12.949664 -3.62699235 16.418970
## 3.844444 -0.099987968 -6.660793719 6.460818 -10.13387449 9.933899
## 3.845833 -2.096014962 -8.663943466 4.471914 -12.14079479 7.948765
## 3.847222 3.407989275 -3.167054265 9.983033 -6.64767207 13.463651
## 3.848611 0.911955508 -5.670195376 7.494106 -9.15457558 10.978487
## 3.850000 -2.084084906 -8.673335469 4.505166 -12.16147401 7.993304
## 3.851389 5.419894532 -1.176448068 12.016237 -4.66834091 15.508130
## 3.852778 3.923856901 -2.679570120 10.527284 -6.17521323 14.022927
## 3.854167 0.427857147 -6.182646702 7.038361 -9.68203606 10.537750
## 3.855556 0.931755819 -5.685817290 7.549329 -9.18894889 11.052461
## 3.856944 0.435747997 -6.188886829 7.060383 -9.69575667 10.567253
## 3.858333 1.439705078 -5.191983945 8.071394 -8.70258805 11.581998
## 3.859722 5.443721652 -1.195014073 12.082457 -4.70934848 15.596792
## 3.861111 3.947679690 -2.698095264 10.593455 -6.21615601 14.111515
## 3.862500 -0.048323525 -6.701130260 6.604483 -10.22291341 10.126266
## 3.863889 -0.544391180 -7.204222272 6.115440 -10.72972389 9.640942
## 3.865278 -2.540365052 -9.207213100 4.126483 -12.73642927 7.655699
## 3.866667 1.463581330 -5.210276296 8.137439 -8.74320312 11.670366
## 3.868056 5.467556840 -1.213303009 12.148417 -4.74993658 15.685050
## 3.869444 7.471542247 0.783687505 14.159397 -2.75664894 17.699733
## 3.870833 0.975489499 -5.719352827 7.670332 -9.26338828 11.214367
## 3.872222 0.979470021 -5.722352603 7.681293 -9.27008320 11.229023
## 3.873611 -2.016545717 -8.725341377 4.692250 -12.27676327 8.243672
## 3.875000 -1.012602053 -7.728363508 5.703159 -11.28347287 9.258269
## 3.876389 -4.008638940 -10.731358973 2.714081 -14.29015199 6.272874
## 3.877778 -0.004622365 -6.734293780 6.725049 -10.29676664 10.287522
## 3.879167 2.999313570 -3.737302055 9.735929 -7.30345095 13.302078
## 3.880556 -0.996667135 -7.740219819 5.746886 -11.31004097 9.316707
## 3.881944 -0.992707887 -7.743190501 5.757775 -11.31668014 9.331264
## 3.883333 1.511222254 -5.246183183 8.268628 -8.82333754 11.845782
## 3.884722 -0.484786010 -7.249107185 6.279535 -10.82992251 9.860350
## 3.886111 1.519190821 -5.252039029 8.290421 -8.83651159 11.874893
## 3.887500 -1.476836553 -8.254968035 5.301295 -11.84309410 8.889421
## 3.888889 -1.472854712 -8.257880807 5.312171 -11.84965666 8.903947
## 3.890278 -3.468891980 -10.260805689 3.323022 -13.85622762 6.918444
## 3.891667 3.035115743 -3.763678603 9.833910 -7.36274293 13.432974
## 3.893056 -2.460918403 -9.266586429 4.344750 -12.86928946 7.947453
## 3.894444 -2.456971255 -9.269506025 4.355564 -12.87584409 7.961902
## 3.895833 -2.452994424 -9.272389024 4.366400 -12.88235846 7.976370
## 3.897222 -0.949065643 -7.775313179 5.877182 -11.38891034 9.490779
## 3.898611 -1.445112868 -8.278206468 5.387981 -11.89542772 9.005202
## 3.900000 -0.941046799 -7.780979611 5.898886 -11.40182132 9.519728
## 3.901389 1.062873045 -5.783892146 7.909638 -9.40835070 11.534097
## 3.902778 -0.433115368 -7.286706128 6.420475 -10.91477792 10.048547
## 3.904167 4.570848044 -2.289561494 11.431258 -5.92124293 15.062939
## 3.905556 3.574831710 -3.292389835 10.442053 -6.92767733 14.077341
## 3.906944 -1.921209485 -8.795236286 4.952817 -12.43412627 8.591707
## 3.908333 5.082776560 -1.798048767 11.963602 -5.44053767 15.606091
## 3.909722 0.586971021 -6.300646122 7.474588 -9.94673039 11.120672
## 3.911111 4.591179310 -2.303222957 11.485582 -5.95289906 15.135258
## 3.912500 -0.904647157 -7.805827877 5.996534 -11.45909227 9.649798
## 3.913889 -0.900475046 -7.808427569 6.007477 -11.46527674 9.664327
## 3.915278 0.103742798 -6.810974895 7.018460 -10.47140533 10.678891
## 3.916667 0.607877094 -6.313599157 7.529353 -9.97760736 11.193362
## 3.918056 0.612104495 -6.316123721 7.540333 -9.98370620 11.207915
## 3.919444 -0.383695068 -7.318668675 6.551279 -10.98982195 10.222432
## 3.920833 1.120521356 -5.821191088 8.062234 -9.49591168 11.736954
## 3.922222 -2.375320692 -9.323765437 4.573124 -13.00204990 8.251409
## 3.923611 4.628889979 -2.326280550 11.584061 -6.00812543 15.265905
## 3.925000 -2.366920158 -9.328809974 4.594970 -13.01421182 8.280372
## 3.926389 -2.362759299 -9.331361923 4.605843 -13.02031732 8.294799
## 3.927778 4.641451158 -2.333857814 11.616760 -6.02636334 15.309266
## 3.929167 0.145640618 -6.836368260 7.127649 -10.53242050 10.823702
## 3.930556 0.649828252 -6.338874108 7.638531 -10.03846966 11.338126
## 3.931944 2.654053613 -4.341335826 9.649443 -8.04447131 13.352579
## 3.933333 -0.841787755 -7.843857887 6.160282 -11.55052991 9.866954
## 3.934722 1.162359685 -5.846384771 8.171104 -9.55658997 11.881309
## 3.936111 -0.833432489 -7.848844919 6.181980 -11.56257993 9.895715
## 3.937500 -1.329243029 -8.351317102 5.692831 -12.06857857 9.410093
## 3.938889 -1.325080005 -8.353809407 5.703649 -12.07459400 9.424434
## 3.940278 -1.320853750 -8.356232184 5.714525 -12.08053656 9.438829
## 3.941667 3.683332928 -3.358688262 10.725354 -7.08650910 14.453175
## 3.943056 2.687515741 -4.361141944 9.736173 -8.09247593 13.467507
## 3.944444 4.691697409 -2.363590528 11.746985 -6.09843436 15.481829
## 3.945833 9.195902452 2.133990487 16.257814 -1.60435989 19.996165
## 3.947222 -3.799904502 -10.868434287 3.268625 -14.61028793 7.010479
## 3.948611 0.204291389 -6.870850026 7.279433 -10.61620365 11.024786
## 3.950000 2.208489785 -4.873257087 9.290237 -8.62210743 13.039087
## 3.951389 -2.287342370 -9.375688544 4.801004 -13.12803235 8.553348
## 3.952778 1.716885029 -5.378054308 8.811824 -9.13388832 12.567658
## 3.954167 1.221040135 -5.880486244 8.322567 -9.63980723 12.081887
## 3.955556 4.225230801 -2.882876516 11.333338 -6.64568124 15.096143
## 3.956944 -0.270597149 -7.385279317 6.844085 -11.15156455 10.610370
## 3.958333 8.233552946 1.112301998 15.354804 -2.65746054 19.124566
## 3.959722 1.237801927 -5.890011748 8.365616 -9.66324838 12.138852
## 3.961111 0.241999950 -6.892370415 7.376370 -10.66907795 11.153078
## 3.962500 0.746195066 -6.394725969 7.887116 -10.17490122 11.667291
## 3.963889 -1.249627165 -8.397092866 5.897839 -12.18073266 9.681478
## 3.965278 1.254567271 -5.899437109 8.408572 -9.68653827 12.195673
## 3.966667 2.758744977 -4.401792110 9.919282 -8.19235147 13.709841
## 3.968056 -0.237040395 -7.404104236 6.930023 -11.19811865 10.724038
## 3.969444 4.266959988 -2.906624668 11.440545 -6.70409100 15.238011
## 3.970833 -1.228986506 -8.409086056 5.951113 -12.21000116 9.752028
## 3.972222 -1.724976908 -8.911585445 5.461632 -12.71594620 9.265992
## 3.973611 0.279018188 -6.914093445 7.472130 -10.72189673 11.279933
## 3.975000 5.283100420 -1.916508437 12.482709 -5.72775115 16.293952
## 3.976389 6.787083395 -0.419016828 13.993184 -4.23369586 17.807863
## 3.977778 -2.208877601 -9.421463347 5.003708 -13.23957561 8.821820
## 3.979167 -0.704864818 -7.923930260 6.514201 -11.74547266 10.335743
## 3.980556 2.299167011 -4.926372317 9.524706 -8.75134179 13.349676
## 3.981944 1.303169282 -5.928838136 8.535177 -9.75723161 12.363570
## 3.983333 -1.192788931 -8.431258660 6.045681 -12.26307307 9.877495
## 3.984722 -1.688763749 -8.933690024 5.556163 -12.76892233 9.391395
## 3.986111 -1.184750286 -8.436127359 6.066627 -12.27477451 9.905274
## 3.987500 1.819301130 -5.438521006 9.077123 -9.28057996 12.919182
## 3.988889 3.323323481 -3.940938002 10.587585 -7.78640574 14.433053
## 3.990278 -2.172653828 -9.443348954 5.098041 -13.29222246 8.946915
## 3.991667 3.831371430 -3.445751650 11.108495 -7.29802791 14.960771
## 3.993056 0.835352995 -6.448192367 8.118898 -10.30386838 11.974574
## 3.994444 2.839404413 -4.450557573 10.129366 -8.30963034 13.988439
## 3.995833 -2.156562597 -9.452935565 5.139810 -13.31540210 9.002277
## 3.997222 1.347471475 -5.955306846 8.650250 -9.82116417 12.516107
## 3.998611 1.851496672 -5.457681389 9.160675 -9.32692653 13.029920
## 4.000000 0.202030989 -7.113541213 7.517603 -10.98617121 11.390233
## 4.001389 1.203425077 -6.118535682 8.525386 -9.99454758 12.401398
## 4.002778 0.871416573 -6.456927174 8.199760 -10.33631802 12.079151
## 4.004167 -1.127274563 -8.461995744 6.207447 -12.34476260 10.090213
## 4.005556 -0.792576855 -8.133669928 6.548516 -12.01980986 10.434656
## 4.006944 -2.791239380 -10.138698821 4.556220 -14.02820890 8.445730
## 4.008333 -2.123214288 -9.477034584 5.230606 -13.36991190 9.123483
## 4.009722 1.878158912 -5.482016743 9.238335 -9.37825838 13.134576
## 4.011111 2.879470698 -4.487054833 10.245996 -8.38665789 14.145599cforecast_arima## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 3.512500 -2.26221812 -6.793723759 2.2692875 -9.19255794 4.668122
## 3.513889 -4.15732261 -8.741079966 0.4264348 -11.16757454 2.852929
## 3.515278 -1.65913354 -6.251891118 2.9336240 -8.68315012 5.364883
## 3.516667 -2.16093390 -6.762674093 2.4408063 -9.19868820 4.876820
## 3.518056 -0.66279579 -5.273501100 3.9479095 -7.71426105 6.388669
## 3.519444 3.33538752 -1.284265504 7.9550405 -3.72976209 10.400537
## 3.520833 -1.16642159 -5.795005032 3.4621619 -8.24522910 5.912386
## 3.522222 -3.16823497 -7.805731631 1.4692617 -10.26067407 3.924204
## 3.523611 -0.67003567 -5.316428455 3.9763571 -7.77608022 6.436009
## 3.525000 0.82812928 -3.827142632 5.4834012 -6.29149472 7.947753
## 3.526389 5.32630561 0.662171473 9.9904397 -1.80687199 12.459483
## 3.527778 5.32449928 0.651519733 9.9974788 -1.82220621 12.471205
## 3.529167 -0.17731234 -4.859120588 4.5044959 -7.33752016 6.982895
## 3.530556 4.32085511 -0.369765224 9.0114754 -2.85282964 11.494540
## 3.531944 -1.68096078 -6.380376671 3.0184551 -8.86809717 5.506176
## 3.533333 0.81719643 -3.890998587 5.5253915 -6.38336648 8.017759
## 3.534722 5.31539004 0.598432230 10.0323478 -1.89857440 12.529354
## 3.536111 -2.68643359 -7.412137937 2.0392708 -9.91377471 4.540908
## 3.537500 -0.18819141 -4.922626146 4.5462433 -7.42888450 7.052502
## 3.538889 3.30992643 -1.433222620 8.0530755 -3.94409405 10.563947
## 3.540278 8.30816412 3.556316740 13.0600115 1.04084069 15.575488
## 3.541667 1.30631408 -3.454215734 6.0668439 -5.97428799 8.586916
## 3.543056 2.30447190 -2.464724546 7.0736683 -4.98938464 9.598328
## 3.544444 -0.69731066 -5.475158021 4.0805367 -8.00439763 6.609776
## 3.545833 1.80084552 -2.985637114 6.5873282 -5.51944796 9.121139
## 3.547222 0.29903525 -4.496067109 5.0941376 -7.03444097 7.632511
## 3.548611 3.79719734 -1.006509280 8.6009040 -3.54943796 11.143833
## 3.550000 -0.70462601 -5.516921502 4.1076695 -8.06439685 6.655145
## 3.551389 2.29358085 -2.527288212 7.1144499 -5.07930214 9.666464
## 3.552778 -0.70824608 -5.537673499 4.1211813 -8.09421794 6.677726
## 3.554167 1.28993084 -3.548039783 6.1279015 -6.10910673 8.688968
## 3.555556 3.78810499 -1.058393788 8.6346038 -3.62397527 11.200185
## 3.556944 6.28630406 1.431292110 11.1413160 -1.13879597 13.711404
## 3.558333 8.28446154 3.420951318 13.1479718 0.84636453 15.722559
## 3.559722 3.28267477 -1.589318897 8.1546684 -4.16839655 10.733746
## 3.561111 2.28084567 -2.599616699 7.1613080 -5.18317742 9.744869
## 3.562500 2.77899820 -2.109918198 7.6679146 -4.69795421 10.255951
## 3.563889 0.27719195 -4.620163890 5.1745478 -7.21266747 7.767051
## 3.565278 -1.22461796 -6.130398714 3.6811628 -8.72736218 6.278126
## 3.566667 0.77358141 -4.140609818 5.6877726 -6.74202552 8.289188
## 3.568056 3.77174894 -1.150838400 8.6943363 -3.75669873 11.300197
## 3.569444 8.76992623 3.838957084 13.7008954 1.22865969 16.311193
## 3.570833 1.26810507 -3.671231662 6.2074418 -6.28595858 8.822169
## 3.572222 0.76630438 -4.181385787 5.7139945 -6.80053475 8.333144
## 3.573611 5.26450415 0.308474633 10.2205337 -2.31508892 12.844097
## 3.575000 -0.23732703 -5.201681892 4.7270278 -7.82965261 7.354999
## 3.576389 16.26082166 11.288155392 21.2334879 8.65578488 23.865858
## 3.577778 1.75899867 -3.221965136 6.7399625 -5.85872810 9.376725
## 3.579167 2.25718001 -2.732067535 7.2464276 -5.37321565 9.887576
## 3.580556 -2.24461567 -7.242133229 2.7529019 -9.88765922 5.398428
## 3.581944 -0.24644964 -5.252223541 4.7593243 -7.90212018 7.409221
## 3.583333 7.25175854 2.237741888 12.2657752 -0.41651820 14.920035
## 3.584722 2.24992173 -2.772324145 7.2721676 -5.43094052 9.930784
## 3.586111 -2.25190933 -7.282370964 2.7785523 -9.94533649 5.441518
## 3.587500 19.74628524 14.707621244 24.7849492 12.04031364 27.452257
## 3.588889 3.74446461 -1.302388425 8.7913176 -3.97403104 11.462960
## 3.590278 -0.25735560 -5.312384399 4.7976732 -7.98835500 7.473644
## 3.591667 8.24080820 3.177616842 13.3039996 0.49732524 15.984291
## 3.593056 2.23903283 -2.832307954 7.3103736 -5.51691360 9.994979
## 3.594444 1.73719881 -3.342278335 6.8166759 -6.03119111 9.505589
## 3.595833 -2.76457811 -7.852178592 2.3230224 -10.54539160 5.016235
## 3.597222 0.73357763 -4.362133246 5.8292885 -7.05963964 8.526795
## 3.598611 -1.26826259 -6.372070959 3.8355458 -9.07386391 6.537339
## 3.600000 1.72988468 -3.382008370 6.8417777 -6.08808109 9.547850
## 3.601389 6.22807808 1.108113121 11.3480430 -1.60223261 14.058389
## 3.602778 11.22626242 6.098238255 16.3542866 3.38362624 19.068899
## 3.604167 4.72449722 -0.411573500 9.8605679 -3.13044510 12.579440
## 3.605556 2.22266081 -2.921443883 7.3667655 -5.64456841 10.089890
## 3.606944 -2.27858655 -7.430712678 2.8735396 -10.15808350 5.600910
## 3.608333 2.72016298 -2.439972126 7.8802981 -5.17158265 10.611909
## 3.609722 4.71888669 -0.449244982 9.8870184 -3.18508863 12.622862
## 3.611111 4.71764339 -0.458472490 9.8937593 -3.19854272 12.633829
## 3.612500 7.21637157 2.032283786 12.4004594 -0.71200652 15.144750
## 3.613889 5.71515768 0.523110220 10.9072051 -2.22539368 13.655709
## 3.615278 2.71388844 -2.486106513 7.9138834 -5.23881756 10.666594
## 3.616667 6.21261208 1.004681771 11.4205424 -1.75223000 14.177454
## 3.618056 -1.78860222 -7.004455814 3.4272514 -9.76556192 6.188357
## 3.619444 2.21017210 -3.013592760 7.4339370 -5.77888684 10.199231
## 3.620833 4.20886031 -1.022803858 9.4405245 -3.79227958 12.210000
## 3.622222 0.70762582 -4.531925743 5.9471774 -7.30557680 8.720828
## 3.623611 5.20638883 -0.041038278 10.4538159 -2.81885839 13.231636
## 3.625000 8.70513374 3.449842900 13.9604246 0.66785997 16.742408
## 3.626389 2.20383305 -3.059309780 7.4669759 -5.84544930 10.253115
## 3.627778 6.20262012 0.931636990 11.4736032 -1.85865293 14.263893
## 3.629167 4.70130771 -0.577504068 9.9801195 -3.37193822 12.774554
## 3.630556 9.20014613 3.913517299 14.4867750 1.11494505 17.285347
## 3.631944 -0.30112284 -5.595557182 4.9933115 -8.39826142 7.796016
## 3.633333 3.19763550 -2.104592868 8.4998639 -4.91142302 11.306694
## 3.634722 6.69636347 1.386352522 12.0063744 -1.42459748 14.817324
## 3.636111 8.19512979 2.877347646 13.5129119 0.06228383 16.327976
## 3.637500 2.19386759 -3.131674402 7.5194096 -5.95084604 10.338581
## 3.638889 3.19264225 -2.140648302 8.5259328 -4.96392178 11.349206
## 3.640278 0.19137905 -5.149648831 5.5324069 -7.97701820 8.359776
## 3.641667 1.69014240 -3.658611606 7.0388964 -6.49007094 9.870356
## 3.643056 -2.31113707 -7.667606062 3.0453319 -10.50314946 5.880875
## 3.644444 1.18767303 -4.176499844 6.5518459 -7.01612144 9.391468
## 3.645833 2.68637291 -2.685492813 8.0582386 -5.52918675 10.901933
## 3.647222 3.68512927 -1.694418286 9.0646768 -4.54217874 11.912437
## 3.648611 7.18387339 1.796654948 12.5710918 -1.05516623 15.422913
## 3.650000 -1.81742370 -7.212302126 3.5774547 -10.06817825 6.433331
## 3.651389 3.68139392 -1.721133623 9.0839215 -4.58105895 11.943847
## 3.652778 2.68015084 -2.730015005 8.0903167 -5.59398380 10.954285
## 3.654167 8.17881635 2.761022973 13.5966097 -0.10698360 16.464616
## 3.655556 4.17760057 -1.247809620 9.6030108 -4.11984829 12.475049
## 3.656944 0.67635733 -4.756658989 6.1093736 -7.63272410 8.985439
## 3.658333 4.67512027 -0.765491544 10.1157321 -3.64557747 12.995818
## 3.659722 0.67383545 -4.774361282 6.1220322 -7.65846241 9.006133
## 3.661111 2.67262922 -2.783141874 8.1284003 -5.67125263 11.016511
## 3.662500 2.67134656 -2.791988401 8.1346815 -5.68410323 11.026796
## 3.663889 0.67013260 -4.800755764 6.1410210 -7.69686912 9.037134
## 3.665278 5.16884601 -0.309585346 10.6472774 -3.20969172 13.547384
## 3.666667 7.66913804 2.183174065 13.1551020 -0.72091984 16.059196
## 3.668056 7.16941551 1.675929241 12.6629018 -1.23214672 15.570978
## 3.669444 -0.33029441 -5.831292683 5.1707039 -8.74334526 8.082756
## 3.670833 2.16997635 -3.338523687 7.6784764 -6.25454746 10.594500
## 3.672222 0.17022645 -5.345765144 5.6862180 -8.26575471 8.606208
## 3.673611 4.67047411 -0.852998884 10.1939471 -3.77694886 13.117897
## 3.675000 -0.82926320 -6.360207474 4.7016811 -9.28811251 7.629586
## 3.676389 1.17107335 -4.367332116 6.7094788 -7.29918687 9.641334
## 3.677778 -3.32865670 -8.874513326 2.2171999 -11.81031249 5.152999
## 3.679167 0.67161081 -4.881686979 6.2249086 -7.82142526 9.164647
## 3.680556 0.17189954 -5.388829464 5.7326285 -8.33250158 8.676301
## 3.681944 -0.32779183 -5.895942119 5.2403585 -8.84354283 8.187959
## 3.683333 1.67246464 -3.903097061 7.2480263 -6.85462114 10.199550
## 3.684722 1.67270951 -3.910253758 7.2556728 -6.86569599 10.211115
## 3.686111 0.67301741 -4.917337639 6.2633725 -7.87669284 9.222728
## 3.687500 3.67326615 -1.924470911 9.2710032 -4.88773391 12.234266
## 3.688889 9.17355317 3.568443824 14.7786625 0.60127817 17.745828
## 3.690278 1.17385253 -4.438619420 6.7863245 -7.40968260 9.757388
## 3.691667 -3.32584133 -8.945666246 2.2939836 -11.92062185 5.268939
## 3.693056 -2.32559577 -7.952764038 3.3015725 -10.93160698 6.280415
## 3.694444 1.17466611 -4.459835934 6.8091682 -7.44256114 9.791893
## 3.695833 3.17495490 -2.466871395 8.8167812 -5.45347383 11.803384
## 3.697222 3.17519341 -2.473947637 8.8243345 -5.46442227 11.814809
## 3.698611 0.17547787 -5.480968473 5.8319242 -8.47531029 8.826266
## 3.700000 -2.32423511 -7.987977318 3.3395071 -10.98618134 6.337711
## 3.701389 5.17606317 -0.494965524 10.8470919 -3.49702677 13.849153
## 3.702778 6.17631612 0.498010291 11.8546219 -2.50790324 14.860535
## 3.704167 6.67658688 0.991013232 12.3621605 -2.01874765 15.371921
## 3.705556 8.17687994 2.484047746 13.8697121 -0.52955558 16.883315
## 3.706944 0.17719955 -5.522881937 5.8772810 -8.54032281 8.894722
## 3.708333 -1.82255353 -7.529875108 3.8847680 -10.55114865 6.906042
## 3.709722 0.17771080 -5.536841697 5.8922633 -8.56194306 8.917365
## 3.711111 -0.82202487 -6.543799148 4.8997494 -9.57272349 7.928674
## 3.712500 3.17826222 -2.550724736 8.9072492 -5.58346723 11.939992
## 3.713889 8.67854262 2.942352062 14.4147332 -0.09420379 17.451289
## 3.715278 -2.82117978 -8.564564914 2.9222053 -11.60492935 5.962570
## 3.716667 3.17908671 -2.571483993 8.9296574 -5.61565224 11.973826
## 3.718056 3.67937339 -2.078373913 9.4371207 -5.12634122 12.485088
## 3.719444 0.17957816 -5.585336810 5.9444931 -8.63709845 8.996255
## 3.720833 7.17987684 1.407803101 12.9519506 -1.64774817 16.007502
## 3.722222 3.68018297 -2.099040668 9.4594066 -5.15837686 12.518743
## 3.723611 3.68047176 -2.105892949 9.4668365 -5.16900940 12.529953
## 3.725000 3.68075133 -2.112745642 9.4742483 -5.17963768 12.541140
## 3.726389 4.18102948 -1.619590989 9.9816499 -4.69025398 13.052313
## 3.727778 2.18128344 -3.626451779 7.9890187 -6.70088109 11.063448
## 3.729167 1.18158577 -4.633255501 6.9964270 -7.71144653 10.074618
## 3.730556 10.68183547 4.859896814 16.5037741 1.77794866 19.585722
## 3.731944 -1.81788070 -7.646908090 4.0111467 -10.73260880 7.096847
## 3.733333 1.68238606 -4.153721453 7.5184936 -7.24317015 10.607942
## 3.734722 -0.81731019 -6.660489254 5.0258689 -9.75368140 8.119061
## 3.736111 0.68296511 -5.167276956 6.5332072 -8.26420803 9.630138
## 3.737500 1.18324894 -4.674047607 7.0405455 -7.77471310 10.141211
## 3.738889 6.68350148 0.819158935 12.5478440 -2.28523648 15.652239
## 3.740278 -1.31621042 -7.187590506 4.5551697 -10.29571137 7.663291
## 3.741667 6.18403973 0.305630537 12.0624489 -2.80621132 15.174291
## 3.743056 2.18429178 -3.701138137 8.0697217 -6.81669654 11.185280
## 3.744444 0.68453409 -5.207908188 6.5769764 -8.32717871 9.696247
## 3.745833 2.68488019 -3.214566102 8.5843265 -6.33754433 11.707305
## 3.747222 3.18515753 -2.721284480 9.0915995 -5.84796601 12.218281
## 3.748611 -1.81455722 -7.727986664 4.0988722 -10.85836712 7.229253
## 3.750000 0.18571381 -5.734694826 6.1061225 -8.86876984 9.240197
## 3.751389 11.18599622 5.258616610 17.1133758 2.12085138 20.251141
## 3.752778 -2.31372279 -8.248065188 3.6206196 -11.38951629 6.762071
## 3.754167 -0.81343611 -6.754733137 5.1278609 -9.89986580 8.272994
## 3.755556 5.68682496 -0.261418558 11.6350685 -3.41022847 14.783878
## 3.756944 3.68709599 -2.268085920 9.6422779 -5.42056880 12.794761
## 3.758333 -1.31018811 -7.272300338 4.6519241 -10.42845190 7.808076
## 3.759722 2.19251214 -3.776522358 8.1615466 -6.93633835 11.321363
## 3.761111 1.69520102 -4.280747738 7.6711498 -7.44422392 10.834626
## 3.762500 7.19789558 1.215040563 13.1807506 -1.95209157 16.347883
## 3.763889 4.20055505 -1.789198268 10.1903084 -4.95998214 13.361092
## 3.765278 2.20326674 -3.793376932 8.1999104 -6.96780834 11.374342
## 3.766667 5.70597656 -0.297549576 11.7095027 -3.47562434 14.887577
## 3.768056 3.70864728 -2.301753425 9.7190480 -5.48346737 12.900762
## 3.769444 -0.28864569 -6.305913115 5.7286217 -9.49126208 8.913971
## 3.770833 6.21401019 0.189883876 12.2381365 -2.99909597 15.427116
## 3.772222 2.21674521 -3.814232204 8.2477226 -7.00683879 11.440329
## 3.773611 1.71941328 -4.318407454 7.7572340 -7.51463668 10.953463
## 3.775000 0.22214424 -5.822512065 6.2668005 -9.02235982 9.466648
## 3.776389 1.22484876 -4.826635390 7.2763329 -8.03009759 10.479795
## 3.777778 7.72752125 1.669216943 13.7858256 -1.53785563 16.992898
## 3.779167 -0.76977769 -6.834894480 5.2953391 -10.04557337 8.506018
## 3.780556 6.73293064 0.661009004 12.8048523 -2.55327215 16.019133
## 3.781944 -0.76441437 -6.843133231 5.3143045 -10.06101262 8.532184
## 3.783333 -2.76169541 -8.847203900 3.3238131 -12.06867750 6.545287
## 3.784722 2.24099455 -3.851296006 8.3332851 -7.07635981 11.558349
## 3.786111 1.24369623 -4.855368850 7.3427613 -8.08401887 10.571411
## 3.787500 0.24632500 -5.859507089 6.3521571 -9.09173935 9.584389
## 3.788889 6.24908035 0.136488751 12.3616720 -3.09932178 15.597482
## 3.790278 5.75177844 -0.367565207 11.8711221 -3.60695006 15.110507
## 3.791667 0.25446813 -5.871620131 6.3805564 -9.11457536 9.623512
## 3.793056 0.75719353 -5.375631922 6.8900190 -8.62215361 10.136541
## 3.794444 -1.74010269 -7.879657933 4.3994526 -11.12974216 7.649537
## 3.795833 -1.23740243 -7.383680100 4.9088752 -10.63732296 8.162518
## 3.797222 0.76526809 -5.387724669 6.9182608 -8.64492228 10.175458
## 3.798611 -0.23204343 -6.391743953 5.9276571 -9.65249244 9.188406
## 3.800000 4.27066116 -1.895739829 10.4370621 -5.16003533 13.701358
## 3.801389 2.77335004 -3.399744141 8.9464442 -6.66758280 12.214283
## 3.802778 3.27603072 -2.903749399 9.4558108 -6.17512739 12.727189
## 3.804167 -1.72126008 -7.907718919 4.4651988 -11.18263241 7.740112
## 3.805556 1.28146458 -4.911665778 7.4745949 -8.19011095 10.753040
## 3.806944 -3.21585962 -9.415654313 2.9839351 -12.69762738 6.265908
## 3.808333 0.28683393 -5.919617942 6.4932858 -9.20511510 9.778783
## 3.809722 -2.21049595 -8.423597878 4.0026060 -11.71261536 7.291623
## 3.811111 3.29220919 -2.927535670 9.5119541 -6.22006971 12.804488
## 3.812500 -0.70512063 -6.931501345 5.5212601 -10.22754819 8.817307
## 3.813889 1.29762746 -4.935382034 7.5306370 -8.23493795 10.830193
## 3.815278 -0.69968440 -6.939315639 5.5399468 -10.24237690 8.843008
## 3.816667 3.80301518 -2.443230781 10.0492611 -5.74979366 13.355824
## 3.818056 4.30698188 -1.945871811 10.5598356 -5.25593261 13.869896
## 3.819444 2.81093775 -3.448516690 9.0703922 -6.76207172 12.383947
## 3.820833 4.31494483 -1.951103408 10.5809931 -5.26814898 13.898039
## 3.822222 -3.68109843 -9.953733535 2.5915367 -13.27426598 5.912069
## 3.823611 -2.17715267 -8.456367729 4.1020624 -11.78038339 7.426078
## 3.825000 0.82681010 -5.458978028 7.1125982 -8.78647327 10.440093
## 3.826389 -1.16921052 -7.461564857 5.1231438 -10.79253603 8.454115
## 3.827778 7.83480299 1.535889297 14.1337167 -1.79855419 17.468160
## 3.829167 1.33876034 -4.966705881 7.6442266 -8.30461807 10.982139
## 3.830556 1.84269893 -4.469313025 8.1547109 -7.81069032 11.496088
## 3.831944 -1.15329860 -7.471849503 5.1652523 -10.81668831 8.510091
## 3.833333 -2.64940948 -8.974492579 3.6756736 -12.32278932 7.023970
## 3.834722 -2.64533456 -8.976943112 3.6862740 -12.32869421 7.038025
## 3.836111 4.35859818 -1.979529100 10.6967255 -5.33473101 14.051927
## 3.837500 3.36257289 -2.982066432 9.7072122 -6.34071561 13.065861
## 3.838889 4.86657848 -1.484566196 11.2177232 -4.84665910 14.579816
## 3.840278 4.87053272 -1.487110664 11.2281761 -4.85264377 14.593709
## 3.841667 2.87452143 -3.489614015 9.2386569 -6.85858382 12.607627
## 3.843056 6.37843021 0.007809316 12.7490511 -3.36459368 16.121454
## 3.844444 -0.11754678 -6.494646533 6.2595530 -9.87047922 9.635386
## 3.845833 -2.11357378 -8.497145807 4.2699983 -11.87640471 7.649257
## 3.847222 3.39043046 -2.999607293 9.7804682 -6.38228895 13.163150
## 3.848611 0.89439669 -5.502100248 7.2908936 -8.88820119 10.676995
## 3.850000 -2.10164372 -8.504593333 4.3013059 -11.89411011 7.690823
## 3.851389 5.40233572 -1.007060071 11.8117315 -4.39998925 15.204661
## 3.852778 3.90629809 -2.509537401 10.3221336 -5.90587555 13.718472
## 3.854167 0.41029833 -6.011970397 6.8325671 -9.41171410 10.232311
## 3.855556 0.91419701 -5.514498529 7.3428925 -8.91764438 10.746038
## 3.856944 0.41818918 -6.016926737 6.8533051 -9.42347133 10.259850
## 3.858333 1.42214626 -5.019383642 7.8636762 -8.42932359 11.273616
## 3.859722 5.42616284 -1.021774674 11.8741003 -4.43510661 15.287432
## 3.861111 3.93012088 -2.524217881 10.3844596 -5.94093843 13.801180
## 3.862500 -0.06588234 -6.526615998 6.3948513 -9.94672180 9.814957
## 3.863889 -0.56194999 -7.029072232 5.9051722 -10.45255995 9.328660
## 3.865278 -2.55792387 -9.031428378 3.9155806 -12.45829467 7.342447
## 3.866667 1.44602252 -5.033857985 7.9259030 -8.46409952 11.356145
## 3.868056 5.44999803 -1.036252195 11.9362482 -4.46986565 15.369862
## 3.869444 7.45398343 0.961369741 13.9465971 -2.47561234 17.383579
## 3.870833 0.95793068 -5.541040248 7.4569016 -8.98138764 10.897249
## 3.872222 0.96191121 -5.543410754 7.4672332 -8.98712018 10.910943
## 3.873611 -2.03410453 -8.545771326 4.4775623 -11.99283951 7.924630
## 3.875000 -1.03016087 -7.548166319 5.4878446 -10.99858998 8.938268
## 3.876389 -4.02619775 -10.550535705 2.4981402 -14.00431159 5.951916
## 3.877778 -0.02218118 -6.552845489 6.5084831 -10.00997035 9.965608
## 3.879167 2.98175476 -3.555229790 9.5187393 -7.01570038 12.979210
## 3.880556 -1.01422595 -7.557524627 5.5290727 -11.02133771 8.992886
## 3.881944 -1.01026670 -7.559873423 5.5393400 -11.02702579 9.006492
## 3.883333 1.49366344 -5.062245257 8.0495721 -8.53273368 11.520061
## 3.884722 -0.50234482 -7.064549444 6.0598598 -10.53837073 9.533681
## 3.886111 1.50163201 -5.066862501 8.0701265 -8.54401346 11.547277
## 3.887500 -1.49439537 -8.069173746 5.0803830 -11.54965118 8.560860
## 3.888889 -1.49041353 -8.071469776 5.0906427 -11.55527051 8.574443
## 3.890278 -3.48645079 -10.073778931 3.1008773 -13.56089981 6.587998
## 3.891667 3.01755693 -3.576037131 9.6111510 -7.06647498 13.101589
## 3.893056 -2.47847722 -9.078331250 4.1213768 -12.57208293 7.615128
## 3.894444 -2.47453007 -9.080638143 4.1315780 -12.57770052 7.628640
## 3.895833 -2.47055324 -9.082909438 4.1418030 -12.58327937 7.642173
## 3.897222 -0.96662446 -7.585222884 5.6519740 -11.08889725 9.155648
## 3.898611 -1.46267168 -8.087506455 5.1621631 -11.59448214 8.669139
## 3.900000 -0.95860561 -7.589670866 5.6724596 -11.09994477 9.182734
## 3.901389 1.04531423 -5.591975654 7.6826041 -9.10554468 11.196173
## 3.902778 -0.45067418 -7.094182867 6.1928345 -10.61104393 9.709696
## 3.904167 4.55328923 -2.096432440 11.2030109 -5.61658246 14.723161
## 3.905556 3.55727290 -3.098655958 10.2132018 -6.62209186 13.736638
## 3.906944 -1.93876830 -8.600898555 4.7233620 -12.12761728 8.250081
## 3.908333 5.06521775 -1.603108143 11.7335436 -5.13310664 15.263542
## 3.909722 0.56941221 -6.105103566 7.2439280 -9.63837879 10.777203
## 3.911111 4.57362050 -2.107079424 11.2543204 -5.64362834 14.790869
## 3.912500 -0.92220597 -7.609084320 5.7646724 -11.14890389 9.304492
## 3.913889 -0.91803386 -7.611084934 5.7750172 -11.15417215 9.318104
## 3.915278 0.08618398 -6.613034128 6.7854021 -10.15938598 10.331754
## 3.916667 0.59031828 -6.115061198 7.2956978 -9.66467468 10.845311
## 3.918056 0.59454568 -6.116989507 7.3060809 -9.66986162 10.858953
## 3.919444 -0.40125388 -7.118939138 6.3164314 -10.67506690 9.872559
## 3.920833 1.10296254 -5.620867158 7.8267922 -9.18024759 11.386173
## 3.922222 -2.39287951 -9.122848040 4.3370890 -12.68547818 7.899719
## 3.923611 4.61133116 -2.124770607 11.3474329 -5.69064749 14.913310
## 3.925000 -2.38447897 -9.126708404 4.3577505 -12.69582907 7.926871
## 3.926389 -2.38031811 -9.128669641 4.3680334 -12.70103115 7.940395
## 3.927778 4.62389234 -2.130575731 11.3783604 -5.70617514 14.953960
## 3.929167 0.12808180 -6.632497284 6.8886609 -10.21133167 10.467495
## 3.930556 0.63226944 -6.134415144 7.3989540 -9.71648158 10.981020
## 3.931944 2.63649480 -4.136289773 9.4092794 -7.72158535 12.994575
## 3.933333 -0.85934657 -7.638225642 5.9195325 -11.22674746 9.508054
## 3.934722 1.14480087 -5.640167229 7.9297690 -9.23191238 11.521514
## 3.936111 -0.85099130 -7.642042970 5.9400604 -11.23700857 9.535026
## 3.937500 -1.34680184 -8.143931633 5.4503279 -11.74211479 9.048511
## 3.938889 -1.34263882 -8.145841301 5.4605637 -11.74723915 9.061962
## 3.940278 -1.33841256 -8.147682321 5.4708572 -11.75229199 9.075467
## 3.941667 3.66577411 -3.149557519 10.4811057 -6.75737615 14.088924
## 3.943056 2.66995693 -4.151431194 9.4913450 -7.76245594 13.102370
## 3.944444 4.67413859 -2.153300641 11.5015778 -5.76752865 15.115806
## 3.945833 9.17834364 2.344858645 16.0118286 -1.27256980 19.629257
## 3.947222 -3.81746332 -10.656988722 3.0220621 -14.27761477 6.642688
## 3.948611 0.18673257 -6.658827914 7.0322931 -10.28264874 10.656114
## 3.950000 2.19093097 -4.660659285 9.0425212 -8.28767208 12.669534
## 3.951389 -2.30490118 -9.162515906 4.5527135 -12.79271786 8.182915
## 3.952778 1.69932621 -5.164307684 8.5629601 -8.79769599 12.196348
## 3.954167 1.20348132 -5.666166482 8.0731291 -9.30273836 11.709701
## 3.955556 4.20767199 -2.667984458 11.0833284 -6.30773712 14.723081
## 3.956944 -0.28815596 -7.169815805 6.5935039 -10.81274647 10.236435
## 3.958333 8.21599413 1.328336126 15.1036521 -2.31776978 18.749758
## 3.959722 1.22024311 -5.673407839 8.1138941 -9.32268622 11.763172
## 3.961111 0.22444114 -6.675197555 7.1240798 -10.32764565 10.776528
## 3.962500 0.72863625 -6.176984987 7.6342575 -9.83260005 11.289873
## 3.963889 -1.26718598 -8.178784588 5.6444126 -11.83756388 9.303192
## 3.965278 1.23700846 -5.680562357 8.1545793 -9.34250314 11.816520
## 3.966667 2.74118616 -4.182351703 9.6647240 -7.84745126 13.329824
## 3.968056 -0.25459921 -7.184098990 6.6749006 -10.85235459 10.343156
## 3.969444 4.24940117 -2.686055397 11.1848577 -6.35746433 14.856267
## 3.970833 -1.24654532 -8.187953569 5.6948629 -11.86251313 9.369422
## 3.972222 -1.74253572 -8.689890550 5.2048191 -12.36759804 8.882527
## 3.973611 0.26145937 -6.691836947 7.2147557 -10.37268968 10.895608
## 3.975000 5.26554161 -1.693691136 12.2247743 -5.37768642 15.908770
## 3.976389 6.76952458 -0.195639523 13.7346887 -3.88277468 17.421824
## 3.977778 -2.22643642 -9.197526833 4.7446540 -12.88779919 8.434926
## 3.979167 -0.72242363 -7.699435331 6.2545881 -11.39284222 9.947995
## 3.980556 2.28160820 -4.701319762 9.2645362 -8.39785853 12.961075
## 3.981944 1.28561047 -5.703228742 8.2744497 -9.40289674 11.974118
## 3.983333 -1.21034775 -8.205093211 5.7843977 -11.90788779 9.487192
## 3.984722 -1.70632256 -8.706969302 5.2943242 -12.41288783 9.000243
## 3.986111 -1.20230910 -8.208852142 5.8042339 -11.91789198 9.513274
## 3.987500 1.80174232 -5.210692070 8.8141767 -8.92285060 12.526335
## 3.988889 3.30576467 -3.712556118 10.3240855 -7.42783072 14.039360
## 3.990278 -2.19021264 -9.214414894 4.8339896 -12.93280296 8.552378
## 3.991667 3.81381262 -3.216266182 10.8438914 -6.93776510 14.565390
## 3.993056 0.81779418 -6.218156255 7.8537446 -9.94276343 11.578352
## 3.994444 2.82184560 -4.219971578 9.8636628 -7.94768442 13.591376
## 3.995833 -2.17412141 -9.221800447 4.8735576 -12.95261637 8.604374
## 3.997222 1.32991266 -5.723623362 8.3834487 -9.45753978 12.117365
## 3.998611 1.83393786 -5.225450292 8.8933260 -8.96246464 12.630340
## 4.000000 0.18447217 -6.880763255 7.2497076 -10.62087297 10.989817
## 4.001389 1.18586626 -5.885211611 8.2569441 -9.62841413 12.000147
## 4.002778 0.85385776 -6.223057737 7.9307733 -9.96935050 11.677066
## 4.004167 -1.14483338 -8.227581683 5.9379149 -11.97696215 9.687295
## 4.005556 -0.81013567 -7.898711985 6.2784406 -11.65117762 10.030906
## 4.006944 -2.80879819 -9.903197733 4.2856013 -13.65874599 8.041150
## 4.008333 -2.14077310 -9.240991088 4.9594449 -12.99961945 8.718073
## 4.009722 1.86060010 -5.245431572 8.9666318 -9.00713751 12.728338
## 4.011111 2.86191188 -4.249928716 9.9737525 -8.01470972 13.738533# visualization
library(ggplot2)
library(gridExtra)##
## Attaching package: 'gridExtra'## The following object is masked from 'package:dplyr':
##
## combinea <- autoplot(cforecast_ets, series = "ETS", fcol = "red") +
autolayer(crime_msts, series = "Actual", color = "black") +
labs(y = "Crime Frequency") +
theme_minimal()
b <- autoplot(cforecast_arima, series = "ARIMA", fcol = "blue") +
autolayer(crime_msts, series = "Actual", color = "black") +
labs(y = "Crime Frequency") +
theme_minimal()
grid.arrange(a,b)
data.frame(ETS = RMSE(cforecast_ets$mean, test), ARIMA = RMSE(cforecast_arima$mean, test))From the table above, it can be seen that the Arima model that produces the best performance with an RMSE of 3.75 although it is not much different from the ETS model.
Assumption check
Normality: Shapiro.test
- H0 : residuals are normally distributed
- H1 : residuals are not normally distributed
shapiro.test(cforecast_arima$residuals)##
## Shapiro-Wilk normality test
##
## data: cforecast_arima$residuals
## W = 0.9888, p-value = 0.0000000001285hist(cforecast_arima$residuals, breaks = 20)
plot(cforecast_arima$residuals)
Autocorrelation: Box.test - Ljng-Box
- H0 : No autocorrelation in the forecast errors
- H1 : there is an autocorrelation in the forecast errors
Box.test(cforecast_arima$residuals, type = "Ljung-Box")##
## Box-Ljung test
##
## data: cforecast_arima$residuals
## X-squared = 0.0022337, df = 1, p-value = 0.9623Based on the assumption check, there is no autocorrelation on our forecast residuals (p-value > 0.05). Still, our forecast’s residuals are not distributed normally, therefore it’s residuals may not be appeared around its mean as seen in the histogram.