Advanced Data Analytics - D213
Thomas N Cole III
8/6/2021
In this task I was presented with data that shows how much revenue was taken in by the telecommunication company I am consulting. A question that I get when looking at the data is: Are there any patterns hidden in the data that will help the stakeholders better understand the company’s revenue trends that can help make more accurate forecasts? Some publicly traded companies are required to report earnings and in doing so they will give forecasts of their expected earnings. It is important for a company to be as accurate as possible when reporting their financial information. The goal for this data set analysis is to find any patterns and make specific, actionable recommendations to the decision makers of the company.
Time series modeling comes with a few assumptions and these assumptions vary depending on the type of series being presented. A white noise series is one that is random with independent variables that are distributed the same. It is also assumed that the mean for a white noise series in zero. A stationary series is one that does not change over time. The variance of the series is fairly the same over time, it may fluctuate but it will behave the same from one period to another. With all of the time series data types, there is an assumption of normality because a time series is a sequence of data in chronological order. Some time series are auto-correlated which assumes that each observation is related to it’s past. The greater the auto-correlation the greater the predictability of the data set.
The first step in the analysis of the data set was to import it into R. The data on the Excel spreadsheet had two columns, “Day” and “Revenue.” When it was imported into R there was a third column added by R that was the row numbers.
Import and view data from Excel.
time_series_data <- read.csv("C:\\Users\\tncol\\teleco_ts.csv", sep = ",")
View(time_series_data)
print(time_series_data)## Day Revenue
## 1 1 0.000000000
## 2 2 0.000793191
## 3 3 0.825541786
## 4 4 0.320332280
## 5 5 1.082554085
## 6 6 0.107653784
## 7 7 0.493901361
## 8 8 0.376698452
## 9 9 0.304074848
## 10 10 0.591747849
## 11 11 0.731018901
## 12 12 1.111586215
## 13 13 0.960902689
## 14 14 0.964013419
## 15 15 0.979731886
## 16 16 1.085546892
## 17 17 1.262549787
## 18 18 1.331183307
## 19 19 1.411429732
## 20 20 1.709116779
## 21 21 1.735783527
## 22 22 1.760959886
## 23 23 2.558601513
## 24 24 1.484204950
## 25 25 2.089944633
## 26 26 2.840568572
## 27 27 2.162204307
## 28 28 3.113269737
## 29 29 2.746580938
## 30 30 3.079242819
## 31 31 2.647838543
## 32 32 3.118230798
## 33 33 2.442888497
## 34 34 3.042407809
## 35 35 3.012894439
## 36 36 3.290825745
## 37 37 3.274029416
## 38 38 3.586805202
## 39 39 3.170426377
## 40 40 3.572077947
## 41 41 2.693130469
## 42 42 2.399656142
## 43 43 2.433776379
## 44 44 2.593848576
## 45 45 3.323496491
## 46 46 2.951387252
## 47 47 3.216948781
## 48 48 2.877026963
## 49 49 3.057523831
## 50 50 2.858495329
## 51 51 2.533082553
## 52 52 3.223959721
## 53 53 2.973617615
## 54 54 3.360834071
## 55 55 2.310401570
## 56 56 1.843596513
## 57 57 1.631977575
## 58 58 1.644604449
## 59 59 1.497246827
## 60 60 1.267859555
## 61 61 1.587253694
## 62 62 2.131085418
## 63 63 1.995083459
## 64 64 2.330776602
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## 72 72 3.888201920
## 73 73 3.938590458
## 74 74 5.138308366
## 75 75 5.482323172
## 76 76 6.133334991
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## 110 110 7.946744172
## 111 111 8.034009599
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## 118 118 7.001823150
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## 122 122 7.110366293
## 123 123 7.131337879
## 124 124 6.764886020
## 125 125 7.260526894
## 126 126 7.544141343
## 127 127 7.330936080
## 128 128 7.568258573
## 129 129 7.486801358
## 130 130 8.117012517
## 131 131 7.415766826
## 132 132 7.780389211
## 133 133 7.830950713
## 134 134 7.922069280
## 135 135 7.769710310
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## 139 139 6.606752799
## 140 140 7.606778130
## 141 141 7.157498503
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## 145 145 7.506441261
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## 153 153 7.146617054
## 154 154 5.698388374
## 155 155 6.419385890
## 156 156 5.351951701
## 157 157 6.013657257
## 158 158 5.676967765
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## 160 160 5.136908683
## 161 161 6.138729310
## 162 162 5.468684959
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## 164 164 6.305916197
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## 176 176 5.609863758
## 177 177 4.907574806
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## 182 182 6.233395622
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## 224 224 5.280124523
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## 226 226 6.708512399
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## 229 229 6.434691767
## 230 230 6.475201270
## 231 231 7.384317349
## 232 232 7.651861894
## 233 233 8.701207113
## 234 234 7.591809144
## 235 235 8.133817813
## 236 236 7.353039585
## 237 237 7.544987314
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## 241 241 6.781578804
## 242 242 6.518598196
## 243 243 6.365795463
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## 247 247 7.092380299
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## 252 252 6.502133156
## 253 253 7.391609269
## 254 254 6.989221405
## 255 255 7.453399290
## 256 256 7.591757865
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## 258 258 8.361602643
## 259 259 8.037333381
## 260 260 8.959983524
## 261 261 9.151138020
## 262 262 8.905577837
## 263 263 9.415349311
## 264 264 9.724684431
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## 275 275 10.898026480
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## 694 694 15.495969540
## 695 695 15.121205360
## 696 696 15.814724790
## 697 697 14.951802430
## 698 698 15.692242060
## 699 699 15.322562730
## 700 700 15.481488980
## 701 701 15.077929900
## 702 702 15.269828820
## 703 703 15.563934860
## 704 704 15.709683550
## 705 705 16.270576130
## 706 706 15.864648010
## 707 707 16.791980880
## 708 708 16.446697680
## 709 709 15.576833140
## 710 710 16.517374880
## 711 711 16.792900340
## 712 712 16.692964830
## 713 713 17.152341380
## 714 714 17.903188990
## 715 715 17.471066730
## 716 716 17.145850130
## 717 717 17.948834030
## 718 718 18.154769090
## 719 719 17.768817030
## 720 720 17.250846320
## 721 721 17.952277840
## 722 722 17.255934220
## 723 723 17.583626510
## 724 724 16.821758070
## 725 725 17.459056140
## 726 726 16.761278250
## 727 727 16.931558660
## 728 728 17.490666180
## 729 729 16.803637980
## 730 730 16.194813500
## 731 731 16.620798500I made sure I installed the packages I would need and added their libraries.
library(astsa)
library(ggplot2)
library(forecast)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo##
## Attaching package: 'forecast'## The following object is masked from 'package:astsa':
##
## gasI ran an auto correlation function on the revenue column using, acf(). This was to see how the data was correlated. I wanted to get the measure of the correlation for different lags.
acf(time_series_data$Revenue, plot = FALSE)##
## Autocorrelations of series 'time_series_data$Revenue', by lag
##
## 0 1 2 3 4 5 6 7 8 9 10 11 12
## 1.000 0.984 0.977 0.966 0.955 0.945 0.935 0.924 0.913 0.901 0.890 0.878 0.868
## 13 14 15 16 17 18 19 20 21 22 23 24 25
## 0.857 0.846 0.834 0.824 0.813 0.803 0.792 0.784 0.775 0.766 0.760 0.751 0.743
## 26 27 28
## 0.735 0.727 0.721The 1st lag has a correlation of .984 and the 28th lag in the series has a correlation of .721. This high correlation through 28 lags indicates the data has high predictability.
Next, I printed the first 10 observations and a plot of all of the observations.
head(time_series_data, 10)## Day Revenue
## 1 1 0.000000000
## 2 2 0.000793191
## 3 3 0.825541786
## 4 4 0.320332280
## 5 5 1.082554085
## 6 6 0.107653784
## 7 7 0.493901361
## 8 8 0.376698452
## 9 9 0.304074848
## 10 10 0.591747849plot(time_series_data, xlab = "Time Series Analysis", ylab = "Revenue (Millions)")
When the data was imported into R, there was a third column added that represented each row numbers. I wanted keep the original two column data set for the analysis.To do that when I converted it to a time series, I only used the “Revenue” column. This gave me two columns, the first is the row number, the second the Revenue column. The integrity of the original data set remained because the number for the rows matched the days in the time sequence of the data set. I used the ts() function with the as.numeric() function to convert time_series_data to time_series that started at 0 because that is the revenue reported in the first year.
time_series <- ts(as.numeric(time_series_data$Revenue), start = 0)
summary(time_series)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 6.873 10.786 9.823 12.567 18.155Next, I ran a quick analysis of the this new data set. I printed the length of the series, the first and last few observations and a plot with labels.
length(time_series)## [1] 731head(time_series)## Time Series:
## Start = 0
## End = 5
## Frequency = 1
## [1] 0.000000000 0.000793191 0.825541786 0.320332280 1.082554085 0.107653784tail(time_series)## Time Series:
## Start = 725
## End = 730
## Frequency = 1
## [1] 16.76128 16.93156 17.49067 16.80364 16.19481 16.62080ts.plot(time_series, xlab = "Days", ylab = "Revenue (Millions)", main = "Telecom Revenue (Years 1&2)")
Next, is an analysis of the data to evaluate the stationarity of the data set. I used the auto correlate function with lags out to 700, the number of observations. I plotted the time_series using the acf() function, acf(time_series, lag = 700, plot =TRUE) and acf(time_series, lag=150, plot=TRUE).
acf(time_series, lag = 700, plot =TRUE)
acf(time_series, lag = 150, plot = TRUE)
These two plots show the correlation tailing off with an increase in number of lags observed. Overall, the predictability will be high because 1 lag is .984 and 50 lags the correlation is still at .599. The correlation reduces to zero, this reduction means the time series is not stationary.
To determine the best model to use I need to test if the time series is Auto regressive or Moving average. AR and MA respectively. To do that I used the acf2() function to see what the data does when plugged into ACF and PACF graphs.Because there is a trend I will use diff(time_series) to eliminate it. I only needed to difference once because there is no seasonality.
acf2(diff(time_series))
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## ACF -0.47 0.22 -0.08 0.05 -0.03 0.03 0.00 0.02 -0.03 0.02 -0.03 0.01 -0.02
## PACF -0.47 0.00 0.04 0.03 -0.01 0.01 0.03 0.04 -0.02 -0.01 -0.02 -0.01 -0.02
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## ACF 0.02 -0.01 0.02 -0.06 0.04 -0.05 -0.02 0.07 -0.13 0.10 -0.05 0.03
## PACF 0.01 0.01 0.02 -0.06 -0.03 -0.03 -0.07 0.05 -0.09 -0.01 0.02 0.02
## [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
## ACF -0.02 -0.01 -0.01 0.00 -0.03 0.01 0.00 -0.02 0.01 0.01 0.05 0.00
## PACF 0.01 -0.02 -0.04 -0.02 -0.04 -0.02 0.01 -0.02 -0.01 0.01 0.08 0.06
## [,38]
## ACF -0.03
## PACF -0.03The ACF and PACF both cut off at 1, so I will use the ARIMA model, sarima(time_series, 1, 1, 1).
sarima(time_series, 1 , 1, 1)## initial value -0.624543
## iter 2 value -0.697696
## iter 3 value -0.740976
## iter 4 value -0.744623
## iter 5 value -0.746188
## iter 6 value -0.749141
## iter 7 value -0.749172
## iter 8 value -0.749175
## iter 8 value -0.749175
## final value -0.749175
## converged
## initial value -0.749687
## iter 2 value -0.749687
## iter 3 value -0.749687
## iter 4 value -0.749687
## iter 5 value -0.749688
## iter 5 value -0.749688
## iter 5 value -0.749688
## final value -0.749688
## converged
## $fit
##
## Call:
## arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S),
## xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc,
## REPORT = 1, reltol = tol))
##
## Coefficients:
## ar1 ma1 constant
## -0.4764 0.0091 0.0226
## s.e. 0.0655 0.0731 0.0120
##
## sigma^2 estimated as 0.2232: log likelihood = -488.55, aic = 985.11
##
## $degrees_of_freedom
## [1] 727
##
## $ttable
## Estimate SE t.value p.value
## ar1 -0.4764 0.0655 -7.2763 0.0000
## ma1 0.0091 0.0731 0.1248 0.9007
## constant 0.0226 0.0120 1.8917 0.0589
##
## $AIC
## [1] 1.349461
##
## $AICc
## [1] 1.349506
##
## $BIC
## [1] 1.374628What this model shows is AR(1), I(1), and MA(1). To test if this is the best model I added some parameters to see if over-fitting would change the numbers much.
sarima(time_series, 2, 1, 1)## initial value -0.625397
## iter 2 value -0.640302
## iter 3 value -0.741117
## iter 4 value -0.747411
## iter 5 value -0.750320
## iter 6 value -0.750321
## iter 7 value -0.750321
## iter 8 value -0.750322
## iter 9 value -0.750322
## iter 10 value -0.750323
## iter 11 value -0.750324
## iter 12 value -0.750324
## iter 13 value -0.750325
## iter 14 value -0.750325
## iter 15 value -0.750328
## iter 16 value -0.750335
## iter 17 value -0.750349
## iter 18 value -0.750414
## iter 19 value -0.750415
## iter 20 value -0.750431
## iter 21 value -0.750433
## iter 22 value -0.750433
## iter 23 value -0.750434
## iter 24 value -0.750435
## iter 25 value -0.750437
## iter 26 value -0.750446
## iter 27 value -0.750475
## iter 28 value -0.750490
## iter 29 value -0.750519
## iter 30 value -0.750526
## iter 31 value -0.750527
## iter 32 value -0.750528
## iter 33 value -0.750531
## iter 34 value -0.750540
## iter 35 value -0.750566
## iter 36 value -0.750646
## iter 37 value -0.750777
## iter 38 value -0.750823
## iter 39 value -0.750845
## iter 40 value -0.750848
## iter 41 value -0.750851
## iter 42 value -0.750860
## iter 43 value -0.750883
## iter 44 value -0.750940
## iter 45 value -0.751018
## iter 46 value -0.751083
## iter 47 value -0.751150
## iter 48 value -0.751169
## iter 49 value -0.751173
## iter 50 value -0.751174
## iter 51 value -0.751178
## iter 52 value -0.751181
## iter 53 value -0.751184
## iter 54 value -0.751184
## iter 55 value -0.751184
## iter 56 value -0.751184
## iter 56 value -0.751184
## iter 56 value -0.751184
## final value -0.751184
## converged
## initial value -0.749674
## iter 2 value -0.749679
## iter 3 value -0.749682
## iter 4 value -0.749682
## iter 5 value -0.749682
## iter 5 value -0.749682
## iter 5 value -0.749682
## final value -0.749682
## converged
## $fit
##
## Call:
## arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S),
## xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc,
## REPORT = 1, reltol = tol))
##
## Coefficients:
## ar1 ar2 ma1 constant
## -1.0735 -0.2856 0.6030 0.0226
## s.e. 3.5140 1.5932 3.5448 0.0119
##
## sigma^2 estimated as 0.2232: log likelihood = -488.56, aic = 987.11
##
## $degrees_of_freedom
## [1] 726
##
## $ttable
## Estimate SE t.value p.value
## ar1 -1.0735 3.5140 -0.3055 0.7601
## ar2 -0.2856 1.5932 -0.1792 0.8578
## ma1 0.6030 3.5448 0.1701 0.8650
## constant 0.0226 0.0119 1.9014 0.0576
##
## $AIC
## [1] 1.352211
##
## $AICc
## [1] 1.352287
##
## $BIC
## [1] 1.38367sarima(time_series, 1, 1, 2)## initial value -0.624543
## iter 2 value -0.648734
## iter 3 value -0.745837
## iter 4 value -0.749127
## iter 5 value -0.750035
## iter 6 value -0.750042
## iter 7 value -0.750071
## iter 8 value -0.750109
## iter 9 value -0.750148
## iter 10 value -0.750166
## iter 11 value -0.750168
## iter 12 value -0.750169
## iter 13 value -0.750169
## iter 14 value -0.750170
## iter 14 value -0.750170
## iter 14 value -0.750170
## final value -0.750170
## converged
## initial value -0.750700
## iter 2 value -0.750700
## iter 3 value -0.750700
## iter 3 value -0.750700
## iter 3 value -0.750700
## final value -0.750700
## converged
## $fit
##
## Call:
## arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S),
## xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc,
## REPORT = 1, reltol = tol))
##
## Coefficients:
## ar1 ma1 ma2 constant
## -0.2982 -0.1708 0.1033 0.0226
## s.e. 0.1675 0.1687 0.0814 0.0126
##
## sigma^2 estimated as 0.2227: log likelihood = -487.81, aic = 985.63
##
## $degrees_of_freedom
## [1] 726
##
## $ttable
## Estimate SE t.value p.value
## ar1 -0.2982 0.1675 -1.7799 0.0755
## ma1 -0.1708 0.1687 -1.0123 0.3117
## ma2 0.1033 0.0814 1.2700 0.2045
## constant 0.0226 0.0126 1.7978 0.0726
##
## $AIC
## [1] 1.350175
##
## $AICc
## [1] 1.350251
##
## $BIC
## [1] 1.381635There are no significant changes so the model fits.
Next I will use the model to forecast. Since my units of time are days I am going to forecast with a day count. I am going to forecast out one year. This gives the stakeholder some indication of potential revenue for the next year so they can plan accordingly. Also it’s not too far out that poor expectations are set should the company have bad month or quarter during the coming year.
sarima.for(time_series, n.ahead = 365, 1, 1, 1)
## $pred
## Time Series:
## Start = 731
## End = 1095
## Frequency = 1
## [1] 16.45229 16.56595 16.54521 16.58848 16.60126 16.62857 16.64896 16.67264
## [9] 16.69475 16.71761 16.74012 16.76279 16.78539 16.80802 16.83063 16.85326
## [17] 16.87587 16.89850 16.92111 16.94373 16.96635 16.98897 17.01159 17.03421
## [25] 17.05683 17.07945 17.10207 17.12469 17.14731 17.16993 17.19255 17.21517
## [33] 17.23779 17.26041 17.28303 17.30565 17.32827 17.35089 17.37351 17.39613
## [41] 17.41875 17.44137 17.46399 17.48661 17.50923 17.53185 17.55447 17.57709
## [49] 17.59971 17.62233 17.64495 17.66757 17.69019 17.71281 17.73543 17.75805
## [57] 17.78067 17.80329 17.82591 17.84853 17.87115 17.89377 17.91639 17.93900
## [65] 17.96162 17.98424 18.00686 18.02948 18.05210 18.07472 18.09734 18.11996
## [73] 18.14258 18.16520 18.18782 18.21044 18.23306 18.25568 18.27830 18.30092
## [81] 18.32354 18.34616 18.36878 18.39140 18.41402 18.43664 18.45926 18.48188
## [89] 18.50450 18.52712 18.54974 18.57236 18.59498 18.61760 18.64022 18.66284
## [97] 18.68546 18.70808 18.73070 18.75332 18.77594 18.79856 18.82118 18.84380
## [105] 18.86642 18.88904 18.91166 18.93427 18.95689 18.97951 19.00213 19.02475
## [113] 19.04737 19.06999 19.09261 19.11523 19.13785 19.16047 19.18309 19.20571
## [121] 19.22833 19.25095 19.27357 19.29619 19.31881 19.34143 19.36405 19.38667
## [129] 19.40929 19.43191 19.45453 19.47715 19.49977 19.52239 19.54501 19.56763
## [137] 19.59025 19.61287 19.63549 19.65811 19.68073 19.70335 19.72597 19.74859
## [145] 19.77121 19.79383 19.81645 19.83907 19.86169 19.88431 19.90693 19.92954
## [153] 19.95216 19.97478 19.99740 20.02002 20.04264 20.06526 20.08788 20.11050
## [161] 20.13312 20.15574 20.17836 20.20098 20.22360 20.24622 20.26884 20.29146
## [169] 20.31408 20.33670 20.35932 20.38194 20.40456 20.42718 20.44980 20.47242
## [177] 20.49504 20.51766 20.54028 20.56290 20.58552 20.60814 20.63076 20.65338
## [185] 20.67600 20.69862 20.72124 20.74386 20.76648 20.78910 20.81172 20.83434
## [193] 20.85696 20.87958 20.90220 20.92482 20.94743 20.97005 20.99267 21.01529
## [201] 21.03791 21.06053 21.08315 21.10577 21.12839 21.15101 21.17363 21.19625
## [209] 21.21887 21.24149 21.26411 21.28673 21.30935 21.33197 21.35459 21.37721
## [217] 21.39983 21.42245 21.44507 21.46769 21.49031 21.51293 21.53555 21.55817
## [225] 21.58079 21.60341 21.62603 21.64865 21.67127 21.69389 21.71651 21.73913
## [233] 21.76175 21.78437 21.80699 21.82961 21.85223 21.87485 21.89747 21.92009
## [241] 21.94270 21.96532 21.98794 22.01056 22.03318 22.05580 22.07842 22.10104
## [249] 22.12366 22.14628 22.16890 22.19152 22.21414 22.23676 22.25938 22.28200
## [257] 22.30462 22.32724 22.34986 22.37248 22.39510 22.41772 22.44034 22.46296
## [265] 22.48558 22.50820 22.53082 22.55344 22.57606 22.59868 22.62130 22.64392
## [273] 22.66654 22.68916 22.71178 22.73440 22.75702 22.77964 22.80226 22.82488
## [281] 22.84750 22.87012 22.89274 22.91536 22.93797 22.96059 22.98321 23.00583
## [289] 23.02845 23.05107 23.07369 23.09631 23.11893 23.14155 23.16417 23.18679
## [297] 23.20941 23.23203 23.25465 23.27727 23.29989 23.32251 23.34513 23.36775
## [305] 23.39037 23.41299 23.43561 23.45823 23.48085 23.50347 23.52609 23.54871
## [313] 23.57133 23.59395 23.61657 23.63919 23.66181 23.68443 23.70705 23.72967
## [321] 23.75229 23.77491 23.79753 23.82015 23.84277 23.86539 23.88801 23.91063
## [329] 23.93325 23.95586 23.97848 24.00110 24.02372 24.04634 24.06896 24.09158
## [337] 24.11420 24.13682 24.15944 24.18206 24.20468 24.22730 24.24992 24.27254
## [345] 24.29516 24.31778 24.34040 24.36302 24.38564 24.40826 24.43088 24.45350
## [353] 24.47612 24.49874 24.52136 24.54398 24.56660 24.58922 24.61184 24.63446
## [361] 24.65708 24.67970 24.70232 24.72494 24.74756
##
## $se
## Time Series:
## Start = 731
## End = 1095
## Frequency = 1
## [1] 0.4724336 0.5353010 0.6433411 0.7127343 0.7856844 0.8480705 0.9080933
## [8] 0.9635224 1.0163219 1.0663334 1.1141829 1.1600235 1.2041372 1.2466831
## [15] 1.2878277 1.3276963 1.3664028 1.4040424 1.4406991 1.4764458 1.5113474
## [22] 1.5454610 1.5788376 1.6115231 1.6435588 1.6749818 1.7058261 1.7361225
## [29] 1.7658992 1.7951820 1.8239948 1.8523595 1.8802963 1.9078241 1.9349604
## [36] 1.9617212 1.9881219 2.0141766 2.0398985 2.0653001 2.0903931 2.1151884
## [43] 2.1396964 2.1639268 2.1878889 2.2115914 2.2350425 2.2582501 2.2812216
## [50] 2.3039641 2.3264843 2.3487886 2.3708831 2.3927735 2.4144655 2.4359644
## [57] 2.4572751 2.4784027 2.4993516 2.5201264 2.5407313 2.5611705 2.5814478
## [64] 2.6015671 2.6215320 2.6413460 2.6610124 2.6805346 2.6999156 2.7191585
## [71] 2.7382662 2.7572414 2.7760870 2.7948055 2.8133994 2.8318712 2.8502234
## [78] 2.8684581 2.8865777 2.9045842 2.9224797 2.9402664 2.9579461 2.9755207
## [85] 2.9929922 3.0103623 3.0276327 3.0448051 3.0618812 3.0788627 3.0957510
## [92] 3.1125476 3.1292541 3.1458719 3.1624023 3.1788468 3.1952067 3.2114832
## [99] 3.2276776 3.2437912 3.2598252 3.2757806 3.2916588 3.3074607 3.3231874
## [106] 3.3388401 3.3544197 3.3699274 3.3853639 3.4007305 3.4160279 3.4312570
## [113] 3.4464189 3.4615144 3.4765444 3.4915096 3.5064110 3.5212493 3.5360253
## [120] 3.5507399 3.5653937 3.5799876 3.5945222 3.6089982 3.6234165 3.6377775
## [127] 3.6520822 3.6663310 3.6805246 3.6946637 3.7087489 3.7227808 3.7367601
## [134] 3.7506872 3.7645628 3.7783874 3.7921617 3.8058861 3.8195612 3.8331875
## [141] 3.8467655 3.8602958 3.8737788 3.8872150 3.9006050 3.9139491 3.9272479
## [148] 3.9405019 3.9537114 3.9668769 3.9799989 3.9930777 4.0061138 4.0191077
## [155] 4.0320597 4.0449702 4.0578396 4.0706684 4.0834568 4.0962054 4.1089143
## [162] 4.1215841 4.1342151 4.1468075 4.1593619 4.1718785 4.1843576 4.1967997
## [169] 4.2092049 4.2215737 4.2339064 4.2462032 4.2584646 4.2706907 4.2828819
## [176] 4.2950386 4.3071609 4.3192492 4.3313037 4.3433248 4.3553128 4.3672678
## [183] 4.3791902 4.3910802 4.4029381 4.4147641 4.4265586 4.4383217 4.4500537
## [190] 4.4617549 4.4734255 4.4850657 4.4966757 4.5082559 4.5198064 4.5313275
## [197] 4.5428193 4.5542821 4.5657162 4.5771217 4.5884988 4.5998478 4.6111689
## [204] 4.6224623 4.6337281 4.6449666 4.6561780 4.6673624 4.6785201 4.6896513
## [211] 4.7007561 4.7118347 4.7228874 4.7339142 4.7449154 4.7558912 4.7668417
## [218] 4.7777671 4.7886675 4.7995432 4.8103944 4.8212211 4.8320235 4.8428018
## [225] 4.8535563 4.8642869 4.8749939 4.8856775 4.8963377 4.9069748 4.9175888
## [232] 4.9281801 4.9387486 4.9492945 4.9598180 4.9703192 4.9807983 4.9912554
## [239] 5.0016906 5.0121041 5.0224960 5.0328665 5.0432156 5.0535435 5.0638504
## [246] 5.0741363 5.0844014 5.0946459 5.1048697 5.1150732 5.1252563 5.1354193
## [253] 5.1455621 5.1556850 5.1657881 5.1758715 5.1859352 5.1959795 5.2060043
## [260] 5.2160100 5.2259964 5.2359638 5.2459123 5.2558419 5.2657528 5.2756451
## [267] 5.2855189 5.2953743 5.3052113 5.3150302 5.3248309 5.3346137 5.3443785
## [274] 5.3541256 5.3638549 5.3735666 5.3832607 5.3929375 5.4025969 5.4122391
## [281] 5.4218641 5.4314721 5.4410631 5.4506373 5.4601946 5.4697353 5.4792593
## [288] 5.4887668 5.4982579 5.5077326 5.5171910 5.5266333 5.5360594 5.5454696
## [295] 5.5548638 5.5642421 5.5736046 5.5829515 5.5922827 5.6015984 5.6108986
## [302] 5.6201834 5.6294529 5.6387072 5.6479463 5.6571703 5.6663793 5.6755734
## [309] 5.6847526 5.6939170 5.7030666 5.7122017 5.7213221 5.7304280 5.7395195
## [316] 5.7485965 5.7576593 5.7667079 5.7757422 5.7847625 5.7937687 5.8027609
## [323] 5.8117392 5.8207037 5.8296543 5.8385913 5.8475146 5.8564243 5.8653205
## [330] 5.8742032 5.8830725 5.8919284 5.9007710 5.9096004 5.9184167 5.9272198
## [337] 5.9360099 5.9447869 5.9535511 5.9623023 5.9710407 5.9797664 5.9884793
## [344] 5.9971796 6.0058673 6.0145424 6.0232051 6.0318553 6.0404931 6.0491185
## [351] 6.0577317 6.0663327 6.0749215 6.0834981 6.0920627 6.1006153 6.1091559
## [358] 6.1176845 6.1262013 6.1347063 6.1431995 6.1516810 6.1601507 6.1686089
## [365] 6.1770555