Georgia tech ISYE 6501 Week 4 homework Exponential Smoothing

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ISYE 6501 Week 4 homework Exponential Smoothing Exponential smoothing assists with change detection as it smoothes out the data. The benfits of exponential smoothing are that it gives you smoother data (less noisy data) and the ability to forecast using trends and seanolaity for time series data. Additionally, data can be made to be less noisy for more confidence in a CUSUM change detection model, which will see in this assignment. Let’s pull in the data again. rm(list = ls()) temps <- read.table("/Users/paris.piotis/Desktop/OMSA/temps.txt", stringsAsFactors = FALSE, header=TRUE) head(temps) ## DAY X1996 X1997 X1998 X1999 X2000 X2001 X2002 X2003 X2004 X2005 X2006 X2007 ## 1 1-Jul 98 86 91 84 89 84 90 73 82 91 93 95 ## 2 2-Jul 97 90 88 82 91 87 90 81 81 89 93 85 ## 3 3-Jul 97 93 91 87 93 87 87 87 86 86 93 82 ## 4 4-Jul 90 91 91 88 95 84 89 86 88 86 91 86 ## 5 5-Jul 89 84 91 90 96 86 93 80 90 89 90 88 ## 6 6-Jul 93 84 89 91 96 87 93 84 90 82 81 87 ## X2008 X2009 X2010 X2011 X2012 X2013 X2014 X2015 ## 1 85 95 87 92 105 82 90 85 ## 2 87 90 84 94 93 85 93 87 ## 3 91 89 83 95 99 76 87 79 ## 4 90 91 85 92 98 77 84 85 ## 5 88 80 88 90 100 83 86 84 ## 6 82 87 89 90 98 83 87 84 tail(temps) ## DAY X1996 X1997 X1998 X1999 X2000 X2001 X2002 X2003 X2004 X2005 X2006 ## 118 26-Oct 75 71 79 69 75 64 68 68 79 61 62 ## 119 27-Oct 75 57 79 75 78 51 69 64 81 63 66 ## 120 28-Oct 81 55 79 73 80 55 75 57 78 62 63 ## 121 29-Oct 82 64 78 72 75 63 75 70 75 64 72 ## 122 30-Oct 82 66 82 75 77 72 68 77 78 69 73 ## 123 31-Oct 81 60 79 75 78 71 60 75 82 70 68 ## X2007 X2008 X2009 X2010 X2011 X2012 X2013 X2014 X2015 ## 118 68 70 65 85 77 80 61 84 67 ## 119 67 59 60 76 79 70 69 84 56 ## 120 70 50 71 74 74 56 64 77 78 ## 121 62 59 75 68 59 56 75 73 70 ## 122 67 65 66 71 61 56 78 68 70 ## 123 71 67 69 75 65 65 74 63 62 Note that there are 123 rows per year.16/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 2/96 Data transformation Holt winters exponential smoothing requires a time series input, so we need to transform the data into time series data. First, we transform the data table into a vector: temps_vec <- as.vector(unlist(temps[,2:21])) head(temps_vec) ## [1] 98 97 97 90 89 93 plot(temps_vec) Then we transform into time series We know from our data exploration that there are 123 days per cycle, that’s why frequency gets set to 123 and starts at 1996. temps_ts <- ts(temps_vec, start = 1996, frequency = 123) head(temps_ts) ## [1] 98 97 97 90 89 9316/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 3/96 plot(temps_ts) Visualise the time series using decompose The function decomposes a time series into seasonal, trend and irregular components using moving averages. Deals with additive or multiplicative seasonal component. This is a nice way to visualise the time series data before running the Holt Winter exponential smoothing. plot(decompose(temps_ts, type = "multiplicative"))16/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 4/96 It is interesting to note that the trend compnent in the data signals that the temperatures in the later years do seem to get hotter. We would need a method of change detection to prove that the increase has breached a valid threshold. Running Holt Winter exponential smoothing A note on additive versus multiplicative exponential smoothing. Additive smoothing is appropriate when faced with seasonality that is fairly constant. For example, every year sales decrease by 100 items in february due to the month being shorter and this is fairly consistent. Conversely, multiplicative seasonality is more appropriate for time series data where the seasonality is changing periodically and it is proportional to the base. For example, sales in December every increases by 15% compared to November, but over the years it has increased by a higher and higher percentage. In this example, it makes more sense to use multiplicative as the seasonality seems to be changing and it is proportional to the base. temps_HW <- HoltWinters(temps_ts, alpha = NULL, beta = NULL, gamma = NULL, seasonal = "multiplicative") head(temps_HW)16/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 5/96 ## $fitted ## Time Series: ## Start = c(1997, 1) ## End = c(2015, 123) ## Frequency = 123 ## xhat level trend season ## 1997.000 87.23653 82.87739 -0.004362918 1.0526529 ## 1997.008 90.42182 82.15059 -0.004362918 1.1007422 ## 1997.016 92.99734 81.91055 -0.004362918 1.1354128 ## 1997.024 90.94030 81.90763 -0.004362918 1.1103378 ## 1997.033 83.99917 81.93634 -0.004362918 1.0252306 ## 1997.041 84.04496 81.93247 -0.004362918 1.0258379 ## 1997.049 75.06333 81.90115 -0.004362918 0.9165601 ## 1997.057 87.04945 81.85429 -0.004362918 1.0635250 ## 1997.065 84.02220 81.82134 -0.004362918 1.0269532 ## 1997.073 87.06445 81.80368 -0.004362918 1.0643665 ## 1997.081 84.05272 81.76208 -0.004362918 1.0280709 ## 1997.089 88.05141 81.72617 -0.004362918 1.0774531 ## 1997.098 86.03178 81.69246 -0.004362918 1.0531740 ## 1997.106 89.92787 81.66954 -0.004362918 1.1011777 ## 1997.114 90.89514 81.70546 -0.004362918 1.1125326 ## 1997.122 90.93680 81.75907 -0.004362918 1.1123128 ## 1997.130 88.91874 81.78965 -0.004362918 1.0872217 ## 1997.138 88.89374 81.83125 -0.004362918 1.0863634 ## 1997.146 88.86666 81.88704 -0.004362918 1.0852925 ## 1997.154 89.83703 81.95824 -0.004362918 1.0961900 ## 1997.163 88.79334 82.04531 -0.004362918 1.0823051 ## 1997.171 83.82979 82.15838 -0.004362918 1.0203980 ## 1997.179 87.02152 82.25660 -0.004362918 1.0579836 ## 1997.187 88.03833 82.23973 -0.004362918 1.0705653 ## 1997.195 89.02771 82.21335 -0.004362918 1.0829438 ## 1997.203 89.19019 82.19325 -0.004362918 1.0851855 ## 1997.211 91.19587 82.08110 -0.004362918 1.1111050 ## 1997.220 91.20560 81.96832 -0.004362918 1.1127526 ## 1997.228 89.13435 81.85032 -0.004362918 1.0890501 ## 1997.236 88.00292 81.77009 -0.004362918 1.0762812 ## 1997.244 71.84081 81.76406 -0.004362918 0.8786824 ## 1997.252 79.85674 81.87112 -0.004362918 0.9754477 ## 1997.260 83.92934 81.95708 -0.004362918 1.0241191 ## 1997.268 88.04492 81.99515 -0.004362918 1.0738392 ## 1997.276 88.94256 81.96506 -0.004362918 1.0851855 ## 1997.285 87.84357 81.99325 -0.004362918 1.0714083 ## 1997.293 83.78960 82.07868 -0.004362918 1.0208992 ## 1997.301 83.73355 82.20106 -0.004362918 1.0186973 ## 1997.309 79.86675 82.35756 -0.004362918 0.9698076 ## 1997.317 72.87791 82.43769 -0.004362918 0.8840830 ## 1997.325 79.86607 82.51826 -0.004362918 0.9679105 ## 1997.333 85.83089 82.59900 -0.004362918 1.0391824 ## 1997.341 87.84293 82.69472 -0.004362918 1.0623117 ## 1997.350 87.87486 82.78128 -0.004362918 1.0615865 ## 1997.358 87.04117 82.84942 -0.004362918 1.0506501 ## 1997.366 88.16633 82.82096 -0.004362918 1.0645974 ## 1997.374 91.26671 82.72051 -0.004362918 1.1033724 ## 1997.382 91.24096 82.56748 -0.004362918 1.1051055 ## 1997.390 89.10351 82.42902 -0.004362918 1.0810298 ## 1997.398 88.98271 82.36577 -0.004362918 1.0803932 ## 1997.407 87.97149 82.37125 -0.004362918 1.068044416/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 6/96 ## 1997.415 82.05789 82.38330 -0.004362918 0.9961028 ## 1997.423 79.16500 82.34320 -0.004362918 0.9614539 ## 1997.431 81.10871 82.23329 -0.004362918 0.9863768 ## 1997.439 82.12695 82.16115 -0.004362918 0.9996368 ## 1997.447 84.02904 82.07869 -0.004362918 1.0238164 ## 1997.455 87.04126 82.05688 -0.004362918 1.0607993 ## 1997.463 90.17163 82.02860 -0.004362918 1.0993291 ## 1997.472 90.29310 81.92822 -0.004362918 1.1021588 ## 1997.480 91.27188 81.76031 -0.004362918 1.1163944 ## 1997.488 91.24459 81.60617 -0.004362918 1.1181688 ## 1997.496 84.25877 81.46728 -0.004362918 1.0343206 ## 1997.504 80.51733 83.68744 -0.004362918 0.9621698 ## 1997.512 77.72033 88.46587 -0.004362918 0.8785779 ## 1997.520 102.39420 97.75726 -0.004362918 1.0474800 ## 1997.528 93.27164 92.23731 -0.004362918 1.0112615 ## 1997.537 88.82479 84.76990 -0.004362918 1.0478880 ## 1997.545 86.01454 80.17319 -0.004362918 1.0729176 ## 1997.553 83.68771 77.86767 -0.004362918 1.0748030 ## 1997.561 85.13833 79.18640 -0.004362918 1.0752229 ## 1997.569 88.92440 80.81884 -0.004362918 1.1003524 ## 1997.577 79.33060 78.06216 -0.004362918 1.0163060 ## 1997.585 81.68431 78.46287 -0.004362918 1.0411147 ## 1997.593 83.80257 78.64499 -0.004362918 1.0656396 ## 1997.602 75.52504 79.90881 -0.004362918 0.9451919 ## 1997.610 83.66133 87.37080 -0.004362918 0.9575911 ## 1997.618 93.22283 89.51067 -0.004362918 1.0415224 ## 1997.626 85.81989 86.42231 -0.004362918 0.9930795 ## 1997.634 87.21610 87.76806 -0.004362918 0.9937606 ## 1997.642 84.63587 89.48655 -0.004362918 0.9458405 ## 1997.650 87.87254 91.66960 -0.004362918 0.9586244 ## 1997.659 89.87110 93.67165 -0.004362918 0.9594715 ## 1997.667 91.86176 96.95482 -0.004362918 0.9475124 ## 1997.675 93.66411 95.09297 -0.004362918 0.9850192 ## 1997.683 82.06845 80.31377 -0.004362918 1.0219033 ## 1997.691 80.83164 79.06457 -0.004362918 1.0224060 ## 1997.699 84.41654 79.76301 -0.004362918 1.0583998 ## 1997.707 70.53082 69.05737 -0.004362918 1.0214010 ## 1997.715 65.96298 68.73340 -0.004362918 0.9597542 ## 1997.724 61.40063 67.47117 -0.004362918 0.9100865 ## 1997.732 62.84325 71.92642 -0.004362918 0.8737688 ## 1997.740 63.61506 81.88630 -0.004362918 0.7769120 ## 1997.748 79.91419 99.60184 -0.004362918 0.8023716 ## 1997.756 83.92780 95.83084 -0.004362918 0.8758309 ## 1997.764 90.12375 88.15305 -0.004362918 1.0224060 ## 1997.772 67.32195 79.05137 -0.004362918 0.8516698 ## 1997.780 69.67696 86.75778 -0.004362918 0.8031610 ## 1997.789 74.35139 95.42379 -0.004362918 0.7792060 ## 1997.797 74.14035 101.45623 -0.004362918 0.7307934 ## 1997.805 102.73294 108.06620 -0.004362918 0.9506866 ## 1997.813 80.78130 94.64961 -0.004362918 0.8535167 ## 1997.821 82.72991 94.08228 -0.004362918 0.8793765 ## 1997.829 78.91909 93.56745 -0.004362918 0.8434853 ## 1997.837 80.82615 95.80944 -0.004362918 0.8436521 ## 1997.846 84.30297 94.47386 -0.004362918 0.8923830 ## 1997.854 88.32901 91.50402 -0.004362918 0.9653480 ## 1997.862 77.73427 78.54847 -0.004362918 0.9896894 ## 1997.870 67.82727 69.38808 -0.004362918 0.9775676 ## 1997.878 62.68712 62.57211 -0.004362918 1.001907816/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 7/96 ## 1997.886 52.09185 64.60130 -0.004362918 0.8064136 ## 1997.894 56.72255 73.67857 -0.004362918 0.7699105 ## 1997.902 69.37749 83.48141 -0.004362918 0.8310966 ## 1997.911 81.04889 83.93770 -0.004362918 0.9656341 ## 1997.919 76.18954 76.89640 -0.004362918 0.9908639 ## 1997.927 57.53302 68.08496 -0.004362918 0.8450721 ## 1997.935 64.48533 72.05921 -0.004362918 0.8949479 ## 1997.943 63.00914 70.34694 -0.004362918 0.8957468 ## 1997.951 72.30786 78.57528 -0.004362918 0.9202878 ## 1997.959 71.50660 77.69691 -0.004362918 0.9203791 ## 1997.967 67.54725 67.99915 -0.004362918 0.9934182 ## 1997.976 60.54091 60.22706 -0.004362918 1.0052839 ## 1997.984 62.64525 62.33887 -0.004362918 1.0049851 ## 1997.992 63.87067 64.38745 -0.004362918 0.9920411 ## 1998.000 65.04516 61.98352 -0.004362918 1.0494684 ## 1998.008 84.87634 77.18905 -0.004362918 1.0996526 ## 1998.016 89.61560 78.93165 -0.004362918 1.1354197 ## 1998.024 88.47600 79.67715 -0.004362918 1.1104920 ## 1998.033 83.11178 81.07061 -0.004362918 1.0252328 ## 1998.041 88.00054 85.79813 -0.004362918 1.0257218 ## 1998.049 79.16627 86.39303 -0.004362918 0.9163964 ## 1998.057 101.73335 95.67263 -0.004362918 1.0633971 ## 1998.065 94.23796 91.77411 -0.004362918 1.0268958 ## 1998.073 98.14237 92.22613 -0.004362918 1.0641997 ## 1998.081 90.55054 88.09417 -0.004362918 1.0279344 ## 1998.089 95.18590 88.35871 -0.004362918 1.0773199 ## 1998.098 87.51833 83.11045 -0.004362918 1.0530916 ## 1998.106 91.83509 83.38738 -0.004362918 1.1013644 ## 1998.114 89.77962 80.68310 -0.004362918 1.1128040 ## 1998.122 90.49864 81.35319 -0.004362918 1.1124762 ## 1998.130 86.35333 79.41470 -0.004362918 1.0874318 ## 1998.138 88.52662 81.47273 -0.004362918 1.0866380 ## 1998.146 89.96005 82.86822 -0.004362918 1.0856367 ## 1998.154 93.99549 85.71895 -0.004362918 1.0966103 ## 1998.163 90.99112 84.03464 -0.004362918 1.0828372 ## 1998.171 85.78181 84.03533 -0.004362918 1.0208357 ## 1998.179 90.94522 85.96976 -0.004362918 1.0579282 ## 1998.187 92.05251 85.99724 -0.004362918 1.0704667 ## 1998.195 92.45980 85.38819 -0.004362918 1.0828725 ## 1998.203 88.63122 81.71507 -0.004362918 1.0846953 ## 1998.211 90.34555 81.35282 -0.004362918 1.1105995 ## 1998.220 84.10078 75.61953 -0.004362918 1.1122211 ## 1998.228 84.66499 77.77124 -0.004362918 1.0887025 ## 1998.236 86.32935 80.21570 -0.004362918 1.0762737 ## 1998.244 72.35333 82.30882 -0.004362918 0.8790938 ## 1998.252 89.62603 91.85149 -0.004362918 0.9758175 ## 1998.260 91.73385 89.56184 -0.004362918 1.0243014 ## 1998.268 89.88008 83.71316 -0.004362918 1.0737233 ## 1998.276 87.19187 80.34082 -0.004362918 1.0853337 ## 1998.285 85.37699 79.66109 -0.004362918 1.0718115 ## 1998.293 84.06971 82.30940 -0.004362918 1.0214407 ## 1998.301 86.92183 85.27353 -0.004362918 1.0193818 ## 1998.309 83.93596 86.52294 -0.004362918 0.9701495 ## 1998.317 77.67002 87.82703 -0.004362918 0.8843960 ## 1998.325 87.94583 90.83371 -0.004362918 0.9682535 ## 1998.333 93.79846 90.22858 -0.004362918 1.0396150 ## 1998.341 92.23253 86.79404 -0.004362918 1.0627131 ## 1998.350 87.09867 82.02543 -0.004362918 1.061906116/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 8/96 ## 1998.358 85.49378 81.38477 -0.004362918 1.0505450 ## 1998.366 83.17559 78.16428 -0.004362918 1.0641720 ## 1998.374 85.43211 77.48052 -0.004362918 1.1026890 ## 1998.382 85.91642 77.79289 -0.004362918 1.1044871 ## 1998.390 82.91302 76.72142 -0.004362918 1.0807639 ## 1998.398 85.39603 79.04273 -0.004362918 1.0804377 ## 1998.407 87.21678 81.65902 -0.004362918 1.0681176 ## 1998.415 76.60801 76.92359 -0.004362918 0.9959540 ## 1998.423 78.30412 81.48379 -0.004362918 0.9610294 ## 1998.431 85.82979 87.04428 -0.004362918 0.9860969 ## 1998.439 87.70479 87.76975 -0.004362918 0.9993096 ## 1998.447 90.03060 87.94707 -0.004362918 1.0237415 ## 1998.455 93.25608 87.92432 -0.004362918 1.0606929 ## 1998.463 95.17176 86.61186 -0.004362918 1.0988859 ## 1998.472 91.58039 83.15341 -0.004362918 1.1014004 ## 1998.480 91.77910 82.26658 -0.004362918 1.1156896 ## 1998.488 92.67801 82.93522 -0.004362918 1.1175336 ## 1998.496 86.74185 83.10805 -0.004362918 1.0437786 ## 1998.504 83.90163 85.61263 -0.004362918 0.9800646 ## 1998.512 79.43400 87.55253 -0.004362918 0.9073177 ## 1998.520 92.98923 90.64312 -0.004362918 1.0259324 ## 1998.528 79.47995 81.05389 -0.004362918 0.9806344 ## 1998.537 90.03013 87.64715 -0.004362918 1.0272394 ## 1998.545 93.68956 88.22344 -0.004362918 1.0620101 ## 1998.553 92.42281 85.50339 -0.004362918 1.0809808 ## 1998.561 91.07385 84.12062 -0.004362918 1.0827141 ## 1998.569 90.14966 82.93826 -0.004362918 1.0870062 ## 1998.577 78.00900 76.62568 -0.004362918 1.0181109 ## 1998.585 79.82644 76.61588 -0.004362918 1.0419640 ## 1998.593 82.82348 77.30419 -0.004362918 1.0714575 ## 1998.602 75.86386 77.97514 -0.004362918 0.9729781 ## 1998.610 83.29172 86.27391 -0.004362918 0.9654824 ## 1998.618 91.17418 88.63168 -0.004362918 1.0287367 ## 1998.626 85.98411 86.13190 -0.004362918 0.9983347 ## 1998.634 87.39493 87.36938 -0.004362918 1.0003423 ## 1998.642 83.07599 87.12222 -0.004362918 0.9536046 ## 1998.650 83.45434 86.42393 -0.004362918 0.9656881 ## 1998.659 81.74469 84.21965 -0.004362918 0.9706633 ## 1998.667 79.40697 84.37705 -0.004362918 0.9411455 ## 1998.675 79.40859 86.06713 -0.004362918 0.9226823 ## 1998.683 93.28728 91.78927 -0.004362918 1.0163684 ## 1998.691 88.35836 86.16519 -0.004362918 1.0255050 ## 1998.699 81.90533 81.74796 -0.004362918 1.0019785 ## 1998.707 83.41424 81.80171 -0.004362918 1.0197671 ## 1998.715 78.33457 82.15060 -0.004362918 0.9535990 ## 1998.724 81.54528 87.73482 -0.004362918 0.9294980 ## 1998.732 78.93023 86.70802 -0.004362918 0.9103449 ## 1998.740 69.29105 84.04851 -0.004362918 0.8244602 ## 1998.748 69.89014 88.30271 -0.004362918 0.7915226 ## 1998.756 85.64960 100.81551 -0.004362918 0.8496044 ## 1998.764 93.54797 95.27383 -0.004362918 0.9819302 ## 1998.772 74.51818 84.90514 -0.004362918 0.8777091 ## 1998.780 74.65890 90.14322 -0.004362918 0.8282655 ## 1998.789 76.00527 95.58976 -0.004362918 0.7951556 ## 1998.797 69.58630 93.26102 -0.004362918 0.7461805 ## 1998.805 93.58477 103.48804 -0.004362918 0.9043433 ## 1998.813 73.89895 86.76469 -0.004362918 0.8517598 ## 1998.821 74.94450 85.38921 -0.004362918 0.877726116/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 9/96 ## 1998.829 71.43955 84.02237 -0.004362918 0.8502886 ## 1998.837 74.51851 88.76310 -0.004362918 0.8395626 ## 1998.846 80.57035 91.30902 -0.004362918 0.8824342 ## 1998.854 81.50619 89.51328 -0.004362918 0.9105930 ## 1998.862 80.40923 85.11472 -0.004362918 0.9447643 ## 1998.870 79.21806 84.19301 -0.004362918 0.9409589 ## 1998.878 84.45198 83.39253 -0.004362918 1.0127573 ## 1998.886 66.28908 78.86291 -0.004362918 0.8406075 ## 1998.894 70.02729 87.42646 -0.004362918 0.8010251 ## 1998.902 80.44384 96.61438 -0.004362918 0.8326657 ## 1998.911 86.58862 92.58922 -0.004362918 0.9352352 ## 1998.919 79.19245 83.64909 -0.004362918 0.9467716 ## 1998.927 62.96712 73.12645 -0.004362918 0.8611231 ## 1998.935 64.91087 73.14557 -0.004362918 0.8874733 ## 1998.943 72.43246 78.05385 -0.004362918 0.9280325 ## 1998.951 73.10585 79.75098 -0.004362918 0.9167266 ## 1998.959 73.25489 83.70082 -0.004362918 0.8752448 ## 1998.967 83.28482 87.73334 -0.004362918 0.9493421 ## 1998.976 86.39494 84.95319 -0.004362918 1.0170234 ## 1998.984 81.14651 79.87233 -0.004362918 1.0160082 ## 1998.992 78.67855 80.38460 -0.004362918 0.9788295 ## 1999.000 90.29613 80.58220 -0.004362918 1.1206074 ## 1999.008 85.44878 77.12245 -0.004362918 1.1080252 ## 1999.016 85.65942 75.20386 -0.004362918 1.1390957 ## 1999.024 84.80741 75.92328 -0.004362918 1.1170788 ## 1999.033 81.14293 77.67658 -0.004362918 1.0446840 ## 1999.041 85.21673 82.88636 -0.004362918 1.0281693 ## 1999.049 81.76000 86.34128 -0.004362918 0.9469877 ## 1999.057 90.62904 86.49278 -0.004362918 1.0478748 ## 1999.065 86.16667 83.77162 -0.004362918 1.0286439 ## 1999.073 88.22532 84.26548 -0.004362918 1.0470467 ## 1999.081 85.96049 83.54140 -0.004362918 1.0290106 ## 1999.089 85.54340 81.16999 -0.004362918 1.0539363 ## 1999.098 80.31334 76.18030 -0.004362918 1.0543137 ## 1999.106 78.28257 71.90992 -0.004362918 1.0886859 ## 1999.114 81.95325 73.44064 -0.004362918 1.1159777 ## 1999.122 80.42732 72.91096 -0.004362918 1.1031556 ## 1999.130 83.37434 76.01333 -0.004362918 1.0969014 ## 1999.138 82.22725 75.23841 -0.004362918 1.0929526 ## 1999.146 85.55693 77.91967 -0.004362918 1.0980760 ## 1999.154 86.34055 79.28361 -0.004362918 1.0890688 ## 1999.163 88.08131 81.34575 -0.004362918 1.0828596 ## 1999.171 84.79696 82.43110 -0.004362918 1.0287555 ## 1999.179 91.13163 86.13499 -0.004362918 1.0580630 ## 1999.187 93.13039 87.21663 -0.004362918 1.0678589 ## 1999.195 92.89644 87.13717 -0.004362918 1.0661477 ## 1999.203 93.17998 86.03885 -0.004362918 1.0830538 ## 1999.211 92.94438 85.93229 -0.004362918 1.0816550 ## 1999.220 96.51293 85.95955 -0.004362918 1.1228284 ## 1999.228 92.44079 84.03107 -0.004362918 1.1001359 ## 1999.236 91.56319 84.33931 -0.004362918 1.0857087 ## 1999.244 79.58941 87.41465 -0.004362918 0.9105268 ## 1999.252 97.22483 100.52091 -0.004362918 0.9672520 ## 1999.260 99.70362 99.73777 -0.004362918 0.9997013 ## 1999.268 101.17300 95.60943 -0.004362918 1.0582390 ## 1999.276 95.17142 87.94948 -0.004362918 1.0821683 ## 1999.285 91.50010 84.43786 -0.004362918 1.0836943 ## 1999.293 86.97874 84.14969 -0.004362918 1.033672916/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 10/96 ## 1999.301 89.86943 87.72778 -0.004362918 1.0244633 ## 1999.309 87.36933 89.60276 -0.004362918 0.9751216 ## 1999.317 83.31669 93.14962 -0.004362918 0.8944812 ## 1999.325 95.08070 98.42794 -0.004362918 0.9660358 ## 1999.333 97.61011 95.18908 -0.004362918 1.0254810 ## 1999.341 98.12269 94.21910 -0.004362918 1.0414792 ## 1999.350 99.69677 94.14228 -0.004362918 1.0590500 ## 1999.358 95.86987 92.57188 -0.004362918 1.0356751 ## 1999.366 99.54801 93.83243 -0.004362918 1.0609619 ## 1999.374 99.41198 90.03241 -0.004362918 1.1042334 ## 1999.382 95.02883 86.45689 -0.004362918 1.0992024 ## 1999.390 94.96893 86.99590 -0.004362918 1.0917031 ## 1999.398 96.88707 88.69907 -0.004362918 1.0923658 ## 1999.407 93.38251 89.32129 -0.004362918 1.0455187 ## 1999.415 88.04839 86.73901 -0.004362918 1.0151467 ## 1999.423 86.93974 88.52281 -0.004362918 0.9821652 ## 1999.431 90.04738 91.06086 -0.004362918 0.9889176 ## 1999.439 91.02447 91.02704 -0.004362918 1.0000197 ## 1999.447 86.23211 84.24273 -0.004362918 1.0236679 ## 1999.455 86.19932 81.69578 -0.004362918 1.0551820 ## 1999.463 90.25019 83.32377 -0.004362918 1.0831832 ## 1999.472 90.02265 82.04181 -0.004362918 1.0973361 ## 1999.480 91.76475 82.02475 -0.004362918 1.1188041 ## 1999.488 91.25276 81.60001 -0.004362918 1.1183533 ## 1999.496 85.87526 81.45665 -0.004362918 1.0543013 ## 1999.504 79.35376 80.35840 -0.004362918 0.9875516 ## 1999.512 78.70182 85.73853 -0.004362918 0.9179750 ## 1999.520 92.48410 93.97341 -0.004362918 0.9841975 ## 1999.528 89.19763 88.66752 -0.004362918 1.0060281 ## 1999.537 93.67320 90.98762 -0.004362918 1.0295653 ## 1999.545 97.02477 92.37315 -0.004362918 1.0504065 ## 1999.553 98.63643 91.76879 -0.004362918 1.0748874 ## 1999.561 94.15698 87.39520 -0.004362918 1.0774239 ## 1999.569 90.39614 85.58881 -0.004362918 1.0562216 ## 1999.577 79.18668 77.78431 -0.004362918 1.0180861 ## 1999.585 86.22223 82.49980 -0.004362918 1.0451758 ## 1999.593 89.14082 82.95309 -0.004362918 1.0746497 ## 1999.602 82.14344 81.72357 -0.004362918 1.0051913 ## 1999.610 81.91655 84.07876 -0.004362918 0.9743341 ## 1999.618 88.89196 87.28309 -0.004362918 1.0184837 ## 1999.626 87.62050 87.34396 -0.004362918 1.0032162 ## 1999.634 83.22530 83.28102 -0.004362918 0.9993833 ## 1999.642 77.88723 81.90725 -0.004362918 0.9509706 ## 1999.650 80.92318 84.56266 -0.004362918 0.9570105 ## 1999.659 80.92710 83.32240 -0.004362918 0.9713035 ## 1999.667 71.18580 75.13295 -0.004362918 0.9475195 ## 1999.675 75.58365 80.20052 -0.004362918 0.9424847 ## 1999.683 77.35234 77.85770 -0.004362918 0.9935649 ## 1999.691 76.88652 76.39727 -0.004362918 1.0064614 ## 1999.699 77.24030 77.07331 -0.004362918 1.0022233 ## 1999.707 81.06037 79.37605 -0.004362918 1.0212756 ## 1999.715 77.89446 79.93752 -0.004362918 0.9744950 ## 1999.724 74.05394 79.99976 -0.004362918 0.9257275 ## 1999.732 75.58507 83.94564 -0.004362918 0.9004518 ## 1999.740 71.16071 84.90766 -0.004362918 0.8381383 ## 1999.748 69.97230 84.78538 -0.004362918 0.8253298 ## 1999.756 72.46503 87.03714 -0.004362918 0.8326177 ## 1999.764 83.62868 88.90520 -0.004362918 0.940696216/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 11/96 ## 1999.772 79.80358 89.14360 -0.004362918 0.8952689 ## 1999.780 70.16830 83.09163 -0.004362918 0.8445133 ## 1999.789 67.12308 85.14941 -0.004362918 0.7883381 ## 1999.797 68.02458 88.16953 -0.004362918 0.7715585 ## 1999.805 77.79141 92.13103 -0.004362918 0.8443962 ## 1999.813 75.07664 88.63692 -0.004362918 0.8470548 ## 1999.821 75.40762 86.39877 -0.004362918 0.8728299 ## 1999.829 72.72821 83.99337 -0.004362918 0.8659253 ## 1999.837 71.35147 84.18204 -0.004362918 0.8476293 ## 1999.846 72.90673 83.19711 -0.004362918 0.8763591 ## 1999.854 68.81487 76.94227 -0.004362918 0.8944208 ## 1999.862 76.41455 81.19080 -0.004362918 0.9412231 ## 1999.870 74.04565 78.95534 -0.004362918 0.9378687 ## 1999.878 80.29879 80.88828 -0.004362918 0.9927659 ## 1999.886 70.11921 80.69882 -0.004362918 0.8689470 ## 1999.894 67.26600 81.31784 -0.004362918 0.8272429 ## 1999.902 65.91991 80.37229 -0.004362918 0.8202265 ## 1999.911 68.39817 75.92920 -0.004362918 0.9008668 ## 1999.919 65.62063 72.92230 -0.004362918 0.8999244 ## 1999.927 67.13764 77.96095 -0.004362918 0.8612182 ## 1999.935 64.11453 70.71722 -0.004362918 0.9066885 ## 1999.943 62.85827 67.24369 -0.004362918 0.9348437 ## 1999.951 63.33752 67.99044 -0.004362918 0.9316249 ## 1999.959 63.76594 71.72410 -0.004362918 0.8890989 ## 1999.967 74.61136 79.49051 -0.004362918 0.9386713 ## 1999.976 78.01719 78.43040 -0.004362918 0.9947869 ## 1999.984 76.06534 74.70605 -0.004362918 1.0182545 ## 1999.992 72.54863 74.05825 -0.004362918 0.9796735 ## 2000.000 83.39938 75.59277 -0.004362918 1.1033356 ## 2000.008 86.44444 78.71021 -0.004362918 1.0983230 ## 2000.016 92.85774 81.25672 -0.004362918 1.1428313 ## 2000.024 91.55309 81.32891 -0.004362918 1.1257743 ## 2000.033 88.80208 83.20757 -0.004362918 1.0672915 ## 2000.041 91.04477 87.35085 -0.004362918 1.0423403 ## 2000.049 85.53361 90.27018 -0.004362918 0.9475747 ## 2000.057 100.56627 97.05880 -0.004362918 1.0361841 ## 2000.065 94.18067 91.37660 -0.004362918 1.0307361 ## 2000.073 96.51614 92.45777 -0.004362918 1.0439436 ## 2000.081 95.66733 93.91669 -0.004362918 1.0186877 ## 2000.089 96.95181 94.11316 -0.004362918 1.0302098 ## 2000.098 94.75432 91.74969 -0.004362918 1.0327972 ## 2000.106 98.14391 89.50973 -0.004362918 1.0965142 ## 2000.114 96.42187 86.62030 -0.004362918 1.1132116 ## 2000.122 94.77465 84.72550 -0.004362918 1.1186659 ## 2000.130 91.53214 83.74550 -0.004362918 1.0930368 ## 2000.138 92.27433 83.44172 -0.004362918 1.1059115 ## 2000.146 95.06253 86.06533 -0.004362918 1.1045952 ## 2000.154 97.56065 88.80999 -0.004362918 1.0985864 ## 2000.163 97.47310 89.61139 -0.004362918 1.0877841 ## 2000.171 90.90420 87.07806 -0.004362918 1.0439915 ## 2000.179 95.70924 90.07558 -0.004362918 1.0625952 ## 2000.187 90.76903 85.03053 -0.004362918 1.0675423 ## 2000.195 84.88694 79.97440 -0.004362918 1.0614844 ## 2000.203 80.37017 74.24173 -0.004362918 1.0826107 ## 2000.211 81.30626 75.16324 -0.004362918 1.0817919 ## 2000.220 87.96008 78.96429 -0.004362918 1.1139839 ## 2000.228 88.82128 80.63819 -0.004362918 1.1015387 ## 2000.236 88.71071 80.73361 -0.004362918 1.098867116/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 12/96 ## 2000.244 75.88916 79.77181 -0.004362918 0.9513801 ## 2000.252 83.24872 86.30342 -0.004362918 0.9646538 ## 2000.260 86.71634 88.05310 -0.004362918 0.9848675 ## 2000.268 86.71767 84.47916 -0.004362918 1.0265508 ## 2000.276 88.36831 82.84665 -0.004362918 1.0667053 ## 2000.285 89.43698 82.62994 -0.004362918 1.0824370 ## 2000.293 87.53390 83.51363 -0.004362918 1.0481939 ## 2000.301 88.26339 85.54292 -0.004362918 1.0318551 ## 2000.309 86.11145 87.16962 -0.004362918 0.9879102 ## 2000.317 82.17564 90.20852 -0.004362918 0.9109961 ## 2000.325 95.02797 99.53683 -0.004362918 0.9547435 ## 2000.333 101.68592 99.51445 -0.004362918 1.0218655 ## 2000.341 95.65618 91.87515 -0.004362918 1.0412035 ## 2000.350 92.58543 87.93921 -0.004362918 1.0528868 ## 2000.358 89.31066 85.84055 -0.004362918 1.0404779 ## 2000.366 89.55161 85.65257 -0.004362918 1.0455749 ## 2000.374 96.07519 88.26473 -0.004362918 1.0885429 ## 2000.382 97.79521 88.78286 -0.004362918 1.1015642 ## 2000.390 98.29599 89.45113 -0.004362918 1.0989328 ## 2000.398 99.59663 90.96003 -0.004362918 1.0950019 ## 2000.407 94.93472 91.74386 -0.004362918 1.0348293 ## 2000.415 95.02640 92.96691 -0.004362918 1.0222009 ## 2000.423 87.38872 88.13349 -0.004362918 0.9915986 ## 2000.431 86.28677 87.26783 -0.004362918 0.9888075 ## 2000.439 85.88100 88.32903 -0.004362918 0.9723331 ## 2000.447 93.36216 92.19495 -0.004362918 1.0127081 ## 2000.455 97.05005 91.36337 -0.004362918 1.0622932 ## 2000.463 94.02634 87.27746 -0.004362918 1.0773806 ## 2000.472 93.23609 84.97474 -0.004362918 1.0972776 ## 2000.480 94.11811 84.27757 -0.004362918 1.1168213 ## 2000.488 92.88335 83.10682 -0.004362918 1.1176968 ## 2000.496 84.38050 80.41544 -0.004362918 1.0493642 ## 2000.504 82.67013 81.94629 -0.004362918 1.0088868 ## 2000.512 75.36957 79.70467 -0.004362918 0.9456623 ## 2000.520 80.35293 83.36200 -0.004362918 0.9639540 ## 2000.528 85.65915 84.40847 -0.004362918 1.0148694 ## 2000.537 88.18573 85.21665 -0.004362918 1.0348944 ## 2000.545 84.82626 80.94206 -0.004362918 1.0480439 ## 2000.553 73.82753 69.89025 -0.004362918 1.0564012 ## 2000.561 69.87250 65.32895 -0.004362918 1.0696202 ## 2000.569 69.61912 68.27276 -0.004362918 1.0197854 ## 2000.577 77.36553 74.52881 -0.004362918 1.0381229 ## 2000.585 80.90943 77.26999 -0.004362918 1.0471594 ## 2000.593 84.54115 79.08074 -0.004362918 1.0691075 ## 2000.602 81.10153 79.91558 -0.004362918 1.0148955 ## 2000.610 82.36720 83.48555 -0.004362918 0.9866559 ## 2000.618 87.34847 85.74558 -0.004362918 1.0187454 ## 2000.626 80.19480 81.30504 -0.004362918 0.9863976 ## 2000.634 77.56064 78.06181 -0.004362918 0.9936354 ## 2000.642 72.31591 75.23467 -0.004362918 0.9612602 ## 2000.650 72.04141 75.66799 -0.004362918 0.9521274 ## 2000.659 77.95553 83.38798 -0.004362918 0.9349023 ## 2000.667 86.48230 89.33330 -0.004362918 0.9681331 ## 2000.675 77.69872 83.30534 -0.004362918 0.9327467 ## 2000.683 79.16001 80.20290 -0.004362918 0.9870506 ## 2000.691 82.11481 81.34498 -0.004362918 1.0095179 ## 2000.699 83.49469 82.48908 -0.004362918 1.0122443 ## 2000.707 83.51060 81.57660 -0.004362918 1.023762516/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 13/96 ## 2000.715 70.42765 72.25458 -0.004362918 0.9747741 ## 2000.724 68.30233 72.61133 -0.004362918 0.9407132 ## 2000.732 69.58934 76.98565 -0.004362918 0.9039774 ## 2000.740 66.43030 79.30166 -0.004362918 0.8377373 ## 2000.748 71.26502 85.58852 -0.004362918 0.8326894 ## 2000.756 75.32378 89.81986 -0.004362918 0.8386501 ## 2000.764 87.10251 92.51137 -0.004362918 0.9415774 ## 2000.772 77.83223 89.17424 -0.004362918 0.8728534 ## 2000.780 77.82944 91.40185 -0.004362918 0.8515490 ## 2000.789 75.30142 94.40953 -0.004362918 0.7976409 ## 2000.797 72.52493 92.63071 -0.004362918 0.7829838 ## 2000.805 72.88139 87.50127 -0.004362918 0.8329596 ## 2000.813 62.36809 74.29446 -0.004362918 0.8395210 ## 2000.821 59.53633 68.89249 -0.004362918 0.8642466 ## 2000.829 62.44792 72.06450 -0.004362918 0.8666084 ## 2000.837 65.95246 78.12926 -0.004362918 0.8441926 ## 2000.846 70.92210 83.25911 -0.004362918 0.8518686 ## 2000.854 78.48328 86.19877 -0.004362918 0.9105380 ## 2000.862 78.14257 83.84170 -0.004362918 0.9320736 ## 2000.870 78.55924 83.08344 -0.004362918 0.9455959 ## 2000.878 83.33821 84.01613 -0.004362918 0.9919825 ## 2000.886 71.38738 81.94217 -0.004362918 0.8712386 ## 2000.894 72.51486 88.01741 -0.004362918 0.8239104 ## 2000.902 71.02641 88.37518 -0.004362918 0.8037316 ## 2000.911 79.82001 89.88097 -0.004362918 0.8881066 ## 2000.919 79.60737 86.53881 -0.004362918 0.9199501 ## 2000.927 71.55955 86.12841 -0.004362918 0.8308893 ## 2000.935 78.96585 88.67058 -0.004362918 0.8905969 ## 2000.943 80.63005 85.92759 -0.004362918 0.9383965 ## 2000.951 79.84462 84.19955 -0.004362918 0.9483276 ## 2000.959 74.48394 81.05339 -0.004362918 0.9189986 ## 2000.967 77.92048 83.40201 -0.004362918 0.9343247 ## 2000.976 82.87582 84.76645 -0.004362918 0.9777463 ## 2000.984 81.01774 79.80819 -0.004362918 1.0152111 ## 2000.992 76.32379 77.36993 -0.004362918 0.9865343 ## 2001.000 87.68863 78.41052 -0.004362918 1.1183896 ## 2001.008 84.78855 76.37778 -0.004362918 1.1101842 ## 2001.016 88.70570 77.59848 -0.004362918 1.1432014 ## 2001.024 86.98750 76.67651 -0.004362918 1.1345385 ## 2001.033 81.40681 75.05270 -0.004362918 1.0847250 ## 2001.041 81.83758 77.65252 -0.004362918 1.0539539 ## 2001.049 78.26787 80.66054 -0.004362918 0.9703890 ## 2001.057 87.39561 86.19033 -0.004362918 1.0140352 ## 2001.065 90.19628 87.15902 -0.004362918 1.0348992 ## 2001.073 91.96891 87.63228 -0.004362918 1.0495390 ## 2001.081 86.35832 84.71626 -0.004362918 1.0194356 ## 2001.089 88.73793 86.90884 -0.004362918 1.0210973 ## 2001.098 89.75739 87.66462 -0.004362918 1.0239235 ## 2001.106 92.56837 85.40344 -0.004362918 1.0839505 ## 2001.114 87.70892 79.40288 -0.004362918 1.1046670 ## 2001.122 84.91834 76.22018 -0.004362918 1.1141826 ## 2001.130 82.64572 75.70891 -0.004362918 1.0916876 ## 2001.138 87.33837 78.15754 -0.004362918 1.1175281 ## 2001.146 87.64849 78.51729 -0.004362918 1.1163574 ## 2001.154 87.94279 79.80838 -0.004362918 1.1019846 ## 2001.163 85.37091 79.27785 -0.004362918 1.0769162 ## 2001.171 82.87834 78.49060 -0.004362918 1.0559603 ## 2001.179 84.19253 80.88673 -0.004362918 1.040925616/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 14/96 ## 2001.187 88.04785 84.31356 -0.004362918 1.0443446 ## 2001.195 84.64987 81.92546 -0.004362918 1.0333097 ## 2001.203 87.34506 80.34396 -0.004362918 1.0871982 ## 2001.211 88.75419 80.71008 -0.004362918 1.0997262 ## 2001.220 91.32532 81.40241 -0.004362918 1.1219595 ## 2001.228 85.27144 77.38267 -0.004362918 1.1020070 ## 2001.236 86.94974 79.45913 -0.004362918 1.0943301 ## 2001.244 78.68154 80.60699 -0.004362918 0.9761660 ## 2001.252 83.37241 85.84341 -0.004362918 0.9712644 ## 2001.260 83.69269 86.23644 -0.004362918 0.9705517 ## 2001.268 88.11725 86.42681 -0.004362918 1.0196107 ## 2001.276 89.45440 83.93903 -0.004362918 1.0657623 ## 2001.285 89.01599 81.94128 -0.004362918 1.0863966 ## 2001.293 85.97577 81.36177 -0.004362918 1.0567663 ## 2001.301 83.29080 80.20758 -0.004362918 1.0384970 ## 2001.309 81.75216 81.80762 -0.004362918 0.9993753 ## 2001.317 80.53764 85.64810 -0.004362918 0.9403798 ## 2001.325 85.79337 89.87008 -0.004362918 0.9546840 ## 2001.333 90.61220 91.28722 -0.004362918 0.9926530 ## 2001.341 90.64831 88.42535 -0.004362918 1.0251900 ## 2001.350 89.40006 85.63250 -0.004362918 1.0440500 ## 2001.358 83.87936 80.68004 -0.004362918 1.0397106 ## 2001.366 87.15778 82.52158 -0.004362918 1.0562374 ## 2001.374 87.99513 80.67858 -0.004362918 1.0907467 ## 2001.382 90.34134 81.80463 -0.004362918 1.1044137 ## 2001.390 90.80802 82.16705 -0.004362918 1.1052221 ## 2001.398 90.34673 82.26952 -0.004362918 1.0982381 ## 2001.407 83.56428 80.39102 -0.004362918 1.0395292 ## 2001.415 82.06333 81.82767 -0.004362918 1.0029334 ## 2001.423 84.45364 85.46370 -0.004362918 0.9882319 ## 2001.431 88.27636 88.91098 -0.004362918 0.9929110 ## 2001.439 87.52209 88.73544 -0.004362918 0.9863747 ## 2001.447 93.02249 92.14654 -0.004362918 1.0095538 ## 2001.455 94.37828 90.30093 -0.004362918 1.0452034 ## 2001.463 94.25224 88.30877 -0.004362918 1.0673560 ## 2001.472 94.56528 86.43049 -0.004362918 1.0941746 ## 2001.480 87.57743 78.80149 -0.004362918 1.1114292 ## 2001.488 86.08974 77.92427 -0.004362918 1.1048491 ## 2001.496 79.29655 75.08675 -0.004362918 1.0561271 ## 2001.504 76.58676 76.65666 -0.004362918 0.9991449 ## 2001.512 75.59514 78.75325 -0.004362918 0.9599519 ## 2001.520 75.86207 78.36760 -0.004362918 0.9680823 ## 2001.528 77.93358 76.54503 -0.004362918 1.0181983 ## 2001.537 79.65148 78.39282 -0.004362918 1.0161124 ## 2001.545 83.89035 84.65190 -0.004362918 0.9910547 ## 2001.553 89.90097 87.19780 -0.004362918 1.0310520 ## 2001.561 92.76624 85.46307 -0.004362918 1.0855095 ## 2001.569 85.64102 81.62525 -0.004362918 1.0492538 ## 2001.577 85.98473 81.83129 -0.004362918 1.0508121 ## 2001.585 88.22035 83.59166 -0.004362918 1.0554277 ## 2001.593 88.91896 82.87620 -0.004362918 1.0729696 ## 2001.602 82.43692 80.05239 -0.004362918 1.0298432 ## 2001.610 80.62238 80.98147 -0.004362918 0.9956193 ## 2001.618 84.26284 84.29892 -0.004362918 0.9996238 ## 2001.626 77.61225 79.82620 -0.004362918 0.9723185 ## 2001.634 77.90599 79.43458 -0.004362918 0.9808105 ## 2001.642 78.36944 81.37027 -0.004362918 0.9631729 ## 2001.650 81.58531 83.04557 -0.004362918 0.982467816/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 15/96 ## 2001.659 79.65821 83.30079 -0.004362918 0.9563220 ## 2001.667 81.26794 86.08860 -0.004362918 0.9440514 ## 2001.675 82.06096 89.16694 -0.004362918 0.9203521 ## 2001.683 91.70378 92.46297 -0.004362918 0.9918361 ## 2001.691 91.45169 90.16202 -0.004362918 1.0143530 ## 2001.699 77.18130 76.54518 -0.004362918 1.0083679 ## 2001.707 68.20737 69.72135 -0.004362918 0.9783466 ## 2001.715 70.39825 72.10109 -0.004362918 0.9764418 ## 2001.724 71.92505 74.99509 -0.004362918 0.9591192 ## 2001.732 72.02505 78.88609 -0.004362918 0.9130765 ## 2001.740 67.15638 78.19130 -0.004362918 0.8589207 ## 2001.748 68.48675 80.93904 -0.004362918 0.8461978 ## 2001.756 72.56235 85.66840 -0.004362918 0.8470573 ## 2001.764 84.63747 91.06412 -0.004362918 0.9294717 ## 2001.772 78.02723 88.65295 -0.004362918 0.8801858 ## 2001.780 77.50004 90.02700 -0.004362918 0.8608950 ## 2001.789 72.17815 91.09418 -0.004362918 0.7923845 ## 2001.797 68.58443 89.39926 -0.004362918 0.7672074 ## 2001.805 69.54055 88.92641 -0.004362918 0.7820394 ## 2001.813 78.71319 96.36105 -0.004362918 0.8168939 ## 2001.821 76.13751 86.78549 -0.004362918 0.8773510 ## 2001.829 75.25418 84.58181 -0.004362918 0.8897665 ## 2001.837 72.75906 84.40176 -0.004362918 0.8621007 ## 2001.846 75.95880 88.13616 -0.004362918 0.8618773 ## 2001.854 80.78218 89.58832 -0.004362918 0.9017483 ## 2001.862 79.56998 85.64044 -0.004362918 0.9291641 ## 2001.870 78.41245 82.61127 -0.004362918 0.9492240 ## 2001.878 70.77160 71.97326 -0.004362918 0.9833637 ## 2001.886 58.17941 65.23226 -0.004362918 0.8919406 ## 2001.894 57.12541 69.24126 -0.004362918 0.8250718 ## 2001.902 64.32625 79.57893 -0.004362918 0.8083771 ## 2001.911 76.84533 87.69501 -0.004362918 0.8763229 ## 2001.919 81.92503 89.20280 -0.004362918 0.9184582 ## 2001.927 73.76098 87.90943 -0.004362918 0.8390982 ## 2001.935 82.09923 93.21079 -0.004362918 0.8808324 ## 2001.943 84.82826 91.04252 -0.004362918 0.9317880 ## 2001.951 77.87388 83.23121 -0.004362918 0.9356822 ## 2001.959 68.76195 74.10786 -0.004362918 0.9279178 ## 2001.967 58.55706 62.33127 -0.004362918 0.9395149 ## 2001.976 57.40645 59.99847 -0.004362918 0.9568681 ## 2001.984 63.85349 63.58922 -0.004362918 1.0042247 ## 2001.992 67.95633 68.57391 -0.004362918 0.9910570 ## 2002.000 78.07509 70.45831 -0.004362918 1.1081721 ## 2002.008 86.02384 77.07192 -0.004362918 1.1162135 ## 2002.016 90.23022 79.25831 -0.004362918 1.1384950 ## 2002.024 87.27931 77.50901 -0.004362918 1.1261171 ## 2002.033 86.06745 78.44437 -0.004362918 1.0972392 ## 2002.041 87.87757 82.32570 -0.004362918 1.0674945 ## 2002.049 84.57088 85.27246 -0.004362918 0.9918232 ## 2002.057 89.58810 88.01448 -0.004362918 1.0179296 ## 2002.065 90.87944 87.65480 -0.004362918 1.0368396 ## 2002.073 90.35935 87.12880 -0.004362918 1.0371299 ## 2002.081 89.97627 87.50433 -0.004362918 1.0283007 ## 2002.089 85.94746 83.92569 -0.004362918 1.0241432 ## 2002.098 79.69193 78.54834 -0.004362918 1.0146155 ## 2002.106 84.39853 79.94300 -0.004362918 1.0557915 ## 2002.114 89.31829 82.03651 -0.004362918 1.0888206 ## 2002.122 92.23933 82.98203 -0.004362918 1.111616416/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 16/96 ## 2002.130 92.02329 83.39851 -0.004362918 1.1034742 ## 2002.138 93.94839 83.93850 -0.004362918 1.1193109 ## 2002.146 93.63384 83.41304 -0.004362918 1.1225911 ## 2002.154 91.31864 83.06144 -0.004362918 1.0994686 ## 2002.163 88.94263 82.87884 -0.004362918 1.0732210 ## 2002.171 92.10373 86.34561 -0.004362918 1.0667408 ## 2002.179 90.45678 85.70492 -0.004362918 1.0554981 ## 2002.187 87.72297 84.85174 -0.004362918 1.0338913 ## 2002.195 86.63572 84.41733 -0.004362918 1.0263320 ## 2002.203 90.19397 82.83358 -0.004362918 1.0889150 ## 2002.211 88.74025 80.46052 -0.004362918 1.1029641 ## 2002.220 88.81200 80.60099 -0.004362918 1.1019320 ## 2002.228 90.97117 81.81778 -0.004362918 1.1119346 ## 2002.236 89.98388 81.82936 -0.004362918 1.0997113 ## 2002.244 80.44218 80.71554 -0.004362918 0.9966673 ## 2002.252 84.24926 86.60891 -0.004362918 0.9728040 ## 2002.260 89.48854 92.13674 -0.004362918 0.9713040 ## 2002.268 93.94451 93.08939 -0.004362918 1.0092333 ## 2002.276 96.47537 91.29071 -0.004362918 1.0568433 ## 2002.285 95.47417 88.10010 -0.004362918 1.0837547 ## 2002.293 91.15649 86.69171 -0.004362918 1.0515547 ## 2002.301 94.20044 90.10493 -0.004362918 1.0455033 ## 2002.309 87.13212 85.86501 -0.004362918 1.0148086 ## 2002.317 81.96715 85.78058 -0.004362918 0.9555930 ## 2002.325 84.81478 88.37168 -0.004362918 0.9597981 ## 2002.333 88.74211 90.40829 -0.004362918 0.9816179 ## 2002.341 91.80235 90.56550 -0.004362918 1.0137057 ## 2002.350 92.05358 90.07436 -0.004362918 1.0220227 ## 2002.358 93.69853 89.43601 -0.004362918 1.0477112 ## 2002.366 90.82563 86.67362 -0.004362918 1.0479567 ## 2002.374 93.16148 85.01101 -0.004362918 1.0959317 ## 2002.382 92.05947 83.23252 -0.004362918 1.1061096 ## 2002.390 91.37052 82.63909 -0.004362918 1.1057158 ## 2002.398 91.00741 83.54105 -0.004362918 1.0894305 ## 2002.407 87.35596 83.53250 -0.004362918 1.0458268 ## 2002.415 88.37378 86.84714 -0.004362918 1.0176297 ## 2002.423 89.76239 89.63862 -0.004362918 1.0014296 ## 2002.431 89.68961 90.39430 -0.004362918 0.9922521 ## 2002.439 93.57880 93.68135 -0.004362918 0.9989518 ## 2002.447 93.54700 93.32065 -0.004362918 1.0024724 ## 2002.455 95.15418 91.75373 -0.004362918 1.0371099 ## 2002.463 92.69983 87.50696 -0.004362918 1.0593951 ## 2002.472 87.20953 82.45215 -0.004362918 1.0577547 ## 2002.480 87.92346 79.41885 -0.004362918 1.1071465 ## 2002.488 83.00918 76.12409 -0.004362918 1.0905082 ## 2002.496 77.95080 73.29475 -0.004362918 1.0635884 ## 2002.504 73.34099 72.74060 -0.004362918 1.0083144 ## 2002.512 75.93270 79.23750 -0.004362918 0.9583452 ## 2002.520 81.04410 84.41020 -0.004362918 0.9601717 ## 2002.528 91.86669 89.50170 -0.004362918 1.0264740 ## 2002.537 95.20562 91.37464 -0.004362918 1.0419759 ## 2002.545 90.15648 90.06845 -0.004362918 1.0010259 ## 2002.553 92.74011 90.58232 -0.004362918 1.0238706 ## 2002.561 93.68932 87.73075 -0.004362918 1.0679720 ## 2002.569 88.07865 83.87427 -0.004362918 1.0501819 ## 2002.577 90.57771 85.58069 -0.004362918 1.0584435 ## 2002.585 92.75240 88.14587 -0.004362918 1.0523124 ## 2002.593 94.81504 89.45508 -0.004362918 1.059969616/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 17/96 ## 2002.602 89.59247 86.65699 -0.004362918 1.0339267 ## 2002.610 78.67906 77.97270 -0.004362918 1.0091156 ## 2002.618 76.02819 77.55448 -0.004362918 0.9803749 ## 2002.626 84.38945 86.94215 -0.004362918 0.9706878 ## 2002.634 88.22661 89.22534 -0.004362918 0.9888550 ## 2002.642 85.18583 87.83617 -0.004362918 0.9698745 ## 2002.650 83.76962 85.17755 -0.004362918 0.9835210 ## 2002.659 80.07822 82.81602 -0.004362918 0.9669921 ## 2002.667 82.70188 86.57788 -0.004362918 0.9552791 ## 2002.675 81.43103 87.40924 -0.004362918 0.9316531 ## 2002.683 83.05168 84.47987 -0.004362918 0.9831451 ## 2002.691 79.81521 83.81763 -0.004362918 0.9522980 ## 2002.699 77.37783 79.41194 -0.004362918 0.9744388 ## 2002.707 73.33562 74.12003 -0.004362918 0.9894752 ## 2002.715 74.35102 75.15016 -0.004362918 0.9894236 ## 2002.724 73.68730 75.54918 -0.004362918 0.9754117 ## 2002.732 71.81394 78.89451 -0.004362918 0.9103030 ## 2002.740 69.24549 79.69145 -0.004362918 0.8689675 ## 2002.748 74.66196 86.59075 -0.004362918 0.8622829 ## 2002.756 79.35970 91.82007 -0.004362918 0.8643369 ## 2002.764 87.57926 95.11742 -0.004362918 0.9207911 ## 2002.772 82.03899 92.72245 -0.004362918 0.8848218 ## 2002.780 80.11634 92.69099 -0.004362918 0.8643786 ## 2002.789 76.81786 97.58433 -0.004362918 0.7872298 ## 2002.797 80.21850 104.75328 -0.004362918 0.7658170 ## 2002.805 83.94891 104.57344 -0.004362918 0.8028081 ## 2002.813 74.38168 94.64938 -0.004362918 0.7859017 ## 2002.821 76.58724 88.08598 -0.004362918 0.8695031 ## 2002.829 74.16946 83.42244 -0.004362918 0.8891294 ## 2002.837 75.27793 86.06763 -0.004362918 0.8746813 ## 2002.846 79.90219 92.19590 -0.004362918 0.8666977 ## 2002.854 81.24465 91.55135 -0.004362918 0.8874640 ## 2002.862 75.56592 82.36858 -0.004362918 0.9174605 ## 2002.870 62.99156 69.91889 -0.004362918 0.9009795 ## 2002.878 68.25243 71.96807 -0.004362918 0.9484285 ## 2002.886 62.95470 69.20625 -0.004362918 0.9097253 ## 2002.894 62.58532 72.61266 -0.004362918 0.8619583 ## 2002.902 65.56891 78.61213 -0.004362918 0.8341277 ## 2002.911 74.11298 84.08671 -0.004362918 0.8814332 ## 2002.919 74.84775 81.91033 -0.004362918 0.9138253 ## 2002.927 63.82339 74.60545 -0.004362918 0.8555290 ## 2002.935 62.14080 71.13376 -0.004362918 0.8736303 ## 2002.943 67.85428 75.25405 -0.004362918 0.9017217 ## 2002.951 62.62527 69.89282 -0.004362918 0.8960745 ## 2002.959 63.83408 73.57730 -0.004362918 0.8676299 ## 2002.967 71.59036 77.23470 -0.004362918 0.9269720 ## 2002.976 77.53892 79.49247 -0.004362918 0.9754782 ## 2002.984 80.16958 77.88742 -0.004362918 1.0293585 ## 2002.992 70.62168 70.61218 -0.004362918 1.0001962 ## 2003.000 73.10059 64.07674 -0.004362918 1.1409065 ## 2003.008 72.13247 64.01815 -0.004362918 1.1268271 ## 2003.016 77.77739 68.85353 -0.004362918 1.1296779 ## 2003.024 83.52416 73.87001 -0.004362918 1.1307578 ## 2003.033 83.86090 75.21222 -0.004362918 1.1150549 ## 2003.041 78.93483 73.07840 -0.004362918 1.0802035 ## 2003.049 76.14107 75.95784 -0.004362918 1.0024698 ## 2003.057 83.97486 82.61530 -0.004362918 1.0165102 ## 2003.065 89.24516 86.25623 -0.004362918 1.034704116/09/2020 ISYE 6501 Week 4 homework file:///Users/paris.piotis/Desktop/ISYE-6501-Week-4-Homework.html 18/96 ## 2003.073 89.43210 86.10615 -0.004362918 1.0386788 ## 2003.081 83.97802 82.88543 -0.004362918 1.0132353 ## 2003.089 82.89367 82.89441 -0.004362918 1.0000437 ## 2003.098 86.55329 84.80037 -0.004362918 1.0207237 ## 2003.106 90.59650 85.06516 -0.004362918 1.0650794 ## 2003.114 88.81225 81.25181 -0.004362918 1.0931082 ## 2003.122 88.70603 79.66522 -0.004362918 1.1135460 ## 2003.130 87.66374 79.27092 -0.004362918 1.1059360 ## 2003.138 88.73723 79.45355 -0.004362918 1.1169054 ## 2003.146 88.60074 79.04325 -0.004362918 1.1209766 ## 2003.154 86.46959 78.70930 -0.004362918 1.0986552 ## 2003.163 86.56331 79.56163 -0.004362918 1.0880629 ## 2003.171 86.11103 80.93455 -0.004362918 1.0640162 ## 2003.179 85.05561 80.86601 -0.004362918 1.0518659 ## 2003.187 81.00385 78.49042 -0.004362918 1.0320794 ## 2003.195 80.62517 79.07965 -0.004362918 1.0196000 ## 2003.203 87.42373 81.11092 -0.004362918 1.0778872 ## 2003.211 89.24129 80.86480 -0.004362918 1.1036459 ## 2003.220 88.17203 79.61149 -0.004362918 1.1075897 ## 2003.228 89.03023 80.06686 -0.004362918 1.1120091 ## 2003.236 87.00091 79.49272 -0.004362918 1.0945113 ## 2003.244 79.35490 77.80216 -0.004362918 1.0200148 ## 2003.252 82.41633 83.01023 -0.004362918 0.9928976 ## 2003.260 81.86096 83.98680 -0.004362918 0.9747391 ## 2003.268 85.53327 85.33204 -0.004362918 1.0024094 ## 2003.276 88.06969 84.38698 -0.004362918 1.0436947 ## 2003.285 87.08129 80.80602 -0.004362918 1.0777167 ## 2003.293 84.19823 79.04331 -0.004362918 1.0652753 ## 2003.301 79.92442 77.76986 -0.004362918 1.0277619 ## 2003.309 81.36144 80.20429 -0.004362918 1.0144827 ## 2003.317 78.95256 81.79948 -0.004362918 0.9652479 ## 2003.325 83.45545 86.28536 -0.004362918 0.9672518 ## 2003.333 86.95630 88.53471 -0.004362918 0.9822203 ## 2003.341 87.69954 86.67930 -0.004362918 1.0118212 ## 2003.350 84.21165 82.60284 -0.004362918 1.0195304 ## 2003.358 87.33046 84.28047 -0.004362918 1.0362423 ## 2003.366 88.75194 85.26696 -0.004362918 1.0409246 ## 2003.374 93.55426 85.99998 -0.004362918 1.0878957 ## 2003.382 90.17042 81.72509 -0.004362918 1.1033973 ## 2003.390 89.96827 81.06836 -0.004362918 1.1098424 ## 2003.398 88.32648 81.08158 -0.004362918 1.0894118 ## 2003.407 86.90390 82.02197 -0.004362918 1.0595761 ## 2003.415 84.41198 82.07339 -0.004362918 1.0285485 ## 2003.423 84.57437 84.21442 -0.004362918 1.0043262 ## 2003.431 86.66974 86.30776 -0.004362918 1.0042449 ## 2003.439 88.12999 88.34286 -0.004362918 0.9976397 ## 2003.447 88.56826 88.87482 -0.004362918 0.9965996 ## 2003.455 90.26920 88.51979 -0.004362918 1.0198132 ## 2003.463 90.99857 87.75003 -0.004362918 1.0370721 ## 2003.472 90.97296 87.15349 -0.004362918 1.0438769 ## 2003.480 95.06476 87.16506 -0.004362918 1.0906838 ## 2003.488 90.10470 83.74096 -0.004362918 1.0760492 ## 2003.496 87.54900 82.53369 -0.004362918 1.0608230 ## 2003.504 86.43188 83.37053 -0.004362918 1.0367742 ## 2003.512 82.48520 84.29635 -0.004362918 0.9785651 ## 2003.520 84.67826 86.50095 -0.004362918 0.9789780 ## 2003.528 90.91723 87.95513 -0.004362918

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