Simplified SC
Christine
November 30, 2016
1/4 CLEAR 96X130, SS06CLR96130
MAM Quarterly (H.W. with mult. error, MAPE = 5%)
A <- ts(c(1074,1381,1181,1143,1220,1176,1379,1667,1413,1830,1400,1114,1217,1203,1324,1805,1474,1370,2239,1817,1884,2188,2118,2026,1626,1526,1099,1422,1979,1752,1909,2196,2275,2613,2327,2649,2008,1520,1695,2076,1725,2184,2591,2627,2590,2975,2349,2800,1926,1870,2327,2273,2418,2596,2578,3146,2656,2386
),
start = c(2012, 1), frequency = 12)Monthly
AData <- window(A)
fit <- ets(AData, model = "MAM")
summary(fit)## ETS(M,A,M)
##
## Call:
## ets(y = AData, model = "MAM")
##
## Smoothing parameters:
## alpha = 0.001
## beta = 8e-04
## gamma = 1e-04
##
## Initial states:
## l = 1243.9225
## b = 24.0393
## s=1.1264 1.0676 1.215 1.0808 1.1453 1.1002
## 0.9214 0.9136 0.9221 0.8216 0.8366 0.8494
##
## sigma: 0.1157
##
## AIC AICc BIC
## 893.6213 908.9213 928.6489
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -20.48308 219.9445 165.3238 -2.657917 9.3926 0.4353865
## ACF1
## Training set 0.071752plot(fit)
plot(forecast(fit,h=8),
ylab="Forecasted Demand")
forecast(fit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 2803.583 2387.803 3219.363 2167.702 3439.463
## Dec 2016 2983.791 2541.285 3426.297 2307.036 3660.545
## Jan 2017 2269.869 1933.238 2606.499 1755.037 2784.700
## Feb 2017 2254.887 1920.477 2589.298 1743.451 2766.324
## Mar 2017 2233.444 1902.211 2564.678 1726.867 2740.022
## Apr 2017 2527.982 2153.063 2902.902 1954.592 3101.372
## May 2017 2525.699 2151.112 2900.286 1952.818 3098.581
## Jun 2017 2568.585 2187.629 2949.540 1985.964 3151.206
## Jul 2017 3092.506 2633.834 3551.177 2391.028 3793.984
## Aug 2017 3245.865 2764.432 3727.297 2509.577 3982.152
## Sep 2017 3087.770 2629.769 3545.771 2387.318 3788.222
## Oct 2017 3499.452 2980.365 4018.540 2705.577 4293.328
## Nov 2017 3099.600 2639.801 3559.399 2396.398 3802.802
## Dec 2017 3296.087 2807.113 3785.061 2548.266 4043.909
## Jan 2018 2505.389 2133.691 2877.087 1936.926 3073.852
## Feb 2018 2486.847 2117.874 2855.821 1922.551 3051.144
## Mar 2018 2461.246 2096.041 2826.450 1902.714 3019.778
## Apr 2018 2783.652 2370.572 3196.733 2151.900 3415.404
## May 2018 2779.003 2366.573 3191.434 2148.245 3409.761
## Jun 2018 2824.055 2404.894 3243.216 2183.004 3465.106
## Jul 2018 3397.556 2893.215 3901.897 2626.233 4168.879
## Aug 2018 3563.432 3034.403 4092.462 2754.351 4372.513
## Sep 2018 3387.427 2884.460 3890.394 2618.205 4156.649
## Oct 2018 3836.337 3266.634 4406.041 2965.051 4707.623Atrain = A[1:48]
Atest = A[49:58]Quarterly
Aquarterly <- aggregate(A, nfrequency=4)
Aquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 3636 3539 4459 4344
## 2013 3744 4649 5940 6332
## 2014 4251 5153 6380 7589
## 2015 5223 5985 7808 8124
## 2016 6123 7287 8380ADataA <- window(Aquarterly)
AfitA <- ets(ADataA)
summary(AfitA)## ETS(M,A,M)
##
## Call:
## ets(y = ADataA)
##
## Smoothing parameters:
## alpha = 1e-04
## beta = 1e-04
## gamma = 1e-04
##
## Initial states:
## l = 3678.1501
## b = 210.4404
## s=1.14 1.1048 0.9198 0.8354
##
## sigma: 0.0627
##
## AIC AICc BIC
## 296.483 316.483 304.983
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -20.25647 318.4145 263.9886 -1.095917 5.166931 0.2841438
## ACF1
## Training set 0.1368467ADecomp <- decompose(Aquarterly)
ADecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 3636 3539 4459 4344
## 2013 3744 4649 5940 6332
## 2014 4251 5153 6380 7589
## 2015 5223 5985 7808 8124
## 2016 6123 7287 8380
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -1090.0521 -477.5000 655.9479 911.6042
## 2013 -1090.0521 -477.5000 655.9479 911.6042
## 2014 -1090.0521 -477.5000 655.9479 911.6042
## 2015 -1090.0521 -477.5000 655.9479 911.6042
## 2016 -1090.0521 -477.5000 655.9479
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 4008.000 4160.250
## 2013 4484.125 4917.750 5229.625 5356.000
## 2014 5474.000 5686.125 5964.750 6190.250
## 2015 6472.750 6718.125 6897.500 7172.750
## 2016 7407.000 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -204.94792 -727.85417
## 2013 349.92708 208.75000 54.42708 64.39583
## 2014 -132.94792 -55.62500 -240.69792 487.14583
## 2015 -159.69792 -255.62500 254.55208 39.64583
## 2016 -193.94792 NA NA
##
## $figure
## [1] -1090.0521 -477.5000 655.9479 911.6042
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"forecast(AfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 8989.612 8267.773 9711.452 7885.654 10093.571
## 2017 Q1 6763.226 6220.159 7306.293 5932.676 7593.776
## 2017 Q2 7639.931 7026.467 8253.395 6701.718 8578.143
## 2017 Q3 9409.621 8654.056 10165.186 8254.084 10565.158
## 2017 Q4 9949.029 9150.151 10747.907 8727.251 11170.808
## 2018 Q1 7466.274 6866.753 8065.794 6549.387 8383.161
## 2018 Q2 8413.997 7738.377 9089.616 7380.725 9447.268
## 2018 Q3 10339.437 9509.210 11169.664 9069.714 11609.160A Line
R-squared: 0.7019 p-value: 4.622e-06 m = 237 int. = 3362
fit.A <- tslm(Aquarterly ~ trend)
a <- forecast(fit.A, h=5,level=c(80,95))
plot(a, ylab="Demand",
xlab="t")
lines(fitted(fit.A),col="blue")
summary(fit.A)##
## Call:
## tslm(formula = Aquarterly ~ trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1270.91 -639.26 32.78 700.09 1380.74
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3362.7 410.5 8.192 2.64e-07 ***
## trend 237.1 36.0 6.586 4.62e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 859.6 on 17 degrees of freedom
## Multiple R-squared: 0.7184, Adjusted R-squared: 0.7019
## F-statistic: 43.38 on 1 and 17 DF, p-value: 4.622e-06SS02CLR7284
MNA 7% quarterly
Monthly
B <- ts( c(468,538,687,932,1162,1067,1131,1201,1183,1082,793,505,498,382,401,752,955,878,999,1053,1007,1283,753,396,397,381,422,742,1028,1168,1216,1179,1329,1161,575,622,347,296,566,820,919,1379,1159,1132,1119,1238,904,699,512,471,744,789,943,1341,973,1255,1146,1045
),
start =c(2012, 1), frequency = 12)
BData <- window(B)
Bfit <- ets(BData)
summary(Bfit)## ETS(M,N,A)
##
## Call:
## ets(y = BData)
##
## Smoothing parameters:
## alpha = 0.1999
## gamma = 0.0016
##
## Initial states:
## l = 923.9891
## s=-292.0386 -96.8391 334.8276 331.2985 286.2546 268.5247
## 302.8817 126.1503 -78.3819 -299.3737 -452.0562 -431.2479
##
## sigma: 0.1271
##
## AIC AICc BIC
## 802.9122 814.3407 833.8188
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -6.318946 106.2753 86.34946 -2.110883 11.21386 0.6453412
## ACF1
## Training set -0.04292431plot(Bfit)
plot(forecast(Bfit,h=8),
ylab="Forecasted Demand")
forecast(Bfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 753.9034 631.0569 876.7499 566.0259 941.7809
## Dec 2016 558.6669 464.3280 653.0057 414.3881 702.9456
## Jan 2017 419.5618 344.5710 494.5526 304.8733 534.2503
## Feb 2017 398.6583 325.4471 471.8695 286.6914 510.6251
## Mar 2017 551.3268 454.4618 648.1919 403.1845 699.4692
## Apr 2017 772.4956 640.2631 904.7282 570.2634 974.7279
## May 2017 976.9179 810.7093 1143.1265 722.7238 1231.1121
## Jun 2017 1153.5547 956.9647 1350.1448 852.8961 1454.2133
## Jul 2017 1118.9171 923.9954 1313.8387 820.8101 1417.0240
## Aug 2017 1137.1131 936.0229 1338.2033 829.5722 1444.6540
## Sep 2017 1181.7198 970.5532 1392.8864 858.7684 1504.6713
## Oct 2017 1185.3213 970.0919 1400.5507 856.1563 1514.4863
## Nov 2017 753.9034 593.8151 913.9917 509.0695 998.7373
## Dec 2017 558.6669 419.2275 698.1062 345.4128 771.9210
## Jan 2018 419.5618 292.3752 546.7484 225.0467 614.0769
## Feb 2018 398.6583 272.4659 524.8506 205.6637 591.6528
## Mar 2018 551.3268 410.0455 692.6082 335.2557 767.3980
## Apr 2018 772.4956 604.9358 940.0554 516.2350 1028.7562
## May 2018 976.9179 781.4143 1172.4215 677.9210 1275.9149
## Jun 2018 1153.5547 931.6278 1375.4817 814.1468 1492.9627
## Jul 2018 1118.9171 898.4471 1339.3870 781.7374 1456.0968
## Aug 2018 1137.1131 911.1555 1363.0707 791.5408 1482.6854
## Sep 2018 1181.7198 946.7656 1416.6741 822.3884 1541.0513
## Oct 2018 1185.3213 946.7037 1423.9389 820.3872 1550.2554Quarterly
Bquarterly <- aggregate(B, nfrequency=4)
Bquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 1693 3161 3515 2380
## 2013 1281 2585 3059 2432
## 2014 1200 2938 3724 2358
## 2015 1209 3118 3410 2841
## 2016 1727 3073 3374BDataA <- window(Bquarterly)
BfitA <- ets(BDataA)
summary(BfitA)## ETS(M,N,A)
##
## Call:
## ets(y = BDataA)
##
## Smoothing parameters:
## alpha = 0.7887
## gamma = 1e-04
##
## Initial states:
## l = 2658.7423
## s=-51.2047 859.2664 391.8974 -1199.959
##
## sigma: 0.0859
##
## AIC AICc BIC
## 273.0032 283.1850 279.6143
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -6.595736 218.2399 187.1359 -0.3109325 7.245141 0.6598585
## ACF1
## Training set -0.08106083BDecomp <- decompose(Bquarterly)
BDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 1693 3161 3515 2380
## 2013 1281 2585 3059 2432
## 2014 1200 2938 3724 2358
## 2015 1209 3118 3410 2841
## 2016 1727 3073 3374
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -1196.58594 384.50781 867.32031 -55.24219
## 2013 -1196.58594 384.50781 867.32031 -55.24219
## 2014 -1196.58594 384.50781 867.32031 -55.24219
## 2015 -1196.58594 384.50781 867.32031 -55.24219
## 2016 -1196.58594 384.50781 867.32031
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 2635.750 2512.250
## 2013 2383.250 2332.750 2329.125 2363.125
## 2014 2490.375 2564.250 2556.125 2579.750
## 2015 2563.000 2584.125 2709.250 2768.375
## 2016 2758.250 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 11.92969 -77.00781
## 2013 94.33594 -132.25781 -137.44531 124.11719
## 2014 -93.78906 -10.75781 300.55469 -166.50781
## 2015 -157.41406 149.36719 -166.57031 127.86719
## 2016 165.33594 NA NA
##
## $figure
## [1] -1196.58594 384.50781 867.32031 -55.24219
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(BDecomp)
forecast(BfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 2508.700 2232.4335 2784.967 2086.1869 2931.213
## 2017 Q1 1359.973 1094.9236 1625.022 954.6151 1765.330
## 2017 Q2 2951.792 2542.1925 3361.391 2325.3637 3578.220
## 2017 Q3 3419.159 2899.1941 3939.125 2623.9411 4214.378
## 2017 Q4 2508.700 1966.1619 3051.238 1678.9596 3338.440
## 2018 Q1 1359.973 822.1228 1897.823 537.4022 2182.543
## 2018 Q2 2951.792 2329.0569 3574.527 1999.4010 3904.183
## 2018 Q3 3419.159 2718.1651 4120.154 2347.0811 4491.238B Line
Adjusted R-squared: -0.02699
p-value: 0.4777 m = 25.05 int. = 2332.56
fit.B <- tslm(Bquarterly ~ trend)
b <- forecast(fit.B, h=5,level=c(80,95))
plot(b, ylab="Demand",
xlab="t")
lines(fitted(fit.B),col="blue")
summary(fit.B)##
## Call:
## tslm(formula = Bquarterly ~ trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1449.2 -469.9 107.7 558.3 1115.9
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2332.56 393.42 5.929 1.65e-05 ***
## trend 25.05 34.51 0.726 0.478
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 823.8 on 17 degrees of freedom
## Multiple R-squared: 0.03007, Adjusted R-squared: -0.02699
## F-statistic: 0.527 on 1 and 17 DF, p-value: 0.4777SS06S6T100144
MAN quarterly, MAPE = 26%
Monthly
C <- ts(c(104,105,99,94,116,113,140,134,97,95,115,61,185,122,125,126,36,85,249,198,185,284,328,392,314,235,120,206,198,332,200,458,437,441,338,330,316,300,294,321,238,415,652,424,469,414,637,612,605,487,478,575,682,784,653,903,612,590
),
start = c(2012, 1), frequency = 12)
CData <- window(C)
Cfit <- ets(CData)
summary(Cfit)## ETS(M,A,N)
##
## Call:
## ets(y = CData)
##
## Smoothing parameters:
## alpha = 0.2046
## beta = 1e-04
##
## Initial states:
## l = 111.6516
## b = 8.5027
##
## sigma: 0.3268
##
## AIC AICc BIC
## 766.5066 767.6605 776.8089
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 6.206277 95.77837 75.41451 -16.69307 34.00243 0.5016728
## ACF1
## Training set 0.2603513plot(Cfit)
plot(forecast(Cfit,h=8),
ylab="Forecasted Demand")
forecast(Cfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 685.6805 398.5074 972.8536 246.4871 1124.874
## Dec 2016 694.2192 396.9600 991.4783 239.6005 1148.838
## Jan 2017 702.7579 395.5120 1010.0037 232.8659 1172.650
## Feb 2017 711.2965 394.1511 1028.4420 226.2644 1196.329
## Mar 2017 719.8352 392.8666 1046.8039 219.7798 1219.891
## Apr 2017 728.3739 391.6490 1065.0988 213.3976 1243.350
## May 2017 736.9126 390.4901 1083.3351 207.1051 1266.720
## Jun 2017 745.4513 389.3824 1101.5202 200.8909 1290.012
## Jul 2017 753.9899 388.3193 1119.6606 194.7449 1313.235
## Aug 2017 762.5286 387.2949 1137.7624 188.6581 1336.399
## Sep 2017 771.0673 386.3038 1155.8308 182.6223 1359.512
## Oct 2017 779.6060 385.3412 1173.8708 176.6300 1382.582
## Nov 2017 788.1447 384.4027 1191.8866 170.6747 1405.615
## Dec 2017 796.6834 383.4844 1209.8823 164.7501 1428.617
## Jan 2018 805.2220 382.5826 1227.8615 158.8508 1451.593
## Feb 2018 813.7607 381.6939 1245.8275 152.9715 1474.550
## Mar 2018 822.2994 380.8153 1263.7835 147.1078 1497.491
## Apr 2018 830.8381 379.9441 1281.7321 141.2552 1520.421
## May 2018 839.3768 379.0775 1299.6761 135.4097 1543.344
## Jun 2018 847.9154 378.2132 1317.6177 129.5678 1566.263
## Jul 2018 856.4541 377.3490 1335.5593 123.7260 1589.182
## Aug 2018 864.9928 376.4828 1353.5028 117.8812 1612.104
## Sep 2018 873.5315 375.6127 1371.4503 112.0304 1635.033
## Oct 2018 882.0702 374.7369 1389.4034 106.1709 1657.969Quarterly
Cquarterly <- aggregate(C, nfrequency=4)
Cquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 308 323 371 271
## 2013 432 247 632 1004
## 2014 669 736 1095 1109
## 2015 910 974 1545 1663
## 2016 1570 2041 2168CDataA <- window(Cquarterly)
CfitA <- ets(CDataA)
summary(CfitA)## ETS(M,A,N)
##
## Call:
## ets(y = CDataA)
##
## Smoothing parameters:
## alpha = 0.1564
## beta = 1e-04
##
## Initial states:
## l = 164.7408
## b = 77.5099
##
## sigma: 0.2613
##
## AIC AICc BIC
## 268.5289 273.1443 273.2511
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 50.27912 238.6125 194.4384 -8.492323 26.31539 0.4614104
## ACF1
## Training set 0.3468974CDecomp <- decompose(Cquarterly)
CDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 308 323 371 271
## 2013 432 247 632 1004
## 2014 669 736 1095 1109
## 2015 910 974 1545 1663
## 2016 1570 2041 2168
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -88.09115 -180.53906 130.37760 138.25260
## 2013 -88.09115 -180.53906 130.37760 138.25260
## 2014 -88.09115 -180.53906 130.37760 138.25260
## 2015 -88.09115 -180.53906 130.37760 138.25260
## 2016 -88.09115 -180.53906 130.37760
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 333.750 339.750
## 2013 362.875 487.125 608.375 699.125
## 2014 818.125 889.125 932.375 992.250
## 2015 1078.250 1203.750 1355.500 1571.375
## 2016 1782.625 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -93.12760 -207.00260
## 2013 157.21615 -59.58594 -106.75260 166.62240
## 2014 -61.03385 27.41406 32.24740 -21.50260
## 2015 -80.15885 -49.21094 59.12240 -46.62760
## 2016 -124.53385 NA NA
##
## $figure
## [1] -88.09115 -180.53906 130.37760 138.25260
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(CDecomp)
forecast(CfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 1864.010 1239.713 2488.306 909.2308 2818.789
## 2017 Q1 1941.615 1283.519 2599.711 935.1437 2948.087
## 2017 Q2 2019.221 1327.350 2711.091 961.0956 3077.346
## 2017 Q3 2096.826 1371.176 2822.476 987.0407 3206.611
## 2017 Q4 2174.431 1414.974 2933.889 1012.9417 3335.921
## 2018 Q1 2252.037 1458.722 3045.351 1038.7672 3465.307
## 2018 Q2 2329.642 1502.404 3156.880 1064.4910 3594.794
## 2018 Q3 2407.248 1546.005 3268.490 1090.0909 3724.405C Line R-squared: 0.7019
Adjusted R-squared: 0.8665 p-value: 4.599e-09 m = 100.3 int. = -52
fit.C <- tslm(Cquarterly ~ trend)
c <- forecast(fit.C, h=5,level=c(80,95))
plot(c, ylab="Demand",
xlab="t")
lines(fitted(fit.C),col="blue")
summary(fit.C)##
## Call:
## tslm(formula = Cquarterly ~ trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -378.15 -132.35 -17.44 148.32 314.34
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -52.070 105.365 -0.494 0.627
## trend 100.302 9.241 10.854 4.6e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 220.6 on 17 degrees of freedom
## Multiple R-squared: 0.8739, Adjusted R-squared: 0.8665
## F-statistic: 117.8 on 1 and 17 DF, p-value: 4.599e-093/16 CLEAR 96X130
MNA quarterly MAPE = 7% ###Monthly
D <- ts(c(278,366,381,393,498,441,381,534,453,534,391,288,302,319,384,372,381,371,407,365,532,445,367,266,250,266,168,323,360,329,432,395,471,549,295,302,219,179,321,389,299,488,449,509,499,561,466,373,262,337,408,437,420,704,492,726,768,519
),
start = c(2012, 1), frequency = 12)
DData <- window(D)
Dfit <- ets(DData)
summary(Dfit)## ETS(M,A,A)
##
## Call:
## ets(y = DData)
##
## Smoothing parameters:
## alpha = 0.0377
## beta = 0.0377
## gamma = 1e-04
##
## Initial states:
## l = 448.7472
## b = -3.5758
## s=-70.4183 2.9142 145.0808 112.7078 74.4152 40.423
## 35.6681 -16.7346 -13.6575 -69.228 -109.8366 -131.3342
##
## sigma: 0.1496
##
## AIC AICc BIC
## 738.4683 753.7683 773.4958
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 7.506794 62.19073 48.89058 0.257838 12.50341 0.6134661
## ACF1
## Training set -0.2634204plot(Dfit)
plot(forecast(Dfit,h=8),
ylab="Forecasted Demand")
forecast(Dfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 584.4604 472.3720 696.5488 413.0359 755.8848
## Dec 2016 523.9533 422.6559 625.2507 369.0323 678.8743
## Jan 2017 475.8636 382.3101 569.4171 332.7858 618.9414
## Feb 2017 510.1922 408.3453 612.0391 354.4308 665.9536
## Mar 2017 563.6246 449.4956 677.7537 389.0793 738.1700
## Apr 2017 632.0166 502.4387 761.5946 433.8443 830.1890
## May 2017 641.7712 506.5636 776.9788 434.9890 848.5534
## Jun 2017 707.0115 555.8197 858.2032 475.7837 938.2393
## Jul 2017 724.5660 564.8276 884.3044 480.2672 968.8649
## Aug 2017 771.4058 597.4355 945.3761 505.3412 1037.4704
## Sep 2017 822.5213 632.8941 1012.1485 532.5115 1112.5311
## Oct 2017 867.6835 662.5580 1072.8091 553.9710 1181.3960
## Nov 2017 738.3658 542.2742 934.4574 438.4696 1038.2620
## Dec 2017 677.8587 477.8936 877.8238 372.0385 983.6789
## Jan 2018 629.7690 422.1774 837.3607 312.2850 947.2531
## Feb 2018 664.0976 438.4351 889.7601 318.9766 1009.2187
## Mar 2018 717.5301 471.1788 963.8813 340.7683 1094.2918
## Apr 2018 785.9221 516.7538 1055.0903 374.2648 1197.5794
## May 2018 795.6766 509.2791 1082.0741 357.6694 1233.6838
## Jun 2018 860.9169 550.9452 1170.8886 386.8561 1334.9778
## Jul 2018 878.4714 549.0920 1207.8509 374.7290 1382.2139
## Aug 2018 925.3113 572.9450 1277.6775 386.4135 1464.2090
## Sep 2018 976.4267 599.9923 1352.8611 400.7200 1552.1335
## Oct 2018 1021.5889 620.9405 1422.2373 408.8501 1634.3278DDecomp <- decompose(D)
DDecomp$seasonal## Jan Feb Mar Apr May
## 2012 -124.091435 -110.247685 -70.528935 -13.653935 -16.921296
## 2013 -124.091435 -110.247685 -70.528935 -13.653935 -16.921296
## 2014 -124.091435 -110.247685 -70.528935 -13.653935 -16.921296
## 2015 -124.091435 -110.247685 -70.528935 -13.653935 -16.921296
## 2016 -124.091435 -110.247685 -70.528935 -13.653935 -16.921296
## Jun Jul Aug Sep Oct
## 2012 30.189815 40.627315 74.596065 112.616898 145.377315
## 2013 30.189815 40.627315 74.596065 112.616898 145.377315
## 2014 30.189815 40.627315 74.596065 112.616898 145.377315
## 2015 30.189815 40.627315 74.596065 112.616898 145.377315
## 2016 30.189815 40.627315 74.596065 112.616898 145.377315
## Nov Dec
## 2012 3.231481 -71.195602
## 2013 3.231481 -71.195602
## 2014 3.231481 -71.195602
## 2015 3.231481 -71.195602
## 2016plot(stl(D, "periodic"))
Quarterly
#for Chris to rememberdiff(D)
#For Chris to remenber DPct <- D/lag(D,1) - 1
Dquarterly <- aggregate(D, nfrequency=4)
Dquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 1025 1332 1368 1213
## 2013 1005 1124 1304 1078
## 2014 684 1012 1298 1146
## 2015 719 1176 1457 1400
## 2016 1007 1561 1986DDataD <- window(Dquarterly)
DfitD <- ets(DDataD)
summary(DfitD)## ETS(M,N,A)
##
## Call:
## ets(y = DDataD)
##
## Smoothing parameters:
## alpha = 0.9984
## gamma = 1e-04
##
## Initial states:
## l = 1227.5876
## s=45.5845 230.6325 27.7691 -303.9861
##
## sigma: 0.0946
##
## AIC AICc BIC
## 248.1520 258.3338 254.7630
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 27.80315 117.4353 94.87023 1.361505 7.727523 0.5178506
## ACF1
## Training set 0.06665997plot(forecast(DfitD, h = 1), ylab= "Quarterly Demand")
forecast(DfitD)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 1800.633 1582.2490 2019.016 1466.6437 2134.621
## 2017 Q1 1451.051 1170.0887 1732.014 1021.3561 1880.746
## 2017 Q2 1782.811 1427.4028 2138.218 1239.2613 2326.360
## 2017 Q3 1985.687 1555.2338 2416.140 1327.3656 2644.009
## 2017 Q4 1800.633 1316.4216 2284.843 1060.0959 2541.169
## 2018 Q1 1451.051 933.9634 1968.139 660.2336 2241.869
## 2018 Q2 1782.811 1220.2859 2345.335 922.5033 2643.118
## 2018 Q3 1985.687 1371.6008 2599.773 1046.5232 2924.851DDDecomp <- decompose(Dquarterly)
DDDecomp$seasonal## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -309.49219 10.44531 226.57031 72.47656
## 2013 -309.49219 10.44531 226.57031 72.47656
## 2014 -309.49219 10.44531 226.57031 72.47656
## 2015 -309.49219 10.44531 226.57031 72.47656
## 2016 -309.49219 10.44531 226.57031forecast(DfitD)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 1800.633 1582.2490 2019.016 1466.6437 2134.621
## 2017 Q1 1451.051 1170.0887 1732.014 1021.3561 1880.746
## 2017 Q2 1782.811 1427.4028 2138.218 1239.2613 2326.360
## 2017 Q3 1985.687 1555.2338 2416.140 1327.3656 2644.009
## 2017 Q4 1800.633 1316.4216 2284.843 1060.0959 2541.169
## 2018 Q1 1451.051 933.9634 1968.139 660.2336 2241.869
## 2018 Q2 1782.811 1220.2859 2345.335 922.5033 2643.118
## 2018 Q3 1985.687 1371.6008 2599.773 1046.5232 2924.851D Line
Adjusted R-squared: 0.07152 p-value: 0.1408 int = 1020.11 m = 18.49
fit.D <- tslm(Dquarterly ~ trend)
d <- forecast(fit.D, h=5,level=c(80,95))
plot(d, ylab="Demand",
xlab="t")
lines(fitted(fit.D),col="blue")
summary(fit.D)##
## Call:
## tslm(formula = Dquarterly ~ trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -541.47 -105.26 -7.04 157.01 614.59
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1020.11 136.46 7.475 9.08e-07 ***
## trend 18.49 11.97 1.545 0.141
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 285.7 on 17 degrees of freedom
## Multiple R-squared: 0.1231, Adjusted R-squared: 0.07152
## F-statistic: 2.386 on 1 and 17 DF, p-value: 0.1408SS10CLR96130
ANN quarterly MAPE = 13.5%
Monthly
E <- ts(c(196,238,282,250,309,269,215,314,185,242,188,151,196,292,244,199,320,384,272,263,225,215,265,219,202,236,274,234,275,298,320,362,426,408,244,280,211,253,410,449,408,450,441,462,470,396,468,544,293,415,467,402,490,598,501,380,529,439
), start = c(2012, 1), frequency = 12)
EData <- window(E)
Efit <- ets(EData)
summary(Efit)## ETS(M,N,N)
##
## Call:
## ets(y = EData)
##
## Smoothing parameters:
## alpha = 0.4002
##
## Initial states:
## l = 225.0354
##
## sigma: 0.2254
##
## AIC AICc BIC
## 731.9426 732.3870 738.1239
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 10.39883 70.30633 56.64873 -0.7338025 18.21559 0.7006835
## ACF1
## Training set 0.07894876plot(Efit)
plot(forecast(Efit,h=8),
ylab="Forecasted Demand")
forecast(Efit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 466.4319 331.6989 601.1649 260.375526 672.4882
## Dec 2016 466.4319 320.8000 612.0638 243.707084 689.1567
## Jan 2017 466.4319 310.5820 622.2818 228.080010 704.7837
## Feb 2017 466.4319 300.9181 631.9456 213.300427 719.5633
## Mar 2017 466.4319 291.7159 641.1479 199.226777 733.6370
## Apr 2017 466.4319 282.9052 649.9586 185.751993 747.1118
## May 2017 466.4319 274.4316 658.4322 172.792793 760.0710
## Jun 2017 466.4319 266.2518 666.6119 160.282898 772.5809
## Jul 2017 466.4319 258.3307 674.5331 148.168547 784.6952
## Aug 2017 466.4319 250.6392 682.2246 136.405425 796.4583
## Sep 2017 466.4319 243.1531 689.7106 124.956501 807.9073
## Oct 2017 466.4319 235.8520 697.0117 113.790453 819.0733
## Nov 2017 466.4319 228.7184 704.1453 102.880517 829.9832
## Dec 2017 466.4319 221.7372 711.1266 92.203605 840.6601
## Jan 2018 466.4319 214.8951 717.9686 81.739644 851.1241
## Feb 2018 466.4319 208.1809 724.6829 71.471045 861.3927
## Mar 2018 466.4319 201.5842 731.2796 61.382302 871.4815
## Apr 2018 466.4319 195.0961 737.7676 51.459657 881.4041
## May 2018 466.4319 188.7086 744.1551 41.690840 891.1729
## Jun 2018 466.4319 182.4145 750.4492 32.064853 900.7989
## Jul 2018 466.4319 176.2074 756.6564 22.571789 910.2920
## Aug 2018 466.4319 170.0812 762.7825 13.202692 919.6611
## Sep 2018 466.4319 164.0309 768.8329 3.949432 928.9143
## Oct 2018 466.4319 158.0514 774.8124 -5.195401 938.0592Quarterly
Equarterly <- aggregate(E, nfrequency=4)
Equarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 716 828 714 581
## 2013 732 903 760 699
## 2014 712 807 1108 932
## 2015 874 1307 1373 1408
## 2016 1175 1490 1410EDataA <- window(Equarterly)
EfitA <- ets(EDataA)
summary(EfitA)## ETS(A,N,N)
##
## Call:
## ets(y = EDataA)
##
## Smoothing parameters:
## alpha = 0.6338
##
## Initial states:
## l = 738.8147
##
## sigma: 170.5384
##
## AIC AICc BIC
## 257.2248 258.8248 260.0582
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 55.44331 170.5384 134.9438 3.377817 13.5182 0.7038097
## ACF1
## Training set -0.07172848EDecomp <- decompose(Equarterly)
EDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 716 828 714 581
## 2013 732 903 760 699
## 2014 712 807 1108 932
## 2015 874 1307 1373 1408
## 2016 1175 1490 1410
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -115.32552 78.82031 77.64323 -41.13802
## 2013 -115.32552 78.82031 77.64323 -41.13802
## 2014 -115.32552 78.82031 77.64323 -41.13802
## 2015 -115.32552 78.82031 77.64323 -41.13802
## 2016 -115.32552 78.82031 77.64323
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 711.750 723.125
## 2013 738.250 758.750 771.000 756.500
## 2014 788.000 860.625 910.000 992.750
## 2015 1088.375 1181.000 1278.125 1338.625
## 2016 1366.125 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -75.39323 -100.98698
## 2013 109.07552 65.42969 -88.64323 -16.36198
## 2014 39.32552 -132.44531 120.35677 -19.61198
## 2015 -99.04948 47.17969 17.23177 110.51302
## 2016 -75.79948 NA NA
##
## $figure
## [1] -115.32552 78.82031 77.64323 -41.13802
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"forecast(EfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 1406.455 1187.9010 1625.008 1072.2057 1740.704
## 2017 Q1 1406.455 1147.7035 1665.206 1010.7288 1802.181
## 2017 Q2 1406.455 1112.9608 1699.949 957.5945 1855.315
## 2017 Q3 1406.455 1081.9163 1730.993 910.1160 1902.793
## 2017 Q4 1406.455 1053.5926 1759.317 866.7987 1946.111
## 2018 Q1 1406.455 1027.3793 1785.530 826.7089 1986.201
## 2018 Q2 1406.455 1002.8650 1810.044 789.2175 2023.692
## 2018 Q3 1406.455 979.7568 1833.153 753.8765 2059.033E Line
Adjusted R-squared: 0.07152 p-value: 0.1408 Intercept = 523.351 m = 45.186
fit.E <- tslm(Equarterly ~ trend)
e <- forecast(fit.E, h=5,level=c(80,95))
plot(e, ylab="Demand",
xlab="t")
lines(fitted(fit.E),col="blue")
summary(fit.E)##
## Call:
## tslm(formula = Equarterly ~ trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -236.77 -128.34 28.12 149.25 214.28
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 523.351 74.397 7.035 2.01e-06 ***
## trend 45.186 6.525 6.925 2.45e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 155.8 on 17 degrees of freedom
## Multiple R-squared: 0.7383, Adjusted R-squared: 0.7229
## F-statistic: 47.95 on 1 and 17 DF, p-value: 2.455e-061/2 CLEAR 102X130
MAN quarterly MAPE = 16%
Monthly
FF <- ts(c(168,81,136,168,148,103,101,194,256,232,160,130,190,231,215,297,222,336,326,436,221,185,255,217,160,153,198,230,263,307,319,330,433,384,275,358,248,307,364,390,384,429,622,498,485,423,291,320,325,465,440,416,475,509,642,677,437,414
), start = c(2012,1), frequency = 12)
FFData <- window(FF)
FFfit <- ets(FFData)
summary(FFfit)## ETS(M,A,N)
##
## Call:
## ets(y = FFData)
##
## Smoothing parameters:
## alpha = 0.6804
## beta = 1e-04
##
## Initial states:
## l = 109.641
## b = 11.8682
##
## sigma: 0.2475
##
## AIC AICc BIC
## 741.7355 742.8893 752.0377
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -8.820438 79.38034 61.34659 -8.419909 23.66188 0.6193905
## ACF1
## Training set 0.1044451plot(FFfit)
plot(forecast(FFfit,h=8),
ylab="Forecasted Demand")
forecast(FFfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 460.5441 314.474335 606.6138 237.149678 683.9385
## Dec 2016 472.3612 290.885315 653.8370 194.817809 749.9045
## Jan 2017 484.1782 271.095019 697.2614 158.295587 810.0608
## Feb 2017 495.9953 253.534285 738.4563 125.183186 866.8074
## Mar 2017 507.8123 237.402879 778.2218 94.256753 921.3679
## Apr 2017 519.6294 222.229762 817.0291 64.795896 974.4629
## May 2017 531.4465 207.711138 855.1818 36.336002 1026.5570
## Jun 2017 543.2635 193.638113 892.8890 8.557591 1077.9695
## Jul 2017 555.0806 179.860014 930.3012 -18.769769 1128.9310
## Aug 2017 566.8977 166.264032 967.5313 -45.818605 1179.6139
## Sep 2017 578.7147 152.763123 1004.6663 -72.722040 1230.1515
## Oct 2017 590.5318 139.288418 1041.7752 -99.585400 1280.6490
## Nov 2017 602.3489 125.784254 1078.9135 -126.493813 1331.1915
## Dec 2017 614.1659 112.204803 1116.1271 -153.517367 1381.8492
## Jan 2018 625.9830 98.511717 1153.4543 -180.714712 1432.6807
## Feb 2018 637.8001 84.672431 1190.9277 -208.135650 1483.7358
## Mar 2018 649.6171 70.658927 1228.5753 -235.823030 1535.0573
## Apr 2018 661.4342 56.446808 1266.4216 -263.814167 1586.6825
## May 2018 673.2513 42.014588 1304.4879 -292.141919 1638.6444
## Jun 2018 685.0683 27.343154 1342.7935 -320.835518 1690.9722
## Jul 2018 696.8854 12.415334 1381.3554 -349.921223 1743.6920
## Aug 2018 708.7024 -2.784432 1420.1893 -379.422836 1796.8277
## Sep 2018 720.5195 -18.270369 1459.3094 -409.362109 1850.4011
## Oct 2018 732.3366 -34.055582 1498.7287 -439.759085 1904.4322Quarterly
FFquarterly <- aggregate(FF, nfrequency=4)
FFquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 385 419 551 522
## 2013 636 855 983 657
## 2014 511 800 1082 1017
## 2015 919 1203 1605 1034
## 2016 1230 1400 1756FFDataA <- window(FFquarterly)
FFfitA <- ets(FFDataA)
summary(FFfitA)## ETS(M,A,N)
##
## Call:
## ets(y = FFDataA)
##
## Smoothing parameters:
## alpha = 0.0012
## beta = 1e-04
##
## Initial states:
## l = 314.7573
## b = 64.0285
##
## sigma: 0.1787
##
## AIC AICc BIC
## 258.3000 262.9154 263.0222
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -30.24154 181.0643 140.5132 -6.581949 16.03091 0.54002
## ACF1
## Training set 0.06677017FFDecomp <- decompose(FFquarterly)
FFDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 385 419 551 522
## 2013 636 855 983 657
## 2014 511 800 1082 1017
## 2015 919 1203 1605 1034
## 2016 1230 1400 1756
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -148.01823 35.61719 208.60677 -96.20573
## 2013 -148.01823 35.61719 208.60677 -96.20573
## 2014 -148.01823 35.61719 208.60677 -96.20573
## 2015 -148.01823 35.61719 208.60677 -96.20573
## 2016 -148.01823 35.61719 208.60677
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 500.625 586.500
## 2013 695.000 765.875 767.125 744.625
## 2014 750.125 807.500 903.500 1004.875
## 2015 1120.625 1188.125 1229.125 1292.625
## 2016 1336.125 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -158.231771 31.705729
## 2013 89.018229 53.507812 7.268229 8.580729
## 2014 -91.106771 -43.117188 -30.106771 108.330729
## 2015 -53.606771 -20.742188 167.268229 -162.419271
## 2016 41.893229 NA NA
##
## $figure
## [1] -148.01823 35.61719 208.60677 -96.20573
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(FFDecomp)
forecast(FFfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 1594.178 1229.038 1959.317 1035.745 2152.610
## 2017 Q1 1658.149 1278.357 2037.941 1077.307 2238.991
## 2017 Q2 1722.120 1327.675 2116.565 1118.869 2325.371
## 2017 Q3 1786.091 1376.993 2195.188 1160.430 2411.752
## 2017 Q4 1850.062 1426.312 2273.812 1201.992 2498.132
## 2018 Q1 1914.033 1475.630 2352.436 1243.553 2584.513
## 2018 Q2 1978.004 1524.948 2431.060 1285.115 2670.894
## 2018 Q3 2041.975 1574.266 2509.684 1326.676 2757.275FF Line
Adjusted R-squared: 0.7718 p-value: 4.597e-07 Intercept = 304.333 m = 62.014
fit.FF <- tslm(FFquarterly ~ trend)
ff <- forecast(fit.FF, h=5,level=c(80,95))
plot(ff, ylab="Demand",
xlab="t")
lines(fitted(fit.FF),col="blue")
summary(fit.FF)##
## Call:
## tslm(formula = FFquarterly ~ trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -351.46 -126.52 -9.36 78.07 370.46
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 304.333 89.898 3.385 0.00352 **
## trend 62.014 7.885 7.865 4.6e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 188.2 on 17 degrees of freedom
## Multiple R-squared: 0.7844, Adjusted R-squared: 0.7718
## F-statistic: 61.86 on 1 and 17 DF, p-value: 4.597e-071/8 ENEG ADV LOW-E 72 X 84
ANN monthly MAPE = 27%
Monthly
G <- ts(c(243,209,163,181,123,121,268,275,360,149,174,346,395,234,243,
185,218,183,281,331,434,428,375,473,536,333,292,259,193,281,299,190,292,365,522,457,398), start = c(2013,10), frequency = 12)
GData <- window(G)
Gfit <- ets(GData)
summary(Gfit)## ETS(A,N,N)
##
## Call:
## ets(y = GData)
##
## Smoothing parameters:
## alpha = 0.8439
##
## Initial states:
## l = 236.7237
##
## sigma: 89.5984
##
## AIC AICc BIC
## 472.2589 472.9862 477.0917
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 5.490188 89.59838 72.8848 -4.62779 27.07409 0.7857352
## ACF1
## Training set 0.03798041plot(Gfit)
plot(forecast(Gfit,h=8),
ylab="Forecasted Demand")
forecast(Gfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 408.1467 293.321764 522.9716 232.537112 583.7563
## Dec 2016 408.1467 257.900060 558.3933 178.364291 637.9291
## Jan 2017 408.1467 229.363759 586.9296 134.721786 681.5716
## Feb 2017 408.1467 204.793247 611.5002 97.144430 719.1490
## Mar 2017 408.1467 182.887041 633.4064 63.641779 752.6516
## Apr 2017 408.1467 162.930059 653.3633 33.120209 783.1732
## May 2017 408.1467 144.479319 671.8141 4.902237 811.3912
## Jun 2017 408.1467 127.237862 689.0555 -21.466295 837.7597
## Jul 2017 408.1467 110.995119 705.2983 -46.307428 862.6008
## Aug 2017 408.1467 95.595346 720.6981 -69.859349 886.1528
## Sep 2017 408.1467 80.919506 735.3739 -92.304108 908.5975
## Oct 2017 408.1467 66.874194 749.4192 -113.784559 930.0780
## Nov 2017 408.1467 53.384512 762.9089 -134.415248 950.7087
## Dec 2017 408.1467 40.389311 775.9041 -154.289693 970.5831
## Jan 2018 408.1467 27.837899 788.4555 -173.485422 989.7788
## Feb 2018 408.1467 15.687694 800.6057 -192.067556 1008.3610
## Mar 2018 408.1467 3.902518 812.3909 -210.091427 1026.3848
## Apr 2018 408.1467 -7.548676 823.8421 -227.604517 1043.8979
## May 2018 408.1467 -18.692770 834.9862 -244.647937 1060.9413
## Jun 2018 408.1467 -29.553220 845.8466 -261.257563 1077.5510
## Jul 2018 408.1467 -40.150642 856.4440 -277.464922 1093.7583
## Aug 2018 408.1467 -50.503269 866.7967 -293.297898 1109.5913
## Sep 2018 408.1467 -60.627319 876.9207 -308.781297 1125.0747
## Oct 2018 408.1467 -70.537296 886.8307 -323.937299 1140.2307Quarterly
gquarterly <- aggregate(G, nfrequency=4)
gquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 615
## 2014 425 903 669 872
## 2015 586 1046 1276 1161
## 2016 733 781 1344gDataA <- window(gquarterly)
gfitA <- ets(gDataA)
summary(gfitA)## ETS(M,N,N)
##
## Call:
## ets(y = gDataA)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 867.6016
##
## sigma: 0.3187
##
## AIC AICc BIC
## 170.7473 173.7473 172.2020
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.04080992 276.4478 232.7689 -11.51011 30.55139 0.9613583
## ACF1
## Training set 0.1748691gDecomp <- decompose(gquarterly)
gDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 615
## 2014 425 903 669 872
## 2015 586 1046 1276 1161
## 2016 733 781 1344
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 100.23438
## 2014 -291.32812 123.23438 67.85938 100.23438
## 2015 -291.32812 123.23438 67.85938 100.23438
## 2016 -291.32812 123.23438 67.85938
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 NA
## 2014 NA 685.125 737.375 775.375
## 2015 869.125 981.125 1035.625 1020.875
## 2016 996.250 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 NA
## 2014 NA 94.640625 -136.234375 -3.609375
## 2015 8.203125 -58.359375 172.515625 39.890625
## 2016 28.078125 NA NA
##
## $figure
## [1] 100.23438 -291.32812 123.23438 67.85938
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(gDecomp)
forecast(gfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 867.6017 513.2971 1221.906 325.7396 1409.464
## 2017 Q1 867.6017 513.2971 1221.906 325.7396 1409.464
## 2017 Q2 867.6017 513.2971 1221.906 325.7396 1409.464
## 2017 Q3 867.6017 513.2971 1221.906 325.7396 1409.464
## 2017 Q4 867.6017 513.2971 1221.906 325.7396 1409.464
## 2018 Q1 867.6017 513.2971 1221.906 325.7396 1409.464
## 2018 Q2 867.6017 513.2971 1221.906 325.7396 1409.464
## 2018 Q3 867.6017 513.2971 1221.906 325.7396 1409.464G Line
Adjusted R-squared: 0.2691 p-value: 0.000595 Intercept = 187.351 m = 5.515
fit.G <- tslm(G ~ trend)
g <- forecast(fit.G, h=5,level=c(80,95))
plot(g, ylab="Demand",
xlab="t")
lines(fitted(fit.G),col="blue")
summary(fit.G)##
## Call:
## tslm(formula = G ~ trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -173.83 -74.02 -11.13 60.80 210.78
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 187.351 31.838 5.885 1.1e-06 ***
## trend 5.515 1.461 3.775 0.000595 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 94.88 on 35 degrees of freedom
## Multiple R-squared: 0.2894, Adjusted R-squared: 0.2691
## F-statistic: 14.25 on 1 and 35 DF, p-value: 0.0005951/4 SOLARBAN 60 100 X 144
MAN quarterly MAPE = 23%
Monthly
H <- ts(c(69,140,81,50,35,56,63,180,141,122,56,153,97,81,43,91,154,63,186,89,67,121,164,259,156,185,84,122,92,96,184,144,191,201,172,224,269,151,152,149,134,212,246,165,232,382,358,336,194,232,272,288,289,239,248,284,298,398
),
start = c(2012, 1), frequency = 12)
HData <- window(H)
Hfit <- ets(HData)
summary(Hfit)## ETS(A,A,N)
##
## Call:
## ets(y = HData)
##
## Smoothing parameters:
## alpha = 1e-04
## beta = 1e-04
##
## Initial states:
## l = 66.3181
## b = 3.7355
##
## sigma: 55.3721
##
## AIC AICc BIC
## 711.1385 712.2924 721.4407
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -4.621318 55.37211 44.55253 -20.06258 36.69403 0.6341016
## ACF1
## Training set 0.3045034plot(Hfit)
plot(forecast(Hfit,h=8),
ylab="Forecasted Demand")
forecast(Hfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 285.0587 214.0964 356.0209 176.5313 393.5860
## Dec 2016 288.7673 217.8051 359.7296 180.2400 397.2947
## Jan 2017 292.4760 221.5138 363.4383 183.9487 401.0034
## Feb 2017 296.1847 225.2225 367.1470 187.6574 404.7121
## Mar 2017 299.8934 228.9312 370.8557 191.3660 408.4208
## Apr 2017 303.6021 232.6398 374.5644 195.0747 412.1295
## May 2017 307.3108 236.3485 378.2731 198.7833 415.8382
## Jun 2017 311.0195 240.0572 381.9818 202.4920 419.5470
## Jul 2017 314.7282 243.7658 385.6905 206.2006 423.2557
## Aug 2017 318.4369 247.4745 389.3992 209.9093 426.9645
## Sep 2017 322.1455 251.1831 393.1080 213.6179 430.6732
## Oct 2017 325.8542 254.8917 396.8167 217.3265 434.3820
## Nov 2017 329.5629 258.6004 400.5255 221.0350 438.0908
## Dec 2017 333.2716 262.3090 404.2343 224.7436 441.7996
## Jan 2018 336.9803 266.0176 407.9430 228.4522 445.5085
## Feb 2018 340.6890 269.7262 411.6518 232.1607 449.2173
## Mar 2018 344.3977 273.4347 415.3606 235.8692 452.9262
## Apr 2018 348.1064 277.1433 419.0695 239.5777 456.6351
## May 2018 351.8151 280.8518 422.7783 243.2862 460.3440
## Jun 2018 355.5238 284.5604 426.4871 246.9946 464.0529
## Jul 2018 359.2324 288.2689 430.1960 250.7031 467.7618
## Aug 2018 362.9411 291.9774 433.9049 254.4115 471.4708
## Sep 2018 366.6498 295.6859 437.6138 258.1198 475.1798
## Oct 2018 370.3585 299.3943 441.3227 261.8282 478.8888Quarterly
Hquarterly <- aggregate(H, nfrequency=4)
Hquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 290 141 384 331
## 2013 221 308 342 544
## 2014 425 310 519 597
## 2015 572 495 643 1076
## 2016 698 816 830HDataA <- window(Hquarterly)
HfitA <- ets(HDataA)
summary(HfitA)## ETS(M,A,N)
##
## Call:
## ets(y = HDataA)
##
## Smoothing parameters:
## alpha = 1e-04
## beta = 1e-04
##
## Initial states:
## l = 237.4554
## b = 26.4435
##
## sigma: 0.2559
##
## AIC AICc BIC
## 248.7225 253.3379 253.4447
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.5849946 129.5789 90.138 -11.16418 23.42515 0.5416947
## ACF1
## Training set 0.07730516HDecomp <- decompose(Hquarterly)
HDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 290 141 384 331
## 2013 221 308 342 544
## 2014 425 310 519 597
## 2015 572 495 643 1076
## 2016 698 816 830
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -51.27865 -101.07031 10.59635 141.75260
## 2013 -51.27865 -101.07031 10.59635 141.75260
## 2014 -51.27865 -101.07031 10.59635 141.75260
## 2015 -51.27865 -101.07031 10.59635 141.75260
## 2016 -51.27865 -101.07031 10.59635
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 277.875 290.125
## 2013 305.750 327.125 379.250 405.000
## 2014 427.375 456.125 481.125 522.625
## 2015 561.250 636.625 712.250 768.125
## 2016 831.625 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 95.528646 -100.877604
## 2013 -33.471354 81.945312 -47.846354 -2.752604
## 2014 48.903646 -45.054688 27.278646 -67.377604
## 2015 62.028646 -40.554688 -79.846354 166.122396
## 2016 -82.346354 NA NA
##
## $figure
## [1] -51.27865 -101.07031 10.59635 141.75260
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(HDecomp)
forecast(HfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 765.7661 514.5936 1016.939 381.6310 1149.901
## 2017 Q1 792.2107 532.3644 1052.057 394.8100 1189.611
## 2017 Q2 818.6553 550.1351 1087.176 407.9890 1229.322
## 2017 Q3 845.0999 567.9058 1122.294 421.1681 1269.032
## 2017 Q4 871.5445 585.6765 1157.413 434.3471 1308.742
## 2018 Q1 897.9891 603.4471 1192.531 447.5260 1348.452
## 2018 Q2 924.4337 621.2178 1227.650 460.7050 1388.162
## 2018 Q3 950.8783 638.9884 1262.768 473.8839 1427.873H line
Adjusted R-squared: 0.6205
p-value: 1.333e-13 Intercept = 46.949 m = 4.2180
fit.H <- tslm(H ~ trend)
h <- forecast(fit.H, h=5,level=c(80,95))
plot(h, ylab="Demand",
xlab="t")
lines(fitted(fit.H),col="blue")
summary(fit.H)##
## Call:
## tslm(formula = H ~ trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -85.89 -37.78 -4.77 24.33 141.02
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.9492 14.7402 3.185 0.00237 **
## trend 4.2180 0.4346 9.706 1.33e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 55.4 on 56 degrees of freedom
## Multiple R-squared: 0.6272, Adjusted R-squared: 0.6205
## F-statistic: 94.21 on 1 and 56 DF, p-value: 1.333e-13Monthly
I <- ts(c(157,149,168,223,230,249,293,317,269,333,209,144,184,151,144,211,253,225,270,214,177,350,213,164,112,115,100,216,226,274,270,260,276,302,216,177,118,101,153,181,233,273,318,298,298,356,298,197,145,143,217,208,243,311,349,306,251,236
),
start = c(2012, 1), frequency = 12)
IData <- window(I)
Ifit <- ets(IData)
summary(Ifit)## ETS(M,N,A)
##
## Call:
## ets(y = IData)
##
## Smoothing parameters:
## alpha = 0.1761
## gamma = 1e-04
##
## Initial states:
## l = 232.0982
## s=-49.4809 12.0742 100.7175 27.2913 51.4044 68.476
## 38.8245 9.8169 -15.3121 -68.2627 -95.186 -80.363
##
## sigma: 0.1287
##
## AIC AICc BIC
## 651.3483 662.7769 682.2550
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.3444858 30.11825 22.99002 -1.813708 10.81167 0.6580841
## ACF1
## Training set 0.1595263plot(Ifit)
plot(forecast(Ifit,h=8),
ylab="Forecasted Demand")
forecast(Ifit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 240.6538 200.96452 280.3430 179.95431 301.3532
## Dec 2016 179.0979 148.73199 209.4639 132.65722 225.5387
## Jan 2017 148.2168 122.24213 174.1914 108.49200 187.9415
## Feb 2017 133.3958 109.31206 157.4795 96.56290 170.2287
## Mar 2017 160.3175 131.84998 188.7850 116.78017 203.8548
## Apr 2017 213.2673 176.24665 250.2879 156.64912 269.8854
## May 2017 238.3987 196.94608 279.8513 175.00239 301.7950
## Jun 2017 267.4063 220.86385 313.9487 196.22578 338.5868
## Jul 2017 297.0598 245.26328 348.8562 217.84388 376.2756
## Aug 2017 279.9853 230.07872 329.8919 203.65977 356.3108
## Sep 2017 255.8717 208.89673 302.8467 184.02967 327.7137
## Oct 2017 329.2912 270.71210 387.8703 239.70221 418.8801
## Nov 2017 240.6538 194.27908 287.0284 169.72981 311.5777
## Dec 2017 179.0979 140.39707 217.7988 119.91007 238.2858
## Jan 2018 148.2168 112.85221 183.5813 94.13135 202.3022
## Feb 2018 133.3958 99.39118 167.4004 81.39023 185.4013
## Mar 2018 160.3175 123.07512 197.5599 103.36018 217.2748
## Apr 2018 213.2673 169.13753 257.3970 145.77667 280.7579
## May 2018 238.3987 190.48689 286.3105 165.12391 311.6735
## Jun 2018 267.4063 215.02589 319.7866 187.29739 347.5151
## Jul 2018 297.0598 239.95752 354.1620 209.72943 384.3901
## Aug 2018 279.9853 224.58855 335.3821 195.26328 364.7073
## Sep 2018 255.8717 203.09838 308.6450 175.16187 336.5816
## Oct 2018 329.2912 265.96542 392.6169 232.44278 426.1396Quarterly
Iquarterly <- aggregate(I, nfrequency=4)
Iquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 474 702 879 686
## 2013 479 689 661 727
## 2014 327 716 806 695
## 2015 372 687 914 851
## 2016 505 762 906IDataA <- window(Iquarterly)
IfitA <- ets(IDataA)
summary(IfitA)## ETS(M,N,A)
##
## Call:
## ets(y = IDataA)
##
## Smoothing parameters:
## alpha = 0.3971
## gamma = 1e-04
##
## Initial states:
## l = 690.2762
## s=64.2873 147.8217 34.67 -246.779
##
## sigma: 0.0951
##
## AIC AICc BIC
## 226.4914 236.6733 233.1025
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 7.032893 65.63119 52.08215 0.137992 8.028488 0.6581569
## ACF1
## Training set -0.05764806IDecomp <- decompose(Iquarterly)
IDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 474 702 879 686
## 2013 479 689 661 727
## 2014 327 716 806 695
## 2015 372 687 914 851
## 2016 505 762 906
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -254.43750 41.78125 145.37500 67.28125
## 2013 -254.43750 41.78125 145.37500 67.28125
## 2014 -254.43750 41.78125 145.37500 67.28125
## 2015 -254.43750 41.78125 145.37500 67.28125
## 2016 -254.43750 41.78125 145.37500
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 685.875 684.875
## 2013 656.000 633.875 620.000 604.375
## 2014 625.875 640.000 641.625 643.625
## 2015 653.500 686.500 722.625 748.625
## 2016 757.000 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 47.75000 -66.15625
## 2013 77.43750 13.34375 -104.37500 55.34375
## 2014 -44.43750 34.21875 19.00000 -15.90625
## 2015 -27.06250 -41.28125 46.00000 35.09375
## 2016 2.43750 NA NA
##
## $figure
## [1] -254.43750 41.78125 145.37500 67.28125
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(IDecomp)
forecast(IfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 807.6320 709.2275 906.0365 657.1353 958.1287
## 2017 Q1 496.5638 424.4427 568.6848 386.2641 606.8634
## 2017 Q2 778.0108 672.5952 883.4265 616.7915 939.2301
## 2017 Q3 891.1666 767.2758 1015.0573 701.6920 1080.6412
## 2017 Q4 807.6320 684.6534 930.6107 619.5524 995.7116
## 2018 Q1 496.5638 393.3662 599.7613 338.7368 654.3908
## 2018 Q2 778.0108 649.2928 906.7289 581.1536 974.8681
## 2018 Q3 891.1666 746.9026 1035.4306 670.5339 1111.7993Monthly
REMOVED 21 MOS OF DATA 15,1,8,4,4,20,32,2,16,3,5,3,3,1,1,10,3,15,9,7,13,
J <- ts(c(90,229,205,171,126,128,161,197,117,297,240,197,227,266,323,105,199,176,180,255,264,244,179,271,245,162,339,225,125,309,130,192,230,236,323,168,192
),
start = c(2013, 10), frequency = 12)
JData <- window(J)
Jfit <- ets(JData)
summary(Jfit)## ETS(M,N,N)
##
## Call:
## ets(y = JData)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 208.7103
##
## sigma: 0.3053
##
## AIC AICc BIC
## 447.0174 447.7447 451.8502
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.03825653 63.70378 52.68033 -11.20274 29.71262 0.7272271
## ACF1
## Training set -0.1003191plot(Jfit)
plot(forecast(Jfit,h=8),
ylab="Forecasted Demand")
forecast(Jfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 208.7104 127.064 290.3569 83.84296 333.5779
## Dec 2016 208.7104 127.064 290.3569 83.84296 333.5779
## Jan 2017 208.7104 127.064 290.3569 83.84296 333.5779
## Feb 2017 208.7104 127.064 290.3569 83.84296 333.5779
## Mar 2017 208.7104 127.064 290.3569 83.84296 333.5779
## Apr 2017 208.7104 127.064 290.3569 83.84296 333.5779
## May 2017 208.7104 127.064 290.3569 83.84296 333.5779
## Jun 2017 208.7104 127.064 290.3569 83.84296 333.5779
## Jul 2017 208.7104 127.064 290.3569 83.84296 333.5779
## Aug 2017 208.7104 127.064 290.3569 83.84296 333.5779
## Sep 2017 208.7104 127.064 290.3569 83.84295 333.5779
## Oct 2017 208.7104 127.064 290.3569 83.84295 333.5779
## Nov 2017 208.7104 127.064 290.3569 83.84295 333.5779
## Dec 2017 208.7104 127.064 290.3569 83.84295 333.5779
## Jan 2018 208.7104 127.064 290.3569 83.84295 333.5779
## Feb 2018 208.7104 127.064 290.3569 83.84295 333.5779
## Mar 2018 208.7104 127.064 290.3569 83.84295 333.5779
## Apr 2018 208.7104 127.064 290.3569 83.84295 333.5779
## May 2018 208.7104 127.064 290.3569 83.84295 333.5779
## Jun 2018 208.7104 127.064 290.3569 83.84295 333.5779
## Jul 2018 208.7104 127.064 290.3569 83.84295 333.5779
## Aug 2018 208.7104 127.064 290.3569 83.84295 333.5779
## Sep 2018 208.7104 127.064 290.3569 83.84295 333.5779
## Oct 2018 208.7104 127.064 290.3569 83.84295 333.5779Quarterly
Jquarterly <- aggregate(J, nfrequency=4)
Jquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 524
## 2014 425 475 734 816
## 2015 480 699 694 746
## 2016 659 552 727JDataA <- window(Jquarterly)
JfitA <- ets(JDataA)
summary(JfitA)## ETS(A,N,N)
##
## Call:
## ets(y = JDataA)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 627.1647
##
## sigma: 123.759
##
## AIC AICc BIC
## 151.4589 154.4589 152.9137
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.4374141 123.759 113.7315 -4.286211 19.63951 0.874858
## ACF1
## Training set 0.1295097JDecomp <- decompose(Jquarterly)
JDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 524
## 2014 425 475 734 816
## 2015 480 699 694 746
## 2016 659 552 727
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 102.3750
## 2014 -113.5000 -43.6875 54.8125 102.3750
## 2015 -113.5000 -43.6875 54.8125 102.3750
## 2016 -113.5000 -43.6875 54.8125
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 NA
## 2014 NA 576.000 619.375 654.250
## 2015 677.250 663.500 677.125 681.125
## 2016 666.875 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 NA
## 2014 NA -57.3125 59.8125 59.3750
## 2015 -83.7500 79.1875 -37.9375 -37.5000
## 2016 105.6250 NA NA
##
## $figure
## [1] 102.3750 -113.5000 -43.6875 54.8125
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(JDecomp)
forecast(JfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 627.1653 468.5618 785.7688 384.6022 869.7284
## 2017 Q1 627.1653 468.5618 785.7688 384.6021 869.7284
## 2017 Q2 627.1653 468.5618 785.7688 384.6021 869.7284
## 2017 Q3 627.1653 468.5618 785.7688 384.6021 869.7284
## 2017 Q4 627.1653 468.5618 785.7688 384.6021 869.7284
## 2018 Q1 627.1653 468.5618 785.7688 384.6021 869.7284
## 2018 Q2 627.1653 468.5618 785.7688 384.6021 869.7284
## 2018 Q3 627.1653 468.5618 785.7688 384.6021 869.7284K <- ts(c(51,118,171,200,160,137,155,142,180,181,196,216,235,239,205,210,171,201,221,169,235,182,209,239,249,385,274,217),
start = c(2014, 7), frequency = 12)
KData <- window(K)
Kfit <- ets(KData)
summary(Kfit)## ETS(M,A,N)
##
## Call:
## ets(y = KData)
##
## Smoothing parameters:
## alpha = 9e-04
## beta = 1e-04
##
## Initial states:
## l = 126.5158
## b = 4.9157
##
## sigma: 0.2093
##
## AIC AICc BIC
## 310.6141 313.3414 317.2751
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.3765047 40.10694 28.41424 -6.218273 17.93991 0.5380212
## ACF1
## Training set 0.2144315plot(Kfit)
plot(forecast(Kfit,h=8),
ylab="Forecasted Demand")
forecast(Kfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 269.0649 196.9039 341.2259 158.7041 379.4256
## Dec 2016 273.9817 200.5020 347.4614 161.6042 386.3592
## Jan 2017 278.8984 204.1000 353.6968 164.5042 393.2927
## Feb 2017 283.8152 207.6981 359.9323 167.4042 400.2263
## Mar 2017 288.7320 211.2962 366.1678 170.3041 407.1599
## Apr 2017 293.6488 214.8942 372.4033 173.2041 414.0935
## May 2017 298.5656 218.4923 378.6388 176.1040 421.0271
## Jun 2017 303.4823 222.0903 384.8744 179.0040 427.9607
## Jul 2017 308.3991 225.6883 391.1099 181.9039 434.8944
## Aug 2017 313.3159 229.2863 397.3455 184.8038 441.8280
## Sep 2017 318.2327 232.8843 403.5810 187.7036 448.7617
## Oct 2017 323.1495 236.4823 409.8166 190.6035 455.6954
## Nov 2017 328.0662 240.0803 416.0522 193.5033 462.6292
## Dec 2017 332.9830 243.6782 422.2878 196.4031 469.5629
## Jan 2018 337.8998 247.2762 428.5234 199.3029 476.4967
## Feb 2018 342.8166 250.8741 434.7591 202.2027 483.4305
## Mar 2018 347.7334 254.4720 440.9947 205.1024 490.3643
## Apr 2018 352.6502 258.0699 447.2304 208.0021 497.2982
## May 2018 357.5669 261.6677 453.4661 210.9018 504.2321
## Jun 2018 362.4837 265.2656 459.7018 213.8014 511.1660
## Jul 2018 367.4005 268.8634 465.9376 216.7011 518.0999
## Aug 2018 372.3173 272.4612 472.1733 219.6007 525.0339
## Sep 2018 377.2341 276.0590 478.4091 222.5002 531.9679
## Oct 2018 382.1508 279.6568 484.6449 225.3997 538.9019Quarterly
kquarterly <- aggregate(K, nfrequency=4)
kquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 340 497
## 2015 477 593 679 582
## 2016 625 630 908kDataA <- window(kquarterly)
kfitA <- ets(kDataA)
summary(kfitA)## ETS(M,N,N)
##
## Call:
## ets(y = kDataA)
##
## Smoothing parameters:
## alpha = 0.6793
##
## Initial states:
## l = 397.2878
##
## sigma: 0.2376
##
## AIC AICc BIC
## 112.2407 117.0407 112.8324
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 68.73648 123.7605 94.06214 9.215167 15.12356 0.56123
## ACF1
## Training set -0.2338792kDecomp <- decompose(kquarterly)
kDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 340 497
## 2015 477 593 679 582
## 2016 625 630 908
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 73.03125 -47.09375
## 2015 -42.09375 16.15625 73.03125 -47.09375
## 2016 -42.09375 16.15625 73.03125
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 NA NA
## 2015 519.125 572.125 601.250 624.375
## 2016 657.625 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 NA NA
## 2015 -0.03125 4.71875 4.71875 4.71875
## 2016 9.46875 NA NA
##
## $figure
## [1] 73.03125 -47.09375 -42.09375 16.15625
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(kDecomp)
forecast(kfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 817.504 568.6104 1066.398 436.854038 1198.154
## 2017 Q1 817.504 513.9507 1121.057 353.259244 1281.749
## 2017 Q2 817.504 466.6086 1168.399 280.855832 1354.152
## 2017 Q3 817.504 423.9086 1211.099 215.551778 1419.456
## 2017 Q4 817.504 384.4509 1250.557 155.206483 1479.802
## 2018 Q1 817.504 347.3955 1287.613 98.535120 1536.473
## 2018 Q2 817.504 312.1917 1322.816 44.695509 1590.313
## 2018 Q3 817.504 278.4555 1356.553 -6.899556 1641.908Monthly
L <- ts(c(64,186,176,185,186,191,141,72,92,62,64,139,208,162,188,198,194,217,124,89,71,84,53,140,203,214,221,226,262,239,115,100,64,45,97,195,192,257,235,250,275,256,208,134,102,82,173,146,232,263,215,262,261,248
),
start = c(2012, 5), frequency = 12)
LData <- window(L)
Lfit <- ets(LData)
summary(Lfit)## ETS(A,A,A)
##
## Call:
## ets(y = LData)
##
## Smoothing parameters:
## alpha = 1e-04
## beta = 1e-04
## gamma = 1e-04
##
## Initial states:
## l = 114.0429
## b = 1.6887
## s=-13.6711 -70.6222 -97.4467 -82.0822 -63.081 -13.6397
## 69.1101 77.0645 59.7065 50.1901 55.6766 28.7951
##
## sigma: 23.6311
##
## AIC AICc BIC
## 590.9618 607.9618 624.7746
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.8834436 23.63106 17.46351 -2.20327 15.02224 0.5924615
## ACF1
## Training set -0.06401832plot(Lfit)
plot(forecast(Lfit,h=8),
ylab="Forecasted Demand")
forecast(Lfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 193.5460 163.2616 223.8305 147.23001 239.8621
## Dec 2016 145.7979 115.5135 176.0824 99.48191 192.1140
## Jan 2017 128.4905 98.2061 158.7749 82.17449 174.8066
## Feb 2017 114.8193 84.5349 145.1038 68.50329 161.1354
## Mar 2017 143.3373 113.0528 173.6217 97.02121 189.6533
## Apr 2017 201.9816 171.6972 232.2661 155.66556 248.2977
## May 2017 246.1370 215.8526 276.4215 199.82093 292.4531
## Jun 2017 274.7159 244.4315 305.0004 228.39986 321.0320
## Jul 2017 270.9200 240.6356 301.2045 224.60392 317.2362
## Aug 2017 282.1328 251.8483 312.4173 235.81669 328.4490
## Sep 2017 301.1805 270.8960 331.4650 254.86435 347.4967
## Oct 2017 294.9199 264.6354 325.2045 248.60370 341.2361
## Nov 2017 213.8671 183.5825 244.1517 167.55086 260.1834
## Dec 2017 166.1190 135.8344 196.4036 119.80270 212.4353
## Jan 2018 148.8116 118.5270 179.0963 102.49523 195.1280
## Feb 2018 135.1404 104.8557 165.4251 88.82397 181.4569
## Mar 2018 163.6583 133.3736 193.9431 117.34182 209.9749
## Apr 2018 222.3027 192.0179 252.5875 175.98611 268.6193
## May 2018 266.4581 236.1732 296.7430 220.14140 312.7748
## Jun 2018 295.0370 264.7521 325.3220 248.72024 341.3538
## Jul 2018 291.2411 260.9561 321.5261 244.92421 337.5580
## Aug 2018 302.4539 272.1688 332.7390 256.13689 348.7710
## Sep 2018 321.5016 291.2164 351.7868 275.18444 367.8188
## Oct 2018 315.2410 284.9557 345.5263 268.92368 361.5583Quarterly
lquarterly <- aggregate(L, nfrequency=4)
lquarterly## Time Series:
## Start = 2012.33333333333
## End = 2016.58333333333
## Frequency = 4
## [1] 426 562 305 265 558 609 284 277 638 727 279 337 684 781 444 401 710
## [18] 771lDataA <- window(lquarterly)
lfitA <- ets(lDataA)
summary(lfitA)## ETS(A,N,A)
##
## Call:
## ets(y = lDataA)
##
## Smoothing parameters:
## alpha = 0.6156
## gamma = 1e-04
##
## Initial states:
## l = 425.6685
## s=-168.0598 -145.1341 197.0304 116.1635
##
## sigma: 53.7266
##
## AIC AICc BIC
## 209.4474 220.6474 215.6800
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 13.79875 53.72657 41.93349 2.257509 9.793262 0.6988915
## ACF1
## Training set 0.01610375lDecomp <- decompose(lquarterly)
lDecomp## $x
## Time Series:
## Start = 2012.33333333333
## End = 2016.58333333333
## Frequency = 4
## [1] 426 562 305 265 558 609 284 277 638 727 279 337 684 781 444 401 710
## [18] 771
##
## $seasonal
## Time Series:
## Start = 2012.33333333333
## End = 2016.58333333333
## Frequency = 4
## [1] 137.2891 204.8307 -159.3568 -182.7630 137.2891 204.8307 -159.3568
## [8] -182.7630 137.2891 204.8307 -159.3568 -182.7630 137.2891 204.8307
## [15] -159.3568 -182.7630 137.2891 204.8307
##
## $trend
## Time Series:
## Start = 2012.33333333333
## End = 2016.58333333333
## Frequency = 4
## [1] NA NA 406.000 428.375 431.625 430.500 442.000 466.750
## [9] 480.875 487.750 501.000 513.500 540.875 569.500 580.750 582.750
## [17] NA NA
##
## $random
## Time Series:
## Start = 2012.33333333333
## End = 2016.58333333333
## Frequency = 4
## [1] NA NA 58.356771 19.388021 -10.914062 -26.330729
## [7] 1.356771 -6.986979 19.835938 34.419271 -62.643229 6.263021
## [13] 5.835938 6.669271 22.606771 1.013021 NA NA
##
## $figure
## [1] 137.2891 204.8307 -159.3568 -182.7630
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(lDecomp)
forecast(lfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016.833 433.4396 364.5862 502.2930 328.1375 538.7417
## 2017.083 410.5180 329.6639 491.3722 286.8623 534.1738
## 2017.333 694.7426 603.4487 786.0366 555.1206 834.3647
## 2017.583 775.6072 674.9534 876.2611 621.6705 929.5440
## 2017.833 433.4396 324.2251 542.6541 266.4105 600.4687
## 2018.083 410.5180 293.3668 527.6693 231.3507 589.6854
## 2018.333 694.7426 570.1569 819.3283 504.2052 885.2800
## 2018.583 775.6072 644.0086 907.2059 574.3444 976.8700Monthly
M <- ts(c(73,79,102,112,122,114,96,125,83,98,80,96,96,101,97,147,145,154,137,122,101,106,92,108,94,133,95,124,145,162,137,123,146,135,96,166,119,125,153,184,175,204,213,150,154,134,148,58,19,29,29,27,42,155,143,184,210,180
),
start = c(2012, 1), frequency = 12)
MData <- window(M)
Mfit <- ets(MData)
summary(Mfit)## ETS(A,N,N)
##
## Call:
## ets(y = MData)
##
## Smoothing parameters:
## alpha = 0.8564
##
## Initial states:
## l = 74.3316
##
## sigma: 30.9283
##
## AIC AICc BIC
## 639.5796 640.0241 645.7610
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 2.200898 30.92831 22.28927 -5.284683 23.16611 0.5677223
## ACF1
## Training set -0.02699559plot(Mfit)
plot(forecast(Mfit,h=8),
ylab="Forecasted Demand")
forecast(Mfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 183.6491 144.01288 223.2853 123.030734 244.2675
## Dec 2016 183.6491 131.46504 235.8332 103.840457 263.4578
## Jan 2017 183.6491 121.39700 245.9012 88.442730 278.8555
## Feb 2017 183.6491 112.74444 254.5538 75.209780 292.0884
## Mar 2017 183.6491 105.03856 262.2597 63.424645 303.8736
## Apr 2017 183.6491 98.02338 269.2748 52.695853 314.6024
## May 2017 183.6491 91.54095 275.7573 42.781835 324.5164
## Jun 2017 183.6491 85.48567 281.8125 33.521089 333.7771
## Jul 2017 183.6491 79.78281 287.5154 24.799318 342.4989
## Aug 2017 183.6491 74.37718 292.9210 16.532115 350.7661
## Sep 2017 183.6491 69.22664 298.0716 8.655043 358.6432
## Oct 2017 183.6491 64.29816 303.0001 1.117587 366.1806
## Nov 2017 183.6491 59.56528 307.7329 -6.120726 373.4189
## Dec 2017 183.6491 55.00641 312.2918 -13.092914 380.3911
## Jan 2018 183.6491 50.60367 316.6946 -19.826338 387.1246
## Feb 2018 183.6491 46.34202 320.9562 -26.343964 393.6422
## Mar 2018 183.6491 42.20872 325.0895 -32.665302 399.9635
## Apr 2018 183.6491 38.19282 329.1054 -38.807085 406.1053
## May 2018 183.6491 34.28486 333.0134 -44.783795 412.0820
## Jun 2018 183.6491 30.47657 336.8216 -50.608068 417.9063
## Jul 2018 183.6491 26.76070 340.5375 -56.291006 423.5892
## Aug 2018 183.6491 23.13082 344.1674 -61.842423 429.1406
## Sep 2018 183.6491 19.58124 347.7170 -67.271049 434.5693
## Oct 2018 183.6491 16.10683 351.1914 -72.584689 439.8829Quarterly
Mquarterly <- aggregate(M, nfrequency=4)
Mquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 254 348 304 274
## 2013 294 446 360 306
## 2014 322 431 406 397
## 2015 397 563 517 340
## 2016 77 224 537MDataA <- window(Mquarterly)
MfitA <- ets(MDataA)
summary(MfitA)## ETS(M,N,N)
##
## Call:
## ets(y = MDataA)
##
## Smoothing parameters:
## alpha = 0.146
##
## Initial states:
## l = 302.3688
##
## sigma: 0.3339
##
## AIC AICc BIC
## 241.0965 242.6965 243.9299
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 21.91731 119.9167 87.44649 -16.40537 39.81236 0.8984228
## ACF1
## Training set 0.268795MDecomp <- decompose(Mquarterly)
MDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 254 348 304 274
## 2013 294 446 360 306
## 2014 322 431 406 397
## 2015 397 563 517 340
## 2016 77 224 537
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -87.82812 85.07812 30.42188 -27.67188
## 2013 -87.82812 85.07812 30.42188 -27.67188
## 2014 -87.82812 85.07812 30.42188 -27.67188
## 2015 -87.82812 85.07812 30.42188 -27.67188
## 2016 -87.82812 85.07812 30.42188
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 300.000 317.250
## 2013 336.500 347.500 355.000 356.625
## 2014 360.500 377.625 398.375 424.250
## 2015 454.625 461.375 414.250 331.875
## 2016 292.000 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -26.421875 -15.578125
## 2013 45.328125 13.421875 -25.421875 -22.953125
## 2014 49.328125 -31.703125 -22.796875 0.421875
## 2015 30.203125 16.546875 72.328125 35.796875
## 2016 -127.171875 NA NA
##
## $figure
## [1] -87.82812 85.07812 30.42188 -27.67188
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(MDecomp)
forecast(MfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 363.1727 207.7856 518.5598 125.5287 600.8167
## 2017 Q1 363.1727 205.9554 520.3900 122.7295 603.6159
## 2017 Q2 363.1727 204.1419 522.2035 119.9561 606.3893
## 2017 Q3 363.1727 202.3447 524.0008 117.2074 609.1380
## 2017 Q4 363.1727 200.5631 525.7823 114.4827 611.8627
## 2018 Q1 363.1727 198.7966 527.5488 111.7812 614.5642
## 2018 Q2 363.1727 197.0448 529.3006 109.1020 617.2434
## 2018 Q3 363.1727 195.3072 531.0382 106.4446 619.9008Monthly
2,26,7,6,3,2,6,9,37,19,11,5,5,15,20,27,26,29,34,54,39,46,38,23
N <- ts(c(6,24,11,45,35,45,40,79,52,62,30,25,48,52,1,78,70,110,91,125,176,155,130,167,95,19,126,137,150,158,150,210,233,148
),
start = c(2014, 1), frequency = 12)
NData <- window(N)
Nfit <- ets(NData)
summary(Nfit)## ETS(M,A,N)
##
## Call:
## ets(y = NData)
##
## Smoothing parameters:
## alpha = 0.002
## beta = 0.002
##
## Initial states:
## l = 5.907
## b = 4.8811
##
## sigma: 0.4098
##
## AIC AICc BIC
## 362.3696 364.5125 370.0014
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.02406572 35.67067 25.01267 -255.9237 277.6084 0.4019566
## ACF1
## Training set 0.3421183plot(Nfit)
plot(forecast(Nfit,h=8),
ylab="Forecasted Demand")
forecast(Nfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 174.4951 82.85932 266.1309 34.35027 314.6399
## Dec 2016 179.3746 85.17447 273.5747 35.30793 323.4413
## Jan 2017 184.2541 87.48818 281.0200 36.26340 332.2448
## Feb 2017 189.1336 89.80009 288.4671 37.21611 341.0511
## Mar 2017 194.0131 92.10984 295.9164 38.16551 349.8607
## Apr 2017 198.8926 94.41709 303.3681 39.11109 358.6741
## May 2017 203.7721 96.72149 310.8227 40.05231 367.4919
## Jun 2017 208.6516 99.02272 318.2805 40.98869 376.3145
## Jul 2017 213.5311 101.32046 325.7418 41.91973 385.1425
## Aug 2017 218.4106 103.61440 333.2068 42.84495 393.9763
## Sep 2017 223.2901 105.90421 340.6760 43.76386 402.8164
## Oct 2017 228.1696 108.18959 348.1497 44.67600 411.6633
## Nov 2017 233.0491 110.47023 355.6280 45.58089 420.5174
## Dec 2017 237.9286 112.74583 363.1114 46.47806 429.3792
## Jan 2018 242.8081 115.01609 370.6002 47.36706 438.2492
## Feb 2018 247.6876 117.28069 378.0946 48.24742 447.1279
## Mar 2018 252.5671 119.53934 385.5949 49.11868 456.0156
## Apr 2018 257.4466 121.79173 393.1016 49.98036 464.9129
## May 2018 262.3261 124.03757 400.6147 50.83202 473.8203
## Jun 2018 267.2056 126.27654 408.1348 51.67318 482.7381
## Jul 2018 272.0852 128.50835 415.6620 52.50338 491.6669
## Aug 2018 276.9647 130.73268 423.1966 53.32215 500.6072
## Sep 2018 281.8442 132.94924 430.7391 54.12902 509.5593
## Oct 2018 286.7237 135.15770 438.2896 54.92353 518.5238Quarterly
Nquarterly <- aggregate(N, nfrequency=4)
Nquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 41 125 171 117
## 2015 101 258 392 452
## 2016 240 445 593NDataA <- window(Nquarterly)
NfitA <- ets(NDataA)
summary(NfitA)## ETS(M,A,N)
##
## Call:
## ets(y = NDataA)
##
## Smoothing parameters:
## alpha = 1e-04
## beta = 1e-04
##
## Initial states:
## l = 5.4186
## b = 43.0232
##
## sigma: 0.3037
##
## AIC AICc BIC
## 128.6572 140.6572 130.6467
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 3.271272 82.43401 66.12098 -12.72291 33.45558 0.3627327
## ACF1
## Training set -0.01109151NDecomp <- decompose(Nquarterly)
NDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 41 125 171 117
## 2015 101 258 392 452
## 2016 240 445 593
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 -119.296875 7.703125 70.515625 41.078125
## 2015 -119.296875 7.703125 70.515625 41.078125
## 2016 -119.296875 7.703125 70.515625
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 NA NA 121.000 145.125
## 2015 189.375 258.875 318.125 358.875
## 2016 407.375 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 NA NA -20.515625 -69.203125
## 2015 30.921875 -8.578125 3.359375 52.046875
## 2016 -48.078125 NA NA
##
## $figure
## [1] -119.296875 7.703125 70.515625 41.078125
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(NDecomp)
forecast(NfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 521.6737 318.6506 724.6969 211.1766 832.1708
## 2017 Q1 564.7006 344.9324 784.4688 228.5941 900.8070
## 2017 Q2 607.7274 371.2142 844.2407 246.0116 969.4432
## 2017 Q3 650.7543 397.4959 904.0126 263.4291 1038.0794
## 2017 Q4 693.7811 423.7777 963.7845 280.8466 1106.7157
## 2018 Q1 736.8080 450.0595 1023.5564 298.2640 1175.3519
## 2018 Q2 779.8348 476.3412 1083.3284 315.6814 1243.9882
## 2018 Q3 822.8617 502.6230 1143.1003 333.0988 1312.6245Monthly
3,8,5,3,22,2,4,3,2,25,2,19,1,7,2,18,12,7,62,43,64,92,57,45,
O <- ts(c(27,70,44,72,78,104,110,73,111,66,51,50,51,28,55,76,96,135,75,92,118,247,156,94,96,71,141,86,124,96,107,104,113,170
),
start = c(2014, 1), frequency = 12)
OData <- window(O)
Ofit <- ets(OData)
summary(Ofit)## ETS(M,N,N)
##
## Call:
## ets(y = OData)
##
## Smoothing parameters:
## alpha = 0.5771
##
## Initial states:
## l = 69.3159
##
## sigma: 0.4301
##
## AIC AICc BIC
## 370.1040 370.9040 374.6831
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 3.826632 39.60797 30.69276 -8.152196 36.42665 0.7519384
## ACF1
## Training set -0.005749268plot(Ofit)
plot(forecast(Ofit,h=8),
ylab="Forecasted Demand")
forecast(Ofit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 144.3987 64.814343 223.9831 22.6849065 266.1126
## Dec 2016 144.3987 50.413970 238.3835 0.6614362 288.1360
## Jan 2017 144.3987 37.223404 251.5741 -19.5117932 308.3093
## Feb 2017 144.3987 24.801893 263.9956 -38.5088557 327.3063
## Mar 2017 144.3987 12.892984 275.9045 -56.7219598 345.5194
## Apr 2017 144.3987 1.331226 287.4663 -74.4041427 363.2016
## May 2017 144.3987 -9.998152 298.7956 -91.7309313 380.5284
## Jun 2017 144.3987 -21.179242 309.9767 -108.8309331 397.6284
## Jul 2017 144.3987 -32.276395 321.0739 -125.8025637 414.6000
## Aug 2017 144.3987 -43.340652 332.1381 -142.7238842 431.5214
## Sep 2017 144.3987 -54.413748 343.2112 -159.6587240 448.4562
## Oct 2017 144.3987 -65.530723 354.3282 -176.6606694 465.4582
## Nov 2017 144.3987 -76.721680 365.5192 -193.7757619 482.5732
## Dec 2017 144.3987 -88.013024 376.8105 -211.0443814 499.8419
## Jan 2018 144.3987 -99.428338 388.2258 -228.5025975 517.3001
## Feb 2018 144.3987 -110.989038 399.7865 -246.1831621 534.9806
## Mar 2018 144.3987 -122.714855 411.5123 -264.1162528 552.9137
## Apr 2018 144.3987 -134.624211 423.4217 -282.3300412 571.1275
## May 2018 144.3987 -146.734504 435.5320 -300.8511350 589.6486
## Jun 2018 144.3987 -159.062335 447.8598 -319.7049253 608.5024
## Jul 2018 144.3987 -171.623693 460.4212 -338.9158658 627.7134
## Aug 2018 144.3987 -184.434104 473.2316 -358.5076987 647.3052
## Sep 2018 144.3987 -197.508748 486.3062 -378.5036400 667.3011
## Oct 2018 144.3987 -210.862560 499.6600 -398.9265342 687.7240Quarterly
Oquarterly <- aggregate(O, nfrequency=4)
Oquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 141 254 294 167
## 2015 134 307 285 497
## 2016 308 306 324ODataA <- window(Oquarterly)
OfitA <- ets(ODataA)
summary(OfitA)## ETS(A,N,N)
##
## Call:
## ets(y = ODataA)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 274.2472
##
## sigma: 97.7784
##
## AIC AICc BIC
## 133.1963 136.6249 134.3900
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.04354118 97.77838 72.94132 -15.32799 34.00356 0.8329351
## ACF1
## Training set 0.2238666ODecomp <- decompose(Oquarterly)
ODecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 141 254 294 167
## 2015 134 307 285 497
## 2016 308 306 324
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 -78.515625 32.109375 8.796875 37.609375
## 2015 -78.515625 32.109375 8.796875 37.609375
## 2016 -78.515625 32.109375 8.796875
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 NA NA 213.125 218.875
## 2015 224.375 264.500 327.500 349.125
## 2016 353.875 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2014 NA NA 72.07812 -89.48438
## 2015 -11.85938 10.39062 -51.29688 110.26562
## 2016 32.64062 NA NA
##
## $figure
## [1] -78.515625 32.109375 8.796875 37.609375
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(ODecomp)
forecast(OfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 274.2472 148.9392 399.5553 82.60512 465.8893
## 2017 Q1 274.2472 148.9392 399.5553 82.60512 465.8893
## 2017 Q2 274.2472 148.9392 399.5553 82.60512 465.8893
## 2017 Q3 274.2472 148.9392 399.5553 82.60512 465.8893
## 2017 Q4 274.2472 148.9392 399.5553 82.60511 465.8893
## 2018 Q1 274.2472 148.9392 399.5553 82.60511 465.8893
## 2018 Q2 274.2472 148.9392 399.5553 82.60511 465.8893
## 2018 Q3 274.2472 148.9392 399.5553 82.60511 465.8893 P <- ts(c(3,9,50,28,45,30,26,28,115,17,5,36,13,1,76,32,92,86,50,65,37,44,79,19,99,78,96,30,50,109,41,46,97,255,175,140,134,193,85,178,50,144,68,111),
start = c(2013, 3), frequency = 12)
PData <- window(P)
Pfit <- ets(PData)
summary(Pfit)## ETS(M,Ad,N)
##
## Call:
## ets(y = PData)
##
## Smoothing parameters:
## alpha = 6e-04
## beta = 1e-04
## phi = 0.9673
##
## Initial states:
## l = 9.862
## b = 4.0461
##
## sigma: 0.658
##
## AIC AICc BIC
## 503.5232 505.7934 514.2283
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 4.2918 45.26952 33.60282 -175.7104 204.759 0.6678821
## ACF1
## Training set 0.1494022plot(Pfit)
plot(forecast(Pfit,h=8),
ylab="Forecasted Demand")
forecast(Pfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 102.8401 16.11376 189.5665 -29.79641 235.4767
## Dec 2016 103.7376 16.25434 191.2208 -30.05649 237.5317
## Jan 2017 104.6057 16.39032 192.8211 -30.30810 239.5196
## Feb 2017 105.4455 16.52185 194.3692 -30.55149 241.4425
## Mar 2017 106.2578 16.64906 195.8666 -30.78695 243.3026
## Apr 2017 107.0436 16.77211 197.3151 -31.01474 245.1020
## May 2017 107.8037 16.89112 198.7163 -31.23511 246.8426
## Jun 2017 108.5390 17.00623 200.0718 -31.44829 248.5263
## Jul 2017 109.2502 17.11756 201.3829 -31.65453 250.1550
## Aug 2017 109.9382 17.22524 202.6512 -31.85406 251.7305
## Sep 2017 110.6038 17.32938 203.8781 -32.04709 253.2546
## Oct 2017 111.2475 17.43011 205.0650 -32.23384 254.7289
## Nov 2017 111.8703 17.52752 206.2130 -32.41451 256.1551
## Dec 2017 112.4727 17.62174 207.3236 -32.58931 257.5346
## Jan 2018 113.0554 17.71285 208.3979 -32.75842 258.8692
## Feb 2018 113.6190 17.80098 209.4371 -32.92204 260.1601
## Mar 2018 114.1643 17.88620 210.4423 -33.08033 261.4089
## Apr 2018 114.6917 17.96862 211.4148 -33.23349 262.6169
## May 2018 115.2019 18.04832 212.3555 -33.38167 263.7854
## Jun 2018 115.6954 18.12540 213.2654 -33.52504 264.9159
## Jul 2018 116.1728 18.19994 214.1457 -33.66375 266.0094
## Aug 2018 116.6346 18.27203 214.9972 -33.79797 267.0672
## Sep 2018 117.0813 18.34173 215.8209 -33.92783 268.0904
## Oct 2018 117.5134 18.40914 216.6177 -34.05348 269.0803Quarterly
Pquarterly <- aggregate(P, nfrequency=4)
Pquarterly## Time Series:
## Start = 2013.16666666667
## End = 2016.41666666667
## Frequency = 4
## [1] 62 103 169 58 90 210 152 142 273 189 184 570 412 372PDataA <- window(Pquarterly)
PfitA <- ets(PDataA)
summary(PfitA)## ETS(M,A,N)
##
## Call:
## ets(y = PDataA)
##
## Smoothing parameters:
## alpha = 1e-04
## beta = 1e-04
##
## Initial states:
## l = 55.7447
## b = 21.6039
##
## sigma: 0.3719
##
## AIC AICc BIC
## 167.3220 174.8220 170.5173
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -4.412721 91.24897 67.96411 -23.74178 40.82991 0.5561712
## ACF1
## Training set 0.01017069PDecomp <- decompose(Pquarterly)
PDecomp## $x
## Time Series:
## Start = 2013.16666666667
## End = 2016.41666666667
## Frequency = 4
## [1] 62 103 169 58 90 210 152 142 273 189 184 570 412 372
##
## $seasonal
## Time Series:
## Start = 2013.16666666667
## End = 2016.41666666667
## Frequency = 4
## [1] 13.067708 -1.869792 -36.869792 25.671875 13.067708 -1.869792
## [7] -36.869792 25.671875 13.067708 -1.869792 -36.869792 25.671875
## [13] 13.067708 -1.869792
##
## $trend
## Time Series:
## Start = 2013.16666666667
## End = 2016.41666666667
## Frequency = 4
## [1] NA NA 101.500 118.375 129.625 138.000 171.375 191.625
## [9] 193.000 250.500 321.375 361.625 NA NA
##
## $random
## Time Series:
## Start = 2013.16666666667
## End = 2016.41666666667
## Frequency = 4
## [1] NA NA 104.36979 -86.04688 -52.69271 73.86979
## [7] 17.49479 -75.29688 66.93229 -59.63021 -100.50521 182.70312
## [13] NA NA
##
## $figure
## [1] 13.067708 -1.869792 -36.869792 25.671875
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(PDecomp)
forecast(PfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016.667 379.6110 198.6769 560.5450 102.8962 656.3257
## 2016.917 401.2086 209.9805 592.4368 108.7504 693.6669
## 2017.167 422.8063 221.2841 624.3286 114.6046 731.0081
## 2017.417 444.4040 232.5876 656.2204 120.4588 768.3492
## 2017.667 466.0017 243.8911 688.1122 126.3129 805.6905
## 2017.917 487.5994 255.1947 720.0041 132.1671 843.0317
## 2018.167 509.1970 266.4982 751.8959 138.0212 880.3729
## 2018.417 530.7947 277.8017 783.7878 143.8753 917.7142Q <- ts(c(24,3,12,25,1,10,22,14,9,10,9,14,9,5,7,16,42,28,20,20,10,27,25,45, 6,10,14,10,41,103,22,82,101,154,105,104,85,41,75,72,59,135,250,84,130,144,139,94,157,52,70,52,56,73,89,123,139,143),
start = c(2012, 1), frequency = 12)
QData <- window(Q)
Qfit <- ets(QData)
summary(Qfit)## ETS(M,A,N)
##
## Call:
## ets(y = QData)
##
## Smoothing parameters:
## alpha = 0.8647
## beta = 1e-04
##
## Initial states:
## l = 12.1355
## b = 4.9559
##
## sigma: 0.5666
##
## AIC AICc BIC
## 616.7643 617.9181 627.0665
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -3.112284 39.1548 25.07897 -102.6211 127.4585 0.6949595
## ACF1
## Training set -0.3194836plot(Qfit)
plot(forecast(Qfit,h=8),
ylab="Forecasted Demand")
forecast(Qfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 147.7859 40.479250 255.0925 -16.32544 311.8972
## Dec 2016 152.7237 -1.143435 306.5908 -82.59578 388.0432
## Jan 2017 157.6615 -40.754542 356.0776 -145.78967 461.1128
## Feb 2017 162.5994 -81.254548 406.4533 -210.34302 535.5418
## Mar 2017 167.5372 -124.078481 459.1529 -278.45050 613.5249
## Apr 2017 172.4750 -170.205584 515.1557 -351.60975 696.5599
## May 2017 177.4129 -220.447558 575.2733 -431.06215 785.8879
## Jun 2017 182.3507 -275.562018 640.2635 -517.96638 882.6678
## Jul 2017 187.2886 -336.307786 710.8849 -613.48295 988.0601
## Aug 2017 192.2264 -403.477049 787.9298 -718.82340 1103.2762
## Sep 2017 197.1642 -477.917292 872.2457 -835.28386 1229.6123
## Oct 2017 202.1021 -560.548532 964.7527 -964.27137 1368.4755
## Nov 2017 207.0399 -652.378511 1066.4583 -1107.32714 1521.4069
## Dec 2017 211.9777 -754.517238 1178.4727 -1266.14877 1690.1042
## Jan 2018 216.9156 -868.191729 1302.0229 -1442.61284 1876.4440
## Feb 2018 221.8534 -994.761479 1438.4683 -1638.79850 2082.5053
## Mar 2018 226.7912 -1135.735075 1589.3176 -1857.01294 2310.5954
## Apr 2018 231.7291 -1292.788262 1756.2464 -2099.81899 2563.2772
## May 2018 236.6669 -1467.783780 1941.1176 -2370.06548 2843.3993
## Jun 2018 241.6048 -1662.793237 2146.0027 -2670.92064 3154.1301
## Jul 2018 246.5426 -1880.121327 2373.2065 -3005.90920 3498.9944
## Aug 2018 251.4804 -2122.332681 2625.2935 -3378.95343 3881.9143
## Sep 2018 256.4183 -2392.281703 2905.1182 -3794.41877 4307.2553
## Oct 2018 261.3561 -2693.145729 3215.8579 -4257.16452 4779.8767Quarterly
Qquarterly <- aggregate(Q, nfrequency=4)
Qquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 39 36 45 33
## 2013 21 86 50 97
## 2014 30 154 205 363
## 2015 201 266 464 377
## 2016 279 181 351QDataA <- window(Qquarterly)
QfitA <- ets(QDataA)
summary(QfitA)## ETS(M,A,N)
##
## Call:
## ets(y = QDataA)
##
## Smoothing parameters:
## alpha = 0.3757
## beta = 1e-04
##
## Initial states:
## l = 14.2506
## b = 13.4415
##
## sigma: 0.5081
##
## AIC AICc BIC
## 223.5831 228.1985 228.3053
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 7.123587 86.4731 62.95245 -29.61569 58.03064 0.6436856
## ACF1
## Training set 0.1128948QDecomp <- decompose(Qquarterly)
QDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 39 36 45 33
## 2013 21 86 50 97
## 2014 30 154 205 363
## 2015 201 266 464 377
## 2016 279 181 351
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -57.11198 -11.88281 27.26302 41.73177
## 2013 -57.11198 -11.88281 27.26302 41.73177
## 2014 -57.11198 -11.88281 27.26302 41.73177
## 2015 -57.11198 -11.88281 27.26302 41.73177
## 2016 -57.11198 -11.88281 27.26302
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 36.000 40.000
## 2013 46.875 55.500 64.625 74.250
## 2014 102.125 154.750 209.375 244.750
## 2015 291.125 325.250 336.750 335.875
## 2016 311.125 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -18.2630208 -48.7317708
## 2013 31.2369792 42.3828125 -41.8880208 -18.9817708
## 2014 -15.0130208 11.1328125 -31.6380208 76.5182292
## 2015 -33.0130208 -47.3671875 99.9869792 -0.6067708
## 2016 24.9869792 NA NA
##
## $figure
## [1] -57.11198 -11.88281 27.26302 41.73177
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(QDecomp)
forecast(QfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 334.0258 116.51419 551.5375 1.370469 666.6812
## 2017 Q1 347.4809 103.33041 591.6314 -25.915055 720.8769
## 2017 Q2 360.9360 90.63328 631.2387 -52.456324 774.3283
## 2017 Q3 374.3911 78.18511 670.5970 -78.616841 827.3989
## 2017 Q4 387.8461 65.82211 709.8701 -104.647105 880.3393
## 2018 Q1 401.3012 53.42577 749.1766 -130.728352 933.3307
## 2018 Q2 414.7563 40.90697 788.6055 -156.996892 986.5094
## 2018 Q3 428.2113 28.19648 828.2262 -183.558593 1039.9812Monthly
R <- ts(c(34,171,64,48,55,39,33,63,18,45,57,15, 51,33,25,51,34,30,39,34,28,81,78,26, 63,32,36,77,41,31,80,55,72,167,118,142,56,33,187,76,67,73,68,87,113,234,102,108,42,49,99,56,57,42,4,1,112,190), start = c(2012, 1), frequency = 12)
RData <- window(R)
Rfit <- ets(RData)
summary(Rfit)## ETS(M,N,M)
##
## Call:
## ets(y = RData)
##
## Smoothing parameters:
## alpha = 0.1086
## gamma = 1e-04
##
## Initial states:
## l = 57.3819
## s=1.113 1.1706 1.772 0.9405 0.7599 0.7493
## 0.5241 0.7079 0.9426 1.3097 1.2882 0.7222
##
## sigma: 0.5212
##
## AIC AICc BIC
## 664.2202 675.6488 695.1268
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 2.675134 36.8921 27.58712 -138.0946 166.866 0.8187146
## ACF1
## Training set 0.1003402plot(Rfit)
plot(forecast(Rfit,h=8),
ylab="Forecasted Demand")
forecast(Rfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 88.27899 29.311742 147.24624 -1.903632 178.46161
## Dec 2016 83.92700 27.447763 140.40624 -2.450540 170.30455
## Jan 2017 54.46436 17.541456 91.38726 -2.004348 110.93307
## Feb 2017 97.13003 30.802025 163.45804 -4.309901 198.56996
## Mar 2017 98.75664 30.830917 166.68237 -5.126791 202.64008
## Apr 2017 71.08472 21.842882 120.32657 -4.224175 146.39362
## May 2017 53.38489 16.142912 90.62687 -3.571801 110.34158
## Jun 2017 39.52488 11.759189 67.29058 -2.939081 81.98885
## Jul 2017 56.50011 16.535209 96.46502 -4.620933 117.62116
## Aug 2017 57.29841 16.491634 98.10519 -5.110169 119.70699
## Sep 2017 70.92679 20.072246 121.78133 -6.848524 148.70210
## Oct 2017 133.63447 37.176548 230.09239 -13.885191 281.15413
## Nov 2017 88.27926 24.135516 152.42300 -9.820129 186.37864
## Dec 2017 83.92726 22.545263 145.30925 -9.948401 177.80292
## Jan 2018 54.46452 14.371739 94.55731 -6.852100 115.78115
## Feb 2018 97.13033 25.169729 169.09092 -12.923909 207.18456
## Mar 2018 98.75694 25.124642 172.38924 -13.853944 211.36783
## Apr 2018 71.08494 17.749831 124.42005 -10.484070 152.65395
## May 2018 53.38505 13.079517 93.69059 -8.256944 115.02705
## Jun 2018 39.52500 9.498725 69.55128 -6.396226 85.44623
## Jul 2018 56.50028 13.314535 99.68603 -9.546619 122.54719
## Aug 2018 57.29858 13.235984 101.36118 -10.089347 124.68651
## Sep 2018 70.92701 16.054987 125.79902 -12.992505 154.84652
## Oct 2018 133.63487 29.631038 237.63871 -25.425267 292.69502Quarterly
Rquarterly <- aggregate(R, nfrequency=4)
Rquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 269 142 114 117
## 2013 109 115 101 185
## 2014 131 149 207 427
## 2015 276 216 268 444
## 2016 190 155 117RDataA <- window(Rquarterly)
RfitA <- ets(RDataA)
summary(RfitA)## ETS(M,N,N)
##
## Call:
## ets(y = RDataA)
##
## Smoothing parameters:
## alpha = 0.9999
##
## Initial states:
## l = 232.2842
##
## sigma: 0.4273
##
## AIC AICc BIC
## 227.6587 229.2587 230.4920
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -6.067994 104.6002 74.87981 -12.11409 35.00823 0.8914263
## ACF1
## Training set -0.2295149RDecomp <- decompose(Rquarterly)
RDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 269 142 114 117
## 2013 109 115 101 185
## 2014 131 149 207 427
## 2015 276 216 268 444
## 2016 190 155 117
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -22.53906 -44.07031 -28.10156 94.71094
## 2013 -22.53906 -44.07031 -28.10156 94.71094
## 2014 -22.53906 -44.07031 -28.10156 94.71094
## 2015 -22.53906 -44.07031 -28.10156 94.71094
## 2016 -22.53906 -44.07031 -28.10156
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 140.500 117.125
## 2013 112.125 119.000 130.250 137.250
## 2014 154.750 198.250 246.625 273.125
## 2015 289.125 298.875 290.250 271.875
## 2016 245.375 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 1.601562 -94.835938
## 2013 19.414062 40.070312 -1.148438 -46.960938
## 2014 -1.210938 -5.179688 -11.523438 59.164062
## 2015 9.414062 -38.804688 5.851562 77.414062
## 2016 -32.835938 NA NA
##
## $figure
## [1] -22.53906 -44.07031 -28.10156 94.71094
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(RDecomp)
forecast(RfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 117.0038 52.925336 181.0823 19.00424 215.0034
## 2017 Q1 117.0038 22.341346 211.6663 -27.76993 261.7775
## 2017 Q2 117.0038 -4.249442 238.2571 -68.43703 302.4446
## 2017 Q3 117.0038 -29.597718 263.6053 -107.20388 341.2115
## 2017 Q4 117.0038 -54.813984 288.8216 -145.76883 379.7764
## 2018 Q1 117.0038 -80.522742 314.5304 -185.08698 419.0946
## 2018 Q2 117.0038 -107.152024 341.1596 -225.81295 459.8206
## 2018 Q3 117.0038 -135.038255 369.0459 -268.46126 502.4689Monthly
S <- ts(c(91,89,84,77,65,59,73,99,108,167,134,158,191,113,114, 79,46,47,68,78,87,122,90,105,99,109,89,57,69,73,73,60,89,104,109,88,89), start = c(2013, 10), frequency = 12)
SData <- window(S)
Sfit <- ets(SData)
summary(Sfit)## ETS(M,N,N)
##
## Call:
## ets(y = SData)
##
## Smoothing parameters:
## alpha = 0.9999
##
## Initial states:
## l = 86.3414
##
## sigma: 0.2403
##
## AIC AICc BIC
## 365.9519 366.6792 370.7847
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.07185931 24.59677 18.17952 -2.970506 19.80661 0.6917627
## ACF1
## Training set -0.1463349plot(Sfit)
plot(forecast(Sfit,h=8),
ylab="Forecasted Demand")
forecast(Sfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 88.9999 61.586425 116.4134 47.074608 130.9252
## Dec 2016 88.9999 49.677525 128.3223 28.861516 149.1383
## Jan 2017 88.9999 40.143983 137.8558 14.281223 163.7186
## Feb 2017 88.9999 31.762640 146.2372 1.463065 176.5367
## Mar 2017 88.9999 24.064061 153.9357 -10.310895 188.3107
## Apr 2017 88.9999 16.808873 161.1909 -21.406748 199.4065
## May 2017 88.9999 9.855191 168.1446 -32.041488 210.0413
## Jun 2017 88.9999 3.110415 174.8894 -42.356732 220.3565
## Jul 2017 88.9999 -3.490156 181.4900 -52.451434 230.4512
## Aug 2017 88.9999 -9.994132 187.9939 -62.398407 240.3982
## Sep 2017 88.9999 -16.438020 194.4378 -72.253483 250.2533
## Oct 2017 88.9999 -22.850778 200.8506 -82.060951 260.0608
## Nov 2017 88.9999 -29.256054 207.2559 -91.856974 269.8568
## Dec 2017 88.9999 -35.673641 213.6734 -101.671827 279.6716
## Jan 2018 88.9999 -42.120481 220.1203 -111.531418 289.5312
## Feb 2018 88.9999 -48.611359 226.6112 -121.458359 299.4582
## Mar 2018 88.9999 -55.159404 233.1592 -131.472730 309.4725
## Apr 2018 88.9999 -61.776465 239.7763 -141.592651 319.5925
## May 2018 88.9999 -68.473382 246.4732 -151.834702 329.8345
## Jun 2018 88.9999 -75.260205 253.2600 -162.214252 340.2141
## Jul 2018 88.9999 -82.146354 260.1462 -172.745708 350.7455
## Aug 2018 88.9999 -89.140754 267.1406 -183.442720 361.4425
## Sep 2018 88.9999 -96.251939 274.2517 -194.318339 372.3181
## Oct 2018 88.9999 -103.488133 281.4879 -205.385142 383.3849Quarterly
Squarterly <- aggregate(S, nfrequency=4)
Squarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 264
## 2014 201 280 459 418
## 2015 172 233 317 297
## 2016 199 222 301SDataA <- window(Squarterly)
SfitA <- ets(SDataA)
summary(SfitA)## ETS(A,N,N)
##
## Call:
## ets(y = SDataA)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 280.2618
##
## sigma: 83.4099
##
## AIC AICc BIC
## 141.9893 144.9893 143.4440
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.01600885 83.40993 65.12899 -8.334272 24.33473 0.9525264
## ACF1
## Training set 0.2052201SDecomp <- decompose(Squarterly)
SDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 264
## 2014 201 280 459 418
## 2015 172 233 317 297
## 2016 199 222 301
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 58.64062
## 2014 -99.85938 -44.17188 85.39062 58.64062
## 2015 -99.85938 -44.17188 85.39062 58.64062
## 2016 -99.85938 -44.17188 85.39062
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 NA
## 2014 NA 320.250 335.875 326.375
## 2015 302.750 269.875 258.125 260.125
## 2016 256.750 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 NA
## 2014 NA 3.921875 37.734375 32.984375
## 2015 -30.890625 7.296875 -26.515625 -21.765625
## 2016 42.109375 NA NA
##
## $figure
## [1] 58.64062 -99.85938 -44.17188 85.39062
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(SDecomp)
forecast(SfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 280.2617 173.3676 387.1559 116.7813 443.7422
## 2017 Q1 280.2617 173.3676 387.1559 116.7813 443.7422
## 2017 Q2 280.2617 173.3676 387.1559 116.7813 443.7422
## 2017 Q3 280.2617 173.3676 387.1559 116.7813 443.7422
## 2017 Q4 280.2617 173.3676 387.1559 116.7813 443.7422
## 2018 Q1 280.2617 173.3676 387.1559 116.7813 443.7422
## 2018 Q2 280.2617 173.3676 387.1559 116.7813 443.7422
## 2018 Q3 280.2617 173.3676 387.1559 116.7813 443.7422Monthly
U <- ts(c(75,31,42,114,33,53,109,152,71,105,88,51,41,59,29,39,187,79,146,81,59,147,65,71, 48,64,36,46,73,64,94,107,107,108,69,114
, 61,89,48,72,60,52,88,99,114,65,61,95, 71,33,90,41,54,61,83,92,99,120), start = c(2012,1), frequency = 12)
UData <- window(U)
Ufit <- ets(UData)
summary(Ufit)## ETS(A,N,N)
##
## Call:
## ets(y = UData)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 77.671
##
## sigma: 32.994
##
## AIC AICc BIC
## 647.0795 647.5240 653.2609
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.002959718 32.99402 26.38711 -19.67271 41.27086 0.9126368
## ACF1
## Training set 0.07840545plot(Ufit)
plot(forecast(Ufit,h=8),
ylab="Forecasted Demand")
forecast(Ufit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 77.67103 35.38749 119.9546 13.00394 142.3381
## Dec 2016 77.67103 35.38749 119.9546 13.00394 142.3381
## Jan 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Feb 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Mar 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Apr 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## May 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Jun 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Jul 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Aug 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Sep 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Oct 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Nov 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Dec 2017 77.67103 35.38749 119.9546 13.00394 142.3381
## Jan 2018 77.67103 35.38749 119.9546 13.00394 142.3381
## Feb 2018 77.67103 35.38749 119.9546 13.00394 142.3381
## Mar 2018 77.67103 35.38749 119.9546 13.00394 142.3381
## Apr 2018 77.67103 35.38749 119.9546 13.00394 142.3381
## May 2018 77.67103 35.38749 119.9546 13.00394 142.3381
## Jun 2018 77.67103 35.38749 119.9546 13.00394 142.3381
## Jul 2018 77.67103 35.38749 119.9546 13.00394 142.3381
## Aug 2018 77.67103 35.38749 119.9546 13.00394 142.3381
## Sep 2018 77.67103 35.38749 119.9546 13.00394 142.3381
## Oct 2018 77.67103 35.38749 119.9546 13.00394 142.3381Quarterly
Uquarterly <- aggregate(U, nfrequency=4)
Uquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 148 200 332 244
## 2013 129 305 286 283
## 2014 148 183 308 291
## 2015 198 184 301 221
## 2016 194 156 274UDataA <- window(Uquarterly)
UfitA <- ets(UDataA)
summary(UfitA)## ETS(A,N,A)
##
## Call:
## ets(y = UDataA)
##
## Smoothing parameters:
## alpha = 2e-04
## gamma = 1e-04
##
## Initial states:
## l = 235.7526
## s=29.616 65.4914 -23.4219 -71.6855
##
## sigma: 34.5914
##
## AIC AICc BIC
## 204.6013 214.7832 211.2124
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -3.406323 34.59141 28.38871 -4.071438 13.48771 0.751024
## ACF1
## Training set -0.2886025UDecomp <- decompose(Uquarterly)
UDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 148 200 332 244
## 2013 129 305 286 283
## 2014 148 183 308 291
## 2015 198 184 301 221
## 2016 194 156 274
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -69.60938 -16.85938 66.76562 19.70312
## 2013 -69.60938 -16.85938 66.76562 19.70312
## 2014 -69.60938 -16.85938 66.76562 19.70312
## 2015 -69.60938 -16.85938 66.76562 19.70312
## 2016 -69.60938 -16.85938 66.76562
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 228.625 239.375
## 2013 246.750 245.875 253.125 240.250
## 2014 227.750 231.500 238.750 245.125
## 2015 244.375 234.750 225.500 221.500
## 2016 214.625 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 36.609375 -15.078125
## 2013 -48.140625 75.984375 -33.890625 23.046875
## 2014 -10.140625 -31.640625 2.484375 26.171875
## 2015 23.234375 -33.890625 8.734375 -20.203125
## 2016 48.984375 NA NA
##
## $figure
## [1] -69.60938 -16.85938 66.76562 19.70312
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(UDecomp)
forecast(UfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 265.3518 221.0211 309.6824 197.55385 333.1497
## 2017 Q1 164.0522 119.7216 208.3829 96.25432 231.8502
## 2017 Q2 212.3126 167.9820 256.6433 144.51472 280.1106
## 2017 Q3 301.2289 256.8982 345.5596 233.43100 369.0269
## 2017 Q4 265.3518 221.0211 309.6825 197.55384 333.1497
## 2018 Q1 164.0522 119.7216 208.3829 96.25431 231.8502
## 2018 Q2 212.3126 167.9820 256.6433 144.51471 280.1106
## 2018 Q3 301.2289 256.8982 345.5596 233.43099 369.0269Monthly
V <- ts(c(19,26,24,13,18,29,21,37,13,49,30,17, 21,18,22,30,25,30,25,30,24,39,20,27, 13,24,22,29,39,33,35,37,46,49,53,37, 28,37,42,36,40,58,84,77,56,78,66,72,81,77,97,135,82,95,75,85,66,141), start = c(2012,1), frequency = 12)
VData <- window(V)
Vfit <- ets(VData)
summary(Vfit)## ETS(M,A,N)
##
## Call:
## ets(y = VData)
##
## Smoothing parameters:
## alpha = 0.2774
## beta = 1e-04
##
## Initial states:
## l = 18.1913
## b = 1.0596
##
## sigma: 0.3097
##
## AIC AICc BIC
## 535.6375 536.7913 545.9397
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 1.29895 15.08606 10.28412 -9.327191 26.71314 0.5339383
## ACF1
## Training set 0.0004804586plot(Vfit)
plot(forecast(Vfit,h=8),
ylab="Forecasted Demand")
forecast(Vfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 101.5053 61.21660 141.7940 39.88904683 163.1216
## Dec 2016 102.5724 60.20987 144.9350 37.78447320 167.3604
## Jan 2017 103.6396 59.25079 148.0284 35.75278064 171.5264
## Feb 2017 104.7067 58.33244 151.0810 33.78337961 175.6301
## Mar 2017 105.7739 57.44915 154.0986 31.86758969 179.6802
## Apr 2017 106.8410 56.59620 157.0858 29.99819580 183.6838
## May 2017 107.9082 55.76961 160.0467 28.16912816 187.6472
## Jun 2017 108.9753 54.96602 162.9846 26.37522632 191.5754
## Jul 2017 110.0424 54.18252 165.9024 24.61206194 195.4728
## Aug 2017 111.1096 53.41662 168.8026 22.87580327 199.3434
## Sep 2017 112.1767 52.66612 171.6873 21.16311009 203.1904
## Oct 2017 113.2439 51.92912 174.5586 19.47105112 207.0167
## Nov 2017 114.3110 51.20392 177.4181 17.79703828 210.8250
## Dec 2017 115.3782 50.48901 180.2673 16.13877387 214.6176
## Jan 2018 116.4453 49.78307 183.1075 14.49420758 218.3964
## Feb 2018 117.5124 49.08487 185.9400 12.86150140 222.1634
## Mar 2018 118.5796 48.39335 188.7658 11.23900048 225.9202
## Apr 2018 119.6467 47.70753 191.5859 9.62520897 229.6683
## May 2018 120.7139 47.02651 194.4013 8.01876972 233.4090
## Jun 2018 121.7810 46.34949 197.2126 6.41844717 237.1436
## Jul 2018 122.8482 45.67573 200.0206 4.82311283 240.8732
## Aug 2018 123.9153 45.00456 202.8261 3.23173284 244.5989
## Sep 2018 124.9825 44.33535 205.6296 1.64335738 248.3215
## Oct 2018 126.0496 43.66754 208.4317 0.05711148 252.0421Quarterly
Vquarterly <- aggregate(V, nfrequency=4)
Vquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 69 60 71 96
## 2013 61 85 79 86
## 2014 59 101 118 139
## 2015 107 134 217 216
## 2016 255 312 226VDataA <- window(Vquarterly)
VfitA <- ets(VDataA)
summary(VfitA)## ETS(M,N,N)
##
## Call:
## ets(y = VDataA)
##
## Smoothing parameters:
## alpha = 0.6639
##
## Initial states:
## l = 62.2937
##
## sigma: 0.2941
##
## AIC AICc BIC
## 191.6433 193.2433 194.4766
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 14.61293 37.44144 29.45422 6.569823 21.69205 0.5867375
## ACF1
## Training set -0.06442327VDecomp <- decompose(Vquarterly)
VDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 69 60 71 96
## 2013 61 85 79 86
## 2014 59 101 118 139
## 2015 107 134 217 216
## 2016 255 312 226
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -16.575521 -3.898438 10.580729 9.893229
## 2013 -16.575521 -3.898438 10.580729 9.893229
## 2014 -16.575521 -3.898438 10.580729 9.893229
## 2015 -16.575521 -3.898438 10.580729 9.893229
## 2016 -16.575521 -3.898438 10.580729
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 73.000 75.125
## 2013 79.250 79.000 77.500 79.250
## 2014 86.125 97.625 110.250 120.375
## 2015 136.875 158.875 187.000 227.750
## 2016 251.125 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -12.580729 10.981771
## 2013 -1.674479 9.898438 -9.080729 -3.143229
## 2014 -10.549479 7.273438 -2.830729 8.731771
## 2015 -13.299479 -20.976562 19.419271 -21.643229
## 2016 20.450521 NA NA
##
## $figure
## [1] -16.575521 -3.898438 10.580729 9.893229
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(VDecomp)
forecast(VfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 246.6143 153.65433 339.5742 104.4443158 388.7842
## 2017 Q1 246.6143 133.56753 359.6610 73.7241995 419.5043
## 2017 Q2 246.6143 115.94156 377.2870 46.7676049 446.4609
## 2017 Q3 246.6143 99.86639 393.3621 22.1827531 471.0458
## 2017 Q4 246.6143 84.85964 408.3689 -0.7680903 493.9966
## 2018 Q1 246.6143 70.62932 422.5992 -22.5314767 515.7600
## 2018 Q2 246.6143 56.98281 436.2457 -43.4020230 536.6305
## 2018 Q3 246.6143 43.78498 449.4435 -63.5863530 556.8149Monthly
W <- ts(c(71,56,40, 35,28,36,68,53,74,73,68,85,95,55,45, 30,12,41,42,71,69,79,76,89,110,66,68,31,40,45,57,73,76,92,125,72,68
), start = c(2013, 10), frequency = 12)
WData <- window(W)
Wfit <- ets(WData)
summary(Wfit)## ETS(A,N,N)
##
## Call:
## ets(y = WData)
##
## Smoothing parameters:
## alpha = 0.8778
##
## Initial states:
## l = 68.9324
##
## sigma: 20.2136
##
## AIC AICc BIC
## 362.0743 362.8016 366.9071
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.008748175 20.21361 15.98652 -8.782722 31.50222 1.141894
## ACF1
## Training set -0.009507915plot(Wfit)
plot(forecast(Wfit,h=8),
ylab="Forecasted Demand")
forecast(Wfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 69.21657 43.3117875 95.12134 29.598626 108.8345
## Dec 2016 69.21657 34.7475928 103.68554 16.500821 121.9323
## Jan 2017 69.21657 27.9229505 110.51018 6.063431 132.3697
## Feb 2017 69.21657 22.0761879 116.35694 -2.878421 141.3116
## Mar 2017 69.21657 16.8785511 121.55458 -10.827520 149.2607
## Apr 2017 69.21657 12.1523883 126.28074 -18.055563 156.4887
## May 2017 69.21657 7.7887796 130.64435 -24.729126 163.1623
## Jun 2017 69.21657 3.7152263 134.71791 -30.959088 169.3922
## Jul 2017 69.21657 -0.1194136 138.55255 -36.823664 175.2568
## Aug 2017 69.21657 -3.7528155 142.18595 -42.380473 180.8136
## Sep 2017 69.21657 -7.2136845 145.64682 -47.673415 186.1065
## Oct 2017 69.21657 -10.5244884 148.95762 -52.736853 191.1700
## Nov 2017 69.21657 -13.7032044 152.13634 -57.598279 196.0314
## Dec 2017 69.21657 -16.7644836 155.19762 -62.280102 200.7132
## Jan 2018 69.21657 -19.7204536 158.15359 -66.800868 205.2340
## Feb 2018 69.21657 -22.5812881 161.01442 -71.176137 209.6093
## Mar 2018 69.21657 -25.3556212 163.78875 -75.419113 213.8522
## Apr 2018 69.21657 -28.0508548 166.48399 -79.541117 217.9742
## May 2018 69.21657 -30.6733921 169.10652 -83.551942 221.9851
## Jun 2018 69.21657 -33.2288160 171.66195 -87.460125 225.8933
## Jul 2018 69.21657 -35.7220296 174.15516 -91.273167 229.7063
## Aug 2018 69.21657 -38.1573667 176.59050 -94.997693 233.4308
## Sep 2018 69.21657 -40.5386798 178.97181 -98.639597 237.0727
## Oct 2018 69.21657 -42.8694124 181.30254 -102.204145 240.6373Quarterly
Wquarterly <- aggregate(W, nfrequency=4)
Wquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 167
## 2014 99 195 226 195
## 2015 83 182 244 244
## 2016 116 206 289WDataA <- window(Wquarterly)
WfitA <- ets(WDataA)
summary(WfitA)## ETS(M,N,N)
##
## Call:
## ets(y = WDataA)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 187.1598
##
## sigma: 0.3196
##
## AIC AICc BIC
## 134.0111 137.0111 135.4658
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.01555291 59.82054 48.14404 -14.54487 33.91747 1.704214
## ACF1
## Training set 0.02068148WDecomp <- decompose(Wquarterly)
WDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 167
## 2014 99 195 226 195
## 2015 83 182 244 244
## 2016 116 206 289
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 32.6875
## 2014 -91.9375 9.3125 49.9375 32.6875
## 2015 -91.9375 9.3125 49.9375 32.6875
## 2016 -91.9375 9.3125 49.9375
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 NA
## 2014 NA 175.250 176.750 173.125
## 2015 173.750 182.125 192.375 199.500
## 2016 208.125 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2013 NA
## 2014 NA 10.4375 -0.6875 -10.8125
## 2015 1.1875 -9.4375 1.6875 11.8125
## 2016 -0.1875 NA NA
##
## $figure
## [1] 32.6875 -91.9375 9.3125 49.9375
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(WDecomp)
forecast(WfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 187.1598 110.4937 263.8259 69.90916 304.4105
## 2017 Q1 187.1598 110.4937 263.8259 69.90916 304.4105
## 2017 Q2 187.1598 110.4937 263.8259 69.90915 304.4105
## 2017 Q3 187.1598 110.4937 263.8259 69.90915 304.4105
## 2017 Q4 187.1598 110.4937 263.8259 69.90915 304.4105
## 2018 Q1 187.1598 110.4937 263.8259 69.90915 304.4105
## 2018 Q2 187.1598 110.4937 263.8259 69.90915 304.4105
## 2018 Q3 187.1598 110.4937 263.8259 69.90915 304.4105Monthly
X <- ts(c(17,33,24,25,17,20,14,14,12,11,15,10
,16,13,18,26,41,16,5,33,47,36,20,24
, 19,25,56,30,48,47,40,24,31,41,35,40, 28,29,32,49,63,84,55,70,70,92,62,49,42,69,79,71,85,72,93,52,43), start = c(2012,1), frequency = 12)
XData <- window(X)
Xfit <- ets(XData)
summary(Xfit)## ETS(A,N,N)
##
## Call:
## ets(y = XData)
##
## Smoothing parameters:
## alpha = 0.4705
##
## Initial states:
## l = 22.4425
##
## sigma: 14.1105
##
## AIC AICc BIC
## 538.2029 538.6557 544.3320
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 1.271077 14.11052 10.99735 -10.6223 34.98885 0.5566712
## ACF1
## Training set 0.1223726plot(Xfit)
plot(forecast(Xfit,h=8),
ylab="Forecasted Demand")
forecast(Xfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Oct 2016 56.53436 38.45100 74.61772 28.878254 84.19047
## Nov 2016 56.53436 36.54906 76.51966 25.969474 87.09925
## Dec 2016 56.53436 34.81301 78.25571 23.314422 89.75430
## Jan 2017 56.53436 33.20580 79.86292 20.856407 92.21231
## Feb 2017 56.53436 31.70240 81.36632 18.557151 94.51157
## Mar 2017 56.53436 30.28496 82.78376 16.389368 96.67935
## Apr 2017 56.53436 28.94024 84.12848 14.332791 98.73593
## May 2017 56.53436 27.65807 85.41065 12.371882 100.69684
## Jun 2017 56.53436 26.43046 86.63826 10.494415 102.57430
## Jul 2017 56.53436 25.25098 87.81773 8.690567 104.37815
## Aug 2017 56.53436 24.11439 88.95433 6.952301 106.11642
## Sep 2017 56.53436 23.01632 90.05240 5.272945 107.79577
## Oct 2017 56.53436 21.95310 91.11562 3.646888 109.42183
## Nov 2017 56.53436 20.92161 92.14711 2.069355 110.99936
## Dec 2017 56.53436 19.91916 93.14956 0.536246 112.53247
## Jan 2018 56.53436 18.94344 94.12528 -0.955994 114.02471
## Feb 2018 56.53436 17.99241 95.07631 -2.410469 115.47919
## Mar 2018 56.53436 17.06429 96.00443 -3.829908 116.89863
## Apr 2018 56.53436 16.15749 96.91123 -5.216728 118.28545
## May 2018 56.53436 15.27062 97.79810 -6.573080 119.64180
## Jun 2018 56.53436 14.40242 98.66630 -7.900887 120.96961
## Jul 2018 56.53436 13.55174 99.51698 -9.201878 122.27060
## Aug 2018 56.53436 12.71758 100.35114 -10.477617 123.54634
## Sep 2018 56.53436 11.89901 101.16971 -11.729519 124.79824Quarterly
Xquarterly <- aggregate(X, nfrequency=4)
Xquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 74 62 40 36
## 2013 47 83 85 80
## 2014 100 125 95 116
## 2015 89 196 195 203
## 2016 190 228 188XDataA <- window(Xquarterly)
XfitA <- ets(XDataA)
summary(XfitA)## ETS(A,N,N)
##
## Call:
## ets(y = XDataA)
##
## Smoothing parameters:
## alpha = 0.7426
##
## Initial states:
## l = 69.4497
##
## sigma: 31.139
##
## AIC AICc BIC
## 192.6059 194.2059 195.4392
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 8.964681 31.13903 23.00844 2.428966 21.52055 0.5023677
## ACF1
## Training set -0.1495378XDecomp <- decompose(Xquarterly)
XDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 74 62 40 36
## 2013 47 83 85 80
## 2014 100 125 95 116
## 2015 89 196 195 203
## 2016 190 228 188
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -17.276042 23.890625 -1.401042 -5.213542
## 2013 -17.276042 23.890625 -1.401042 -5.213542
## 2014 -17.276042 23.890625 -1.401042 -5.213542
## 2015 -17.276042 23.890625 -1.401042 -5.213542
## 2016 -17.276042 23.890625 -1.401042
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 49.625 48.875
## 2013 57.125 68.250 80.375 92.250
## 2014 98.750 104.500 107.625 115.125
## 2015 136.500 159.875 183.375 200.000
## 2016 203.125 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -8.223958 -7.661458
## 2013 7.151042 -9.140625 6.026042 -7.036458
## 2014 18.526042 -3.390625 -11.223958 6.088542
## 2015 -30.223958 12.234375 13.026042 8.213542
## 2016 4.151042 NA NA
##
## $figure
## [1] -17.276042 23.890625 -1.401042 -5.213542
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(XDecomp)
forecast(XfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 195.9366 156.0303 235.8428 134.90518 256.9680
## 2017 Q1 195.9366 146.2303 245.6429 119.91733 271.9558
## 2017 Q2 195.9366 138.0667 253.8064 107.43223 284.4409
## 2017 Q3 195.9366 130.9202 260.9529 96.50261 295.3705
## 2017 Q4 195.9366 124.4849 267.3882 86.66075 305.2124
## 2018 Q1 195.9366 118.5832 273.2899 77.63484 314.2383
## 2018 Q2 195.9366 113.1009 278.7722 69.25037 322.6228
## 2018 Q3 195.9366 107.9596 283.9136 61.38737 330.4858Quarterly
Y <- ts(c(12,11,17,17,43,20,17,23,12,18,11,29, 13,30,17,6,21,35,28,26,18,101,52,18, 23,23,19,39,35,27,47,39,54,38,34,34,
12,32,27,43,56,60,51,31,40,52,39,29,47,42,62,54,107,81,96,112,110,60), start = c(2012,12), frequency = 12)
YData <- window(Y)
Yfit <- ets(YData)
summary(Yfit)## ETS(M,N,A)
##
## Call:
## ets(y = YData)
##
## Smoothing parameters:
## alpha = 0.2394
## gamma = 1e-04
##
## Initial states:
## l = 21.8664
## s=0.6057 -4.8189 23.0302 -7.6226 -3.226 3.0836
## 1.1993 8.0777 -5.0708 -5.39 -2.0166 -7.8515
##
## sigma: 0.4419
##
## AIC AICc BIC
## 567.0208 578.4494 597.9275
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 4.065376 18.45713 13.35556 -5.672447 37.36407 0.6366381
## ACF1
## Training set 0.1042809plot(Yfit)
plot(forecast(Yfit,h=8),
ylab="Forecasted Demand")
forecast(Yfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Oct 2017 73.48865 31.8711962 115.1061 9.8402459 137.1371
## Nov 2017 78.90777 32.9134474 124.9021 8.5655235 149.2500
## Dec 2017 70.45297 27.4581675 113.4478 4.6980952 136.2078
## Jan 2018 76.28808 29.0118559 123.5643 3.9853345 148.5908
## Feb 2018 72.91684 25.9523721 119.8813 1.0908828 144.7428
## Mar 2018 73.23710 24.8321391 121.6421 -0.7918993 147.2661
## Apr 2018 86.38863 30.3433342 142.4339 0.6747472 172.1025
## May 2018 79.50836 25.2195176 133.7972 -3.5192585 162.5360
## Jun 2018 81.39228 24.8801791 137.9044 -5.0355184 167.8201
## Jul 2018 75.08419 19.9978352 130.1705 -9.1631193 159.3315
## Aug 2018 70.68879 16.2217980 125.1558 -12.6112842 153.9889
## Sep 2018 101.32799 32.1763337 170.4797 -4.4303430 207.0863
## Oct 2018 73.48865 14.6391723 132.3381 -16.5138605 163.4912
## Nov 2018 78.90777 16.7297068 141.0858 -16.1853718 174.0009
## Dec 2018 70.45297 10.2968072 130.6091 -21.5479405 162.4539
## Jan 2019 76.28808 12.8447331 139.7314 -20.7401462 173.3163
## Feb 2019 72.91684 9.5469420 136.2867 -23.9990574 169.8327
## Mar 2019 73.23710 8.6356015 137.8386 -25.5623660 172.0366
## Apr 2019 86.38863 15.7354482 157.0418 -21.6660866 194.4433
## May 2019 79.50836 10.0910415 148.9257 -26.6562662 185.6730
## Jun 2019 81.39228 10.0758741 152.7087 -27.6767491 190.4613
## Jul 2019 75.08419 4.7408243 145.4276 -32.4967040 182.6651
## Aug 2019 70.68879 0.6769568 140.7006 -36.3850672 177.7626
## Sep 2019 101.32799 19.2388960 183.4171 -24.2164465 226.8724Quarterly
Yquarterly <- aggregate(Y, nfrequency=4)
Yquarterly## Time Series:
## Start = 2012.91666666667
## End = 2017.41666666667
## Frequency = 4
## [1] 40 80 52 58 60 62 72 171 65 101 140 106 71 159 122 120 151
## [18] 242 318YDataA <- window(Yquarterly)
YfitA <- ets(YDataA)
summary(YfitA)## ETS(M,A,N)
##
## Call:
## ets(y = YDataA)
##
## Smoothing parameters:
## alpha = 1e-04
## beta = 1e-04
##
## Initial states:
## l = 37.3191
## b = 7.6491
##
## sigma: 0.3466
##
## AIC AICc BIC
## 202.6238 207.2392 207.3460
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 1.47732 45.94174 33.84957 -11.64143 29.47274 0.6323083
## ACF1
## Training set 0.155054YDecomp <- decompose(Yquarterly)
YDecomp## $x
## Time Series:
## Start = 2012.91666666667
## End = 2017.41666666667
## Frequency = 4
## [1] 40 80 52 58 60 62 72 171 65 101 140 106 71 159 122 120 151
## [18] 242 318
##
## $seasonal
## Time Series:
## Start = 2012.91666666667
## End = 2017.41666666667
## Frequency = 4
## [1] -27.231771 9.664062 4.424479 13.143229 -27.231771 9.664062
## [7] 4.424479 13.143229 -27.231771 9.664062 4.424479 13.143229
## [13] -27.231771 9.664062 4.424479 13.143229 -27.231771 9.664062
## [19] 4.424479
##
## $trend
## Time Series:
## Start = 2012.91666666667
## End = 2017.41666666667
## Frequency = 4
## [1] NA NA 60.000 60.250 60.500 77.125 91.875 97.375
## [9] 110.750 111.125 103.750 111.750 116.750 116.250 128.000 148.375
## [17] 183.250 NA NA
##
## $random
## Time Series:
## Start = 2012.91666666667
## End = 2017.41666666667
## Frequency = 4
## [1] NA NA -12.424479 -15.393229 26.731771 -24.789062
## [7] -24.299479 60.481771 -18.518229 -19.789062 31.825521 -18.893229
## [13] -18.518229 33.085938 -10.424479 -41.518229 -5.018229 NA
## [19] NA
##
## $figure
## [1] -27.231771 9.664062 4.424479 13.143229
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(YDecomp)
Z <- ts(c(26,88,31,85,33,28,81,60,38,26,35,32, 15,17,65,84,102,33,12,48,133,402,34,53, 90,17,25,22,135,52,76,11,39,72,155,78, 30,5,93,245,34,56,56,45,40,2,1,1,28,19,92,54,55,98,63,16,48,63), start = c(2012, 1), frequency = 12)
ZData <- window(Z)
Zfit <- ets(ZData)
summary(Zfit)## ETS(M,N,M)
##
## Call:
## ets(y = ZData)
##
## Smoothing parameters:
## alpha = 0.0305
## gamma = 1e-04
##
## Initial states:
## l = 76.5537
## s=0.5866 1.2566 2.2297 1.0044 0.5173 0.7202
## 0.7931 1.0685 1.7132 0.9385 0.557 0.6149
##
## sigma: 0.6497
##
## AIC AICc BIC
## 696.3234 707.7519 727.2300
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -10.20286 60.41428 43.53493 -419.5878 445.5997 0.6989902
## ACF1
## Training set 0.1419605plot(Zfit)
plot(forecast(Zfit,h=8),
ylab="Forecasted Demand")
forecast(Zfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 78.88435 13.206074 144.56263 -21.561908 179.33061
## Dec 2016 36.82846 6.145189 67.51173 -10.097553 83.75447
## Jan 2017 38.60457 6.420295 70.78885 -10.617033 87.82618
## Feb 2017 34.96724 5.796124 64.13835 -9.646130 79.58060
## Mar 2017 58.92708 9.735248 108.11891 -16.305335 134.15950
## Apr 2017 107.55905 17.710480 197.40761 -29.852477 244.97057
## May 2017 67.08725 11.009566 123.16493 -18.676165 152.85066
## Jun 2017 49.79950 8.145121 91.45387 -13.905373 113.50437
## Jul 2017 45.22130 7.371460 83.07115 -12.665037 103.10764
## Aug 2017 32.48102 5.276834 59.68521 -9.124194 74.08624
## Sep 2017 63.06102 10.210179 115.91185 -17.767364 143.88940
## Oct 2017 139.97723 22.586757 257.36771 -39.556007 319.51048
## Nov 2017 78.88446 12.685196 145.08372 -22.358577 180.12750
## Dec 2017 36.82851 5.902074 67.75495 -10.469393 84.12641
## Jan 2018 38.60462 6.165522 71.04373 -11.006702 88.21595
## Feb 2018 34.96728 5.565417 64.36915 -9.998992 79.93356
## Mar 2018 58.92716 9.346560 108.50776 -16.899824 134.75414
## Apr 2018 107.55919 17.001198 198.11718 -30.937307 246.05569
## May 2018 67.08734 10.567285 123.60739 -19.352623 153.52730
## Jun 2018 49.79956 7.816898 91.78223 -14.407384 114.00651
## Jul 2018 45.22136 7.073488 83.36924 -13.120778 103.56351
## Aug 2018 32.48107 5.062866 59.89927 -9.451453 74.41359
## Sep 2018 63.06110 9.794874 116.32733 -18.402563 144.52476
## Oct 2018 139.97742 21.665139 258.28971 -40.965601 320.92045Quarterly
Zquarterly <- aggregate(Z, nfrequency=4)
Zquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 145 146 179 93
## 2013 97 219 193 489
## 2014 132 209 126 305
## 2015 128 335 141 4
## 2016 139 207 127ZDataA <- window(Zquarterly)
ZfitA <- ets(ZDataA)
summary(ZfitA)## ETS(M,N,M)
##
## Call:
## ets(y = ZDataA)
##
## Smoothing parameters:
## alpha = 1e-04
## gamma = 0.001
##
## Initial states:
## l = 192.2752
## s=1.5985 0.721 1.0845 0.5959
##
## sigma: 0.3771
##
## AIC AICc BIC
## 228.8473 239.0291 235.4584
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -6.542397 102.1583 59.38205 -405.3691 425.84 0.6246358
## ACF1
## Training set -0.01503622ZDecomp <- decompose(Zquarterly)
ZDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 145 146 179 93
## 2013 97 219 193 489
## 2014 132 209 126 305
## 2015 128 335 141 4
## 2016 139 207 127
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -62.71615 51.36719 -24.96615 36.31510
## 2013 -62.71615 51.36719 -24.96615 36.31510
## 2014 -62.71615 51.36719 -24.96615 36.31510
## 2015 -62.71615 51.36719 -24.96615 36.31510
## 2016 -62.71615 51.36719 -24.96615
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 134.750 137.875
## 2013 148.750 200.000 253.875 257.000
## 2014 247.375 216.000 192.500 207.750
## 2015 225.375 189.625 153.375 138.750
## 2016 121.000 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 69.21615 -81.19010
## 2013 10.96615 -32.36719 -35.90885 195.68490
## 2014 -52.65885 -58.36719 -41.53385 60.93490
## 2015 -34.65885 94.00781 12.59115 -171.06510
## 2016 80.71615 NA NA
##
## $figure
## [1] -62.71615 51.36719 -24.96615 36.31510
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(ZDecomp)
forecast(ZfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 307.0338 158.64747 455.4201 80.09649 533.9711
## 2017 Q1 114.6572 59.24454 170.0699 29.91084 199.4036
## 2017 Q2 208.6000 107.78573 309.4143 54.41788 362.7822
## 2017 Q3 138.7137 71.67476 205.7526 36.18650 241.2409
## 2017 Q4 307.0338 158.64737 455.4203 80.09633 533.9713
## 2018 Q1 114.6572 59.24450 170.0699 29.91078 199.4036
## 2018 Q2 208.6000 107.78566 309.4144 54.41777 362.7823
## 2018 Q3 138.7137 71.67471 205.7527 36.18642 241.2409AA <- ts(c(12,16,34,17,13,7,16,7,46,48,34,16
,25,50,14,32,39,26,47,61,30,44,26,7
,5,30,18,28,49,44,48,70,87,66,45,55
, 53,30,29,44,68,63,67,56,73,54,28,48,39,64,36,22,55,91,44,44,10,24), start = c(2012,1), frequency = 12)
AAData <- window(AA)
AAfit <- ets(AAData)
summary(AAfit)## ETS(A,N,N)
##
## Call:
## ets(y = AAData)
##
## Smoothing parameters:
## alpha = 0.4678
##
## Initial states:
## l = 16.4249
##
## sigma: 18.2727
##
## AIC AICc BIC
## 578.5333 578.9777 584.7146
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.4244408 18.27274 15.04225 -30.07152 59.85113 0.7496677
## ACF1
## Training set 0.1418948plot(AAfit)
plot(forecast(AAfit,h=8),
ylab="Forecasted Demand")
forecast(AAfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 27.94074 4.5232800 51.35819 -7.873173 63.75465
## Dec 2016 27.94074 2.0877489 53.79373 -11.597996 67.47947
## Jan 2017 27.94074 -0.1373094 56.01878 -15.000929 70.88240
## Feb 2017 27.94074 -2.1985461 58.08002 -18.153319 74.03479
## Mar 2017 27.94074 -4.1275665 60.00904 -21.103501 76.98497
## Apr 2017 27.94074 -5.9469567 61.82843 -23.886018 79.76749
## May 2017 27.94074 -7.6735225 63.55500 -26.526573 82.40805
## Jun 2017 27.94074 -9.3201699 65.20164 -29.044902 84.92638
## Jul 2017 27.94074 -10.8970652 66.77854 -31.456556 87.33803
## Aug 2017 27.94074 -12.4123865 68.29386 -33.774040 89.65551
## Sep 2017 27.94074 -13.8728287 69.75430 -36.007593 91.88907
## Oct 2017 27.94074 -15.2839547 71.16543 -38.165724 94.04720
## Nov 2017 27.94074 -16.6504468 72.53192 -40.255594 96.13707
## Dec 2017 27.94074 -17.9762900 73.85776 -42.283296 98.16477
## Jan 2018 27.94074 -19.2649096 75.14638 -44.254069 100.13554
## Feb 2018 27.94074 -20.5192750 76.40075 -46.172456 102.05393
## Mar 2018 27.94074 -21.7419810 77.62346 -48.042423 103.92390
## Apr 2018 27.94074 -22.9353100 78.81678 -49.867462 105.74894
## May 2018 27.94074 -24.1012831 79.98276 -51.650664 107.53214
## Jun 2018 27.94074 -25.2416996 81.12317 -53.394781 109.27625
## Jul 2018 27.94074 -26.3581696 82.23964 -55.102274 110.98375
## Aug 2018 27.94074 -27.4521412 83.33362 -56.775360 112.65683
## Sep 2018 27.94074 -28.5249220 84.40640 -58.416036 114.29751
## Oct 2018 27.94074 -29.5776979 85.45917 -60.026118 115.90759Quarterly
AAquarterly <- aggregate(AA, nfrequency=4)
AAquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 62 37 69 98
## 2013 89 97 138 77
## 2014 53 121 205 166
## 2015 112 175 196 130
## 2016 139 168 98AADataA <- window(AAquarterly)
AAfitA <- ets(AADataA)
summary(AAfitA)## ETS(M,A,N)
##
## Call:
## ets(y = AADataA)
##
## Smoothing parameters:
## alpha = 1e-04
## beta = 1e-04
##
## Initial states:
## l = 63.0154
## b = 5.7287
##
## sigma: 0.296
##
## AIC AICc BIC
## 200.3156 204.9310 205.0378
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -2.900727 36.6818 28.36591 -14.77333 29.59656 0.6229702
## ACF1
## Training set 0.1232476AADecomp <- decompose(AAquarterly)
AADecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 62 37 69 98
## 2013 89 97 138 77
## 2014 53 121 205 166
## 2015 112 175 196 130
## 2016 139 168 98
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -30.471354 1.664062 34.778646 -5.971354
## 2013 -30.471354 1.664062 34.778646 -5.971354
## 2014 -30.471354 1.664062 34.778646 -5.971354
## 2015 -30.471354 1.664062 34.778646 -5.971354
## 2016 -30.471354 1.664062 34.778646
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 69.875 80.750
## 2013 96.875 102.875 95.750 94.250
## 2014 105.625 125.125 143.625 157.750
## 2015 163.375 157.750 156.625 159.125
## 2016 146.000 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -35.653646 23.221354
## 2013 22.596354 -7.539062 7.471354 -11.278646
## 2014 -22.153646 -5.789062 26.596354 14.221354
## 2015 -20.903646 15.585938 4.596354 -23.153646
## 2016 23.471354 NA NA
##
## $figure
## [1] -30.471354 1.664062 34.778646 -5.971354
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(AADecomp)
forecast(AAfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 177.5405 110.1891 244.8918 74.53550 280.5454
## 2017 Q1 183.2637 113.7412 252.7862 76.93824 289.5891
## 2017 Q2 188.9869 117.2933 260.6805 79.34096 298.6328
## 2017 Q3 194.7101 120.8453 268.5749 81.74369 307.6766
## 2017 Q4 200.4333 124.3974 276.4693 84.14641 316.7203
## 2018 Q1 206.1566 127.9495 284.3637 86.54912 325.7640
## 2018 Q2 211.8798 131.5015 292.2581 88.95182 334.8077
## 2018 Q3 217.6030 135.0536 300.1525 91.35452 343.8515BB <- ts(c(55,19,11,25,15,51,9,9,44,65,21,22
, 24,11,18,4,35,22,26,29,26,30,8,24
, 29,7,14,9,12,12,9,15,7,10,22,14
, 39,16,16,42,80,38,24,24,31,24,40,166,50,30,47,74,24,11,25,27,46,63), start = c(2012,1), frequency = 12)
BBData <- window(BB)
BBfit <- ets(BBData)
summary(BBfit)## ETS(M,N,M)
##
## Call:
## ets(y = BBData)
##
## Smoothing parameters:
## alpha = 0.275
## gamma = 1e-04
##
## Initial states:
## l = 26.25
## s=1.7786 0.8197 1.1797 0.9376 0.6391 0.6134
## 1.0326 1.515 0.9358 0.581 0.5011 1.4664
##
## sigma: 0.5803
##
## AIC AICc BIC
## 575.1591 586.5877 606.0658
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.2839452 22.4258 14.40288 -36.10825 67.12377 0.6795205
## ACF1
## Training set 0.06920825plot(BBfit)
plot(forecast(BBfit,h=8),
ylab="Forecasted Demand")
forecast(BBfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 37.62017 9.6413128 65.59903 -5.169799 80.41014
## Dec 2016 81.61879 17.9235966 145.31399 -15.794605 179.03219
## Jan 2017 67.29452 12.3618761 122.22716 -16.717709 151.30675
## Feb 2017 22.99910 3.4142470 42.58396 -6.953350 52.95156
## Mar 2017 26.66476 3.0334264 50.29609 -9.476244 62.80576
## Apr 2017 42.94172 3.4158164 82.46763 -17.507934 103.39138
## May 2017 69.51887 3.1791806 135.85856 -31.938931 170.97668
## Jun 2017 47.38798 0.5808543 94.19510 -24.197341 118.97330
## Jul 2017 28.15212 -0.5892103 56.89345 -15.803950 72.10819
## Aug 2017 29.33062 -1.5802379 60.24148 -17.943458 76.60470
## Sep 2017 43.02654 -3.7272126 89.78030 -28.477155 114.53024
## Oct 2017 54.13703 -6.4539027 114.72796 -38.528803 146.80286
## Nov 2017 37.62052 -5.7069486 80.94799 -28.643124 103.88416
## Dec 2017 81.61955 -15.0229966 178.26210 -66.182471 229.42157
## Jan 2018 67.29514 -14.5596317 149.14992 -57.890932 192.48122
## Feb 2018 22.99932 -5.7177142 51.71635 -20.919592 66.91822
## Mar 2018 26.66500 -7.4883498 60.81836 -25.568043 78.89805
## Apr 2018 42.94212 -13.4433320 99.32757 -43.291987 129.17622
## May 2018 69.51952 -24.0052640 163.04430 -73.514292 212.55333
## Jun 2018 47.38842 -17.8931766 112.67001 -52.451165 147.22800
## Jul 2018 28.15238 -11.5403055 67.84506 -32.552343 88.85710
## Aug 2018 29.33089 -12.9739653 71.63575 -35.368805 94.03059
## Sep 2018 43.02694 -20.4302684 106.48415 -54.022486 140.07637
## Oct 2018 54.13753 -27.4704175 135.74548 -70.671055 178.94611Quarterly
BBquarterly <- aggregate(BB, nfrequency=4)
BBquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 85 91 62 108
## 2013 53 61 81 62
## 2014 50 33 31 46
## 2015 71 160 79 230
## 2016 127 109 98BBDataA <- window(BBquarterly)
BBfitA <- ets(BBDataA)
summary(BBfitA)## ETS(M,N,N)
##
## Call:
## ets(y = BBDataA)
##
## Smoothing parameters:
## alpha = 0.7746
##
## Initial states:
## l = 65.2583
##
## sigma: 0.5856
##
## AIC AICc BIC
## 205.8992 207.4992 208.7325
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 2.511442 48.90745 35.81641 -8.922444 38.82135 0.7359535
## ACF1
## Training set -0.4489179BBDecomp <- decompose(BBquarterly)
BBDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 85 91 62 108
## 2013 53 61 81 62
## 2014 50 33 31 46
## 2015 71 160 79 230
## 2016 127 109 98
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -12.640625 8.421875 -21.078125 25.296875
## 2013 -12.640625 8.421875 -21.078125 25.296875
## 2014 -12.640625 8.421875 -21.078125 25.296875
## 2015 -12.640625 8.421875 -21.078125 25.296875
## 2016 -12.640625 8.421875 -21.078125
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 82.500 74.750
## 2013 73.375 70.000 63.875 60.000
## 2014 50.250 42.000 42.625 61.125
## 2015 83.000 112.000 142.000 142.625
## 2016 138.625 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 0.578125 7.953125
## 2013 -7.734375 -17.421875 38.203125 -23.296875
## 2014 12.390625 -17.421875 9.453125 -40.421875
## 2015 0.640625 39.578125 -41.921875 62.078125
## 2016 1.015625 NA NA
##
## $figure
## [1] -12.640625 8.421875 -21.078125 25.296875
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(BBDecomp)
forecast(BBfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 102.219 25.5019703 178.9360 -15.10957 219.5476
## 2017 Q1 102.219 -0.8714502 205.3094 -55.44423 259.8822
## 2017 Q2 102.219 -25.6307243 230.0687 -93.31028 297.7483
## 2017 Q3 102.219 -50.2277686 254.6658 -130.92821 335.3662
## 2017 Q4 102.219 -75.4130130 279.8510 -169.44572 373.8837
## 2018 Q1 102.219 -101.6849156 306.1229 -209.62512 414.0631
## 2018 Q2 102.219 -129.4337949 333.8718 -252.06337 456.5014
## 2018 Q3 102.219 -159.0015284 363.4395 -297.28332 501.7213CC <- ts(c(12,31,10,14,25,25,12,34,19,26,9,13
, 12,44,25,17,19,26,37,31,13,27,8,23
, 11,8,9,18,32,36,30,42,24,14,20,1,
5,9,23,8,49,54,58,47,49,103,26,47,33,35,32,35,45,44,67,74,44,59), start = c(2012,1), frequency = 12)
CCData <- window(CC)
CCfit <- ets(CCData)
summary(CCfit)## ETS(M,N,A)
##
## Call:
## ets(y = CCData)
##
## Smoothing parameters:
## alpha = 0.1806
## gamma = 1e-04
##
## Initial states:
## l = 20.6524
## s=-5.9888 -9.5895 12.3898 -3.7328 7.6394 4.7185
## 6.7458 7.9288 -7.3201 -3.9303 2.4739 -11.3347
##
## sigma: 0.4641
##
## AIC AICc BIC
## 543.1273 554.5559 574.0340
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 2.604013 13.07638 9.226494 -34.10777 62.72762 0.6142094
## ACF1
## Training set 0.2055978plot(CCfit)
plot(forecast(CCfit,h=8),
ylab="Forecasted Demand")
forecast(CCfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 38.34598 15.536617 61.15535 3.4620680 73.22990
## Dec 2016 41.94738 16.585652 67.30912 3.1599619 80.73481
## Jan 2017 36.60156 13.809396 59.39372 1.7439544 71.45916
## Feb 2017 50.40866 19.365591 81.45173 2.9323814 97.88494
## Mar 2017 44.00505 15.971691 72.03841 1.1317266 86.87838
## Apr 2017 40.61637 13.927561 67.30518 -0.2006401 81.43338
## May 2017 55.86537 20.412430 91.31831 1.6447778 110.08596
## Jun 2017 54.68435 19.250082 90.11861 0.4923144 108.87638
## Jul 2017 52.65947 17.701837 87.61710 -0.8036154 106.12255
## Aug 2017 55.58047 18.506036 92.65491 -1.1199844 112.28093
## Sep 2017 44.20507 12.295141 76.11500 -4.5969561 93.00710
## Oct 2017 60.32726 19.777725 100.87680 -1.6879019 122.34242
## Nov 2017 38.34598 7.843369 68.84860 -8.3037387 84.99570
## Dec 2017 41.94738 9.447199 74.44757 -7.7573587 91.65213
## Jan 2018 36.60156 6.016891 67.18623 -10.1736539 83.37677
## Feb 2018 50.40866 13.226255 87.59107 -6.4569230 107.27425
## Mar 2018 44.00505 9.253128 78.75698 -9.1434302 97.15354
## Apr 2018 40.61637 6.895703 74.33703 -10.9549392 92.18768
## May 2018 55.86537 14.820454 96.91029 -6.9074123 118.63815
## Jun 2018 54.68435 13.618863 95.74983 -8.1198914 117.48859
## Jul 2018 52.65947 11.967527 93.35141 -9.5734854 114.89242
## Aug 2018 55.58047 13.020067 98.14088 -9.5100502 120.67099
## Sep 2018 44.20507 6.019263 82.39088 -14.1950830 102.60523
## Oct 2018 60.32726 14.640561 106.01396 -9.5445166 130.19904Quarterly
CCquarterly <- aggregate(CC, nfrequency=4)
CCquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 53 64 65 48
## 2013 81 62 81 58
## 2014 28 86 96 35
## 2015 37 111 154 176
## 2016 100 124 185CCDataA <- window(CCquarterly)
CCfitA <- ets(CCDataA)
summary(CCfitA)## ETS(M,N,N)
##
## Call:
## ets(y = CCDataA)
##
## Smoothing parameters:
## alpha = 0.3068
##
## Initial states:
## l = 45.0414
##
## sigma: 0.5358
##
## AIC AICc BIC
## 198.2023 199.8023 201.0357
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 15.98379 38.86408 30.67573 0.9510692 39.28377 0.9004618
## ACF1
## Training set 0.1845802CCDecomp <- decompose(CCquarterly)
CCDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 53 64 65 48
## 2013 81 62 81 58
## 2014 28 86 96 35
## 2015 37 111 154 176
## 2016 100 124 185
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -25.971354 8.070312 20.497396 -2.596354
## 2013 -25.971354 8.070312 20.497396 -2.596354
## 2014 -25.971354 8.070312 20.497396 -2.596354
## 2015 -25.971354 8.070312 20.497396 -2.596354
## 2016 -25.971354 8.070312 20.497396
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 61.000 64.250
## 2013 66.000 69.250 63.875 60.250
## 2014 65.125 64.125 62.375 66.625
## 2015 77.000 101.875 127.375 136.875
## 2016 142.375 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -16.4973958 -13.6536458
## 2013 40.9713542 -15.3203125 -3.3723958 0.3463542
## 2014 -11.1536458 13.8046875 13.1276042 -29.0286458
## 2015 -14.0286458 1.0546875 6.1276042 41.7213542
## 2016 -16.4036458 NA NA
##
## $figure
## [1] -25.971354 8.070312 20.497396 -2.596354
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(CCDecomp)
forecast(CCfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 138.2269 43.320256 233.1336 -6.920288 283.3741
## 2017 Q1 138.2269 37.734376 238.7194 -15.463154 291.9170
## 2017 Q2 138.2269 32.303717 244.1501 -23.768632 300.2224
## 2017 Q3 138.2269 27.001954 249.4519 -31.876978 308.3308
## 2017 Q4 138.2269 21.807960 254.6458 -39.820505 316.2743
## 2018 Q1 138.2269 16.704457 259.7494 -47.625641 324.0794
## 2018 Q2 138.2269 11.677082 264.7767 -55.314347 331.7682
## 2018 Q3 138.2269 6.713733 269.7401 -62.905134 339.3589DD <- ts(c(15,25,22,20,30,35,30,48,27,23,25,13
,23,17,26,35,34,39,41,29,27,25,29,19
,20,20,21,35,32,52,58,42,42,30,30,39
,25,26,48,47,33,56,48,58,81,48,56,30,15,24,30,16,16,39,41,58,53,46), start = c(2012,1), frequency = 12)
DDData <- window(DD)
DDfit <- ets(DDData)
summary(DDfit)## ETS(M,N,M)
##
## Call:
## ets(y = DDData)
##
## Smoothing parameters:
## alpha = 0.171
## gamma = 1e-04
##
## Initial states:
## l = 28.0393
## s=0.7311 0.9579 0.9549 1.3279 1.3438 1.2214
## 1.3056 0.9407 0.9978 0.8951 0.6837 0.64
##
## sigma: 0.263
##
## AIC AICc BIC
## 510.2532 521.6818 541.1599
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 1.180512 9.056773 6.863269 -4.473593 22.83321 0.6482759
## ACF1
## Training set 0.319248plot(DDfit)
plot(forecast(DDfit,h=8),
ylab="Forecasted Demand")
forecast(DDfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 35.84138 23.76095 47.92182 17.365951 54.31681
## Dec 2016 27.35681 17.99309 36.72052 13.036237 41.67738
## Jan 2017 23.94722 15.62697 32.26746 11.222497 36.67194
## Feb 2017 25.58263 16.56384 34.60142 11.789577 39.37569
## Mar 2017 33.49338 21.51722 45.46954 15.177420 51.80933
## Apr 2017 37.33403 23.79894 50.86912 16.633891 58.03416
## May 2017 35.19717 22.26379 48.13056 15.417266 54.97708
## Jun 2017 48.85398 30.66491 67.04306 21.036198 76.67177
## Jul 2017 45.70560 28.46902 62.94217 19.344525 72.06667
## Aug 2017 50.28323 31.08119 69.48527 20.916240 79.65022
## Sep 2017 49.68628 30.47834 68.89423 20.310269 79.06230
## Oct 2017 35.73198 21.75206 49.71190 14.351537 57.11242
## Nov 2017 35.84143 21.65312 50.02973 14.142286 57.54056
## Dec 2017 27.35684 16.40225 38.31143 10.603246 44.11043
## Jan 2018 23.94725 14.24956 33.64493 9.115908 38.77858
## Feb 2018 25.58266 15.10792 36.05740 9.562927 41.60240
## Mar 2018 33.49342 19.63071 47.35613 12.292232 54.69460
## Apr 2018 37.33407 21.71716 52.95098 13.450070 61.21807
## May 2018 35.19722 20.32031 50.07412 12.444950 57.94948
## Jun 2018 48.85404 27.99298 69.71510 16.949806 80.75828
## Jul 2018 45.70565 25.99244 65.41886 15.556901 75.85440
## Aug 2018 50.28329 28.38119 72.18539 16.786921 83.77966
## Sep 2018 49.68634 27.83392 71.53876 16.265949 83.10674
## Oct 2018 35.73202 19.86669 51.59736 11.468091 59.99595Quarterly
DDquarterly <- aggregate(DD, nfrequency=4)
DDquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 62 85 105 61
## 2013 66 108 97 73
## 2014 61 119 142 99
## 2015 99 136 187 134
## 2016 69 71 152DDDataA <- window(DDquarterly)
DDfitA <- ets(DDDataA)
summary(DDfitA)## ETS(M,N,M)
##
## Call:
## ets(y = DDDataA)
##
## Smoothing parameters:
## alpha = 0.4606
## gamma = 2e-04
##
## Initial states:
## l = 85.0421
## s=0.8576 1.3287 1.0887 0.725
##
## sigma: 0.1998
##
## AIC AICc BIC
## 181.7044 191.8862 188.3155
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 2.367837 22.07567 16.89398 -1.675061 17.82908 0.6351122
## ACF1
## Training set 0.1128718DDDecomp <- decompose(DDquarterly)
DDDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 62 85 105 61
## 2013 66 108 97 73
## 2014 61 119 142 99
## 2015 99 136 187 134
## 2016 69 71 152
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -30.55208 12.81250 29.26042 -11.52083
## 2013 -30.55208 12.81250 29.26042 -11.52083
## 2014 -30.55208 12.81250 29.26042 -11.52083
## 2015 -30.55208 12.81250 29.26042 -11.52083
## 2016 -30.55208 12.81250 29.26042
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 78.750 82.125
## 2013 84.000 84.500 85.375 86.125
## 2014 93.125 102.000 110.000 116.875
## 2015 124.625 134.625 135.250 123.375
## 2016 110.875 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -3.010417 -9.604167
## 2013 12.552083 10.687500 -17.635417 -1.604167
## 2014 -1.572917 4.187500 2.739583 -6.354167
## 2015 4.927083 -11.437500 22.489583 22.145833
## 2016 -11.322917 NA NA
##
## $figure
## [1] -30.55208 12.81250 29.26042 -11.52083
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(DDDecomp)
EE<- ts(c(32,16,21,23,41,72,42,37,62,32,40,33
, 54,28,57,28,31,44,25,27,38,32,31,33
, 32,19,24,21,31,27,44,40,50,38,21,32, 17,19,28,21,31,25,56,32,49,36,34,23,25,34,39,39,74,65,40,47,35
), start = c(2012,1), frequency = 12)
EEData <- window(EE)
EEfit <- ets(EEData)
summary(EEfit)## ETS(M,N,N)
##
## Call:
## ets(y = EEData)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 35.5607
##
## sigma: 0.3692
##
## AIC AICc BIC
## 529.9763 530.4291 536.1054
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.002482662 13.12759 10.03836 -13.12327 31.20605 0.7952925
## ACF1
## Training set 0.2986171plot(EEfit)
plot(forecast(EEfit,h=8),
ylab="Forecasted Demand")
forecast(EEfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Oct 2016 35.56071 18.73548 52.38593 9.828742 61.29267
## Nov 2016 35.56071 18.73548 52.38593 9.828742 61.29267
## Dec 2016 35.56071 18.73548 52.38593 9.828741 61.29267
## Jan 2017 35.56071 18.73548 52.38593 9.828741 61.29267
## Feb 2017 35.56071 18.73548 52.38593 9.828741 61.29267
## Mar 2017 35.56071 18.73548 52.38593 9.828741 61.29267
## Apr 2017 35.56071 18.73548 52.38593 9.828741 61.29267
## May 2017 35.56071 18.73548 52.38593 9.828741 61.29267
## Jun 2017 35.56071 18.73548 52.38593 9.828741 61.29267
## Jul 2017 35.56071 18.73548 52.38593 9.828740 61.29267
## Aug 2017 35.56071 18.73548 52.38593 9.828740 61.29267
## Sep 2017 35.56071 18.73548 52.38593 9.828740 61.29267
## Oct 2017 35.56071 18.73548 52.38593 9.828740 61.29267
## Nov 2017 35.56071 18.73548 52.38593 9.828740 61.29267
## Dec 2017 35.56071 18.73548 52.38593 9.828740 61.29267
## Jan 2018 35.56071 18.73548 52.38593 9.828740 61.29267
## Feb 2018 35.56071 18.73548 52.38593 9.828739 61.29267
## Mar 2018 35.56071 18.73548 52.38593 9.828739 61.29267
## Apr 2018 35.56071 18.73548 52.38594 9.828739 61.29267
## May 2018 35.56071 18.73548 52.38594 9.828739 61.29267
## Jun 2018 35.56071 18.73548 52.38594 9.828739 61.29267
## Jul 2018 35.56071 18.73548 52.38594 9.828739 61.29267
## Aug 2018 35.56071 18.73548 52.38594 9.828739 61.29267
## Sep 2018 35.56071 18.73548 52.38594 9.828738 61.29267Quarterly
EEquarterly <- aggregate(EE, nfrequency=4)
EEquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 69 136 141 105
## 2013 139 103 90 96
## 2014 75 79 134 91
## 2015 64 77 137 93
## 2016 98 178 122EEDataA <- window(EEquarterly)
EEfitA <- ets(EEDataA)
summary(EEfitA)## ETS(A,N,N)
##
## Call:
## ets(y = EEDataA)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 106.6806
##
## sigma: 29.8062
##
## AIC AICc BIC
## 190.9436 192.5436 193.7769
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.005122846 29.80621 25.28547 -7.812693 25.1084 0.8104318
## ACF1
## Training set 0.06838643EEDecomp <- decompose(EEquarterly)
EEDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 69 136 141 105
## 2013 139 103 90 96
## 2014 75 79 134 91
## 2015 64 77 137 93
## 2016 98 178 122
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -9.182292 -9.859375 25.255208 -6.213542
## 2013 -9.182292 -9.859375 25.255208 -6.213542
## 2014 -9.182292 -9.859375 25.255208 -6.213542
## 2015 -9.182292 -9.859375 25.255208 -6.213542
## 2016 -9.182292 -9.859375 25.255208
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 121.500 126.125
## 2013 115.625 108.125 99.000 88.000
## 2014 90.500 95.375 93.375 91.750
## 2015 91.875 92.500 97.000 113.875
## 2016 124.625 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -5.755208 -14.911458
## 2013 32.557292 4.734375 -34.255208 14.213542
## 2014 -6.317708 -6.515625 15.369792 5.463542
## 2015 -18.692708 -5.640625 14.744792 -14.661458
## 2016 -17.442708 NA NA
##
## $figure
## [1] -9.182292 -9.859375 25.255208 -6.213542
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(EEDecomp)
FFF <- ts(c(14,6,13,23,18,14,11,47,72,21,23,17
,19,16,19,14,32,18,19,17,23,34,36,12
,24,14,26,22,18,19,35,24,37,57,33,68
,34,24,35,16,17,58,28,38,20,56,17,22,16,18,22,23,18,78,56,117,36,43), start = c(2012,1), frequency = 12)
FFData <- window(FF)
FFfit <- ets(FFData)
summary(FFfit)## ETS(M,A,N)
##
## Call:
## ets(y = FFData)
##
## Smoothing parameters:
## alpha = 0.6804
## beta = 1e-04
##
## Initial states:
## l = 109.641
## b = 11.8682
##
## sigma: 0.2475
##
## AIC AICc BIC
## 741.7355 742.8893 752.0377
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -8.820438 79.38034 61.34659 -8.419909 23.66188 0.6193905
## ACF1
## Training set 0.1044451plot(FFfit)
plot(forecast(FFfit,h=8),
ylab="Forecasted Demand")
forecast(FFfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 460.5441 314.474335 606.6138 237.149678 683.9385
## Dec 2016 472.3612 290.885315 653.8370 194.817809 749.9045
## Jan 2017 484.1782 271.095019 697.2614 158.295587 810.0608
## Feb 2017 495.9953 253.534285 738.4563 125.183186 866.8074
## Mar 2017 507.8123 237.402879 778.2218 94.256753 921.3679
## Apr 2017 519.6294 222.229762 817.0291 64.795896 974.4629
## May 2017 531.4465 207.711138 855.1818 36.336002 1026.5570
## Jun 2017 543.2635 193.638113 892.8890 8.557591 1077.9695
## Jul 2017 555.0806 179.860014 930.3012 -18.769769 1128.9310
## Aug 2017 566.8977 166.264032 967.5313 -45.818605 1179.6139
## Sep 2017 578.7147 152.763123 1004.6663 -72.722040 1230.1515
## Oct 2017 590.5318 139.288418 1041.7752 -99.585400 1280.6490
## Nov 2017 602.3489 125.784254 1078.9135 -126.493813 1331.1915
## Dec 2017 614.1659 112.204803 1116.1271 -153.517367 1381.8492
## Jan 2018 625.9830 98.511717 1153.4543 -180.714712 1432.6807
## Feb 2018 637.8001 84.672431 1190.9277 -208.135650 1483.7358
## Mar 2018 649.6171 70.658927 1228.5753 -235.823030 1535.0573
## Apr 2018 661.4342 56.446808 1266.4216 -263.814167 1586.6825
## May 2018 673.2513 42.014588 1304.4879 -292.141919 1638.6444
## Jun 2018 685.0683 27.343154 1342.7935 -320.835518 1690.9722
## Jul 2018 696.8854 12.415334 1381.3554 -349.921223 1743.6920
## Aug 2018 708.7024 -2.784432 1420.1893 -379.422836 1796.8277
## Sep 2018 720.5195 -18.270369 1459.3094 -409.362109 1850.4011
## Oct 2018 732.3366 -34.055582 1498.7287 -439.759085 1904.4322Quarterly
FFFquarterly <- aggregate(FFF, nfrequency=4)
FFFquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 33 55 130 61
## 2013 54 64 59 82
## 2014 64 59 96 158
## 2015 93 91 86 95
## 2016 56 119 209FFFDataA <- window(FFFquarterly)
FFFfitA <- ets(FFFDataA)
summary(FFFfitA)## ETS(M,N,N)
##
## Call:
## ets(y = FFFDataA)
##
## Smoothing parameters:
## alpha = 0.1584
##
## Initial states:
## l = 71.6644
##
## sigma: 0.5329
##
## AIC AICc BIC
## 202.1124 203.7124 204.9457
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 12.03473 41.14466 27.19462 -1.934132 29.64645 0.7131456
## ACF1
## Training set 0.0998291FFFDecomp <- decompose(FFFquarterly)
FFFDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 33 55 130 61
## 2013 54 64 59 82
## 2014 64 59 96 158
## 2015 93 91 86 95
## 2016 56 119 209
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -20.098958 -9.609375 13.088542 16.619792
## 2013 -20.098958 -9.609375 13.088542 16.619792
## 2014 -20.098958 -9.609375 13.088542 16.619792
## 2015 -20.098958 -9.609375 13.088542 16.619792
## 2016 -20.098958 -9.609375 13.088542
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 72.375 76.125
## 2013 68.375 62.125 66.000 66.625
## 2014 70.625 84.750 97.875 105.500
## 2015 108.250 99.125 86.625 85.500
## 2016 104.375 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 44.536458 -31.744792
## 2013 5.723958 11.484375 -20.088542 -1.244792
## 2014 13.473958 -16.140625 -14.963542 35.880208
## 2015 4.848958 1.484375 -13.713542 -7.119792
## 2016 -28.276042 NA NA
##
## $figure
## [1] -20.098958 -9.609375 13.088542 16.619792
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(FFFDecomp)
forecast(FFFfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 107.8765 34.20139 181.5516 -4.799854 220.5528
## 2017 Q1 107.8765 33.02452 182.7285 -6.599719 222.3527
## 2017 Q2 107.8765 31.85768 183.8953 -8.384245 224.1372
## 2017 Q3 107.8765 30.70036 185.0526 -10.154216 225.9072
## 2017 Q4 107.8765 29.55208 186.2009 -11.910363 227.6633
## 2018 Q1 107.8765 28.41239 187.3406 -13.653372 229.4064
## 2018 Q2 107.8765 27.28086 188.4721 -15.383888 231.1369
## 2018 Q3 107.8765 26.15711 189.5959 -17.102513 232.8555GG <- ts(c(17,23,32,65,16,33,8,11,40,36,27,25
,16,39,22,72,25,14,32,39,32,20,9,32
,9,27,16,12,31,18,41,21,30,56,25,37
,8,18,28,34,23,20,44,36,64,44,28,28,38,40,36,29,43,47,27,45,24,15), start = c(2012,1), frequency = 12)
GGData <- window(GG)
GGfit <- ets(GGData)
summary(GGfit)## ETS(A,N,N)
##
## Call:
## ets(y = GGData)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 29.7751
##
## sigma: 13.9551
##
## AIC AICc BIC
## 547.2638 547.7082 553.4451
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.004685413 13.95511 10.82969 -29.56022 52.07473 0.7315212
## ACF1
## Training set 0.0249865plot(GGfit)
plot(forecast(GGfit,h=8),
ylab="Forecasted Demand")
forecast(GGfit)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 29.77516 11.89097 47.65935 2.423646 57.12667
## Dec 2016 29.77516 11.89097 47.65935 2.423646 57.12667
## Jan 2017 29.77516 11.89097 47.65935 2.423646 57.12667
## Feb 2017 29.77516 11.89097 47.65935 2.423646 57.12667
## Mar 2017 29.77516 11.89097 47.65935 2.423646 57.12667
## Apr 2017 29.77516 11.89097 47.65935 2.423646 57.12667
## May 2017 29.77516 11.89097 47.65935 2.423646 57.12667
## Jun 2017 29.77516 11.89097 47.65935 2.423645 57.12667
## Jul 2017 29.77516 11.89097 47.65935 2.423645 57.12667
## Aug 2017 29.77516 11.89097 47.65935 2.423645 57.12667
## Sep 2017 29.77516 11.89097 47.65935 2.423645 57.12667
## Oct 2017 29.77516 11.89097 47.65935 2.423645 57.12667
## Nov 2017 29.77516 11.89096 47.65935 2.423645 57.12667
## Dec 2017 29.77516 11.89096 47.65935 2.423645 57.12667
## Jan 2018 29.77516 11.89096 47.65935 2.423644 57.12667
## Feb 2018 29.77516 11.89096 47.65935 2.423644 57.12667
## Mar 2018 29.77516 11.89096 47.65935 2.423644 57.12667
## Apr 2018 29.77516 11.89096 47.65935 2.423644 57.12667
## May 2018 29.77516 11.89096 47.65935 2.423644 57.12667
## Jun 2018 29.77516 11.89096 47.65935 2.423644 57.12667
## Jul 2018 29.77516 11.89096 47.65935 2.423644 57.12667
## Aug 2018 29.77516 11.89096 47.65935 2.423643 57.12667
## Sep 2018 29.77516 11.89096 47.65935 2.423643 57.12667
## Oct 2018 29.77516 11.89096 47.65935 2.423643 57.12667Quarterly
GGquarterly <- aggregate(GG, nfrequency=4)
GGquarterly## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 72 114 59 88
## 2013 77 111 103 61
## 2014 52 61 92 118
## 2015 54 77 144 100
## 2016 114 119 96GGDataA <- window(GGquarterly)
GGfitA <- ets(GGDataA)
summary(GGfitA)## ETS(A,N,N)
##
## Call:
## ets(y = GGDataA)
##
## Smoothing parameters:
## alpha = 1e-04
##
## Initial states:
## l = 90.0927
##
## sigma: 25.6423
##
## AIC AICc BIC
## 185.2256 186.8256 188.0589
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.01737616 25.64232 22.10157 -9.09305 27.71113 0.7207035
## ACF1
## Training set 0.0915364GGDecomp <- decompose(GGquarterly)
GGDecomp## $x
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 72 114 59 88
## 2013 77 111 103 61
## 2014 52 61 92 118
## 2015 54 77 144 100
## 2016 114 119 96
##
## $seasonal
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 -14.78125 -2.50000 13.25000 4.03125
## 2013 -14.78125 -2.50000 13.25000 4.03125
## 2014 -14.78125 -2.50000 13.25000 4.03125
## 2015 -14.78125 -2.50000 13.25000 4.03125
## 2016 -14.78125 -2.50000 13.25000
##
## $trend
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA 83.875 84.125
## 2013 89.250 91.375 84.875 75.500
## 2014 67.875 73.625 81.000 83.250
## 2015 91.750 96.000 101.250 114.000
## 2016 113.250 NA NA
##
## $random
## Qtr1 Qtr2 Qtr3 Qtr4
## 2012 NA NA -38.12500 -0.15625
## 2013 2.53125 22.12500 4.87500 -18.53125
## 2014 -1.09375 -10.12500 -2.25000 30.71875
## 2015 -22.96875 -16.50000 29.50000 -18.03125
## 2016 15.53125 NA NA
##
## $figure
## [1] -14.78125 -2.50000 13.25000 4.03125
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"plot(GGDecomp)
forecast(GGfitA)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2016 Q4 90.09275 57.23079 122.9547 39.83472 140.3508
## 2017 Q1 90.09275 57.23079 122.9547 39.83472 140.3508
## 2017 Q2 90.09275 57.23079 122.9547 39.83472 140.3508
## 2017 Q3 90.09275 57.23079 122.9547 39.83472 140.3508
## 2017 Q4 90.09275 57.23079 122.9547 39.83472 140.3508
## 2018 Q1 90.09275 57.23079 122.9547 39.83472 140.3508
## 2018 Q2 90.09275 57.23079 122.9547 39.83472 140.3508
## 2018 Q3 90.09275 57.23079 122.9547 39.83472 140.3508# Data for modeling
A.mod <- window(A,end = c(2015,11))
#Data for testing
coil.test <- window(A,start=c(2015,12))
# Model using multiple methods - arima, expo smooth, theta, random walk, structural time series
#HW
coil.arima <- forecast(HoltWinters(A.mod, alpha = 0.001, beta = 8e-04, gamma = 1e-04),h=11)
#exponential smoothing
coil.ets <- forecast(ets(A.mod),h=11)
#random walk
coil.rwf <- rwf(A.mod, h=11)
#structts
coil.struc <- forecast(StructTS(A.mod),h=11)
#snaive
coil.naiv <- forecast(snaive(A.mod))
#exp sm
coil.ses <- forecast(ses(A.mod))
##accuracy
arm.acc <- accuracy(coil.arima,coil.test)
ets.acc <- accuracy(coil.ets,coil.test)
rwf.acc <- accuracy(coil.rwf,coil.test)
str.acc <- accuracy(coil.struc,coil.test)
naiv.acc <- accuracy(coil.naiv, coil.test)
ses.acc <- accuracy(coil.ses, coil.test)
arm.acc## ME RMSE MAE MPE MAPE MASE
## Training set 107.2353 324.2152 256.6859 4.145135 13.91524 0.6527652
## Test set -124.3202 403.0288 311.7312 -6.392579 13.14547 0.7927481
## ACF1 Theil's U
## Training set 0.3467567 NA
## Test set -0.1046235 0.9334408ets.acc## ME RMSE MAE MPE MAPE MASE
## Training set 40.62123 323.9361 258.4090 0.2007258 14.51994 0.6571472
## Test set -46.05273 353.2799 279.4924 -4.0465514 12.07098 0.7107632
## ACF1 Theil's U
## Training set -0.009873886 NA
## Test set 0.335816969 0.9718275rwf.acc## ME RMSE MAE MPE MAPE MASE
## Training set 27.71739 339.1818 282.8478 -0.08880283 15.92658 0.7192962
## Test set 103.36364 365.1985 285.1818 2.17589329 11.60619 0.7252317
## ACF1 Theil's U
## Training set -0.2334444 NA
## Test set 0.3358170 0.9149744str.acc## ME RMSE MAE MPE
## Training set 0.08646065 0.9892994 0.7885775 -0.001839906
## Test set -166.78099246 364.4296394 302.4948699 -8.796129458
## MAPE MASE ACF1 Theil's U
## Training set 0.04535497 0.002005392 0.02388733 NA
## Test set 13.33059724 0.769259642 0.17317454 1.038241ses.acc## ME RMSE MAE MPE MAPE MASE
## Training set 40.63588 323.9361 258.4108 0.2010311 14.52003 0.6571516
## Test set -39.51199 368.8241 296.2000 -3.9840363 12.80742 0.7532515
## ACF1 Theil's U
## Training set -0.009660471 NA
## Test set 0.349437049 0.988784Trend Additive: A Multiplicative:
Seasonal Additive: Multiplicative:A