Forecasting with multiple predictors and with multiple seasonalities in R

Question

I have half-hourly electricity data of several homes for a duration of one month. Also, I have ambient temperature at same sampling rate. Now, I need to make half-hourly forecasts using historical electricity data and forecasted temperature for coming day.

Currently, I am facing a problem with seasonality, i.e., in some homes I observe (via visual inspection) day-wise seasonality and in some other homes no pattern is found. Stephan suggested to check for weekly seasonality as well, but visually I do not find any. So, I thought to try models with different forced seasonalities to find the prediction accuracy. I can think of two options:

1. Model_1 with daily seasonality (frequency = 48 observations)
2. Model_2 with weekly and daily seasonality (48, 7*48)

Keeping the above approach in mind, I am facing the following issue:

1. For Model_1, I can use auto.arima() with xreg option to specify an extra temperature predictor. However, this only works for a single seasonality.
2. For Model_2, I tried to use the tbats() forecasting function, which models multiple seasonalities, but does not allow extra predictor variables.

Is there a forecasting function which allows both multiple predictors and multiple seasonalities?

Here is the electricity data of one month.

    data <- structure(c(1642.8, 1467.1, 165.57, 1630.99, 1618.65, 1629.29, 
        1598.93, 1839.9, 1604.52, 1606.73, 1473.82, 1669.17, 1698.9, 
        2111.21, 2056.41, 3671.29, 2808.01, 1336.15, 794.11, 1212.15, 
        377.36, 888.54, 174.58, 218.54, 420.76, 389.58, 397.77, 395.31, 
        359.11, 364.8, 376.13, 389.37, 929.5, 1702.38, 519.65, 2452.28, 
        1354.45, 1842.96, 725.41, 661.11, 528.44, 733.4, 429.51, 310.47, 
        279.72, 407.83, 1791.1, 1754.53, 1536.73, 1608.37, 1432.23, 1401.72, 
        1582.14, 1558.75, 1536.24, 1745.59, 1375.61, 1556.71, 1671.12, 
        1206.77, 1391.84, 876.23, 1617.3, 1638.99, 1833.61, 1591.42, 
        1455, 183.87, 177.55, 184.36, 332.99, 352.95, 425.1, 945.67, 
        342.3, 348.45, 227.18, 382.15, 268.91, 335.88, 326.94, 233.23, 
        169.71, 179.51, 195.3, 207.23, 1681.9, 1493.32, 941.52, 980.36, 
        924.31, 379.02, 1229.89, 1590.21, 1250.92, 1149.24, 1124.04, 
        993.78, 883.98, 860.69, 934.17, 969.31, 1049.55, 1104.94, 904.3, 
        1220.23, 1183.9, 891.26, 825.33, 787.77, 1060.93, 1029.1, 982.25, 
        193.13, 182.65, 181.75, 167.68, 165.89, 291.02, 300.2, 418.4, 
        297.66, 231.89, 305.66, 701.18, 338.5, 337.24, 332.11, 332.66, 
        187.8, 179.03, 130.22, 177.73, 172.24, 173.45, 334.23, 810.53, 
        359.41, 330.23, 333.29, 568.85, 2462.46, 1660.32, 1156.13, 1136.2, 
        1189.07, 832.73, 181.91, 185.93, 1076.77, 672.78, 1376.71, 1020.17, 
        382.18, 1160.74, 791.36, 1569.4, 817.78, 850.71, 747.21, 826.12, 
        1306.46, 506.23, 140.05, 132.6, 304.19, 308.14, 406.13, 290.73, 
        188.9, 165.59, 174.56, 145.97, 151.83, 142.29, 443.58, 799.9, 
        279.36, 223.88, 221.03, 291.26, 374.97, 431.36, 598.98, 625.82, 
        1052.02, 2036.83, 1230.03, 1429.81, 1099.34, 1646.03, 1668.56, 
        1631.79, 1604.04, 2849.49, 2998.63, 2476.96, 1601.04, 1216.42, 
        2004.2, 1868.51, 1961.91, 1813.35, 1500.22, 1276.94, 1369.29, 
        632.43, 238.15, 488.76, 467.6, 330.27, 144.67, 153.36, 924.12, 
        1348.18, 799.01, 524.3, 420.5, 264.34, 283.86, 198.95, 206.52, 
        217.33, 356.16, 207.83, 197.6, 194.03, 193.01, 249.85, 271.22, 
        244.25, 442.5, 660.49, 245.66, 356.13, 443.32, 336.15, 849.74, 
        1709.21, 1542.09, 1315.69, 2628.6, 2261.8, 1576.42, 1776.48, 
        1239.64, 1401.12, 1106.17, 1378.55, 1315.57, 1141, 1642.63, 2484.13, 
        1968.94, 3059.42, 1317.32, 905.05, 484.42, 486.86, 96.79, 183.45, 
        173.94, 342, 255.96, 320.54, 106.19, 147.88, 150.77, 176.17, 
        344.75, 371.73, 309.46, 237.86, 187.32, 202.61, 292.5, 248.28, 
        259.3, 283.67, 365.09, 230.47, 326.15, 350.92, 335.42, 419.39, 
        345.31, 1093.22, 1392.87, 1298.11, 919.16, 1654.53, 1045.99, 
        558.42, 437.29, 857.9, 758.34, 1220.04, 1390.62, 956.74, 909.93, 
        584.67, 409.87, 387.3, 387.93, 1276.28, 871.06, 413.8, 313.2, 
        199.5, 330.21, 210.9, 358.28, 352.13, 233.62, 259.18, 123.57, 
        255.58, 411.78, 427.65, 318.95, 298.5, 283.23, 279.85, 200.45, 
        205.97, 254.24, 307.98, 1090.53, 289.71, 215.14, 286.63, 328.55, 
        288.96, 1281.19, 1354.8, 1302.05, 1254.46, 261.95, 270.09, 243.28, 
        696.61, 314.27, 241.73, 245.68, 157.74, 222.55, 294.36, 185.46, 
        203.49, 182.14, 246.69, 178.26, 397.5, 330.2, 212.02, 248.72, 
        265.48, 249.37, 130.59, 248.97, 279.94, 319.07, 358.5, 278.98, 
        251.92, 304.66, 455.05, 365.95, 340.93, 287.51, 264.82, 260.18, 
        34.35, 35.11, 184.09, 247.18, 160.9, 139.27, 284.96, 296.31, 
        252.3, 342.65, 353.03, 380.52, 346.19, 350.06, 218.52, 133.94, 
        173.7, 128.26, 167.8, 112.77, 147.8, 129, 170.54, 89.88, 243.08, 
        97.61, 190.31, 193.94, 268.17, 233.5, 205.27, 92.29, 167.43, 
        168.34, 151.99, 193.84, 379.1, 318.69, 327.28, 487.39, 414.01, 
        336.06, 278.02, 168.05, 155.6, 236.4, 264.94, 296.05, 326.46, 
        357.43, 356.31, 340.29, 319.81, 312.79, 341.53, 317.36, 309.62, 
        440.6, 285.5, 282.06, 288.99, 334.48, 196.54, 144.24, 218.55, 
        173.64, 242.29, 251.78, 186.81, 184.36, 141.62, 208.91, 157.53, 
        154.03, 139.44, 137.66, 256.75, 1202.05, 177.36, 177.93, 72.83, 
        252.9, 231.35, 1090.39, 442.91, 363.12, 248.96, 478.75, 249.64, 
        297.29, 227.28, 365.82, 879.7, 488.93, 184.79, 138.13, 151.77, 
        123.18, 175.76, 251.84, 208.06, 126.68, 246.3, 307.34, 319.79, 
        324.3, 379.6, 309.53, 253.17, 221.91, 228.42, 150.24, 148.59, 
        118.79, 86.89, 140.51, 200.43, 212.15, 276.14, 441.81, 125.77, 
        152.42, 329.28, 269.21, 177.35, 1106.29, 128.92, 96.35, 63.53, 
        520.62, 940.25, 1014.34, 314.99, 390.2, 330.1, 377.04, 341.35, 
        342.79, 241.79, 249.9, 391.92, 292.68, 105.02, 179.99, 118.53, 
        154.17, 90.53, 206.7, 345.33, 244.75, 291.68, 820.57, 1777.84, 
        1805.83, 1753.73, 1416.7, 279.2, 262.82, 1345.88, 467.98, 1136.66, 
        170.02, 159.96, 1478.8, 1414.12, 1347.93, 1505.59, 1341.69, 445.53, 
        277.59, 1609.61, 1476.45, 244.08, 192.57, 213.55, 439.02, 112.86, 
        128.54, 376.09, 251.15, 116.27, 254.82, 302.56, 304.6, 198.5, 
        240.05, 219.35, 70.3, 190.96, 211.95, 328.53, 714.62, 3176.56, 
        2604.09, 191.65, 145.25, 93.93, 83.6, 81.13, 146.12, 331.75, 
        250.24, 1144.53, 1616.47, 1008.7, 316.65, 311.46, 1152.99, 1504.86, 
        1543.21, 1081.54, 1428.07, 1358.23, 1349.75, 190.23, 2398.92, 
        2196.11, 1466.94, 2249.77, 2150.2, 2542.25, 618.03, 453.22, 880.99, 
        1497.86, 440.96, 161.85, 324.88, 434.11, 316.33, 444.66, 359.14, 
        277.41, 1237.28, 761.41, 183.53, 309.44, 213.48, 121.64, 346.7, 
        149.86, 2060.39, 1102.13, 347.97, 600.24, 912.6, 590.77, 1805.76, 
        1673.93, 1573.91, 505.74, 446.76, 1033.41, 1668.68, 1293.9, 383.81, 
        1419.99, 1349.4, 711.55, 218.63, 182, 401.93, 1876, 1486.34, 
        1543.11, 2313.8, 478.57, 615.19, 542.68, 971.98, 531.11, 766.21, 
        489.76, 344.47, 319.86, 321.26, 311.41, 288.67, 310.67, 305.15, 
        419.33, 422.84, 950.08, 2188.88, 3454.92, 1989.54, 590.33, 327.05, 
        354.78, 578.41, 1583.29, 2016.66, 1481.03, 293.21, 1864.84, 399.65, 
        366.76, 357.7, 2074.97, 1626.86, 1133, 1624.61, 1506.93, 628.4, 
        1405.68, 217.8, 1223.11, 1356.97, 1171.72, 1182.86, 1642.11, 
        2289.02, 814.39, 595.76, 542.78, 1596.41, 884.97, 235.25, 1540.68, 
        781.95, 115.71, 1204.98, 718.66, 452.09, 305.58, 444.67, 356.76, 
        182.54, 674.47, 153.8, 862.25, 1322.88, 323.33, 1659.64, 496.72, 
        304.74, 246.6, 327.12, 239.31, 246.72, 225.72, 234.7, 324.07, 
        304.27, 171.86, 97.64, 242.69, 295, 324.53, 513.81, 1100.65, 
        1151.77, 231.56, 189.88, 786.3, 1164.87, 676.09, 882.82, 1496.3, 
        1027.91, 872.92, 809.1, 840.31, 1302.18, 2055.87, 677.74, 934.66, 
        263.91, 186.68, 248.5, 214.62, 371.54, 298, 294.52, 304.86, 1295.77, 
        942.5, 305, 265.78, 255.89, 255.63, 151.54, 108.16, 116.81, 100.19, 
        224.75, 84.11, 1143.89, 262.15, 784.21, 1728.29, 1506.79, 434.94, 
        374.29, 265.43, 560.74, 1651.49, 1063.07, 1054.69, 1298.4, 1261.59, 
        1132.75, 692.82, 660.57, 198.25, 97.81, 1258.67, 833.64, 796.35, 
        868.76, 999.86, 2240.02, 885.72, 1317.52, 1267.18, 167.93, 133.22, 
        364.44, 267.17, 406.13, 412.52, 1036.04, 779.34, 655.43, 1901.2, 
        270.18, 266.31, 284.21, 288.66, 135.38, 176.11, 154.86, 160.21, 
        146.28, 163.72, 139.75, 278.12, 253.51, 319.62, 396.39, 1662.69, 
        1577.09, 1059.71, 241.64, 407.54, 290.49, 846.17, 1325.31, 1418.23, 
        1432.5, 1412.6, 1015.87, 1619.88, 1426.58, 1333.32, 1963.35, 
        1638.11, 1081.89, 285.56, 1084.58, 2038.77, 1022.39, 1145.92, 
        513.87, 107.7, 176.19, 143.77, 374.3, 373.99, 221.76, 148.3, 
        331.01, 2323.12, 1502.09, 347.17, 296.7, 306.06, 313.03, 221.69, 
        295.35, 301.95, 250.92, 231.54, 140.6, 717.63, 863.5, 402.15, 
        1337.78, 1575.44, 1738.49, 1675.57, 1617.66, 1365.58, 242.8, 
        286.29, 712.34, 1559.59, 1600.34, 3447.88, 3432.89, 3337.23, 
        1472.21, 1323.76, 1265.31, 1221.65, 1312.63, 2016.09, 2972.13, 
        1451.67, 735.67, 130.13, 379.87, 162.21, 226.48, 417.18, 357.51, 
        346.37, 200.89, 190.15, 276.05, 942.74, 1471.99, 1047.53, 1240.06, 
        742.62, 169.34, 144.28, 220.58, 165.23, 344.8, 227.47, 254.64, 
        742.37, 1809.01, 436.11, 1692.87, 1697.74, 1474.91, 1635.68, 
        1664.2, 1489.85, 1316.4, 1364.07, 1510.02, 1497.89, 1709.17, 
        2846.71, 2736.08, 1015.43, 1017.08, 1195.21, 751.18, 523.89, 
        199.69, 2148.71, 1151.39, 1182.84, 788.91, 259.31, 146.29, 141.09, 
        299.38, 349.53, 404.08, 449.9, 391.77, 251.89, 222.25, 281.04, 
        565.4, 371.24, 219.75, 324.45, 227, 157.67, 212.48, 201.69, 140.84, 
        220.67, 187.64, 399.79, 157.92, 275.49, 326.99, 1340.51, 1578.68, 
        1599.41, 1667.74, 1129.07, 1312.55, 1393.09, 1368.69, 1130.85, 
        968.71, 1130.2, 1223.1, 1124.5, 1077.09, 1052.42, 1255.31, 918.21, 
        1263.93, 706.21, 3080.29, 1620.18, 1122.48, 750.06, 262.89, 110.02, 
        236.83, 413, 227.53, 355, 277.51, 258.03, 368.44, 656.68, 1808.17, 
        1493.41, 1272.21, 330.67, 1674.18, 719.45, 1089.68, 784.01, 275.73, 
        312.69, 345.11, 761.8, 1020.11, 259.16, 345.98, 270.23, 580.15, 
        1303.68, 1659.7, 1732.94, 204.9, 373.58, 373.36, 1381.52, 1437.74, 
        1262.78, 1264.6, 1184.54, 1175.12, 857.09, 1428.34, 841.92, 232.47, 
        223.22, 473.24, 382.7, 189.84, 1737.48, 1689.34, 378.48, 872.56, 
        180.12, 363.03, 301.38, 412.57, 401.17, 387.35, 417.63, 300.61, 
        376.82, 284.31, 232.31, 269.96, 188.41, 203.79, 134.88, 193.66, 
        57.86, 89.71, 167.66, 60.84, 197.03, 703.66, 1638.7, 1467.03, 
        347.22, 1397.23, 1511.92, 1362.25, 1397.18, 1106.19, 826.5, 1033.04, 
        1039.46, 584.98, 706.35, 548.07, 373.39, 681.6, 1231.28, 288.94, 
        649.36, 79.28, 209.23, 290.75, 304.12, 132.57, 91.2, 355.37, 
        197.45, 343.17, 339.2, 284.65, 229, 234.73, 322.36, 323.43, 295.1, 
        197.03, 308.17, 223.88, 235.08, 225.16, 172.83, 236.26, 135.17, 
        394.89, 479.2, 315.34, 280.9, 282.36, 204.78, 367.24, 1712.29, 
        1521, 1686.09, 960.19, 1019.12, 1062.77, 851.88, 1369.98, 689.2, 
        580.5, 751.74, 547.67, 556.64, 493.85, 404.15, 428.07, 716.2, 
        1442.05, 1045.81, 1497.07, 567.59, 155.07, 537.72, 446.03, 282.57, 
        642.88, 409.37, 338.91, 173.89, 358.63, 195.42, 209.95, 186.44, 
        152.34, 105.51, 132.26, 82.14, 122.88, 149.38, 211.42, 350.17, 
        429.72, 336.4, 982.21, 1436.25, 1726.87, 1830.42, 1282.05, 1293.4, 
        1121.01, 946.2, 707.1, 154.53, 767.56, 607.78, 448, 288.23, 270.32, 
        223.93, 166.22, 262.45, 223.74, 159.31, 210.44, 257.94, 183.99, 
        151.38, 206.11, 193.43, 388.95, 577.98, 304.64, 285.13, 256.59, 
        420.26, 289.34, 356.02, 358.08, 325.22, 275.95, 164.46, 213.23, 
        142.99, 221.66, 270.61, 206.56, 213.68, 254.33, 250.15, 267.99, 
        403.95, 671.2, 1574.11, 396.34, 477.88, 631.08, 618.25, 1366.88, 
        298.19, 287.05, 290.38, 332.44, 235.9, 229.79, 831.6, 1320.86, 
        477.31, 944.84, 547.33, 411.21, 705.43, 873.49, 572.81, 585.36, 
        1229.69, 701.01, 653.49, 74.81, 162.47, 179.54, 330.27, 544.51, 
        332.41, 296.66, 130.66, 1055.61, 556.79, 265.43, 383.44, 398.22, 
        362.66, 223.99, 130.35, 193.67, 217.68, 273.3, 247.84, 161.66, 
        320.08, 322.52, 274.61, 811.44, 353.85, 323.41, 383.61, 389.5
        ), .Dim = c(1248L, 1L), .Dimnames = list(NULL, "power"))

Temperature predictor values are:

temp<-structure(c(31, 31, 31, 31, 30, 29, 28, 27.5, 27, 26, 26, 26, 
26, 26, 26, 28, 29, 29, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36, 
36.5, 37, 38, 38, 39, 39, 39, 38, 37, 36, 36, 35, 34, 33, 33, 
33, 32.5, 32, 32, 31, 30, 26, 24, 24, 24, 24, 24, 24, 25, 25, 
25, 25, 26, 26, 26, 27, 28.3, 29.7, 31, 32, 33, 33, 34, 34, 34, 
35, 35, 35, 36, 37, 38, 39, 39, 39, 39, 38.5, 38, 37, 36, 35, 
34, 34, 33, 32.5, 32, 30, 30, 30, 28, 28, 27, 25, 25, 24, 24, 
24, 24, 24, 25, 25, 25, 25, 25, 27, 28, 30, 31, 33, 34, 37, 37, 
38, 38, 39, 39.3, 39.7, 40, 40, 40, 40, 41, 40, 40, 38, 38, 36, 
35, 34, 33, 31, 31, 30, 31, 30, 30, 29, 28, 28, 28, 27, 27, 28, 
27, 24, 24, 24, 23, 23, 23, 24, 24, 26, 27, 28, 29.7, 31.3, 33, 
34, 35.3, 36.7, 38, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 
39, 38, 37, 36, 34, 33, 32, 32, 30, 28.5, 27, 27, 27, 27, 27, 
26, 26, 26, 25.3, 24.7, 24, 24, 24, 24, 23, 23, 24.5, 26, 28, 
29.7, 31.3, 33, 34.5, 36, 37, 38, 39, 40, 40, 40.3, 40.7, 41, 
41, 41, 41, 41, 41, 41, 39.5, 38, 37, 37, 35, 34, 34, 33, 33, 
33, 32, 31, 31, 30, 30, 30, 29, 29, 28.5, 28, 28, 28, 29, 29, 
28.5, 28, 29, 29, 30, 31, 31, 32.5, 34, 35, 37, 39.5, 42, 42, 
42, 42, 42, 43, 44, 45, 45, 44, 43, 42, 42, 41, 39, 36.5, 34, 
34, 33, 33, 33, 33, 33, 33, 32, 32, 32, 32, 31.5, 31, 30, 30, 
30, 29, 29, 28, 27, 28, 29, 30, 31, 32.5, 34, 34, 35, 37, 40, 
41, 41, 42, 42, 42.5, 43, 43, 43, 42, 42, 43, 42, 42, 41, 40, 
39, 37.5, 36, 35, 35, 34, 34, 33, 32, 32, 32, 32, 31, 31, 31, 
31, 31, 28, 28, 28, 28, 28, 28.5, 29, 30, 30.5, 31, 32, 33.5, 
35, 36, 37, 38, 39, 38, 39, 40, 41, 42, 42, 43, 42, 42, 42, 41, 
41, 40, 39, 38, 37, 36, 35, 34, 34, 33, 32, 32, 31, 30.5, 30, 
30, 30, 30, 29, 29, 29, 28, 27, 27, 27, 27, 28.5, 30, 31.5, 33, 
34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 42, 42, 42, 42, 42, 42, 
42, 42, 42, 42, 40, 40, 39, 37, 37, 36.5, 36, 35, 34, 34, 33, 
33, 32, 32, 32, 32, 32, 32, 31, 31, 30, 30, 30, 30, 31, 31, 31, 
32, 33.5, 35, 35.5, 36, 37, 37, 38, 39, 40, 41, 41, 42, 42, 42, 
43, 43, 42, 41, 41, 40, 38, 36, 35, 34, 34, 35, 36, 36, 35, 35, 
33, 33, 32, 31, 31, 30, 30, 29, 29, 29, 29, 29, 31, 31.5, 32, 
31, 30, 31, 32, 33, 34, 34, 35, 36, 36, 37, 37, 38, 38, 38, 39, 
39, 39, 39, 39, 39, 39, 38, 37, 37, 37, 37, 35, 33, 31, 31, 31, 
30, 30, 30, 30, 30, 30, 30, 30, 29, 29, 28, 28, 27, 27, 26, 29, 
29, 30, 31, 32, 32, 33, 33, 33, 34, 34, 35, 35, 37, 37, 37, 37, 
37, 38, 38, 38, 38, 37, 37, 36, 35, 34.5, 34, 34, 33, 33, 32, 
32, 32, 32, 32, 31, 31, 31, 31, 31, 31, 31, 30, 30, 29.5, 29, 
29, 30, 30, 31, 32, 32, 33, 33.5, 34, 34, 34, 30.7, 27.3, 24, 
24, 24, 24.7, 25.3, 26, 27, 28, 29, 29, 29, 28, 28, 27, 27, 26, 
25, 25, 25, 24.5, 24, 24, 24, 24, 24, 24, 24, 24, 23, 23, 23, 
23, 23, 23, 24, 24, 24, 25, 25, 26, 28, 29, 30, 31, 31, 32, 33, 
34, 35, 35, 36, 37, 37, 37, 37, 37, 37, 37, 35, 34, 33, 33, 33, 
32, 31.7, 31.3, 31, 31, 31, 31, 30, 30, 30, 30, 28, 28, 27, 27, 
26, 26, 25, 25, 25, 25, 25, 26, 27, 28, 29.7, 31.3, 33, 34, 34, 
35, 36, 37, 36, 36, 35, 35, 34, 32, 32, 33, 31.3, 29.7, 28, 28, 
29, 29, 29, 29, 28, 27.5, 27, 26.5, 26, 26, 26, 26, 26, 26, 25, 
25, 25, 25, 25, 24, 24, 23, 23, 23, 24.5, 26, 27, 28, 29, 30, 
31, 32, 32, 32, 33, 34, 35, 36, 36, 36, 36, 36, 36, 36, 37, 37, 
37, 36, 36, 34, 34, 33, 32, 32, 31, 31, 30, 28, 28, 28, 28, 27, 
27, 27, 27, 27, 27, 27, 27, 26.5, 26, 26, 26, 26, 27, 28.8, 30.5, 
32.2, 34, 34, 34, 35, 36, 37, 38, 38, 38, 39, 40, 39, 39, 39, 
40, 40, 40, 39, 39, 38, 37, 35, 35, 34, 34, 34, 33, 32, 32, 32, 
32, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 32, 33, 34, 
35, 37, 37, 37.3, 37.7, 38, 39, 40, 43, 43, 43, 44, 44, 43, 43, 
43, 43, 43, 43, 42, 41, 40, 39, 38, 37, 36, 35, 35, 34, 34, 34, 
34, 34, 33, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 33, 33, 34, 
35, 36, 37, 38, 38, 40, 40.5, 41, 42, 42, 42, 42, 41, 40, 40, 
39, 39, 37, 32, 27, 26, 26, 27, 28, 27, 27, 27, 26, 25, 25, 25, 
25, 25, 25, 25, 25, 25, 25, 25, 24, 23, 23, 22.5, 22, 22, 23, 
24, 25, 26.3, 27.7, 29, 30, 32, 33, 34, 36, 37, 37, 39, 40, 40.5, 
41, 42, 42, 42, 43, 43, 42, 41.5, 41, 39, 38, 35, 35, 35, 34, 
33, 33, 33, 33, 32, 32, 31, 31, 30, 30, 30, 29, 29, 28, 28, 28, 
28.5, 29, 29, 29, 30, 32, 32, 34, 35, 37, 38, 39, 40, 41, 42, 
42, 43, 44, 44, 44, 45, 45, 45, 44, 43, 43, 42, 40, 38.5, 37, 
35, 35, 34, 34, 33, 34, 33, 33, 33, 33, 32, 32, 31, 32, 31, 30, 
29, 29, 29, 29, 30, 30, 31, 32.7, 34.3, 36, 37, 38, 39, 40, 41, 
41, 42, 42, 43, 44, 44, 44, 44, 44, 43.5, 43, 43, 42, 42, 40, 
38, 38, 37, 36, 36, 36, 36, 35, 35, 34, 34, 34, 34, 32, 32, 31.5, 
31, 31, 30, 30, 30, 30, 31, 33, 34, 35, 37, 38, 39, 40, 40, 42, 
42, 43, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 43, 42, 40, 40, 
40, 39, 38, 38, 38, 37, 37, 36, 36, 35.5, 35, 34.5, 34, 33, 33, 
32, 32, 31, 30, 30, 30, 31, 32, 33.5, 35, 36.5, 38, 38, 39, 40, 
41, 41, 41, 42, 42, 42, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 
42, 40.5, 39, 38, 38, 37, 37, 37, 36, 36, 36, 36, 36, 36, 36, 
35, 35, 34, 34, 34, 32, 32, 31, 31, 33, 34, 35, 35, 36, 38, 39, 
40, 41, 42, 42.5, 43, 43, 44, 44, 45, 45, 45, 45, 45, 45, 45, 
44, 43.5, 43, 43, 42, 41, 40, 40, 39, 39, 38, 38, 37, 36, 36, 
35, 34, 34, 34, 32, 32, 32, 32, 32, 32, 32, 34, 35, 35, 37, 38, 
38, 39, 39, 40, 41, 42, 43, 44, 44, 44, 45, 45, 46, 46, 46, 45, 
45, 45, 44, 42, 40, 38, 37, 37, 37, 37, 37, 36, 35), .Dim = c(1248L, 
1L), .Dimnames = list(NULL, "temperature"))

Forecasted temperature values are:

forecast_temp <- structure(c(35, 34, 34, 33, 33, 30, 29.7, 29.3, 29, 28, 28, 27, 
27.3, 27.7, 28, 29, 30.5, 32, 33.5, 35, 36, 36, 37, 39, 39.5, 
40, 40, 41, 42, 42, 43, 43, 43, 44, 44, 44, 43, 42, 40, 39, 39, 
38, 37, 37, 36, 35, 35, 34), .Dim = c(48L, 1L), .Dimnames = list(
    NULL, "temperature"))

UPDATE-1

From @Stephan's suggestion, I followed the approach mentioned by Prof. Rob at link1, link2 as

library(forecast)
tsob <- ts(train_power,frequency = 48) #training electriciy data at daily frequency
tsob_weekly <- fourier(ts(train_power,frequency = 7*48),K=3) #training data at weekly frequency
tempob <- ts(train_temperature,frequency = 48) #Temperature, another predictor variable
fit <- auto.arima(tsob,xreg=cbind(tsob_weekly,tempob),seasonal=FALSE)
tempob_forecast <- ts(test_temperature,frequency = 48) #forecasted temperauture values
forecast_val <- forecast(fit,xreg=tempob_forecast,h=48*5) #forecast for coming 5 days

As evident from the code, I have used Fourier transformation suggested at above links to show weekly seasonal affect. Up to auto.arima(), it works properly but at forecast function an error is thrown as:

Error in forecast.Arima(fit, xreg = tempob_forecast, h = 48 * 5) : 
  Number of regressors does not match fitted model

The error is clear, i.e., I do not provide the forecasted regressor values of tsob_weekly, which I don't have. How should I handle this issue? Prof. Rob has used the same value of regressor for both the auto.arima and forecast functions.

Answer 1

This is a nice question.

First, note that the arima(), Arima() and auto.arima() functions in R fit regressions with ARIMA errors, not ARIMAX models, and yes, there is a difference.

This immediately suggests one possible way forward: regress your historical data on your regressors, using lm() or similar (possibly accounting for heteroskedasticity). Afterwards, fit a tbats model to the residuals from that initial regression. For forecasting, forecast both models separately and add the predictions.

Alternatively, you could "roll your own model". Use lm() or similar to regress your observations on temperature etc., and also add periodic dummies to capture the seasonality. For instance, periodic B-splines or "bumps" are often nice for time seasonality, and you could use six Boolean dummies for the days of the week, since adjacent days will be less similar than adjacent hours.

Answer 2

Use regression with ARMA errors and include Fourier terms and dummy variables to account for the seasonalities. This has been suggested in Rob J. Hyndman's blog posts "Forecasting with daily data" and "Forecasting with long seasonal periods". Intuitively, it may be a good alternative to the suggestions by @StephanKolassa. In his approach, fitting a regression first and then modelling its errors with TBATS would mean the point estimates of the regression coefficients would not account for the TBATS patterns in the residuals. Regression with ARMA errors is a "cleaner" approach in this respect as the regression coefficients include the information from the residual modelling.

Regarding the error you are getting, perhaps you need newxreg in place of xreg?