Obtaining vehicle forecast (Interpretation of output)

Question

Hello, I am starting in the use of the programming in R for the realization of a project in which I am involved. The question is the following: I have data of counting vehicles hour by hour from 2017 until 2019 and I would like to make projections hour by hour in a regression containing explanatory variables such as time, day, holiday, etc. For this, I am using the following code:

library(forecast)

Pedagiados_Pira=data[,4]
Pedagiados_Pira=as.numeric(unlist(Pedagiados_Pira)
Pedagiados_Pira_TS=ts(Pedagiados_Pira,start = c(2017,1),frequency=8760)

#Criando séries temporais
Pedagiados=ts(Pedagiados_Pira_TS,start=c(2017,1),end=c(2019,4380),frequency=8760)

Feriado_Prolongado=as.numeric(unlist(data[,7]))
Feriado_Prolongado_TS=ts(Feriado_Prolongado,start=c(2017,1),end=c(2019,4380),frequency = 8760)
Feriado_Prolongado_TS

Vespera_Feriado=as.numeric(unlist(data[,8]))
Vespera_Feriado_TS=ts(Vespera_Feriado,start=c(2017,1),end=c(2019,4380),frequency=8760)
Vespera_Feriado_TS

Feriado=as.numeric(unlist(data[,9]))
Feriado_TS=ts(Feriado,start=c(2017,1),end=c(2019,4380),frequency=8760)
Feriado_TS

Dia=as.numeric(unlist(data[,18]))
Dia_TS=ts(Dia,start=c(2017,1),end=c(2019,4380),frequency=8760)
Dia_TS

Hora=as.numeric(unlist(data[,19]))
Hora_TS=ts(Hora,start=c(2017,1),end=c(2019,4380),frequency=8760)
Hora_TS

#Base treino e teste
Volume_Treino=window(Pedagiados,start=c(2017,1),end=c(2019,4380),frequency=8760)
Volume_teste=window(Pedagiados,start=c(2019,4380),frequency=8760)

frequency(Pedagiados)
frequency(Feriado_Prolongado_TS)
frequency(Vespera_Feriado_TS)
frequency(Feriado_TS)
frequency(Dia_TS)
frequency(Hora_TS)
frequency(Volume_Treino)

length(Pedagiados)
length(Feriado_Prolongado_TS)
length(Vespera_Feriado_TS)
length(Feriado_TS)
length(Dia_TS)
length(Hora_TS)
length(Volume_Treino)

#Modelo ARIMA
library(forecast)
Pedagiados_Modelo=auto.arima(Volume_Treino,xreg = cbind(Feriado_Prolongado_TS,Vespera_Feriado_TS,Feriado_TS,Dia_TS,Hora_TS),trace = T, stepwise = T, approximation = T, seasonal = T)
Volume_Prev=forecast(Pedagiados_Modelo,xreg=cbind(Feriado_Prolongado_TS,Vespera_Feriado_TS,Feriado_TS,Dia_TS,Hora_TS),h=8760)
Volume_Prev

However, I am getting as an answer for next week (h=168=24*7):

           Point Forecast     Lo 80    Hi 80      Lo 95    Hi 95
2019.50000       1639.295 1308.8097 1969.780 1133.86137 2144.728
2019.50011       1613.166 1023.5096 2202.822  711.36437 2514.968
2019.50023       1589.458  773.1267 2405.790  340.98677 2837.930
2019.50034       1572.230  575.6854 2568.774   48.14677 3096.313
2019.50046       1552.210  422.4699 2681.951 -175.57864 3279.999
2019.50057       1526.521  310.4231 2742.619 -333.34010 3386.382
2019.50068       1500.311  236.2215 2764.400 -432.94684 3433.568
2019.50080       1478.352  190.8950 2765.810 -490.64377 3447.349
2019.50091       1462.639  164.8258 2760.452 -522.19483 3447.472
2019.50103       1452.861  150.6538 2755.067 -538.69284 3444.414
2019.50114       1447.625  143.5283 2751.722 -546.81896 3442.069
2019.50126       1445.410  140.4502 2750.371 -550.35400 3441.175
2019.50137       1445.028  139.6426 2750.413 -551.38664 3441.442
2019.50148       1445.715  140.1077 2751.321 -551.03885 3442.468
2019.50160       1447.044  141.3188 2752.769 -549.89028 3443.978
2019.50171       1448.796  143.0085 2754.583 -548.23362 3445.826
2019.50183       1450.856  145.0363 2756.676 -546.22291 3447.935
2019.50194       1453.153  147.3169 2758.989 -543.95092 3450.257
2019.50205       1455.632  149.7883 2761.476 -541.48370 3452.748
2019.50217       1458.249  152.4010 2764.097 -538.87307 3455.371
2019.50228       1460.965  155.1151 2766.815 -536.15997 3458.090
2019.50240       1463.751  157.9000 2769.601 -533.37556 3460.877
2019.50251       1466.585  160.7335 2772.436 -530.54233 3463.712
2019.50263       1469.451  163.6001 2775.303 -527.67584 3466.579
2019.50274       1438.889  133.0372 2744.740 -558.23882 3436.016
2019.50285       1441.794  135.9424 2747.645 -555.33363 3438.922
2019.50297       1444.710  138.8588 2750.562 -552.41724 3441.838
2019.50308       1447.635  141.7832 2753.486 -549.49288 3444.762
2019.50320       1450.565  144.7133 2756.416 -546.56282 3447.693
2019.50331       1453.499  147.6474 2759.351 -543.62870 3450.627
2019.50342       1456.436  150.5844 2762.288 -540.69169 3453.564
2019.50354       1459.375  153.5234 2765.227 -537.75264 3456.503
2019.50365       1462.316  156.4639 2768.167 -534.81216 3459.443
2019.50377       1465.257  159.4054 2771.109 -531.87068 3462.385
2019.50388       1468.199  162.3476 2774.051 -528.92848 3465.327
2019.50400       1471.142  165.2903 2776.993 -525.98578 3468.270
2019.50411       1474.085  168.2333 2779.937 -523.04274 3471.213
2019.50422       1477.028  171.1766 2782.880 -520.09944 3474.156
2019.50434       1479.972  174.1201 2785.823 -517.15597 3477.099
2019.50445       1482.915  177.0637 2788.767 -514.21237 3480.043
2019.50457       1485.859  180.0074 2791.711 -511.26868 3482.987
2019.50468       1488.803  182.9511 2794.654 -508.32494 3485.930
2019.50479       1491.747  185.8949 2797.598 -505.38114 3488.874
2019.50491       1494.690  188.8388 2800.542 -502.43732 3491.818
2019.50502       1497.634  191.7826 2803.486 -499.49348 3494.762
2019.50514       1500.578  194.7265 2806.430 -496.54962 3497.706
2019.50525       1503.522  197.6703 2809.374 -493.60575 3500.650
2019.50537       1506.466  200.6142 2812.317 -490.66187 3503.593
2019.50548       1437.271  131.4193 2743.122 -559.85679 3434.399
2019.50559       1440.215  134.3632 2746.066 -556.91291 3437.342
2019.50571       1443.159  137.3071 2749.010 -553.96901 3440.286
2019.50582       1446.103  140.2510 2751.954 -551.02512 3443.230
2019.50594       1449.046  143.1948 2754.898 -548.08123 3446.174
2019.50605       1451.990  146.1387 2757.842 -545.13733 3449.118
2019.50616       1454.934  149.0826 2760.786 -542.19344 3452.062
2019.50628       1457.878  152.0265 2763.730 -539.24954 3455.006
2019.50639       1460.822  154.9704 2766.674 -536.30565 3457.950
2019.50651       1463.766  157.9143 2769.618 -533.36175 3460.894
2019.50662       1466.710  160.8582 2772.561 -530.41785 3463.837
2019.50674       1469.654  163.8021 2775.505 -527.47396 3466.781
2019.50685       1472.598  166.7460 2778.449 -524.53006 3469.725
2019.50696       1475.542  169.6899 2781.393 -521.58616 3472.669
2019.50708       1478.485  172.6338 2784.337 -518.64226 3475.613
2019.50719       1481.429  175.5777 2787.281 -515.69837 3478.557
2019.50731       1484.373  178.5216 2790.225 -512.75447 3481.501
2019.50742       1487.317  181.4655 2793.169 -509.81057 3484.445
2019.50753       1490.261  184.4094 2796.113 -506.86668 3487.389
2019.50765       1493.205  187.3533 2799.056 -503.92278 3490.333
2019.50776       1496.149  190.2972 2802.000 -500.97888 3493.276
2019.50788       1499.093  193.2411 2804.944 -498.03499 3496.220
2019.50799       1502.037  196.1850 2807.888 -495.09109 3499.164
2019.50811       1504.980  199.1289 2810.832 -492.14719 3502.108
2019.50822       1435.786  129.9340 2741.637 -561.34210 3432.913
2019.50833       1438.729  132.8779 2744.581 -558.39821 3435.857
2019.50845       1441.673  135.8218 2747.525 -555.45431 3438.801
2019.50856       1444.617  138.7657 2750.469 -552.51041 3441.745
2019.50868       1447.561  141.7096 2753.413 -549.56651 3444.689
2019.50879       1450.505  144.6535 2756.357 -546.62262 3447.633
2019.50890       1453.449  147.5974 2759.301 -543.67872 3450.577
2019.50902       1456.393  150.5413 2762.244 -540.73482 3453.521
2019.50913       1459.337  153.4852 2765.188 -537.79093 3456.464
2019.50925       1462.281  156.4290 2768.132 -534.84703 3459.408
2019.50936       1465.225  159.3729 2771.076 -531.90313 3462.352
2019.50947       1468.168  162.3168 2774.020 -528.95924 3465.296
2019.50959       1471.112  165.2607 2776.964 -526.01534 3468.240
2019.50970       1474.056  168.2046 2779.908 -523.07144 3471.184
2019.50982       1477.000  171.1485 2782.852 -520.12754 3474.128
2019.50993       1479.944  174.0924 2785.796 -517.18365 3477.072
2019.51005       1482.888  177.0363 2788.740 -514.23975 3480.016
2019.51016       1485.832  179.9802 2791.683 -511.29585 3482.959
2019.51027       1488.776  182.9241 2794.627 -508.35196 3485.903
2019.51039       1491.720  185.8680 2797.571 -505.40806 3488.847
2019.51050       1494.664  188.8119 2800.515 -502.46416 3491.791
2019.51062       1497.607  191.7558 2803.459 -499.52026 3494.735
2019.51073       1500.551  194.6997 2806.403 -496.57637 3497.679
2019.51084       1503.495  197.6436 2809.347 -493.63247 3500.623
2019.51096       1434.300  128.4487 2740.152 -562.82738 3431.428
2019.51107       1437.244  131.3926 2743.096 -559.88348 3434.372
2019.51119       1440.188  134.3365 2746.040 -556.93959 3437.316
2019.51130       1443.132  137.2804 2748.984 -553.99569 3440.260
2019.51142       1446.076  140.2243 2751.927 -551.05179 3443.204
2019.51153       1449.020  143.1682 2754.871 -548.10790 3446.147
2019.51164       1451.964  146.1121 2757.815 -545.16400 3449.091
2019.51176       1454.908  149.0560 2760.759 -542.22010 3452.035
2019.51187       1457.851  151.9999 2763.703 -539.27620 3454.979
2019.51199       1460.795  154.9438 2766.647 -536.33231 3457.923
2019.51210       1463.739  157.8877 2769.591 -533.38841 3460.867
2019.51221       1466.683  160.8316 2772.535 -530.44451 3463.811
2019.51233       1469.627  163.7755 2775.479 -527.50062 3466.755
2019.51244       1472.571  166.7194 2778.423 -524.55672 3469.699
2019.51256       1475.515  169.6633 2781.366 -521.61282 3472.643
2019.51267       1478.459  172.6072 2784.310 -518.66893 3475.586
2019.51279       1481.403  175.5510 2787.254 -515.72503 3478.530
2019.51290       1484.347  178.4949 2790.198 -512.78113 3481.474
2019.51301       1487.290  181.4388 2793.142 -509.83723 3484.418
2019.51313       1490.234  184.3827 2796.086 -506.89334 3487.362
2019.51324       1493.178  187.3266 2799.030 -503.94944 3490.306
2019.51336       1496.122  190.2705 2801.974 -501.00554 3493.250
2019.51347       1499.066  193.2144 2804.918 -498.06165 3496.194
2019.51358       1502.010  196.1583 2807.862 -495.11775 3499.138
2019.51370       1432.815  126.9634 2738.667 -564.31266 3429.943
2019.51381       1435.759  129.9073 2741.611 -561.36876 3432.887
2019.51393       1438.703  132.8512 2744.554 -558.42487 3435.830
2019.51404       1441.647  135.7951 2747.498 -555.48097 3438.774
2019.51416       1444.591  138.7390 2750.442 -552.53707 3441.718
2019.51427       1447.534  141.6829 2753.386 -549.59317 3444.662
2019.51438       1450.478  144.6268 2756.330 -546.64928 3447.606
2019.51450       1453.422  147.5707 2759.274 -543.70538 3450.550
2019.51461       1456.366  150.5146 2762.218 -540.76148 3453.494
2019.51473       1459.310  153.4585 2765.162 -537.81759 3456.438
2019.51484       1462.254  156.4024 2768.106 -534.87369 3459.382
2019.51495       1465.198  159.3463 2771.049 -531.92979 3462.326
2019.51507       1468.142  162.2902 2773.993 -528.98590 3465.269
2019.51518       1471.086  165.2341 2776.937 -526.04200 3468.213
2019.51530       1474.030  168.1780 2779.881 -523.09810 3471.157
2019.51541       1476.973  171.1219 2782.825 -520.15420 3474.101
2019.51553       1479.917  174.0658 2785.769 -517.21031 3477.045
2019.51564       1482.861  177.0097 2788.713 -514.26641 3479.989
2019.51575       1485.805  179.9536 2791.657 -511.32251 3482.933
2019.51587       1488.749  182.8975 2794.601 -508.37862 3485.877
2019.51598       1491.693  185.8414 2797.545 -505.43472 3488.821
2019.51610       1494.637  188.7853 2800.488 -502.49082 3491.765
2019.51621       1497.581  191.7292 2803.432 -499.54692 3494.708
2019.51632       1500.525  194.6731 2806.376 -496.60303 3497.652
2019.51644       1431.330  125.4781 2737.181 -565.79794 3428.457
2019.51655       1434.274  128.4220 2740.125 -562.85404 3431.401
2019.51667       1437.218  131.3659 2743.069 -559.91014 3434.345
2019.51678       1440.161  134.3098 2746.013 -556.96625 3437.289
2019.51689       1443.105  137.2537 2748.957 -554.02235 3440.233
2019.51701       1446.049  140.1976 2751.901 -551.07845 3443.177
2019.51712       1448.993  143.1415 2754.845 -548.13456 3446.121
2019.51724       1451.937  146.0854 2757.789 -545.19066 3449.065
2019.51735       1454.881  149.0293 2760.733 -542.24676 3452.009
2019.51747       1457.825  151.9732 2763.676 -539.30287 3454.952
2019.51758       1460.769  154.9171 2766.620 -536.35897 3457.896
2019.51769       1463.713  157.8610 2769.564 -533.41507 3460.840
2019.51781       1466.656  160.8049 2772.508 -530.47117 3463.784
2019.51792       1469.600  163.7488 2775.452 -527.52728 3466.728
2019.51804       1472.544  166.6927 2778.396 -524.58338 3469.672
2019.51815       1475.488  169.6366 2781.340 -521.63948 3472.616
2019.51826       1478.432  172.5805 2784.284 -518.69559 3475.560
2019.51838       1481.376  175.5244 2787.228 -515.75169 3478.504
2019.51849       1484.320  178.4683 2790.171 -512.80779 3481.448
2019.51861       1487.264  181.4122 2793.115 -509.86389 3484.391
2019.51872       1490.208  184.3561 2796.059 -506.92000 3487.335
2019.51884       1493.152  187.3000 2799.003 -503.97610 3490.279
2019.51895       1496.095  190.2439 2801.947 -501.03220 3493.223
2019.51906       1499.039  193.1878 2804.891 -498.08831 3496.167
2019.51918       1429.844  123.9929 2735.696 -567.28322 3426.972
2019.51929       1432.788  126.9368 2738.640 -564.33932 3429.916
2019.51941       1435.732  129.8807 2741.584 -561.39542 3432.860
2019.51952       1438.676  132.8246 2744.528 -558.45153 3435.804
2019.51963       1441.620  135.7684 2747.472 -555.50763 3438.748
2019.51975       1444.564  138.7123 2750.416 -552.56373 3441.692
2019.51986       1447.508  141.6562 2753.359 -549.61983 3444.636
2019.51998       1450.452  144.6001 2756.303 -546.67594 3447.579
2019.52009       1453.396  147.5440 2759.247 -543.73204 3450.523
2019.52021       1456.340  150.4879 2762.191 -540.78814 3453.467
2019.52032       1459.283  153.4318 2765.135 -537.84425 3456.411
2019.52043       1462.227  156.3757 2768.079 -534.90035 3459.355
2019.52055       1465.171  159.3196 2771.023 -531.95645 3462.299
2019.52066       1468.115  162.2635 2773.967 -529.01256 3465.243
2019.52078       1471.059  165.2074 2776.911 -526.06866 3468.187
2019.52089       1474.003  168.1513 2779.855 -523.12476 3471.131
2019.52100       1476.947  171.0952 2782.798 -520.18086 3474.074
2019.52112       1479.891  174.0391 2785.742 -517.23697 3477.018
2019.52123       1482.835  176.9830 2788.686 -514.29307 3479.962
2019.52135       1485.779  179.9269 2791.630 -511.34917 3482.906
2019.52146       1488.722  182.8708 2794.574 -508.40528 3485.850
2019.52158       1491.666  185.8147 2797.518 -505.46138 3488.794
2019.52169       1494.610  188.7586 2800.462 -502.51748 3491.738
2019.52180       1497.554  191.7025 2803.406 -499.57358 3494.682
2019.52192       1428.359  122.5076 2734.211 -568.76850 3425.487
2019.52203       1431.303  125.4515 2737.155 -565.82460 3428.431
2019.52215       1434.247  128.3954 2740.099 -562.88070 3431.375
2019.52226       1437.191  131.3393 2743.042 -559.93680 3434.319
2019.52237       1440.135  134.2832 2745.986 -556.99291 3437.262
2019.52249       1443.079  137.2271 2748.930 -554.04901 3440.206
2019.52260       1446.023  140.1710 2751.874 -551.10511 3443.150
2019.52272       1448.966  143.1149 2754.818 -548.16122 3446.094

Therefore, I would like to understand if I am declaring some variable incorrectly because I verified that the forecast is not returning me values consistent with the historical volume hour by hour (it is noticed that the values obtained rotate around 1420, with no great characteristic variation along the hours of the same day and variation between the days of the week). The vector I got as answer > Volume_prev is returning me the vehicle counts hour by hour over 7 days? If so, there is some error in my code, as my volume forecast has remained "constant" throughout the 168 steps. I thank you in advance!

EDIT:

Summary of Pedagiados_modelo

Series: Volume_Treino 
Regression with ARIMA(3,0,5) errors 

Coefficients:
         ar1      ar2     ar3      ma1     ma2     ma3     ma4     ma5  intercept
      1.6672  -1.0226  0.2429  -0.1895  0.2674  0.1498  0.1148  0.0310  1441.7267
s.e.  0.0864   0.1121  0.0501   0.0866  0.0557  0.0412  0.0302  0.0251    29.5701
      Feriado_Prolongado_TS  Vespera_Feriado_TS  Feriado_TS   Dia_TS  Hora_TS
                   136.7165            -11.7244    -38.6865  -1.4853   2.9439
s.e.                46.6111             35.8038     40.9769   1.2445   0.3418

sigma^2 estimated as 66502:  log likelihood=-152668.9
AIC=305367.8   AICc=305367.9   BIC=305487.8

Training set error measures:
                     ME     RMSE      MAE       MPE     MAPE      MASE        ACF1
Training set 0.03133539 257.7965 170.8875 -7.276662 18.21747 0.3270411 0.001809384
> 

Answer 1

The adjusted model is an ARIMA(3,0,5). In other words, it is a ARMA model(3,5). It is known that the prediction m forward steps of a ARMA model(p, q) is given by

As the ARMA(p, q) model is stationary (i.e., its conditional hope does not change in time), the weights psi are given by

With that, it’s easy to see that psi tends to zero very fast when m tends to infinity. Therefore, the prediction of the time series, when m grow, will tend to average the series very quickly. So the forecasts found are all the same.

In summary, there is nothing wrong with the modeling performed. Constant prediction is an intrinsic characteristic of the ARMA model(p, q). There is nothing that can be done to change that, other than to change the modeling used, which is something that is not the scope of this question.

This text by Professor Frederic Schoenberg makes this situation quite clear, showing the math behind this result.