Negative forecast value using ARIMA from fable and NaNs warning using ARIMA

[arvind]

My data set is a weekly data that contains two variables Production and Shipment. Production is the independent variable and Shipment is the dependent variable. First I'm trying to forecast Production values and use that as a regressor to forecast Shipment variable.

I had two issues

1. When generating Production forecasts - the last step forecast (h=4) was a negative value. If it was zero then it would be more accurate than having a huge negative value as on that week the actual would most likely be zero.
2. When fitting the forecast model for Shipment variable- using Production data and fourier term I get a NaNs warning despite using Automated ARIMA with stepwise=False, approx= False.

Could you anyone please help me with these two queries.
Thank you for the support

Original.df<-structure(list(YearWeek = c("201901", "201902", "201903", "201904", 
"201905", "201906", "201907", "201908", "201909", "201910", "201911", 
"201912", "201913", "201914", "201915", "201916", "201917", "201918", 
"201919", "201920", "201921", "201922", "201923", "201924", "201925", 
"201926", "201927", "201928", "201929", "201930", "201931", "201932", 
"201933", "201934", "201935", "201936", "201937", "201938", "201939", 
"201940", "201941", "201942", "201943", "201944", "201945", "201946", 
"201947", "201948", "201949", "201950", "201951", "201952", "202001", 
"202002", "202003", "202004", "202005", "202006", "202007", "202008", 
"202009", "202010", "202011", "202012", "202013", "202014", "202015", 
"202016", "202017", "202018", "202019", "202020", "202021", "202022", 
"202023", "202024", "202025", "202026", "202027", "202028", "202029", 
"202030", "202031", "202032", "202033", "202034", "202035", "202036", 
"202037", "202038", "202039", "202040", "202041", "202042", "202043", 
"202044", "202045", "202046", "202047", "202048", "202049", "202050", 
"202051", "202052", "202053", "202101", "202102", "202103", "202104", 
"202105", "202106", "202107", "202108", "202109", "202110", "202111", 
"202112", "202113", "202114", "202115", "202116", "202117", "202118", 
"202119", "202120", "202121", "202122", "202123", "202124", "202125", 
"202126", "202127", "202128", "202129", "202130", "202131", "202132", 
"202133", "202134", "202135", "202136", "202137", "202138", "202139", 
"202140", "202141", "202142", "202143", "202144", "202145", "202146", 
"202147", "202148", "202149", "202150", "202151", "202152", "202201", 
"202202", "202203"), Shipment = c(399, 1336, 1018, 1126, 1098, 
1235, 1130, 1258, 897, 1333, 1221, 1294, 1628, 1611, 1484, 1238, 
1645, 1936, 1664, 1482, 2060, 1964, 1875, 1645, 2039, 1640, 733, 
1764, 1639, 1968, 1692, 1677, 1542, 1299, 1328, 1130, 1741, 1929, 
1843, 1427, 1467, 1450, 1041, 1238, 1721, 1757, 1813, 1001, 1208, 
1916, 1435, 540, 681, 1436, 1170, 938, 1206, 1648, 1169, 1311, 
1772, 1333, 1534, 1365, 1124, 846, 732, 753, 1266, 1652, 1772, 
1814, 1649, 1191, 1298, 986, 1296, 1066, 777, 1041, 1388, 1289, 
1097, 1356, 1238, 1732, 1109, 1104, 1155, 1334, 1094, 770, 1411, 
1304, 1269, 1093, 1096, 1121, 943, 695, 1792, 2033, 1586, 768, 
685, 993, 1406, 1246, 1746, 1740, 938, 160, 1641, 1373, 1023, 
1173, 1611, 928, 1038, 1009, 1274, 1369, 1231, 1053, 1163, 880, 
870, 1131, 882, 1143, 632, 394, 510, 543, 535, 824, 874, 591, 
512, 448, 247, 452, 470, 747, 545, 639, 326, 414, 604, 640, 458, 
272, 524, 589, 666, 217, 215, 348, 537, 466), Production = c(794, 
1400, 1505, 1055, 1396, 1331, 1461, 1623, 1513, 1667, 1737, 1264, 
1722, 1587, 2094, 1363, 2007, 1899, 1749, 1693, 1748, 1455, 2078, 
1702, 1736, 1885, 860, 1372, 1716, 1290, 1347, 1451, 1347, 1409, 
1203, 1235, 1397, 1557, 1406, 1451, 1704, 670, 1442, 1336, 1611, 
1401, 1749, 744, 1558, 1665, 1317, 0, 441, 1351, 1392, 1180, 
1447, 1265, 1485, 1494, 1543, 1581, 1575, 1597, 1191, 1386, 889, 
1002, 1573, 1380, 1346, 1243, 1009, 965, 1051, 905, 1094, 1194, 
891, 1033, 921, 880, 1135, 1058, 1171, 1022, 956, 880, 902, 983, 
1014, 945, 1021, 1058, 1191, 1139, 1292, 573, 1173, 514, 1292, 
1310, 1239, 0, 0, 1182, 1028, 1028, 1196, 1214, 1045, 256, 1451, 
1344, 1352, 1257, 1444, 786, 1369, 1185, 1262, 1025, 949, 1051, 
941, 727, 911, 951, 987, 1136, 884, 770, 959, 1102, 1109, 1098, 
988, 983, 1002, 904, 1147, 1149, 919, 1058, 1112, 479, 1028, 
1154, 1126, 1155, 1208, 536, 839, 1178, 1225, 539, 0, 862, 839, 
873)), row.names = c(NA, 160L), class = "data.frame")

# Converting the df to accomodate leap year for weekly observations
Original.df <- Original.df %>%
  mutate(
    isoweek =stringr::str_replace(YearWeek, "^(\\d{4})(\\d{2})$", "\\1-W\\2-1"),
    date = ISOweek::ISOweek2date(isoweek)
  )

#creating test and train data
Original.train.df <- Original.df %>%
  filter(date >= "2018-12-31", date <= "2021-11-22")

Original.test.df <- Original.df %>%
  filter(date >= "2021-11-29", date <= "2021-12-27")

Shipment.Test.df<- Original.test.df %>%
  dplyr::select(-YearWeek, -Production, -date,-isoweek) %>% as_tibble()

# splitting the original train data to contain only Week, Dependent and Independent variables
Total.train.df<-Original.train.df %>%
  mutate(Week.1 = yearweek(ISOweek::ISOweek(date))) %>%
  dplyr::select(-YearWeek,-date,-isoweek) %>%
  as_tsibble(index = Week.1)

#Model.1-Fitting forecast model(Arima with Fourier terms) to Production.qty 

bestfit.Prod.1.AICc <- Inf

for(K in seq(25)){
  fit.Prod.1 <- Total.train.df %>% 
    model(ARIMA(Production ~ fourier(K = K), stepwise = FALSE, approximation = FALSE))
  
  if(purrr::pluck(glance(fit.Prod.1), "AICc") < bestfit.Prod.1.AICc)
  {
    bestfit.Prod.1.AICc <- purrr::pluck(glance(fit.Prod.1), "AICc")
    bestfit.Prod.1<- fit.Prod.1
    bestK.Prod.1 <- K
  }
}

bestK.Prod.1
glance(bestfit.Prod.1)

#Model.1-Forecasting Net.Production.Qty for 4 steps using the fitted model above
Forecast.Prod.1<-bestfit.Prod.1 %>% 
  forecast(h = 4)

#Here I get the 4th step forecasted point as -146
Final.Prod.1<-Forecast.Prod.1$.mean

#Model.1-Fitting forecast model(Arima with Fourier terms) and 
#Production training data(actuals) as regressors to Shipment Qty

bestfit.Shipment.1.AICc <- Inf

for(K in seq(25)){
  fit.Shipment.1 <- Total.train.df %>% 
    model(ARIMA(Shipment ~Production + fourier(K = K),stepwise = FALSE))
  
  if(purrr::pluck(glance(fit.Shipment.1), "AICc") < bestfit.Shipment.1.AICc)
  {
    bestfit.Shipment.1.AICc <- purrr::pluck(glance(fit.Shipment.1), "AICc")
    bestfit.Shipment.1<- fit.Shipment.1
    bestK.Shipment.1 <- K
  }
}

#above model shows NaNs warning
bestK.Shipment.1
glance(bestfit.Shipment.1)

[arvind]

Taking log transformations solved this issue- log(x + 1) on both the variables resolved the negative forecast outputs. Since I'm using fable it helped me back transform the forecast to have the forecasted mean