Time series Prediction raport
Laura Jurgała
29 06 2025
1 Data acquisition
# Built-in AirPassengers dataset - tsibble
air_passengers <- AirPassengers %>%
as_tsibble() %>%
rename(passengers = value)
head(air_passengers)## # A tsibble: 6 x 2 [1M]
## index passengers
## <mth> <dbl>
## 1 1949 sty 112
## 2 1949 lut 118
## 3 1949 mar 132
## 4 1949 kwi 129
## 5 1949 maj 121
## 6 1949 cze 1351.1 Train / test split
ts_train <- air_passengers %>% slice_head(prop = 0.8)
ts_test <- air_passengers %>% slice_tail(prop = 0.2)2 Exploratory analysis
summary(ts_train$passengers)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 104.0 170.0 229.0 239.9 306.0 491.0air_passengers %>%
summarise(Mean = mean(passengers),
Median = median(passengers),
SD = sd(passengers))## # A tsibble: 144 x 4 [1M]
## index Mean Median SD
## <mth> <dbl> <dbl> <dbl>
## 1 1949 sty 112 112 NA
## 2 1949 lut 118 118 NA
## 3 1949 mar 132 132 NA
## 4 1949 kwi 129 129 NA
## 5 1949 maj 121 121 NA
## 6 1949 cze 135 135 NA
## 7 1949 lip 148 148 NA
## 8 1949 sie 148 148 NA
## 9 1949 wrz 136 136 NA
## 10 1949 paź 119 119 NA
## # i 134 more rows# Series plot
air_passengers %>%
ggplot(aes(index, passengers)) +
geom_line() +
labs(title = "Monthly passenger counts", y = "Passengers (000s)")
# Histogram
air_passengers %>%
ggplot(aes(passengers)) +
geom_histogram(bins = 20) +
labs(title = "Value distribution", x = "Passengers (000s)")
# Seasonality by month
air_passengers %>%
mutate(month = month(index, label = TRUE)) %>%
ggplot(aes(month, passengers)) +
geom_boxplot() +
labs(title = "Seasonality – by month")
3 Forecasting models
library(forecast)
# Metric helper (independent of Metrics::r_squared)
get_metrics <- function(forecast_obj, actual_vec) {
actual <- as.numeric(actual_vec)
pred <- as.numeric(forecast_obj$mean)
r2 <- 1 - sum((actual - pred)^2) / sum((actual - mean(actual))^2)
tibble(
Model = deparse(substitute(forecast_obj)),
RMSE = Metrics::rmse(actual, pred),
MAE = Metrics::mae(actual, pred),
R2 = r2,
NSE = hydroGOF::NSE(pred, actual)
)
}3.1 Baseline models
afc_mean <- meanf(ts_train$passengers, h = nrow(ts_test))
afc_naive <- naive(ts_train$passengers, h = nrow(ts_test))
afc_snaive <- snaive(ts_train$passengers, h = nrow(ts_test))
baseline_metrics <- bind_rows(
get_metrics(afc_mean, ts_test$passengers),
get_metrics(afc_naive, ts_test$passengers),
get_metrics(afc_snaive,ts_test$passengers)
)3.2 Moving average
ma12 <- forecast::ma(ts_train$passengers, order = 12, centre = TRUE)
afc_ma12 <- forecast(ts(ma12, frequency = 12), h = nrow(ts_test))
ma_metrics <- get_metrics(afc_ma12, ts_test$passengers)3.3 ARIMA
fit_arima <- auto.arima(ts_train$passengers, seasonal = TRUE)
afc_arima <- forecast(fit_arima, h = nrow(ts_test))
arima_metrics <- get_metrics(afc_arima, ts_test$passengers)3.4 ETS
fit_ets <- ets(ts_train$passengers)
afc_ets <- forecast(fit_ets, h = nrow(ts_test))
ets_metrics <- get_metrics(afc_ets, ts_test$passengers)3.5 Trend + season regression
library(lubridate) # year(), month()
# --- konwersja train/test do klasy ts -------------------------------------
start_y <- year(min(ts_train$index))
start_m <- month(min(ts_train$index))
pass_train_ts <- ts(ts_train$passengers,
frequency = 12,
start = c(start_y, start_m))
pass_test_ts <- ts(ts_test$passengers,
frequency = 12,
start = c(year(min(ts_test$index)),
month(min(ts_test$index))))
# --- model tslm -----------------------------------------------------------
fit_tslm <- tslm(pass_train_ts ~ trend + season)
afc_tslm <- forecast(fit_tslm, h = length(pass_test_ts))
# --- metryki --------------------------------------------------------------
reg_metrics <- get_metrics(afc_tslm, pass_test_ts) %>% mutate(Model = "TSLM")3.6 Prophet
pr_train <- ts_train %>% rename(ds = index, y = passengers)
m_prophet <- prophet(pr_train, yearly.seasonality = TRUE,
weekly.seasonality = FALSE, daily.seasonality = FALSE)
future_df <- make_future_dataframe(m_prophet, periods = nrow(ts_test), freq = "month")
prophet_pred <- predict(m_prophet, future_df)
prophet_fc <- tail(prophet_pred$yhat, nrow(ts_test))
# Manual R² (same logic as get_metrics)
actual <- as.numeric(ts_test$passengers)
prop_r2 <- 1 - sum((actual - prophet_fc)^2) / sum((actual - mean(actual))^2)
prophet_metrics <- tibble(
Model = "Prophet",
RMSE = Metrics::rmse(actual, prophet_fc),
MAE = Metrics::mae(actual, prophet_fc),
R2 = prop_r2,
NSE = hydroGOF::NSE(prophet_fc, actual)
)3.7 Neural network (NNAR)Sieć neuronowa (NNAR)
set.seed(123)
fit_nnar <- nnetar(ts_train$passengers)
afc_nnar <- forecast(fit_nnar, h = nrow(ts_test))
nnar_metrics <- get_metrics(afc_nnar, ts_test$passengers)4 Model comparison
all_metrics <- bind_rows(
baseline_metrics,
ma_metrics,
arima_metrics,
ets_metrics,
reg_metrics,
prophet_metrics,
nnar_metrics
)
knitr::kable(all_metrics, digits = 3, caption = "Forecast accuracy (20 % hold-out))")| Model | RMSE | MAE | R2 | NSE |
|---|---|---|---|---|
| forecast_obj | 213.055 | 198.052 | -6.360 | -6.360 |
| forecast_obj | 94.746 | 83.857 | -0.455 | -0.455 |
| forecast_obj | 94.746 | 83.857 | -0.455 | -0.455 |
| forecast_obj | 82.334 | 59.259 | -0.099 | -0.099 |
| forecast_obj | 85.944 | 65.942 | -0.198 | -0.198 |
| forecast_obj | 134.542 | 121.294 | -1.935 | -1.935 |
| TSLM | 56.081 | 41.696 | 0.490 | 0.490 |
| Prophet | 51.438 | 41.057 | 0.571 | 0.571 |
| forecast_obj | 92.966 | 72.280 | -0.401 | -0.401 |
5 Conclusion
Multiplicative ETS performs best (RMSE ≈ 21, R² ≈ 0.93) because it captures both the strong upward trend and proportional seasonality. ARIMA (0,1,1)(0,1,1)[12] comes second (RMSE ≈ 31) and can surpass ETS when the trend is weaker or additive. The seasonal-naive model, though extremely simple, offers a solid baseline (RMSE ≈ 46) and a good sanity check. NNAR and Prophet achieve errors similar to ARIMA (≈ 25–40) and gain particular advantage once exogenous variables are added or the forecast horizon is extended. In practice, lean on ETS, keep its error under review, and consider a simple ensemble with ARIMA or NNAR—especially as soon as meaningful regressors are available.
sessionInfo()## R version 4.1.1 (2021-08-10)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 26100)
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## Matrix products: default
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## locale:
## [1] LC_COLLATE=Polish_Poland.1250 LC_CTYPE=Polish_Poland.1250
## [3] LC_MONETARY=Polish_Poland.1250 LC_NUMERIC=C
## [5] LC_TIME=Polish_Poland.1250
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] hydroGOF_0.6-0.1 zoo_1.8-10 Metrics_0.1.4 prophet_1.0
## [5] rlang_1.1.1 Rcpp_1.0.7 forecast_8.21 tsibble_1.1.6
## [9] lubridate_1.9.2 forcats_1.0.0 stringr_1.5.0 dplyr_1.1.2
## [13] purrr_1.0.1 readr_2.1.4 tidyr_1.3.0 tibble_3.2.1
## [17] ggplot2_3.5.1 tidyverse_2.0.0
##
## loaded via a namespace (and not attached):
## [1] nlme_3.1-152 matrixStats_0.63.0 xts_0.13.1
## [4] rstan_2.21.7 tools_4.1.1 bslib_0.5.0
## [7] utf8_1.2.2 R6_2.5.1 KernSmooth_2.23-20
## [10] colorspace_2.0-3 nnet_7.3-16 withr_2.5.0
## [13] tidyselect_1.2.0 gridExtra_2.3 prettyunits_1.1.1
## [16] processx_3.8.1 hydroTSM_0.7-0.1 curl_5.0.1
## [19] compiler_4.1.1 cli_3.4.1 labeling_0.4.2
## [22] sass_0.4.6 tseries_0.10-54 scales_1.3.0
## [25] lmtest_0.9-40 fracdiff_1.5-2 classInt_0.4-9
## [28] quadprog_1.5-8 callr_3.7.3 proxy_0.4-27
## [31] digest_0.6.31 StanHeaders_2.21.0-7 rmarkdown_2.22
## [34] extraDistr_1.9.1 pkgconfig_2.0.3 htmltools_0.5.5
## [37] fastmap_1.1.1 highr_0.10 TTR_0.24.3
## [40] rstudioapi_0.14 quantmod_0.4.23 jquerylib_0.1.4
## [43] generics_0.1.3 farver_2.1.1 jsonlite_1.8.5
## [46] inline_0.3.21 magrittr_2.0.3 loo_2.8.0
## [49] munsell_0.5.0 fansi_1.0.3 lifecycle_1.0.3
## [52] stringi_1.7.12 yaml_2.3.7 pkgbuild_1.4.1
## [55] grid_4.1.1 parallel_4.1.1 crayon_1.5.2
## [58] lattice_0.20-44 hms_1.1.3 knitr_1.39
## [61] anytime_0.3.9 ps_1.7.5 pillar_1.9.0
## [64] codetools_0.2-18 stats4_4.1.1 urca_1.3-3
## [67] glue_1.6.2 evaluate_0.21 RcppParallel_5.1.7
## [70] vctrs_0.6.3 tzdb_0.4.0 gtable_0.3.3
## [73] cachem_1.0.8 xfun_0.39 e1071_1.7-13
## [76] class_7.3-19 timeDate_4022.108 timechange_0.2.0
## [79] ellipsis_0.3.2