## 4.2 Time Domain TSA

• Prophet
• Prophet is a procedure for forecasting time series data based on an additive model where non-linear trends are fit with yearly, weekly, and daily seasonality, plus holiday effects. It works best with time series that have strong seasonal effects and several seasons of historical data.
• Prophet is robust to missing data and shifts in the trend, and typically handles outliers well.
• Forecasting: Principles and Practice, V3
• Machine Learning with R, the tidyverse, and mlr
• Some R Time Series Issues, TSA2

### 4.2.1tsibbles, feasts, fable

The fable ARIMA() function uses an alternate parameterisation of constants to stats::arima() and forecast::Arima(). While the parameterisations are equivalent, the coefficients for the constant/mean will differ.

In fable, the parameterisation used is:

$(1-φ_1B - \cdots - φ_p B^p)(1-B)^d y_t = c + (1 + θ_1 B + \cdots + θ_q B^q)\varepsilon_t$

In stats and forecast, an ARIMA model is parameterised as:

$(1-φ_1B - \cdots - φ_p B^p)(y_t' - μ) = (1 + θ_1 B + \cdots + θ_q B^q)\varepsilon_t$

where μ is the mean of $$(1-B)^d y_t$$ and $$c = μ(1-φ_1 - \cdots - φ_p)$$.

### 4.2.3 Autoregressions (AR)

For example 4.1, it can be visualised:

When using other functions in feasts, the univaraite time series must be specified.

tsi_fulton %>% feasts::gg_tsdisplay(p.log)

tsi_fulton %>% ACF(p.log) %>% autoplot()

#### 4.2.3.1 Models

mods_fulton[[1]] %>% glance() %>% tab_ti()
.model sigma2 log_lik AIC AICc BIC ar_roots ma_roots
fable::ARIMA(p.log ~ 1 + pdq(1, 0, 0)) 0.06436 -4.67 15.34 15.56 23.47 1.31917441155036+0i complex(0)
mods_fulton[[1]] %>% tidy() %>% tab_ti()
.model term estimate std.error statistic p.value
fable::ARIMA(p.log ~ 1 + pdq(1, 0, 0)) ar1 0.75805 0.06287 12.058 6.601e-22
fable::ARIMA(p.log ~ 1 + pdq(1, 0, 0)) constant -0.04351 0.02324 -1.872 6.379e-02

The model residuals can be analysed using function gg_tsresiduals().

and function qqPlot().

#> [1]  6 47

### 4.2.4 Inference

#### Log Likelihood Ratio Test

whi stat df1 df2 p_value prob if_reject
logLik 90.98 1 110 0 0.05 TRUE

### 4.2.5 Mis-Specification Analysis (MSA)

mods_fulton[[1]] %>% test_jb_tsi() %>% tab_ti()
#> Registered S3 method overwritten by 'propagate':
#>   method         from
#>   print.interval tsibble
whi stat df1 df2 p_value prob if_reject
Jarque-Bera 0.3879 2 109 0.8237 0.05 FALSE
mods_fulton[[1]] %>%
residuals() %>%
features(.resid, ljung_box) %>%
tab_ti()
.model lb_stat lb_pvalue
fable::ARIMA(p.log ~ 1 + pdq(1, 0, 0)) 2.381 0.1228