Chapter overview and Task

Chapter 1

This is the overview of the chapter 4. I will add info from the index file here. I will filter from there to what will go here. My goal is to first have the website have a map of the concepts in time series. I want to first avoid doing it by chapters since it seperates topics and limit my learning to that lesson. Bigger picture first.

code for lesson 1.5

the following does the multiplicative model

Code 
source("../common_functions.R")
Code 
# Set random seed for reproducibility
set.seed(123) 

# Set parameters & initialize vectors
num_years <- 10
n <- 12 * num_years
sigma <- .75
a <- 0.03
b <- 1
c <- 0.5 
trend <- seasonal <- x_t <- rep(0,n)
time_seq <- seq(1,n)

# Generate correlated error terms
w <- rnorm(n + 4, 0.2, 0.1) # Changed to a mean of 1 and sd of 0.03
z = w + lead(w,1) + lead(w,2) + lead(w,3) + lead(w,4)
z  = head(z, n)

# Get date
year_seq <- lubridate::year(today()) - num_years  + (time_seq - 1) %/% 12
month_seq <- (time_seq - 1) %% 12 + 1
date_seq <- ymd(paste0(year_seq,"-",month_seq,"-01"))

# Get data
for (t in 1:n) {
  trend[t] <- exp(a * t)
  seasonal[t] <- exp( b * sin(t / 12 * 2 * pi * 1)  + c * cos(t / 12 * 2 * pi * 3) + 1 )
  x_t[t] <- trend[t] * seasonal[t] * z[t] # Note R's definition of the mult. model
}

x_df <- data.frame(x_t = x_t, trend = trend, seasonal = seasonal)

start_year <- lubridate::year(today()) - num_years
start_date <- lubridate::ymd(paste0(start_year,"-01-01"))

# start_date <- lubridate::ymd("1958-01-01")
date_seq <- seq(start_date,
    start_date + months(nrow(x_df)-1),
    by = "1 months")

x_df_ts <- x_df |>
  mutate(
    date = date_seq,
    month = tsibble::yearmonth(date)
  ) |>
  select(date, month, trend, seasonal, x_t) |>
  as_tsibble(index = month)
Date Month Trend, $$m_t$$ Seasonal, $$s_t$$ Data, $$x_t$$
2014-01-01 2014 Jan 1.03 4.482 5.065
2014-02-01 2014 Feb 1.062 3.92 5.512
2014-03-01 2014 Mar 1.094 7.389 11.266
2014-04-01 2014 Apr 1.127 10.655 13.348
2014-05-01 2014 May 1.162 4.482 5.391
2014-06-01 2014 Jun 1.197 1.649 1.93
⋮ ⋮ ⋮ ⋮ ⋮
2023-11-01 2023 Nov 35.517 1.649 39.853
2023-12-01 2023 Dec 36.598 4.482 121.366
Code 
trend_plot <- ggplot(x_df_ts, aes(x=month, y=trend)) + 
  geom_line() +
  labs(
    title="Plot of Trend", 
    x="Month", 
    y="Trend"
    ) +
  theme(plot.title = element_text(hjust = 0.5))

seasonal_plot <- ggplot(x_df_ts, aes(x=month, y=seasonal)) + 
  geom_line() +
  labs(
    title="Plot of Seasonal Effect", 
    x="Month", 
    y="Seasonal"
    ) +
  theme(plot.title = element_text(hjust = 0.5))

error_plot <- ggplot(x_df_ts, aes(x = month, y = x_t / trend / seasonal)) + 
  geom_line() +
  labs(
    title="Plot of Random Error Term", 
    x="Month", 
    y="Random"
  ) +
  theme(plot.title = element_text(hjust = 0.5))

x_plot <- ggplot(x_df_ts, aes(x=month, y=x_t)) + 
  geom_line() +
  labs(
    title="Plot of Simulated Time Series", 
    x="Month", 
    y="x_t"
  ) +
  theme(plot.title = element_text(hjust = 0.5))

x_plot <- x_plot  + labs(title = "True (Simulated) Values", x = NULL)
trend_plot <- trend_plot + labs(title = NULL, x = NULL)
seasonal_plot <- seasonal_plot + labs(title = NULL, x = NULL)
error_plot <- error_plot + labs(title = NULL)

x_plot / trend_plot / seasonal_plot / error_plot

Random Error Term (Error Plot): \[ e_t = \frac{x_t}{m_t \cdot s_t} \]

Trend: \[ m_t = e^{a \cdot t} \]

Seasonal: \[ s_t = e^{b \cdot \sin\left(\frac{2 \pi t}{12}\right) + c \cdot \cos\left(\frac{6 \pi t}{12}\right) + 1} \]

Combined Time Series (Multiplicative Model): \[ x_t = m_t \cdot s_t \cdot z_t \]

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