Chapter 1: Introduction to Time Series

1 Introduction

1.1 Learning Objectives

By the end of this chapter, students should be able to:

  1. Understand Time Series Fundamentals.
  2. Identify Time Series Components.
  3. Understand the Relationship Among Components.
  4. Measure Forecast Performance.
  5. Apply these concepts using R.

1.2 Definition

A time series is fundamentally defined as a time-oriented or chronological sequence of observations measured on a variable of interest. These observations are typically collected sequentially over time at fixed, equally spaced intervals, referred to as the sampling interval.

Depending on the context, time series data can take several forms:

  • Instantaneous measurements: A reading taken at a specific point in time, such as the viscosity of a chemical product at the moment it is measured.
  • Cumulative quantities: An accumulated total over the interval, such as the total product demand or sales during a month.
  • Summary statistics: A metric that reflects activity of the variable during the time period, such as the daily closing price of a stock.

1.2.1 Statistical and Mathematical Definition

A time series is modelled as a discrete-time stochastic process — a collection of random variables indexed according to the discrete time order in which they are obtained.

A time series of length \(n\) is represented as:

\[\{y_t : t = 1, 2, \ldots, n\}\]

where the subscript \(t\) denotes the discrete time period.

An important distinction:

Concept Description
The Time Series Model The theoretical sequence of underlying random variables — the ensemble of possibilities
The Realization The actual historical sequence of data observed; a single realization of the stochastic process

1.2.2 Arrangement of Time Series Data

Show R Code 
df_example <- data.frame(
  t     = 1:6,
  Year  = 2018:2023,
  y_t   = c(3.4, 3.3, 3.9, 4.5, 3.7, 3.5)
)
knitr::kable(df_example, col.names = c("$t$", "Year", "$y_t$ (Unemployment Rate, %)"),
             align = "c")
Table 1: Example arrangement of time series data
\(t\) Year \(y_t\) (Unemployment Rate, %)
1 2018 3.4
2 2019 3.3
3 2020 3.9
4 2021 4.5
5 2022 3.7
6 2023 3.5

1.3 Categorization of Forecasting Models

Source: Wang, Shouyi & Chaovalitwongse, Wanpracha (2011). Evaluating and Comparing Forecasting Models.

1.4 Visualization of Time Series

A time series can be visualized via a time series plot.

1.4.1 Example 1: Malaysia Unemployment Rate (1982–2020)

Show R Code 
unrate   <- read.csv("data/employment.csv")
unratets <- ts(unrate$u_rate, start = 1982)

plot(unratets,
     ylab = "Unemployment Rate (%)", xlab = "Year", xaxt = "n",
     col = "steelblue", lwd = 1.5)
years <- seq(1982, 2020, by = 2)
axis(1, at = years, labels = years, las = 2)
title(sub = "Figure 1: Malaysia Unemployment Rate from 1982 to 2020",
      col.sub = "grey40")
Figure 1: Malaysia Unemployment Rate, 1982–2020

1.4.2 Example 2: Monthly Oil Price (Jan 1986 – Jan 2006)

Show R Code 
data(oil.price)
plot(oil.price, ylab = "Price per Barrel (USD)", type = "l",
     col = "darkred", lwd = 1.5)
title(sub = "Figure 2: Monthly Price of Oil: Jan 1986 – Jan 2006",
      col.sub = "grey40")
Figure 2: Monthly price of oil, January 1986 – January 2006

1.4.3 Example 3: Monthly Sales of Specialty Oil Filters (1983–1987)

Show R Code 
data(oilfilters)
plot(oilfilters, type = "l", ylab = "Sales", col = "darkgreen", lwd = 1.5)
points(y = oilfilters, x = time(oilfilters),
       pch = as.vector(season(oilfilters)))
title(sub = "Figure 3: Monthly Sales of Specialty Oil Filters, Jul 1983 – Jun 1987",
      col.sub = "grey40")
Figure 3: Monthly sales of specialty oil filters, 1983–1987

1.5 Interpretation of Time Series Plot

When interpreting a time series plot, always include the following:

Element Description
Minimum The smallest value in the plot
Maximum The largest value in the plot
Outliers Values much further away from the rest of the data points
Trends Patterns such as upward/downward movements or clusters

Example interpretation (Figure 1): Malaysia’s unemployment rate ranged from a minimum of 2.4% (1997) to a maximum of 7.4% (2020). The data shows a general downward trend from 1986 to 1997 and exhibits considerable fluctuation thereafter, with a sharp spike in 2020 likely attributed to the COVID-19 pandemic.

2 Components of Time Series

A time series is typically composed of four components:

Component Notation Description
Trend \(T_t\) Long-run direction of the series
Seasonal \(S_t\) Regular, repeating fluctuations within a fixed period
Cyclical \(C_t\) Long-wave rises and falls around the trend
Irregular/Random \(I_t\) Unpredictable, residual variation

2.1 Trend

The trend component describes the general upward or downward movements that characterise all economic and business activities.

2.1.4 Trend Using Simple Linear Regression

In simple linear regression for trend, the independent variable is time \(t\) where \(t = 1, 2, \ldots, n\):

\[\hat{T}_t = \hat{\beta}_0 + \hat{\beta}_1 t\]

Show R Code 
prod    <- read.csv("data/electric.csv")
electric <- data.frame(t = seq(1:108), index = prod[1:108, 3])

ggplot(data = electric, aes(x = t, y = index)) +
  geom_line(colour = "grey50", linewidth = 0.7) +
  geom_smooth(method = "lm", colour = "#d73027", se = TRUE) +
  stat_regline_equation(label.x = 30, label.y = 130) +
  labs(title = "Output of Electricity Index with Linear Trend",
       x = "Time (months, Jan 2015 = 1)",
       y = "Output of Electricity (Index)") +
  theme_minimal()
Figure 5: Output of electricity index with fitted linear trend (Jan 2015 – Dec 2023)
Show R Code 
lm_fit <- lm(index ~ t, data = electric)
# Linear regression output
summary(lm_fit)

Call:
lm(formula = index ~ t, data = electric)

Residuals:
     Min       1Q   Median       3Q      Max 
-18.2613  -2.7849   0.3872   3.8791  10.5326 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 103.05133    1.08696   94.81   <2e-16 ***
t             0.22672    0.01731   13.10   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.609 on 106 degrees of freedom
Multiple R-squared:  0.618, Adjusted R-squared:  0.6144 
F-statistic: 171.5 on 1 and 106 DF,  p-value: < 2.2e-16

Excel Tutorial: https://youtu.be/n6e71G9PImc

2.1.5 Trend Using Moving Average

\[M_t^{(k)} = \frac{1}{k}\sum_{i=0}^{k-1} y_{t-i}\]

  • If \(k = 3\): \(M_1 = \dfrac{y_1 + y_2 + y_3}{3}\) — 3-period moving average
  • If \(k = 5\): \(M_1 = \dfrac{y_1 + y_2 + y_3 + y_4 + y_5}{5}\) — 5-period moving average
Note
  • The larger \(k\) is, the smoother the series, but more observations are lost at both ends.
  • The first \(\lfloor k/2 \rfloor\) and last \(\lfloor k/2 \rfloor\) observations are lost from the moving average series.
Show R Code 
ma3 <- ma(electric$index, 3)
ma5 <- ma(electric$index, 5)

matplot(electric$t,
        cbind(electric$index, ma3, ma5),
        type = "l", lty = 1, col = c("red", "blue", "green"),
        xlab = "Time (months)", ylab = "Output Index",
        main = "Moving Average Trend")
legend("bottom", legend = c("Actual", "3-period MA", "5-period MA"),
       col = c("red", "blue", "green"), lty = 1, bty = "n")
Figure 6: Electricity output index: actual vs. 3-period and 5-period moving averages

Excel Tutorial: https://youtu.be/FODxTjhY5TM

2.2 Seasonal

The seasonal component refers to a recurring pattern that repeats at regular, fixed intervals (daily, weekly, monthly, or yearly).

2.2.1 Causes of Seasonal Patterns

Factor Example
Weather Palm oil production influenced by rainfall (November–March, east coast Malaysia)
Holidays Tourist numbers peak during school holidays
Cultural Events Zakat collection highest during Hari Raya Eidulfitri

2.2.2 Identifying Seasonal Components

  • Visual Inspection: Plotting the time series and observing recurring patterns.
  • Seasonal Decomposition: STL or classical decomposition methods.
  • Statistical Tests: Autocorrelation function (ACF) analysis (Chapter 5).
Show R Code 
electric$date <- seq(from = as.Date("2015-01-01"),
                     to   = as.Date("2023-12-01"),
                     by   = "month")

ggplot(data = electric, aes(x = date, y = index)) +
  geom_line(colour = "steelblue", linewidth = 0.8) +
  scale_x_date(date_labels = "%b %Y", date_breaks = "6 months") +
  labs(title = "Monthly Electricity Output Index (Jan 2015 to Dec 2023)",
       x = "Year", y = "Electric Consumption (Index)") +
  theme_minimal(base_size = 11) +
  theme(axis.text.x = element_text(angle = 90, hjust = 1))
Figure 7: Monthly electricity output index showing seasonal pattern (2015–2023)
Show R Code 
elec_ts <- ts(electric$index, start = c(2015, 1), frequency = 12)
decomp  <- decompose(elec_ts, type = "additive")
autoplot(decomp) +
  labs(title = "Additive Decomposition: Electricity Output Index") +
  theme_bw(base_size = 11)
Figure 8: Classical additive decomposition of the electricity index

Excel Tutorial: https://youtu.be/Wwlx2IgB7uw

2.3 Cyclical

Cyclical variation refers to rises and falls of the series over an unspecified period, usually around the long-run trend line.

Important
  • Requires sufficiently long data — typically more than 5 years.
  • Easier to detect with yearly data.
  • There is no fixed period for cyclical recurrence (unlike seasonal).
  • Identification is challenging because data patterns are typically not stable.

Source: https://www.mathworks.com/help/econ/choose-time-series-filter-for-business-cycle-analysis.html

2.3.1 Methods to Identify Cyclical Variation

2.3.1.1 Residual Method

\[\text{Percent of Trend} = \frac{y_t}{\hat{T}_t} \times 100\]

Value Interpretation
\(< 100\) Economy contracting (recession)
\(> 100\) Economy expanding

2.3.1.2 Relative Cyclical Residual Method

\[\text{Relative Cyclical Residual} = \frac{y_t - \hat{T}_t}{\hat{T}_t} \times 100\]

Value Interpretation
Negative Trend pulled down by recession
Positive Output above average (expansion)

2.4 Irregular / Random

The irregular or random component is the portion that cannot be explained by trend, seasonal, or cyclical components. It has four distinct sub-types:

Sub-type Key Characteristic Visual Pattern
Turning Point Trend permanently changes direction Sustained shift in slope
Random Shock Sudden, temporary spike or dip Sharp impulse, then recovery
Outlier One extreme isolated value Single point far from neighbours
Pure Noise Small unpredictable fluctuations Irregular scatter around zero

2.4.1 Turning Point

A turning point is where the long-run direction permanently reverses — from upward to downward or vice versa. Unlike a shock, the series does not recover to its prior direction.

Examples:

  • Introduction of robotic manufacturing permanently raises production.
  • Sudden change in consumer preference shifts demand to a new sustained level.
  • Policy change (e.g., new minimum wage legislation) permanently alters labour market dynamics.
Show R Code 
unrate <- read.csv("data/employment.csv")
df_u   <- data.frame(
  Year   = as.numeric(format(as.Date(unrate$date), "%Y")),
  u_rate = unrate$u_rate
)

ggplot(df_u, aes(x = Year, y = u_rate)) +
  geom_line(colour = "steelblue", linewidth = 0.9) +
  geom_point(size = 1.8, colour = "steelblue") +
  geom_point(data = df_u[df_u$Year == 1986, ],
             aes(x = Year, y = u_rate),
             colour = "#d73027", size = 4, shape = 17) +
  annotate("label", x = 1986, y = df_u$u_rate[df_u$Year == 1986] + 0.5,
           label = "Peak (1986)\nTurning Point ↓",
           colour = "#d73027", fill = "white", size = 3.2, label.size = 0.3) +
  geom_point(data = df_u[df_u$Year == 1997, ],
             aes(x = Year, y = u_rate),
             colour = "#1a9641", size = 4, shape = 25) +
  annotate("label", x = 1997, y = df_u$u_rate[df_u$Year == 1997] - 0.6,
           label = "Trough (1997)\nTurning Point ↑",
           colour = "#1a9641", fill = "white", size = 3.2, label.size = 0.3) +
  geom_point(data = df_u[df_u$Year == 2020, ],
             aes(x = Year, y = u_rate),
             colour = "#fc8d59", size = 4, shape = 17) +
  annotate("label", x = 2019, y = df_u$u_rate[df_u$Year == 2020] + 0.5,
           label = "COVID-19 (2020)\nShock / Turning Point?",
           colour = "#fc8d59", fill = "white", size = 3.2, label.size = 0.3) +
  labs(title = "Turning Points in Malaysia's Unemployment Rate (1982–2020)",
       subtitle = "▲ Peak = trend reverses downward  |  ▽ Trough = trend reverses upward",
       x = "Year", y = "Unemployment Rate (%)") +
  theme_ts()
Figure 9: Turning points in Malaysia’s unemployment rate (1982–2020)
Show R Code 
t_tp  <- 1:80
# Before turning point: upward; after: downward
y_tp  <- ifelse(t_tp <= 45,
                20 + 0.5 * t_tp,
                20 + 0.5 * 45 - 0.4 * (t_tp - 45)) +
         rnorm(80, 0, 1.2)

df_tp <- data.frame(t = t_tp, y = y_tp)

ggplot(df_tp, aes(x = t, y = y)) +
  geom_line(colour = "#4575b4", linewidth = 0.9) +
  geom_vline(xintercept = 45, linetype = "dashed", colour = "#d73027",
             linewidth = 1) +
  annotate("label", x = 45, y = max(y_tp) - 1,
           label = "Turning Point\n(t = 45)", colour = "#d73027",
           fill = "white", size = 3.5, label.size = 0.3) +
  annotate("segment", x = 5, xend = 42, y = 22, yend = 40,
           arrow = arrow(length = unit(0.15, "cm")),
           colour = "#1a9641", linewidth = 0.8) +
  annotate("text", x = 15, y = 26, label = "Upward trend",
           colour = "#1a9641", size = 3.5) +
  annotate("segment", x = 48, xend = 75, y = 42, yend = 30,
           arrow = arrow(length = unit(0.15, "cm")),
           colour = "#d73027", linewidth = 0.8) +
  annotate("text", x = 62, y = 38, label = "Downward trend",
           colour = "#d73027", size = 3.5) +
  labs(title = "Turning Point in a Time Series",
       subtitle = "Before t = 45: upward trend  |  After t = 45: downward trend",
       x = "Time", y = "Value") +
  theme_ts()
Figure 10: Turning point — the trend permanently reverses direction at the marked point
Note

Turning point vs. Random Shock: A turning point results in a permanent direction change. After a random shock, the series recovers toward its previous level.

2.4.2 Random Shock

A random shock is an abrupt, temporary spike or drop caused by an unexpected external event. The series typically recovers toward its prior level once the event passes.

Examples:

  • Crude oil price sudden drop due to COVID-19.
  • Production of masks increasing tremendously during the pandemic.
  • Stock market crash following a geopolitical crisis or natural disaster.
Show R Code 
data(oil.price)
oil_df <- data.frame(Date = as.numeric(time(oil.price)),
                     Price = as.numeric(oil.price))

ggplot(oil_df, aes(x = Date, y = Price)) +
  geom_line(colour = "steelblue", linewidth = 0.8) +
  annotate("rect", xmin = 1990, xmax = 1991.5,
           ymin = -Inf, ymax = Inf, alpha = 0.15, fill = "#d73027") +
  annotate("label", x = 1990.75, y = 40,
           label = "Gulf War\nshock (1990–91)",
           colour = "#d73027", fill = "white", size = 3, label.size = 0.3) +
  annotate("rect", xmin = 1997.5, xmax = 1999.5,
           ymin = -Inf, ymax = Inf, alpha = 0.15, fill = "#4575b4") +
  annotate("label", x = 1998.5, y = 44,
           label = "Asian Financial\nCrisis (1998)",
           colour = "#4575b4", fill = "white", size = 3, label.size = 0.3) +
  annotate("rect", xmin = 2003, xmax = 2006,
           ymin = -Inf, ymax = Inf, alpha = 0.12, fill = "#fc8d59") +
  annotate("label", x = 2004.5, y = 20,
           label = "Iraq War &\nprice surge",
           colour = "#a63603", fill = "white", size = 3, label.size = 0.3) +
  labs(title = "Random Shocks in Monthly Oil Prices (Jan 1986 – Jan 2006)",
       subtitle = "Each shaded region marks an unexpected external event",
       x = "Year", y = "Price per Barrel (USD)") +
  theme_ts()
Figure 11: Random shocks in monthly oil prices (Jan 1986 – Jan 2006)
Show R Code 
oil_sub <- oil_df[oil_df$Date >= 1989.5 & oil_df$Date <= 1995, ]
lm_pre  <- lm(Price ~ Date, data = oil_df[oil_df$Date < 1990, ])
oil_sub$trend <- predict(lm_pre, newdata = data.frame(Date = oil_sub$Date))

ggplot(oil_sub, aes(x = Date)) +
  geom_line(aes(y = Price, colour = "Observed"), linewidth = 0.9) +
  geom_line(aes(y = trend, colour = "Pre-shock trend"),
            linetype = "dashed", linewidth = 1) +
  annotate("segment", x = 1990.25, xend = 1991.5, y = 38, yend = 22,
           arrow = arrow(length = unit(0.15, "cm")), colour = "#d73027") +
  annotate("text", x = 1990, y = 39, label = "Spike then\nrecovery",
           colour = "#d73027", size = 3.2) +
  scale_colour_manual(values = c("Observed" = "steelblue",
                                 "Pre-shock trend" = "#d73027")) +
  labs(title = "Gulf War Shock: Price Recovers to Pre-shock Level",
       x = "Year", y = "Price per Barrel (USD)", colour = NULL) +
  theme_ts()
Figure 12: Gulf War shock (1990–91): price spikes then recovers to prior trend
Show R Code 
t_sh  <- 1:80
trend_sh <- 30 + 0.2 * t_sh
noise_sh <- rnorm(80, 0, 1.2)
shock    <- rep(0, 80)
shock[40] <- -18   # sudden sharp downward shock
shock[41] <- -10
shock[42] <- -4    # gradual recovery

y_sh  <- trend_sh + shock + noise_sh
df_sh <- data.frame(t = t_sh, y = y_sh)

ggplot(df_sh, aes(x = t, y = y)) +
  geom_line(colour = "#4575b4", linewidth = 0.9) +
  annotate("rect", xmin = 39, xmax = 43, ymin = -Inf, ymax = Inf,
           alpha = 0.15, fill = "#d73027") +
  annotate("label", x = 41, y = min(y_sh) + 3,
           label = "Shock period\n(e.g. COVID-19)", colour = "#d73027",
           fill = "white", size = 3.5, label.size = 0.3) +
  annotate("segment", x = 43, xend = 55, y = y_sh[42] + 1, yend = 35,
           arrow = arrow(length = unit(0.15, "cm")), colour = "#1a9641",
           linewidth = 0.8) +
  annotate("text", x = 58, y = 35.5, label = "Recovery",
           colour = "#1a9641", size = 3.5) +
  labs(title = "Random Shock in a Time Series",
       subtitle = "Sharp temporary drop followed by recovery to original trend",
       x = "Time", y = "Value") +
  theme_ts()
Figure 13: Random shock — a sudden spike followed by recovery to the previous level
Show R Code 
t2    <- 1:60
base  <- 20 + 0.15 * t2

# Single-period impulse
y_imp <- base + rnorm(60, 0, 0.8)
y_imp[30] <- y_imp[30] + 12

# Multi-period sustained shock (e.g. supply chain disruption for 5 months)
shock_sus <- rep(0, 60)
shock_sus[30:34] <- c(8, 10, 9, 7, 4)
y_sus <- base + shock_sus + rnorm(60, 0, 0.8)

df_imp <- data.frame(t = t2, y = y_imp, type = "Single-period Impulse")
df_sus <- data.frame(t = t2, y = y_sus, type = "Sustained Shock (5 periods)")
df_cmp <- rbind(df_imp, df_sus)

ggplot(df_cmp, aes(x = t, y = y)) +
  geom_line(colour = "#4575b4", linewidth = 0.8) +
  facet_wrap(~type, scales = "free_y") +
  labs(title = "Types of Random Shock",
       x = "Time", y = "Value") +
  theme_ts()
Figure 14: Comparison: impulse shock (single period) vs. sustained shock (multiple periods)

2.4.3 Outliers

An outlier is an isolated data point that significantly deviates from the overall pattern. Unlike a random shock (which spans several periods), an outlier is typically a single observation.

Outliers arise from: errors in data collection, measurement instrument failure, or genuinely unusual one-off events.

Show R Code 
t_out  <- 1:60
y_out  <- 50 + 0.3 * t_out + rnorm(60, 0, 2)
# Inject outliers
y_out[15] <- y_out[15] + 20   # high outlier
y_out[42] <- y_out[42] - 18  # low outlier

# Z-score to flag outliers
z_scores  <- abs(scale(y_out))
is_outlier <- z_scores > 2.5

df_out <- data.frame(t = t_out, y = y_out, outlier = as.vector(is_outlier))

ggplot(df_out, aes(x = t, y = y)) +
  geom_line(colour = "grey50", linewidth = 0.7) +
  geom_point(aes(colour = outlier, size = outlier)) +
  scale_colour_manual(values = c("FALSE" = "#4575b4", "TRUE" = "#d73027"),
                      labels = c("FALSE" = "Normal", "TRUE" = "Outlier")) +
  scale_size_manual(values = c("FALSE" = 1.5, "TRUE" = 3.5)) +
  annotate("label", x = 15, y = y_out[15] + 2,
           label = "High outlier\n(data entry error?)",
           colour = "#d73027", fill = "white", size = 3.2, label.size = 0.3) +
  annotate("label", x = 42, y = y_out[42] - 3,
           label = "Low outlier\n(instrument failure?)",
           colour = "#d73027", fill = "white", size = 3.2, label.size = 0.3) +
  labs(title = "Outliers in a Time Series",
       subtitle = "Flagged using Z-score threshold > 2.5",
       x = "Time", y = "Value", colour = NULL, size = NULL) +
  theme_ts()
Figure 15: Outliers in a time series — isolated extreme data points

How to detect outliers:

  1. Visual inspection of the time series plot.
  2. Z-score method — flag where \(|z_t| = |\frac{y_t - \bar{y}}{s}| > 2.5\).
  3. IQR method — flag outside \([Q_1 - 1.5 \times IQR,\ Q_3 + 1.5 \times IQR]\).
  4. Decomposition residuals — inspect the irregular component after removing trend and seasonal effects.

How to deal with outliers:

Strategy When to Use
Remove Confirmed data entry errors or instrument failures
Winsorize Replace with boundary value (5th/95th percentile)
Impute Replace with average of neighbours
Robust models Use outlier-resistant forecasting techniques

2.4.4 Random (Pure Noise)

The random sub-component refers to small, unpredictable fluctuations that remain after all systematic effects are removed. Also called the residual or error term.

Random errors are expected to be \(I_t \sim N(0, \sigma^2)\) and uncorrelated across time.

Show R Code 
elec_ts <- ts(electric$index, start = c(2015, 1), frequency = 12)
decomp  <- decompose(elec_ts, type = "additive")
resid_df <- data.frame(Date     = as.numeric(time(elec_ts)),
                       Residual = as.numeric(decomp$random))

p1 <- ggplot(resid_df, aes(x = Date, y = Residual)) +
  geom_line(colour = "grey40", linewidth = 0.7, na.rm = TRUE) +
  geom_hline(yintercept = 0, linetype = "dashed", colour = "#d73027") +
  geom_hline(yintercept =  2 * sd(decomp$random, na.rm = TRUE),
             linetype = "dotted", colour = "#4575b4") +
  geom_hline(yintercept = -2 * sd(decomp$random, na.rm = TRUE),
             linetype = "dotted", colour = "#4575b4") +
  annotate("text", x = 2023.5, y =  2 * sd(decomp$random, na.rm = TRUE) + 0.2,
           label = "+2σ", colour = "#4575b4", size = 3.2, hjust = 1) +
  annotate("text", x = 2023.5, y = -2 * sd(decomp$random, na.rm = TRUE) - 0.3,
           label = "−2σ", colour = "#4575b4", size = 3.2, hjust = 1) +
  labs(title = "Irregular Component (Random Noise)", x = "Year",
       y = expression(I[t])) + theme_ts()

p2 <- ggplot(na.omit(resid_df), aes(x = Residual)) +
  geom_histogram(aes(y = after_stat(density)), bins = 12,
                 fill = "#4575b4", alpha = 0.7, colour = "white") +
  stat_function(fun  = dnorm,
                args = list(mean = mean(decomp$random, na.rm = TRUE),
                            sd   = sd(decomp$random, na.rm = TRUE)),
                colour = "#d73027", linewidth = 1) +
  labs(title = "Distribution of Residuals",
       subtitle = expression("Should approximate " * N(0, sigma^2)),
       x = expression(I[t]), y = "Density") + theme_ts()

grid.arrange(p1, p2, ncol = 2)
Figure 16: Irregular (residual) component from the electricity index decomposition

2.4.5 Summary: Distinguishing Irregular Sub-types

Sub-type Duration Reversible? Real Example
Turning Point Permanent No Malaysia unemployment 1986, 1997
Random Shock Few periods Yes — recovers Gulf War oil spike 1990
Outlier Single period N/A Flagged oil filter observation
Pure Noise Throughout N/A Electricity residuals
Show R Code 
t_all <- 1:80
base_all <- 30 + 0.2 * t_all

# 1. Turning Point
y1 <- ifelse(t_all <= 45,
             base_all,
             base_all[45] - 0.35 * (t_all - 45)) + rnorm(80, 0, 1)

# 2. Random Shock
shock2 <- rep(0, 80); shock2[c(35, 36, 37)] <- c(-15, -8, -3)
y2 <- base_all + shock2 + rnorm(80, 0, 1)

# 3. Outlier
y3 <- base_all + rnorm(80, 0, 1)
y3[25] <- y3[25] + 22

# 4. Pure Noise
y4 <- base_all + rnorm(80, 0, 3.5)

make_plot <- function(t, y, title, subtitle, highlight = NULL,
                      vline = NULL) {
  df <- data.frame(t = t, y = y)
  g  <- ggplot(df, aes(x = t, y = y)) +
    geom_line(colour = "#4575b4", linewidth = 0.8) +
    labs(title = title, subtitle = subtitle, x = "Time", y = "Value") +
    theme_ts() +
    theme(plot.subtitle = element_text(size = 9))
  if (!is.null(highlight)) {
    g <- g +
      geom_point(data = df[highlight, ], aes(x = t, y = y),
                 colour = "#d73027", size = 3)
  }
  if (!is.null(vline)) {
    g <- g +
      geom_vline(xintercept = vline, linetype = "dashed",
                 colour = "#d73027", linewidth = 0.9)
  }
  g
}

pa <- make_plot(t_all, y1, "Turning Point",
                "Permanent change in trend direction", vline = 45)
pb <- make_plot(t_all, y2, "Random Shock",
                "Sharp temporary impulse, then recovery", highlight = 35:37)
pc <- make_plot(t_all, y3, "Outlier",
                "Single isolated extreme value", highlight = 25)
pd <- make_plot(t_all, y4, "Pure Random Noise",
                "Small unpredictable fluctuations throughout")

grid.arrange(pa, pb, pc, pd, nrow = 2)
Figure 17: Side-by-side comparison of the four irregular component sub-types
Tip

Key distinctions at a glance:

  • Turning point → slope changes permanently after one point.
  • Random shock → sharp deviation for a few periods, then the series recovers.
  • Outlier → exactly one data point is extreme; neighbours are normal.
  • Pure noise → small irregular deviations with no pattern throughout the whole series.
Tip

Quick visual check:

  • Does the trend permanently change direction? → Turning Point
  • Does the series spike then return to where it was? → Random Shock
  • Is one single point unusually extreme? → Outlier
  • Are there small unpatterned wiggles throughout? → Pure Noise

3 Relationship Among Components

3.1 Additive Effect

In the additive model, seasonal variation is constant — its magnitude is independent of the level of the series.

\[\hat{Y}_t = T_t + S_t + C_t + I_t\]

3.2 Multiplicative Effect

In the multiplicative model, seasonal variation grows proportionally with the level of the data series.

\[\hat{Y}_t = T_t \times S_t \times C_t \times I_t\]

3.3 Additive vs. Multiplicative

Show R Code 
elec_ts <- ts(electric$index, start = c(2015, 1), frequency = 12)
p_add <- autoplot(decompose(elec_ts, "additive")) +
  labs(title = "Additive Decomposition") + theme_bw(base_size = 10)
p_mul <- autoplot(decompose(elec_ts, "multiplicative")) +
  labs(title = "Multiplicative Decomposition") + theme_bw(base_size = 10)
grid.arrange(p_add, p_mul, ncol = 2)
Figure 18: Additive vs. multiplicative decomposition of the electricity output index
Feature Additive Multiplicative
Seasonal magnitude Constant Grows with level
Use when Seasonal swings roughly constant Seasonal swings widen as series grows
Linearisation Not needed Apply log transform
R function decompose(ts, type = "additive") decompose(ts, type = "multiplicative")
Tip

If seasonal fluctuations grow proportionally with the level of the series → use multiplicative. If they remain roughly constant → use additive.

3.3.1 Examples

(a) Example 1 — Airline passengers (top): seasonal swings widen as the series grows → multiplicative. Milk production (bottom): seasonal swings remain roughly constant → additive.
(b) Example 2 — CO₂ concentration (left) and Australian electricity production (right): both exhibit seasonal fluctuations that grow proportionally with the level → multiplicative.
(c) Example 3 — Additive seasonality (left): the band of variation around the trend (dashed lines) stays constant. Multiplicative seasonality (right): the band widens as the level rises.
Figure 19: Real-world examples illustrating additive and multiplicative seasonal patterns.

4 Measuring Performance

4.1 Data Partition

Part Purpose Rule of Thumb
Estimation (Training) Fit the forecasting model 80% of data
Evaluation (Test) Assess forecast accuracy 20% of data
Show R Code 
cpi   <- read.csv("data/CPI Malaysia.csv", check.names = FALSE)
cpits <- ts(cpi$`Consumer price index (2010 = 100)`,
            start = 1960, frequency = 1)

est_part <- head(cpits, 0.8 * length(cpits))
eva_part <- tail(cpits, 0.2 * length(cpits))

str(est_part)
 Time-Series [1:50] from 1960 to 2009: 21.3 21.3 21.3 22 21.9 21.8 22.1 23.1 23 22.9 ...
Show R Code 
str(eva_part)
 Time-Series [1:13] from 2010 to 2022: 100 103 105 107 110 ...
Show R Code 
n_est <- length(est_part)
n_eva <- length(eva_part)

df_cpi <- data.frame(
  Year  = as.numeric(time(cpits)),
  Value = as.numeric(cpits),
  Part  = c(rep("Estimation (80%)", n_est), rep("Evaluation (20%)", n_eva))
)

ggplot(df_cpi, aes(x = Year, y = Value, colour = Part)) +
  geom_line(linewidth = 0.9) +
  geom_vline(xintercept = as.numeric(time(cpits))[n_est],
             linetype = "dashed", colour = "black") +
  annotate("label",
           x = as.numeric(time(cpits))[n_est],
           y = max(as.numeric(cpits)) * 0.7,
           label = "Partition\ncut-off", size = 3.2,
           fill = "white", label.size = 0.3) +
  scale_colour_manual(values = c("Estimation (80%)" = "#4575b4",
                                 "Evaluation (20%)" = "#d73027")) +
  labs(title = "Malaysia CPI: Data Partition for Forecasting Evaluation",
       x = "Year", y = "CPI (2010 = 100)", colour = NULL) +
  theme_ts()
Figure 20: Data partition: 80% estimation, 20% evaluation — Malaysia CPI

4.2 Error Measures

The criterion used to differentiate between a poor and a good forecast model is called an error measure.

4.2.1 Mean Square Error (MSE)

\[\text{MSE} = \frac{1}{n} \sum_{t=1}^{n} (y_t - \hat{y}_t)^2\]

Advantage Easy to compute; penalises large errors more heavily
Disadvantage Highly sensitive to large forecast errors; not in original units

4.2.2 Mean Absolute Percentage Error (MAPE)

\[\text{MAPE} = \frac{1}{n} \sum_{t=1}^{n} \left| \frac{y_t - \hat{y}_t}{y_t} \right| \times 100\]

Advantage Easy to interpret (%); scale-independent — allows comparison across datasets
Disadvantage Undefined when \(y_t = 0\); not suitable for negative values

4.2.3 Mean Absolute Error (MAE)

\[\text{MAE} = \frac{1}{n} \sum_{t=1}^{n} |y_t - \hat{y}_t|\]

Advantage Robust to outliers; in original units; easy to compute
Disadvantage Treats all forecast errors equally regardless of magnitude
Note
  • The best model produces the lowest error measure value.
  • A truly good model gives consistently low values across multiple error measures.

4.3 Example: Malaysia CPI Data

4.3.1 Model 1 — Linear Trend

Show R Code 
n          <- length(cpits)
n_train    <- length(est_part)   # derived from actual 80% split
n_test     <- n - n_train
time_train <- 1:n_train
time_test  <- (n_train + 1):n

model1 <- lm(est_part ~ time_train)
summary(model1)

Call:
lm(formula = est_part ~ time_train)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.2836 -3.3028 -0.2493  2.3069 10.7824 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  8.82882    1.11813   7.896 3.16e-10 ***
time_train   1.68883    0.03816  44.255  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.894 on 48 degrees of freedom
Multiple R-squared:  0.9761,    Adjusted R-squared:  0.9756 
F-statistic:  1959 on 1 and 48 DF,  p-value: < 2.2e-16
Show R Code 
pred1      <- predict(model1, newdata = data.frame(time_train = time_test))
actual     <- as.numeric(eva_part)
predicted1 <- as.numeric(pred1)

mse1  <- mean((actual - predicted1)^2)
mape1 <- mean(abs((actual - predicted1) / actual)) * 100
mae1  <- mean(abs(actual - predicted1))

cat("=== Model 1: Linear Trend ===\n")
=== Model 1: Linear Trend ===
Show R Code 
cat("MSE  :", round(mse1,  4), "\n")
MSE  : 90.1995
Show R Code 
cat("MAE  :", round(mae1,  4), "\n")
MAE  : 9.2001
Show R Code 
cat("MAPE :", round(mape1, 4), "%\n")
MAPE : 7.959 %

4.3.2 Model 2 — Quadratic Trend

Show R Code 
model2 <- lm(est_part ~ time_train + I(time_train^2))
summary(model2)

Call:
lm(formula = est_part ~ time_train + I(time_train^2))

Residuals:
    Min      1Q  Median      3Q     Max 
-5.4353 -1.2541 -0.1506  1.3311  4.3873 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)     16.03953    1.05250  15.239  < 2e-16 ***
time_train       0.85682    0.09520   9.000 8.57e-12 ***
I(time_train^2)  0.01631    0.00181   9.014 8.17e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.382 on 47 degrees of freedom
Multiple R-squared:  0.9912,    Adjusted R-squared:  0.9909 
F-statistic:  2657 on 2 and 47 DF,  p-value: < 2.2e-16
Show R Code 
pred2      <- predict(model2, newdata = data.frame(time_train = time_test))
predicted2 <- as.numeric(pred2)

mse2  <- mean((actual - predicted2)^2)
mape2 <- mean(abs((actual - predicted2) / actual)) * 100
mae2  <- mean(abs(actual - predicted2))

cat("=== Model 2: Quadratic Trend ===\n")
=== Model 2: Quadratic Trend ===
Show R Code 
cat("MSE  :", round(mse2,  4), "\n")
MSE  : 21.6595
Show R Code 
cat("MAE  :", round(mae2,  4), "\n")
MAE  : 3.8183
Show R Code 
cat("MAPE :", round(mape2, 4), "%\n")
MAPE : 3.2507 %

4.3.3 Model Comparison

Show R Code 
library(kableExtra)

knitr::kable(
  data.frame(Model = c("Linear", "Quadratic"),
             MSE   = round(c(mse1, mse2),  4),
             MAE   = round(c(mae1, mae2),  4),
             MAPE  = round(c(mape1, mape2), 4)),
  col.names = c("Model", "MSE", "MAE", "MAPE (%)"),
  align = "c"
) %>%
  kable_styling() %>%
  row_spec(2, bold = TRUE, background = "#FFFFCC")
Table 2: Error measure comparison: linear vs. quadratic trend
Model MSE MAE MAPE (%)
Linear 90.1995 9.2001 7.9590
Quadratic 21.6595 3.8183 3.2507

4.3.4 4-Year Forecast

Show R Code 
h             <- 4
time_all      <- 1:n
time_forecast <- (n + 1):(n + h)

final_model     <- lm(cpits ~ time_all + I(time_all^2))
forecast_values <- predict(final_model,
                           newdata  = data.frame(time_all = time_forecast),
                           interval = "prediction")
print(forecast_values)
       fit      lwr      upr
1 132.2664 127.1072 137.4256
2 134.9212 129.7168 140.1255
3 137.6011 132.3471 142.8551
4 140.3063 134.9979 145.6146
Figure 21: Malaysia CPI: observed data and 4-year quadratic trend forecast

5 Summary

  • A time series is defined as \(\{y_t : t = 1, 2, \ldots, n\}\) — data indexed in ascending time order.
  • A time series plot interpretation should include minimum, maximum, outliers, and overall trends.
  • Four components: Trend, Seasonal, Cyclical, and Irregular/Random.
  • The Irregular component has four sub-types: Turning Point, Random Shock, Outlier, and Pure Noise.
  • Component relationships are modelled as Additive or Multiplicative.
  • Performance is measured via MSE, MAPE, and MAE on an 80/20 data split — lowest value wins.

6 References