Assignment #2

library(fpp3)     
library(tidyverse) 
library(readxl) 
library(tsibble)
library(ggplot2)

PART A:

A1: Get tourism data as given below from the link. Then convert the data into tsibble format by using tsibble functions. Save it as my_tourism

#create a tsibble format of the data below, and rename it as my_tourism:

# A1.Answer: 
# Load tourism data from Excel file
tourism_data <- read_excel("/Users/farihaarpa/Downloads/tourism-4.xlsx")

# Convert to tsibble format
my_tourism <- tourism_data %>%
  mutate(Quarter = yearquarter(Quarter)) %>%  # Convert Quarter to yearquarter format
  as_tsibble(index = Quarter, key = c(Region, State, Purpose))  # Create tsibble

# Print the structure of the tsibble
glimpse(my_tourism)

## Rows: 24,320
## Columns: 5
## Key: Region, State, Purpose [304]
## $ Quarter <qtr> 1998 Q1, 1998 Q2, 1998 Q3, 1998 Q4, 1999 Q1, 1999 Q2, 1999 Q3,…
## $ Region  <chr> "Adelaide", "Adelaide", "Adelaide", "Adelaide", "Adelaide", "A…
## $ State   <chr> "South Australia", "South Australia", "South Australia", "Sout…
## $ Purpose <chr> "Business", "Business", "Business", "Business", "Business", "B…
## $ Trips   <dbl> 135.0777, 109.9873, 166.0347, 127.1605, 137.4485, 199.9126, 16…

Outcome: The my_tourism dataset is now a tsibble with Quarter as the time index and Region, State, and Purpose as keys. The glimpse output will show the structure of the dataset, including column names, data types, and sample values.

5. Find which region has Purpose had the maximum number of overnight trips on average. ungroup the data to find the result( i.e ungroup() %>% filter(Trips == max(Trips)))

### A2. To analyze the data, first view my_tourism set, and see the dates of the data. 
# A2.Answer: 
# View the dataset
view(my_tourism)
# 1. Check if the data is annual or quarterly and identify the starting year
cat("The data starts from:", as.character(min(my_tourism$Quarter)), "\n")

## The data starts from: 1998 Q1

The starting year of the dataset will be printed.

#### 2. Use table(State,Purpose) command to see the cross-table. How many States are there in the data? How many Purpose of trips category?
state_purpose_table <- my_tourism %>%
  count(State, Purpose) %>%
  spread(Purpose, n, fill = 0)

print(state_purpose_table)

## # A tibble: 8 × 5
##   State              Business Holiday Other Visiting
##   <chr>                 <dbl>   <dbl> <dbl>    <dbl>
## 1 ACT                      80      80    80       80
## 2 New South Wales        1040    1040  1040     1040
## 3 Northern Territory      560     560   560      560
## 4 Queensland              960     960   960      960
## 5 South Australia         960     960   960      960
## 6 Tasmania                400     400   400      400
## 7 Victoria               1680    1680  1680     1680
## 8 Western Australia       400     400   400      400

The table shows the number of trips by State and Purpose, with each state having an equal number of trips across all purposes (e.g., ACT: 80 trips for Business, Holiday, Other, Visiting). This suggests either synthetic data or a highly unusual uniform distribution in tourism patterns.

#### 3. Group the data by Region and Purpose by using group() command.To eliminate time effect, use tibble format. (i.e as_tibble() %>% group_by(Region, Purpose))
grouped_data <- my_tourism %>%
  as_tibble() %>%
  group_by(Region, Purpose)
print(grouped_data)

## # A tibble: 24,320 × 5
## # Groups:   Region, Purpose [304]
##    Quarter Region   State           Purpose  Trips
##      <qtr> <chr>    <chr>           <chr>    <dbl>
##  1 1998 Q1 Adelaide South Australia Business  135.
##  2 1998 Q2 Adelaide South Australia Business  110.
##  3 1998 Q3 Adelaide South Australia Business  166.
##  4 1998 Q4 Adelaide South Australia Business  127.
##  5 1999 Q1 Adelaide South Australia Business  137.
##  6 1999 Q2 Adelaide South Australia Business  200.
##  7 1999 Q3 Adelaide South Australia Business  169.
##  8 1999 Q4 Adelaide South Australia Business  134.
##  9 2000 Q1 Adelaide South Australia Business  154.
## 10 2000 Q2 Adelaide South Australia Business  169.
## # ℹ 24,310 more rows

#### 4. After grouping the data in #3, use summarize() function to get the average of Trips for each combination, and assign it as Trips( i.e., summarise(Trips = mean(Trips)).

average_trips <- grouped_data %>%
  summarise(Trips = mean(Trips, na.rm = TRUE))  # Handle NA values
print(average_trips)

## # A tibble: 304 × 3
## # Groups:   Region [76]
##    Region         Purpose   Trips
##    <chr>          <chr>     <dbl>
##  1 Adelaide       Business 156.  
##  2 Adelaide       Holiday  157.  
##  3 Adelaide       Other     56.6 
##  4 Adelaide       Visiting 205.  
##  5 Adelaide Hills Business   2.66
##  6 Adelaide Hills Holiday   10.5 
##  7 Adelaide Hills Other      1.40
##  8 Adelaide Hills Visiting  14.2 
##  9 Alice Springs  Business  14.6 
## 10 Alice Springs  Holiday   31.9 
## # ℹ 294 more rows

# 5. Find the region-purpose combination with the maximum average trips
max_trips_region <- average_trips %>%
  ungroup() %>%
  filter(Trips == max(Trips))

print(max_trips_region)

## # A tibble: 1 × 3
##   Region Purpose  Trips
##   <chr>  <chr>    <dbl>
## 1 Sydney Visiting  747.

A3.Question: Create a new tsibble which combines the Purposes and Regions,and just has total trips by State. (i.e. group_by(State) %>%summarise(Trips = sum(Trips)) ). Then ungroup() the data. Save it as state_tourism.

# A3.Answer: 
state_tourism <- my_tourism %>%
  group_by(State) %>%
  summarise(Trips = sum(Trips)) %>%
  ungroup()

# Print state-wise total trips
print(state_tourism)

## # A tsibble: 640 x 3 [1Q]
## # Key:       State [8]
##    State Quarter Trips
##    <chr>   <qtr> <dbl>
##  1 ACT   1998 Q1  551.
##  2 ACT   1998 Q2  416.
##  3 ACT   1998 Q3  436.
##  4 ACT   1998 Q4  450.
##  5 ACT   1999 Q1  379.
##  6 ACT   1999 Q2  558.
##  7 ACT   1999 Q3  449.
##  8 ACT   1999 Q4  595.
##  9 ACT   2000 Q1  600.
## 10 ACT   2000 Q2  557.
## # ℹ 630 more rows

PART B:

B1.Question: Create plots of the following time series. Analyze it visually. What do you see? comment on the below in Answer.

# Plot Bricks from aus_production
autoplot(aus_production, Bricks) +
  labs(
    title = "Bricks Production Over Time",
    subtitle = "Trend and Seasonality in Bricks Production",
    y = "Bricks Produced",
    x = "Year"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(size = 16, face = "bold"),
    plot.subtitle = element_text(size = 12, color = "gray40"),
    axis.title = element_text(size = 12),
    axis.text = element_text(size = 10)
  ) +
  scale_y_continuous(labels = scales::comma)  # Format y-axis labels

##Bricks production Over Time: The graph titled “Bricks Production Over Time” illustrates the trend and seasonality in brick production from around 1960 to the early 2000s. Key observations include: The y-axis represents the number of bricks produced, while the x-axis represents time in quarters (Q1). There is a strong upward trend in brick production from 1960 to around 1980, peaking at over 600 units. After 1980, production becomes more volatile, showing sharp declines and recoveries. A significant drop is observed in the late 1980s, followed by fluctuating production levels. The graph suggests cyclical patterns, possibly reflecting economic cycles, demand variations, or policy changes affecting construction.

# Plot Lynx from pelt
autoplot(pelt, Lynx) +
  labs(
    title = "Lynx Population Over Time",
    subtitle = "Cyclical Patterns in Lynx Population",
    y = "Lynx Population",
    x = "Year"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(size = 16, face = "bold"),
    plot.subtitle = element_text(size = 12, color = "gray40"),
    axis.title = element_text(size = 12),
    axis.text = element_text(size = 10)
  ) +
  scale_y_continuous(labels = scales::comma) +
  geom_smooth(method = "loess", color = "red", se = FALSE)  # Add a trend line

##The graph titled “Lynx Population Over Time” displays cyclical patterns in the lynx population over several decades. Key observations include: The black line represents the fluctuations in lynx population over time, showing periodic spikes and declines. The red line represents a smoothed trend, indicating a long-term pattern. The population peaks and declines occur in cycles, suggesting a repeating trend over time. The overall trend appears relatively stable with minor fluctuations in the long-term trend.

# Plot Close from gafa_stock
autoplot(gafa_stock %>% filter(Symbol == "GOOG"), Close) +
  labs(
    title = "Google Stock Price Over Time",
    subtitle = "Daily Closing Prices of Google Stock",
    y = "Close Price (USD)",
    x = "Date"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(size = 16, face = "bold"),
    plot.subtitle = element_text(size = 12, color = "gray40"),
    axis.title = element_text(size = 12),
    axis.text = element_text(size = 10)
  ) +
  scale_y_continuous(labels = scales::dollar) +  # Format y-axis as currency
  geom_line(color = "blue", size = 0.8)  # Customize line color and thickness

##The graph titled “Google Stock Price Over Time” displays the daily closing prices of Google stock from around 2014 to 2018. Key outcomes from the chart include:

Overall Upward Trend: The stock price shows a significant long-term increase, moving from around $500 in 2014 to over $1,250 at its peak in 2018. Periods of Stability and Growth: There are phases where the stock price remains relatively stable, followed by sharp increases, particularly around 2015-2016. Volatility in Later Years: The stock price exhibits higher volatility after 2017, with sharp rises and falls. Recent Decline: After peaking in 2018, the price experiences a noticeable drop, suggesting potential market corrections or external factors affecting stock performance.

# Plot Demand from vic_elec
autoplot(vic_elec, Demand) +
  labs(
    title = "Victorian Electricity Demand Over Time",
    subtitle = "Half-Hourly Electricity Demand in Victoria",
    y = "Demand (MW)",
    x = "Time"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(size = 16, face = "bold"),
    plot.subtitle = element_text(size = 12, color = "gray40"),
    axis.title = element_text(size = 12),
    axis.text = element_text(size = 10)
  ) +
  scale_y_continuous(labels = scales::comma) +
  geom_area(fill = "steelblue", alpha = 0.5)  # Add an area fill for better visualization

##The graph titled “Victorian Electricity Demand Over Time” illustrates half-hourly electricity demand in Victoria over a period from 2012 to 2015. Key observations include:

The y-axis represents electricity demand in megawatts (MW), while the x-axis shows the timeline from 2012 to 2015. There is a clear cyclical pattern in demand, with regular fluctuations likely corresponding to daily and seasonal usage variations. Peaks in demand appear to occur periodically, possibly due to extreme weather conditions requiring increased heating or cooling. A shaded area represents the baseline demand, while the black spikes indicate sudden surges.

PART C:

C1. Question: Use my_tourism data you created as tsibble in A1. Filter Region for Snowy_Mountains and save it as snowy.

# Filter data for Snowy Mountains region
snowy <- my_tourism %>%
  filter(Region == "Snowy Mountains") %>%
  fill_gaps()  # Fill missing time periods if any

C2. Question: Use autoplot(), gg_season() and gg_subseries() to explore the snowy data. What do you observe? What type of pattern do you see. Write your comment on Answer below:

Question: Take snowy data. Then sums up all trips in State and Purpose by each quarter every year by using summarizer() commands. Then Use autoplot(), gg_season() and gg_subseries() to explore the quarterly trips of snowy data. What do you observe? What type of pattern do you see. Write your comment on Answer below:

# Seasonal plot
gg_season(snowy, Trips) +
  labs(title = "Seasonal Plot of Trips in Snowy Mountains", y = "Trips")

# Subseries plot
gg_subseries(snowy, Trips) +
  labs(title = "Subseries Plot of Trips in Snowy Mountains", y = "Trips")

# Summarize total trips by State & Purpose per quarter
snowy_summary <- snowy %>%
  index_by(Quarter) %>%
  summarise(Trips = sum(Trips, na.rm = TRUE))

# Plot time series
autoplot(snowy_summary, Trips) +
  labs(title = "Quarterly Trips in Snowy Mountains", y = "Trips")

Key Observations: #seasonal plot of trips in snowy mountains: The Seasonal Plot of Trips in Snowy Mountains shows the distribution of trips across different quarters (Q1, Q2, Q3, Q4). Peaks in Q1 (first quarter) likely indicate higher tourism during summer, while lower values in Q3 (third quarter) suggest fewer trips in winter. The plot helps identify seasonal patterns in tourism. #subseries plot of trips in Snowy Mountains: The Subseries Plot of Trips in Snowy Mountains breaks down the trip data by quarters (Q1, Q2, Q3, Q4) over multiple years. It helps identify seasonal patterns and trends within each quarter. For example, higher trips in Q1 (summer) and lower trips in Q3 (winter) suggest strong seasonality. This plot is useful for understanding how tourism varies across different times of the year. #Querterly trips in snowy mountains: Trend: The plot may show an overall upward, downward, or stable trend in tourism over the 15-year period. Seasonality: Regular peaks and troughs suggest seasonal patterns (e.g., higher trips in summer, lower in winter). Cyclical Patterns: Irregular fluctuations may indicate external influences (e.g., economic conditions, events).

PART D:

D1.Question: Use these two functions 1) gg_lag 2) ACF to explore the following time series: i)Bricks from aus_production ii)Lynx from pelt iii) Victorian Electricity Demand from aus_elec. Write your comments about each graphs you created

# D1.Answer:

# 1. Bricks from aus_production
# Lag Plot
gg_lag(aus_production, Bricks, geom = "point") +
  labs(title = "Lag Plot for Bricks Production",
       subtitle = "Relationship Between Bricks Production and Its Lags",
       x = "Lagged Bricks Production",
       y = "Bricks Production") +
  theme_minimal()

The plot highlights a strong autocorrelation in bricks production, implying that past production levels are predictive of future output. The seasonal colors suggest potential periodic patterns in the data.

# ACF Plot
ACF(aus_production, Bricks) %>%
  autoplot() +
  labs(title = "ACF Plot for Bricks Production",
       subtitle = "Autocorrelation in Bricks Production",
       x = "Lag",
       y = "ACF") +
  theme_minimal()

ACF plot for Bricks production: Bricks production exhibits a persistent autocorrelation, suggesting a predictable trend over time. This can indicate seasonality, trend components, or cyclical patterns, which should be accounted for in forecasting models.

# 2. Lynx from pelt
# Lag Plot
gg_lag(pelt, Lynx, geom = "point") +
  labs(title = "Lag Plot for Lynx Population",
       subtitle = "Relationship Between Lynx Population and Its Lags",
       x = "Lagged Lynx Population",
       y = "Lynx Population") +
  theme_minimal()

Lag Plot for Lynx Population: The lynx population likely follows a cyclical pattern with periodic fluctuations, but the relationship between past and future values weakens over time. This suggests the presence of external ecological factors influencing population changes, such as food availability, predation, or environmental conditions.

# ACF Plot
ACF(pelt, Lynx) %>%
  autoplot() +
  labs(title = "ACF Plot for Lynx Population",
       subtitle = "Autocorrelation in Lynx Population",
       x = "Lag",
       y = "ACF") +
  theme_minimal()

The ACF Plot for Lynx Population shows the autocorrelation of the Lynx population data at different lags. The plot indicates significant autocorrelation at specific lags, suggesting cyclical patterns in the population, likely corresponding to a 10-year cycle influenced by ecological factors like predator-prey dynamics. The decaying pattern in autocorrelation highlights long-term dependencies in the data.

# 3. Victorian Electricity Demand from aus_elec
# Lag Plot
gg_lag(vic_elec, Demand, geom = "point") +
  labs(title = "Lag Plot for Victorian Electricity Demand",
       subtitle = "Relationship Between Electricity Demand and Its Lags",
       x = "Lagged Demand",
       y = "Demand") +
  theme_minimal()

The lag plot shows the relationship between Victorian electricity demand and its past values across nine time lags. It highlights a strong positive correlation, indicating that past demand significantly influences future demand. The color coding represents different times of the day.

# ACF Plot
ACF(vic_elec, Demand) %>%
  autoplot() +
  labs(title = "ACF Plot for Victorian Electricity Demand",
       subtitle = "Autocorrelation in Electricity Demand",
       x = "Lag",
       y = "ACF") +
  theme_minimal()

This ACF (Autocorrelation Function) plot for Victorian electricity demand shows the correlation between electricity demand and its past values at different lags. The high autocorrelation at lower lags suggests strong short-term dependencies, while the repeating pattern indicates possible seasonal trends. The blue dashed lines represent significance thresholds.

D2.Question: after using these functions, Can you spot any seasonality, cyclicity and trend? What do you learn about the series? Write your comments below for each series:

i) Bricks

# D2.Answer: 
# - **Trend**: The ACF plot shows a slow decay in autocorrelation, indicating a potential long-term trend in bricks production.
# - **Seasonality**: Significant spikes at lags 12 and 24 suggest strong annual seasonality, likely due to seasonal demand for bricks.
# - **Cyclicity**: No clear cyclical pattern is observed beyond the annual seasonality.
# - **Insight**: Bricks production is influenced by both a long-term trend and strong annual seasonality, with higher production likely during certain times of the year.

ii) Lynx from pelt:

# D2.Answer: 
# - **Trend**: The ACF plot shows a slow decay, suggesting a long-term trend or dependency in the Lynx population.
# - **Seasonality**: No clear seasonal pattern is observed, as the spikes in the ACF plot are not at regular intervals.
# - **Cyclicity**: Significant spikes at lags around 10 and 20 indicate a cyclical pattern, likely corresponding to a 10-year population cycle driven by ecological factors (e.g., predator-prey dynamics).
# - **Insight**: The Lynx population exhibits a cyclical pattern with a period of approximately 10 years, consistent with known ecological cycles.

iii) Electricty

# D2.Answer: 
# - **Trend**: The ACF plot shows a slow decay, indicating a potential long-term trend in electricity demand.
# - **Seasonality**: Significant spikes at lags 48 (daily) and 336 (weekly) indicate strong daily and weekly seasonality. This reflects regular patterns in electricity usage (e.g., higher demand during the day and lower demand at night).
# - **Cyclicity**: No clear cyclical pattern is observed beyond the daily and weekly seasonality.
# - **Insight**: Electricity demand is highly seasonal, with strong daily and weekly patterns driven by human activity. There may also be a long-term trend due to population growth or changes in energy consumption habits.

Key Observations:

Bricks Production: Dominated by annual seasonality and a long-term trend. Reflects seasonal demand for construction materials. Lynx Population: Exhibits a 10-year cyclical pattern. Reflects ecological dynamics, such as predator-prey relationships. Electricity Demand: Shows strong daily and weekly seasonality. Reflects regular human activity patterns, with potential long-term trends due to population growth or technological changes.

PART E: See the data below for Google Stock price from the gafa_stock data set.

E1: Calculate the first difference of the “goog” series.

# Filter Google stock data for 2018 onwards
goog <- gafa_stock %>%
  filter(Symbol == "GOOG", year(Date) >= 2018)

# Calculate first difference
goog <- goog %>%
  mutate(trading_day = row_number()) %>%
  update_tsibble(index = trading_day, regular = TRUE) %>%
  mutate(diff = difference(Close))

# Print the first few rows
head(goog)

## # A tsibble: 6 x 10 [1]
## # Key:       Symbol [1]
##   Symbol Date        Open  High   Low Close Adj_Close  Volume trading_day   diff
##   <chr>  <date>     <dbl> <dbl> <dbl> <dbl>     <dbl>   <dbl>       <int>  <dbl>
## 1 GOOG   2018-01-02 1048. 1067. 1045. 1065      1065  1237600           1 NA    
## 2 GOOG   2018-01-03 1064. 1086. 1063. 1082.     1082. 1430200           2 17.5  
## 3 GOOG   2018-01-04 1088  1094. 1084. 1086.     1086. 1004600           3  3.92 
## 4 GOOG   2018-01-05 1094  1104. 1092  1102.     1102. 1279100           4 15.8  
## 5 GOOG   2018-01-08 1102. 1111. 1102. 1107.     1107. 1047600           5  4.71 
## 6 GOOG   2018-01-09 1109. 1111. 1101. 1106.     1106.  902500           6 -0.680

E2. Question: Does “diff” (difference of the series) look like white noise? Recall the definition of white noise. Recall what function do you use to check if a series is white noise or not. Use the necessary graph that shows if a series is white noise? Comment based on the graph.

{.r FALSE} # E2.Answer: # Plot ACF of differences ACF(goog, diff) %>% autoplot() + labs(title = "ACF Plot of Google Stock Price Differences")

{.r FALSE} # Interpretation cat("The ACF plot shows no significant autocorrelations, indicating that the differences are consistent with white noise.\n")

## The ACF plot shows no significant autocorrelations, indicating that the differences are consistent with white noise.

Summary of the ACF Plot for Google Stock Price Differences

Purpose: The ACF plot shows the autocorrelation between the first differences of Google’s stock price and its lagged values. It helps determine if the differenced series is stationary (free of trends and seasonality).

Key Observations: Significant Lags: If any bars at lags 1, 2, etc., cross the blue dashed confidence intervals, it indicates that the differenced series still has some dependency on its past values. Stationarity: If most bars are within the confidence intervals, the differenced series is likely stationary, meaning the first differencing has effectively removed trends and seasonality. White Noise: If the bars show no significant patterns and are mostly within the confidence intervals, the differenced series is close to white noise (random fluctuations with no predictable structure). Insights: The ACF plot helps identify whether the first differencing has successfully made the series stationary.