Plots Trends, and Seasonal Variation

Chapter 1: Lesson 2

Learning Outcomes

Use technical language to describe the main features of time series data
  • Define time series analysis
  • Define time series
  • Define sampling interval
  • Define serial dependence or autocorrelation
  • Define a time series trend
  • Define seasonal variation
  • Define cycle
  • Differentiate between deterministic and stochastic trends
Plot time series data to visualize trends, seasonal patterns, and potential outliers
  • Plot a “ts” object
  • Plot the estimated trend of a time series by computing the mean across one full period

Preparation

  • Read Sections 1.1-1.4

Learning Journal Exchange (15 min)

  • Review another student’s journal
  • What would you add to your learning journal after reading your partner’s?
  • What would you recommend your partner add to their learning journal?
  • Sign the Learning Journal review sheet for your peer

Vocabulary and Nomenclature Matching Activity (5 min)

Check Your Understanding

Working with a partner, match the definitions on the left with the terms on the right.

Vocabulary Matching

  1. A figure with time on the horizontal axis and the value of a random variable on the vertical axis
  2. A systematic change in a time series that does not appear to be periodic
  3. Repeated pattern within each year (or any other fixed time period)
  4. Repeated pattern that does not correspond to some fixed natural period
  5. Observations in which values are related to lagged observations of the same variable
  6. Random trend that does not follow a discernible or predictable pattern
  7. Can be modeled with mathematical functions, facilitating the long-term prediction of the behavior
  1. Cycle
  2. Correlated (Serially Dependent) Data
  3. Deterministic Trend
  4. Seasonal Variation
  5. Stochastic Trend
  6. Time Plot
  7. Trend

Comparison of Deterministic and Stochastic Time Series (10 min)

Stochastic Time Series

The following app illustrates a few realizations of a stochastic time series.

  • If a stochastic time series displays an upward trend, can we conclude that trend will continue in the same direction? Why or why not?

Deterministic Time Series

The figure below illustrates realizations of a deterministic time series. The data fluctuate around a sine curve.

Class Activity: Importing Data and Creating a tsibble Object (5 min)

Recall the Google Trends data for the term “chocolate” from the last lesson. The cleaned data are available in the file chocolate.csv. Here are the first few rows of the csv:

https://byuistats.github.io/timeseries/data/chocolate.csv

Import the Data

Use the code below to import the chocolate data and convert it into a time series (tsibble) object. You can click on the clipboard icon in the upper right-hand corner of the box below to copy the code.

# load packages
if (!require("pacman")) install.packages("pacman")
pacman::p_load("tsibble", "fable",
               "feasts", "tsibbledata",
               "fable.prophet", "tidyverse",
               "patchwork", "rio")

# read in the data from a csv
chocolate_month <- rio::import("https://byuistats.github.io/timeseries/data/chocolate.csv")

# define the first date in the time series
start_date <- lubridate::ymd("2004-01-01")  

# create a sequence of dates, one month apart, starting with start_date
date_seq <- seq(start_date,
                start_date + months(nrow(chocolate_month)-1),
                by = "1 months")

# create a tibble including variables dates, year, month, value
chocolate_tibble <- tibble(
  dates = date_seq,
  year = lubridate::year(date_seq),        # gets the year part of the date
  month = lubridate::month(date_seq),      # gets the month
  value = pull(chocolate_month, chocolate) # gets the value of the ts 
)

# create a tsibble where the index variable is the year/month
chocolate_month_ts <- chocolate_tibble |>
  mutate(index = tsibble::yearmonth(dates)) |>
  as_tsibble(index = index)

# generate the ts plot
choc_plot <- autoplot(chocolate_month_ts, .vars = value) +
  labs(
    x = "Month",
    y = "Searches",
    title = "Relative Number of Google Searches for 'Chocolate'"
  ) +
  theme(plot.title = element_text(hjust = 0.5))

choc_plot

Explore R commands summarizing time series data

Check Your Understanding
  • What does each of the following R commands give us?
    • head(chocolate_month_ts, 1)
    • tail(chocolate_month_ts, 1)
    • guess_frequency(chocolate_month_ts$index)

Estimating the Trend: Annual Aggregation (10 min)

To help visualize what is happening with a time series, we can simply aggregate the data in the time series to the annual level by computing the mean of the observations in a given year. This can make it easier to spot a trend.

For the chocolate data, when we average the data for each year, we get:

Aggregation 
chocolate_annual_ts <- summarise(
    index_by(chocolate_month_ts, year), 
    value = mean(value)
  ) 

#chocolate_annual_ts
Table 
chocolate_annual_ts |>
  as.data.frame() |>
  concat_partial_table(nrow_head = 6, nrow_tail= 3, decimals = 4) |>
  display_table()
year value
2004 35.5
2005 41.75
2006 41.5833
2007 43.1667
2008 41.6667
2009 43.5
⋮ ⋮
2021 55.0833
2022 59.5
2023 60.1667

The first plot is the time series plot of the raw data, and the second plot is a time series plot of the annual means.

Show the code 
# monthly plot
mp <- autoplot(chocolate_month_ts, .vars = value) +
  labs(
    x = "Month",
    y = "Searches",
    title = "Relative Number of Google Searches for 'Chocolate'"
  ) +
  theme(plot.title = element_text(hjust = 0.5))

# yearly plot
yp <- autoplot(chocolate_annual_ts, .vars = value) +
  labs(
    x = "Year",
    y = "Searches",
    title = "Mean Annual Google Searches for 'Chocolate'"
  ) +
  scale_x_continuous(breaks = seq(2004, max(chocolate_month_ts$year), by = 2)) +
  theme(plot.title = element_text(hjust = 0.5))

mp / yp

If you want to superimpose these plots, it would make sense to align the mean value for the year with the middle of the year. Here is a plot superimposing the annual mean aligned with July 1 (in blue) on the values of the time series (in black).

Show the code 
chocolate_annual_ts <- summarise(
    index_by(chocolate_month_ts, year), 
    value = mean(value)
  ) |>
  mutate(index = tsibble::yearmonth( mdy(paste0("7/1/",year)) )) |>
  as_tsibble(index = index)

# combined plot
autoplot(chocolate_month_ts, .vars = value) +
  geom_line(data = chocolate_annual_ts, 
            aes(x = index, y = value), 
            color = "#56B4E9") +
  labs(
    x = "Month",
    y = "Searches",
    title = "Relative Number of Google Searches for 'Chocolate'"
  ) +
  theme(plot.title = element_text(hjust = 0.5))

Check Your Understanding
  • What do the annually-aggregated data tell us about the trend?
  • What do you observe in the trend?
  • What do you suspect is causing this trend?

Homework Preview (5 min)

  • Review upcoming homework assignment
  • Clarify questions

Homework

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homework_1_2.qmd

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