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Introduction to Applied Time Series

Time series analysis involves studying data points collected or recorded over time. Its applications range from economics to biology, where understanding patterns, trends, and predicting future events is crucial.

Mathematically, a time series \(\left\{X_t\right\}\) is a sequence of random variables indexed by time \(t\). A central concept is stationarity. A time series is (weakly) stationary if its statistical properties do not change over time.

Formally, \(\left\{X_t\right\}\) is weakly stationary if: \(\mathbb{E}\left[X_t\right]=\mu, \quad \operatorname{Var}\left(X_t\right)=\sigma^2, \quad\) and \(\quad \operatorname{Cov}\left(X_t, X_{t+k}\right)=\gamma_k\) for all \(t\). Stationarity is crucial for reliable modeling and forecasting, as many models assume constant mean and variance over time.

Recap: Multiple Linear Regression

Multiple linear regression models a response variable \(Y\) as a linear function of predictors \(X_1, X_2, \ldots, X_p:\)

\[ Y=\beta_0+\beta_1 X_1+\beta_2 X_2+\cdots+\beta_p X_p+\varepsilon \]

where \(\varepsilon\) is the error term. Regression analysis serves as a foundation for understanding relationships between variables, which extends into time series when considering trends and seasonality.

Using `Tidyverts` in the Tidyverse Ecosystem

The Tidyverts collection integrates time series analysis into the tidyverse framework, allowing seamless data manipulation and visualization with familiar syntax.

Understanding Time Series Data

Time series data for a single entity over time can be represented as a tsibble, which is a tidy data structure that inherently understands time.

example1 <- tsibble(
1   year = 2015:2019,
2   y = c(123, 39, 78, 52, 110),
3   index = year
)

4str(example1)

1: Define a time index from 2015 to 2019.
2: Create corresponding observations for each year.
3: Construct a tsibble with year as the index.
4: Use str() to inspect the structure of the tsibble.

tbl_ts [5 × 2] (S3: tbl_ts/tbl_df/tbl/data.frame)
 $ year: int [1:5] 2015 2016 2017 2018 2019
 $ y   : num [1:5] 123 39 78 52 110
 - attr(*, "key")= tibble [1 × 1] (S3: tbl_df/tbl/data.frame)
  ..$ .rows: list<int> [1:1] 
  .. ..$ : int [1:5] 1 2 3 4 5
  .. ..@ ptype: int(0) 
 - attr(*, "index")= chr "year"
  ..- attr(*, "ordered")= logi TRUE
 - attr(*, "index2")= chr "year"
 - attr(*, "interval")= interval [1:1] 1Y
  ..@ .regular: logi TRUE

Working with Real Datasets

Example: Google Stock Data

1google <- read_csv("data/google.csv")
google %>%  
2  as_tsibble() %>%
3  autoplot(Close) +
4  ylab("USD")

1: Read Google’s stock data from a CSV file.
2: Convert the data frame into a tsibble.
3: Use autoplot() to plot the closing prices over time.
4: Label the y-axis as “USD” for clarity.

Figure 1: Daily closing stock prices of Google (in USD)

google %>%  
1  as_tsibble() %>%
2  autoplot(Volume)

1: Convert the Google dataset into a tsibble.
2: Plot the trading volume over time using autoplot().

Figure 2: Daily trading volume of Google shares

Example: Global Temperature Data

1astsa::gtemp_both %>%
2  as_tsibble() %>%
3  autoplot() +
4  labs(x = "Year", y = "Global Temperature Deviation (in Celsius)")

1: Use gtemp_both dataset containing historical temperature data.
2: Convert it into a tsibble format.
3: Create a time series plot with autoplot().
4: Label the axes to contextualize the data.

Figure 3: Global mean land–ocean temperature from 1850 to 2023, measured as deviations from the 1991-2020 average.