Visualizing trends and time series


36-315: Statistical Graphics and Visualization, Summer 2026

Trend and time series data

A time series is a sequence of data points indexed, listed, or graphed in chronological order. — Wikipedia


Main topic

  • What already happened?

  • Extract meaningful statistics and other characteristics of the data

  • What will happen in the future?

  • Predict future values based on previously observed values

Longitudinal data and time series structure

  • Consider a single observation measured across time

  • Time series data usually has the following structure:

Variable \(T_1\) \(T_2\) \(\dots\) \(T_J\)
\(X_1\) \(x_{11}\) \(x_{12}\) \(\dots\) \(x_{1J}\)
\(X_2\) \(x_{21}\) \(x_{22}\) \(\dots\) \(x_{2J}\)
\(\vdots\) \(\vdots\) \(\vdots\) \(\dots\) \(\vdots\)
\(X_P\) \(x_{P1}\) \(x_{P2}\) \(\dots\) \(x_{PJ}\)

  • With \(N\) observations, there are \(N\) of these matrices

  • Time intervals may be regularly or irregularly spaced

Example: statistics PhDs over time

library(tidyverse)
theme_set(theme_light())
stats_phd <- read_csv("https://raw.githubusercontent.com/qntkhvn/36-315-summer26/refs/heads/master/data/stats-phd.csv")
head(stats_phd)
# A tibble: 6 × 5
  broad_field                       major_field               field  year n_phds
  <chr>                             <chr>                     <chr> <dbl>  <dbl>
1 Life sciences                     Biological and biomedica… Biom…  2008    122
2 Mathematics and computer sciences Mathematics and statisti… Stat…  2008    303
3 Mathematics and computer sciences Mathematics and statisti… Math…  2008    140
4 Mathematics and computer sciences Mathematics and statisti… Math…  2008     63
5 Psychology and social sciences    Other social sciences     Stat…  2008     20
6 Education                         Education research        Educ…  2008     66
stats_phd_year <- stats_phd |> 
  group_by(year) |> 
  summarize(n_phds = sum(n_phds))
head(stats_phd_year)
# A tibble: 6 × 2
   year n_phds
  <dbl>  <dbl>
1  2008    870
2  2009    922
3  2010    896
4  2011    885
5  2012   1020
6  2013   1066

Time series plot

Scatterplot with n_phds on the y-axis and year on the x-axis + Add line to emphasize order + Manually set x-axis breaks to show every year

Code 
stats_phd_year |> 
  ggplot(aes(x = year, y = n_phds)) +
  geom_point() +
  geom_line() +
  scale_x_continuous(breaks = unique(stats_phd_year$year),
                     labels = unique(stats_phd_year$year))

Drop points to emphasize trend

Code 
stats_phd_year |> 
  ggplot(aes(x = year, y = n_phds)) +
  geom_line() +
  scale_x_continuous(breaks = unique(stats_phd_year$year),
                     labels = unique(stats_phd_year$year))

Fill area under the line

Code 
stats_phd_year |> 
  ggplot(aes(x = year, y = n_phds)) +
  geom_area(fill = "cyan") +
  scale_x_continuous(breaks = unique(stats_phd_year$year),
                     labels = unique(stats_phd_year$year))

Plotting multiple time series: use lines with colors

Code 
stats_phd |> 
  ggplot(aes(x = year, y = n_phds, color = field)) +
  geom_line() +
  scale_x_continuous(breaks = unique(stats_phd$year),
                     labels = unique(stats_phd$year))

Highlight specific lines

Code 
stats_phd |> 
  filter(!(field %in% c("Biometrics and biostatistics", "Statistics (mathematics)"))) |> 
  ggplot() +
  geom_line(aes(x = year, y = n_phds, group = field),
            color = "gray", linewidth = 1, alpha = 0.5) +
  geom_line(data = filter(stats_phd, field %in% c("Biometrics and biostatistics", "Statistics (mathematics)")),
            aes(x = year, y = n_phds, color = field),
            linewidth = 1.5, alpha = 1) +
  scale_x_continuous(breaks = unique(stats_phd$year),
                     labels = unique(stats_phd$year))

Other packages to check out

Stacked area charts



Characteristics of time series

  • Trends: Does the variable increase or decrease over time, on average?
  • Seasonality: Are there changes in the variable that regularly happen (e.g., every winter, every hour, etc.)?
  • Noise: Variation in the variable beyond average trends and seasonality

Moving average

  • Moving/rolling averages are a starting point for visualizing how a trend changes over time

  • Intuition

    • Divide the data into small subsets (“windows”)

    • Compute the average within each window

    • Connect the averages together to make a trend line

  • This is sometimes called a simple moving average
  • Have we seen this before?

Smoothing

  • Moving averages are extremely similar to loess smoothing

    • Moving average: average within a sliding window

    • Loess: weighted linear regression within a sliding window

  • Simple moving average: local constant regression (degree 0, intercept-only), rectangular kernel

  • Loess: local linear or quadratic regression, smooth kernel (e.g., tri-cube)

  • Loess can also be used for understand the overall trend of a time series

Computing a moving average

Formally, a moving average can be written as

\[\mu_k = \frac{1}{h}\sum_{t=k - h + 1}^k X_t\]

  • Large \(h\): smooth line; captures global trends

  • Small \(h\): jagged/volatile line; captures local trends

Time series visualization example

Data: Mauna Loa monthly atmospheric \(\text{CO}_2\) concentration 1959 to 1997

Code 
co2 <- read_csv("https://raw.githubusercontent.com/qntkhvn/36-315-summer26/refs/heads/master/data/co2.csv")
co2 |>
  ggplot(aes(x = obs_date, y = co2_val)) +
  geom_line() +
  labs(x = "Time index", y = "CO2 (ppm)")

Use ggseas package to plot moving averages

Code 
library(ggseas)
co2 |>
  ggplot(aes(x = obs_date, y = co2_val)) + 
  geom_line(color = "red") +
  stat_rollapplyr(width = 12, align = "right") +
  labs(x = "Year", y = "CO2 (ppm)", title = "Width = 12")

Time series shading

Code 
library(ggseas)
co2 |>
  ggplot(aes(x = obs_date, y = co2_val)) + 
  geom_line(color = "red") +
  stat_rollapplyr(width = 12, align = "right") +
  annotate(geom = "rect", fill = "gray", alpha = 0.3,
           xmin = as.Date("1980-01-01"), xmax = as.Date("1989-12-31"), ymin = -Inf, ymax = Inf) +
  labs(x = "Year", y = "CO2 (ppm)", title = "Width = 12")

Moving window

  • Setting width = 12 and align = "right" means for each point, plot the average of a moving window consisting of that point and the 11 points behind it

    • Since this is monthly data, this is the average of a year’s worth of data

    • Note that there is no trend line for the first 11 points on the left-hand side of the graph

  • Choose a width value that is “just right”

    • Want the trend to be smooth enough, but also don’t want to deviate from the data too much

Moving window

Moving window

Other moving averages

  • Cumulative moving average: The average at time \(k\) is the average of all points at and before \(k\). Mathematically:

\[\mu_k^{\textsf{(CMA)}} = \frac{1}{k} \sum_{t=1}^k X_t\]

  • Weighted moving average: Same as simple moving average, but different measurements get different weights for the average.

\[\mu_k^{\textsf{(WMA)}} = \frac{\sum_{t=k - h + 1}^k X_t \cdot w_t}{ \sum_{t=k - h + 1}^k w_t}\]

Lags

  • Time series data are not independent and identically distributed (iid)

    • Example: today’s temperature is correlated with yesterday’s temperature

  • Lags: a time offset used to compare observations separated by a fixed number of time steps

    • Lag 1: compare \(X_t\) with \(X_{t - 1}\) (i.e., the time series itself and the time series shifted back one time point)

    • Lag 2: compare \(X_t\) with \(X_{t - 2}\)

    • and so on…

  • Given time measurements \((X_1, X_2, X_3, X_4, X_5)\)

    • Lag 1 compares \((X_2, X_3, X_4, X_5)\) with \((X_1, X_2, X_3, X_4)\)

    • Lag 2 compares \((X_3, X_4, X_5)\) vs \((X_1, X_2, X_3)\).

Autocorrelation

  • Autocorrelation: correlation between a time series and a lagged version of itself

  • Define \(r_{\ell}\) as the correlation between a time series and lag \(\ell\) of that time series

    • Lag 1: \(r_1\) is correlation between \((X_2, X_3, X_4, X_5)\) and \((X_1,X_2,X_3,X_4)\)

    • Lag 2: \(r_2\) is correlation between \((X_3, X_4, X_5)\) and \((X_1,X_2,X_3)\)

    • and so on…

  • Common diagnostic: plot \(\ell\) on x-axis, \(r_{\ell}\) on y-axis

    • Reveals if correlations are significantly large/small for certain lags

Autocorrelation plots

Code 
library(ggfortify)
auto_corr <- acf(co2$co2_val, plot = FALSE)
autoplot(auto_corr)

Autocorrelation plots and seasonality

  • Global trends: very high positive autocorrelations at many lags; hard to distinguish trend from seasonality

  • Autocorrelations after removing global trend: Which lags have large positive/negative values?

Code 
wid12 <- co2 |>
  ggplot(aes(x = obs_date, y = co2_val)) +
  geom_line(color = "red") +
  stat_rollapplyr(width = 12, align = "right") +
  labs(x = "Year", y = "CO2 (ppm)")

co2 <- co2 |>
  mutate(moving_avg = zoo::rollapply(co2_val, width = 12, FUN = "mean", align = "right", fill = NA),
         resid = co2_val - moving_avg)

resid_plot <- co2 |>
  ggplot(aes(x = obs_date, y = resid)) +
  geom_line() +
  labs(x = "Year", y = "Residuals of CO2 (ppm)")

library(patchwork)
wid12 + resid_plot

Autocorrelation plots and seasonality

Code 
# ACF of the residuals (except the first 11 points, which are NA)
autoplot(acf(tail(co2$resid, -11), plot = FALSE))

Seasonal decomposition

  • STL decomposition (Seasonal decomposition of Time series by Loess)

  • Break down a time series into three characteristics: trend, seasonality, noise

  • Use ggsdc() (from ggseas) to visualize seasonal decomposition

    • observed: original time series

    • trend: loess curve; global trend

    • seasonality: another loess curve as the seasonality (observed - trend)

    • irregular: irregular/residual noise (observed - trend - seasonality)

    • In other words: trend + seasonality + irregular = observed

Seasonal decomposition

Code 
co2 |>
  ggsdc(aes(obs_date, co2_val), frequency = 12, method = "stl", s.window = 12) +
  geom_line() + 
  labs(x = "Year", y = "CO2 (ppm)")