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23  Time Series Analysis

23.1 Getting Started

23.1.1 Load Packages

Code
library("petersenlab")
library("xts")
library("zoo")
library("forecast")
library("brms")
library("rstan")
library("plotly")
library("tidyverse")

23.1.2 Load Data

Code
load(file = "./data/player_stats_weekly.RData")
load(file = "./data/player_stats_seasonal.RData")

The following code loads the Bayesian model object that was fit in Section 12.3.5.

Code
load(url("https://osf.io/download/q6rjf/")) # Bayesian model object

23.2 Overview of Time Series Analysis

Time series analysis is useful when trying to generate forecasts from longitudinal data. That is, time series analysis seeks to evaluate change over time to predict future values.

There many different types of time series analyses. For simplicity, in this chapter, we use autoregressive integrated moving average (ARIMA) models to demonstrate one approach to time series analysis. We also leverage Bayesian mixed models to generate forecasts of future performance and plots of individuals model-implied performance by age and position.

23.3 Autoregressive Integrated Moving Average (ARIMA) Models

Hyndman & Athanasopoulos (2021) provide a nice overview of ARIMA models. As noted by Hyndman & Athanasopoulos (2021), ARIMA models aim to describe how a variable is correlated with itself over time (autocorrelation)—i.e., how earlier levels of a variable are correlated with later levels of the same variable. ARIMA models perform best when there is a clear pattern where later values are influenced by earlier values. ARIMA models incorporate autoregression effects, moving average effects, and differencing.

ARIMA models can have various numbers of terms and model complexity. They are specified in the following form: \(\text{ARIMA}(p,d,q)\), where:

  • \(p =\) the number of autoregressive terms
  • \(d =\) the number of differences between consecutive scores (to make the time series stationary by reducing trends and seasonality)
  • \(q =\) the number of moving average terms

ARIMA models assume that the data are stationary (i.e., there are no long-term trends), are non-seasonal (i.e., there is no consistency of the timing of the peaks or troughs in the line), and that earlier values influence later values. This may not strongly be the case in fantasy football, so ARIMA models may not be particularly useful in forecasting fantasy football performance. Other approaches, such as exponential smoothing, may be useful for data that show longer-term trends and seasonality (Hyndman & Athanasopoulos, 2021). Nevertheless, ARIMA models are widely used in forecasting financial markets and economic indicators. Thus, it is a useful technique to learn.

Adapted from: https://rc2e.com/timeseriesanalysis [Long & Teetor (2019); archived at https://perma.cc/U5P6-2VWC].

23.3.1 Create the Time Series Objects

Code
weeklyFantasyPoints_tomBrady <- player_stats_weekly %>% 
  filter(
    player_id == "00-0019596" | player_display_name == "Tom Brady")

weeklyFantasyPoints_peytonManning <- player_stats_weekly %>% 
  filter(
    player_id == "00-0010346" | player_display_name == "Peyton Manning")

ts_tomBrady <- xts::xts(
  x = weeklyFantasyPoints_tomBrady["fantasyPoints"],
  order.by = weeklyFantasyPoints_tomBrady$gameday)

ts_peytonManning <- xts::xts(
  x = weeklyFantasyPoints_peytonManning["fantasyPoints"],
  order.by = weeklyFantasyPoints_peytonManning$gameday)

ts_tomBrady
           fantasyPoints
2000-11-23          0.24
2001-09-23          2.74
2001-09-30          6.92
2001-10-07          4.34
2001-10-14         22.56
2001-10-21         19.88
2001-10-28         10.02
2001-11-04         22.00
2001-11-11          8.18
2001-11-18          9.00
       ...              
2022-10-27         17.10
2022-11-06         15.20
2022-11-13         17.02
2022-11-27         18.04
2022-12-05         17.14
2022-12-11         10.12
2022-12-18         20.58
2022-12-25         11.34
2023-01-01         37.68
2023-01-08          7.36
Code
ts_peytonManning
           fantasyPoints
1999-09-12         15.06
1999-09-19         17.22
1999-09-26         29.56
1999-10-10         20.66
1999-10-17         10.10
1999-10-24         17.86
1999-10-31         18.52
1999-11-07         20.60
1999-11-14         15.18
1999-11-21         22.80
       ...              
2015-09-13          4.90
2015-09-17         20.24
2015-09-27         18.86
2015-10-04          8.32
2015-10-11          6.64
2015-10-18          9.60
2015-11-01         11.60
2015-11-08         15.24
2015-11-15         -6.60
2016-01-03          2.56
Code
ts_combined <- merge(
  ts_tomBrady,
  ts_peytonManning
)

names(ts_combined) <- c("Tom Brady","Peyton Manning")

23.3.2 Plot the Time Series

Code
plot(
  ts_tomBrady,
  main = "Tom Brady's Fantasy Points by Game")
Tom Brady's Historical Fantasy Points by Game.
Figure 23.1: Tom Brady’s Historical Fantasy Points by Game.
Code
plot(
  ts_combined,
  legend,
  legend.loc = "topright",
  main = "Fantasy Points by Game")
Historical Fantasy Points by Game for Tom Brady and Peyton Manning.
Figure 23.2: Historical Fantasy Points by Game for Tom Brady and Peyton Manning.

23.3.3 Rolling Mean/Median

Code
zoo::rollmean(
  x = ts_tomBrady,
  k = 5)
           fantasyPoints
2001-09-30         7.360
2001-10-07        11.288
2001-10-14        12.744
2001-10-21        15.760
2001-10-28        16.528
2001-11-04        13.816
2001-11-11        15.384
2001-11-18        15.084
2001-11-25        11.568
2001-12-02        11.888
       ...              
2022-10-16        18.332
2022-10-23        15.892
2022-10-27        15.348
2022-11-06        16.212
2022-11-13        16.900
2022-11-27        15.504
2022-12-05        16.580
2022-12-11        15.444
2022-12-18        19.372
2022-12-25        17.416
Code
zoo::rollmedian(
  x = ts_tomBrady,
  k = 5)
           fantasyPoints
2001-09-30          4.34
2001-10-07          6.92
2001-10-14         10.02
2001-10-21         19.88
2001-10-28         19.88
2001-11-04         10.02
2001-11-11         10.02
2001-11-18          9.00
2001-11-25          8.52
2001-12-02          9.00
       ...              
2022-10-16         17.10
2022-10-23         15.20
2022-10-27         15.20
2022-11-06         17.02
2022-11-13         17.10
2022-11-27         17.02
2022-12-05         17.14
2022-12-11         17.14
2022-12-18         17.14
2022-12-25         11.34

23.3.4 Autocorrelation

The autocorrelation function (ACF) plot depicts the autocorrelation of scores as a function of the length of the lag. Significant autocorrelation is detected when the autocorrelation exceeds the dashed blue lines, as is depicted in Figure 23.3.

Code
acf(ts_tomBrady)
Autocorrelation Function (ACF) Plot of Tom Brady's Historical Fantasy Points by Game.
Figure 23.3: Autocorrelation Function (ACF) Plot of Tom Brady’s Historical Fantasy Points by Game.
Code
Box.test(ts_tomBrady)

    Box-Pierce test

data:  ts_tomBrady
X-squared = 4.6744, df = 1, p-value = 0.03062

23.3.5 Fit an Autoregressive Integrated Moving Average Model

Using the forecast package (Hyndman et al., 2024; Hyndman & Khandakar, 2008):

Code
forecast::auto.arima(ts_tomBrady)
Series: ts_tomBrady 
ARIMA(2,1,2) 

Coefficients:
         ar1     ar2      ma1     ma2
      0.5827  0.0703  -1.5041  0.5161
s.e.  0.2615  0.0658   0.2577  0.2496

sigma^2 = 62.76:  log likelihood = -1164.4
AIC=2338.8   AICc=2338.99   BIC=2357.86
Code
forecast::auto.arima(ts_peytonManning)
Series: ts_peytonManning 
ARIMA(1,0,1) with non-zero mean 

Coefficients:
         ar1      ma1     mean
      0.8659  -0.7311  18.1529
s.e.  0.0973   0.1287   0.9575

sigma^2 = 57.94:  log likelihood = -860.72
AIC=1729.43   AICc=1729.59   BIC=1743.52
Code
arima_tomBrady <- arima(
  ts_tomBrady,
  order = c(5, 1, 4))

summary(arima_tomBrady)

Call:
arima(x = ts_tomBrady, order = c(5, 1, 4))

Coefficients:
         ar1      ar2     ar3      ar4     ar5      ma1     ma2     ma3
      0.3416  -0.0655  -0.205  -0.0959  0.1961  -1.2442  0.3689  0.2403
s.e.     NaN   0.2256     NaN   0.0475  0.0571      NaN     NaN     NaN
          ma4
      -0.3361
s.e.      NaN

sigma^2 estimated as 59.78:  log likelihood = -1158.36,  aic = 2336.72

Training set error measures:
                    ME     RMSE      MAE       MPE     MAPE     MASE
Training set 0.6879023 7.720194 6.223572 -21.21763 50.46479 0.732154
                      ACF1
Training set -0.0007899074
Code
confint(arima_tomBrady)
          2.5 %       97.5 %
ar1         NaN          NaN
ar2 -0.50779428  0.376718457
ar3         NaN          NaN
ar4 -0.18900813 -0.002859993
ar5  0.08427103  0.307929276
ma1         NaN          NaN
ma2         NaN          NaN
ma3         NaN          NaN
ma4         NaN          NaN
Code
forecast::checkresiduals(arima_tomBrady)

    Ljung-Box test

data:  Residuals from ARIMA(5,1,4)
Q* = 4.3247, df = 3, p-value = 0.2285

Model df: 9.   Total lags used: 12
Model Summary of Autoregressive Integrated Moving Average Model fit to Tom Brady's Historical Performance by Game.
Figure 23.4: Model Summary of Autoregressive Integrated Moving Average Model fit to Tom Brady’s Historical Performance by Game.
Code
arima_tomBrady_removeNonSigTerms <- arima(
  ts_tomBrady,
  order = c(5, 1, 4),
  fixed = c(NA, NA, 0, NA, NA, NA, NA, NA, NA))

summary(arima_tomBrady_removeNonSigTerms)

Call:
arima(x = ts_tomBrady, order = c(5, 1, 4), fixed = c(NA, NA, 0, NA, NA, NA, 
    NA, NA, NA))

Coefficients:
          ar1      ar2  ar3     ar4     ar5     ma1      ma2      ma3      ma4
      -0.9051  -0.7933    0  0.0987  0.1551  0.0137  -0.0185  -0.6539  -0.2529
s.e.   0.3144   0.1769    0  0.1013  0.0929  0.3217   0.2181   0.1528   0.1079

sigma^2 estimated as 59.38:  log likelihood = -1157.29,  aic = 2332.58

Training set error measures:
                    ME     RMSE      MAE       MPE     MAPE      MASE
Training set 0.6609654 7.694434 6.168512 -20.93448 50.25172 0.7256766
                     ACF1
Training set -0.006363576
Code
confint(arima_tomBrady_removeNonSigTerms)
          2.5 %      97.5 %
ar1 -1.52137219 -0.28875955
ar2 -1.13999746 -0.44652063
ar3          NA          NA
ar4 -0.09983263  0.29728144
ar5 -0.02703935  0.33727487
ma1 -0.61686843  0.64426701
ma2 -0.44596328  0.40905685
ma3 -0.95344054 -0.35438261
ma4 -0.46435785 -0.04145554
Code
forecast::checkresiduals(arima_tomBrady_removeNonSigTerms)

    Ljung-Box test

data:  Residuals from ARIMA(5,1,4)
Q* = 3.3999, df = 3, p-value = 0.334

Model df: 9.   Total lags used: 12
Model Summary of modified Autoregressive Integrated Moving Average Model fit to Tom Brady's Historical Performance by Game.
Figure 23.5: Model Summary of modified Autoregressive Integrated Moving Average Model fit to Tom Brady’s Historical Performance by Game.
Code
arima_peytonManning <- arima(
  ts_peytonManning,
  order = c(1, 0, 1))

summary(arima_peytonManning)

Call:
arima(x = ts_peytonManning, order = c(1, 0, 1))

Coefficients:
         ar1      ma1  intercept
      0.8659  -0.7311    18.1529
s.e.  0.0973   0.1287     0.9575

sigma^2 estimated as 57.24:  log likelihood = -860.72,  aic = 1729.43

Training set error measures:
                      ME     RMSE      MAE       MPE     MAPE      MASE
Training set 0.002008698 7.565751 5.932631 -101.5589 126.2153 0.7578856
                   ACF1
Training set 0.02467125
Code
confint(arima_peytonManning)
               2.5 %     97.5 %
ar1        0.6752778  1.0565343
ma1       -0.9833206 -0.4789367
intercept 16.2763161 20.0295532
Code
forecast::checkresiduals(arima_peytonManning)

    Ljung-Box test

data:  Residuals from ARIMA(1,0,1) with non-zero mean
Q* = 9.7829, df = 8, p-value = 0.2806

Model df: 2.   Total lags used: 10
Model Summary of Autoregressive Integrated Moving Average Model fit to Peyton Manning's Historical Performance by Game.
Figure 23.6: Model Summary of Autoregressive Integrated Moving Average Model fit to Peyton Manning’s Historical Performance by Game.

23.3.6 Generate the Model Forecasts

Code
forecast_tomBrady <- forecast::forecast(
  arima_tomBrady,
  level = c(80, 95)) # 80% and 95% confidence intervals

forecast_peytonManning <- forecast::forecast(
  arima_peytonManning,
  level = c(80, 95)) # 80% and 95% confidence intervals

forecast_tomBrady
    Point Forecast     Lo 80    Hi 80      Lo 95    Hi 95
336       16.89941  6.990778 26.80803  1.7454680 32.05334
337       21.84148 11.885926 31.79704  6.6157730 37.06719
338       14.62678  4.629175 24.62438 -0.6632356 29.91679
339       22.63793 12.473318 32.80254  7.0924968 38.18336
340       17.97384  7.804932 28.14275  2.4218376 33.52584
341       18.73072  8.416107 29.04533  2.9558825 34.50555
342       19.31420  8.980909 29.64748  3.5107975 35.11759
343       18.23659  7.893548 28.57964  2.4182709 34.05491
344       19.69349  9.341389 30.04559  3.8613185 35.52566
345       19.15502  8.802780 29.50726  3.3226357 34.98740
Code
forecast_peytonManning
    Point Forecast    Lo 80    Hi 80       Lo 95    Hi 95
251       12.03144 2.335537 21.72734 -2.79716238 26.86004
252       12.85229 3.068726 22.63586 -2.11038095 27.81497
253       13.56308 3.714290 23.41186 -1.49934216 28.62550
254       14.17855 4.281143 24.07595 -0.95822712 29.31533
255       14.71149 4.777786 24.64519 -0.48079979 29.90378
256       15.17297 5.212133 25.13380 -0.06081402 30.40675
257       15.57256 5.591435 25.55369  0.30774583 30.83738
258       15.91857 5.922259 25.91489  0.63052902 31.20662
259       16.21819 6.210500 26.22588  0.91274926 31.52363
260       16.47763 6.461418 26.49383  1.15915808 31.79610

23.3.7 Plot the Model Forecasts

Code
forecast::autoplot(forecast_tomBrady) + 
  labs(
    x = "Game Number",
    y = "Fantasy Points",
    title = "Tom Brady's Historical and Projected Fantasy Points by Game",
    subtitle = "(if he were to have continued playing additional seasons)"
  ) +
  theme_classic()
Tom Brady's Historical and Projected Fantasy Points by Game.
Figure 23.7: Tom Brady’s Historical and Projected Fantasy Points by Game.
Code
forecast::autoplot(forecast_peytonManning) + 
  labs(
    x = "Game Number",
    y = "Fantasy Points",
    title = "Peyton Manning's Historical and Projected Fantasy Points by Game",
    subtitle = "(if he were to have continued playing additional seasons)"
  ) +
  theme_classic()
Peyton Manning's Historical and Projected Fantasy Points by Game.
Figure 23.8: Peyton Manning’s Historical and Projected Fantasy Points by Game.

23.4 Bayesian Mixed Models

The Bayesian longitudinal mixed models were estimated in Section 12.3.5.

23.4.1 Prepare New Data Object

Code
player_stats_seasonal_offense_subset <- player_stats_seasonal %>% 
  dplyr::filter(position_group %in% c("QB","RB","WR","TE") | position %in% c("K"))

player_stats_seasonal_offense_subset$position[which(player_stats_seasonal_offense_subset$position == "HB")] <- "RB"

player_stats_seasonal_offense_subset$player_idFactor <- factor(player_stats_seasonal_offense_subset$player_id)
player_stats_seasonal_offense_subset$positionFactor <- factor(player_stats_seasonal_offense_subset$position)
Code
player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subset %>%
  filter(
    !is.na(player_idFactor),
    !is.na(fantasyPoints),
    !is.na(positionFactor),
    !is.na(ageCentered20),
    !is.na(ageCentered20Quadratic),
    !is.na(years_of_experience))

player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subsetCC %>% 
  filter(player_id %in% bayesianMixedModelFit$data$player_idFactor) %>% 
  mutate(positionFactor = droplevels(positionFactor))

player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subsetCC %>%
  group_by(player_id) %>% 
  group_modify(~ add_row(.x, season = max(player_stats_seasonal_offense_subsetCC$season) + 1)) %>% 
  fill(player_display_name, player_idFactor, position, position_group, positionFactor, team, .direction = "downup") %>% 
  ungroup

player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subsetCC %>% 
  left_join(
    player_stats_seasonal_offense_subsetCC %>% 
      filter(season == max(player_stats_seasonal_offense_subsetCC$season) - 1) %>% 
      select(player_id, age_lastYear = age, years_of_experience_lastYear = years_of_experience),
    by = "player_id") %>%
  mutate(
    age = if_else(season == max(player_stats_seasonal_offense_subsetCC$season), age_lastYear + 1, age), # increment age by 1
    ageCentered20 = age - 20,
    years_of_experience = if_else(season == max(player_stats_seasonal_offense_subsetCC$season), years_of_experience_lastYear + 1, years_of_experience)) # increment experience by 1

activePlayers <- unique(player_stats_seasonal_offense_subsetCC[c("player_id","season")]) %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season) - 1) %>% 
  select(player_id) %>% 
  pull()

inactivePlayers <- player_stats_seasonal_offense_subsetCC$player_id[which(player_stats_seasonal_offense_subsetCC$player_id %ni% activePlayers)]

player_stats_seasonal_offense_subsetCC <- player_stats_seasonal_offense_subsetCC %>% 
  filter(player_id %in% activePlayers | (player_id %in% inactivePlayers & season < max(player_stats_seasonal_offense_subsetCC$season) - 1)) %>% 
  mutate(
    player_idFactor = droplevels(player_idFactor) 
  )

23.4.2 Generate Predictions

Code
player_stats_seasonal_offense_subsetCC$fantasyPoints_bayesian <- predict(
  bayesianMixedModelFit,
  newdata = player_stats_seasonal_offense_subsetCC
)[,"Estimate"]

23.4.3 Table of Next Season Predictions

Code
player_stats_seasonal_offense_subsetCC %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season), position == "QB") %>%
  arrange(-fantasyPoints_bayesian) %>% 
  select(player_display_name, fantasyPoints_bayesian)
Code
player_stats_seasonal_offense_subsetCC %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season), position == "RB") %>%
  arrange(-fantasyPoints_bayesian) %>% 
  select(player_display_name, fantasyPoints_bayesian)
Code
player_stats_seasonal_offense_subsetCC %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season), position == "WR") %>%
  arrange(-fantasyPoints_bayesian) %>% 
  select(player_display_name, fantasyPoints_bayesian)
Code
player_stats_seasonal_offense_subsetCC %>% 
  filter(season == max(player_stats_seasonal_offense_subsetCC$season), position == "TE") %>%
  arrange(-fantasyPoints_bayesian) %>% 
  select(player_display_name, fantasyPoints_bayesian)

23.4.4 Plot of Individuals’ Model-Implied Predictions

23.4.4.1 Quarterbacks

Code
plot_individualFantasyPointsByAgeQB <- ggplot(
  data = player_stats_seasonal_offense_subsetCC %>% filter(position == "QB"),
  mapping = aes(
    x = round(age, 2),
    y = round(fantasyPoints_bayesian, 2),
    group = player_id)) +
  geom_smooth(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    se = FALSE,
    linewidth = 0.5,
    color = "black") +
  geom_point(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    size = 1,
    color = "transparent" # make points invisible but keep tooltips
  ) +
  labs(
    x = "Player Age (years)",
    y = "Fantasy Points (Season)",
    title = "Fantasy Points (Season) by Player Age: Quarterbacks"
  ) +
  theme_classic()

plotly::ggplotly(
  plot_individualFantasyPointsByAgeQB,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)
Figure 23.9: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Quarterbacks.

23.4.4.2 Running Backs

Code
plot_individualFantasyPointsByAgeRB <- ggplot(
  data = player_stats_seasonal_offense_subsetCC %>% filter(position == "RB"),
  mapping = aes(
    x = age,
    y = fantasyPoints_bayesian,
    group = player_id)) +
  geom_smooth(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    se = FALSE,
    linewidth = 0.5,
    color = "black") +
  geom_point(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    size = 1,
    color = "transparent" # make points invisible but keep tooltips
  ) +
  labs(
    x = "Player Age (years)",
    y = "Fantasy Points (Season)",
    title = "Fantasy Points (Season) by Player Age: Running Backs"
  ) +
  theme_classic()

plotly::ggplotly(
  plot_individualFantasyPointsByAgeRB,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)
Figure 23.10: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Running Backs.

23.4.4.3 Wide Receivers

Code
plot_individualFantasyPointsByAgeWR <- ggplot(
  data = player_stats_seasonal_offense_subsetCC %>% filter(position == "WR"),
  mapping = aes(
    x = age,
    y = fantasyPoints_bayesian,
    group = player_id)) +
  geom_smooth(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    se = FALSE,
    linewidth = 0.5,
    color = "black") +
  geom_point(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    size = 1,
    color = "transparent" # make points invisible but keep tooltips
  ) +
  labs(
    x = "Player Age (years)",
    y = "Fantasy Points (Season)",
    title = "Fantasy Points (Season) by Player Age: Wide Receivers"
  ) +
  theme_classic()

plotly::ggplotly(
  plot_individualFantasyPointsByAgeWR,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)
Figure 23.11: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Wide Receivers.

23.4.4.4 Tight Ends

Code
plot_individualFantasyPointsByAgeTE <- ggplot(
  data = player_stats_seasonal_offense_subsetCC %>% filter(position == "TE"),
  mapping = aes(
    x = age,
    y = fantasyPoints_bayesian,
    group = player_id)) +
  geom_smooth(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    se = FALSE,
    linewidth = 0.5,
    color = "black") +
  geom_point(
    aes(
      x = age,
      y = fantasyPoints_bayesian,
      text = player_display_name, # add player name for mouse over tooltip
      label = season # add season for mouse over tooltip
    ),
    size = 1,
    color = "transparent" # make points invisible but keep tooltips
  ) +
  labs(
    x = "Player Age (years)",
    y = "Fantasy Points (Season)",
    title = "Fantasy Points (Season) by Player Age: Tight Ends"
  ) +
  theme_classic()

plotly::ggplotly(
  plot_individualFantasyPointsByAgeTE,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)
Figure 23.12: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Tight Ends.

23.5 Conclusion

23.6 Session Info

Code
sessionInfo()
R version 4.5.0 (2025-04-11)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.2 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0

locale:
 [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
 [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
 [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
[10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   

time zone: UTC
tzcode source: system (glibc)

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] lubridate_1.9.4     forcats_1.0.0       stringr_1.5.1      
 [4] dplyr_1.1.4         purrr_1.0.4         readr_2.1.5        
 [7] tidyr_1.3.1         tibble_3.2.1        tidyverse_2.0.0    
[10] plotly_4.10.4       ggplot2_3.5.2       rstan_2.32.7       
[13] StanHeaders_2.32.10 brms_2.22.0         Rcpp_1.0.14        
[16] forecast_8.24.0     xts_0.14.1          zoo_1.8-14         
[19] petersenlab_1.1.5  

loaded via a namespace (and not attached):
  [1] DBI_1.2.3            mnormt_2.1.1         gridExtra_2.3       
  [4] inline_0.3.21        sandwich_3.1-1       rlang_1.1.6         
  [7] magrittr_2.0.3       multcomp_1.4-28      tseries_0.10-58     
 [10] matrixStats_1.5.0    compiler_4.5.0       mgcv_1.9-1          
 [13] loo_2.8.0            vctrs_0.6.5          reshape2_1.4.4      
 [16] quadprog_1.5-8       pkgconfig_2.0.3      fastmap_1.2.0       
 [19] backports_1.5.0      labeling_0.4.3       pbivnorm_0.6.0      
 [22] cmdstanr_0.9.0.9000  rmarkdown_2.29       tzdb_0.5.0          
 [25] ps_1.9.1             xfun_0.52            jsonlite_2.0.0      
 [28] psych_2.5.3          parallel_4.5.0       lavaan_0.6-19       
 [31] cluster_2.1.8.1      R6_2.6.1             stringi_1.8.7       
 [34] RColorBrewer_1.1-3   rpart_4.1.24         lmtest_0.9-40       
 [37] estimability_1.5.1   knitr_1.50           base64enc_0.1-3     
 [40] bayesplot_1.12.0     timechange_0.3.0     Matrix_1.7-3        
 [43] splines_4.5.0        nnet_7.3-20          tidyselect_1.2.1    
 [46] rstudioapi_0.17.1    abind_1.4-8          yaml_2.3.10         
 [49] timeDate_4041.110    codetools_0.2-20     processx_3.8.6      
 [52] curl_6.2.3           pkgbuild_1.4.7       lattice_0.22-6      
 [55] plyr_1.8.9           withr_3.0.2          quantmod_0.4.27     
 [58] bridgesampling_1.1-2 urca_1.3-4           posterior_1.6.1     
 [61] coda_0.19-4.1        evaluate_1.0.3       foreign_0.8-90      
 [64] survival_3.8-3       RcppParallel_5.1.10  pillar_1.10.2       
 [67] tensorA_0.36.2.1     checkmate_2.3.2      stats4_4.5.0        
 [70] distributional_0.5.0 generics_0.1.4       TTR_0.24.4          
 [73] hms_1.1.3            mix_1.0-13           rstantools_2.4.0    
 [76] scales_1.4.0         xtable_1.8-4         glue_1.8.0          
 [79] lazyeval_0.2.2       emmeans_1.11.1       Hmisc_5.2-3         
 [82] tools_4.5.0          data.table_1.17.2    mvtnorm_1.3-3       
 [85] grid_4.5.0           mitools_2.4          crosstalk_1.2.1     
 [88] QuickJSR_1.7.0       colorspace_2.1-1     nlme_3.1-168        
 [91] fracdiff_1.5-3       htmlTable_2.4.3      Formula_1.2-5       
 [94] cli_3.6.5            viridisLite_0.4.2    Brobdingnag_1.2-9   
 [97] V8_6.0.3             gtable_0.3.6         digest_0.6.37       
[100] TH.data_1.1-3        htmlwidgets_1.6.4    farver_2.1.2        
[103] htmltools_0.5.8.1    lifecycle_1.0.4      httr_1.4.7          
[106] MASS_7.3-65

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