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21  Time Series Analysis

21.1 Getting Started

21.1.1 Load Packages

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

21.1.2 Load Data

Code
load(file = "./data/player_stats_weekly.RData")
load(file = "./data/player_stats_seasonal.RData")
load(url("https://osf.io/download/q6rjf/"))

21.2 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.

21.3 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 (archived at https://perma.cc/U5P6-2VWC).

21.3.1 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          6.02
2001-11-04         22.00
2001-11-11          7.18
2001-11-18          7.00
       ...              
2022-10-27         17.10
2022-11-06         15.20
2022-11-13         16.02
2022-11-27         18.04
2022-12-05         16.14
2022-12-11          8.12
2022-12-18         18.58
2022-12-25          9.34
2023-01-01         37.68
2023-01-08          7.36
Code
ts_peytonManning
           fantasyPoints
1999-09-12         13.06
1999-09-19         15.22
1999-09-26         28.56
1999-10-10         19.66
1999-10-17          9.10
1999-10-24         16.86
1999-10-31         18.52
1999-11-07         19.60
1999-11-14         14.18
1999-11-21         22.80
       ...              
2015-09-13          3.90
2015-09-17         19.24
2015-09-27         17.86
2015-10-04          6.32
2015-10-11          4.64
2015-10-18          6.60
2015-11-01         10.60
2015-11-08         13.24
2015-11-15        -10.60
2016-01-03          2.56
Code
ts_combined <- merge(
  ts_tomBrady,
  ts_peytonManning
)

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

21.3.2 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 21.1: 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 Peyone Manning.
Figure 21.2: Historical Fantasy Points by Game for Tom Brady and Peyone Manning.

21.3.3 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        11.944
2001-10-21        14.960
2001-10-28        15.528
2001-11-04        12.416
2001-11-11        13.984
2001-11-18        14.484
2001-11-25        10.568
2001-12-02        10.888
       ...              
2022-10-16        18.332
2022-10-23        15.892
2022-10-27        15.148
2022-11-06        16.012
2022-11-13        16.500
2022-11-27        14.704
2022-12-05        15.380
2022-12-11        14.044
2022-12-18        17.972
2022-12-25        16.216
Code
zoo::rollmedian(
  x = ts_tomBrady,
  k = 5)
           fantasyPoints
2001-09-30          4.34
2001-10-07          6.92
2001-10-14          6.92
2001-10-21         19.88
2001-10-28         19.88
2001-11-04          7.18
2001-11-11          7.18
2001-11-18          8.52
2001-11-25          7.18
2001-12-02          8.52
       ...              
2022-10-16         17.10
2022-10-23         15.20
2022-10-27         15.20
2022-11-06         16.02
2022-11-13         16.14
2022-11-27         16.02
2022-12-05         16.14
2022-12-11         16.14
2022-12-18         16.14
2022-12-25          9.34

21.3.4 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 21.3.

Code
acf(ts_tomBrady)
Autocorrelation Function (ACF) Plot of Tom Brady's Historical Fantasy Points by Game.
Figure 21.3: 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 = 3.8957, df = 1, p-value = 0.04841

21.3.5 Fit an Autoregressive Integrated Moving Average Model

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

Coefficients:
         ar1      ma1     ma2
      0.8007  -1.7041  0.7111
s.e.  0.1381   0.1527  0.1479

sigma^2 = 67.3:  log likelihood = -1176.57
AIC=2361.13   AICc=2361.25   BIC=2376.38
Code
forecast::auto.arima(ts_peytonManning)
Series: ts_peytonManning 
ARIMA(1,0,1) with non-zero mean 

Coefficients:
         ar1      ma1     mean
      0.8795  -0.7465  17.2154
s.e.  0.0861   0.1152   1.0481

sigma^2 = 62.99:  log likelihood = -871.18
AIC=1750.36   AICc=1750.53   BIC=1764.45
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
      0.6376  -0.0319  -0.4470  -0.1250  0.2078  -1.5598  0.6323  0.5117
s.e.  0.2066   0.2223   0.1676   0.0706  0.0667   0.2078  0.3149  0.3352
          ma4
      -0.5574
s.e.   0.1461

sigma^2 estimated as 63.92:  log likelihood = -1169.59,  aic = 2359.17

Training set error measures:
                    ME     RMSE      MAE       MPE     MAPE      MASE
Training set 0.6798849 7.983157 6.455929 -9.382029 62.84247 0.7287981
                    ACF1
Training set 0.001251287
Code
confint(arima_tomBrady)
         2.5 %      97.5 %
ar1  0.2326528  1.04261876
ar2 -0.4676647  0.40383415
ar3 -0.7755181 -0.11841000
ar4 -0.2633125  0.01336914
ar5  0.0771301  0.33839489
ma1 -1.9670637 -1.15248737
ma2  0.0150989  1.24940961
ma3 -0.1454023  1.16874068
ma4 -0.8438263 -0.27099009
Code
forecast::checkresiduals(arima_tomBrady)

    Ljung-Box test

data:  Residuals from ARIMA(5,1,4)
Q* = 3.1415, df = 3, p-value = 0.3703

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 21.4: 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
      -0.8409  -0.7345    0  0.0804  0.1507  -0.0637  -0.0167  -0.6063  -0.2275
s.e.   0.3120   0.2164    0  0.1024  0.0840   0.3197   0.2207   0.1873   0.1068

sigma^2 estimated as 63.88:  log likelihood = -1169.47,  aic = 2356.93

Training set error measures:
                    ME     RMSE      MAE       MPE     MAPE      MASE
Training set 0.6574145 7.980413 6.385955 -9.143247 62.73514 0.7208988
                     ACF1
Training set -0.007136381
Code
confint(arima_tomBrady_removeNonSigTerms)
          2.5 %      97.5 %
ar1 -1.45240974 -0.22948037
ar2 -1.15869850 -0.31030272
ar3          NA          NA
ar4 -0.12027200  0.28102794
ar5 -0.01390496  0.31531240
ma1 -0.69025865  0.56278078
ma2 -0.44934933  0.41594092
ma3 -0.97335654 -0.23917503
ma4 -0.43672063 -0.01826288
Code
forecast::checkresiduals(arima_tomBrady_removeNonSigTerms)

    Ljung-Box test

data:  Residuals from ARIMA(5,1,4)
Q* = 3.1606, df = 3, p-value = 0.3675

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 21.5: 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
      0.8795  -0.7465    17.2154
s.e.  0.0861   0.1152     1.0481

sigma^2 estimated as 62.24:  log likelihood = -871.18,  aic = 1750.36

Training set error measures:
                      ME     RMSE     MAE       MPE     MAPE      MASE
Training set 0.004669414 7.888962 6.28244 -89.05348 136.3438 0.7615486
                   ACF1
Training set 0.02492534
Code
confint(arima_peytonManning)
               2.5 %     97.5 %
ar1        0.7106760  1.0482992
ma1       -0.9722609 -0.5206973
intercept 15.1612132 19.2695648
Code
forecast::checkresiduals(arima_peytonManning)

    Ljung-Box test

data:  Residuals from ARIMA(1,0,1) with non-zero mean
Q* = 8.4306, df = 8, p-value = 0.3926

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 21.6: Model Summary of Autoregressive Integrated Moving Average Model fit to Peyton Manning’s Historical Performance by Game.

21.3.6 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       16.79157  6.545441 27.03770  1.121467 32.46168
337       21.31309 11.035944 31.59023  5.595554 37.03062
338       13.69041  3.371785 24.00903 -2.090564 29.47138
339       21.48471 10.979805 31.98961  5.418846 37.55057
340       17.19892  6.693126 27.70471  1.131696 33.26614
341       19.01892  8.370031 29.66780  2.732851 35.30498
342       18.72444  8.060966 29.38792  2.416061 35.03283
343       17.83641  7.157114 28.51571  1.503835 34.16899
344       18.62106  7.938214 29.30391  2.283057 34.95907
345       18.16347  7.480427 28.84651  1.825168 34.50177
Code
forecast_peytonManning
    Point Forecast     Lo 80    Hi 80      Lo 95    Hi 95
251       10.55320 0.4430843 20.66331 -4.9088854 26.01528
252       11.35607 1.1569232 21.55522 -4.2421805 26.95433
253       12.06219 1.7947024 22.32968 -3.6405790 27.76497
254       12.68322 2.3631750 23.00326 -3.0999254 28.46636
255       13.22940 2.8688924 23.58991 -2.6156293 29.07443
256       13.70976 3.3180615 24.10146 -2.1829722 29.60250
257       14.13223 3.7164703 24.54800 -1.7973016 30.06177
258       14.50379 4.0694543 24.93813 -1.4541504 30.46173
259       14.83057 4.3818907 25.27926 -1.1493076 30.81045
260       15.11797 4.6582086 25.57774 -0.8788563 31.11480

21.3.7 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 21.7: 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 21.8: Peyton Manning’s Historical and Projected Fantasy Points by Game.

21.4 Bayesian Mixed Models

The Bayesian longitudinal mixed models were estimated in Section 12.3.5.

21.4.1 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) 
  )

21.4.2 Generate Predictions

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

21.4.3 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)

21.4.4 Plot of Individuals’ Model-Implied Predictions

21.4.4.1 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()

ggplotly(
  plot_individualFantasyPointsByAgeQB,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)
Figure 21.9: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Quarterbacks.

21.4.4.2 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()

ggplotly(
  plot_individualFantasyPointsByAgeRB,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)
Figure 21.10: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Running Backs.

21.4.4.3 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()

ggplotly(
  plot_individualFantasyPointsByAgeWR,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)
Figure 21.11: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Wide Receivers.

21.4.4.4 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: Wide Receivers"
  ) +
  theme_classic()

ggplotly(
  plot_individualFantasyPointsByAgeTE,
  tooltip = c("age","fantasyPoints_bayesian","text","label")
)
Figure 21.12: Plot of Individuals’ Implied Trajectories of Fantasy Points by Age, from a Bayesian Generalized Additive Model, for Tight Ends.

21.5 Conclusion

21.6 Session Info

Code
sessionInfo()
R version 4.4.2 (2024-10-31)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.1 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_1.0.2         readr_2.1.5        
 [7] tidyr_1.3.1         tibble_3.2.1        tidyverse_2.0.0    
[10] plotly_4.10.4       ggplot2_3.5.1       rstan_2.32.6       
[13] StanHeaders_2.32.10 brms_2.22.0         Rcpp_1.0.13-1      
[16] forecast_8.23.0     xts_0.14.1          zoo_1.8-12         
[19] petersenlab_1.1.0  

loaded via a namespace (and not attached):
  [1] DBI_1.2.3            mnormt_2.1.1         gridExtra_2.3       
  [4] inline_0.3.20        rlang_1.1.4          magrittr_2.0.3      
  [7] tseries_0.10-58      matrixStats_1.4.1    compiler_4.4.2      
 [10] mgcv_1.9-1           loo_2.8.0            vctrs_0.6.5         
 [13] reshape2_1.4.4       quadprog_1.5-8       pkgconfig_2.0.3     
 [16] fastmap_1.2.0        backports_1.5.0      labeling_0.4.3      
 [19] pbivnorm_0.6.0       cmdstanr_0.8.1.9000  rmarkdown_2.29      
 [22] tzdb_0.4.0           ps_1.8.1             xfun_0.49           
 [25] jsonlite_1.8.9       psych_2.4.12         parallel_4.4.2      
 [28] lavaan_0.6-19        cluster_2.1.6        R6_2.5.1            
 [31] stringi_1.8.4        RColorBrewer_1.1-3   rpart_4.1.23        
 [34] lmtest_0.9-40        estimability_1.5.1   knitr_1.49          
 [37] base64enc_0.1-3      bayesplot_1.11.1     splines_4.4.2       
 [40] timechange_0.3.0     Matrix_1.7-1         nnet_7.3-19         
 [43] tidyselect_1.2.1     rstudioapi_0.17.1    abind_1.4-8         
 [46] yaml_2.3.10          timeDate_4041.110    codetools_0.2-20    
 [49] processx_3.8.4       curl_6.0.1           pkgbuild_1.4.5      
 [52] lattice_0.22-6       plyr_1.8.9           quantmod_0.4.26     
 [55] withr_3.0.2          bridgesampling_1.1-2 urca_1.3-4          
 [58] posterior_1.6.0      coda_0.19-4.1        evaluate_1.0.1      
 [61] foreign_0.8-87       RcppParallel_5.1.9   pillar_1.10.0       
 [64] tensorA_0.36.2.1     checkmate_2.3.2      stats4_4.4.2        
 [67] distributional_0.5.0 generics_0.1.3       TTR_0.24.4          
 [70] hms_1.1.3            mix_1.0-13           rstantools_2.4.0    
 [73] munsell_0.5.1        scales_1.3.0         xtable_1.8-4        
 [76] glue_1.8.0           emmeans_1.10.6       Hmisc_5.2-1         
 [79] lazyeval_0.2.2       tools_4.4.2          data.table_1.16.4   
 [82] mvtnorm_1.3-2        grid_4.4.2           mitools_2.4         
 [85] crosstalk_1.2.1      QuickJSR_1.4.0       colorspace_2.1-1    
 [88] nlme_3.1-166         fracdiff_1.5-3       htmlTable_2.4.3     
 [91] Formula_1.2-5        cli_3.6.3            viridisLite_0.4.2   
 [94] Brobdingnag_1.2-9    V8_6.0.0             gtable_0.3.6        
 [97] digest_0.6.37        farver_2.1.2         htmlwidgets_1.6.4   
[100] htmltools_0.5.8.1    lifecycle_1.0.4      httr_1.4.7

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