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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("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_offense %>% 
  filter(
    player_id == "00-0019596" | player_display_name == "Tom Brady")

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

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

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

ts_tomBrady
           fantasy_points
2000-11-23           0.24
2001-09-23           2.74
2001-09-30           6.92
2001-10-07           0.34
2001-10-14          22.56
2001-10-21          19.88
2001-10-28           8.02
2001-11-04          22.00
2001-11-11           4.18
2001-11-18           8.00
       ...               
2022-11-06          15.20
2022-11-13          16.02
2022-11-27          18.04
2022-12-05          17.14
2022-12-11          10.12
2022-12-18          16.58
2022-12-25          11.34
2023-01-01          37.68
2023-01-08           7.36
2023-01-16          22.04
Code
ts_peytonManning
           fantasy_points
1999-09-12          15.06
1999-09-19          16.40
1999-09-26          29.56
1999-10-10          19.66
1999-10-17          10.10
1999-10-24          17.86
1999-10-31          17.76
1999-11-07          17.76
1999-11-14          15.18
1999-11-21          22.00
       ...               
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
2016-01-17          10.78
2016-01-24          14.14
2016-02-07           3.64
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)
           fantasy_points
2001-09-30          6.560
2001-10-07         10.488
2001-10-14         11.544
2001-10-21         14.560
2001-10-28         15.328
2001-11-04         12.416
2001-11-11         13.984
2001-11-18         14.084
2001-11-25         10.568
2001-12-02         11.488
       ...               
2022-10-23         15.492
2022-10-27         14.748
2022-11-06         15.612
2022-11-13         16.700
2022-11-27         15.304
2022-12-05         15.580
2022-12-11         14.644
2022-12-18         18.572
2022-12-25         16.616
2023-01-01         19.000
Code
zoo::rollmedian(
  x = ts_tomBrady,
  k = 5)
           fantasy_points
2001-09-30           2.74
2001-10-07           6.92
2001-10-14           8.02
2001-10-21          19.88
2001-10-28          19.88
2001-11-04           8.02
2001-11-11           8.02
2001-11-18           8.52
2001-11-25           8.00
2001-12-02           8.52
       ...               
2022-10-23          15.20
2022-10-27          15.20
2022-11-06          16.02
2022-11-13          17.10
2022-11-27          16.02
2022-12-05          16.58
2022-12-11          16.58
2022-12-18          16.58
2022-12-25          11.34
2023-01-01          16.58

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 = 6.9664, df = 1, p-value = 0.008306

21.3.5 Fit an Autoregressive Integrated Moving Average Model

Code
forecast::auto.arima(ts_tomBrady)
Series: ts_tomBrady 
ARIMA(5,1,4) 

Coefficients:
         ar1      ar2      ar3      ar4     ar5      ma1     ma2      ma3
      0.4516  -0.4604  -0.1763  -0.0835  0.2286  -1.3632  0.8840  -0.1503
s.e.  0.2437   0.2276   0.3552   0.0585  0.0535   0.2497  0.4107   0.5232
          ma4
      -0.3315
s.e.   0.3413

sigma^2 = 59.26:  log likelihood = -1318.42
AIC=2656.85   AICc=2657.44   BIC=2696.3
Code
forecast::auto.arima(ts_peytonManning)
Series: ts_peytonManning 
ARIMA(1,0,1) with non-zero mean 

Coefficients:
         ar1      ma1     mean
      0.9122  -0.8173  17.4824
s.e.  0.0617   0.0801   0.9618

sigma^2 = 60.49:  log likelihood = -959.81
AIC=1927.62   AICc=1927.76   BIC=1942.11
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
      0.4516  -0.4604  -0.1763  -0.0835  0.2286  -1.3632  0.8840  -0.1503
s.e.  0.2437   0.2276   0.3552   0.0585  0.0535   0.2497  0.4107   0.5232
          ma4
      -0.3315
s.e.   0.3413

sigma^2 estimated as 57.87:  log likelihood = -1318.42,  aic = 2656.85

Training set error measures:
                    ME     RMSE      MAE       MPE     MAPE      MASE
Training set 0.6139671 7.597044 6.060271 -26.43586 54.72858 0.7239426
                     ACF1
Training set -0.005813919
Code
confint(arima_tomBrady)
          2.5 %      97.5 %
ar1 -0.02608874  0.92929800
ar2 -0.90644893 -0.01428465
ar3 -0.87244587  0.51981406
ar4 -0.19810559  0.03107464
ar5  0.12370355  0.33351988
ma1 -1.85261459 -0.87375464
ma2  0.07904646  1.68893613
ma3 -1.17576451  0.87522599
ma4 -1.00046565  0.33737979
Code
forecast::checkresiduals(arima_tomBrady)

    Ljung-Box test

data:  Residuals from ARIMA(5,1,4)
Q* = 2.2789, df = 3, p-value = 0.5166

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
      0.5193  -0.5543    0  -0.0923  0.2250  -1.4318  1.0416  -0.4116  -0.1650
s.e.  0.2241   0.2086    0   0.0579  0.0559   0.2309  0.3702   0.2138   0.0657

sigma^2 estimated as 57.92:  log likelihood = -1318.6,  aic = 2655.2

Training set error measures:
                    ME    RMSE     MAE       MPE    MAPE      MASE         ACF1
Training set 0.6227877 7.60082 6.07869 -26.33863 54.9103 0.7261428 -0.005557294
Code
confint(arima_tomBrady_removeNonSigTerms)
          2.5 %       97.5 %
ar1  0.07998006  0.958521477
ar2 -0.96317880 -0.145398938
ar3          NA           NA
ar4 -0.20567506  0.021095338
ar5  0.11546585  0.334451330
ma1 -1.88434856 -0.979298817
ma2  0.31604537  1.767195595
ma3 -0.83059876  0.007494026
ma4 -0.29370838 -0.036243035
Code
forecast::checkresiduals(arima_tomBrady_removeNonSigTerms)

    Ljung-Box test

data:  Residuals from ARIMA(5,1,4)
Q* = 2.3778, df = 3, p-value = 0.4978

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
      0.9122  -0.8173    17.4824
s.e.  0.0617   0.0801     0.9618

sigma^2 estimated as 59.84:  log likelihood = -959.81,  aic = 1927.62

Training set error measures:
                       ME     RMSE      MAE       MPE     MAPE      MASE
Training set -0.004427934 7.735376 6.029051 -99.44356 124.3119 0.7310316
                     ACF1
Training set -0.008441419
Code
confint(arima_peytonManning)
               2.5 %     97.5 %
ar1        0.7913291  1.0330331
ma1       -0.9744194 -0.6602769
intercept 15.5973175 19.3675413
Code
forecast::checkresiduals(arima_peytonManning)

    Ljung-Box test

data:  Residuals from ARIMA(1,0,1) with non-zero mean
Q* = 3.8482, df = 8, p-value = 0.8705

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
384       21.04468 11.295940 30.79342 6.1352694 35.95409
385       15.83560  6.048829 25.62238 0.8680246 30.80318
386       22.88606 13.050490 32.72162 7.8438557 37.92826
387       18.55059  8.525819 28.57536 3.2190255 33.88216
388       17.70447  7.678403 27.73053 2.3709241 33.03801
389       18.28265  8.159424 28.40587 2.8005125 33.76479
390       17.91802  7.758353 28.07768 2.3801515 33.45588
391       19.61025  9.448326 29.77218 4.0689260 35.15158
392       19.51992  9.353826 29.68602 3.9722189 35.06763
393       18.52265  8.355983 28.68932 2.9740740 34.07123
Code
forecast_peytonManning
    Point Forecast    Lo 80    Hi 80       Lo 95    Hi 95
278       12.73528 2.821995 22.64856 -2.42577939 27.89634
279       13.15217 3.194408 23.10993 -2.07691104 28.38125
280       13.53245 3.537830 23.52706 -1.75300026 28.81789
281       13.87933 3.854149 23.90451 -1.45286154 29.21152
282       14.19575 4.145208 24.24629 -1.17522778 29.56673
283       14.48438 4.412787 24.55598 -0.91879401 29.88756
284       14.74767 4.658587 24.83675 -0.68224942 30.17759
285       14.98783 4.884225 25.09144 -0.46430054 30.43996
286       15.20690 5.091228 25.32258 -0.26368781 30.67750
287       15.40674 5.281029 25.53245 -0.07919712 30.89267

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_offense %>% 
  filter(position_group %in% c("QB","RB","WR","TE"))

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

player_stats_seasonal_kicking_subset <- player_stats_seasonal_kicking %>% 
  filter(position == "K")

player_stats_seasonal_offense_subset <- bind_rows(
  player_stats_seasonal_offense_subset,
  player_stats_seasonal_kicking_subset
)

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(fantasy_points),
    !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 22.04.5 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-r0.3.20.so;  LAPACK version 3.10.0

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] rstan_2.32.6        StanHeaders_2.32.10 lubridate_1.9.3    
 [4] forcats_1.0.0       stringr_1.5.1       dplyr_1.1.4        
 [7] purrr_1.0.2         readr_2.1.5         tidyr_1.3.1        
[10] tibble_3.2.1        tidyverse_2.0.0     plotly_4.10.4      
[13] ggplot2_3.5.1       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       utf8_1.2.4           cmdstanr_0.8.1.9000 
 [22] rmarkdown_2.29       tzdb_0.4.0           ps_1.8.1            
 [25] xfun_0.49            jsonlite_1.8.9       psych_2.4.6.26      
 [28] parallel_4.4.2       lavaan_0.6-19        cluster_2.1.6       
 [31] R6_2.5.1             stringi_1.8.4        RColorBrewer_1.1-3  
 [34] rpart_4.1.23         lmtest_0.9-40        estimability_1.5.1  
 [37] knitr_1.49           base64enc_0.1-3      bayesplot_1.11.1    
 [40] splines_4.4.2        timechange_0.3.0     Matrix_1.7-1        
 [43] nnet_7.3-19          tidyselect_1.2.1     rstudioapi_0.17.1   
 [46] abind_1.4-8          yaml_2.3.10          timeDate_4041.110   
 [49] codetools_0.2-20     processx_3.8.4       curl_6.0.1          
 [52] pkgbuild_1.4.5       lattice_0.22-6       plyr_1.8.9          
 [55] quantmod_0.4.26      withr_3.0.2          bridgesampling_1.1-2
 [58] urca_1.3-4           posterior_1.6.0      coda_0.19-4.1       
 [61] evaluate_1.0.1       foreign_0.8-87       RcppParallel_5.1.9  
 [64] pillar_1.9.0         tensorA_0.36.2.1     checkmate_2.3.2     
 [67] stats4_4.4.2         distributional_0.5.0 generics_0.1.3      
 [70] TTR_0.24.4           mix_1.0-12           hms_1.1.3           
 [73] rstantools_2.4.0     munsell_0.5.1        scales_1.3.0        
 [76] xtable_1.8-4         glue_1.8.0           emmeans_1.10.5      
 [79] Hmisc_5.2-0          lazyeval_0.2.2       tools_4.4.2         
 [82] data.table_1.16.2    mvtnorm_1.3-2        grid_4.4.2          
 [85] mitools_2.4          crosstalk_1.2.1      QuickJSR_1.4.0      
 [88] colorspace_2.1-1     nlme_3.1-166         fracdiff_1.5-3      
 [91] htmlTable_2.4.3      Formula_1.2-5        cli_3.6.3           
 [94] fansi_1.0.6          viridisLite_0.4.2    Brobdingnag_1.2-9   
 [97] V8_6.0.0             gtable_0.3.6         digest_0.6.37       
[100] farver_2.1.2         htmlwidgets_1.6.4    htmltools_0.5.8.1   
[103] lifecycle_1.0.4      httr_1.4.7

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