9.5 Stochastic Processes

9.5.1 Multiple Transitions

after three plate appearances

P_matrix_3 <- P_matrix %*% P_matrix %*% P_matrix

state	new_state	Prob
After 3 Plate Appearances
starting with bases empty and zero outs
000 0	3	0.372
000 0	100 2	0.241
000 0	110 1	0.081
000 0	010 2	0.074
000 0	000 2	0.053
000 0	001 2	0.029
000 0	101 1	0.028
000 0	100 1	0.026
000 0	000 1	0.019
000 0	011 1	0.019
000 0	010 1	0.016
000 0	111 0	0.011
000 0	110 0	0.007
000 0	001 1	0.007
000 0	101 0	0.004
000 0	000 0	0.004
000 0	100 0	0.003
000 0	011 0	0.003
000 0	010 0	0.002
000 0	001 0	0.001

table code

P_matrix_3 |>
  as_tibble(rownames = "state") |>
  filter(state == "000 0") |>
  pivot_longer(
    cols = -state, 
    names_to = "new_state", 
    values_to = "Prob" 
  ) |>
  filter(Prob > 0) |>
  arrange(desc(Prob)) |>
  gt() |>
  cols_align(align = "center") |>
  data_color(columns = Prob,
             palette = "inferno") |>
  fmt_number(columns = Prob,
             decimals = 3) |>
  tab_header(title = "After 3 Plate Appearances",
             subtitle = "starting with bases empty and zero outs")

9.5.2 Fundamental Matrix

\[N = (I - Q)^{-1}\]

Q <- P_matrix[-25, -25] #without 3-out states
N <- solve(diag(rep(1, 24)) - Q)

N_0000 <- round(N["000 0", ], 2)
head(N_0000, n = 6)

## 000 0 000 1 000 2 001 0 001 1 001 2 
##  1.05  0.75  0.60  0.01  0.03  0.05

Starting at the beginning of the inning (the “000 0” state)

the average number of times the inning will be in the “000 0” state is 1.05
the average number of times in the “000 1” state is 0.75
the average number of times in the “000 2” state is 0.6
etc.

sum(N_0000)

## [1] 4.27

the average number of plate appearances in a half-inning (before three outs) is 4.27.

9.5.3 Visit Frequency

the length of the remainder of the inning, on average, starting with each possible state

\[N\vec{1}\]

avg_num_plays <- N %*% rep(1, 24) |> t() |> round(2)
avg_num_plays[,1:8]

## 000 0 000 1 000 2 001 0 001 1 001 2 010 0 010 1 
##  4.27  2.87  1.46  4.33  2.99  1.53  4.34  2.93