Markov Assignment

1) Consider the following Markov model of a batter based on Gray, 2002.

a)

b) Let Txy be the temporal swing error for pitch y when expecting pitch x. For example, Tfs is the error when a slow pitch is thrown and a fast pitch is expected. Tff, Tsf, and Tss are defined likewise.

2) Modify the function markov_at_the_bat.

a) Implement the analytic equations of the expected temporal swing error for all possible three pitch sequences: FFF, FFS, FSF, FSS, SFF, SFS, SSF, SSS. For example, for the pitch sequence SFS, the equation is ((start_in_s*Tss + start_in_f*Tfs) + (start_in_s*Tsf + start_in_f*(1-as)*Tff + start_in_f*as*Tsf) + (start_in_s*(1-af)*Tss + start_in_s*af*Tfs + start_in_f*(1-as)*Tfs + start_in_f*as*(1-af)*Tss + start_in_f*as*af*Tfs)/3.

i) These equations should be in general terms of as, af, Tff, Tfs, Tsf, Tss, the probability of expecting a fast pitch on the first pitch, and the probability of a fast pitch (not all of these terms may be needed).

ii) Assume that the batter has a .65 probability of expecting a fast pitch on the first pitch.

iii) Assume that the probability of a fast pitch is always .55.

iv) Assume that [af as Tff Tfs Tsf Tss] = [.7 .5 10 60 40 20].

b) Fill in the code that simulates a three-pitch sequence.

i) Start with the same assumptions as in 2.a.

ii) What you do here is to

(1) Assume that the batter starts in some state selected probabilistically.

(2) Simulate a pitch, again selected probabilistically

(3) Calculate swing error from the parameters (for simplicity, assume a fixed error, e.g., Tff is a constant).

(4) Change states as necessary, probabilistically.

(5) Repeat for a total of 3 pitches.

(6) Repeat the simulation 1000 times.

(7) Calculate the average swing error and the standard deviation of the swing errors over all simulations.

3) Hand in

a) The analytic and simulated plots.

b) The numbers returned from markov_at_the_bat.

c) The equation for part 2.a. for FSF. (Code not necessary.)