Explain

Interpreting the Results

What does $K$ represent?

$K$ represents the number of "clusters" or distinct types of regions.

• $K=1$: Means the whole city was treated equally. The bombs fell randomly across the entire map with a single average rate $\lambda$. Think of it like rain falling uniformly over a city.
• $K=2$: Means there are two types of regions. Perhaps one "Target Zone" with a high $\lambda_{target}$ and one "Non-Target Zone" (accidental misses) with a low $\lambda_{miss}$.

Evidence of Targeting

Standard statistical theory suggests that for a purely random process (like uniformly dropping bombs), the number of hits per square should follow a single Poisson distribution.

1. London:

   • The counts (229 zeros, 211 ones...) match the mathematical prediction of a Single Poisson almost perfectly.
   • Running the EM algorithm with $K=2$ will likely result in two very similar $\lambda$ values, or one $\pi$ being very close to 0. This means the model doesn't need extra complexity to explain the data.
   • Interpretation: The Germans likely intended to target, but their guidance systems were not precise enough to hit specific squares, resulting in an effectively random distribution over London.

2. Antwerp:

   • Look at the "Zero" count (325) and the "High" count (21 with 5+).
   • A single Poisson with a low mean (to explain the many zeros) shouldn't have that many 5s.
   • A single Poisson with a high mean (to explain the 5s) shouldn't have that many zeros.
   • EM with $K=2$ will separate these into:
     1. A large group of squares with low $\lambda$ (the misses/outskirts).
     2. A small group of squares with high $\lambda$ (the actual target/port).
   • Interpretation: The accuracy was sufficient (or the target concentrated enough) to create a statistically distinct "Hot Zone" vs. the rest.