Exercise 4.15 - Sequential(recursive) updating of covariance matrix

Answers

For question (a), note that:

n 𝐂_{n + 1} - (n - 1) 𝐂_{n} = \sum_{i = 1}^{n + 1} (𝐱_{i} - 𝐦_{n + 1}) {(𝐱_{i} - 𝐦_{n + 1})}^{T} - \sum_{i = 1}^{n} (𝐱_{i} - 𝐦_{n}) {(𝐱_{i} - 𝐦_{n})}^{T} .

Making use of:

𝐦_{n + 1} = \frac{n 𝐦_{n} + 𝐱_{n + 1}}{n + 1},

we have:

\begin{aligned} n 𝐂_{n + 1} - (n - 1) 𝐂_{n} & = 𝐱_{n + 1} 𝐱_{n + 1}^{T} - (n + 1) 𝐦_{n + 1} 𝐦_{n + 1}^{T} + n 𝐦_{n} 𝐦_{n}^{T} \\ = \frac{n}{n + 1} (𝐱_{n + 1} - 𝐦_{n}) {(𝐱_{n + 1} - 𝐦_{n})}^{T} . \end{aligned}

For question (b), the complexity is $𝒪 (d^{2})$ .

For question (c), plugging (4.281) directly into (4.279) yields (4.280).

For question (d), the complexity remains $𝒪 (d^{2})$ .

solour_lfq

2021-03-24 13:42