Subset Pruning

We have seen how retrospective pruning can be used to remove previous changepoint vectors from future considerations. However, supposing we are at some current time-

CHAPTER 4. MULTIVARIATE CHANGEPOINT DETECTION 102 point τ∗ _{within the algorithm, this method of pruning does not prune any of the}

cτ∗ ∈ ¯C_τ∗ which each have to be considered at τ∗. Pruning these vectors would

reduce the amount of vectors cτ∗ ∈ ¯C_τ∗ for which h_c

τ ∗(c) has to be calculated for

each c ∈ Cτ∗₋₁(c_τ∗). Within this section we introduce further theory which allows for

the pruning of such vectors at each time-point τ∗_{, which we refer to herein as subset}

pruning.

Before continuing, we define some new notation in order to accommodate this theory. We use fj(t) to denote the minimum cost from time 0 up to time t in variable

j, including the α penalties but not the β penalties. We exclude these because fj(t)

represents a univariate cost, whereas β represents a multivariate penalty. Also, recall that for some changepoint vector c ∈ Cn, M(c) is the number of changepoint locations

occurring in any variable up to and including those in c. Hence, for some changepoint vector (t1, t2, . . . , tp), we can decompose F (·) as follows:

F  (t1, t2, . . . , tp)  = p X j=1 fj(tj) + βM  (t1, t2, . . . , tp)  .

Further, for a given J ∈ {1, . . . , p}, we use ¯CJ

τ∗ to denote the distinct subsets of ¯C_τ∗

such that ¯C_τJ∗ contains only the c_τ∗ ∈ ¯C_τ∗ which have J variables changing at time

τ∗, so that Pp

j=1I(c

j

τ∗ = τ∗) = J. This can be expressed by

¯ C_τJ∗ =    cτ∗ ∈ ¯C_τ∗ : p X j=1 I(cjτ∗ = τ∗) = J    . (4.7.6)

Note that ¯C_τp∗ = {(τ∗, τ∗, . . . , τ∗)}. For ease of notation, we define P to be the set of

all variables, so that P = {1, . . . , p}.

The motivation behind subset pruning is the consideration of the following sce- nario. Suppose that we have some p-variate series X of length n, time-points w and

τ∗ such that τ∗ < w, and some cw ∈ ¯Cw. Suppose further that we make the assump-

tion that the minimum cost to cw from the changepoint vector (τ∗, τ∗, . . . , τ∗) is lower

than the minimum cost from all changepoint vectors cJ ∈ ¯CτJ∗, for some J ∈ P with

(τ∗_{, τ}∗_{, . . . , τ}∗_{) to c}

w is lower that the minimum cost from all ci ∈ ¯Cτi∗, for i < J, to

cw. If such a property holds true, then this would allow for the pruning of different

subsets of affected variables, depending on the number of variables they contain which are changing at τ∗_.

We will see in the following proposition that this characteristic does indeed hold under certain conditions. Before examining this result, it is necessary to introduce some further notation. For a given time-point τ∗ _{and changepoint vector c}

τ∗, define

Pτ∗(c_τ∗) to be the set of variable indices of c_τ∗ such that cj_τ∗ = τ∗, so that |P_τ∗(c_J)| = J

for each cJ ∈ ¯CτJ∗. That is,

Pτ∗(c_τ∗) =

n

j ∈ P : cj_τ∗ = τ∗

o

. (4.7.7)

Finally, for a given cτ∗ ∈ ¯CJ ∗

τ∗, for J < J∗ define the following set:

E_τJ∗(c_τ∗) = n c ∈ ¯C_τJ∗ : cj ≤ cj_τ∗ ∀ j ∈ P o , (4.7.8) so that EJ

τ∗(c_τ∗) is the set of previous time-point vectors which are ‘viable’ for being

changepoint vectors prior to cτ∗. Proposition 4.7.2 establishes that, under certain

conditions regarding the changepoint vectors with one variable changing at some time- point τ∗_{, then we can prune the changepoint vectors which have i variables changing}

at τ∗_.

Proposition 4.7.2. Suppose that for some J ∈ {1, . . . , p} and each cJ ∈ ¯CτJ∗, we

have for every cJ −1 ∈

n

E_τJ −1∗ (c_J) : cj_{J −1} = cj_J ∀ j ∈ P \ P_τ∗(c_J)} that

hcw(cJ) < hcw(cJ −1) (4.7.9)

for some future vector cw ∈ ¯Cw, where w > τ∗.

Suppose further that we have changepoint vectors {cJ −1,j₁∗, cJ −1,j₂∗, . . . , cJ −1,j_i∗} ∈

E_τJ −1∗ (c_J) such that for each x = 1, . . . , i, we have cj ∗ x J −1,j∗ x = tj ∗ x and c j∗ x J = τ ∗ _(with tj∗ x < τ ∗_{), and c}j J −1,j∗ x = c j J for all j ∈ {P \ Pτ∗(c_J)}.

CHAPTER 4. MULTIVARIATE CHANGEPOINT DETECTION 104

Then if it holds that (i − 1)M(cJ) ≥Pix=1M(cJ −1,j∗

x), we have

hcw(cJ) < hcw(cJ −i) (4.7.10)

for every cJ −i ∈ {EτJ −i∗ (c_J) : c_{J −i}j = cj_J ∀ j ∈ P \ P_τ∗(c_J)}, i = 2, . . . , J − 1.

Proof. See Appendix A.3 for a full proof.

Proposition 4.7.2 implies that we do not need to calculate any of the hcw(cJ −i) for

any cJ −i. Hence, these cJ −i can be ‘pruned’ from our considerations for cw. Otherwise,

it is not necessarily true that hcw(cJ) < hcw(cJ −i), and so we are not able to use such

an inequality for pruning purposes.