Page 230 - Special Topic Session (STS) - Volume 2

STS486 Alessandro F. et al.
() be the set of time steps corresponding to the − ℎ period. If Δ is
constant, than () = Δ( − 1) + 1: Δ].
Now, the maximum likelihood estimate for − ℎ period is given by
̂
̂
= () = arg maxlog ( () )

and the fitted values for (), () are obtained by
(~)
0
( |) = () ( ()| −;+ ).

(~)
In the last equation, −;+ are the data in a neighborhood of , excluded.

If +1 ≠ , we may have an artificial discontinuity of ̂ around the

border of the − ℎ and ( + 1) − ℎ periods. For change detection, this
discontinuity is not a concern as long as it does not propagate to the residuals.
In any case, this issue may be substantially mitigated using a smooth update
̂
of .

In particular, an approach similar to the recursive estimation of dynamical
models of Grillenzoni (1994 and 1997) considers a single step of a Newton-
Raphson algorithm at each Δ-period:
̂
′
′′ −1
= ̂ −1 + ( ) (4)

where ′ and ′′ are Jacobian and Hessian of log ( () ).

5. Fused LASSO change detection
Let us denote the change model for a fixed station and altitude ℎ, by

0
= +

where is the zero mean GP of Equation (2) and defines isolated,
0

temporary or permanent changes as discussed in Section 1. Formally, is a

deterministic, piece-wise constant function with change points at an unknown
number ≥ 0, of unknown change points , … , ∈ {1, … , }.
∗
∗
1

Once the fitted values of the previous section are available for the station
at , the following (sign changed) residuals are computed:

= ( , , ℎ) = ( , , ℎ) − ̂ ( , , ℎ)

for = 1, … , , ℎ ∈ [ℎ , … , ℎ ] . Now, conditionally on the GP parameter
1

estimates of the previous section, we have that
( ) = .

We than test for changes the station in using the fused LASSO approach,

Tibshirani et al. (2005). In particular, we assume that
= +

where ≡ (0, ).
2

The model (5) has parameters , plus = ( ). Hence, we regularise
2

the estimation using the following penalised criterion:

2
∑( − ) + ∑| | + ∑| − −1 |
1

2

=1 =1 =2
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