The Procedure

Multivariate Online Filtering With Fixed Window Width

The filtering procedure described by Lanius and Gather (2007) estimates the signal in a time window (yt+i; i = −m, . . . , m) of fixed odd width n = 2m + 1 centered around time t. Since this means a delay of m time units for each estimation, this filter is more suitable for retrospective analyses. Here, we describe an online version of this filter which uses a moving time window (yt+i; i = −n + 1, . . . , 0) with a width n ∈ N which estimates the signal at the most recent time t.

To achieve a multivariate online signal estimate by the Trimmed Repeated Median-Least Squares procedure (TRM-LS), the following steps have to be performed within each time window (yt−n+1, . . . , yt) of length n and dimension k:

1. Use univariate RM regression to find the signal estimate ˆµt(j) and the slope ˆβt(j) according to (2.3) and (2.2) for each component j = 1, . . . , k, and combine these to the k-dimensional level and slope estimates

ˆ µ_t = µˆt(1), . . . , ˆµt(k)
0 and βˆ_t = βˆt(1), . . . , ˆβt(k)
0 .

3. Use a robust method to estimate the local error covariance matrix Σt∈ Rk×k based on the sample of residuals rt+i ∈ Rk, i = −n + 1, . . . , 0.

4. Determine St := i = −n + 1, . . . , 0 : r>t+iΣ −1

t rt+i ≤ dn

the set of time points within the window at which the residuals rt+i have a squared Mahalanobis distance w.r.t. the local covariance structure which is not larger than a specified value dn.

5. Perform multivariate Least Squares regression on the trimmed sample {(i, yt+i) : i ∈ St}

to obtain the signal and slope estimates ˆµT RM −LS_{t ∈ R}k and ˆβT RM −LS_{t ∈ R}k.

As a suitable estimator for the local error covariance matrix Σt in step 3,Lanius and Ga-

ther(2007) suggest to apply a slightly modified version of the fast computable orthogona- lised Gnanadesikan-Kettenring estimator (OGK) byMaronna and Zamar(2002) to the re- siduals rt+iin the window. This estimator is based on the fact that the covariance between two variables X and Y can be expressed as Cov(X, Y ) = (σ(X + Y )2− σ(X − Y )2_{) /4} where σ(·) denotes the standard deviation. Since the multivariate OGK estimator inherits the explosion breakdown point of the univariate method, used for estimating the standard deviation, a high breakdown method should be applied to guarantee robustness against outliers. In a comparison study, Lanius and Gather (2007) find the Qn scale estimator (Rousseeuw and Croux, 1993) to be a suitable candidate which possesses a maximum asymptotic breakdown point of 50% if the data are in general position.

Due to the discrete measurement scale of the considered intensive care data, window samples might contain collinear data. This might cause the estimate ˆσQn

t (·) to be close to, or even equal to zero. To prevent the singularity of the estimated covariance matrix

b

Σt (which is required for the inversion at step 4), Lanius and Gather (2007) propose to use a lower bound of ϑ = 0.02 for the univariate estimates of the standard deviation, i.e.

ˆ

σt(·) = max{ˆσQtn(·), ϑ} . (3.25) At step 4 an upper bound dnfor the squared Mahalanobis distance of each k-dimensional residual vector determines whether an observation is regarded as an outlier or not. If the residuals were independently normal, the scaled squared distances had a χ2_{-distribution.} Therefore, Lanius and Gather (2007) follow Maronna and Zamar (2002) in choosing a scaled χ2-quantile for ’hard’ outlier rejection, i.e.

dn =

χ2_k,α· med {di; i = −n + 1, . . . , 0} χ2

k,0.5

where χ2_k,α is the α-quantile of a χ2-distribution with k degrees of freedom and the term di = r>t+iΣ

−1

t rt+i, i ∈ {−n + 1, . . . , 0}, stands for the residual squared distances. Lani-

us and Gather (2007) choose this bound because for independent normal residuals the distribution of the squared distances asymptotically tends to a χ2_{-distribution.}

Multivariate Online Filtering With Adaptive Choice of the Window Width In the following let nmin and nmax specify the extreme values for the possible window widths, let nt(j) denote the window width used to evaluate the fit at time t for compo- nent j, and let nIt(j) denote the number of most recent residual signs which are considered

for testing the adequacy of the current signal estimate in the jth component, analogous to the notation used in Section 3.2, with j ∈ {1, . . . , k}.

Here, we propose a multivariate online filter with adaptive choice of the window width which replaces the univariate Repeated Median regression in the first step of the multi- variate TRM-LS procedure, described in the previous section, by the adaptive univariate RM regression proposed in Section 3.2.

For the first step of this multivariate adaptive online Repeated Median filter the following has to be carried out:

1. (a) Use the adaptive univariate RM procedure to find the signal estimate ˆµt(j), the slope ˆβt(j) and the window width nt(j) for each component j = 1, . . . , k, and use nt,0 = min{Nt−1 + 1, nmax} as the starting window width for all k components.

(b) Set the overall window width for all components to Nt= minnt(j) : j = 1, . . . , k .

(c) Re-estimate ˆµt(j) and ˆβt(j) by univariate RM regression using the window width Nt for all j = 1, . . . , k.

(d) Combine these values to the k-dimensional level and slope estimates ˆ µ_t = µˆt(1), . . . , ˆµt(k)
0 and βˆ_t = βˆt(1), . . . , ˆβt(k)
0 .

The remaining steps of the procedure correspond to steps 2 to 5 of the TRM-LS procedure described above, with the difference that the fixed window width n is replaced by the time- dependent overall window width Nt.

Similar to Section 3.2the first online estimation takes place at time t = nmin in a window of width Nt = nmin. For all other times t, the overall window width Nt is set to the

minimum of the individual window widths nt(j), j = 1, . . . , k, derived from the adaptive RM procedure in step 1.(a), to ensure that the local linearity assumption (3.23) holds for all components and thus, the remaining steps 2 to 5 can be applied.