robust innovations

A robust innovation limits how much an unusual observation changes the state of a state space model.

In Kalman filtering, the time update is

with expected observation

The measurement update is

The innovation is new information from the observation. An outlier creates a large state update which is then propagated into future predictions.

A robust update uses

Robustness changes the measurement update. Persistence remains in ; for a scalar AR(1) state, .

raw innovations

The standard kalman filter uses

where denotes one scalar innovation. Its influence is unbounded:

This learns genuine changes quickly, but one bad observation can substantially change the state.

bounded innovations

Huber introduced bounded M-estimation in 1964:

It is linear near zero, bounded and monotone.

This limits outlier influence while continuing to learn structural changes. However, the clipping threshold must be chosen, and even an arbitrarily extreme observation moves the state by .

A smooth alternative is

robust exponential smoothing

Gelper, Fried and Croux (2010) applied bounded innovations to exponential and Holt–Winters smoothing).

Construct a cleaned observation:

Then apply ordinary exponential smoothing (can instead be applied for level, trend and seasonality updates):

Substitution gives

This is the Huber innovation applied to an exponential-smoothing state.

This prevents an observation outlier from becoming a persistent forecast change, but may delay adaptation to a genuine level shift.

score-based robust filtering

Masreliez (1975) introduced score-based non-Gaussian filtering).

Before observing , the filter has the predictive density

The score with respect to its predicted location is

It points in the direction that would make more likely, with magnitude measuring the strength of that evidence.

For a Gaussian predictive density,

the score is

Scaling by recovers the raw innovation:

For a scalar innovation ,

because .

This derives the update from the predictive distribution. Robustness depends on the chosen density. Masreliez filtering is also an approximation to the generally intractable non-Gaussian filter.

GAS and DCS

Creal, Koopman and Lucas (2013) introduced Generalised Autoregressive Score models). Harvey (2013) developed the equivalent Dynamic Conditional Score framework). GAS and DCS are two names for essentially the same score-driven model class.

Let be a time-varying parameter of the predictive density:

Calculate the score

then optionally scale it:

The parameter recursion is

Here controls persistence. In a scalar model, and :

When is the predicted location, a Gaussian score gives a raw innovation. A heavy-tailed score gives a robust innovation.

Student-t score innovations

Harvey and Luati (2014) used the Student-t score for robust filtering.

For one standardised innovation,

the Student-t log density is

Its rescaled location score is

For small innovations,

Its derivative is

so the update is largest at

After this point, larger innovations produce smaller updates:

For a GAS/DCS model, this score enters through .

The score is redescending, strongly rejecting isolated outliers. It can also reject genuine structural breaks: when the prediction is very wrong, its correction approaches zero.