This article considers a robust Kalman filter from the M-estimation point of view. Pak (
The Kalman filter, named after Rudolf E. Kalman (1960), has been an important algorithm in the fields of control theory and time series analysis. The Kalman filter leads the estimators of the signal or the state of an state-space model. Its usefulness and versatility have prevailed with adequate performance when observations are made online. Recently, it has become a major tool for artificial intelligence and robotics utilizing big data (Thrun, 2002) as well as for space time series analysis (Lee and Kim, 2010; Lee,
The Kalman filter uses the mean squared error (MSE) as a criterion of optimality but this criterion exaggerates the magnitude of errors. Therefore, the estimates of the signals are heavily influenced by abnormal observations or outliers. There have been numerous attempts to make the Kalman filter robust against abnormal observations based on M-estimation methodology (Ruckdeschel, 2000; Gandhi and Mili, 2010).
In order to use M-estimating functions, we need the values of shaping constants such as ‘cut-off constant’, ‘bending constant’, or ‘tuning constant’. In the previous works by Ruckdeschel (2000) and by Gandhi and Mili (2010), these constants are usually fixed or predetermined while updating the estimates for the signals themselves. However, the M-estimating function proposed by Pak (1998) is actually based on data-driven shaping constants, so that those constants can be easily updated as an observation comes in or as the iteration continues. As a result, it is found out that the Kalman filter with the proposed M-estimating function performs very stably under contaminated situations.
It should be noted that this article is not actually talking about M-estimation on the Kalman filter, rather it uses the M-estimating function themselves to handle unusual observations. In order to carry out the Kalman filtering stably with unusual observations, this article concerns to utilize the M-estimating functions to treat those observations. Robustly estimating the Kalman filter is another difficult problem that needs to be solved.
This section is based on the references by Brockwell and Davis (1986) and by Kay (1993). Assume that the
where
where
The estimator of
can be sequentially obtained in the following manner:
Step 0. Initialization:
Step 1. Prediction:
Step 2. Mean squared error matrix for prediction:
Step 3. Kalman gain:
Step 4. Correction:
Step 5. Mean squared error matrix for correction:
M-estimation is a representative statistical method to estimate parameters robustly. The content in this section are based mainly on the books by Huber and Ronchetti (2009) and by Hampel
when the random samples {
The minimization problem is actually turned to solve
For example, a representative
for a given cutoff constant
However, Pak (1998) introduced a new type of M-estimating function as
where the
Let
where
is called the minimum
which is also equivalent to solve
because
The
and then an M-estimating function
For example, if
where the
For simplicity, we assume that the signals follow the scalar Gauss-Markov signal model (or state model);
where
where
We can summarize the algorithm for the scalar state-scalar observation Kalman filter according to the following steps.
The robust version of the Kalman filter can be proposed by replacing (4.3) by
in order to bound the influence of the one-stop prediction error. The signal estimate is then
Suppose that the data are from
The true signal (- - -), the ten examples of the signal estimates (· · ·), the mean of the all signal estimates (—) and an sample of observations (●) are plotted in Figure 2. We can observe that the
We have demonstrated to run the Kalman filter with a special M-estimating function as well as indicated that the estimated signal can cope with unusual observations. This article utilized the M-estimating functions to treat those observations in order to conduct the Kalman filtering stably with unusual observations. The robustly estimating the Kalman filter is another difficult problem that needs to be solved. In this article, only the scalar Kalman filter has been treated, though an idea how to extend the proposed methodology to the multivariate situation, but has to be fully studied in the future.
minimum squared error statistics when
Contamination | Min | 1st quarter | Median | Mean | 3rd quarter | Max |
---|---|---|---|---|---|---|
Without |
||||||
0% | 4.422 | 5.563 | 5.946 | 5.989 | 6.365 | 8.050 |
10% | 9.913 | 10.843 | 11.175 | 11.211 | 11.564 | 12.969 |
20% | 16.810 | 18.170 | 18.540 | 18.550 | 18.930 | 20.180 |
With |
||||||
0% | 1.015 | 1.780 | 2.060 | 2.058 | 2.314 | 3.162 |
10% | 1.438 | 2.294 | 2.564 | 2.585 | 2.859 | 3.942 |
20% | 1.615 | 2.558 | 2.892 | 2.946 | 3.274 | 4.995 |
With |
||||||
0% | 0.573 | 1.440 | 1.720 | 1.736 | 2.009 | 3.280 |
10% | 0.679 | 1.491 | 1.779 | 1.775 | 2.017 | 3.335 |
20% | 0.301 | 1.047 | 1.261 | 1.2897 | 1.526 | 2.794 |
bw.nrd = bandwidth by Scott (2009); MAD = median absolute deviance; bw.SJ = bandwidth by Sheather and Jones (1991).
minimum squared error statistics when
Contamination | Min | 1st quarter | Median | Mean | 3rd quarter | Max |
---|---|---|---|---|---|---|
Without |
||||||
0% | 4.422 | 5.563 | 5.946 | 5.989 | 6.365 | 8.050 |
10% | 9.913 | 10.843 | 11.175 | 11.211 | 11.564 | 12.969 |
20% | 16.810 | 18.170 | 18.540 | 18.550 | 18.930 | 20.180 |
With |
||||||
0% | 0.811 | 1.846 | 2.335 | 2.358 | 2.793 | 4.775 |
10% | 1.227 | 2.444 | 2.900 | 2.977 | 3.432 | 6.523 |
20% | 2.549 | 5.180 | 6.647 | 6.754 | 8.242 | 12.537 |
With |
||||||
0% | 0.540 | 1.872 | 2.396 | 2.436 | 2.954 | 5.614 |
10% | 0.508 | 1.399 | 1.843 | 1.945 | 2.288 | 7.145 |
20% | 0.537 | 1.381 | 1.865 | 2.374 | 2.773 | 10.664 |
bw.nrd = bandwidth by Scott (2009); MAD = median absolute deviance; bw.SJ = bandwidth by Sheather and Jones (1991).