(625i) Data Reconciliation with Inequality Constraints Induces Bias: A Cause for Concern?

Data
reconciliation comprises an important set of tools for data quality
improvement. By removing always-present random measurement noise, one seeks to
process the data so that the transformed data are an improved representation of
the measured variables. Most often, this involves projection, filtering, or
smoothing steps (e.g., Narasimhan & Jordache, 1999; Vachhani et al., 2005, 2006). Data reconciliation
can be based on mechanistic models (white box, Venkatasubramanian et al., 2003a), empirical models (black
box, Venkatasubramanian et al., 2003b,
or a combination of both (gray box). This work is focused on data
reconciliation methods that rely on mechanistic process understanding (white
box data reconciliation).

A majority
of the data reconciliation literature is concerned with the application of
equality constraints as a way to improve the accuracy of the available
measurements. In this case, data reconciliation is interpreted geometrically as
a projection to a plane or manifold. The application of linear and nonlinear
equality constraints for data reconciliation has been studied in great detail
already (Tamhane & Mah, 1985; Crowe, 1996; Veverka & Madron, 1997;
Romagnoli & Sanchez, 1999).

The
application of inequality constraints has been studied as well. An early
example is found in Narasimhan & Harikumar (1993) where knowledge about the
feasible range of values for a measured process state is accounted for.
Vachhani et al. (2006) proposed a
method for dynamic data reconciliation based on the unscented Kalman filter,
thereby enabling the application of inequality constraints into an online data
reconciliation method. More recently, special attention has been given to the
incorporation of prior knowledge about function and signal shapes to optimally
describe or reconcile time series data (Villez et al., 2013; Villez & Habermacher,  2016; Derlon et al., 2017; Vertis et al.,
2017; Masic et al, 2017; Srinivasan et
al., 2017). Most typical is that the data analysis is enhanced by incorporation
of prior knowledge about the signs of the first and second derivative of the
fitted function or the data series.

The
application of inequality constraints offers several advantages. For instance,
in Narasimhan & Harikumar (1993) and Srinivasan et al. (2017) the improved accuracy of the reconciled data is the
principal motivation to consider inequality constraints. In Villez &
Habermacher (2016), shape constraints lead to the formulation of a lack-of-fit
statistic. Vertis et al. (2016)
mainly motivate the application of shape constraints as a way to differentiate
time series data without risk of noise amplification (Bhatt et al., 
2012) in turn enabling a reasonable guess of parameter values in kinetic
models. A similar strategy to initialize parameter estimation is deployed in
Masic et al. (2017).

The data
reconciliation literature implicitly incorporates the idea that any prior
knowledge should be applied during the data reconciliation process. This is
based on the observation that application of knowledge-based equality and
inequality constraints invariably improves the accuracy of the reconciled data
as long as the applied constraints are correctly assumed for the
data-generating process. The impact of using reconciled data, e.g. during model
parameter estimation, has not been studied in detail yet. In this work it is
demonstrated that data reconciliation with inequality constraints can induce
significant bias during model parameter estimation.

2. Methods

The utility
of data reconciliation is studied by means of a simple model parameter
estimation problem. To this end, a batch experiment involving a single
equilibrium reaction with two species, A and B, is repeatedly simulated. The
simulated rate laws are linear in the consumed species concentrations for both
directions of the reaction. Consequentially, the net reaction can be described
by a kinetic rate law which exhibits two parameters: the reaction rate
coefficient (k1=1 mol/L.h) and the equilibrium parameter (k2=0.2). The
concentration of A is measured at regular intervals. The data collected in each
experiment are first reconciled based on a monotonicity constraint and then
used for parameter estimation. In parallel, the same data are also used for
parameter estimation without any data reconciliation. In both cases, the
initial process conditions and the kinetic rate law structure are assumed
known. This complete procedure is repeated 10000 times to investigate the
effects of data reconciliation in the parameter estimates.

3. Results

3.1 Simulation

Figure 1
displays the simulation results for a single batch experiment. One can see that
the concentration of species A decreases with time from its initial value (1
mol/L) towards its equilibrium value (0.2 mol/L). Noisy measurements of the
concentration of species A are obtained every 5 minutes during the 7-hour
experiment.

Figure 1: A
single batch experiment – Black line: simulated concentration; Blue circles:
Measurements before data reconciliation; Red crosses: Measurements after data
reconciliation.

3.2 Data reconciliation

Reconciled
measurements are obtained by computing concentrations that are close as
possible to the original measurements in the least-squares (WLS) sense while
satisfying isotonicity (non-strictly increasing). This monotonicity constraint
is implemented as a positivity constraint for the point-wise differences
between consecutive concentration measurements. Thus, the data reconciliation
problem is a convex quadratic program with linear inequality constraints.
Figure 1 displays the reconciled concentration measurements of species A. One
observes that the reconciled measurements are generally closer to their true
values and that –as expected- they increase with time.

Figure 2
shows the cumulative density of the obtained root mean squared error (RMSE)
with error defined as the deviation between a measurement and its true
noise-free value. As one can see, data reconciliation improves the obtained
accuracy.

Figure 2:
Accuracy of the measurements – Empirical cumulative density of the RMSE before
and after data reconciliation.

3.3 Parameter estimation

Both
kinetic rate law parameters are estimated simultaneously with both the
unreconciled and reconciled measurements and for each simulated experiment
separately. To this end, the parameter estimates are adjusted so that the
simulated concentrations are as close as possible to the (unreconciled or
reconciled) measurements in the least-squares sense. The results of this can be
seen in Figure 3. Individual parameter estimates are shown for the first 250
experiments. The results of the complete Monte Carlo simulation are summarized
by the mean parameter vector and the variance-covariance matrix computed from
all parameter estimates. Figure 3 shows the 99% confidence ellipsoids according
to an assumed multivariate normal distribution for the parameter estimates.
Most importantly, one can see that the application of data reconciliation leads
to a shift of the parameter estimates away from their ground truth values. In
contrast, fitting the model to the unreconciled data directly does not induce
such an effect and delivers parameters that are practically unbiased, i.e.
their average is very close to the true parameter values.

Figure 3:
Accuracy of the parameter estimates – Blue circles: Parameter estimates with
raw data; Blue dashed lines: mean and 99% confidence region for the parameter
estimates with raw data; Red crosses: Parameter estimates obtained with
reconciled data; Red full lines: mean and 99% confidence region for the
parameter estimates obtained with reconciled data.

4. Conclusions

Data
reconciliation with inequality constraints has been advocated as a generally
applicable method to improve the accuracy of experimental data. In this study,
it is shown with a simple simulation study that improved accuracy can come at a
price when using the reconciled data for their ultimate purpose. Indeed, the
parameter estimation study show that data reconciliation induces a biasing
effect during parameter estimation. This is explained by the fact that (i) the
reconciled data are distributed according to truncated multivariate normal
distribution and (ii) the least-squares parameter estimation procedure assumes
a multivariate normal distribution. So far, no data reconciliation method has
been proposed that automatically accounts for this discrepancy. This, in part,
explains why using the unreconciled data during parameter estimations leads to
the best parameter estimates.

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