38
D. Krpelík and T. Basu
Example 2.1 (continued)
d t u(t) = −au(t),
(2.1)
where a denotes a model parameter.
We will further explore how the predictions on the pollution level and the
decisions about the time for sending people to the area change by varying the quality
of knowledge about the model parameter a.
2.2.1 A Point Estimate
A common scientific practice is to identify unknown quantities by point estimates.
These represent our best guesses. Model parameters may sometimes be known
without a doubt. In other cases, such form might come from statistical procedures.
If the parameter a from Example 2.1 is regarded as known exactly (is identified as
a single real number), the prediction about the pollutant concentration at any nonnegative time t is given by the unique solution of equation (2.1),
u(t; a) = exp(−at),
(2.2)
which is, again, a single, precise value, u(t; a) ∈ R, for each time t ≥ 0.
Thus, what would we do if we were to make a decision with this predictive
model? To guarantee personnel safety, the pollution level must be below the critical
level u s . We are looking for the smallest t, which is the earliest time, for which these
criteria are met. This question can be translated into a mathematical optimisation
problem,
min
t≥0
t
s.t. u(t)≤ u s .
(2.3)
Our model gives us the answer:
• “The smallest safe time t for visiting the area is −
ln(u s )
a ”.
2.2.2 An Interval
Point estimates may be overly optimistic in many cases. For example, in manufacturing processes, the geometry of the final device can be specified only up to
known tolerances and allowed deviations. Similarly, due to discretisation of scales
on our measuring devices, even direct measurement actually provides only bounds
D. Krpelík and T. Basu
Example 2.1 (continued)
d t u(t) = −au(t),
(2.1)
where a denotes a model parameter.
We will further explore how the predictions on the pollution level and the
decisions about the time for sending people to the area change by varying the quality
of knowledge about the model parameter a.
2.2.1 A Point Estimate
A common scientific practice is to identify unknown quantities by point estimates.
These represent our best guesses. Model parameters may sometimes be known
without a doubt. In other cases, such form might come from statistical procedures.
If the parameter a from Example 2.1 is regarded as known exactly (is identified as
a single real number), the prediction about the pollutant concentration at any nonnegative time t is given by the unique solution of equation (2.1),
u(t; a) = exp(−at),
(2.2)
which is, again, a single, precise value, u(t; a) ∈ R, for each time t ≥ 0.
Thus, what would we do if we were to make a decision with this predictive
model? To guarantee personnel safety, the pollution level must be below the critical
level u s . We are looking for the smallest t, which is the earliest time, for which these
criteria are met. This question can be translated into a mathematical optimisation
problem,
min
t≥0
t
s.t. u(t)≤ u s .
(2.3)
Our model gives us the answer:
• “The smallest safe time t for visiting the area is −
ln(u s )
a ”.
2.2.2 An Interval
Point estimates may be overly optimistic in many cases. For example, in manufacturing processes, the geometry of the final device can be specified only up to
known tolerances and allowed deviations. Similarly, due to discretisation of scales
on our measuring devices, even direct measurement actually provides only bounds