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Fig. 6. (a) Examples of the input data suitable for ANN application; (b) Coefficients of Discrete Wavelet Transform (DWT) of QRS complex for different
arrhythmias ([10], used with permission).

Use of Bayesian Inference in Water
Quality Forecasting
Application of nonlinear methods is not narrowed only to human being condition monitoring, interior investigation and
exploration. Observing close and distant neighborhoods,
one will notice that the required main ingredient of human
existence is water. There is plenty of water on Earth, as 71%
of its surface is covered with water. However, only about
0.01% of water on Earth is held in rivers, lakes, swamps and
clouds as fresh water. Maintenance of clear water reservoirs is the essential responsibility of administrative bodies
and requires continuous monitoring of water quality. Linear
methods of water quality estimation can hardly represent the
stochastic nature of measured parameters, and therefore, more
advanced, nonlinear approaches are anticipated [12].
Fresh water reservoirs may contain cyanobacteria that
are more commonly known as blue-green algae. Blooms of
the algae pose a significant threat for the environment, water treatment processes and for human health [13]. During
bloom, more chemicals should be used in water purification
to prevent contamination of the water system. Laboratory
methods with high-quality instrumentation are employed
to reliably quantify bacteria but they require a water sample
to be taken and delivered to the lab facility. One drawback
of such a method is significant time lag. In-situ measurements yield results immediately but require probes with a
fluorometer module that increases the cost of the remote
measurement system.
An alternative approach may be derived from the fact that
cyanobacteria are living organisms which depend on nutrients available in the water (potassium and nitrogen), dissolved
oxygen and light irradiance. This technique is based on the
assumption that these parameters are easier to measure and
more widely available. Some processes use these measurements as input data and forecast cyanobacteria amount. There
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are many algorithms that could be applied to solve this computational problem.
Estimation of the amount of cyanobacteria in freshwater
was implemented on the basis of Bayesian inference and provided forecasts five days in advance [14]. It was also used to
improve spatial estimates of the Particulate Matter concentration based on low accuracy sensors [15].
Bayesian inference is an approach that tries to answer the
question of how probable is something (an "event") when we
observe (measure) some evidence of that "event." From the
metrology viewpoint it has an interesting feature that is an intrinsic evaluation of measurement uncertainty. The method is
useful when direct measurement is problematic (e.g., costly,
infeasible, impractical) and measured value is particularly affected by systemic influences of random nature. [25]. In its
most basic form, it is given by:

P ( A| B) =

P ( B| A ) ∗ P ( A )
P ( B)

For those who have not used Bayes' approach we may analyze a simple example with artificial numbers. Let us assume
that during a regular day at the office, we sit in a way that we
cannot directly observe what happens outside the window.
Suddenly, there is less light in the room. So, we may start asking questions such as: Is there a storm coming? The probability
that a day becomes darker during the storm is P(B|A) = 0.8 as it
sometimes happens that there is shining sun during the storm.
The probability of a storm during that day thanks to ceilometer
measurements (altitude at which clouds are) is at the moment
P(A) = 0.05. The long-term probability that the amount of light
decreases during the day P(B) = 0.3 because regular clouds often do that as well without a storm. After simple calculation we
have the probability that a storm occurs P(A|B) = 0.13. However, the same evidence may support a different hypothesis
like (jokingly) that an alien spaceship as large as a city floats

IEEE Instrumentation & Measurement Magazine

June 2020

Instrumentation & Measurement Magazine 23-4

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