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Are fluctuating measurement results
instable, drifting, nonstationary?
A survey on related terms in Metrology.
Karl H. Ruhm
Swiss Federal Institute of Technology (ETH), Zurich, Switzerland
Institute of Machine Tools and Manufacturing (IWF)
[email protected]
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ETH
2013 JOINT IMEKO TC1 - TC7 - TC13 SYMPOSIUM
September 4 – 6, 2013, Genova, Italy
Introductionary Paper: www.mmm.ethz.ch/dok01/e0001047.pdf
Slides: www.mmm.ethz.ch/dok/e0001046.pdf
15.08.2013
In everyday language,
Terminology
is not important,
however,
if we have to rely on it quantitatively in
Science and Technology,
it is important.
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Ambiguity and Misunderstanding
is avoided by
Systematic Structures
and
Consistent Terminology.
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Today, we have a discussion,
may be controversial,
on
selected terms, definitions, models
around the term
«Dynamics».
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Preview
0 Motivation
I Description (Model) of a Dynamical Process
II State of a Dynamical System
III Description (Model) of Signals
IV Instability, Drift, and Nonstationarity
V Conclusion
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0
Motivation
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There are two approaches to terminology,
the bottom up approach,
from everyday language, from history, from habits
in small fields, from individual, very special needs,
from background knowledge,
from specialised standards
• the usual approach,
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the top down approach,
systematical use of
tools of Signal and System and other Theories,
in order to gain a deeper insight
and a consistent concept
• the rare approach.
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Well Known Term
What does the adjective
«dynamical»
mean?
Are there
dynamical quantities and / or dynamical processes?
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A qualitative statement
in everyday language from NPL:
"Dynamic measurement …,
where (a)
physical quantity being measured
varies with time
and where this variation may have
significant effect on measurement result
… and the associated uncertainty."
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http://www.npl.co.uk/server.php?show=ConWebDoc.2131
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Another qualitative, more abstract statement
in everyday language:
A dynamical system
is the description of a process
in form of an abstract model,
which evolves
(changes its state)
in time and space.
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We attribute the term
«dynamical»
to such a modelled process.
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A quantitative statement:
If we are able to describe a process
by
differential (difference) equations
and / or
differential (difference) inequations
in time and space,
we call the developed mathematical model a
«dynamical system».
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We attribute the term
«dynamical»
to such a modelled process.
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What about other
well known and related terms
like
static
stationary
stable
drifting
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and so on?
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Some of these terms refer to signals
(models of real-world physical or other quantities)
and
some of them refer to systems
(models of real-world physical or other processes).
All terms are well defined properties
of signals or systems,
which we describe mathematically by
Signal and System Theory.
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Signals
are for example
models of temperatures, velocities, humidities,
but also
efficiencies, usefulness, measurement errors,
probability density functions,
etc.
Systems
are for example
models of sensors, amplifiers, modulators, tomographs,
but also
transfer response functions, cross correlation functions,
etc.
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I
Description
(Model)
of a
Dynamical Process
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Description (Model) of a Dynamical Process
The kernel of a dynamical system,
the intrinsic model of a dynamical process,
describes its dynamical properties concerning
continuous time t
or discrete time k respectively.
x (t) = f {x(t)}
An important Property of a Dynamical System is
«Stability»
It is determined exclusively
by the kernel of the dynamical system.
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Description (Model) of a Dynamical Process
x (t) = f {x(t)}
If we want to describe
relations between signals and systems,
we expand the kernel of the dynamical system
x (t) = f {x(t); u(t) }
x(0) = x 0
y(t) = g{x(t); u(t) }
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The important terms
controllability and observability
are properties of the input-output-description of
a dynamical system.
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Description (Model) of a Dynamical Process
x (t) = f {x(t)}
And, if we want to describe
sensitivities of a system to parameter
variations,
we expand the kernel even further
x (t) = f {x(t); u(t); p(t) }
x(0) = x 0
y(t) = g{x(t); u(t); p(t) }
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Since signals come from sources, called systems,
signal properties depend on system properties and
on parameters of the system indeed,
but they don´t have the same properties.
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Description (Model) of a Nondynamical Process
If we are able to describe a process exclusively
by algebraic and / or transcendent equations,
we call it
«nondynamical».
y(t) = g{u(t) }
(In the real world there is
no nondynamical process)
There may be a particular situation, a
«nondynamical, time variant system»,
y(t) = g{u(t); p(t) }
in contrast to the
Linear Time Invariant System (LTI)
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Description (Model) of a Dynamical Process
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II
State of a Dynamical System
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State of a Dynamical System
The
vector of
«State Signals»,
denoted in the kernel of the description as
x(t),
represents the
«State of a System»
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State of a Dynamical System
We say, as soon as input signals u(t) vary in time,
a stable dynamical system evolves in time.
We say, the dynamic system is in a
«Transient State».
This transient state may be
stationary or nonstationary
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State of a Dynamical System
As soon as all input signals u(t) are constant,
all derivatives in the differential equations are zero,
and the set of differential equations has degraded
to a set of
algebraic and / or transcendent equations.
0 = f {x; u }
y = g{x; u }
We say, the dynamic system is in a
«Static State»
(steady state, equilibrium state, rest state).
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This particular state is
constant.
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State of a Dynamical System
This is a correct statement:
A Dynamical System is
(momentarily) in a Static State.
The antonym of
«Dynamical System»
is
«Nondynamical System»
and not
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«Static System».
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State of a Dynamical System
As soon as the
probability density functions (pdf) of the input signals u(t)
are time independent,
the mean values of the derivatives in the
differential equations are zero,
and the set of differential equations concerning the mean
values has degraded to a
set of algebraic and / or transcendent equations.
We say, the dynamic system is in a
«Stationary State»
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This particular state is
stationary.
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III
Description (Model) of Signals
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Description (Model) of Signals
As we have seen, the term
«dynamical»
concerns
processes and systems
respectively.
It is not applicable to
time and space dependent
quantities and signals respectively.
(state signal, trajectory, transition, walk, motion, orbit, event)
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Description (Model) of Signals
Quantities and Signals
respectively
and their
characteristic values and functions
are
independent or time and space dependent
constant or transient
stationary or nonstationary
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deterministic or random
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Description (Model) of Signals
Quantities and Signals
respectively
and their
characteristic values and functions
are
not
stable or unstable
linear or nonlinear
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IV
Instability, Drift, and Nonstationarity
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Instability and Drift
Are
Instability and Drift
Synonyms?
Not at all!
A system may be stable and may nevertheless drift.
There exists
no quantitative definition or mathematical formalism
for an
"instable signal"!
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Instability Concerning a System
Qualitative
Once a system is instable, at least one output signal
will run away unbounded or in a limit cycle,
- even if all input signals remain bounded
and
- even if all system parameters remain bounded.
This property is noticed by output signals, indeed,
however, the definition is based on system properties.
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Instability Concerning a System
Quantitative
The stability of a linear dynamical system depends on the
structure of the system model
and on the
magnitudes of the parameters,
of the differential equations of the system:
A linear dynamical system is asymptotically stable, if all
poles (eigen-values) of the set of characteristic equations
of the differential equations have negative real parts and
are therefore located in the left half of the complex plane.
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Instability Concerning a System
Citation 1
"A process is said to be stable,
when all of the response parameters that we use
to measure the process
have both constant means and constant variances
over time,
and also have a constant distribution."
Obviously, not stability is meant here, but stationarity.
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NIST: Engineering Statistics Handbook
http://www.itl.nist.gov/div898/handbook/ppc/section4/ppc45.htm
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Instability Concerning a System
Citation 2
"Stability of a measuring instrument:
property of a measuring instrument,
whereby its metrological properties remain
constant in time"
Obviously, not stability is meant here, but stationarity.
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International Vocabulary of Metrology (VIM); Definition 4.19
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Drift Concerning Signals and Systems
Qualitative
When we are measuring or collecting data from a process,
we may occasionally say:
"Our data drift away".
Drift means that an observed signal changes
on average in time versus a certain direction.
Drift is perceived as a slow movement of behaviour
compared to all other temporal patterns
in a dynamic system.
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Drift Concerning Signals and Systems
Quantitative
Drift is the result of
temporal parameter variations p(t) of a system
and excludes
drift as a result of
varying input quantities u(t) of this system.
Since we assume constant or stationary input signals
for the consideration of drift,
we are allowed to use the tool
System Sensitivity Description
p(t) → y(t).
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Paul M. Frank, Introduction to System Sensitivity Theory
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Drift Concerning Signals and Systems
The parameter deviation ∆pdrift(t) makes the
dynamical system a
«time variant dynamical system».
We usually assume that the parameter drift proceeds
linearly with time t.
∆pdrift (t) =
vpdrift ⋅ t
The constant (average) drift velocity (drift rate) of the
parameter p is
vpdrift = p drift
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It may change with time t too.
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Stationarity Concerning Signals
The term
«Stationary»
refers to signals (time series data) and parameters,
but not to systems.
Signals are (weakly) stationary concerning time and space,
if characteristic values (like mean values, standard
deviations values, covariance values and so on) and
characteristic functions (like probability density functions,
correlation functions, spectral power density functions and
so on) are independent concerning time and space, or
frequency and wavelength.
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V
Conclusion
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Systems
may be unstable
Signals
(data)
cannot be unstable.
Constant parameters
make a system
time invariant;
varying parameters make it
time variant.
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Signals
are
constant and stationary
on the one side
and are
drifting, shifting, varying, fluctuating, oscillating,
limit cycling, transient
as well as
nonconstant, nonstationary, unsteady, unbounded
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on the other side.
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Terms discussed here
are standard terms of
Signal and System Theory
and
Stochastics and Statistics.
However, their treatment in everyday use
and the statements concerned in
International Guides and Standards
of other fields
are more or less inconsistent.
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