1. No category

Universal Journal of Industrial and Business Management 2(4): 103-110, 2014
DOI: 10.13189/ ujibm.2014.020402
http://www.hrpub.org
Efficient Determination of Sampling Rate and Sample Size
in Statistical Process Control
Alessio Gerardo Maugeri1,*, Gabriele Arcidiacono2
1
Leanprove – A&C srl, Via Alfonso La Marmora, 45, 50121 Florence, Italy
Università degliStudiGuglielmo Marconi, Dipartimento per le Tecnologie, e i Processi di Innovazione (DTPI)
Via Plinio, 44, 00193 Rome, Italy
*Corresponding Author: [email protected]
2
Copyright © 2014 Horizon Research Publishing All rights reserved.
Abstract We propose a simple method for the
determination of minimum efficacious sampling rate and
sample size in Shewhart-like control charts, a controversial
topic both in the academic and industrial fields dealing with
Statistical Process Control (SPC). By modeling the control
procedure as the sampling of a stochastic process and
analyzing data in the frequency realm, it is possible to
identify meaningful system time scales and apply the
well-known Nyquist–Shannon sampling theorem in order to
define conditions for an efficient quality control practice.
Such conditions express the minimal requirements for an
efficacious monitoring of quality indices, indicating the
minimal efficacious sampling rate and the minimal effective
size of rational subgroups in Xbar and R (or S) charts, which
are useful both in the set-up phase and in the on-line control
phase of the Shewhart’s control charts. Results can be
applied also to I-MR charts. No statistical assumptions are
made on the monitored data; in particular neither statistical
independence nor Gaussianity is assumed in the derivation of
the results. We focus on continuous processes like those
typical in, but not limited to, e.g. refining, chemical
processing and mining.
Keywords Sampling Rate, Sample Size, Control Charts,
Statistical Process Control (SPC), Nyquist-Shannon
Sampling Theorem, Correlation Time, Lean Six Sigma
(LSS)
1. Introduction
Improving quality by reducing variability is the
fundamental idea at the very heart of successful managerial
approaches like the well-known Lean Six Sigma (LSS) [1]
and the Total Quality Management (TQM) [2], which
emphasize the role of Statistical Process Control (SPC) in
process performance improvements and quality control.
New tools aimed at these purposes are continually
developed. Undoubtedly, however, industrial SPC still
heavily relies on the Shewhart’s pioneering work [3], whose
main tool, the Control Charts (CCs), is widespread and
ubiquitously utilized both in manufacturing and service
companies.
CCs allow monitoring the variability of strategic features,
e.g. numerical quality indices or Critical To Quality (CTQ)
variables, as they are referred to in LSS, of the system under
control, and allow comparing such variability with specific
control limits in order to asses if it is acceptable or not
(Shewhart’s approach).
Repeated observations of a CTQ are therefore needed in
the time domain, in order to assess if the process is in control
or not; control limits being determined from the CTQ
observations themselves.
1.1. SPC Means Sampling
Quality indices and CTQs routinely monitored in all kind
of businesses by CC-like procedures can be modeled as
sequences of random variables, i.e. as discrete-time
stochastic processes. Whenever such indices or CTQs may
be measured continuously in time, we might think of their
monitoring as the sampling of a continuous-time (or
piecewise continuous-time) stochastic quality process (or
simply quality process or quality signal). For general
definitions on stochastic processes we refer to [4-5].
Due to randomness (fluctuations of both controllable and
uncontrollable degrees of freedom involved in the process
dynamics), we cannot predict what the outcome of the
quality control will be, but the process will be under
statistical control (or stable as is customary to say in LSS) if
it may (at least) be modeled by a wide-sense-stationary
(WSS) stochastic process. Such processes are sometimes
also referred to as weak-sense stationary. Finally, in SPC
practices it is customary to assume that the sampled process
is stable and ergodic, so we use the temporal averages
obtained through the sampling to evaluate statistical
moments of interest for the quality monitoring purposes.
1.2. Limits of Sample-Size Formulas
A non-trivial problem often encountered when adopting
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Efficient Determination of Sampling Rate and Sample Size in Statistical Process Control
CCs is how to choose a proper sampling rate and a proper
numerosity of the sample (i.e. sample size), especially if no
assumptions are made on the distribution function (usually
normal data are assumed) and on the independence (usually
statistical independence is assumed) of process data. Indeed,
real SPC activities very often need to deal with various forms
of correlation. Despite of this, the subgroup size of Xbar CCs
is often determined by formulas relying on the Central Limit
Theorem, which in turn relies on the statistical independence
of the sampling data. I-MR CCs instead heavily rely on
Gaussianity of the monitored variable.
Nonetheless, the Shewhart’s concept of stability relies
neither on statistical distribution nor on statistical
independence hypothesis of the sample observations, but
only on predictability of process variation [3]. This means
that in principle it is possible to control also processes
exhibiting correlation or processes not featuring Gaussianity.
Indeed Shewhart-type CCs accounting for different types of
correlation [6] and for non-normality [7] have been
developed.
Optimum sampling intervals, optimum sample sizes and
optimum control limits are the topics of many research
works in the Operational Excellence realm and in the context
of Optimization and Design of Industrial Processes [7-12].
Choosing a proper sampling rate and a proper sample size
is an essential task to correctly and efficiently assess
ordinary process variability, whose evaluation is in turn
essential to set up appropriate control limits in Shewhart CCs.
Control limits are subsequently used to assess if the future
process variability (i.e. during on-line control), and temporal
behavior in general, is acceptable or not.
1.3. A New Perspective from Spectral Analysis
Bearing in mind that CTQ monitoring is equivalent to the
sampling of a stochastic process, with this paper we would
like to open a new perspective in SPC, addressing in
principle data sets exhibiting generic correlation. We
propose to determine efficient sampling rates and sample
sizes in Shewhart CCs by addressing the quality process in
the spectral domain, without any constraints on the statistical
distribution function of sampled data nor assuming their
statistical independence. Such new perspective being
potentially useful also for Shewhart-type and non-Shewhart
CCs, in that it indicates a method to properly assess ordinary
process variability.
We will derive our results thanks to the stochastic
sampling theory as customary in many engineering fields
dealing with signal analysis. This will require identifying
both the bandwidth and the correlation time of the quality
signal.
2. Theoretical Framework
2.1. The Nyquist-Shannon Sampling Theorem
An effective sampling is a faithful sampling. This means
we must be able to exactly reconstruct the original signal at
each instant of time from the sampling.
According to the Nyquist–Shannon sampling theorem
[13], a result very well-known in Communications and
Information Theory, a sampled bandlimited (BL)
deterministic signal can be perfectly reconstructed from an
infinite sequence of samples, if the sampling rate fs exceeds
2B samples per unit of time (ut), where B is the fastest rate in
the original signal and 2B is the so called Nyquist rate. Such
a choice allows not incurring in the well-known aliasing
error.
A similar result also holds for BL WSS stochastic
processes [5], i.e. finite-power processes with a vanishing
power spectral density S(f) = 0, ∀ |f| > B. So, we can apply
the theorem also to the sampling of the quality process.
Moreover, being the quality process a random signal, it
can be approximated with arbitrary accuracy by a sum
involving only its past samples, provided that the Nyquist
criterion is satisfied (i.e. fs > 2B), and therefore we do not
need its future history [5,14].
The reconstruction of stochastic processes from their
samples is still a hot topic in many engineering fields and by
now many results have been presented in scientific literature.
For a general overview of stochastic-process sampling
theorems, we refer to [15-16].
2.2. Process Variability: Common and Special Causes
Each industrial process is supposed to have its own
standard working conditions. Stability of the process is
assured by a proper set up of internal degrees of freedom, i.e.
what we usually call best practices. Operational or
managerial best practices concern - but they are not limited
to - persons involved into the process, machines used into the
process, state of life or degree of wear of such machines,
materials used, operating conditions (e.g. temperature,
pressure, or even type of organizational structure), methods
and procedures adopted.
Within the boundaries defined by operational or
managerial best practices, the process will only suffer from
an ordinary variability. These intrinsic and unavoidable
fluctuations of process internal degrees of freedom are due to
so called common causes, as they are usually indicated in the
LSS approach [1]. Common causes cannot be annihilated
(given the process we are monitoring), but only reduced.
Exit from or violation of the best practice boundaries
determines non-standard behaviors of the process dynamics,
due to so called special causes as they are referred to in LSS
methodology [1], and therefore unusual dynamics of internal
degrees of freedom.
Both common and special causes are triggered by external
(with respect to the monitored process) degrees of freedom
that are coupled to process dynamics. Provided such external
degrees of freedom stay in a stability region of the parameter
space (stability being defined with respect to the
best-practice prescriptions and to process robustness to
Universal Journal of Industrial and Business Management 2(4): 103-110, 2014
external fluctuations), no special causes can occur.
Monitoring of the process is aimed at identifying and
eliminating special causes, and reducing common variability
through external degrees decoupling (or reduction of the
strength of their coupling to internal dynamics) or removal.
We may also refer to the ensemble of all the external
degrees of freedom as the environment, which is responsible
of noisy (both common and special) fluctuations of the
quality signal, as shown in fig. 1.
105
c(t). Such variability can only be reduced, but not eliminated
from the given system, and in general is not easily
attributable to specific and identifiable external degrees of
freedom. Being a WSS process, the average value of c(t) is
constant and its autocorrelation function depends only on
time differences [4-5].
No hypothesis are a priori made on s(t), which accounts
for the non-stationary behavior of system dynamics (due to
special causes of variability). It describes not-ordinary
features of system dynamics, chargeable to specific and
addressable external degrees of freedom.
For sake of clarity, without affecting the overall value of
the study, from now on expression (1) will indicate a generic
CTQ, measured on process outputs.
2.4. Continuous-Time Vs Piecewise Continuous-Time
CTQs
Figure 1. System-environment coupling: the system/process we are
monitoring interacts with external degrees of freedom, which are coupled to
system/process dynamics. Red wavy lines represent such coupling. The time
evolution of a CTQ with a natural variability is naively depicted. Interaction
with the environment determines aleatory modifications (dashed-line circle)
of the isolated system/process dynamics (full-line circle), and, consequently,
randomizes the CTQ signal (full-line deterministic signal vs dashed-line
stochastic signal).
2.3. Modeling the Quality Signal
The CTQ to be monitored could be a process input or
output variable, or even a generic process variable, but in any
case we write the CTQ dynamics as a random signal made of
three different components
Q(t)=q(t)+c(t)+s(t)
(1)
where q(t) is the quality function in absence of
environmental noise sources, i.e. in absence of external
degrees of freedom coupled to system dynamics; c(t) is a BL
WSS stochastic process responsible for common variability;
and s(t) is the process accountable for out-of-ordinary
variability and lack of statistical control.
The deterministic function q(t) accounts for the dynamics
of the process degrees of freedom in absence of any kind of
perturbation stemming from external degrees of freedom
coupled to the process ones. Such isolated (or natural, i.e. not
due to any environmental noise sources) system dynamics is
a pure abstraction, representing a regulative ideal, which
cannot be met in practice, due to the unavoidable interaction
with noisy degrees of freedom.
The stochastic process c(t) accounts for the effect of all the
unavoidable fluctuations of the internal degrees of freedom
(triggered by coupling to external ones), which are
responsible for mean-reverting shifts of the system
deterministic quality signal. Normal, i.e. ordinary, process
variability due to common causes is therefore explained by
The stochastic process described by (1) may properly
represent different control practices in both service and
manufacturing industries.
We now specify (1) distinguishing between
continuous-time and piecewise continuous-time CTQs [4-5].
Such continuous processes are frequent in e.g. refining,
chemical processing and mining.
We first address an ideal case, where (1) represents an
infinite, analog (i.e. continuous-time) signal related to a CTQ
measurable on an infinite and continuous output
(continuous-time processes). Although ideal, this situation is
a good starting point to understand the modelling of more
realistic occurrences; furthermore it may well approximate
such occurrences when the discrete nature of the process
outputs do not hinder the sampling procedure, as will be
more clear later on.
We secondly address the case of signals related to CTQs
that are only piecewise continuously measurable (piecewise
continuous-time processes, related to discrete outputs). This
is the case of all real control practices. Due to discreteness,
periodic time intervals exist where it is intrinsically not
possible to measure any CTQ.
In this paper we do not address quality signals related to
CTQs which are uniquely measurable at specified time
instants (i.e. on a countable set of the time domain): this is
for example the case of measures made once per single
output (as e.g. in destructive testing or in some
conformance-to-requirement measures involving the output
as a whole), giving rise to a sequence of random variables
each one related to a unique output (discrete-time processes).
2.4.1. Continuous-Time Processes
Let’s first suppose that the process we are monitoring
generates an output featuring an infinite and continuous
dynamics with respect to time. In this case, monitoring of
process quality can be modeled as the sampling of an infinite,
analog BL stochastic signal.
A continuous-time process describes a CTQ related to an
output whose features vary continuously in time. For
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Efficient Determination of Sampling Rate and Sample Size in Statistical Process Control
example this could be the case of an infinite extruded tube, of
an infinite liquid flow or of the maximum amount of
information that can be reliably transmitted over a
communication channel, respectively featuring a
continuously variable thickness, viscosity and channel
capacity (which are all continuous CTQs with respect to
some parameters). A continuous-time process may also
describe conformance to some requirements or number of
non-compliant features per reference unit (which are instead
discrete CTQs).
Process stability and dispersion around the target are
assessed by monitoring one or more of such CTQs related to
the output.
Finally, both uncountable, i.e. continuous (e.g. inches of
thickness, centipoise of viscosity or bits conveyed per
second), and countable, i.e. discrete (e.g. good/no-good
indices or number of defects per reference unit), CTQs may
be modeled by a continuous-time process.
2.4.2. Piecewise Continuous-Time Processes
In practice, we never deal with continuous-time processes,
because our outputs are discrete, e.g. parts produced or
documents written up. This means that the quality signal (1),
although (in principle) not time-bounded, is only piecewise
defined, for both continuous and discrete CTQs.
Whatever CTQ variable is chosen to be monitored, it will
be at best piecewise continuously varying in time, due to the
discrete nature of the outputs realized. In comparison with
the previous case, here a limitation is imposed on the number
of subsequent observations we can draw from the process at
a given sampling rate as better outlined in section 3.
2.5. Rational Subgroups: Fast Noise Vs Slow Noise
In the Shewhart’s approach, appropriate evaluation of
system ordinary (i.e. only due to common causes and,
eventually, to natural system dynamics) variability is
fundamental to set up an effective CC featuring proper
control limits.
Shewhart explains that if observations can be grouped in
rational subgroups, then appropriate control limits can be
specified. Rational subgroups are defined as groups of
observations whose variability is only due to common causes
[3].
In other words, the underlying argument for rational
subgroups is that special causes of variability have a slow
dynamics compared to ordinary process dynamics: we might
therefore refer to the effect of special causes as slow noise.
On the contrary, the effect of common causes might be
referred to as fast noise.
Bearing in mind results in section 2.1, set up of proper
rational subgroups could therefore be achieved by analyzing
the spectral content of the monitored system dynamics. In
particular, by identifying the bandwidth upper limit of the
monitored CTQ, it is possible to determine a proper sampling
rate; by identifying the time scale of the CTQ dynamics (as
will be clearer in section 2.6), it is possible to set a proper
sample size for the control procedure. Identifying the
bandwidth upper limit would therefore be useful for both the
phase of set up and on-line control of the CC.
Of course this might require a thorough knowledge of the
underlying monitored process. However we do not need to
know anything about slow noise in order to properly set up
our Shewhart’s sampling rate and sample size. In fact, when
the process is stable (by definition we assess control limits in
condition of statistical control and from data not affected by
special causes) the CTQ dynamics is not affected by s(t),
which operates at frequencies much slower than the CTQ
slowest ordinary dynamics. Setting up of control limits in
Shewhart-like CCs will therefore be based on a spectral
analysis of the BL WSS stochastic process accounting for
common system variability
Qfast(t)=q(t)+c(t)
(2)
Although bandlimitedness assumption is often invoked in
this paper concerning industrial applications (both
manufacturing and service), it is worth saying that sampling
theorems have been developed also for non-BL signals [17].
Furthermore it may also be possible filtering out stray
external (i.e. from sources other than the system, e.g. the
quality measurement system) high frequencies. Finally,
almost sure convergence of the Shannon sampling series has
been demonstrated also for generalized BL processes [18].
2.6. Time Scale of the CTQ Dynamics
The ordinary evolution of each CTQ dynamics is
characterized by a typical time scale accounting for its
common variability, due to both fast noise and, if present, to
system natural variability. In case of a system with e.g. a
natural periodic time evolution, as in fig. 1, such time scale
would be the period of the natural dynamical fluctuations.
Instead, in case of a naturally static CTQ (that is a CTQ stuck
to a target value in absence of any environmental noise
sources), the CTQ time scale would be determined by the
fast-noise component into (2). Fast noise is typically
responsible for fluctuations tending rapidly to zero and, in
this case, we trivially identify the CTQ time scale with the
width of the relative autocorrelation function.
As usual in Signal Theory, the time scale of the CTQ
dynamics is the correlation time τc, which can be defined also
for arbitrary stochastic processes. A formal standard
definition for the correlation time can for example be found
in [4-5]
A practical identification of the correlation time can be
achieved through the observation of the power spectrum of
(2), as customary in Signal Analysis and in many physical
problems [19-21]. Depending on the shape of the power
spectrum, identification of the correlation time may vary,
reflecting the possibly different nature of the dynamical
regimes of the various CTQs. In most cases (e.g. a static
CTQ suffering from fast noise) the width of the
autocorrelation function of (2) is inversely proportional to
the width of the power spectrum (in this case the power
Universal Journal of Industrial and Business Management 2(4): 103-110, 2014
spectrum would have a bell shape). In some cases the power
spectrum may have a peak structure (if e.g. the CTQ has a
natural oscillatory frequency) and the correlation time is
drawn from the inverse of the peak frequency [21].
In any case, the correlation time of (2) defines the time
needed to gain complete knowledge of the monitored CTQ
ordinary variability, and, finally, together with the fastest
rate in the power spectrum of (2), the number of observations
we need to gather to realize proper rational subgroups, as
explained in section 3.
3. Results and Discussions
3.1. Nyquist-Shannon Theorem Applied To Quality
Control
If we monitored our system continuously, we would know
exactly a realization of the random quality signal (1).
Consequently, under the ergodicity assumption, we would be
able to appraise quality dynamics instant by instant.
Unfortunately, due to many practical reasons, it is not
possible or it is not economic a continuous monitoring of the
system, and indeed in practice we just measure (1) at
specified time points (sampling). However we should accept
the discrete quality index stemming from such sampling only
provided it is a faithful approximation of the original signal.
As we have seen, we can borrow results from
stochastic-process sampling theory [14-20].
3.2. Sampling Rate
According to section 2.1, the sampling rate is given by the
fastest frequency (where appropriate, filtering can prevent
aliasing due to external high frequencies) in the power
spectrum of (2).
Mathematically [5], if Sfast(f) is the spectral density of (2)
and Sfast(f) = 0, ∀ |f| > B, then must be
fs > 2B
(3)
B being the fastest rate in the bandwidth of the quality signal
(2).
B being the fastest rate in the bandwidth of the quality
signal (2). Expression (3) holds for both continuous-time
processes and piecewise continuous-time processes
3.3. Sample Size
By virtue of the Nyquist-Shannon theorem, knowing B we
are able to determine the minimal efficacious sampling rate,
which is just above the Nyquist rate as expressed by
condition (3). Acquiring also knowledge of τc allows
determining the number of single observations we need to
gather, because this time scale fully accounts for common
variability of the monitored CTQ, thus setting the minimal
time required to acquire a complete knowledge of process
ordinary dynamics. We can then easily estimate the minimal
107
meaningful sample size as the product of the minimum
efficacious sampling rate and the correlation time, as better
outlined in subsections 3.4.1 and 3.4.2 for Shewhart's CCs.
This information (sampling rate and sample size) will then
be used both to set up proper control limits and to properly
manage the on-line control of the monitored process.
In the case of piecewise continuous-time processes, due to
the fact that we are likely monitoring some CTQ variable on
a discrete output, we may not have enough time to gather
clusters of subsequent observations. Condition (3) will also
apply to such processes, but differently from
continuous-time processes, here sampling of the quality
signal will need to take into account the ratios between τc, tav
(total available time for taking measures of a given CTQ on a
single output, i.e. time interval from the first measurable part
of an output and its last measurable part) and Δt (the time
interval between the last measured part of an output and the
first measurable part of the subsequent output); measure
ability being defined in relation to the specific CTQ to be
monitored.
We will pursue our analysis in section 3.4, highlighting
such relations.
3.4. Shewhart-Type Ccs: Set Up Of Control Limits and
On-Line Control
In order to specify the concepts previously illustrated, let’s
focus on Shewhart-type CCs routinely used in SPC.
Shewhart-type CCs only rely on current observations to
assess the process control state, i.e. each point on the chart
only relies on the sample gathered at the moment of
monitoring (other well known charts, e.g. CUSUM or
EWMA, also utilize previous observations). When adopting
Shewart-type CCs, particular care should therefore be
devoted to the time scales of the monitored CTQ dynamics,
in order to properly estimate sampling rate and sample size,
and, finally, control limits [3,6]. It is worth noticing that
usual formulas to estimate e.g. the required sample size for
subgroups of an Xbar chart may not be riskless, if used when
data presents a certain degree of correlation.
For sake of clarity, let’s adopt the set of definitions in table
1, to be utilized in subsections 3.4.1 and 3.4.2, where we will
indicate how to compute efficient sampling rate and sample
size both in the case of individual (I-MR charts) and
gathered-in-subgroup observations (Xbar-R or Xbar-S
charts). In the following we will assume to know the CTQ
correlation time and its upper rate B, whose empirical
determination will be outlined by the practical procedure in
section 3.5.
3.4.1. I-MR Charts
In order for a sampling to be faithful and to obtain an
unbiased estimate of the variability of the CTQ normal
dynamics, the Overall Sampling Period must be at least as
long as the correlation time, that is T ≳ τc.
According to the Nyquist-Shannon theorem and relation
(3) in particular, for both continuous-time and piecewise
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Efficient Determination of Sampling Rate and Sample Size in Statistical Process Control
continuous-time processes, a necessary condition for a
faithful sampling is
(4)
fs,b> 2B
which sets the more efficient sampling rate for the I-MR
chart of a continuously (in time) measurable CTQ.
Once we are provided with the sampling rate, we can
estimate the minimal meaningful sample size for the I-MR
chart, made of subsequent measures taken at a rate given by
(4) for a sampling time T~ τc.
For continuous-time processes, the more efficient sample
size, aimed at properly assess control limits, would be the
higher nearest integer to
n = 2B ∙ τc
(5)
For piecewise continuous-time processes, the discrete
nature of the output can be an obstacle to the sampling
procedure, because, given a rate set by (4), we run the risk to
miss subsequent measures within the τc time scale. That is, if
tav≤τc, we cannot reach the value set by (5) via subsequent
measures on a single output (even if the equality holds it is
not possible to take the higher nearest integer to (5) because
the last measure is not feasible). In other words, we are not
allowed monitoring slow components of the CTQ ordinary
variability.
Furthermore, if it also holds Δt≥1/2B, we miss
information even if we take measures from the subsequent
output. This happens due to the discrete nature of the process
outcomes, which acts as a high-pass filter of the CTQ
ordinary dynamics. In this case, in order to avoid
underestimating the process variability and incurring in the
aliasing error, sampling should be extended for time periods
longer than τc, which are to be evaluated taking also into
account the length of the time interval Δt.
From the gathered observations it will then be possible to
determine the CTQ ordinary variability and therefore
establishing proper control limits. On-line control will be
carried out at a rate set by (4).
3.4.2. Xbar-R (S) Charts
We make the reasonable assumptions TS ≥ τc(choosing a
shorter Subgroup Sampling Period would undermine the
validity of the results, as it would not be possible to gather
enough observations) and TW ≠ 0 (i.e. sampling within the
subgroup is not instantaneous).
Within the subgroups we have operating conditions
analogous to the one described in subsection 3.4.1 for I-MR
CCs. According to the Nyquist-Shannon theorem, in order to
properly assess the CTQ ordinary dynamics (including
eventual short- or long-term correlations), a faithful
sampling of a continuous-time process would at least require
TW< 1/2B, i.e.
fs,w > 2B
(6)
and a subgroup size equal to the higher nearest integer to (5).
The rate set by (6) can then be used to evaluate proper
control limits and, subsequently, to carry out on-line control.
Relation (6) also holds for piecewise continuous-time
process, however we may miss slow components of our CTQ
ordinary variability, if tav ≤ τc and sampling is not extended
for time periods longer than τc, analogously to the case in
subsection 3.4.1. Missing slow components of ordinary
dynamics would in general result in a systematic
underestimate of the process variability, which would be
incorrectly transferred to variability between subgroups. The
variability usually observed between subgroups in absence
of special causes may therefore be attributed also to the
intrinsic process filtering determined by the output
discreteness when tav ≤ τc.
As far as the Between Sampling Period is concerned, it
should be short enough to detect the eventual occurrence of
special causes, but not too short in order to reduce
unnecessary sampling effort. If there is a clue of special
cause dynamics, once again the Nyquist-Shannon theorem
suggests how to choose Tb. If for example slow noise is
known to affect the natural dynamics at most with frequency
fslow<< 1 / τc, then we will choose fs,b> 2 fslow. In general,
however, slow noise dynamics is unknown, therefore it is not
possible to determine a priori a smart Between Sampling
Period. A prudential choice could be setting Tb ~ 10 τc,
eventually relaxing this conditions if appropriate from
observations gradually accumulated. In any case a trade-off
need to be found between the risk of missing special cause
detection and the sampling effort.
Table 1. Relevant time scales and parameters: definition of sampling periods, sampling rates and sample size for I-MR CCs, Xbar-R CCs, Xbar-S CCs
Between Sampling
Period
Tb
Time interval between two single observations (I-MR charts) or between two rational subgroups (Xbar-R or
Xbar-S charts)
Between Sampling Rate
fs,b
Number of single observations (I-MR charts) or rational subgroups (Xbar-R or Xbar-S charts) in the ut
Within Sampling Period
Tw
Time interval between two observations within a rational subgroup
Within Sampling Rate
fs,w
Number of observations within the subgroup in the ut
n
Overall numerosity of the sample (I-MR charts) or number of observations in a single rational subgroup (Xbar-R
or Xbar-S charts)
Ts
Duration of the sampling within a rational subgroup
T
Overall duration of the sampling procedure
Sample Size
Subgroup Sampling
Period
Overall Sampling
Period
Universal Journal of Industrial and Business Management 2(4): 103-110, 2014
3.5. Practical Determination of Correlation Time and
Nyquist Rate
We outline a practical procedure in order to determine the
stable process power spectral density, and finally the CTQ
correlation time and the Nyquist rate.
The procedure encompasses the following steps:
Qualitatively estimating the order of magnitude of the
fastest time scale of the ordinary (which includes fast noise)
dynamics of the CTQ to be monitored;
Sampling at a reasonable rate based on the previous
estimation (e.g. three times faster) and compute the
corresponding power spectral density until a valid result is
obtained and rapid changes in the spectra stop (stabilization
in the frequency domain);
Sampling again increasing the sampling rate and
re-computing power spectral density, till no observable
differences appear in the stabilized spectra.
Once the stable spectrum has been computed, it is possible
to observe its shape and bandwidth in order to properly
identify the correlation time (see section 2.6) and the upper
rate.
3.6. Future Research
109
the CTQ common variability allows determining lower
bounds for both the sampling rate and the sample size of
usual ShewhartCCs, for both countable and uncountable
CTQs. The sampling rate lower bound is the Nyquist rate,
while the sample size lower bound is the product of the
Nyquist rate times the correlation time.
The illustrated approach is potentially extensible to other
CCs and provides a general method to properly assess
ordinary process variability.
A spectral characterization of common and special causes
of process variability has also been illustrated in terms of fast
and slow noise.
Acknowledgements
We would like to thank dr. Giuseppe Mangano for very
fruitful and stimulating discussions and Alessandro Costa for
proofreading. Dr. Maugeri acknowledges financial support
from Leanprove – A&C srl.
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