Chapter 5
Single Variable Detection
In this chapter methods to extract information from single measure-ment variables are presented. Detection of deviations in single meas-urements time series can be carried out both in the time domain and the frequency domain. Time domain detection involves signal amp-litude, mean and variance. Frequency domain detection is based on changes in the measurement signal frequency.
44 5. Single Variable Detection
Amplitude
The amplitude is the basic information content of the measurement.
Usually, a normal range with high and low limits is de ned in order to make qualitative comparisons.
Mean and Median
The arithmetic mean or justmean is the most basic statistical feature of a signal. The mean is used in many statistical analyses and is expressed by:
mean(
y
) = 1m Xmk=1y(k) (5.1)
A second useful measure of the center of location is the median. As described in Chapter 4, the median is the middle measurement if the measurements are sorted in descending or ascending order:
median(
y
) =yi; i = 12(m+ 1) (5.2) if m is a odd number and:median(
y
) = y(i) +y(i+ 1)2 ; i = 12m (5.3)
if m is an even number of measurements in the data set. If the meas-urements are symmetrically spread and there are no outliers, the mean and the median agree well. However, if there are outliers present the results can di er signi cantly. The median is una ected by single outlying values and, thus, a more robust measure of the center of location. In order to make the mean measurement more robust, the trimmed mean can be used. As the denotation indicates, trimmed mean implies that some of the largest and smallest measurements are removed prior to the calculation (Chapman 1992).
Spread Measures
The variability of a measurement series can be expressed by its stand-ard deviation. The standard deviation of m independent samples is
5.1. Monitoring Measurement Variables 45
estimated by:
=
v
u
u
t
m
X
k=1
(y(k),mean(
y
))2m,1 (5.4)
Sometimes it is more convenient to use the square of . This gives the estimated variance: the variance.
2 = Xm
k=1
(y(k),mean(
y
))2m,1 (5.5)
Standard deviation as presented above can be used as a measure on the spread if the measurements are normally distributed. If this is not the case, other measures can be used to describe the spread.
A simple and intuitive measure of spread is the range (R). This is simply the distance between the largest and smallest value in the data set. The range can also be calculated as the interquartile range (IQR).
The IQR is the distance between the 0.75 and 0.25quantile. A quantile corresponds to the value below which a certain percentage of the data set is located. Thus, the 0.90 quantile is larger than 90 percent of the data. This implies that the 0.50 quantile is the same as the median, i.e. the middle value in a data series. The calculation of quantiles is based on the rank of the measurements, that is the measurements are sorted in ascending order. The smallest value gets rank r = 1 and the largest value gets rank r = m (m is the number of values in the data series). The corresponding fracture can be calculated as (Chapman 1992):
fi = r, 12
m (5.6)
From the fractures quantiles can be calculated. Range and quantile are often referred to as non-parametric measures.
46 5. Single Variable Detection
Charts and Plots
The most straightforward type of plot technique is time series plots.
In time series plots the individual measurement is plotted on the y-axis and time on the x-y-axis. This is an intuitive way of presenting the variable state and an example of time series plots has already been shown in Figure 4.4. Typical ways to present di erent aspects of the measurement signal are, for instance:
raw measurement signal;
ltered measurement signal;
cumulative sum of the measurement signal.
The evaluation of the measurements is done by using control and alarm limits, indicating normal and abnormal ranges for the measurement variable.
Alarm Limits
In order to decide when a variable is inside its normal range, adequate limits must be determined. In the process industry it is common with two types of limits: warning limits and actions limits. Warning limits do not call for immediate action. Instead the purpose is to warn the operator that the measurement variable may be drifting away. If a variable exceeds an action limit, there is need for action in order to bring the variable back into the normal range. For the detection purposes, alarm limits are used. A violation of an alarm limit triggers a detection alarm. The limits may be derived from a previous period when the process is operating in a desired manner or when the product quality is acceptable. They can also be derived from a desired target value, from which the process variable should not deviate signi cantly.
If the measurements are normally distributed, the measurement mean plus two and three standard deviations are a common choice for upper warning and action limits, respectively, while the mean minus two and three standard deviations de ne the lower limits.
However, if the normal distribution is not applicable, there may be ways around this problem by transformation of data to an
approxim-5.1. Monitoring Measurement Variables 47
ately normal distribution. A usable transformation for this purpose is the logarithm of the measurement. Measurements from wastewa-ter treatment plants not seldom tend to be more log-normal than normal distributed (Chapman 1992). If a transformation is not pos-sible then the normal distribution constraint can be avoided by using non-parametric methods. Non-parametric methods do not make any assumptions about the shape of the distribution from which the data are taken (Miller and Miller 1993).
In some cases the limits can be derived from physical limitations. For example, the upper level of a tank is limited by its height (and a safety margin). Empirical information and desired targets may also be the basis for decision of limits.
Alarm and Detection Rules
When a limit is exceeded, it must be decided whether it is a true disturbance situation or simply a false alarm. To determine this, a set of rules can be worked out. Depending on the monitored variable and the use of limits, the rules will have di erent appearances. In addition to violation of the action or detection limit, rules for more than one consecutive sample above (or below) the warning limit can trigger an alarm or detection. More sophisticated rules, such as more than a certain number of consecutive samples on the same side of the target value or more than a certain number of samples above (or below) the warning limits within a speci ed time range (or sample range), is discussed by Bissel (1994).
Statistical Process Control
Monitoring process operation in time series is often referred to as statistical process control (SPC). The rst ideas of SPC for quality improvement go back as far as to the beginning of the century when, for instance, Vilfredo Pareto and Walter Shewart made some import-ant contributions to SPC (Thompson and Koronacki 1993). The ideas were further developed during the 1950s, but it is not until the 1970s that SPC has become a standard tool for quality improvement in the
48 5. Single Variable Detection
process industry. SPC involves many methods for monitoring and presenting measurement variables, but perhaps the most common ones are:
'x'-charts, that is measurement values plotted against the time;
MA charts, i.e. a moving average of the measurement series plot-ted against time;
EWMA charts, that is exponentially weighted moving average ltered measurements;
CUSUM charts, cumulative sum of the di erence between the measurement and a target value.
These methods have great similaritiesto conventional signal processing techniques. There are many references to SPC in the literature, such as Bissel (1994), Thompson and Koronacki (1993) and Box and Lu-ceno (1997). SPC in wastewater treatment applications is described in Chapman (1992). In this thesis, only some basic aspects of SPC is discussed.