Instrumentation & Measurement Magazine 23-5 - 8

The ICDF function for a Weibull distribution is:
	





1
β

x F 1  p|α,β  α ln  1  p   	(6)

Therefore, the centerline can be found by setting p equal to
0.50 and solving for x. The upper and lower yellow threshold
levels can be found by setting p to 0.977 and 0.023, respectively.
Similarly, the upper and lower red threshold levels can be
found by setting p to 0.999 and 0.001, respectively. This ensures
that the probability of an alarm detection is the same regardless of the PDF selected for modeling [11]. Each parameter
monitored by the NILM dashboard can be analyzed individually and the gauges adjusted to provide diagnostic warnings
at appropriate levels.
Next consider a discrete variable, in which the variable has
finite values, such as the number of daily actuations. Because
these can only occur as integer values, the CDF is not continuous and increasing; thus, the generalized inverse distribution
function will be used instead of the ICDF. The generalized inverse distribution function can be expressed as:
	





x
F 1  p  
inf x   : F  x   p ,	(7)

where inf is the infimum, or the greatest lower bound. Similar to a continuous function, centerline and upper and lower
limits can be determined by setting p to the appropriate values
and solving for x.
The SPC method is visualized in Fig. 9 for the MEC CPP
pumps. Histograms for each metric are created using historical NILM data. Then, probability densities are fit to the
data. The probability densities visualized in Fig. 9 are normalized to match the total area of the histograms. For continuous

functions, the histograms are modeled with multiple probability density functions (PDF), while a discrete function is
modeled with probability mass functions (PMF), and the
best fit function is selected. For continuous functions, the
Anderson-Darling (AD) test returns a decision for the null
hypotheses that the data is from a population with a specific
distribution [15]. The test rejects the null hypothesis at the
5% significance level. For the example shown in Fig. 9, the
AD-test did not reject the null hypothesis for a normal distribution for power and power factor, with p-values, or probability
values, of 0.2446 and 0.6561, respectively. The AD-test did not
reject the null hypothesis for a Weibull distribution for average run duration and total run time with p-values of 0.4430 and
0.7205, respectively. To test the discrete models, the chi-square
goodness-of-fit test was used since it is applicable for discrete
distributions [15]. The chi-square test did not reject the null
hypothesis for a Poisson distribution for number of daily actuations with a p-value of 0.0511. Depending on the PDF or PMF
selected, the fault detection thresholds are set as shown in Fig.
8 or using a quantile function, as described in (6) and (7). This
process was repeated to determine control limits for each of the
monitored loads.

Detection of Changing Load Signatures
If a user knows that a piece of equipment should be operating,
but the dashboard Metrics View shows the load has not been
operational, this would alert the user that a fault is present.
There are several reasons the daily actuations parameter could
drop to zero, even if the load should be operating. First, it could
be indicative of a broken sensor, such as a tank-level sensor or
temperature sensor, resulting in the load not turning on even
when it should be. Second, it could be indicative of a complete

Fig. 9. Different probability density functions fit to each parameter of the controllable pitch propeller (CPP) pumps.
8	

IEEE Instrumentation & Measurement Magazine	

August 2020
Instrumentation & Measurement Magazine 23-5

Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 23-5
