ASHRAE Journal - February 2021 - 40

TECHNICAL FEATURE

Calculation Method of N and k

N and k are the main factors influencing the detection
results. The interval number, N, determines the distance between the abnormal points and normal points,
minD. The abnormal number threshold, k, represents
the maximum number of allowed abnormal points in
clusters.
The instantaneous heat data at a thermal substation
from Dec. 11, 2017, to Dec. 21, 2017, which was measured
every minute, was taken as an example to illustrate the
influence of the selection of N and k on the detection
results. The data is shown in Figure 2.
Figure 2 shows obvious abnormalities around 12,500
minutes, which changes the granularity of the data
interval. This kind of granularity change caused by
abnormal data has little effect on the method proposed
in this study. Because the range of detected data used
in this study determines the minimum interval size,
all data will be placed into different intervals. Thus
the different granularity of data will have little effect
on the detection performance of the algorithm. After
deleting the abnormal data, the next step is to supplement the data, so the final granularity of the data is
consistent.
To evaluate the detection results under different N
and k, three evaluation parameters need to be defined,
namely P, TP and FP. P is the total number of data points
detected as abnormal. TP, true positive, is the number
of abnormal data points correctly detected as abnormal.
FP, false positive, is the number of normal data points
incorrectly detected as abnormal. The relationship
between the three parameters is as follows:
P = TP + FP

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FIGURE 2 Instantaneous heat at a thermal substation varying with time.

2,500
Heat (kW)

2,000
1,500
1,000
500
0

6
8
10
Minutes (In Thousands)

FIGURE 3 The changes of three evaluation parameters varying with N.

300
Number of Detected Data Points

is adjacent to a normal interval, it will be determined a
normal interval. The rest of the candidate intervals will
be classified as abnormal intervals, and the data in it
will be seen as the exceptions. Note that if a candidate
interval has been decided to be a normal interval, the
new normal interval won't be involved in the judgment
of the other candidate interval.
Also note that this algorithm is given for the outlier
abnormal data and spike abnormal data, and it cannot
be applied to other kinds of abnormal data. When using
the detection algorithm, the two parameters, N and k,
still need to be specified.

P
TP
FP

250
200
150
100
50
0
10

100

Three regions of the figure are defined as follows.
The first is the rising region, where the increase of P is
mainly contributed by TP. The second is the transition
region, where TP and FP contribute to the increase of P.
The third is the over-detection region, where FP makes
the main contribution to the increase of P.
The changes of the three evaluation parameters varying with N are shown in Figure 3.
Figure 3 shows that the rising region is from N = 10 to
N = 40, where the abnormal data points detected in this
region are all correct. N from 40 to 70 is the transition
zone. At this time, the newly detected abnormal data
points are the sum of data correctly detected and incorrectly detected. N from 70 to 100 is the over-detection
region, where the newly detected abnormal data points
are mainly from incorrectly detected data points. The
abnormal point detected at N = 20, 60 and 90 are taken
as the typical situations of the three regions, and the
abnormal detection results are shown in Figure 4. In all
three cases, k is set to 50.
In Figure 4, the number of intervals with data points
lower than the threshold value k is very small. Therefore,
a large number of abnormal data points cannot be

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ASHRAE Journal - February 2021

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