entropy
Article
Entropy and Compression Capture Different
Complexity Features: The Case of Fetal Heart Rate
João Monteiro-Santos 1,2, * ID , Hernâni Gonçalves 1,2 , João Bernardes 2,3,4 , Luís Antunes 5,6 ,
Mohammad Nozari 6 ID and Cristina Costa-Santos 1,2
1
2
3
4
5
6
*
Department of Community Medicine, Information and Health Decision Sciences—MEDCIDS,
Faculty of Medicine, University of Porto, Rua Dr. Plácido da Costa, 4200-450 Porto, Portugal;
hernanigoncalves@med.up.pt (H.G.); csantos.cristina@gmail.com (C.C.-S.)
Center for Health Technology and Services Research—CINTESIS, Faculty of Medicine, University of Porto,
Rua Dr. Plácido da Costa, 4200-450 Porto, Portugal; jbernardes59@gmail.com
Department of Obstetrics and Gynecology, Faculty of Medicine of University of Porto, Rua Dr. Plácido da Costa,
4200-450 Porto, Portugal
Pedro Hispano Hospital, Local Health Unit of Matosinhos, 4464-513 Matosinhos, Portugal
Computer Science Department, Faculty of Science, University of Porto, Rua do Campo Alegre 1021/1055,
4169-007 Porto, Portugal; lfa@dcc.fc.up.pt
CRACS/INESC-TEC, University of Porto, Rua Dr. Roberto Frias, 4200 Porto, Portugal;
mz.nozary@gmail.com
Correspondence: joaomms90@gmail.com; Tel.: +351-2-2551-3622
Received: 21 September 2017; Accepted: 11 December 2017; Published: 14 December 2017
Abstract: Entropy and compression have been used to distinguish fetuses at risk of hypoxia
from their healthy counterparts through the analysis of Fetal Heart Rate (FHR). Low correlation
that was observed between these two approaches suggests that they capture different complexity
features. This study aims at characterizing the complexity of FHR features captured by entropy
and compression, using as reference international guidelines. Single and multi-scale approaches
were considered in the computation of entropy and compression. The following physiologic-based
features were considered: FHR baseline; percentage of abnormal long (%abLTV) and short (%abSTV)
term variability; average short term variability; and, number of acceleration and decelerations. All of
the features were computed on a set of 68 intrapartum FHR tracings, divided as normal, mildly,
and moderately-severely acidemic born fetuses. The correlation between entropy/compression
features and the physiologic-based features was assessed. There were correlations between
compressions and accelerations and decelerations, but neither accelerations nor decelerations were
significantly correlated with entropies. The %abSTV was significantly correlated with entropies
(ranging between −0.54 and −0.62), and to a higher extent with compression (ranging between −0.80
and −0.94). Distinction between groups was clearer in the lower scales using entropy and in the
higher scales using compression. Entropy and compression are complementary complexity measures.
Keywords: fetal heart rate; entropy; data compression; complexity analysis; nonlinear analysis
1. Introduction
In developed countries, clinical decisions during labor are strongly based on Fetal Heart Rate
(FHR) monitoring [1,2], and cardiotocography is the tool that is routinely used for FHR and uterine
contractions recordings. FHR is generally assessed in beats per minute to evaluate fetal wellbeing
allowing for an obstetrician to intervene and prevent potentially irreversible fetal brain damage
or death. Despite the importance of FHR monitoring, poor reproducibility of visual analysis of
cardiotocograms have been reported [1,3], and consequently computerized FHR analysis and new
Entropy 2017, 19, 688; doi:10.3390/e19120688
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signal processing and pattern recognition techniques have been developed [4–6]. In this setting,
complexity analysis of FHR recordings remains one of the most challenging tasks. Actually, FHR during
labor seems to be part of a complex system, where, most of the times, individual agents behave in
unpredictable ways, and whose actions are connected, inducing changes to one another [7]. In cases like
this, a high degree of uncertainty is known to be present, leading to a poor interrater agreement. As a
result, uncritical adherence to conventional guidelines might become more harmful than beneficial [7]
and other approaches may be more appropriate, such as nonlinear models and scan of patterns [8].
Complexity is a property of systems that quantifies the amount of structured information and
may be assessed using both entropy and compression. Approximate entropy (ApEn) is a measure of
complexity, introduced by Pincus, used to quantify the amount of regularity and the unpredictability
of fluctuations over time-series [9]. Later, Sample Entropy (SampEn) was presented by Richman and
Moorman with the same goal as ApEn to assess biological time series [10]. In the particular case of
FHR analysis, ApEn and SampEn are the most used measures of complexity, and are known to be used
in the detection of different pathologies. On the other hand, the Kolmogorov complexity of an object is
the length of the shortest computer program that can output it. Although Kolmogorov complexity
is a non-computable measure, compressors do a very good job approximating it. This approach has
led to positive results in very different subjects, as in literature [11], music [12], and computer virus
and Internet traffic analysis [13]. Despite the successful application of compressors to FHR tracings in
pathology detection, they have been used only to a limited extent in the analysis of biological signals
to date [6,14].
Although both entropy and compression were able to distinguish fetuses at risk of hypoxia from
their healthy counterparts through the analysis of the FHR signal, their low correlation suggests that
these measures capture different features [15]. Henrique’s work also suggested further research in
order to study how physiological features are captured by entropy and compression.
The small computational time that is associated with both measures, namely with compressors,
is particularly convenient if their inclusion in existing FHR monitoring systems is justified. Hopefully,
the information on the fetus complexity obtained from the FHR signal may provide important auxiliary
information to clinicians in fetal assessment, supporting clinical decisions. However, as entropy and
compression seem to capture different features, it is important to study which features are captured
by these measures. Following previous work’s suggestion [15], this study aims at characterizing
the complexity FHR features that are captured by entropy and by compression having as reference
international clinical guidelines, and exploring the multiscale approach for entropy and compression.
2. Materials and Methods
Sixty eight FHR intrapartum tracings consecutively selected from a pre-existing database of
term singleton gestations, with at most 60 min of tracing, were analyzed according to FIGO (The
International Federation of Gynecology and Obstetrics) guidelines [16] using Omniview-SisPorto,
version 4.0.9 [17] and the following FHR features, from the last 60 min of tracings, were registered:
FHR baseline, which is the mean level of the most horizontal and less oscillatory FHR segments;
percentage of abnormal short term variability (%abSTV), subsequent FHR signals differing < 1 bpm;
percentage of abnormal long term variability (%abLTV), FHR signals with difference between minimum
and maximum values in surrounding 1-min window < 5 bpm; mean short term variability (mean STV);
number of acceleration (Acc), i.e., abrupt increases in FHR above the baseline, of more than 15 bpm in
amplitude, and lasting more than 15 s, but less than 10 min per minute; number of decelerations (Dec),
i.e., decreases in the FHR below the baseline, of more than 15 bpm in amplitude, and lasting more than
15 s per minute.
Newborn umbilical artery blood (UAB) pH was used as measure of fetal oxygenation, as it
represents an active measure of fetal oxygenation. A low UAB pH indicates the presence of acidemia
occurring during labor and delivery, presenting a higher risk of perinatal death or neurological injuries
from hypoxia.
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Of the 68 cases, 48 delivered fetuses with pH in the normal range, pH ≥ 7.20 (N), 10 delivered with
UAB pH between 7.10 and 7.20, mildly acidemic fetuses (MA), and 10 moderate-to-severe acidemic
fetuses with UAB pH ≤ 7.10 (MSA).
All of the tracings were resampled at a frequency of 2Hz after pre-processing, based on an
algorithm described in previous studies [18]. A more detailed description of the data is presented
elsewhere [5].
Spearman Correlation Coefficients were used to compare each complexity measure with different
parameters. For entropy, two measures were used: Approximate Entropy (ApEn) and Sample Entropy
(SampEn), and for each, 3 different tolerances were used (0.10, 0.15 and 0.2). For compression,
six different compressors were used, namely brotli, bzip2, gzip, paq8l, ppmd, and lzma. For the first
five, the lowest and highest levels of compression were tested. The latter, lzma, only one level of
compression was possible. Compression was measured as compression rate, this is the compressed
size of the trace divided by the original size of the same trace.
Regarding the physiological features that were captured by SisPorto, Spearman Correlation
Coefficients between the percentage of abnormal short term variability (%abSTV), mean value of the
STV, baseline, percentage of abnormal long term variability (%abLTV), number of Accelerations (Acc)
per 10 min, and Decelerations (Dec) per 10 min were computed.
Spearman Correlation Coefficients between the physiological features and Compressors and
between the physiological features and entropies were computed.
Multiscale analysis was also performed to study the effect of entropy and compression, up to
scale 20, in the three different groups (N, MA, and MSA). For entropy, tolerance 0.2 was used
in both Approximate Entropy and Sample Entropy, while in compression paq8l, brotli, and gzip,
with maximum level of compression were used.
3. Results
The 68 tracings were acquired in singleton pregnancies and were over 32 to 60 min
long (mean = 55, standard deviation = 7) with gestational age, for groups MSA (mean = 39.8,
standard deviation = 1.3), MA (39.4, 1.6), and N (39.3, 2). Median time between the end of the
tracings and the delivery was 0.0 min for all of the groups. The mean (standard deviation) Apgar score
at the first minute was 6.2 (2.7) in MSA group, 8.5 (0.7) in MA group, and 8.8 (0.7) in N group.
Correlations between Approximate and Sample entropies with different tolerances were almost
perfect, as the lowest Spearman Correlation Coefficient was 0.918. All were significant at the 0.01 level
(2-tailed).
As for compression, six different compressors were applied, with five of them having two versions,
one as the lowest level of compression (faster) and another with highest level of compression (slower).
All of the Spearman Correlation Coefficients are over 0.819. All correlations are significant at the
0.01 level (2-tailed).
Table 1 shows the correlations between different compressors (for distinct levels of compression)
with entropies (with distinct tolerances). Paq8l_8 has the highest correlation with entropies with
Spearman’s Coefficient over 0.549 (p < 0.01). The lowest correlation coefficients were found between
Bzip2 and entropies, as values were under 0.143, with no statistically significant difference found.
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Table 1. Spearman Correlation Coefficients between different entropies and compressors using
different values for tolerance and levels of compression. (ApEn: Approximate entropy; SampEn:
Sample Entropy).
Brotli_1
Brotli_11
Gzip1
Gzip_9
Bzip2_1
Bzip2_9
Ppmd_2
Ppmd_16
Paq8l_1
Paq8l_8
Lzma_6
ApEn (0.1)
ApEn (0.15)
SampEn (0.2)
SampEn (0.1)
SampEn (0.15)
SampEn (0.2)
0.285 *
0.506 **
0.407 **
0.242 *
0.093
0.093
0.256 *
0.172
0.531 **
0.573 **
0.355 **
0.267 *
0.486 **
0.393 **
0.227
0.072
0.072
0.247 *
0.152
0.511 **
0.549 **
0.356 **
0.298 *
0.508 **
0.424 **
0.258 *
0.105
0.105
0.28 *
0.182
0.538 **
0.573 **
0.382 **
0.341 **
0.543 **
0.452 **
0.285 *
0.143
0.143
0.293 *
0.221
0.57 **
0.606 **
0.366 **
0.294 *
0.496 **
0.406 **
0.24 *
0.09
0.09
0.247 *
0.17
0.52 **
0.556 **
0.331 **
0.331 **
0.531 **
0.448 **
0.281 *
0.132
0.132
0.288 *
0.21
0.559 **
0.592 **
0.36 **
* Correlation is significant at the 0.05 level (2-tailed). ** Correlation is significant at the 0.01 level (2-tailed).
Table 2 describes the correlation between physiological features captured by SisPorto, using the
FIGO guidelines. The highest correlations were found between %abSTV and Mean STV (r = −0.796),
and also between %abSTV and %abLTV (r = 0.706). Note that %abSTV was the only variable
significantly correlated with all others.
Table 2. Spearman Correlation Coefficients between physiological features captured by SisPorto,
using the The International Federation of Gynecology and Obstetrics (FIGO) guidelines [19]. (%abSTV:
percentage of abnormal; mean STV: mean short term variability; %abLTV: percentage of abnormal long;
Acc: number of acceleration; Dec: number of decelerations).
%abSTV
Mean STV
%abLTV
baseline
Acc
Dec
%abSTV
Mean STV
%abLTV
Baseline
Acc
Dec
1
−0.796 **
0.706 **
0.324 **
−0.480 **
−0.375 **
1
−0.444 **
−0.201
0.357 **
0.539 **
1
0.27 *
−0.651 **
−0.101
1
−0.138
−0.015
1
−0.035
1
* Correlation is significant at the 0.05 level (2-tailed). ** Correlation is significant at the 0.01 level (2-tailed).
Both entropy and compression captured complexity in the tracings, as seen in Tables 3 and 4,
but compression was correlated with more physiologic features. Actually, there were significant
correlations between compressors and number of accelerations and decelerations per minute,
but neither were correlated with entropies. Despite that there was a significant fair correlation
between entropies and %abSTV, its magnitude was rather median, when compared with compression
(Tables 3 and 4).
Table 3. Spearman Correlation Coefficients between physiological features and Entropies.
%abSTV
Mean STV
%abLTV
baseline
Acc
Dec
ApEn (0.1)
ApEn (0.15)
ApEn (0.2)
SampEn (0.1)
SampEn (0.15)
SampEn (0.2)
−0.557 **
0.34 **
−0.319 **
−0.268 *
0.093
−0.024
−0.541 **
0.331 **
−0.321 **
−0.29 *
0.077
−0.020
−0.561 **
0.353 **
−0.328 *
−0.284 *
0.080
0.029
−0.617 **
0.339 **
−0.451 **
−0.312 **
0.199
−0.058
−0.586 **
0.302 *
−0.441 **
−0.309 *
0.178
−0.103
−0.624 **
0.338 **
−0.459 **
−0.313 **
0.206
−0.072
* Correlation is significant at the 0.05 level (2-tailed). ** Correlation is significant at the 0.01 level (2-tailed).
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Table 4. Spearman Correlation Coefficients between physiological features and Compressors.
%abSTV
Mean STV
%abLTV
baseline
Acc
Dec
Brotli_1
Brotli_11
Gzip_1
Gzip_9
Bzip2_1
Bzip2_9
Ppmd_2
Ppmd_16
Paq8l_1
Paq8l_8
Lzma_6
−0.886 **
0.783 **
−0.622 **
−0.236
0.564 **
0.410 **
−0.934 **
0.829 **
−0.6 **
−0.295 *
0.472 **
0.387 **
−0.908 **
0.794 **
−0.625 **
−0.249 *
0.542 **
0.397 **
−0.851 **
0.774 **
−0.575 **
−0.193
0.562 **
0.424 **
−0.796 **
0.729 **
−0.532 **
−0.168
0.574 **
0.441 **
−0.796 **
0.729 **
−0.532 **
−0.168
0.574 **
0.441 **
−0.838 **
0.774 **
−0.591 **
−0.288 *
0.548 **
0.487 **
−0.808 **
0.733 **
−0.539 **
−0.175
0.570 **
0.423 **
−0.935 **
0.8 **
−0.646
−0.3 *
0.527 **
0.366 **
−0.931 **
0.783 **
−0.606
−0.296 *
0.485 **
0.339 **
−0.882 **
0.839 **
−0.556 **
−0.359 *
0.432 **
0.544 **
* Correlation is significant at the 0.05 level (2-tailed). ** Correlation is significant at the 0.01 level (2-tailed).
A multiscale analysis of the tracings using either entropy or compression was also performed.
Using Approximate Entropy with tolerance of 0.2, it was possible to distinguish MSA tracings from
the other two groups up to scale 7. With Sample Entropy (0.2) the distinction was shown up to scale 4
(Figures 1 and 2).
Figure 1. Multiscale analysis of tracings using Approximate entropy (ApEn) with tolerance 0.2.
Plotted 95% Confidence Intervals of the mean for each group in each scale. (MA: mildly acidemic
fetuses; MSA: moderate-to-severe acidemic fetuses; N: the normal range).
Figure 2. Multiscale analysis of tracings using Sample entropy (SampEn) with tolerance 0.2. Plotted 95%
Confidence Intervals of the mean for each group in each scale.
Regarding compression, the opposite happens. Using all of the compressors, it was possible to
distinguish groups N and MA in lower scales. Actually, paq8l distinguished them in all scales used.
In all compressors, as scale increases, group N starts to diverge from group MSA, with bzip2 and gzip
getting statistically significant results for scale higher than 2 and 7, respectively (Figures 3–6).
Entropy 2017, 19, 688
Figure 3. Multiscale analysis of tracings using compressor Brotli with maximum level of compression.
Plotted 95% Confidence Intervals of the mean for each group in each scale.
Figure 4. Multiscale analysis of tracings using compressor Gzip with maximum level of compression.
Plotted 95% Confidence Intervals of the mean for each group in each scale.
Figure 5. Multiscale analysis of tracings using compressor Paq8l with maximum level of compression.
Plotted 95% Confidence Intervals of the mean for each group in each scale.
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Figure 6. Multiscale analysis of tracings using compressor Bzip2 with maximum level of compression.
Plotted 95% Confidence Intervals of the mean for each group in each scale.
4. Discussion
In this study, the different entropies and tolerances used in FHR analysis were highly correlated as
were the different used compressors. This suggests that different entropy and compressor estimators
are internally valid, i.e., despite some discrepancies in results, they seem to absorb similar information
from the tracings.
Despite the high correlation within similar complexity measures, the use of different values for
certain parameters, such as the threshold r when using ApEn, might lead to different performances
in the characterization of different fetal behavioral patterns or acute and chronic conditions [20].
Besides, different complexity measures seem to capture different information since correlation between
entropies and compressors is low, with bzip2 being perfectly uncorrelated, while ppmd and gzip having
some non-significant correlations with entropies. On the other hand, paq8l is moderately correlated.
It is still not totally clear why this happens, but maybe some characteristics of each compressor
algorithm may provide us some answers. For example, in order to optimize the compression process,
bzip2 performs a block sort (reversible), and, consequently, does not take advantage from the initial
structure (related with entropy).
As to the main objective of this study of comparing the two different complexity measures,
entropy and compression, with the FIGO international guidelines for fetal monitoring, we have
observed a good correlation between percentage of abnormal short term variability and entropy.
However, compression has seemed to be the measure that has captured most of the information out of
abnormal short term variability. Higher values of entropy and compression mean that the presence of
more complex structures and these structures were characteristic from healthier status. Tracings with
lower percentage of abnormal STV were typical from healthy babies, which explains the negative
correlation between the variables (compression/entropy and abnormal short term variability).
There is no uniform definition of STV. The one that is used in this work was defined in SisPorto as
subsequent FHR signals differing <1 bpm, closely following the FIGO guidelines, but other definitions
exist in the literature, with no concrete agreement on which one should be used [21]. It would be
interesting to see how compression behaves with these other variants of STV. In an affirmative case,
compression could be used as a universal measure to capture STV. Moreover, compression seems to
capture more features from tracings since numbers of accelerations and decelerations per minute were
correlated with compression but not with entropies.
Actually, it is interesting to notice that the accelerations and decelerations did not correlate with
any entropy, but correlated with all complexities. As in Baumert’s study [22], our results suggest
that acceleration and decelerations are not purely random but follow some deterministic structures
that can be explored by compression algorithms. The use of different parameters in the entropies,
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considering for instance variants of the threshold such as fuzzy functions, might allow for capturing
accelerations and decelerations.
With multiscale analysis, it was observed that entropy and compression captured distinct clinical
information from the tracings. Entropy distinguishes MSA tracings from the other two groups based
on some features that compression cannot capture, and, on the other hand, compression distinguishes
groups N and MA possibly by using all SisPorto features. Probably, entropy is correlated with some
other features (such as the dynamics) of the physiological data that is not captured by SisPorto. As in
Voss study [23] these results show that several nonlinear indices should be combined in order to
improve the performance of FHR analysis. Moreover, lower scales in entropy and higher scales
in compression can distinguish groups (MSA, MA and N). Compression is better correlated with
physiological features that are captured by SisPorto system, and, in fact, these physiological features
also do not distinguish groups at scale one. On the other hand, entropy is able to distinguish groups in
lower scales, probably because entropy is correlated with some other features, such as the dynamics of
the physiological data that are not captured by SisPorto. Notice that, in higher scales, the averaging of
the points can justify the absence of these dynamics, lowering the ability to distinguish the classes.
One limitation of this work is the low number of cases, particularly in acidemic groups. This is
particularly due to the low prevalence of these cases, which limits the obtention of datasets with higher
sample size. It would be interesting, as a future work, and with a bigger dataset, to try to build a
predictive model and evaluate how much better would entropy/compression plus SisPorto features
perform compared to SisPorto features alone.
Our results enhance the idea that entropy and compression are complementary complexity
measures. More research in this area should be done, regarding higher scale values (with long duration
tracings), and the possibility of building a model combining both measures.
Acknowledgments: João Monteiro-Santos acknowledges the support of the Digi-NewB project funded
from the European Union’s Horizon 2020 research and innovation programme under grant agreement
No 689260. Luís Antunes, Mohammad Nozari and Cristina Costa-Santos acknowledge the support of
Project “NanoSTIMA: Macro-to-Nano Human Sensing: Towards Integrated Multimodal Health Monitoring
and Analytics/NORTE-01-0145-FEDER-000016” is financed by the North Portugal Regional Operational
Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, and through the European
Regional Development Fund (ERDF). Hernâni Gonçalves acknowledges the support by ERDF through
the operation POCI-01-0145-FEDER-007746 funded by the Programa Operacional Competitividade e
Internacionalização—COMPETE2020 and by National Funds through FCT—Fundação para a Ciência e a
Tecnologia within CINTESIS, R&D Unit (reference UID/IC/4255/2013).
Author Contributions: João Monteiro-Santos and Cristina Costa-Santos substantial contribution to conception and
design; João Monteiro-Santos, Cristina Costa-Santos, João Bernardes and Hernâni Gonçalves for interpretation of
data; Mohammad Nozari analyzed data; João Bernardes, Hernâni Gonçalves, Luís Antunes, Cristina Costa-Santos
revise critically for important intellectual content; João Monteiro-Santos and Cristina Costa-Santos wrote the paper.
Conflicts of Interest: João Bernardes has been involved in the development of the commercially available SisPorto
system for FHR monitoring.
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