Measure of regularity of nonlinear time series
For this example, generate two signals for comparison a random signal
xRand and a perfectly regular signal
default for reproducibility of the random signal.
rng('default'); xRand = double(randn(100,1)>0); xReg = repmat([1;0],50,1);
Visualize the random and regular signals.
figure; subplot(2,1,1); plot(xRand); title('Random signal'); subplot(2,1,2); plot(xReg); title('Perfectly regular signal');
The plots show that the regular signal is more predictable than the random signal.
Find approximate entropy of the two signals.
valueReg = approximateEntropy(xReg)
valueReg = 5.1016e-05
valueIrreg = approximateEntropy(xRand)
valueIrreg = 0.6849
The approximate entropy of the perfectly regular signal is significantly smaller than the random signal. Hence, the perfectly regular signal containing many repetitive patterns has a relatively small value of approximate entropy while the less predictable random signal has a higher value of approximate entropy.
In this example, consider the position data of a quadcopter, following a circular path. The file
uavPositionData.mat contains the x, y and z-direction position data traversed by the copter.
Load the data set and visualize the quadcopter path in 3D.
For this example, use only x-direction position data for computation. Since
Lag is unknown, estimate the delay using
'Dimension' to 3. The
Lag parameters are required to compute the approximate entropy of the data.
dim = 3; [~,lag] = phaseSpaceReconstruction(xv,,dim)
lag = 10
Find the approximate entropy using the
Lag value obtained in the previous step.
approxEnt = approximateEntropy(xv,lag,dim)
approxEnt = 0.0386
Since the quadcopter is traversing a pre-defined circular trajectory of fixed radius, the position data is regular and hence, the value of approximate entropy is low.
X— Uniformly sampled time-domain signal
Uniformly sampled time-domain signal, specified as either a vector, array,
or timetable. If
X has multiple columns,
approximateEntropy computes the approximate entropy
X as a multivariate signal.
X is specified as a row vector,
approximateEntropy treats it as a univariate
dim— Embedding dimension
Embedding dimension, specified as a scalar or vector.
equivalent to the '
Dimension' name-value pair.
lag— Time delay
Time delay, specified as a scalar or vector.
lag is equivalent to
Lag' name-value pair.
comma-separated pairs of
the argument name and
Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
'Dimension'— Embedding dimension
Embedding dimension, specified as the comma-separated pair consisting
Dimension' and a scalar or vector. When
Dimension is scalar, every column in
X is reconstructed using
Dimension is a
vector having same length as the number of columns in
X, the reconstruction dimension for column
Dimension based on the dimension of your
system. For more information on embedding dimension, see
'Lag'— Delay in phase space reconstruction
Delay in phase space reconstruction, specified as the comma-separated
pair consisting of '
Lag' and a scalar. When
Lag is scalar, every column in
X is reconstructed using
Lag is a vector having
same length as the number of columns in
reconstruction delay for column
If the delay is too small, random noise is introduced in the data. In
contrast, if the lag is too large, the reconstructed dynamics does not
represent the true dynamics of the time series. For more information on
calculating optimal delay, see
'Radius'— Similarity criterion
X) | 0.2*sqrt(trace(cov(
X))) | scalar
Similarity criterion, specified as the comma-separated pair consisting
Radius' and a scalar. The similarity criterion,
also called radius of similarity, is a tuning parameter that is used to
identify a meaningful range in which fluctuations in data are to be
The default value of
X has a single column.
X has multiple columns.
approxEnt— Approximate entropy of nonlinear time series
Approximate entropy of nonlinear times series, returned as a scalar. Approximate entropy is a regularity statistic that quantifies the unpredictability of fluctuations in a time series. A relatively higher value of approximate entropy reflects the likelihood that similar patterns of observations are not followed by additional similar observations.
For example, consider two binary signals
S1 = [0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1];
S2 = [1 1 0 1 1 1 1 0 1 0 1 0 0 0 0 1];
S1 is perfectly regular since it alternates
1, that is, you can
predict the next value with knowledge of the previous value. Signal
S2 however offers no insight into the next value,
even with prior knowledge of the previous value. Hence, signal
S2 is random and less predictable. Therefore, a
signal containing highly repetitive patterns has a relatively small value of
approxEnt while a less predictable signal has a
relatively larger value of
approximateEntropy as a measure of regularity to
quantify levels of complexity within a time series. The ability to discern
levels of complexity within data sets is useful in the field of engineering
to estimate component failure by studying their vibration and acoustic
signals, or in the clinical domain where, for instance, the chance of a
seizure is predicted by observing Electroencephalography (EEG)
Approximate entropy is computed in the following way,
approximateEntropy function first generates a
delayed reconstruction Y1:N for N data points with embedding dimension m, and lag τ.
The software then calculates the number of within range points, at point i, given by,
where 1 is the indicator function, and R is the radius of similarity.
The approximate entropy is then calculated as where,
 Pincus, Steven M. "Approximate entropy as a measure of system complexity." Proceedings of the National Academy of Sciences. 1991 88 (6) 2297-2301; doi:10.1073/pnas.88.6.2297.
 U. Rajendra Acharya, Filippo Molinari, S. Vinitha Sree, Subhagata Chattopadhyay, Kwan-Hoong Ng, Jasjit S. Suri. "Automated diagnosis of epileptic EEG using entropies." Biomedical Signal Processing and Control Volume 7, Issue 4, 2012, Pages 401-408, ISSN 1746-8094.
 Caesarendra, Wahyu & Kosasih, P & Tieu, Kiet & Moodie, Craig. "An application of nonlinear feature extraction-A case study for low speed slewing bearing condition monitoring and prognosis." IEEE/ASME International Conference on Advanced Intelligent Mechatronics: Mechatronics for Human Wellbeing, AIM 2013.1713-1718. 10.1109/AIM.2013.6584344.
 Kantz, H., and Schreiber, T. Nonlinear Time Series Analysis. Cambridge: Cambridge University Press, 2003.