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This paper presents a new algorithm making use of kurtosis, which is a statistical parameter, to distinguish the seismic signal generated by a person's footsteps from other signals. It is adaptive to any environment and needs no machine study or training. As persons or other targets moving on the ground generate continuous signals in the form of seismic waves, we can separate different targets based on the seismic waves they generate. The parameter of kurtosis is sensitive to impulsive signals, so it's much more sensitive to the signal generated by person footsteps than other signals generated by vehicles, winds, noise, etc. The parameter of kurtosis is usually employed in the financial analysis, but rarely used in other fields. In this paper, we make use of kurtosis to distinguish person from other targets based on its different sensitivity to different signals. Simulation and application results show that this algorithm is very effective in distinguishing person from other targets.
Persons or other targets moving on the ground generate continuous impacts which propagate in the form of seismic waves that can be measured by geophones or seismic sensors. The signal generated by a person's footsteps can be distinguished from the signals generated by other targets, based on their impulsive nature.
Many previous papers have focused on feature extraction and classifier design. These methods are so complicated and lacking in robustness, that it is impractical to apply them to common applications. In [
From above, we can see that there are more or less faults in the existing methods used in person recognition. In this paper, we provide an algorithm using the parameter of kurtosis which shows more simpleness and robustness. The remainder of this paper is organized as follows: section 2 describes the statistical meaning of kurtosis. Section 3 lists and discusses the simulation results of the algorithm using kurtosis which is applied to recognize person footsteps. Section 4 gives the conclusion and predicts future work.
Karl Pearson [
X denotes the sequence of inputs,
An unbiased estimator [
For large sample sizes (n>1000),
Pearson [
Moors [
Let us denote p(x) the probability density function (pdf) of a random process x(t) and E() the mean. The kurtosis k[x(t)] is:
Assume the mean E() is zero, and k[p(x)] can be written as:
Clearly, the kurtosis sign ks(x) is equal to the fourthorder cumulant sign. Some properties can be easily derived.
Let
Therefore, in the following, the study may be restricted to a zeromean process x(t) whose the pdf p(x) is even and has a variance
It is well known that the kurtosis of a Gaussian distribution is equal to zero. Intuitively, the sign of the kurtosis seems related to the comparison between p(x) and Gaussian distribution, by considering the asymptotic properties of the distribution and the following definition:
A pdf p(x) is said overGaussian (respectively subGaussian), if ∀
Let us consider that for x>0, the equation p(x)=g(x) only has one sulotion
Let us consider that the pdf p(x) is an overGaussian signal. Then, the sign of p(x)g(x) remains constant on each interval [0,
Where 0 <
Taking into account that p(x) is overGaussian, we deduce
Using the above two equation, we remark that:
Finally, if p(x) is an overGaussian pdf, then its kurtosis is positive. Using the same reason and under the same condition, we can claim that a subGaussian pdf has a negative kurtosis.
There are some basic results about kurtosis given by Richard [
For standard scores,
Assume the two points of the distribution are at 0 and 1, with p being the frequency at 1. Then
As
So we have
For a threepoint distribution in which the density is p, then
So
Starting again with a normal distribution, moving scores from the tails and the center to the shoulders will decrease kurtosis. A uniform distribution certainly has a flat top, with
Kurtosis is usually of interest only when dealing with approximately symmetric distributions. Skewed distributions are always leptokurtic [
There is much confusion about how kurtosis is related to the shape of distributions. Many people have asserted that kurtosis is a measure of the peakedness of distributions, which is not strictly true.
It is easy to confuse low kurtosis with high variance, but distributions with identical kurtosis can differ in variance, and distributions with identical variances can also differ in kurtosis. Here are some simple distributions that may explain what kurtosis is, in part, a measure of tail heaviness relatives to the total variance in the distribution.
A has the least kurtosis (–2 is the smallest possible value of kurtosis) and G the most. In the maximally platykurtic distribution A, which initially appears to have all its scores in its tails, no score is more than one σ away from the mean, that is, it has no tails! In the leptokurtic distribution G, which seems only to have a few scores in its tails, one must remember that those scores (5 and 15) are much farther away from the mean (3.3 σ) than are the 5's & 15's in distribution A. In fact, in G nine percent of the scores are more than three σ from the mean, much more than you would expect in a mesokurtic distribution (like a normal distribution), thus G does indeed have fat tails.
Kurtosis is the degree of peakedness of a distribution, defined as a normalized form of the fourth central moment of a distribution. The kurtosis for a number of some common distributions is shown below.
The following example makes it quite clear that a higher kurtosis implies that there are more extreme observations (or that the extreme observations are more extreme). It is also evident that a higher kurtosis also implies that the distribution is more ‘singlepeaked’ (this would be even more evident if the sum of the frequencies was constant).
We may define mesokurtic as “having
From the discussion above, the statistical meanings of kurtosis is given: kurtosis is a kind of measure of data's degree of outlier or data's peakedness.
The kurtosis of a random variable X is defined:
Where
The seismic signals of persons, trucks and tracklayers are collected at the sample rate of 1Ksps with the resolution of 16 bits, and the kurtosis extracted from each target signal is calculated every 512 samples. For each 512 samples of the signal, the kurtosis is calculated by the following formulation:
Where E denotes the mean of input signal,
Why are tailedness and peakedness both components of kurtosis? It is basically because kurtosis represents a movement of mass that does not affect the variance. Consider the case of positive kurtosis, where heavier tails are often accompanied by a higher peak. Note that if mass is simply moved from the shoulders of a distribution to its tails, then the variance will also be larger. To leave the variance unchanged, one must also move mass from the shoulders to the center, which gives a compensating decrease in the variance and a peak. For negative kurtosis, the variance will be unchanged if mass is moved from the tails and center of the distribution to its shoulders, thus resulting in light tails and flatness [
The kurtosis of several typical distributions, including normal distribution, rayleigh distribution and beta distribution, is given in
In this section, we will simulate the results of kurtosis. First, we collect the seismic signal by the seismic sensors. The raw seismic signal is then divided into N blocks with 512 samples each. The parameter of kurtosis is calculated every block. That is to say, we can get only one value from 512 samples. In order to make the simulation results clearer and easier to understand, we add 511 zeros to each kurtosis to form the final simulation results.
In
From
From
After comparing the results from
In
It can be seen from
From above, we can make the following conclusions:
The kurtosis of impulsive signals is far beyond 5;
The kurtosis of nonimpulsive signals is below 5;
The values of kurtosis are independent of the geologic features and are only dependent on the feature of signals.
From the analysis above, it is clear that we can distinguish person from other targets depending on the value of kurtosis in any atmosphere and needs no machine study and training.
From the discussion above, it is clear that walker can be detected and distinguished from other targets by comparing the kurtosis of the seismic signal. The value of kurtosis depends on the features of the signals and is independent of the geologic features.
This work was supported by the vital item of Shanghai Science and Technology Committee (NO. 054SGA1001). The author thanks Dr. Jianming Wei and Dr. Maolin Hu for their kindly help and constructive advice. He also thanks his mentor Prof. Haitao Liu.
the kurtosis of several distributions, including normal distribution, rayleigh distribution and beta distribution. The left figure is the samples of the distribution, and the right is the kurtosis respectively. (a). the samples of normal distribution and its kurtosis. (b). the samples of rayleigh distribution and its kurtosis. (c). the samples of beta distribution and its kurtosis.
The seismic signal, collected in gravelly clay region, of target(left) and its kurtosis every 512 samples. The left figure of (a) is the seismic signal of tracklayer and the right figure of (a) is the kurtosis calculated every 512 samples. The left figure of (b) is the seismic signal of truck and the right figure of (b) is the kurtosis calculated every 512 samples too. The left of (c) is the seismic signal of truck and the right is the kurtosis calculated every 512 samples too.
The seismic signal, collected in loessal soil region, of target(left) and its kurtosis every 512 samples. The left figure of (a) is the seismic signal of tracklayer and the right figure of (a) is the kurtosis calculated every 512 samples. The left figure of (b) is the seismic signal of truck and the right figure of (b) is the kurtosis calculated every 512 samples too. The left of (c) is the seismic signal of truck and the right is the kurtosis calculated every 512 samples too.
The simulation result of the seismic signal of person in gravelly clay region.
The simulation result of the seismic signal of person in loessal soil region.
Kurtosis for 7 Simple Distributions Also Differing in Variance.
05  20  20  20  10  05  03  01 
10  00  10  20  20  20  20  20 
15  20  20  20  10  05  03  01 
 
Kurtosis  −2.0  −1.75  −1.5  −1.0  0.0  1.33  8.0 
 
Variance  25  20  16.6  12.5  8.3  5.77  2.27 
The kurtosis for a number of common distributions.
Bernoulli distribution 

Beta distribution 

Binomial distribution 

Chisquared distribution 

FisherTippett distribution 

Gamma distribution 

Geometric distribution 

Halfnormal distribution 

Laplace distribution  3 
Log normal distribution  
Maxwell distribution 

Negative binomial distribution 

Normal distribution  0 
Poisson distribution 

Rayleigh distribution 

Student's tdistribution 

Continuous uniform distribution 

Discrete uniform distribution 

Kurtosis for Seven Simple Distributions Not Differing in Variance.
–6.6  0  0  0  0  0  0  1 
–0.4  0  0  0  0  0  3  0 
1.3  0  0  0  0  5  0  0 
2.9  0  0  0  10  0  0  0 
3.9  0  0  20  0  0  0  0 
4.4  0  20  0  0  0  0  0 
5  20  0  0  0  0  0  0 
10  0  10  20  20  20  20  20 
15  20  0  0  0  0  0  0 
15.6  0  20  0  0  0  0  0 
16.1  0  0  20  0  0  0  0 
17.1  0  0  0  10  0  0  0 
18.7  0  0  0  0  5  0  0 
20.4  0  0  0  0  0  3  0 
26.6  0  0  0  0  0  0  1 
 
Kurtosis  −2.0  −1.75  −1.5  −1.0  0.0  1.33  8.0 
 
Variance  25  25.1  24.8  25.2  25.2  25.0  25.1 