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Kurtosis and Kurtosis Excess are unitless parameters or statistics that quantify the distribution shape of a signal relative to a Gaussian distribution.
*** Note: "Kurtosis" and "Kurtosis Excess" are often used interchangeably, but are different quantities. Kurtosis excess has a value of 3 subtracted from it compared to Kurtosis. By subtracting 3, Kurtosis Excess will have a value of 0 for a signal with Gaussian distribution.***
Distributions can be “sharper”, “flatter”, or equal to the Gaussian distribution as shown in Figure 1.
By looking at the distribution of a signal, kurtosis can be used to quantify the ‘nature’ of the distribution:
The kurtosis statistic can be used in a number of different applications when analyzing signals. For example, it can be used to analyze signals for anomalies like nicks in a gear pair, the probability of a damaging spikes in a vibration signal, the sound quality indicator of transient sounds, or even the volatility of stock prices.
This article covers:1. Histograms and Kurtosis Excess Background
To better understand kurtosis, it is helpful to understand histograms and how to calculate the distribution of a signal.
1. Histograms and Kurtosis Excess Background
A histogram can be used to determine the distribution of a time domain signal. Where kurtosis produces only a single value, a histogram is a XY plot showing the number of times a certain amplitude value occurs.
To calculate a histogram, the y-axis of the time signal is divided into discrete steps, called either “bins” or “classes”. In Figure 2, the Y axis was divided into four “classes”. The number of data points that fall into each class is counted.
The histogram distribution of a square wave is distinct. There are large amounts of data points at the extremes (classes 1 and 4), and very little in the center (classes 2 and 3).
Other signals, like transients as shown in Figure 3, also have a distinct histogram distribution.
The histogram for a transient signal typically has one class with a large number of data points (in this case, class 2). The other classes contain very few data points (classes 1, 3 and 4).
These histograms can be compared to a Gaussian random histogram as seen in Figure 4.
Each histogram has a unique kurtosis excess value. Typically histograms are calculated with many more classes or bins than the four used in the example.
2. Kurtosis versus Kurtosis Excess
The kurtosis value of a time history is calculated by Equation 1.
The formula takes the current data point minus the mean, divided by the standard deviation, to the fourth power, and normalized by the number of data points. The result is then subtracted by 3 to ensure the kurtosis excess of a Gaussian distribution is equal to zero.
Many time signals have a specific kurtosis excess number:
The “-3” may be included or it may not be included in different implementations of kurtosis calculations. Be sure to know whether the “-3” term is being used in the calculation of of what may be called "kurtosis" in any software being used. The terms “kurtosis” and “kurtosis excess” are often used to distinguish the difference:
Kurtosis excess: Kurtosis value has 3 subtracted:
Kurtosis: Kurtosis value does not have 3 subtracted.
Depending on the software module used in Simcenter Testlab (formerly called LMS Test.Lab), either “kurtosis” or “kurtosis excess” is used. In random vibration control, “kurtosis”, also known as "kurtosis control" is available. In throughput processing, “kurtosis excess” is utilized.
3. Kurtosis Excess as a Sound Metric
Kurtosis excess can be used to differentiate between similar types of transient sound signals like “clicks”, “clunks”, and “pings”.
In Figure 5, there are two transient clicking sounds (Pascals versus time). Both signals have identical peak amplitudes, with one click being slightly longer in duration.
Listen to how the two clicks differ in sound in the video below:
Video: Two Clicking Sounds
Listening to the clicks, it is clear the “short click” is of a higher pitch. The “long click” is of a lower pitch. (Note: Headsets may be required to hear the “low click”).
Despite the similar time waveforms, the kurtosis excess values are very different (red and green trace with scale on the left). For the short duration click, the kurtosis excess peak value is over 120. For the long duration click, the kurtosis excess value is less than 40. Kurtosis excess shows a big difference between the two signals that is not visible from viewing the peak values of the time waveforms as shown in Figure 6.
Simcenter Testlab Throughput Processing was used to calculate kurtosis excess. For this analysis, a sliding frame analysis with an increment of 0.005 seconds and a frame size of 0.05 second was used. There are two main settings for creating this analysis (Figure 7):
The framed kurtosis excess will be calculated. Check out the 'Throughput Processing Tips article' for more information about processing data.
Wavelets are also a helpful method for analyzing the time-frequency content of transients.
4. Kurtosis in Vibration Control
It is common to apply random vibration excitation to a product with an electrodynamic or hydraulic shaker system. The goal is to simulate real life environments and test the product to make sure it will function properly and not fail during the desired lifetime. The equivalent lifetime of vibration is recreated using a shaker system as shown below in Figure 9.
The target random vibration profile is typically represented using a Power Spectral Density (PSD).
4.1 Power Spectral Density versus Kurtosis
The PSD is typically output by the SCADAS frontend hardware and controlled at some location on the structure. The PSD can preserve things like frequency content and a RMS acceleration value of the original time data but is not good at preserving things like the “spikiness” of the original time data.
Spikes in time data are very important when doing vibration testing; transient spikes are highly damaging events. Kurtosis preserves the time data spikes that the PSD cannot.
To illustrate the importance of kurtosis control, look at the four PSDs below in Figure 10. They are nearly identical. They all have similar frequency content with a RMS value of 0.25g.
However, looking at the original time histories of these calculated PSDs reveals something more! The time histories are not all equal. The red time history has much higher amplitude peaks than the magenta time history in Figure 11 below. This is due to the red trace having a higher kurtosis value (k = 12) than the magenta (k=3), which means there are more amplitude peaks and highly damaging events in the red time history.
Knowing that an increase in kurtosis can produce time data with more amplitude spikes (Figure 11), kurtosis can be used in addition to the Gaussian distributed random to preserve the most important aspects of the original time history.
4.2 Setting Kurtosis in Random Control
To utilize kurtosis control within Simcenter Testlab Random Control, turn on “Kurtosis Control” under “Tools -> Add-ins” as shown in Figure 12.
Enter the desired kurtosis value in the “Random Control Setup” worksheet as shown in Figure 13. The desired kurtosis value in this menu assumes that a Gaussian random signal has a value equal to 3. A value of 9 would have higher amplitude peaks in the time history. Only values between 3 and 12 can be used.
After setting the kurtosis value, “Sigma Clipping” can also be set (explained below). Sigma clipping can be defined by clicking the “Advanced…” button and going to the kurtosis tab in the “Advanced Control Setup” pop-up window (Figure 14).
Turn on ‘Use Kurtosis Control’ and set sigma clipping.
4.3 Sigma Clipping
Given a Gaussian amplitude distribution, an amplitude value could theoretically reach unlimited levels. Looking at a Gaussian normal distribution, the number of samples is on the y axis and the amplitude is on the x axis. Most of the samples will fall at some median amplitude (which will be an appropriate amplitude for the test).
However, there is a chance (a small chance), that an extremely damaging, high amplitude event could occur (Figure 15). Looking at the Gaussian random distribution, there are a very small number of samples at this high amplitude value. Without sigma clipping, there is a risk that one of these events could take place during the test.
Sigma clipping allows the test designer to protect against these rare and very damaging events. Sigma clipping is a technique in which the dynamic range of the drive signal in a random test control is reduced.
Sigma is defined as the ratio of the peak value to the RMS value of the signal (also known as crest factor). By default, this sigma clipping value is set to 20 which means that 99.999999999% of the amplitude values are passed.
By setting a smaller sigma value (for example 3), the drive signal is generated such that no peaks exceed this user defined threshold. Of course, by altering the sigma value, the resulting distribution may not be perfectly Gaussian anymore.
By setting a sigma clipping value of 3, the highly damaging event will be removed from the histogram and will not be included in the drive profile as shown in Figure 16 below.
The histograms in Figure 17 show the effect sigma clipping has on the distribution.
In this example, two time histories were captured. Both had a kurtosis of three (Gaussian random). One signal had a sigma clipping value of 3 and the other with a sigma clipping value of 20.
The overall distribution of the two signals appear to be nearly identical. However, when zoomed in to the very upper region of the histogram, it is clear that the signal with a sigma clipping of 20 (pink) contains higher amplitude events than the signal whose signal clipping was set to three.
By using sigma clipping, the likelihood of a highly damaging event can be reduced.
When performing random control tests, keep kurtosis and sigma clipping in mind. Kurtosis can preserve the peak amplitude (ie, spikey) nature of the signal while sigma clipping can protect against rare but extremely damaging events.
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