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Kurtosis

Siemens Genius Siemens Genius
Siemens Genius

Kurtosis

 

Kurtosis (k) is a unitless parameter or statistic that quantifies the distribution shape of a signal relative to a Gaussian distribution. The distribution could be “sharper”, “flatter”, or equal to the Gaussian distribution as shown in Figure 1.

Figure 1: Kurtosis values are negative, positive, or zero depending on the distribution of the signalFigure 1: Kurtosis values are negative, positive, or zero depending on the distribution of the signal

By looking at the distribution of a signal, kurtosis can be used to quantify the ‘nature’ of the distribution:

 

  • Random: A time domain signal with a Gaussian random distribution has a kurtosis value equal to 0 (also called a mesokurtic distribution)
  • Transient: A time domain transient or ‘spikey’ signal has a kurtosis value greater than 0 (called a leptokurtic distribution)
  • Deterministic: A fixed signal, like a sine or square wave, has a kurtosis value less than 0 (called a platykurtic 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:

  • Histograms 
  • Kurtosis versus kurtosis excess
  • Use of kurtosis as a sound metric
  • Use of kurtosis in vibration control

To better understand kurtosis, it is helpful to understand histograms and how to calculate the distribution of a signal.

 

Histograms and Kurtosis

 

Understanding histograms helps to understand kurtosis. 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.
Figure 2: Histogram distribution of square wave with 4 classesFigure 2: Histogram distribution of square wave with 4 classes

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. 
Figure 3: Transient signal histogram distributionFigure 3: Transient signal 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.

Figure 4: Square, Gaussian, and transient histograms and their corresponding kurtosis values.Figure 4: Square, Gaussian, and transient histograms and their corresponding kurtosis values.

Each histogram has a unique kurtosis value.  Typically histograms are calculated with many more classes or bins than the four used in the example.

 

Kurtosis Value and Kurtosis Excess

 

The kurtosis value of a time history is calculated by Equation 1.

 

Equation 1: Kurtosis calculationEquation 1: Kurtosis calculation

Where:

  • k = kurtosis value
  • n = the number of data points in the time signal
  • x = the amplitude value at each data point n
  • x̄ = mean of all data points in the set 
  • i is used as a counter for the samples

 

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 of a Gaussian distribution is equal to zero.

Many time signals have a specific kurtosis number:

  • Square Wave: k = -2.0
  • Sine Wave: k = -1.5
  • Gaussian random: k = 0
  • Transient = Positive number (higher, more transient)

The “-3” may be included or it may not be included in different implementations of kurtosis.  Be sure to know whether the “-3” term is being used in the calculation of kurtosis. The terms “kurtosis” and “kurtosis excess” can be used to distinguish the difference:

Kurtosis excess: Kurtosis value has 3 subtracted:

  • Gaussian random has a kurtosis equal to 0
  • Square wave has kurtosis equal to -2
  • Sine wave has kurtosis equal to -1.5

Kurtosis: Kurtosis value does not have 3 subtracted. 

  • Gaussian random has a kurtosis equal to 3
  • Square wave has kurtosis equal to 1
  • Sine wave has kurtosis equal to 1.5

Depending on the software module used in 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 utlized.

 

Application: Kurtosis as a Sound Metric

 

Kurtosis can be used to idifferentiante between different 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.

Figure 5: Two clicking sound time histories. Blue: “Short” duration click time history. Red: “Long” duration click time history.Figure 5: Two clicking sound time histories. Blue: “Short” duration click time history. Red: “Long” duration click time history.

Listen to how the two clicks differ in sound in the video below:

 

(view in My Videos)

 

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 values are very different (red and green trace with scale on the left). For the short duration click, the kurtosis peak value is over 120.  For the long duration click, the kurtosis value is less than 40. Kurtosis (k) 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.

Figure 6: Kurtosis framed calculation and time waveforms for the low click and high clickFigure 6: Kurtosis framed calculation and time waveforms for the low click and high click

 LMS Test.Lab 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):

 

  • Acquisition Parameters: Set the Frame and Increment for the desired overlap. Increment is set in the ‘Tracking and Triggering’ tab.  Framesize is set in the ‘FS Acquisition’ tab. To understand the relationship between the frame and increment see the ‘LMS Test.Lab Throughput Processing Tips’ article. 

Figure 7: ‘Acquisition parameters’ dialog sets frame size and incrementFigure 7: ‘Acquisition parameters’ dialog sets frame size and increment

  • Section Settings: Find the ‘Frame Statistics’ tab and turn on “Kurtosis (excess)”. Make sure to also select the appropriate ‘Channel Group’ for your data (turn them all on if uncertain) as shown in Figure 8.

Figure 8: Throughput Processing -> Section Settings -> Frame Statistics -> Kurtosis (excess)Figure 8: Throughput Processing -> Section Settings -> Frame Statistics -> Kurtosis (excess)

The framed kurtosis excess will be calculated. Check out the 'Throughput Processing Tips article' for more information about processing data.

 

Application: 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.

Figure 9: Random shaker control test setupFigure 9: Random shaker control test setup

The target random vibration profile is typically represented using a Power Spectral Density (PSD). The PSD is typically output by the SCADAS 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.

 

Figure 10: Four PSDs with similar frequency content and the same RMS amplitudeFigure 10: Four PSDs with similar frequency content and the same RMS amplitude

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.

 

Figure 11: Higher the kurtosis for a time history, the higher and more frequent presence of damaging peaksFigure 11: Higher the kurtosis for a time history, the higher and more frequent presence of damaging peaksKnowing 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.

 

To utilize kurtosis control within LMS Test.Lab Random Control, turn on “Kurtosis Control” under “Tools -> Add-ins” as shown in Figure 12.

 

Figure 12: Tools -> Add-ins -> Kurtosis ControlFigure 12: Tools -> Add-ins -> Kurtosis Control

 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.

 

Figure 13: Random Control setup, the “Desired Kurtosis value” controls the kurtosisFigure 13: Random Control setup, the “Desired Kurtosis value” controls the kurtosis

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).

 

Figure 14: Sigma ClippingFigure 14: Sigma Clipping

Turn on ‘Use Kurtosis Control’ and set sigma clipping.

 

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.

 

Figure 15: Different sigma values will effect which values are included in the Gaussian random distributionFigure 15: Different sigma values will effect which values are included in the Gaussian random distributionSigma 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.

 

Figure 16: Different sigma values will effect which values are included in the Gaussian random distributionFigure 16: Different sigma values will effect which values are included in the Gaussian random distribution

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.

Figure 17: Both signals have a kurtosis of three (Gaussian random). The blue trace has a sigma clipping of 3. The pink trace has a sigma clipping of 20. The upper limit of amplitude values show that the signal with a sigma clipping of 20 has more high amplitude events.Figure 17: Both signals have a kurtosis of three (Gaussian random). The blue trace has a sigma clipping of 3. The pink trace has a sigma clipping of 20. The upper limit of amplitude values show that the signal with a sigma clipping of 20 has more high amplitude events.

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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