Analysis of periodic information in a signal

A “periodic signal parameter” (PSP) indicates periodic patterns in an autocorrelated vibration waveform and potential faults in a monitored machine. The PSP is calculated based on statistical measures derived from an autocorrelation waveform and characteristics of an associated vibration waveform. The PSP provides an indication of periodicity and a generalization of potential fault, whereas characteristics of the associated waveform indicate severity. A “periodic information plot” (PIP) is derived from a vibration signal processed using two analysis techniques to produce two X-Y graphs of the signal data that share a common X-axis. The PIP is created by correlating the Y-values on the two graphs based on the corresponding X-value. The amplitudes of Y-values in the PIP is derived from the two source graphs by multiplication, taking a ratio, averaging, or keeping the maximum value.

FIELD

This invention relates to analysis of signals. More particularly, this invention relates to methods for extracting periodic information from a vibration waveform or other signal containing periodic information.

BACKGROUND

By some estimates, up to half of all mechanical failures in process plants are induced by process conditions. Therefore, providing feedback to an operator that the process machines are being operated in a non-optimal configuration provides a way for the operator to avoid harmful operating states, thereby substantially extending mean time between failures (MTBF) or mean time between repairs (MTBR) on production assets.

Vibration analysis is a well proven technology for detecting faults in rotating machinery. The process of determining the severity and specifics of a fault can be very involved. Part of the analysis process involves determining whether periodic signals are present. While maintenance personnel are concerned with detailed analyses of faults, operations personnel only want to know if a problem exists. Providing a few fault-related parameters to the operator can be sufficient in accomplishing this task. Fault-related parameters can be related to amplitudes of energy from particular vibration frequencies (bandwidth), signal processing techniques such as PeakVue™, and the presence of periodic signals. Parameters calculated from bandwidth and signal processing techniques are well defined. However, a parameter indicating the presence of periodic signals has not been defined.

Further, the ability to detect mechanical faults in industrial machinery is a task requiring skilled analytical personnel with years of training and experience. Because of budgetary and personnel constraints, a qualified analyst may be pressed to analyze most or all of the equipment in a plant. Any technology, technique or tool that can simplify the analyst's job is valuable. Although the Fast Fourier Transform (FFT) is a technique that may be used to simplify the analyst's job, identifying important peaks in an FFT plot can be difficult due to low amplitude and noise issues. The analysis could be made easier with the derivation of a graph that reflects only periodic signals present in the measurement.

What is needed, therefore, is a system for calculating a periodic signal parameter based on an autocorrelation waveform derived form a vibration waveform. Those skilled in the art will see that autocorrelation is one of several ways to quantify the periodicity in a given signal. What is also needed is a system for deriving a graph, also referred to herein as a “periodic information plot,” that reflects only periodic signals present in a measurement waveform.

SUMMARY

Periodic Signal Parameter

The autocorrelation coefficient function is a mathematical process that determines how much of the energy in a waveform is periodic. The pattern of the periodic peaks can be very helpful in identifying fault types. Recognizing these patterns and how to apply them requires an experienced analyst. Preferred embodiments of the present invention calculate a value that is representative of general periodic patterns, which in turn signify potential faults. This value, referred to herein as a “periodic signal parameter” (PSP), is calculated based on statistical measures derived from an autocorrelation waveform along with characteristics of the associated vibration waveform. While the PSP derived from the autocorrelation function produces an indication of periodicity and a generalization of potential fault, characteristics of the associated vibration waveform afford a measure of severity. The combination of these two identities provide further indication as to potential problems associated with machines on the plant floor. This is a significant advantage for a machine operator on the plant floor who may have little-to-no vibration analysis experience.

The process of calculating the PSP begins with taking the autocorrelation of a vibration waveform. Once this is accomplished, several statistical calculations are performed. In a preferred embodiment, these statistical calculations include the maximum absolute waveform peak, standard deviation of the waveform, maximum absolute peak after the first 3% of the waveform, crest factor of both the waveform and positive waveform values, and a sorted mean of positive waveform peak values. The sorted mean is preferably calculated from a subset of values, in this case the larger set is the positive waveform peak values. The sorted subset preferably comprises all peak values from the positive waveform, excluding outliers. The outliers are peak values that exceed a statistically defined standard deviation about the mean. Therefore, the sorted mean is the mean value of the sorted positive waveform peak subset.

Because the autocorrelation of a waveform is normalized to ±1, the maximum standard deviation of the waveform is 1. Therefore, the base value of the PSP ranges from 0 to 1. Mathematical operations can be performed on the base value to achieve a desired scaling. An example would be to multiply the base value by 10 to achieve a PSP range from 0 to 10. Additionally, taking the square root of the PSP base value will accentuate variations in the lower end of the scale, which can then be multiplied by 10 to achieve a PSP range from 0 to 10. As discussed in more detail hereinafter, the PSP is calculated based on the value of the standard deviation of the autocorrelated waveform plus contributions centered on empirical observations from the other calculated statistical parameters mentioned above. Examples of autocorrelated waveforms along with the associated PSP values are provided in the detailed description.

The PSP may apply to autocorrelated waveforms derived from filtered and unfiltered acceleration, velocity or displacement waveforms as well as processed waveforms. Two examples of processed waveforms are results of the PeakVue™ signal processing and demodulation techniques.

Periodic Information Plot

As discussed above, the autocorrelation coefficient function is a mathematical process that indicates whether there is periodicity in a signal. When viewing an autocorrelation waveform, periodic signals are typically evident in the data. However, it is not easy to distinguish the exact frequency or amplitude of these periodic signals from the autocorrelation waveform. By taking a Fast Fourier Transform (FFT) of the autocorrelation waveform, distinct frequency values are evident. By comparing the autocorrelation spectrum to the standard spectrum, the true amplitude of each signal at these frequencies can be obtained.

Preferred embodiments described herein provide a method for analyzing and displaying data to reveal periodicity in a signal. The embodiments include processing the raw signal using two different sets of analysis techniques, thereby producing two X-Y graphic representations of the signal data that share a common X-axis. A third graph is created by correlating the Y-values on the first two graphs based on the corresponding X-value. The amplitude of each Y-value can be derived from the two source graphs using a variety of techniques, including multiplication, taking a ratio, averaging, or keeping the maximum value. The resulting synthesized graph, also referred to herein as a Periodic Information Plot (PIP), accentuates signal components that are pertinent to a given diagnosis while eliminating other undesired signal components. This provides for visualizing the data in a way that simplifies the recognition and quantification of desired characteristics present in the raw signal. The diagnosis may be accomplished either by a human or a computerized expert system. For a human analyst, the technique reduces training requirements while bringing increased efficiency and accuracy. With a computerized expert system, the technique provides new methods for diagnostic software to recognize significant patterns contained in the original signal.

Thus, the analysis process is made easier by providing the analyst with a spectrum showing only the periodic signals present in the data. While the same periodic information is present in the original spectrum generated from the original data, it is often difficult to recognize the periodic information because the noise levels are similar in magnitude to the periodic information.

DETAILED DESCRIPTION

FIG. 1depicts an exemplary system100for deriving and analyzing periodic information in a vibration signal. In the embodiment ofFIG. 1, a sensor104, such as an accelerometer, is attached to a machine102to monitor its vibration. Although an accelerometer is depicted in the exemplary embodiment ofFIG. 1, it should be appreciated that other types of sensors could be used, such as a velocity sensor, a displacement probe, an ultrasonic sensor, or a pressure sensor. The sensor104generates a vibration signal (or other type of signal for a sensor other than an accelerometer) that contains periodic information. The vibration signal is provided to a data collector106preferably comprising an analog-to-digital converter (ADC)108for sampling the vibration signal, a low-pass anti-aliasing filter110(or other type of filter), and buffer memory112. For example, the data collector106may be a digital data recorder manufactured by TEAC or a vibration data collector. In the embodiment ofFIG. 1, the vibration signal data is transferred from the data collector106to a periodic information processor114that performs the information processing tasks described herein. In an alternative embodiment, the processing tasks are performed by a processor in the data collector106.

Periodic Signal Parameter

FIG. 2depicts a flowchart of a method for calculating a periodic signal parameter (PSP) according to a preferred embodiment of the invention. A time-domain vibration waveform is measured, such as using the accelerometer104or other sensor attached to the machine102being monitored (step12). An autocorrelation function is performed on the vibration waveform to determine how much of the energy in the waveform is periodic (step14). In a preferred embodiment, the autocorrelation function cross-correlates the vibration waveform with itself to find repeating patterns within the waveform. The autocorrelation function outputs an autocorrelation waveform16, examples of which are depicted inFIGS. 3-7. Several statistical characteristics of the autocorrelation waveform are calculated, including the standard deviation (σ), the maximum absolute peak amplitude in the waveform (MaxPeak), the maximum absolute peak after the first 3% of the waveform (MaxPeak (after first 3%)), and the crest factor (CF1) (step18). The positive waveform peaks are sorted out (step32), any of those peaks that are statistically too large are discarded (step34), and the mean amplitude (sorted μ) and the crest factor (CF2) of the remaining peaks are calculated (step35). Methods for sorting and discarding peaks that are statistically too large are described hereinafter.

If MaxPeak is greater than or equal to 0.3 (step20) and

MaxPeak⁢⁢(after⁢⁢first⁢⁢3⁢%)sorted⁢⁢μ≥4,(step⁢⁢22)
then Y=0.025 (step24). If MaxPeak is greater than or equal to 0.3 (step20) and

If MaxPeak is less than 0.3 (step20) and CF1 less than 4 and σ is less than or equal to 0.1 (step26), then Z=0.025 (step28). If MaxPeak is less than 0.3 (step20) and CF1 is not less than 4 or a is greater than 0.1 (step26), then Z=0 (step30).

If CF2 is greater than or equal to 4 and the number of discarded peaks is greater than 2 (step36), then W=0.025 (step38). If CF2 is less than 4 or the number of discarded peaks is not greater than 2 (step36), then W=0 (step40).

MaxPeakMaxPeak⁢⁢(after⁢⁢first⁢⁢3⁢%)>1(step⁢⁢42)
and σ is between 0.1 and 0.9 (step44), then X=0.1 (step46). If

MaxPeakMaxPeak⁢⁢(after⁢⁢first⁢⁢3⁢%)≤1(step⁢⁢42)
or σ is not between 0.1 and 0.9 (step44), then X=σ (step48).

The PSP is the sum of the values of X, W, Y and Z (step50).

In general, smaller PSP values are indicative of more noise and less distinctive frequencies, while larger PSP values are symptomatic of more periodic (i.e. sinusoidal) signals relating to large single frequencies. As shown inFIG. 3, PSP values of less than a first threshold, such as 0.1, indicate that the vibration waveform is mostly noise. As shown inFIG. 4, the algorithm for the PSP assigns a value of 0.1 to signals having low amplitude, higher frequency data. This data may also prove to be bad data. As shown inFIG. 5, PSP values between first and second thresholds, such as between about 0.10 and 0.14, indicate that distinct frequencies are present but there is still a significant amount of random noise. As shown inFIG. 6, PSP values greater than the second threshold, such as greater than about 0.14, indicate very distinctive frequencies, such as vane pass or ball pass frequencies, along with small amplitude signals indicative of lower frequencies, such as RPM or cage along with their harmonics. As shown inFIG. 7, PSP values greater than a third threshold, such as greater than 0.5 and above, indicate large dominant single frequencies in the spectrum taken from the vibration waveform. The closer the PSP value is to 1.0, the waveform has more periodic (i.e. sinusoidal) signal components and less random noise.

Following are some advantages of generating a PSP.The PSP provides a single number indicative of the periodic frequencies in a waveform.Statistical values are calculated from the autocorrelated waveform and one or more of these values are combined to produce the PSP.Indication of bad or noisy data is provided.Information about periodicity can be extracted from a large data set and broadcast via a small bandwidth protocol such as HART, wireless HART, and other similar protocols.The PSP value may be applied specifically to PeakVue™ data in order to distinguish between periodic and non-periodic faults, such as lubrication, cavitation, bearing, gear and rotor faults.The PSP value can be used in conjunction with other information to generate an indication of machine condition (i.e. nature of mechanical fault, severity of the fault). The other information may include:the original waveform;processed versions of the waveform;information (i.e. peak value, crest factor, kurtosis, skewness) obtained from the original vibration waveform;information obtained from a processed version of the original waveform (i.e. PeakVue™ processed, rectified, or demodulated waveform); and/orone or more rule sets.
A simple example is illustrated in Table 1 below, where derived values representing PSP output and Stress Wave Analysis output (for example, maximum peak in the PeakVue™ waveform or another derivative of PeakVue™ type analysis or another form of stress wave analysis) are used to distinguish between different types of faults. In the majority of cases, severity of the defect increases as the level of PeakVue™ impacting increases. Although the example below refers to a Stress Wave value, other embodiments may use other vibration waveform information indicative of an impacting or other fault condition.

TABLE 1PSP and Stress Wave Analyses OutputsPeriodic [right]PSP - LowPSP - HighStress Wave [below](PSP < PSP threshold)(PSP > PSP threshold)PeakVue ™ or other stressNo fault indication:Early stage periodic fault related defect:wave analysis - Lowno action called forlook for early indication of one of the(Stress Wave value <based on this findingperiodic fault types such as those listedStress Wave threshold)belowPeakVue ™ or other stressNon-periodic fault:Periodic fault:wave analysis - Highlook for further orlook for rolling element bearing defect or(Stress Wave value >confirming evidence ofgear defect or other source of repetitiveStress Wave threshold)inadequate lubrication orperiodic mechanical impacting - useleak or contact friction orfrequency information and other informationpump cavitationto distinguish among multiple possiblecauses

A further embodiment of the present invention employs a programmable central processing unit programmed with program logic to assist a user with an interpretation of waveform information. The program logic compares the Periodic Signal Parameter and Stress Wave analysis information with expected or historical or empirically-derived experiential values to discern a relative ranking from low to high. Then discrete or graduated outputs, such as those portrayed in Table 1 above, are employed to select logically arrayed observations, findings, and recommendations. In addition to evaluating PSP and Stress Wave Analysis information, program logic sometimes prompts a user to supply additional information or obtains additional information from another source such as from a knowledge base, to enable the logic to distinguish between two or more possible logical results. For example, program logic that returns a high PSP and a high Stress Wave Analysis finding may select a rolling element defect finding rather than other possible findings within that category because a similarity is calculated when program logic compares a periodic frequency finding and a bearing fault frequency for a machine component identified in a knowledge base.

Another technique to differentiate between lubrication and pump cavitation is to look at the trend of the impacting. If it increases slowly, then insufficient lubrication should be suspected. If it increases suddenly on a pump, then it is likely pump cavitation. If combined with logic or inputs on a control system, then the logic could look for process configuration changes that occurred at the same time as the increase in impacting—along with a low PSP—to confirm pump cavitation. In some embodiments, the system suggests to the operator what action caused the cavitation, so that the operator can remove the cause and stop the machine from wearing excessively and failing prematurely.

Periodic Information Plot

A preferred embodiment of the invention creates a new type of vibration spectrum, referred to herein as a Periodic Information Plot (PIP). In this embodiment, a signal is collected from plant equipment (i.e. rotating or reciprocating equipment) and is processed using two different sets of analysis techniques as depicted inFIG. 8.

First, a waveform is acquired (step60ofFIG. 8), such as a vibration waveform acquired using the system depicted inFIG. 1. If employing a high-pass filter and peak-hold decimation to an oversampled waveform to capture impacting information (such as using the PeakVue™ process), this may be a calculated waveform. An FFT of the waveform is taken (step62), resulting in a vibration spectrum (VS)64with frequency on the X-axis and amplitude on the Y-axis, an example of which is shown inFIG. 9.

The waveform from step60is also autocorrelated (step66) to generate a waveform referred to herein as the autocorrelation waveform68, having time on the X-axis and the correlation factor on the Y-axis. The autocorrelation process accentuates periodic components of the original waveform, while diminishing the presence of random events in the original signal. As a result of the autocorrelation calculations, the associated waveform produced has half the x-axis (time) values as that of the original vibration waveform. Therefore, the timespan of the autocorrelation waveform will be half of that of the original vibration waveform. An optional step (70) takes the square root of the correlation factor (Y-axis values) to provide better differentiation between lower amplitude values.

An FFT of the autocorrelation waveform is taken (step72), resulting in an autocorrelation spectrum (AS)74. Since random events have largely been removed from the autocorrelation waveform, the remaining signal in the autocorrelation spectrum is strongly related to periodic events. As shown inFIG. 10, the autocorrelation spectrum has frequency on the X-axis and amplitude related to the correlation factor on the Y-axis. Because the autocorrelation waveform's duration is half that of the vibration waveform, the associated autocorrelation spectrum has half the lines of resolution compared to the vibration spectrum.

In a preferred embodiment, the vibration spectrum and the autocorrelation spectrum are processed to derive a graph referred to herein as the Periodic Information Plot (PIP) (step76). Several methods for processing the vibration spectrum and the autocorrelation spectrum may be used, three of which are described herein.

Because the vibration spectrum is twice the resolution of the autocorrelation spectrum, a point-to-point comparison for values on the x-axis (frequency) between the two spectra is not possible. However, a point-to-point comparison can be made by mathematically combining the amplitude values of two x-axis values in the vibration spectrum (step65) for each associated x-axis value in the autocorrelation spectrum. Each XAS(n) value of the autocorrelation spectrum (where n=1 . . . N, and N is the number of lines of resolution for the autocorrelation spectrum) is mapped to the XVS(2n) value on the vibration spectrum. The mathematically combined x-axis value is defined such that XMCVS(n)=XVS(2n). The mathematically combined amplitude values YVS(2n) and YVS(2n−1) (herein termed YMCVS(n)) associated with the XMCVS(n) value from the vibration spectrum are calculated from the amplitudes of both the XVS(2n) and XVS(2n−1) frequencies from the x-axis. The calculation for deriving the mathematically combined amplitude value associated with the XMCVS(n) value from the vibration spectrum is:
YMCVS(n)=√{square root over ((YVS(2n−1))2+(YVS(2n))2)},  Eq. (0)
where n=1 . . . N and N is the number of lines of resolution found in the autocorrelation spectrum.

In a first method (step76a), for each X-value in the PIP (XPIP1), the Y-value in the PIP (YPIP1) is determined by multiplying the mathematically combined Y-value in the vibration spectrum (YMCVS) by the corresponding Y-value in the autocorrelation spectrum (YAS), according to:
YPIP1(n)=YMCVS(n)×YAS(n)  Eq. (1)
for n=1 to N, where N is the number of X-values (frequency values) in the autocorrelation spectrum. Since amplitudes of periodic signals in the autocorrelation spectrum are higher than the amplitudes of random signals, the multiplication process will accentuate the periodic peaks while decreasing non-periodic peaks. An example of a PIP formed by the first method is depicted inFIG. 11. In all of the examples depicted herein, N=1600.

In a second method (step76b), for each X-value in the PIP (XPIP2), the Y-value in the PIP (YPIP2) is determined by comparing the corresponding Y-value in the autocorrelation spectrum (YAS) to a predetermined threshold value (YTHR). For each autocorrelation spectrum amplitude greater than this threshold value, the associated amplitude for PIP (YPIP2(n)) will be set to the corresponding mathematically combined value from the vibration spectrum (YMCVS(n)). YASvalues above the predetermined threshold indicate data that is largely periodic. Thus, the YPIP2values are determined according to:
IfYAS(n)>YTHR,YPIP2(n)=YMCVS(n)  Eq. (2a)
IfYAS(n)≦YTHR,YPIP2(n)=0 (or some other default level)  Eq. (2b)
for n=1 to N.

In one preferred embodiment of the second method, YTHRis set to only include a percentage of the largest peaks from the autocorrelation spectrum. The percentage may be calculated based on the percent periodic signal in the autocorrelation waveform. The percent periodic signal is calculated based on the autocorrelation coefficient, which is the square root of the Y-value of the largest peak in the autocorrelation waveform. For this method, only the percent periodic signal of the total number of autocorrelation spectrum peaks will be evaluated. An example of a PIP formed by this method, with YTHRset to 59%, is depicted inFIG. 12.

In another preferred embodiment of the second method, YTHRis set to include only peaks with values that are within the “percent periodic signal” of the largest peak value in the autocorrelation spectrum. These peaks, along with their harmonics that appear in the autocorrelation spectrum, will be utilized as the group of peaks to be intersected with those in the vibration spectrum to form the PIP. An example of a PIP formed by this method, with YTHRset to 59%, is depicted inFIG. 13.

In a third method (step76c), the PIP is determined according to the first method described above, and then the threshold of the second method is applied to the PIP according to:
IfYPIP1(n)>YTHR,YPIP3(n)=YPIP1(n)  Eq. (3a)
IfYPIP1(n)≦YTHR,YPIP3(n)=0 (or some other default level)  Eq. (3b)
for n=1 to N. An example of a PIP formed by this method is depicted inFIG. 14.

Some embodiments also derive a Non-periodic Information Plot (NPIP) that consists of only the Y-values of the autocorrelation spectrum that are less than a predetermined threshold (step78). Thus, the NPIP includes only non-periodic components. An example of an NPIP formed by this method is depicted inFIG. 15.

Some embodiments also derive a Periodicity Map from the vibration spectrum and the autocorrelation spectrum (step82). The Periodicity Map is created by pairing the mathematically combined Y-values from the vibration spectrum and the autocorrelation spectrum corresponding to any given X-value of the autocorrelation spectrum. These pairs are plotted with the mathematically combined Y-value from the vibration spectrum YMCVS(n) as the X-value of the point on the map XPM(n), and the Y-value from the autocorrelation spectrum YAS(n) as the corresponding Y-value on the map YPM(n), according to:
XPM(n)=YMCVS(n)  Eq. (4a)
YPM(n)=YAS(n)  Eq. (4b)
for n=1 to N. As shown inFIG. 16, the resulting graph resembles a probability mapping. A specific software implementation would allow the user to run a cursor over each point to view the values creating that point.

Some embodiments also derive a Circular Information Plot from any of the Periodic Information Plots described above (step80). Once a linear PIP is calculated, an inverse FFT can be applied to generate an “information waveform.” A Circular Information Plot can then be generated from this information waveform. An example of a Circular Information Plot formed by this method is depicted inFIG. 17.

Although preferred embodiments of the invention operate on vibration signals, the invention is not limited to only vibration signals. Periodic Signal Parameters and Periodic Information Plots may be derived from any signal containing periodic components.

Methods for Sorting and Discarding Statistically Outlying Peaks in the Autocorrelation Waveform (Step34inFIG. 2).

The following routine takes an array of data values, such as values of positive peaks in the autocorrelation waveform, and discards values outside the statistically calculated boundaries. In a preferred embodiment, there are four methods or criteria for setting the boundaries.

Method 1: Non-Conservative, Using Minimum and Maximum Statistical Boundaries

Consider an array of P values (or elements) where P0represents the number of values in the present array under evaluation. Now let P−1represent the number of values in the array evaluated a single step before P0, let P−2represent the number of values in the array evaluated a single step before P−1, and let P−3represent the number of values in the array evaluated a single step before P−2.

While evaluating the array of values for either the first time or P0≠ P−1,{Calculate the mean (μ) and standard deviation ( ) for P0If⁢⁢n⁢⁢σμ≥x,where⁢⁢x=0.1⁢⁢and⁢⁢n=1,2⁢⁢or⁢⁢3⁢⁢in⁢⁢the⁢⁢preferred⁢⁢embodiment,thenInclude array values such thatμ − n < values < μ + nElseSTOP, values are within statistical boundaries.Endif}

If P0= P−1, thenWhile P−1≠ P−2, and P0= P−1{Calculate the mean (μ) and standard deviation ( ) for P0If⁢⁢n⁢⁢σ2⁢μ≥x,where⁢⁢x=0.1⁢⁢and⁢⁢n=1,2⁢⁢or⁢⁢3⁢⁢in⁢⁢the⁢⁢preferred⁢⁢embodiment,thenInclude array values such thatμ-n⁢⁢σ2<values<μ+n⁢⁢σ2ElseSTOP, values are within statistical boundaries.Endif}Endif

If P0= P−1= P−2, and P−2≠ P−3, thenCalculate the mean (μ) and standard deviation (σ) for P0Include array values such that0.9μ < values < 1.1μElseSTOP, values are within statistical boundaries.Endif

Method 2: Non-Conservative, Using Maximum Statistical Boundary Only (No Minimum Boundary)

Use the same procedure as in Method 1 except only values exceeding the upper statistical boundaries are discarded. The minimum boundary is set to zero.

Method 3: Conservative, Using Minimum and Maximum Statistical Boundaries

Discard values based on Method 1, Step 1 only.

Method 4: Conservative, Using Maximum Statistical Boundary Only (No Minimum Boundary)

Discard values based on Method 1, Step 1 only and based on values exceeding the upper statistical boundaries. The minimum boundary is set to zero.

Example of Method 1 for Sorting Out Statistical Outliers

As an example of the sorting Method 1, consider an original set of values, P0, containing the 21 values listed below in Table 2 below, with n=1.

The mean (μ) of this original set, P0, is 0.54955 and standard deviation (σ) is 0.13982. Therefore, in Step 1 of Method 1,

n⁢⁢σμ=1*0.139820.54955=0.25442.
Since 0.25442 is greater than 0.1, calculate
μ−nσ=0.54955−1*0.13982=0.409735
and
μ+nσ=0.54955+1*0.13982=0.689373.

Next, define the set P−1=P0and define a new set P0, the values of which are all the values of P−1that are between the values μ+σ=0.689343 and μ−σ=0.409735. The set P0now contains the values listed below in Table 3, wherein three outlier values have been eliminated.

Since P0≠P−1, Step 1 is repeated, where for the set P0:
μ=0.50234,
σ=0.06946,
σ/μ=0.138263,
μ+σ=0.571797, and
μ−σ=0.432887.

Now define the set P−2=P−1, and P−1=P0and define a new set P0, the values of which are all the values of P−1that are between the values μ+σ=0.571797 and μ−σ=0.432887. The set P0now contains the values listed below in Table 4, wherein four more outlier values have been eliminated.

Since P0≠ P−1, Step 1 is repeated, where for the set P0:
μ=0.481311,
σ=0.037568, and
σ/μ=0.078053.
Since
σ/μ=0.078053≦0.1,
all the members of the array P0are statistically close in value and need no more sorting.

If at any point in the calculations P0=P−1and P−1≠P−2, then Step 2 would be executed instead of Step 1. In the example above, since P0≠P−1for every iteration, only Step 1 was necessary for the calculations.