tftmoment

Plot the conditional temporal moments of a time series using a plot-only approach and a return-data approach.

Load and plot the data, which consists of simulated vibration measurements for a system with a fault that causes periodic resonances. x is the vector of measurements, and fs is the sampling frequency.

load tftmoment_example x fs

ts=0:1/fs:(length(x)-1)/fs;
figure
subplot(1,2,1)
plot(ts,x)
xlabel('Time in Seconds')
ylabel('Measurement')
title('Simulated Vibration Measurements')

Use the function pspectrum with the 'spectrogram' option to show the frequency content versus time.

subplot(1,2,2)
pspectrum(x,ts,'spectrogram')

The spectrogram shows that the first burst is at 100 Hz, and the second burst is at 300 Hz. The 300-Hz burst is stronger than the 100-Hz burst by 70 dB.

Plot the second temporal moment (variance), using the plot-only approach with no output arguments and specifying fs.

figure
order = 2;
tftmoment(x,fs,order);title('Second Temporal Moment')

There are two distinct features in the plot at 100 and 300 Hz corresponding to the induced resonances shown by the spectrogram. The moments are much closer in magnitude than the spectral results were.

Now find the first four temporal moments, using the timeline ts that you already constructed. This time, use the form that returns both the moment vectors and the associated frequency vectors. Embed the order array as part of the input argument.

[momentT,f] = tftmoment(x,ts,[1 2 3 4]);

Each column of momentT contains the moment corresponding to one of the input orders.

momentT_1 = momentT(:,1);
momentT_2 = momentT(:,2);
momentT_3 = momentT(:,3);
momentT_4 = momentT(:,4);

Plot the four moments separately to compare the shapes.

figure
subplot(2,2,1)
plot(f,momentT_1)
title('First Temporal Moment — Mean')
xlabel('Frequency in Hz')

subplot(2,2,2)
plot(f,momentT_2)
title('Second Temporal Moment — Variance')
xlabel('Frequency in Hz')

subplot(2,2,3)
plot(f,momentT_3)
title('Third Temporal Moment — Skewness')
xlabel('Frequency in Hz')

subplot(2,2,4)
plot(f,momentT_4)
title('Fourth Temporal Moment — Kurtosis')
xlabel('Frequency in Hz')

For the data in this example, the second and fourth temporal moments show the clearest features for the faulty resonance.

By default, tfsmoment calls the function pspectrum internally to generate the power spectrogram that tftmoment uses for the moment computation. You can also import an existing power spectrogram for tftmoment to use instead. This capability is useful if you already have a power spectrogram as a starting point, or if you want to customize the pspectrum options by generating the spectrogram explicitly first.

Input a power spectrogram that has already been generated using default options. Compare the resulting temporal-moment plot with one that tftmoment generates using its own pspectrum default options. The results should be the same.

Load the data, which consists of simulated vibration measurements for a system with a fault that causes periodic resonances. p is the previously computed spectrogram, fp and tp are the frequency and time vectors associated with p, x is the original vector of measurements, and fs is the sampling frequency.

load tftmoment_example p fp tp x fs

Determine the second temporal moment using the spectrogram and its associated frequency and time vectors. Plot the moment.

[momentT_p,f_p] = tftmoment(p,fp,tp,2);
figure
subplot(2,1,1)
plot(f_p,momentT_p)
title('Second Temporal Moment using Input Spectrogram ')

Now find and plot the second temporal moments using the original data and sampling rate.

[momentT,f] = tftmoment(x,fs,2);
subplot(2,1,2)
plot(f,momentT)
title('Second Temporal Moment using Measurement Data')

As expected, the plots match since the default pspectrum options were used for both. This result demonstrates the equivalence between the two approaches when there is no customization.

Real-world measurements often come packaged as part of a time-stamped table that records actual time and readings rather than relative times. You can use the timetable format for capturing this data. This example shows how tftmoment operates with a timetable input, in contrast to the data vector inputs used for the other tftmoment examples, such as Plot the Conditional Temporal Moments of a Time Series Vector.

Load the data, which consists of a single timetable (xt_inner1) containing measurement readings and time information for a piece of machinery. Examine the properties of the timetable.

load tfmoment_tdata.mat xt_inner1;
xt_inner1.Properties

ans = 
  TimetableProperties with properties:

             Description: ''
                UserData: []
          DimensionNames: {'Time'  'Variables'}
           VariableNames: {'x_inner1'}
           VariableTypes: "double"
    VariableDescriptions: {}
           VariableUnits: {}
      VariableContinuity: []
                RowTimes: [146484x1 duration]
               StartTime: 0 sec
              SampleRate: 4.8828e+04
                TimeStep: 2.048e-05 sec
                  Events: []
        CustomProperties: No custom properties are set.
      Use addprop and rmprop to modify CustomProperties.

This table consists of dimensions Time and the Variables, where the only variable is x_inner1.

Find the second and fourth conditional temporal moments (order = [2 4]) for the data in the timetable.

order = [2 4];
[momentT_xt_inner1,f] = tftmoment(xt_inner1,order);
size(momentT_xt_inner1)

ans = 1×2

        1024           2

The temporal moments are represented by the columns of momentT_xt_inner1, just as they would be for a moment taken from a time series vector input.

Plot the moments versus returned frequency vector f.

momentT_inner1_2 = momentT_xt_inner1(:,1);
momentT_inner1_4 = momentT_xt_inner1(:,2);

figure
subplot(2,1,1)
plot(f,momentT_inner1_2)
title("Second Temporal Moment")

subplot(2,1,2)
plot(f,momentT_inner1_4)
title("Fourth Temporal Moment")
xlabel('Frequency in Hz')