Extremes and Statistics Extremes and Statistics

There are several options which can be used on more than one command:


     -HARD

     -BOTH

     -HEADING, HEAD

     -RECORD, BEG_RNUM, END_RNUM

     -VALUES, CV,  VAL_MIN, VAL_MAX

     -MAG_USE
 

They will be defined here and then listed for the commands for which they are applicable.

By default, the results for commands that produce reports (except for the REPORT command discussed above) is to write the results to the terminal. The -HARD option instructs MOSES to produce a report on the OUTPUT channel and the -BOTH option writes the results to both the OUTPUT channel and the terminal. When these reports are written, they have a single line generic heading. The -HEADING allows one to replace the generic heading with one you specify. You can specify as many of these options as you wish. The will appear on the page in the order you specify them.

The -RECORD and -VALUES options defines the records which will be considered. Here, a "RECORD" is simply a row of the matrix of data. With the -RECORD option the beginning and end record numbers are simply specified. With -VALUES the records considered are defined with the values of a column of data. Here CV is the column number for which the values will be obtained and VAL_MIN and VAL_MAX are two numbers (VAL_MIN is less than VAL_MAX). BEG_REC is then the largest record number which the values of column CV is less than or equal VAL_MIN and END_REC is the greatest record number where the value of this column is greater than VAL_MAX. If neither -VALUES nor -RECORD are specified, all records will be considered.

The -MAG_USE option instructs MOSES to add a second heading line based on the definition of magnitude defined with the -MAG_DEFINE option.

The EXTREME command offers the user the opportunity of obtaining a report on the extremes of the data. The form of this command is:


     EXTREME, CS(1), CS(2), .....  -OPTIONS

and the available options are:


     -HARD

     -BOTH

     -HEADING, "HEAD(1)", "HEAD(2)"

     -RECORD, BEG_RNUM, END_RNUM

     -VALUES, CV, VAL_MIN, VAL_MAX

     -MAG_USE

With this command, one will obtain a report of the extremes of the data selected. Here, the first value entered will become the "independent" variable, and the remainder the dependent ones. MOSES will search through the results from BEG_RNUM to END_RNUM to find the minimum and maximum value of each type of data selected. It will then issue a report for each value of the independent variable at which an extreme occurred. This report will contain the values of all of the variables and a remark as to which variables have suffered an extreme. The report will be written to the terminal unless the -HARD option was used, in which case it will be written to the output file.

The STATISTIC command generates a report on the statistics of the data. It produces statistics for the results from BEG_RNUM to END_RNUM for each type of data selected. The specific form of this command is:


     STATISTIC, CS(1), CS(2), .....  -OPTIONS

and the available options are:


     -HARD

     -BOTH

     -RECORD, BEG_RNUM, END_RNUM

     -VALUES, CV, VAL_MIN, VAL_MAX

     -MAG_USE

     -HEADING, "HEAD(1)", "HEAD(2)"

     -TYPE, STYPE

     -EXTREMES, TIME, DEVIATION, MULTIPLIER

Where the report is written depends on the use of the -HARD and -BOTH options. Here, CS(1) is the independent variable against which the statistics will be computed. Normally, it is "event" so that the remaining columns of data can be considered to be time samples. If this is the case, MOSES will compute the following quantities:

     Mean
     Variance
     RMS
     Std. Deviation
     Skewness
     Kurtosis
     Av of 1/3 Highest
     Av of 1/3 Lowest
     Av of 1/100 Highest
     Av of 1/100 Lowest
     Av of 1/1000 Highest
     Av of 1/1000 Lowest
     Maximum
     Minimum
     Pred. Max
     Pred. Min
     Av of 1/3 Highest-Mean
     Av of 1/3 Lowest-Mean
     Av of 1/100 Highest-Mean
     Av of 1/100 Lowest-Mean
     Av of 1/1000 Highest-Mean
     Av of 1/1000 Lowest-Mean
     Maximum - Mean
     Minimum - Mean
     Pred. Max - Mean
     Pred. Min - Mean

of the variables selected. These quantities are calculated for the It will also compute averages for the peaks encountered. Notice these peaks are computed from the samples themselves and not by assuming any type of probability distribution. Extreme values of the maximum and minimum are also predicted. This prediction is controlled by the -EXTREMES option. Here TIME is the time in seconds for the extreme. If, for example, TIME is 3600, then the predicted value will be the probable maximum in one hour. The default is three hours.

In general, the predicted extreme is of the form

     PE  = MEAN +- DEVIAT * FACTOR

here MEAN is the mean and the plus is used for the maximum and the minus for the minimum. Traditionally, the standard deviation is used for DEVIAT and FACTOR is given by

     FACTOR = sqrt { 2 Log ( r * Np ) }

where Np is the number of peaks in the sample, and r is the ratio of the length of the sample to TIME. The values of DEVIATION and MULTIPLIER can be used to change this behavior. In particular, the value of DEVIATION is used to change DEVIAT. Here, a value of STANDARD will use the standard deviation while a value of PEAKS will use the largest peak and smallest peak values minus the mean. PEAKS is the default and normally gives better predictions than the traditional method. The final value, MULTIPLIER can be either GAUSSIAN or WINTERSTEIN. GAUSSIAN is the default. If WINTERSTEIN is used, then FACTOR will be computed according to the paper "Nonlinear Vibration Models for Extremes and Fatigue" by S.R. Winterstein. If one is using both PEAKS and GAUSSIAN, then factor is different than that given above. Here it is

     FACTOR = sqrt { 2 Log ( r * Np ) } / sqrt { 2 Log ( Np ) }

In other words, here the given peak is scaled up based on the ratio of the predicted extreme in three hours to that predicted by the current sample.

In some cases, however, the independent variable is not time but frequency, and the other columns are either Fourier Coefficients or spectral ordinates. For frequency data different statistics are computed. If the frequency data resulted from either a FFT or SPECTRUM command in the Disposition Menu, then MOSES automatically knows how to treat the data. If, however, the original data was frequency type, then one must use the -TYPE option to define how to treat it. STYPE can be either Fourier or SPECTRUM. With frequency data, the report consists of:

     0th Moment
     1st Moment
     2nd Moment
     3rd Moment
     4th Moment
     Root Mean Square
     Significant
     Ave of 1/10 Peaks
     Ave of 1/100 Peaks
     Ave of 1/1000 Peaks
     3 Hour Max.
     TP Peak Period
     TV Visual Period
     TZ Zero Up-Crossing
     TC Crest Period

moments of the spectrum, averages of the peaks, and several periods of the data. Here, in contrast to the time statistics, the statistics are derived assuming a Raleigh distribution.