benchmark_steady_state_detection

Definition

Provides methods for detecting steady-state conditions based on t-test.

Examples

The following example demonstrates some of the methods found in the benchmark_steady_state_detection module.

   use benchmark_steady_state_detection, only: ssd
   use benchmark_workflow, only: workflow
   use benchmark_options
   use benchmark_method
   use benchmark_timer, only: clock
   use benchmark_statistics, only: stats   
   use benchmark_warning, only: display_maxcall_warning
   use benchmark_string
   use benchmark_kinds
   use benchmark_output_unit
 
   class(benchmark_run), intent(inout) :: step
   !private
   integer :: k
   real(r8) :: start, finish
   type(stats) :: s
   real(r8), allocatable :: times(:)
   real(r8) :: treshold
      
      
   maxcall_reached = .false.
   treshold = step%options%ssd_threshold
   if (first_call)  write (output_unit, '(A)') new_line('A'), step%header, new_line('A')
 
   step%options%count = step%options%count + 1
   allocate(times(step%options%sampling_window), source=0.0_r8)
   
   block
      integer     :: icount, jcount, count
      real(r8)    :: crit, t
      
      crit = 0.0_r8; icount = 0; jcount = 0; count = 0
      call clock(start)
      do while (crit <= (1.0_r8 - treshold))
         call step%method%invoke(); call clock(finish)
         t = finish - start
         icount = icount + 1
         if (t > step%options%mintime) then 
               times(1 + modulo(jcount, step%options%sampling_window)) =   &
                  t / real(icount, r8)
               jcount = jcount + 1
               
               if (jcount >= step%options%sampling_window) crit =          &
                  ssd(times, modulo(jcount, step%options%sampling_window), &
                  treshold)
               
               icount = 0
               call clock(start)
         end if
         count = count + 1
 
         if (count >= step%options%maxcalls) then 
               maxcall_reached = .true.
               display_maxcall_warning = .true.
               exit
         end if
      end do
   end block
 
   call s%compute(times)
   if (step%options%csv_unit /= 0) then
      call summary(step, s, step%options%csv_unit)
   else
      call summary(step, s)
   end if
   deallocate(times)
   nullify(step%method)
   nullify(step%options)

Remarks

The steady-state detection algorithm is based on the work by J. D. Kelly and J. D. Hedengren, “A steady-state detection (SSD) algorithm to detect non-stationary drifts in processes,” J. Process Control, vol. 23, no. 3, pp. 326–331, 2013. The original algorithm has been altered to be used with dynamic real-time data.

Methods

ssd()

real(r8) function, public ssd	(	real(r8), dimension(:), intent(in)	x,
		integer, intent(in)	offset,
		real(r8), intent(in)	alpha )

Compute the probability that a sample x reached steady-state. The SSD algorithm presented in this work is also window-based and utilizes the Student-t test to determine if the difference between the process signal value minus its mean is above or below the standard-deviation times its statistical critical value. If less than, then that time instant or point is steady and if greater than, then it is unsteady where the aggregation is computed over the window approximating a probability or frequency of being at steady-state.

Parameters

[in]	x	Sample of real(r8) values
[in]	offset	Index of the first values inside the periodic array x
[in]	alpha	Significance level for the student test.

The mean is defined as:

\[ \mu = \frac{1}{n}\left(\sum x-m\sum i\right) \]

where \(m\) is the slope of a linear drift. In the present case, \(m=0\) It comes that the standard deviation is given by

\[ \sigma = \sqrt{\frac{1}{n-2}\sum (x-\mu)^2} \]

At this point along with a specified Student-t critical or threshold value at a particu100 lar significance level α and degrees-of-freedom n, all of the necessary information is available to test the null-hypothesis that the process signal is steady or is stationary about \(\mu\)

\[ if\ x - \mu < t_{crit}-\sigma\ , then\ 1,\ else\ 0 \]

\(t_{crit}\) is evaluated for a given significance level and a given degree of freedom using the t-distribution.

Remarks

Definition at line 59 of file SteadyStateDetection.f90.