function [retval, s, errorb, tau] = allan_overlap(data,tau,name,verbose)
  % ALLAN_OVERLAP  Compute the overlapping Allan deviation for a set of
  %   time-domain frequency data
  % [RETVAL, S, ERRORB, TAU] = ALLAN_OVERLAP(DATA,TAU,NAME,VERBOSE)
  %
  % Inputs:
  % DATA should be a struct and have the following fields:
  %  DATA.freq or DATA.phase
  %               A vector of fractional frequency measurements (df/f) in
  %               DATA.freq *or* phase offset data (seconds) in DATA.phase
  %               If phase data is not present, it will be generated by
  %                integrating the fractional frequency data.
  %               If both fields are present, then DATA.phase will be used.
  %
  %  DATA.rate or DATA.time
  %               The sampling rate in Hertz (DATA.rate) or a vector of
  %                timestamps for each measurement in seconds (DATA.time).
  %               DATA.rate is used if both fields are present.
  %               If DATA.rate == 0, then the timestamps are used.
  %
  % TAU is an array of tau values for computing Allan deviation.
  %     TAU values must be divisible by 1/DATA.rate (data points cannot be
  %     grouped in fractional quantities!). Invalid values are ignored.
  % NAME is an optional label that is added to the plot titles.
  % VERBOSE sets the level of status messages:
  %     0 = silent & no data plots; 1 = status messages; 2 = all messages 
  %
  % Outputs:
  % RETVAL is the array of overlapping Allan deviation values at each TAU.
  % S is an optional output of other statistical measures of the data (mean, std, etc).
  % ERRORB is an optional output containing the error estimates for a 1-sigma
  %   confidence interval. Error bars are plotted as vertical lines at each point.
  % TAU is an optional output containing the array of tau values used in the
  %   calculation (which may be a truncated subset of the input or default values).
  %
  % Example:
  %
  % To compute the overlapping Allan deviation for the data in the variable "lt":
  % >> lt
  % lt = 
  %     freq: [1x86400 double]
  %     rate: 0.5
  %
  % Use:
  %
  % >> ado = allan_overlap(lt,[2 10 100],'lt data',1);
  %
  % The Allan deviation will be computed and plotted at tau = 2,10,100 seconds.
  %  1-sigma confidence intervals will be indicated by vertical lines.
  % You can also use the default settings, which are usually a good starting point:
  %
  % >> ado = allan_overlap(lt);
  %
  %
  % Notes:
  %  This function calculates the overlapping Allan deviation (ADEV), *not* the
  %   standard ADEV. Use "allan.m" for standard ADEV.
  %  The calculation is performed using phase data. If only frequency data is
  %   provided, phase data is generated by integrating the frequency data.
  %   However, the timestamp-based calculation is performed using frequency
  %   data. Phase data is differentiated to generate frequency data if necessary.
  %  No pre-processing of the data is performed, except to remove any
  %   initial offset in the time record. 
  %  For rate-based data, ADEV is computed only for tau values greater than the
  %   minimum time between samples and less than the half the total time. For
  %   time-stamped data, only tau values greater than the maximum gap between
  %   samples and less than half the total time are used.
  %  The calculation for fixed sample rate data is *much* faster than for
  %   time-stamp data. You may wish to run the rate-based calculation first,
  %   then compare with time-stamp-based. Often the differences are insignificant.
  %  The error bars at each point are calculated using the 1-sigma intervals
  %   based on the size of the data set. This is usually an overestimate for
  %   overlapping ADEV; a more accurate (and usually smaller uncertainty)
  %   value can be determined from chi-squared statistics, but that is not
  %   implemented in this version.
  %  You can choose between loglog and semilog plotting of results by
  %   commenting in/out the appropriate line. Search for "#PLOTLOG".
  %  This function has been validated using the test data from NBS Monograph
  %   140, the 1000-point test data set given by Riley [1], and the example data
  %   given in IEEE standard 1139-1999, Annex C.
  %   The author welcomes other validation results, see contact info below.
  %
  % For more information, see:
  % [1] W. J. Riley, "Addendum to a test suite for the calculation of time domain
  %  frequency stability," presented at IEEE Frequency Control Symposium,
  %  1996.
  % Available on the web:
  %  http://www.ieee-uffc.org/frequency_control/teaching.asp?name=paper1ht
  %
  %
  % M.A. Hopcroft
  %      mhopeng at gmail dot com
  %
  % I welcome your comments and feedback!
  %
  % MH Mar2014
  % v2.24 fix bug related to generating freq data from phase with timestamps
  %       (thanks to S. David-Grignot for finding the bug)
  % MH Oct2010
  % v2.22 tau truncation to integer groups; tau sort
  %       plotting bugfix
  % v2.20 update to match allan.m (dsplot.m, columns)
  %       discard tau values with timestamp irregularities
  
  versionstr = 'allan_overlap v2.24';
  
  %
  % MH MAR2010
  % v2.1  bugfixes for irregular sample rates
  %        (thanks to Ryad Ben-El-Kezadri for feedback and testing)
  %       handle empty rate field
  %       fix integer comparisons for fractional sample rates
  %       update consistency check
  %
  % MH FEB2010
  % v2.0  use phase data for calculation- much faster
  %       Consistent code behaviour for all "allan_x.m" functions:
  %       accept phase data
  %       verbose levels
  %
  % MH JAN2010
  % v1.0  based on allan v1.84
  %
  
  %#ok<*AGROW>
  
  
  % defaults
  if nargin < 4, verbose = 2; end
  if nargin < 3, name=''; end
  if nargin < 2 || isempty(tau), tau=2.^(-10:10); end
  if isfield(data,'rate') && isempty(data.rate), data.rate=0; end % v2.1 
  
  % Formatting for plots
  FontName = 'Arial';
  FontSize = 14;
  plotlinewidth=2;
  
  if verbose >= 1, fprintf(1,'allan_overlap: %s
  
  ',versionstr); end
  
  %% Data consistency checks v2.1
  if ~(isfield(data,'phase') || isfield(data,'freq'))
      error('Either ''phase'' or ''freq'' must be present in DATA. See help file for details. [con0]');
  end
  if isfield(data,'time')
      if isfield(data,'phase') && (length(data.phase) ~= length(data.time))
          if isfield(data,'freq') && (length(data.freq) ~= length(data.time))
              error('The time and freq vectors are not the same length. See help for details. [con2]');
          else
              error('The time and phase vectors are not the same length. See help for details. [con1]');
          end
      end
      if isfield(data,'phase') && (any(isnan(data.phase)) || any(isinf(data.phase)))
          error('The phase vector contains invalid elements (NaN/Inf). [con3]');
      end
      if isfield(data,'freq') && (any(isnan(data.freq)) || any(isinf(data.freq)))
          error('The freq vector contains invalid elements (NaN/Inf). [con4]');
      end
      if isfield(data,'time') && (any(isnan(data.time)) || any(isinf(data.time)))
          error('The time vector contains invalid elements (NaN/Inf). [con5]');
      end
  end
  
  % sort tau vector
  tau=sort(tau);
  
  %% Basic statistical tests on the data set
  if ~isfield(data,'freq')
      if isfield(data,'rate') && data.rate ~= 0
          data.freq=diff(data.phase).*data.rate;
      elseif isfield(data,'time')
          data.freq=diff(data.phase)./diff(data.time);
      end
      if verbose >= 1, fprintf(1,'allan_overlap: Fractional frequency data generated from phase data (M=%g).
  ',length(data.freq)); end
  end
  if size(data.freq,2) > size(data.freq,1), data.freq=data.freq'; end % ensure columns
      
  s.numpoints=length(data.freq);
  s.max=max(data.freq);
  s.min=min(data.freq);
  s.mean=mean(data.freq);
  s.median=median(data.freq);
  if isfield(data,'time')
      if size(data.time,2) > size(data.time,1), data.time=data.time'; end % ensure columns
      s.linear=polyfit(data.time(1:length(data.freq)),data.freq,1);
  elseif isfield(data,'rate') && data.rate ~= 0;
      s.linear=polyfit((1/data.rate:1/data.rate:length(data.freq)/data.rate)',data.freq,1);
  else
      error('Either "time" or "rate" must be present in DATA. Type "help allan_overlap" for details. [err1]');
  end
  s.std=std(data.freq);
  
  if verbose >= 2
      fprintf(1,'allan_overlap: fractional frequency data statistics:
  ');
      disp(s);
  end
  
  
  % scale to median for plotting
  medianfreq=data.freq-s.median;
  sm=[]; sme=[];
  
  % Screen for outliers using 5x Median Absolute Deviation (MAD) criteria
  MAD = median(abs(medianfreq)/0.6745);
  if verbose >= 1 && any(abs(medianfreq) > 5*MAD)
      fprintf(1, 'allan_overlap: NOTE: There appear to be outliers in the frequency data. See plot.
  ');
  end
  
  %%%%
  % There are four cases, freq or phase data, using timestamps or rate:
  
  %% Fixed Sample Rate Data
  %   If there is a regular interval between measurements, calculation is much
  %   easier/faster
  if isfield(data,'rate') && data.rate > 0 % if data rate was given
      if verbose >= 1
          fprintf(1, 'allan_overlap: regular data ');
          if isfield(data,'freq')
              fprintf(1, '(%g freq data points @ %g Hz)
  ',length(data.freq),data.rate);
          elseif isfield(data,'phase')
              fprintf(1, '(%g phase data points @ %g Hz)
  ',length(data.phase),data.rate);
          else
              error('
   phase or freq data missing [err10]');
          end
      end
    
      % string for plot title
      name=[name ' (' num2str(data.rate) ' Hz)'];
  
      % what is the time interval between data points?
      tmstep = 1/data.rate;      
      
      % Is there time data? Just for curiosity/plotting, does not impact calculation
      if isfield(data,'time')
          % adjust time data to remove any starting gap; first time step
          %  should not be zero for comparison with freq data
          dtime=data.time-data.time(1)+mean(diff(data.time)); 
          dtime=dtime(1:length(medianfreq)); % equalize the data vector lengths for plotting (v2.1)
          if verbose >= 2
              fprintf(1,'allan_overlap: End of timestamp data: %g sec.
  ',dtime(end));
              if (data.rate - 1/mean(diff(dtime))) > 1e-6
                  fprintf(1,'allan_overlap: NOTE: data.rate (%f Hz) does not match average timestamped sample rate (%f Hz)
  ',data.rate,1/mean(diff(dtime)));
              end
          end
      else
          % create time axis data using rate (for plotting only)
          dtime=(tmstep:tmstep:length(data.freq)*tmstep);
      end
  
    
      % is phase data present? If not, generate it
      if ~isfield(data,'phase')
          nfreq=data.freq-s.mean;
          dphase=zeros(1,length(nfreq)+1);
          dphase(2:end) = cumsum(nfreq)./data.rate;
          if verbose >= 1, fprintf(1,'allan_overlap: phase data generated from fractional frequency data (N=%g).
  ',length(dphase)); end
      else
          dphase=data.phase;
      end
      
      % check the range of tau values and truncate if necessary
      % find halfway point of time record
      halftime = round(tmstep*length(data.freq)/2);
      % truncate tau to appropriate values
      tau = tau(tau >= tmstep & tau <= halftime);
      if verbose >= 2, fprintf(1, 'allan_overlap: allowable tau range: %g to %g sec. (1/rate to total_time/2)
  ',tmstep,halftime); end
      
      % number of samples
      N=length(dphase);
      % number of samples per tau period
      m = data.rate.*tau;
      % only integer values allowed for m (no fractional groups of points)
      %tau = tau(m-round(m)<1e-8); % numerical precision issues (v2.1)
      tau = tau(m==round(m));  % The round() test is only correct for values < 2^53
      %m = m(m-round(m)<1e-8); % change to round(m) for integer test v2.22
      m = m(m==round(m));
      %m=round(m);
      %fprintf(1,'m: %.50f
  ',m)
          
      if verbose >= 1, fprintf(1,'allan_overlap: calculating overlapping Allan deviation...
         '); end
      
      % calculate the Allan deviation for each value of tau
      k=0; tic;
      for i = tau
          k=k+1;
          if verbose >= 2, fprintf(1,'%d ',i); end
  
  
          % pad phase data set length to an even multiple of this tau value
          mphase=zeros(ceil(length(dphase)./m(k))*m(k),1);
          mphase(1:N)=dphase;
          % group phase values
          mp=reshape(mphase,m(k),[]);
          % compute second differences of phase values (x_k+m - x_k)
          md1=diff(mp,1,2);
          md2=diff(md1,1,2);
          md1=reshape(md2,1,[]);
          
          % compute overlapping ADEV from phase values
          %  only the first N-2*m(k) samples are valid
          sm(k)=sqrt((1/(2*(N-2*m(k))*i^2))*sum(md1(1:N-2*m(k)).^2));
          
          % estimate error bars
          sme(k)=sm(k)/sqrt(N-2*m(k));
          
  
      end % repeat for each value of tau
      
      if verbose >= 2, fprintf(1,'
  '); end
      calctime=toc; if verbose >= 2, fprintf(1,'allan_overlap: Elapsed time for calculation: %g seconds
  ',calctime); end
  
          
      
  %% Irregular data, no fixed interval    
  elseif isfield(data,'time')
      % the interval between measurements is irregular
      %  so we must group the data by time
      if verbose >= 1, fprintf(1, 'allan_overlap: irregular rate data (no fixed sample rate)
  '); end
  
      
      % string for plot title
      name=[name ' (timestamp)'];
      
  
      % adjust time to remove any starting offset
      dtime=data.time-data.time(1)+mean(diff(data.time));
      
      % save the freq data for the loop
      dfreq=data.freq;
      dtime=dtime(1:length(dfreq));
      
      dfdtime=diff(dtime); % only need to do this once (v2.1)
      % where is the maximum gap in time record?
      gap_pos=find(dfdtime==max(dfdtime));
      % what is average data spacing?
      avg_gap = mean(dfdtime);
      s.avg_rate = 1/avg_gap; % save avg rate for user (v2.1)
      
      if verbose >= 2
          fprintf(1, 'allan_overlap: WARNING: irregular timestamp data (no fixed sample rate).
  ');
          fprintf(1, '       Calculation time may be long and the results subject to interpretation.
  ');
          fprintf(1, '       You are advised to estimate using an average sample rate (%g Hz) instead of timestamps.
  ',1/avg_gap);
          fprintf(1, '       Continue at your own risk! (press any key to continue)
  ');
          pause;
      end
      
      if verbose >= 1
          fprintf(1, 'allan_overlap: End of timestamp data: %g sec
  ',dtime(end));
      	fprintf(1, '       Average rate: %g Hz (%g sec/measurement)
  ',1/avg_gap,avg_gap);
          if max(diff(dtime)) ~= 1/mean(diff(dtime))
              fprintf(1, '       Max. gap in time record: %g sec at position %d
  ',max(dfdtime),gap_pos(1));
          end
          if max(diff(dtime)) > 5*avg_gap
              fprintf(1, '       WARNING: Max. gap in time record is suspiciously large (>5x the average interval).
  ');
          end
      end
      
  
      % find halfway point
      halftime = fix(dtime(end)/2);
      % truncate tau to appropriate values
      tau = tau(tau >= max(dfdtime) & tau <= halftime);
      if isempty(tau)
          error('allan_overlap: ERROR: no appropriate tau values (> %g s, < %g s)
  ',max(dfdtime),halftime);
      end
      
  
      % number of samples
      M=length(dfreq);
      % number of samples per tau period
      m=round(tau./avg_gap);
  
      if verbose >= 1, fprintf(1,'allan_overlap: calculating overlapping Allan deviation...
  '); end
  
      k=0; tic;
      for i = tau
          k=k+1;
          fa=[];
  
          if verbose >= 2, fprintf(1,'%d ',i); end
          
          freq = dfreq; time = dtime;
         
          
          % compute overlapping samples (y_k) for this tau
          %for j = 1:i
          for j = 1:m(k) % (v2.1)
              km=0;
              %fprintf(1,'j: %d ',j);
  
              % (v2.1) truncating not correct for overlapping samples
              % truncate data set to an even multiple of this tau value
              %freq = freq(time <= time(end)-rem(time(end),i));
              %time = time(time <= time(end)-rem(time(end),i));
                          
              % break up the data into overlapping groups of tau length
              while i*km <= time(end)
                  km=km+1;
                  %i*km
  
                  % progress bar
                  if verbose >= 2
                      if rem(km,100)==0, fprintf(1,'.'); end
                      if rem(km,1000)==0, fprintf(1,'%g/%g
  ',km,round(time(end)/i)); end
                  end
  
                  f = freq(i*(km-1) < (time) & (time) <= i*km);
  
                  if ~isempty(f)
                      fa(j,km)=mean(f);
                  else
                      fa(j,km)=0;
                  end
  
              end
              %fa
              
              % shift data vector by -1 and repeat
              freq=circshift(dfreq,(size(freq)>1)*-j);
              freq(end-j+1:end)=[];
              time=circshift(dtime,(size(time)>1)*-j);
              time(end-j+1:end)=[];
              time=time-time(1)+avg_gap; % remove time offset
              
          end
          
          % compute second differences of fractional frequency values (y_k+m - y_k)
          fd1=diff(fa,1,2);
          fd1=reshape(fd1,1,[]);
          % compute overlapping ADEV from fractional frequency values
          %  only the first M-2*m(k)+1 samples are valid
          if length(fd1) >= M-2*m(k)+1
              sm(k)=sqrt((1/(2*(M-2*m(k)+1)))*sum(fd1(1:M-2*m(k)+1).^2));
  
              % estimate error bars
              sme(k)=sm(k)/sqrt(M+1);
              
              if verbose >= 2, fprintf(1,'
  '); end
              
          else
              if verbose >=2, fprintf(1,' tau=%g dropped due to timestamp irregularities
  ',tau(k)); end
              sm(k)=0; sme(k)=0;
          end
          
  
      end
  
      if verbose >= 2, fprintf(1,'
  '); end
      calctime=toc; if verbose >= 1, fprintf(1,'allan_overlap: Elapsed time for calculation: %g seconds
  ',calctime); end
  
      % remove any points that were dropped
      tau(sm==0)=[];
      sm(sm==0)=[];
      sme(sme==0)=[];
  
  
  
  else
      error('allan_overlap: WARNING: no DATA.rate or DATA.time! Type "help allan" for more information. [err2]');
  end
  
  
  %%%%%%%%
  %% Plotting
  
  if verbose >= 2 % show all data
      
      % plot the frequency data, centered on median
      if size(dtime,2) > size(dtime,1), dtime=dtime'; end % this should not be necessary, but dsplot 1.1 is a little bit brittle
      try
          % dsplot makes a new figure
          hd=dsplot(dtime,medianfreq);
      catch ME
          figure;
          hd=plot(dtime,medianfreq);
          if verbose >= 1, fprintf(1,'allan_overlap: Note: Install dsplot.m for improved plotting of large data sets (File Exchange File ID: #15850).
  '); end
          if verbose >= 2, fprintf(1,'             (Message: %s)
  ',ME.message); end
      end
      set(hd,'Marker','.','LineStyle','none','Color','b'); % equivalent to '.-'
      hold on;
  
      fx = xlim;
      % plot([fx(1) fx(2)],[s.median s.median],'-k');
      plot([fx(1) fx(2)],[0 0],':k');
      % show 5x Median Absolute deviation (MAD) values
      hm=plot([fx(1) fx(2)],[5*MAD 5*MAD],'-r');
      plot([fx(1) fx(2)],[-5*MAD -5*MAD],'-r');
      % show linear fit line
      hf=plot(xlim,polyval(s.linear,xlim)-s.median,'-g');    
      title(['Data: ' name],'FontSize',FontSize+2,'FontName','Arial');
      %set(get(gca,'Title'),'Interpreter','none');
      xlabel('Time [sec]','FontSize',FontSize,'FontName',FontName);
      if isfield(data,'units')
          ylabel(['data - median(data) [' data.units ']'],'FontSize',FontSize,'FontName',FontName);
      else
          ylabel('freq - median(freq)','FontSize',FontSize,'FontName',FontName);
      end
      set(gca,'FontSize',FontSize,'FontName',FontName);
      legend([hd hm hf],{'data (centered on median)','5x MAD outliers',['Linear Fit (' num2str(s.linear(1),'%g') ')']},'FontSize',max(10,FontSize-2));
      % tighten up
      xlim([dtime(1) dtime(end)]);
  
      
  end % end plot raw data
  
  
  if verbose >= 1 % show analysis results
  
      % plot Allan deviation results
      if ~isempty(sm)
          figure
  
          % Choose loglog or semilogx plot here    #PLOTLOG
          %semilogx(tau,sm,'.-b','LineWidth',plotlinewidth,'MarkerSize',24);
          loglog(tau,sm,'.-b','LineWidth',plotlinewidth,'MarkerSize',24);
  
          % in R14SP3, there is a bug that screws up the error bars on a semilog plot.
          %  When this is fixed, uncomment below to use normal errorbars
          %errorbar(tau,sm,sme,'.-b'); set(gca,'XScale','log');
          % this is a hack to approximate the error bars
          hold on; plot([tau; tau],[sm+sme; sm-sme],'-k','LineWidth',max(plotlinewidth-1,2));
  
          grid on;
          title(['Overlapping Allan Deviation: ' name],'FontSize',FontSize+2,'FontName',FontName);
          %set(get(gca,'Title'),'Interpreter','none');
          xlabel('\tau [sec]','FontSize',FontSize,'FontName','Arial');
          ylabel(' Overlapping \sigma_y(\tau)','FontSize',FontSize,'FontName',FontName);
          set(gca,'FontSize',FontSize,'FontName',FontName);
          % expand the x axis a little bit so that the errors bars look nice
          adax = axis;
          axis([adax(1)*0.9 adax(2)*1.1 adax(3) adax(4)]);
          
          % display the minimum value
          fprintf(1,'allan: Minimum overlapping ADEV value: %g at tau = %g seconds
  ',min(sm),tau(sm==min(sm)));        
          
      elseif verbose >= 1
          fprintf(1,'allan_overlap: WARNING: no values calculated.
  ');
          fprintf(1,'       Check that TAU > 1/DATA.rate and TAU values are divisible by 1/DATA.rate
  ');
          fprintf(1,'Type "help allan_overlap" for more information.
  
  ');
      end
      
  end % end plot analysis
          
  retval = sm;
  errorb = sme;
  
  return