function [retval, s, errorb, tau] = allan_cov(data,tau,name,verbose)
  % ALLAN  Compute the Allan deviation for a set of time-domain frequency data
  % [RETVAL, S, ERRORB, TAU] = ALLAN(DATA,TAU,NAME,VERBOSE)
  %
  % Inputs:
  % DATA should be a structure 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 frequency data is not present, it will be generated by
  %                differentiating the phase data.
  %               If both fields are present, then DATA.freq will be used.
  %               Note: for general-purpose calculations of Allan deviation,
  %                (i.e. a two-sample variance) use DATA.freq .
  %
  %  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.
  %
  %  DATA.units (optional)
  %               The units for the data. If present, the string DATA.units
  %               is added to the plot y-axis label.
  %
  % 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!) and invalid values are ignored.
  %     Leave empty to use default values.
  % 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 & minimum plots;
  %     2 = all messages and plots (default)
  %
  % Outputs:
  % RETVAL is the array of 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. These values are shown on the figure for 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 Allan deviation for the data in the variable "lt":
  % >> lt
  % lt = 
  %     freq: [1x86400 double]
  %     rate: 0.5
  %
  % Use:
  %
  % >> ad = allan(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 at each point.
  % You can also use the default settings, which are usually a good starting point:
  %
  % >> ad = allan(lt);
  %
  %
  % Notes:
  %  This function calculates the standard Allan deviation (ADEV), *not* the
  %   overlapping ADEV. Use "allan_overlap.m" for overlapping ADEV.
  %  The calculation is performed using fractional frequency data. If only
  %   phase data is provided, frequency data is generated by differentiating
  %   the phase data.
  %  No pre-processing of the data is performed, except to remove any
  %   initial offset (i.e., starting gap) 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.
  %  To show the "tau bins" (y_k samples) on the data plot, set the variable
  %   TAUBIN to 1 (search for "#TAUBIN").
  %  You can choose between loglog and semilog plotting of results by
  %   commenting in/out the appropriate line. Search for "#PLOTLOG".
  %  I recommend installing "dsplot.m", which improves the performance of
  %   plotting large data sets. Download from File Exchange, File ID: #15850.
  %   allan.m will use dsplot.m if it is present on your MATLAB path.
  %  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, "The Calculation of Time Domain Frequency Stability,"
  % 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 sychronize updates across allan, allan_overlap, allan_modified
  % v2.16 add TAU as output, fixed unusual error with dsplot v1.1
  % v2.14 update plotting behaviour, default tau values
  %
  
  versionstr = 'allan v2.24';
  
  % MH Jun2010
  % v2.12 bugfix for rate data row/col orientation
  %       add DATA.units for plotting
  %       use dsplot.m for plotting
  %
  % MH MAR2010
  % v2.1  minor interface and bugfixes
  %       update data consistency check
  %
  % MH FEB2010
  % v2.0  Consistent code behaviour for all "allan_x.m" functions:
  %       accept phase data
  %       verbose levels
  %
  %
  % MH JAN2010
  % v1.84  code cleanup
  % v1.82  typos in comments and code cleanup
  %        tau bin plotting changed for performance improvement
  % v1.8   Performance improvements:
  %        vectorize code for rate data
  %        logical indexing for irregular rate data
  % MH APR2008
  % v1.62  loglog plot option
  % v1.61  improve error handling, plotting
  %        fix bug in regular data calc for high-rate data
  %        fix bug in timestamp data calc for large starting gap
  %         (thanks to C. B. Ruiz for identifying these bugs)
  %        uses timestamps for DATA.rate=0
  %        progress indicator for large timestamp data processing
  % MH JUN2007
  % v1.54 Improve data plotting and optional bin plotting
  % MH FEB2007
  % v1.5  use difference from median for plotting
  %       added MAD calculation for outlier detection
  % MH JAN2007
  % v1.48 plotting typos fixes
  % MH DEC2006
  % v1.46 hack to plot error bars
  % v1.44 further validation (Riley 1000-pt)
  %       plot mean and std
  % MH NOV2006
  % v1.42 typo fix comments
  % v1.4  fix irregular rate algorithm
  %       irregular algorithm rejects tau less than max gap in time data
  %       validate both algorithms using test data from NBS Monograph 140
  % v1.3  fix time calc if data.time not present
  %       add error bars (not possible due to bug in MATLAB R14SP3)
  %       remove offset calculation
  % v1.24 improve feedback
  % MH SEP2006
  % v1.22 updated comments
  % v1.2  errors and warnings
  % v1.1  handle irregular interval data
  %#ok<*AGROW>
  
  % defaults
  if nargin < 4, verbose=2; end
  if nargin < 3, name=''; end
  if nargin < 2 || isempty(tau), tau=2.^(-10:10); end
  
  % plot "tau bins"? #TAUBIN
  TAUBIN = 0; % set 0 or 1 % WARNING: this has a significant impact on performance
  
  % Formatting for plots
  FontName = 'Arial';
  FontSize = 14;
  plotlinewidth=2;
  
  if verbose >= 1, fprintf(1,'allan: %s
  
  ',versionstr); end
  
  %% Data consistency checks
  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: Fractional frequency data generated from phase data (M=%g).
  ',length(data.freq)); end
      data.time(1)=[]; % make time stamps correspond to freq data 
  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" for details. [err1]');
  end
  s.std=std(data.freq);
  
  if verbose >= 2
      fprintf(1,'allan: input data statistics:
  ');
      disp(s);
  end
  
  
  % center at median for plotting
  medianfreq=data.freq-s.median;
  sm=[]; sme=[];
  
  % Screen for outliers using 5x Median Absolute Deviation (MAD) criteria
  s.MAD = median(abs(medianfreq)/0.6745);
  if verbose >= 2
      fprintf(1, 'allan: 5x MAD value for outlier detection: %g
  ',5*s.MAD);
  end
  if verbose >= 1 && any(abs(medianfreq) > 5*s.MAD)
      fprintf(1, 'allan: NOTE: There appear to be outliers in the frequency data. See plot.
  ');
  end
  
  
  %%%%
  % There are two cases, either using timestamps or fixed sample 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: regular data (%g data points @ %g Hz)
  ',length(data.freq),data.rate); 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));
          if verbose >= 2
              fprintf(1,'allan: End of timestamp data: %g sec.
  ',dtime(end));
              if (data.rate - 1/mean(diff(dtime))) > 1e-6
                  fprintf(1,'allan: 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)'; % column oriented
      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: allowable tau range: %g to %g sec. (1/rate to total_time/2)
  ',tmstep,halftime); end  
      
      % save the freq data for the loop
      dfreq=data.freq;
      dfreq2=data.freq2;
      % find the number of data points in each tau group
      m = data.rate.*tau;
      % only integer values allowed (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);
      
      if verbose >= 1, fprintf(1,'allan: calculating Allan deviation...
         '); end
      
      % calculate the Allan deviation for each value of tau
      k=0; tic;
      for i = tau
          if verbose >= 2, fprintf(1,'%g ',i); end
          k=k+1;
  
          % truncate frequency set to an even multiple of this tau value
          freq=dfreq(1:end-rem(length(dfreq),m(k)));
          freq2=dfreq2(1:end-rem(length(dfreq2),m(k)));
          % group the data into tau-length groups or bins
          f = reshape(freq,m(k),[]); % Vectorize!     
          f2 = reshape(freq2,m(k),[]); % Vectorize!     
          % find average in each "tau group", y_k (each colummn of f)
          fa=mean(f,1);
          fa2=mean(f2,1);
          % first finite difference
          fd=diff(fa);
          fd2=diff(fa2);
          % calculate two-sample variance for this tau
          M=length(fa);

          sm(k)=sqrt(0.5/(M-1)*(abs(sum(fd.*fd2))));

  
          % estimate error bars
          sme(k)=sm(k)/sqrt(M+1);
          
          if TAUBIN == 1
              % save the binning points for plotting
              fs(k,1:length(freq)/m(k))=m(k):m(k):length(freq); fval{k}=mean(f,1);
          end
          
      end % repeat for each value of tau
      
      if verbose >= 2, fprintf(1,'
  '); end
      calctime=toc; if verbose >= 2, fprintf(1,'allan: Elapsed time for calculation: %e seconds
  ',calctime); end
      
         
      
  %% Irregular data (timestamp)   
  elseif isfield(data,'time')
      % the interval between measurements is irregular
      %  so we must group the data by time
      if verbose >= 1, fprintf(1, 'allan: irregular rate data (no fixed sample rate)
  '); end
      
      % string for plot title
      name=[name ' (timestamp)'];
      
      % adjust time to remove any initial offset or zero
      dtime=data.time-data.time(1)+mean(diff(data.time));
      %dtime=data.time;
      % where is the maximum gap in time record?
      gap_pos=find(diff(dtime)==max(diff(dtime)));
      % what is average data spacing?
      avg_gap = mean(diff(dtime));
      
      if verbose >= 2
          fprintf(1, 'allan: 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: 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: %g sec at position %d
  ',max(diff(dtime)),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(diff(dtime)) & tau <= halftime);
      if isempty(tau)
          error('allan: ERROR: no appropriate tau values (> %g s, < %g s)
  ',max(diff(dtime)),halftime);
      end
      
      % save the freq data for the loop
      dfreq=data.freq;
      dtime=dtime(1:length(dfreq));
  
      if verbose >= 1, fprintf(1,'allan: calculating Allan deviation...
  '); end
  
      k=0; tic;
      for i = tau
          if verbose >= 2, fprintf(1,'%d ',i); end
          
          k=k+1; fa=[]; %f=[];
          km=0;
          
          % truncate data set to an even multiple of this tau value
          freq=dfreq(dtime <= dtime(end)-rem(dtime(end),i));
          time=dtime(dtime <= dtime(end)-rem(dtime(end),i));
          %freq=dfreq;
          %time=dtime;
          
          % break up the data into groups of tau length in sec
          while i*km < time(end)
              km=km+1;
                          
              % 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);
              f = f(~isnan(f)); % make sure values are valid
              
              if ~isempty(f)
                  fa(km)=mean(f);
              else
                  fa(km)=0;
              end
  
              if TAUBIN == 1 % WARNING: this has a significant impact on performance
                  % save the binning points for plotting
                  %if find(time <= i*km) > 0
                      fs(k,km)=max(time(time <= i*km));
                  %else
                  if isempty(fs(k,km))
                      fs(k,km)=0;
                  end
                  fval{k}=fa;
              end % save tau bin plot points
              
          end
          
          if verbose >= 2, fprintf(1,'
  '); end
  
          % first finite difference of the averaged results
          fd=diff(fa);
          % calculate Allan deviation for this tau
          M=length(fa);
          sm(k)=sqrt(0.5/(M-1)*(sum(fd.^2)));
  
          % estimate error bars
          sme(k)=sm(k)/sqrt(M+1);
          
  
      end
  
      if verbose == 2, fprintf(1,'
  '); end
      calctime=toc; if verbose >= 2, fprintf(1,'allan: Elapsed time for calculation: %e seconds
  ',calctime); end
      
  
  else
      error('allan: 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;
          if length(dtime) ~= length(medianfreq)
              fprintf(1,'allan: Warning: length of time axis (%d) is not equal to data array (%d)
  ',length(dtime),length(medianfreq));
          end
          hd=plot(dtime,medianfreq);
          if verbose >= 1, fprintf(1,'allan: 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;
  
      % show center (0)
      plot(xlim,[0 0],':k');
      % show 5x Median Absolute Deviation (MAD) values
      hm=plot(xlim,[5*s.MAD 5*s.MAD],'-r');
      plot(xlim,[-5*s.MAD -5*s.MAD],'-r');
      % show linear fit line
      hf=plot(xlim,polyval(s.linear,xlim)-s.median,'-g');
      title(['Data: ' name],'FontSize',FontSize+2,'FontName',FontName);
      %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)]);
  
  
      % Optional tau bin (y_k samples) plot
      if TAUBIN == 1
          % plot the tau divisions on the data plot
          rfs=size(fs,1);
          colororder=get(gca,'ColorOrder');
          axis tight; kc=2;
          %ap=axis;
          for j=1:rfs
              kc=kc+1; if rem(kc,length(colororder))==1, kc=2; end
              %for b=1:max(find(fs(j,:))); % new form of "find" in r2009a
              for b=1:find(fs(j,:), 1, 'last' );
                  % plot the tau division boundaries
                  %plot([fs(j,b) fs(j,b)],[ap(3)*1.1 ap(4)*1.1],'-','Color',colororder(kc,:));
                  % plot tau group y values
                  if b == 1
                      plot([dtime(1) fs(j,b)],[fval{j}(b)-s.median fval{j}(b)-s.median],'-','Color',colororder(kc,:),'LineWidth',4);
                  else
                      plot([fs(j,b-1) fs(j,b)],[fval{j}(b)-s.median fval{j}(b)-s.median],'-','Color',colororder(kc,:),'LineWidth',4);
                  end
              end
          end
          axis auto
      end % End optional bin plot
      
  end % end plot raw data
  
  
  if verbose >= 1 % show ADEV 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 in a future release, 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(['Allan Deviation: ' name],'FontSize',FontSize+2,'FontName',FontName);
          %set(get(gca,'Title'),'Interpreter','none');
          xlabel('\tau [sec]','FontSize',FontSize,'FontName',FontName);
          if isfield(data,'units')
              ylabel(['\sigma_y(\tau) [' data.units ']'],'FontSize',FontSize,'FontName',FontName);
          else
              ylabel('\sigma_y(\tau)','FontSize',FontSize,'FontName',FontName);
          end
          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 ADEV value: %g at tau = %g seconds
  ',min(sm),tau(sm==min(sm)));
          
      elseif verbose >= 1
          fprintf(1,'allan: 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" for more information.
  
  ');
      end
  
  end % end plot ADEV data
      
  retval = sm;
  errorb = sme;
  
  return