estimatenoise.m

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function noisevar = estimatenoise(X,varargin)
% estimatenoise: additive noise estimation from a time series
% usage: noisevar = estimatenoise(X)
% usage: noisevar = estimatenoise(X,dim)
% usage: noisevar = estimatenoise(X,t)
% usage: noisevar = estimatenoise(X,t,dim)
%
%
% Assume the time series has additive, Gaussian noise on top
% of a smoothly varying signal. The noise component variance
% is extracted. I'd see this code as useful for Kalman filter
% estimation, where one needs an estimate of the noise variance
% in data. It can also be applied to spline fitting problems,
% where one can control the regularization parameter to match
% a known noise variance.
%
%
% arguments: (input)
%  X - (REQUIRED) (numeric vector of length n, or any n-dimensional
%        numeric array)
%
%        X must be at least of length 5 in the specified dimension.
%
%        WARNING: Estimation of any variance will be suspect with
%        few data points in your sample.
%
%        If X is complex, then I treat the problem as two parallel
%        sequences in parallel. The variance returned will be the
%        sum of the variances for the real and complex parts.
%
%  t - (OPTIONAL) - list of "time" steps if X was sampled at an
%        non-uniform spacing. If supplied, then t must be a vector
%        of the same length as X is in dimension dim.
%        
%  dim - specifies the dimension of the array to act on
%        If dim is not provided, then the first non-singleton
%        dimension will be used.
%
%
% arguments: (output)
%  noisevar - scalar (or vector or array as appropriate)
%        estimated variance of additive noise in the data
%        along dimension dim.
%
%
% Example:
%  Simple linear data, with purely additive N(0,1) gaussian noise 
%
%  t = 0:10000;
%  x = t + randn(size(t));
%  mv = estimatenoise(x)
%  mv =
%        1.0166
%
% Example: 
%  Gaussian noise added to a sine wave (Nominal variance = 0.01)
%  t = linspace(0,1,1000)';
%  x = sin(t*50) + randn(size(t))/10;
%
%  mv = estimatenoise(x)
%
%  mv =
%      0.0096887
%
% Example:
%  Pure gaussian noise, with a nominal variance of 9. (Note that var
%  would have been a better estimator for this particular case...)
%
%  mv = estimatenoise(3*randn(2,3,1000),3)
%  mv =
%       9.6584       8.2696        8.632
%       9.2404       8.5346       9.7725
%
% Example:
%  A piecewise constant function with multiple discontinuities.
%  The true noise variance should be 0.01.
%
%  t = linspace(0,1,1000);
%  X = round(cos(t*6*pi)) + randn(size(t))/10;
%  plot(t,X)
%
%  var(X)
%  ans =
%       0.68256
%
%  estimatenoise(X)
%  ans =
%       0.010882
%
% Example:
%  Test if estimatenoise is able to recover the variance of a
%  normally distributed random sample with unit variance. (Yes,
%  it will be much slower than var.)
%
%  mean(estimatenoise(randn(1000,1000)))
%  ans =
%       1.0002
%
% Example: 
%  Use of extimatenoise on a problem with non-uniform spacing
%  The actual noise variance was 1.0 here. Perform the
%  operation 1000 times, then look at the median estimate of
%  the variance. How well did we do?
%
%  t = sort([randn(1,100) , randn(1,100)+5]);
%  X = repmat(sin(t*5)*100,1000,1) + randn(1000,length(t));
%
% Estimatenoise is clearly wrong when the spacing is ignored
%  median(estimatenoise(X,2))
%  ans =
%       16.438
%
% Supplying the sampling "times", we get quite a reasonable result.
%  median(estimatenoise(X,2,t))
%  ans =
%       1.1307
%
% Example: 
%  Use of extimatenoise on a problem with replicates. In this
%  example, each point will be replicated up to 3 times.
%  The actual noise variance was again 1.0 here. Look at the
%  increase in times required for estimatenoise when t is
%  supplied.
%
%  t = sort([0:1:100, 1:2:100, 1:4:100]);
%  X = repmat(sin(t/10)*100,1000,1) + randn(1000,length(t));
%
% Estimatenoise is clearly wrong when the non-uniform spacing
% is ignored.
%
%  tic,median(estimatenoise(X,2)),toc
%  ans =
%       4.2056
%  Elapsed time is 2.690135 seconds.
%
% Supplying the sampling "times", we get quite a reasonable
% result. The cost in time is not quite 2x.
%
%  tic,median(estimatenoise(X,2,t)),toc
%  ans =
%       1.0116
%  Elapsed time is 4.864486 seconds.
%
%  
% Methodology behind estimatenoise:
%  Assume that the time series X is composed of a smoothly varying
%  function, plus additive gaussian noise with mean zero and
%  variance sigma^2. Then at any point it can be represented
%  by a low order polynomial (think of it as a truncated local
%  Taylor series approximation). The finite differences applied
%  using filter was designed to kill off low order polynomial
%  terms, while leaving behind a correlated series of points,
%  but the variance of that series will be unchanged at sigma^2.
%
%  Estimatenoise is designed to be robust to a few arbitrary
%  spikes or discontinuities in the function or its derivatives.
%  This was achieved by trimming off the tails of the distributions,
%  then using percentiles to back out the desired variance.
%
%  This robustness has a penalty however, as it takes a bit more
%  time to run, and there is also an observed bias in the results.
%  This bias is predictable as a function of the series length, so
%  one of the last steps in estimatenoise backs out that predicted
%  bias from the estimated variance. As a test of this,
%
%   X = randn(100000,20);
%   nv = estimatenoise(X,2);
%   mean(nv)
%   ans =
%      0.98355
%
%  Yes, var would have been a better estimator in this particular
%  case, but estimatenoise was designed to solve a more difficult
%  problem.
%
%  When a non-uniform time series spacing is supplied, estimatenoise
%  works the same way, but filter cannot be used to do the
%  computations efficiently, since the filter coefficients vary
%  along that series. Here null is used to compute the time
%  varying filter coefficients.
%
%
% See also: std, var
%
%
% Author: John D'Errico
% e-mail: [e-mail]
% Release: 2.0
% Release date: 8/25/07


% if no input arguments provided, just dump out the help
if nargin<1
  help estimatenoise
  return
end

% complex sequences are split into real and imaginary parts.
if ~isreal(X)
  noisevar = estimatenoise(real(X)) + estimatenoise(imag(X));
  return
end

% get the size of X, so we can pick the value for
% dim if it was not supplied. We will also need to
% make sure the length in that dimension is large
% enough.
sx = size(X);

% were t and/or dim supplied? In which order?
if (nargin>3)
  error('Too many inputs supplied. Maximum of 3 inputs are allowed.')
elseif nargin == 1
  % use defaults
  dim = [];
  t = [];
elseif (nargin == 2) && isempty(varargin{1})
  % use defaults
  dim = [];
  t = [];
elseif (nargin==2) && (length(varargin{1}) == 1)
  % dim was supplied
  dim = varargin{1};
  t = [];
elseif (nargin==2)
  % t must have been specified
  t = varargin{1};
  t = t(:);
  dim = [];
elseif (nargin==3) && (length(varargin{1}) == 1)
  % dim and t were supplied, in that order
  dim = varargin{1};
  t = varargin{2};
  t = t(:);
elseif (nargin==3) && (length(varargin{2}) == 1)
  % t and dim were supplied, in that order
  dim = varargin{2};
  t = varargin{1};
  t = t(:);
else
  % there is a problem with the arguments
  error('A and t arguments are of inconsistent size')
end

% get dim if it was not supplied
if isempty(dim)
  dim = find(sx~=1,1,'first');
  if isempty(dim)
    error('X did not have at least one dimension of length >1')
  end
end

% check the length in the dim dimension
nX = sx(dim);
if nX<5
  error('The length of X in the specified dimension was less than 5')
end

% permute the dimensions so that the chosen dimension
% of X is moved to the end of the line.
ndim = length(sx);
nx = 1:ndim;
nx(dim) = [];
nx = [nx,dim];
Xp = permute(X,nx);
% then just reshape it to be a 2-d array
Xp = reshape(Xp,[],sx(dim));

% was t actually equally spaced anyway? If it was, then we
% want to use the equal spaced code.
equispaced = true;
if ~isempty(t)
  % first sort t, just in case
  [t,tags] = sort(t);
  % and shuffle Xp in the first dimension
  Xp = Xp(:,tags);
  
  % check for equal spacing in t
  dt = diff(t);
  
  % average spacing
  avespace = (t(end) - t(1))/(nX-1);
  if (avespace == 0)
    error('Invalid t vector: t was identically zero.')
  end
  tol = 10*eps(avespace);
  if tol < (max(dt) - min(dt))
    % unequal spacing
    equispaced = false;
  end
end

% estimate the measurement variability in the (now) second dimension
% later we will undo this reshape.

% The idea here is to form a linear combination of successive elements
% of the series. If the underlying form is locally nearly linear, then
% a [1 -2 1] combination (for equally spaced data) will leave only
% the noise remaining. Next, if we assume the measurement noise was
% iid, N(0,s^2), then we can try to back out the noise variance.
fda{1} = [1 -1];
fda{2} = [1 -2 1];
fda{3} = [1 -3 3 -1];
fda{4} = [1 -4 6 -4 1];
fda{5} = [1 -5 10 -10 5 -1];
fda{6} = [1 -6 15 -20 15 -6 1];
nfda = length(fda);
for i = 1:nfda
  % normalize to unit norm
  fda{i} = fda{i}/norm(fda{i});
end

% compute an interquantile range, like the distance between the 25%
% and 75% points. This trims off the trash at each end, potentially
% corrupted if there are discontinuities in the curve. It also deals
% simply with a non-zero mean in this data. Actually do this for
% several different interquantile ranges, then take a median.
% NOTE: While I could have used other methods for the final variance
% estimation, this method was chosen to avoid outlier issues when
% the curve may have isolated discontinuities in function value or
% a derivative.

% The following points correspond to the central 90, 80, 75, 70, 65,
% 60, 55, 50, 45, 40, 35, 30, 25, and 20 percent ranges.
perc = [0.05, 0.1:0.025:0.40];
z = erfinv((1-perc)*2-1)*sqrt(2);

sigmaest = nan(size(Xp,1),nfda);
for i = 1:nfda
  % Apply each difference to the data series with convn
  % the 'valid' option will trim off the junk at the ends
  % from the convolution.
  
  % did we have equal spacing?
  if equispaced
    % If so, then conv will do the trick.
    
    % These convolutions will yield noise of the desired variance,
    % although it will be colored noise.
    noisedata = conv2(Xp,fda{i},'valid');
    ntrim = size(noisedata,2);
  else
    % unequal spacing. do it the hard way
    F = 1./gamma(1:i);
    ntrim = nX - i;
    noisedata = zeros(size(Xp,1),ntrim);
    for j = 1:ntrim
      tj = t(j+(0:i));
      tj = tj - mean(tj);
      if all(abs(tj) <= tol)
        % be careful if all points were reps!
        coef = rem((0:i)',2)*2-1;
        coef = coef - mean(coef);
        % and normalize
        coef = coef/norm(coef);
      else
        % use null to choose the linear combination
        % of the elements of Xp.
        tj = tj/norm(tj);
        A = repmat(tj,1,i).^repmat(0:(i-1),i+1,1);
        A = A.*repmat(F,i+1,1);
        nullvecs = null(A');
        % already normalized to have unit norm by null
        coef = nullvecs(:,1);
      end
      
      % form the appropriate linear combination of Xp
      noisedata(:,j) = Xp(:,j+(0:i))*coef;
    end
  end  % if equispaced
  
  % were there enough points to even try anything?
  if ntrim >= 2
    % sorting will provide the necessary percentiles after
    % interpolation.
    noisedata = sort(noisedata,2);
    
    p = 0.5 + (1:ntrim)';
    p = p/(ntrim + 0.5);
    
    Q = zeros(size(noisedata,1),length(perc));
    for j = 1:length(perc)
      Qj = interp1q(p,noisedata',1-perc(j)) - interp1q(p,noisedata',perc(j));
      Qj = Qj/(2*z(j));
      Q(:,j) = Qj';
    end

    % Trim off any nans first, since if the series was short enough,
    % some of those percentiles were not present.
    wasnan = isnan(Q(1,:));
    Q(:,wasnan) = [];
    
    % Our noise std estimate is given by the median of the interquantile
    % range(s). This is an ad hoc, but hopefully effective, way of
    % estimating the measurement noise present in the signal.
    sigmaest(:,i) = median(Q,2);
    
  end
  
end % for i = 1:nfda

% drop those estimates which failed for lack of enough data
sigmaest(:,isnan(sigmaest(1,:))) = [];

% use median of these estimates to get a noise estimate.
noisevar = median(sigmaest,2).^2;

% Use an adhoc correction to remove the bias in the noise estimate.
% This correction was determined by examination of a large number of
% random samples.
noisevar = noisevar/(1+15*(sx(dim)+1.225)^-1.245);

% The first order difference might be used to guesstimate the
% process noise.
% if nargout>1
%   processvar = max(0,sigmaest(:,1).^2 - noisevar);
% end

% finally, reshape the result to be consistent with the input shape
sx(dim) = 1;
noisevar = reshape(noisevar,sx);
% if nargout>1
%   processvar = reshape(processvar,sx);
% end