octave-nan 3.1.4-3 / usr / share / octave / packages / nan-3.1.4 / doc-cache

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bland_altman


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 BLAND_ALTMANN shows the Bland-Altman plot of two columns of measurements
   and computes several summary results. 

   bland_altman(m1, m2 [,group])
   bland_altman(data [, group])
   R = bland_altman(...)
 
   m1,m2 are two colums with the same number of elements
	containing the measurements. m1,m2 can be also combined 
       in a single two column data matrix. 
   group [optional] indicates which measurements belong to the same group
	This is useful to account for repeated measurements.  


 References:
 [1] JM Bland and DG Altman, Measuring agreement in method comparison studies. 
       Statistical Methods in Medical Research, 1999; 8; 135. 
       doi:10.1177/09622802990080204
 [2] P.S. Myles, Using the Bland– Altman method to measure agreement with repeated measures
	British Journal of Anaesthesia 99(3):309–11 (2007)
	doi:10.1093/bja/aem214



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 BLAND_ALTMANN shows the Bland-Altman plot of two columns of measurements
   and



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cat2bin


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 CAT2BIN converts categorial into binary data 
   each category of each column in D is converted into a logical column
 
   B = cat2bin(C); 
   [B,BinLabel] = cat2bin(C,Label); 
   [B,BinLabel] = cat2bin(C,Label,MODE)

  C        categorial data 
  B        binary data 
  Label    description of each column in C
  BinLabel description of each column in B
  MODE     default [], ignores NaN
           'notIgnoreNAN' includes binary column for NaN 
           'IgnoreZeros'  zeros do not get a separate category 
           'IgnoreZeros+NaN' zeros and NaN are ignored

  example: 
     cat2bin([1;2;5;1;5]) results in 
             1     0     0
             0     1     0
             0     0     1
             1     0     0
             0     0     1



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 CAT2BIN converts categorial into binary data 
   each category of each column i



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cdfplot


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 CDFPLOT plots empirical commulative distribution function

   cdfplot(X)
   cdfplot(X, FMT)
   cdfplot(X, PROPERTY, VALUE,...)
   h = cdfplot(...)
   [h,stats] = cdfplot(X)

  X contains the data vector
 	(matrix data is currently changed to a vector, this might change in future) 
  FMT,PROPERTY,VALUE 
	are used for formating; see HELP PLOT for more details 
  h 	graphics handle to the cdf curve
  stats 
	a struct containing various summary statistics including
	mean, std, median, min, max.

 see also: ecdf, median, statistics, hist2res, plot

 References: 



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 CDFPLOT plots empirical commulative distribution function



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center


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 CENTER removes the mean 

 [z,mu] = center(x,DIM,W)
   removes mean x along dimension DIM

 x	input data 
 DIM	dimension
	1: column
	2: row
	default or []: first DIMENSION, with more than 1 element
 W	weights to computed weighted mean (default: [], all weights = 1)
	numel(W) must be equal to size(x,DIM)

 features:
 - can deal with NaN's (missing values)
 - weighting of data 
 - dimension argument 
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, MEAN, STD, DETREND, ZSCORE

 REFERENCE(S):



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 CENTER removes the mean 



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classify


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 CLASSIFY classifies sample data into categories 
 defined by the training data and its group information 

  CLASS = classify(sample, training, group) 
  CLASS = classify(sample, training, group, TYPE) 
  [CLASS,ERR,POSTERIOR,LOGP,COEF] = CLASSIFY(...) 

  CLASS contains the assigned group. 
  ERR is the classification error on the training set weighted by the 
	prior propability of each group. 

  The same classifier as in TRAIN_SC are supported. 

 ATTENTION: no cross-validation is applied, therefore the 
    classification error is too optimistic (overfitting). 
    Use XVAL instead to obtain cross-validated performance. 
 
 see also: TRAIN_SC, TEST_SC, XVAL

 References: 
 [1] R. Duda, P. Hart, and D. Stork, Pattern Classification, second ed. 
       John Wiley & Sons, 2001. 



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 CLASSIFY classifies sample data into categories 
 defined by the training data 



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coefficient_of_variation


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 COEFFICIENT_OF_VARIATION returns STD(X)/MEAN(X)
 
 cv=coefficient_of_variation(x [,DIM])
  cv=std(x)/mean(x) 

 see also: SUMSKIPNAN, MEAN, STD

   REFERENCE(S):
   http://mathworld.wolfram.com/VariationCoefficient.html



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 COEFFICIENT_OF_VARIATION returns STD(X)/MEAN(X)
 
 cv=coefficient_of_variation(



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cor


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 COR calculates the correlation matrix
 X and Y can contain missing values encoded with NaN.
 NaN's are skipped, NaN do not result in a NaN output. 
 (Its assumed that the occurence of NaN's is uncorrelated) 
 The output gives NaN only if there are insufficient input data

 COR(X);
      calculates the (auto-)correlation matrix of X
 COR(X,Y);
      calculates the crosscorrelation between X and Y

 c = COR(...);
 	c is the correlation matrix

 W	weights to compute weighted mean (default: [])
	if W=[], all weights are 1. 
	number of elements in W must match size(x,DIM) 



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 COR calculates the correlation matrix
 X and Y can contain missing values encod



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corrcoef


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 CORRCOEF calculates the correlation matrix from pairwise correlations.
   The input data can contain missing values encoded with NaN.
   Missing data (NaN's) are handled by pairwise deletion [15]. 
   In order to avoid possible pitfalls, use case-wise deletion or 
   or check the correlation of NaN's with your data (see below). 
   A significance test for testing the Hypothesis  
   'correlation coefficient R is significantly different to zero' 
   is included. 

 [...] = CORRCOEF(X);
      calculates the (auto-)correlation matrix of X
 [...] = CORRCOEF(X,Y);
      calculates the crosscorrelation between X and Y
      NOTE: matlab's CORRCOEF(X,Y) returns the result of CORRCOEF([X,Y])
            use CORRCOEF([X,Y]) if your software should be compatible with both.

 [...] = CORRCOEF(..., Mode);
       Mode='Pearson' or 'parametric' [default]
               gives the correlation coefficient  
               also known as the 'product-moment coefficient of correlation' 
               or 'Pearson''s correlation' [1]
       Mode='Spearman' 	gives 'Spearman''s Rank Correlation Coefficient'
               This replaces SPEARMAN.M
       Mode='Rank' 		gives a nonparametric Rank Correlation Coefficient
               This is the "Spearman rank correlation with proper handling of ties"
               This replaces RANKCORR.M

 [...] = CORRCOEF(..., param1, value1, param2, value2, ... );
	param 		value
	'Mode'		type of correlation 
		'Pearson','parametric'
		'Spearman'
		'rank'
	'rows'		how do deal with missing values encoded as NaN's. 	
		'complete': remove all rows with at least one NaN
		'pairwise': [default]
	'alpha'		0.01	: significance level to compute confidence interval

 [R,p,ci1,ci2,nan_sig] = CORRCOEF(...);
 	R is the correlation matrix
	R(i,j) is the correlation coefficient r between X(:,i) and Y(:,j)
  p    gives the significance of R
	It tests the null hypothesis that the product moment correlation coefficient is zero 
       using Student's t-test on the statistic t = r*sqrt(N-2)/sqrt(1-r^2) 
       where N is the number of samples (Statistics, M. Spiegel, Schaum series).
  p > alpha: do not reject the Null hypothesis: 'R is zero'.
  p < alpha: The alternative hypothesis 'R is larger than zero' is true with probability (1-alpha).
  ci1	lower (1-alpha) confidence interval 
  ci2	upper (1-alpha) confidence interval
	If no alpha is provided, the default alpha is 0.01. This can be changed with function flag_implicit_significance. 
  nan_sig 	p-value whether H0: 'NaN''s are not correlated' could be correct
       if nan_sig < alpha, H1 ('NaNs are correlated') is very likely. 
 
 The result is only valid if the occurence of NaN's is uncorrelated. In
 order to avoid this pitfall, the correlation of NaN's should be checked 
 or case-wise deletion should be applied. 
   Case-Wise deletion can be implemented 
    ix = ~any(isnan([X,Y]),2);
    [...] = CORRCOEF(X(ix,:),Y(ix,:),...); 

  Correlation (non-random distribution) of NaN's can be checked with 
       [nan_R,nan_sig]=corrcoef(X,isnan(X))
   or  [nan_R,nan_sig]=corrcoef([X,Y],isnan([X,Y]))
   or  [R,p,ci1,ci2] = CORRCOEF(...);

 Further recommandation related to the correlation coefficient: 
 + LOOK AT THE SCATTERPLOTS to make sure that the relationship is linear
 + Correlation is not causation because 
	it is not clear which parameter is 'cause' and which is 'effect' and
       the observed correlation between two variables might be due to the action of other, unobserved variables.

 see also: SUMSKIPNAN, COVM, COV, COR, SPEARMAN, RANKCORR, RANKS,
       PARTCORRCOEF, flag_implicit_significance

 REFERENCES:
 on the correlation coefficient 
 [ 1] http://mathworld.wolfram.com/CorrelationCoefficient.html
 [ 2] http://www.geography.btinternet.co.uk/spearman.htm
 [ 3] Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 5th ed.  New York: Macmillan, pp. 338 and 400, 1995.
 [ 4] Lehmann, E. L. and D'Abrera, H. J. M. Nonparametrics: Statistical Methods Based on Ranks, rev. ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 292, 300, and 323, 1998.
 [ 5] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 634-637, 1992
 [ 6] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
 on the significance test of the correlation coefficient
 [11] http://www.met.rdg.ac.uk/cag/STATS/corr.html
 [12] http://www.janda.org/c10/Lectures/topic06/L24-significanceR.htm
 [13] http://faculty.vassar.edu/lowry/ch4apx.html
 [14] http://davidmlane.com/hyperstat/B134689.html
 [15] http://www.statsoft.com/textbook/stbasic.html%Correlations
 others
 [20] http://www.tufts.edu/~gdallal/corr.htm
 [21] Fisher transformation http://en.wikipedia.org/wiki/Fisher_transformation



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 CORRCOEF calculates the correlation matrix from pairwise correlations.



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cov


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 COV covariance matrix
 X and Y can contain missing values encoded with NaN.
 NaN's are skipped, NaN do not result in a NaN output. 
 The output gives NaN only if there are insufficient input data
 The mean is removed from the data. 
 
 Remark: for data contains missing values, the resulting 
 matrix might not be positiv definite, and its elements have magnitudes
 larger than one. This ill-behavior is more likely for small sample 
 sizes, but there is no garantee that the result "behaves well" for larger
 sample sizes. If you want the a "well behaved" result (i.e. positive 
 definiteness and magnitude of elements not larger than 1), use CORRCOEF. 
 However, COV is faster than CORRCOEF and might be good enough in some cases.

 C = COV(X [,Mode]);
      calculates the (auto-)correlation matrix of X
 C = COV(X,Y [,Mode]);
      calculates the crosscorrelation between X and Y. 
      C(i,j) is the correlation between the i-th and jth 
      column of X and Y, respectively. 
   NOTE: Octave and Matlab have (in some special cases) incompatible implemenations. 
       This implementation follows Octave. If the result could be ambigous or  
       incompatible, a warning will be presented in Matlab. To avoid this warning use: 
       a) use COV([X(:),Y(:)]) if you want the traditional Matlab result. 
       b) use C = COV([X,Y]), C = C(1:size(X,2),size(X,2)+1:size(C,2)); if you want to be compatible with this software.  

 Mode = 0 [default] scales C by (N-1)
 Mode = 1 scales C by N. 

 see also: COVM, COR, CORRCOEF, SUMSKIPNAN

 REFERENCES:
 http://mathworld.wolfram.com/Covariance.html



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 COV covariance matrix
 X and Y can contain missing values encoded with NaN.



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covm


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 COVM generates covariance matrix
 X and Y can contain missing values encoded with NaN.
 NaN's are skipped, NaN do not result in a NaN output. 
 The output gives NaN only if there are insufficient input data

 COVM(X,Mode);
      calculates the (auto-)correlation matrix of X
 COVM(X,Y,Mode);
      calculates the crosscorrelation between X and Y
 COVM(...,W);
	weighted crosscorrelation 

 Mode = 'M' minimum or standard mode [default]
 	C = X'*X; or X'*Y correlation matrix

 Mode = 'E' extended mode
 	C = [1 X]'*[1 X]; % l is a matching column of 1's
 	C is additive, i.e. it can be applied to subsequent blocks and summed up afterwards
 	the mean (or sum) is stored on the 1st row and column of C

 Mode = 'D' or 'D0' detrended mode
	the mean of X (and Y) is removed. If combined with extended mode (Mode='DE'), 
 	the mean (or sum) is stored in the 1st row and column of C. 
 	The default scaling is factor (N-1). 
 Mode = 'D1' is the same as 'D' but uses N for scaling. 

 C = covm(...); 
 	C is the scaled by N in Mode M and by (N-1) in mode D.
 [C,N] = covm(...);
	C is not scaled, provides the scaling factor N  
	C./N gives the scaled version. 

 see also: DECOVM, XCOVF



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 COVM generates covariance matrix
 X and Y can contain missing values encoded wi



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cumsumskipnan


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 CUMSUMSKIPNAN  Cumulative sum while skiping NaN's. 
 If DIM is omitted, it defaults to the first non-singleton dimension.
 
 Y = cumsumskipnan(x [,DIM])
 
 x	input data 	
 DIM	dimension (default: [])
 y	resulting sum

 see also: CUMSUM, SUMSKIPNAN



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 CUMSUMSKIPNAN  Cumulative sum while skiping NaN's.



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decovm


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 decompose extended covariance matrix into mean (mu), 
 standard deviation, the (pure) Covariance (COV), 
 correlation (xc) matrix and the correlation coefficients R2.
 NaN's are condsidered as missing values. 
 [mu,sd,COV,xc,N,R2]=decovm(ECM[,NN])

 ECM 	is the extended covariance matrix
 NN	is the number of elements, each estimate (in ECM) is based on 

 see also: MDBC, COVM, R2



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 decompose extended covariance matrix into mean (mu), 
 standard deviation, the 



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detrend


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 DETREND removes the trend from data, NaN's are considered as missing values
 
 DETREND is fully compatible to previous Matlab and Octave DETREND with the following features added:
 - handles NaN's by assuming that these are missing values
 - handles unequally spaced data
 - second output parameter gives the trend of the data
 - compatible to Matlab and Octave 

 [...]=detrend([t,] X [,p])
	removes trend for unequally spaced data
	t represents the time points
	X(i) is the value at time t(i)
	p must be a scalar

 [...]=detrend(X,0)
 [...]=detrend(X,'constant')
	removes the mean

 [...]=detrend(X,p)
	removes polynomial of order p (default p=1)

 [...]=detrend(X,1) - default
 [...]=detrend(X,'linear')
	removes linear trend 

 [X,T]=detrend(...) 

 X is the detrended data
 T is the removed trend
 
 see also: SUMSKIPNAN, ZSCORE		



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 DETREND removes the trend from data, NaN's are considered as missing values
 
 



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ecdf


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 ECDF empirical cumulative function  
  NaN's are considered Missing values and are ignored. 

  [F,X] = ecdf(Y)
	calculates empirical cumulative distribution functions (i.e Kaplan-Meier estimate)
  ecdf(Y)
  ecdf(gca,Y)
	without output arguments plots the empirical cdf, in axis gca. 

 Y 	input data
	must be a vector or matrix, in case Y is a matrix, the ecdf for every column is computed. 

 see also: HISTO2, HISTO3, PERCENTILE, QUANTILE



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 ECDF empirical cumulative function  
  NaN's are considered Missing values and 



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flag_accuracy_level


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 FLAG_ACCURACY_LEVEL sets and gets accuracy level 
   used in SUMSKIPNAN_MEX and COVM_MEX
   The error margin of the naive summation is N*eps (N is the number of samples),
   the error margin is only 2*eps if Kahan's summation is used [1].    

	0: maximum speed [default]
	   accuracy of double (64bit) with naive summation (error = N*2^-52) 
	1: accuracy of extended (80bit) with naive summation (error = N*2^-64) 
	2: accuracy of double (64bit) with Kahan summation (error = 2^-52) 
	3: accuracy of extended (80bit) with Kahan summation  (error = 2^-64)  

   Please note, level 3 might be equally accurate but slower than 1 or 2 on
   some platforms. In order to determine what is good for you, you might want
   to run ACCTEST. 

 FLAG = flag_accuracy_level()
 	gets current level
 flag_accuracy_level(FLAG) 
 	sets accuracy level  
 
 see also: ACCTEST
 
 Reference:
 [1] David Goldberg, 
       What Every Computer Scientist Should Know About Floating-Point Arithmetic
       ACM Computing Surveys, Vol 23, No 1, March 1991. 



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 FLAG_ACCURACY_LEVEL sets and gets accuracy level 
   used in SUMSKIPNAN_MEX an



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flag_implicit_significance


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 The use of FLAG_IMPLICIT_SIGNIFICANCE is in experimental state. 
 flag_implicit_significance might even become obsolete.

 FLAG_IMPLICIT_SIGNIFICANCE sets and gets default alpha (level) of any significance test
 The default alpha-level is stored in the global variable FLAG_implicit_significance
 The idea is that the significance must not be assigned explicitely. 
 This might yield more readable code. 

 Choose alpha low enough, because in alpha*100% of the cases, you will 
 reject the Null hypothesis just by change. For this reason, the default
 alpha is 0.01. 
 
   flag_implicit_significance(0.01) 
	sets the alpha-level for the significance test
 
 alpha = flag_implicit_significance()
 	gets default alpha

 flag_implicit_significance(alpha)
 	sets default alpha-level

 alpha = flag_implicit_significance(alpha)
	gets and sets alpha 

 features:
 - compatible to Matlab and Octave

 see also: CORRCOEF, PARTCORRCOEF



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 The use of FLAG_IMPLICIT_SIGNIFICANCE is in experimental state.



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flag_implicit_skip_nan


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 FLAG_IMPLICIT_SKIP_NAN sets and gets default mode for handling NaNs
	1 skips NaN's (the default mode if no mode is set)
 	0 NaNs are propagated; input NaN's give NaN's at the output
 
 FLAG = flag_implicit_skip_nan()
 	gets current mode

 flag_implicit_skip_nan(FLAG)
 sets mode 

 prevFLAG = flag_implicit_skip_nan(nextFLAG)
	gets previous set FLAG and sets FLAG for the future
 flag_implicit_skip_nan(prevFLAG)
	resets FLAG to previous mode

 It is used in: 
	SUMSKIPNAN, MEDIAN, QUANTILES, TRIMEAN
 and affects many other functions like: 
	CENTER, KURTOSIS, MAD, MEAN, MOMENT, RMS, SEM, SKEWNESS, 
	STATISTIC, STD, VAR, ZSCORE etc. 

 The mode is stored in the global variable FLAG_implicit_skip_nan
 It is recommended to use flag_implicit_skip_nan(1) as default and
 flag_implicit_skip_nan(0) should be used for exceptional cases only.
 This feature might disappear without further notice, so you should really not
 rely on it. 



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 FLAG_IMPLICIT_SKIP_NAN sets and gets default mode for handling NaNs
	1 skips Na



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flag_nans_occured


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 FLAG_NANS_OCCURED checks whether the last call(s) to sumskipnan or covm 
 contained any not-a-numbers in the input argument. Because many other 
 functions like mean, std, etc. are also using sumskipnan, 
 also these functions can be checked for NaN's in the input data. 
 
 A call to FLAG_NANS_OCCURED() resets also the flag whether NaN's occured. 
 Only sumskipnan or covm can set the flag again. 

 see also: SUMSKIPNAN, COVM



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 FLAG_NANS_OCCURED checks whether the last call(s) to sumskipnan or covm 
 conta



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fss


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 FSS - feature subset selection and feature ranking 
   the method is motivated by the max-relevance-min-redundancy (mRMR) 
   approach [1]. However, the default method uses partial correlation,
   which has been developed from scratch. PCCM [3] describes
   a similar idea, but is more complicated. 
   An alternative method based on FSDD is implemented, too. 
    
  [idx,score] = fss(D,cl) 
  [idx,score] = fss(D,cl,MODE) 
  [idx,score] = fss(D,cl,MODE) 
    
 D 	data - each column represents a feature 
 cl	classlabel   
 Mode 	'Pearson' [default] correlation
	'rank' correlation 
       'FSDD' feature selection algorithm based on a distance discriminant [2]
       %%% 'MRMR','MID','MIQ' max-relevance, min redundancy [1] - not supported yet. 

 score score of the feature
 idx	ranking of the feature    
       [tmp,idx]=sort(-score)

 see also: TRAIN_SC, XVAL, ROW_COL_DELETION

 REFERENCES:
 [1] Peng, H.C., Long, F., and Ding, C., 
   Feature selection based on mutual information: criteria of max-dependency, max-relevance, and min-redundancy, 
   IEEE Transactions on Pattern Analysis and Machine Intelligence, 
   Vol. 27, No. 8, pp.1226-1238, 2005.
 [2] Jianning Liang, Su Yang, Adam Winstanley, 
   Invariant optimal feature selection: A distance discriminant and feature ranking based solution, 
   Pattern Recognition, Volume 41, Issue 5, May 2008, Pages 1429-1439.
   ISSN 0031-3203, DOI: 10.1016/j.patcog.2007.10.018.
 [3] K. Raghuraj Rao and S. Lakshminarayanan
   Partial correlation based variable selection approach for multivariate data classification methods
   Chemometrics and Intelligent Laboratory Systems
   Volume 86, Issue 1, 15 March 2007, Pages 68-81 
   http://dx.doi.org/10.1016/j.chemolab.2006.08.007



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 FSS - feature subset selection and feature ranking 
   the method is motivated 



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geomean


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 GEOMEAN calculates the geomentric mean of data elements. 
 
 	y = geomean(x [,DIM [,W]])   is the same as 
 	y = mean(x,'G' [,DIM]) 

 DIM	dimension
	1 STD of columns
	2 STD of rows
	default or []: first DIMENSION, with more than 1 element
 W	weights to compute weighted mean (default: [])
	if W=[], all weights are 1. 
	number of elements in W must match size(x,DIM) 

 features:
 - can deal with NaN's (missing values)
 - weighting of data 
 - dimension argument also in Octave
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, MEAN, HARMMEAN

    This program is free software; you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation; either version 3 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with this program; If not, see <http://www.gnu.org/licenses/>.



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 GEOMEAN calculates the geomentric mean of data elements.



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gscatter


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 GSCATTER scatter plot of groups 

  gscatter(x,y,group)
  gscatter(x,y,group,clr,sym,siz)
  gscatter(x,y,group,clr,sym,siz,doleg)
  gscatter(x,y,group,clr,sym,siz,doleg,xname,yname)
  h = gscatter(...) 

  x,y, group: 	vectors with equal length 
  clf: 	color vector, default 'bgrcmyk'
  sym:		symbol, default '.'
  siz: 	size of Marker
  doleg:  'on' (default) shows legend, 'off' turns of legend 
  xname, yname: name of axis


 see also: ecdf, cdfplot

 References: 



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 GSCATTER scatter plot of groups 



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harmmean


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 HARMMEAN calculates the harmonic mean of data elements. 
 The harmonic mean is the inverse of the mean of the inverse elements.
 
 	y = harmmean(x [,DIM [,W]]) is the same as 
 	y = mean(x,'H' [,DIM [,W]]) 

 DIM	dimension
	1 STD of columns
	2 STD of rows
	default or []: first DIMENSION, with more than 1 element
 W	weights to compute weighted mean (default: [])
	if W=[], all weights are 1. 
	number of elements in W must match size(x,DIM) 

 features:
 - can deal with NaN's (missing values)
 - weighting of data 
 - dimension argument also in Octave
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, MEAN, GEOMEAN




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 HARMMEAN calculates the harmonic mean of data elements.



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hist2res


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 Evaluates Histogram data
 [R]=hist2res(H)

 [y]=hist2res(H,fun)
	estimates fun-statistic

 fun	'mean'	mean
	'std'	standard deviation
	'var'	variance
	'sem'	standard error of the mean
	'rms'	root mean square
	'meansq' mean of squares
	'sum'	sum
	'sumsq'	sum of squares
	'CM#'	central moment of order #
	'skewness' skewness 
	'kurtosis' excess coefficient (Fisher kurtosis)

 see also: NaN/statistic

 REFERENCES:
 [1] C.L. Nikias and A.P. Petropulu "Higher-Order Spectra Analysis" Prentice Hall, 1993.
 [2] C.E. Shannon and W. Weaver "The mathematical theory of communication" University of Illinois Press, Urbana 1949 (reprint 1963).
 [3] http://www.itl.nist.gov/
 [4] http://mathworld.wolfram.com/



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 Evaluates Histogram data
 [R]=hist2res(H)



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iqr


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 IQR calculates the interquartile range  
  Missing values (encoded as NaN) are ignored. 

  Q = iqr(Y)
  Q = iqr(Y,DIM)
     returns the IQR along dimension DIM of sample array Y.

  Q = iqr(HIS)
     returns the IQR from the histogram HIS. 
     HIS must be a HISTOGRAM struct as defined in HISTO2 or HISTO3.

 see also: MAD, RANGE, HISTO2, HISTO3, PERCENTILE, QUANTILE



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 IQR calculates the interquartile range  
  Missing values (encoded as NaN) are 



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kappa


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 KAPPA estimates Cohen's kappa coefficient
   and related statistics 

 [...] = kappa(d1,d2);
	NaN's are handled as missing values and are ignored
 [...] = kappa(d1,d2,'notIgnoreNAN');
	NaN's are handled as just another Label.
 [kap,sd,H,z,ACC,sACC,MI] = kappa(...);
 X = kappa(...);

 d1    data of scorer 1 
 d2    data of scorer 2 

 kap	Cohen's kappa coefficient point
 se	standard error of the kappa estimate
 H	Concordance matrix, i.e. confusion matrix
 z	z-score
 ACC	overall agreement (accuracy) 
 sACC	specific accuracy 
 MI 	Mutual information or transfer information (in [bits])
 X 	is a struct containing all the fields above
       For two classes, a number of additional summary statistics including 
         TPR, FPR, FDR, PPV, NPF, F1, dprime, Matthews Correlation coefficient (MCC) or 
	Phi coefficient (PHI=MCC), Specificity and Sensitivity 
       are provided. Note, the positive category must the larger label (in d and c), otherwise 
       the confusion matrix becomes transposed and the summary statistics are messed up. 


 Reference(s):
 [1] Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20, 37-46.
 [2] J Bortz, GA Lienert (1998) Kurzgefasste Statistik f|r die klassische Forschung, Springer Berlin - Heidelberg. 
        Kapitel 6: Uebereinstimmungsmasze fuer subjektive Merkmalsurteile. p. 265-270.
 [3] http://www.cmis.csiro.au/Fiona.Evans/personal/msc/html/chapter3.html
 [4] Kraemer, H. C. (1982). Kappa coefficient. In S. Kotz and N. L. Johnson (Eds.), 
        Encyclopedia of Statistical Sciences. New York: John Wiley & Sons.
 [5] http://ourworld.compuserve.com/homepages/jsuebersax/kappa.htm
 [6] http://en.wikipedia.org/wiki/Receiver_operating_characteristic



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 KAPPA estimates Cohen's kappa coefficient
   and related statistics 



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kurtosis


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 KURTOSIS estimates the kurtosis

 y = kurtosis(x,DIM)
   calculates kurtosis of x in dimension DIM

 DIM	dimension
	1: STATS of columns
	2: STATS of rows
	default or []: first DIMENSION, with more than 1 element

 features:
 - can deal with NaN's (missing values)
 - dimension argument 
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, VAR, STD, VAR, SKEWNESS, MOMENT, STATISTIC, 
    IMPLICIT_SKIP_NAN

 REFERENCE(S):
 http://mathworld.wolfram.com/



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 KURTOSIS estimates the kurtosis



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load_fisheriris


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 LOAD_FISHERIRIS 
        loads famous iris data set from Fisher, 1936 [1]. 

 References: 
 [1] Fisher,R.A. "The use of multiple measurements in taxonomic problems" 
        Annual Eugenics, 7, Part II, 179-188 (1936); also in "Contributions to Mathematical Statistics" (John Wiley, NY, 1950).
 [2] Duda,R.O., & Hart,P.E. (1973) Pattern Classification and Scene Analysis. 
        (Q327.D83) John Wiley & Sons. ISBN 0-471-22361-1. See page 218.



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 LOAD_FISHERIRIS 
        loads famous iris data set from Fisher, 1936 [1].



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mad


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 MAD estimates the Mean Absolute deviation
 (note that according to [1,2] this is the mean deviation; 
 not the mean absolute deviation)

 y = mad(x,DIM)
   calculates the mean deviation of x in dimension DIM

 DIM	dimension
	1: STATS of columns
	2: STATS of rows
	default or []: first DIMENSION, with more than 1 element

 features:
 - can deal with NaN's (missing values)
 - dimension argument 
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, VAR, STD, 

 REFERENCE(S):
 [1] http://mathworld.wolfram.com/MeanDeviation.html
 [2] L. Sachs, "Applied Statistics: A Handbook of Techniques", Springer-Verlag, 1984, page 253.

 [3] http://mathworld.wolfram.com/MeanAbsoluteDeviation.html
 [4] Kenney, J. F. and Keeping, E. S. "Mean Absolute Deviation." �6.4 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 76-77 1962. 



# name: <cell-element>
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 MAD estimates the Mean Absolute deviation
 (note that according to [1,2] this i



# name: <cell-element>
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mahal


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 MAHAL return the Mahalanobis' D-square distance between the 
 multivariate samples x and y, which must have the same number 
 of components (columns), but may have a different number of observations (rows). 
 
  d = mahal(X,Y)

   d(k) = (X(k,:)-MU)*inv(SIGMA)*(X(k,:)-MU)'

   where MU and SIGMA are the mean and the covariance matrix of Y 


 see also: TRAIN_SC, TEST_SC, COVM

 References: 



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 MAHAL return the Mahalanobis' D-square distance between the 
 multivariate samp



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make


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# length: 46
 This make.m is used for Matlab under Windows



# name: <cell-element>
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# elements: 1
# length: 46
 This make.m is used for Matlab under Windows




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mean


# name: <cell-element>
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 MEAN calculates the mean of data elements. 
 
  y = mean(x [,DIM] [,opt] [, W])

 DIM	dimension
	1 MEAN of columns
	2 MEAN of rows
 	N MEAN of  N-th dimension 
	default or []: first DIMENSION, with more than 1 element

 opt	options 
	'A' arithmetic mean
	'G' geometric mean
	'H' harmonic mean

 W	weights to compute weighted mean (default: [])
	if W=[], all weights are 1. 
	number of elements in W must match size(x,DIM) 

 usage: 
	mean(x)
	mean(x,DIM)
	mean(x,opt)
	mean(x,opt,DIM)
	mean(x,DIM,opt)
	mean(x,DIM,W)
	mean(x,DIM,opt,W); '

 features:
 - can deal with NaN's (missing values)
 - weighting of data 
 - dimension argument also in Octave
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, MEAN, GEOMEAN, HARMMEAN




# name: <cell-element>
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 MEAN calculates the mean of data elements.



# name: <cell-element>
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# elements: 1
# length: 7
meandev


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 MEANDEV estimates the Mean deviation
 (note that according to [1,2] this is the mean deviation; 
 not the mean absolute deviation)

 y = meandev(x,DIM)
   calculates the mean deviation of x in dimension DIM

 DIM	dimension
	1: STATS of columns
	2: STATS of rows
	default or []: first DIMENSION, with more than 1 element

 features:
 - can deal with NaN's (missing values)
 - dimension argument 
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, VAR, STD, MAD

 REFERENCE(S):
 [1] http://mathworld.wolfram.com/MeanDeviation.html
 [2] L. Sachs, "Applied Statistics: A Handbook of Techniques", Springer-Verlag, 1984, page 253.
 [3] http://mathworld.wolfram.com/MeanAbsoluteDeviation.html
 [4] Kenney, J. F. and Keeping, E. S. "Mean Absolute Deviation." �6.4 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 76-77 1962. 



# name: <cell-element>
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 MEANDEV estimates the Mean deviation
 (note that according to [1,2] this is the



# name: <cell-element>
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# elements: 1
# length: 6
meansq


# name: <cell-element>
# type: sq_string
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# length: 527
 MEANSQ calculates the mean of the squares

 y = meansq(x,DIM,W)

 DIM	dimension
	1 STD of columns
	2 STD of rows
 	N STD of  N-th dimension 
	default or []: first DIMENSION, with more than 1 element
 W	weights to compute weighted mean (default: [])
	if W=[], all weights are 1. 
	number of elements in W must match size(x,DIM) 

 features:
 - can deal with NaN's (missing values)
 - weighting of data  
 - dimension argument also in Octave
 - compatible to Matlab and Octave

 see also: SUMSQ, SUMSKIPNAN, MEAN, VAR, STD, RMS



# name: <cell-element>
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 MEANSQ calculates the mean of the squares



# name: <cell-element>
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# elements: 1
# length: 9
medAbsDev


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# type: sq_string
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# length: 373
 medAbsDev calculates the median absolute deviation 

 Usage:  D = medAbsDev(X, DIM)  
    or:  [D, M] = medAbsDev(X, DIM)
 Input:  X  : data
         DIM: dimension along which mad should be calculated (1=columns, 2=rows) 
               (optional, default=first dimension with more than 1 element
 Output: D  : median absolute deviations
         M  : medians (optional)



# name: <cell-element>
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 medAbsDev calculates the median absolute deviation 



# name: <cell-element>
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# elements: 1
# length: 6
median


# name: <cell-element>
# type: sq_string
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# length: 366
 MEDIAN data elements, 
 [y]=median(x [,DIM])

 DIM	dimension
	1: median of columns
	2: median of rows
 	N: median of  N-th dimension 
	default or []: first DIMENSION, with more than 1 element

 features:
 - can deal with NaN's (missing values)
 - accepts dimension argument like in Matlab in Octave, too. 
 - compatible to Matlab and Octave 

 see also: SUMSKIPNAN



# name: <cell-element>
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 MEDIAN data elements, 
 [y]=median(x [,DIM])



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
moment


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 627
 MOMENT estimates the p-th moment 
 
 M = moment(x, p [,opt] [,DIM])
 M = moment(H, p [,opt])
   calculates p-th central moment from data x in dimension DIM
	of from Histogram H

 p	moment of order p
 opt   'ac': absolute 'a' and/or central ('c') moment
	DEFAULT: '' raw moments are estimated
 DIM	dimension
	1: STATS of columns
	2: STATS of rows
	default or []: first DIMENSION, with more than 1 element

 features:
 - can deal with NaN's (missing values)
 - dimension argument 
 - compatible to Matlab and Octave

 see also: STD, VAR, SKEWNESS, KURTOSIS, STATISTIC, 

 REFERENCE(S):
 http://mathworld.wolfram.com/Moment.html



# name: <cell-element>
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 MOMENT estimates the p-th moment 
 
 M = moment(x, p [,opt] [,DIM])
 M = moment



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nanconv


# name: <cell-element>
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 NANCONV computes the convolution for data with missing values. 
  X and Y can contain missing values encoded with NaN.
  NaN's are skipped, NaN do not result in a NaN output. 
  The output gives NaN only if there are insufficient input data

 [...] = NANCONV(X,Y);
      calculates 2-dim convolution between X and Y
 [C]   = NANCONV(X,Y);

 WARNING: missing values can introduce aliasing - causing unintended results.
    Moreover, the behavior of bandpass and highpass filters in case of missing values 
    is not fully understood, and might contain some pitfalls.  

 see also: CONV, NANCONV2, NANFFT, NANFILTER



# name: <cell-element>
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 NANCONV computes the convolution for data with missing values.



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# length: 6
nanfft


# name: <cell-element>
# type: sq_string
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# length: 618
 NANFFT calculates the Fourier-Transform of X for data with missing values. 
  NANFFT is the same as FFT but X can contain missing values encoded with NaN.
  NaN's are skipped, NaN do not result in a NaN output. 

   Y = NANFFT(X)
   Y = NANFFT(X,N)
   Y = NANFFT(X,[],DIM)
 
   [Y,N] = NANFFT(...)
       returns the number of valid samples N


 WARNING: missing values can introduce aliasing - causing unintended results.
    Moreover, the behavior of bandpass and highpass filters in case of missing values 
    is not fully understood, and might contain some pitfalls.  

 see also: FFT, XCORR, NANCONV, NANFILTER



# name: <cell-element>
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# length: 75
 NANFFT calculates the Fourier-Transform of X for data with missing values.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
nanfilter


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 519
 NANFILTER is able to filter data with missing values encoded as NaN. 
       
      [Y,Z] = nanfilter(B,A,X [, Z]);  

 If X contains no missing data, NANFILTER should behave like FILTER. 
 NaN-values are handled gracefully. 

 WARNING: missing values can introduce aliasing - causing unintended results.
    Moreover, the behavior of bandpass and highpass filters in case of missing values 
    is not fully understood, and might contain some pitfalls.  

 see also: FILTER, SUMSKIPNAN, NANFFT, NANCONV, NANFILTER1UC



# name: <cell-element>
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 NANFILTER is able to filter data with missing values encoded as NaN.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 12
nanfilter1uc


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 257
 NANFILTER1UC is an adaptive filter for data with missing values encoded as NaN. 
       
      [Y,Z] = nanfilter1uc(uc,X [, Z]);  

 if X contains no missing data, NANFILTER behaves like FILTER(uc,[1,uc-1],X[,Z]).

 see also: FILTER, NANFILTER, SUMSKIPNAN



# name: <cell-element>
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# length: 80
 NANFILTER1UC is an adaptive filter for data with missing values encoded as NaN.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
naninsttest


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 112
 NANINSTTEST checks whether the functions from NaN-toolbox have been
 correctly installed. 

 see also: NANTEST



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
 NANINSTTEST checks whether the functions from NaN-toolbox have been
 correctly 



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nanmean


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 333
 NANMEAN same as SUM but ignores NaN's. 
 NANMEAN is OBSOLETE; use MEAN instead. NANMEAN is included 
    to provide backward compatibility 

 Y = nanmean(x [,DIM])
 
 DIM	dimension
	1 sum of columns
	2 sum of rows
	default or []: first DIMENSION with more than 1 element
 Y	resulting mean

 
 see also: MEAN, SUMSKIPNAN, NANSUM 



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 39
 NANMEAN same as SUM but ignores NaN's.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nanstd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 518
 NANSTD same as STD but ignores NaN's. 
 NANSTD is OBSOLETE; use NaN/STD instead. NANSTD is included 
    to fix a bug in alternative implementations and to 
    provide some compatibility. 

 Y = nanstd(x, FLAG, [,DIM])
 
 x     data
 FLAG  0: [default] normalizes with (N-1), N = sample size
 FLAG  1: normalizes with N, N = sample size
 DIM	dimension
	1 sum of columns
	2 sum of rows
	default or []: first DIMENSION with more than 1 element
 Y	resulting standard deviation
 
 see also: SUM, SUMSKIPNAN, NANSUM, STD



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 38
 NANSTD same as STD but ignores NaN's.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nansum


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 333
 NANSUM same as SUM but ignores NaN's. 
 NANSUM is OBSOLETE; use SUMSKIPNAN instead. NANSUM is included 
    to fix a bug in some other versions. 

 Y = nansum(x [,DIM])
 
 DIM	dimension
	1 sum of columns
	2 sum of rows
	default or []: first DIMENSION with more than 1 element
 Y	resulting sum

 
 see also: SUM, SUMSKIPNAN, NANSUM 



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 38
 NANSUM same as SUM but ignores NaN's.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nantest


# name: <cell-element>
# type: sq_string
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# length: 366
 NANTEST checks several mathematical operations and a few 
 statistical functions for their correctness related to NaN's.
 e.g. it checks norminv, normcdf, normpdf, sort, matrix division and multiplication.


 see also: NANINSTTEST

 REFERENCE(S): 
 [1] W. Kahan (1996) Lecture notes on the Status of "IEEE Standard 754 for 
     Binary Floating-point Arithmetic. 




# name: <cell-element>
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 NANTEST checks several mathematical operations and a few 
 statistical function



# name: <cell-element>
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# length: 7
normcdf


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 NORMCDF returns normal cumulative distribtion function

 cdf = normcdf(x,m,s);

 Computes the CDF of a the normal distribution 
    with mean m and standard deviation s
    default: m=0; s=1;
 x,m,s must be matrices of same size, or any one can be a scalar. 

 see also: NORMPDF, NORMINV 



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 NORMCDF returns normal cumulative distribtion function



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norminv


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 NORMINV returns inverse cumulative function of the normal distribution

 x = norminv(p,m,s);

 Computes the quantile (inverse of the CDF) of a the normal 
    cumulative distribution with mean m and standard deviation s
    default: m=0; s=1;
 p,m,s must be matrices of same size, or any one can be a scalar. 

 see also: NORMPDF, NORMCDF 



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 NORMINV returns inverse cumulative function of the normal distribution



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normpdf


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 NORMPDF returns normal probability density 

 pdf = normpdf(x,m,s);

 Computes the PDF of a the normal distribution 
    with mean m and standard deviation s
    default: m=0; s=1;
 x,m,s must be matrices of same size, or any one can be a scalar. 

 see also: NORMCDF, NORMINV 



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 NORMPDF returns normal probability density 



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partcorrcoef


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 PARTCORRCOEF calculates the partial correlation between X and Y
 after removing the influence of Z.
 X, Y and Z can contain missing values encoded with NaN.
 NaN's are skipped, NaN do not result in a NaN output. 
 (Its assumed that the occurence of NaN's is uncorrelated) 
 The output gives NaN, only if there are insufficient input data.

  The partial correlation  is defined as 
  pcc(xy|z)=(cc(x,y)-cc(x,z)*cc(y,z))/sqrt((1-cc(x,y)�)*((1-cc(x,z)�)))


 PARTCORRCOEF(X [,Mode]);
      calculates the (auto-)correlation matrix of X
 PARTCORRCOEF(X,Y,Z);
 PARTCORRCOEF(X,Y,Z,[]);
 PARTCORRCOEF(X,Y,Z,'Pearson');
 PARTCORRCOEF(X,Y,Z,'Rank');
 PARTCORRCOEF(X,Y,Z,'Spearman');

 Mode=[] [default]
	removes from X and Y the part that can be explained by Z
	and computes the correlation of the remaining part. 
 	Ideally, this is equivalent to Mode='Pearson', however, in practice
	this is more accurate.
 Mode='Pearson' or 'parametric'
 Mode='Spearman'
 Mode='Rank'
	computes the partial correlation based on cc(x,y),cc(x,z) and cc(y,z) 
	with the respective mode. 

 [R,p,ci1,ci2] = PARTCORRCOEF(...);
  r is the partialcorrelation matrix
	r(i,j) is the partial correlation coefficient r between X(:,i) and Y(:,j) 
	when influence of Z is removed. 
  p    gives the significance of PCC
	It tests the null hypothesis that the product moment correlation coefficient is zero 
       using Student's t-test on the statistic t = r sqrt(N-Nz-2)/sqrt(1-r^2) 
       where N is the number of samples (Statistics, M. Spiegel, Schaum series).
  p > alpha: do not reject the Null hypothesis: "R is zero".
  p < alpha: The alternative hypothesis "R2 is larger than zero" is true with probability (1-alpha).
  ci1	lower 0.95 confidence interval 
  ci2	upper 0.95 confidence interval 

 see also: SUMSKIPNAN, COVM, COV, COR, SPEARMAN, RANKCORR, RANKS, CORRCOEF

 REFERENCES:
 on the partial correlation coefficient 
 [1] http://www.tufts.edu/~gdallal/partial.htm
 [2] http://www.nag.co.uk/numeric/fl/manual/pdf/G02/g02byf.pdf



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 PARTCORRCOEF calculates the partial correlation between X and Y
 after removing



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percentile


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 PERCENTILE calculates the percentiles of histograms and sample arrays.  

  Q = percentile(Y,q)      
  Q = percentile(Y,q,DIM)      
     returns the q-th percentile along dimension DIM of sample array Y.
     size(Q) is equal size(Y) except for dimension DIM which is size(Q,DIM)=length(Q)

  Q = percentile(HIS,q)
     returns the q-th percentile from the histogram HIS. 
     HIS must be a HISTOGRAM struct as defined in HISTO2 or HISTO3.
     If q is a vector, the each row of Q returns the q(i)-th percentile 

 see also: HISTO2, HISTO3, QUANTILE



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 PERCENTILE calculates the percentiles of histograms and sample arrays.



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prctile


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 PRCTILE calculates the percentiles of histograms and sample arrays.  
 (its the same than PERCENTILE.M)

  Q = prctile(Y,q)      
  Q = prctile(Y,q,DIM)      
     returns the q-th percentile along dimension DIM of sample array Y.
     size(Q) is equal size(Y) except for dimension DIM which is size(Q,DIM)=length(Q)

  Q = prctile(HIS,q)
     returns the q-th percentile from the histogram HIS. 
     HIS must be a HISTOGRAM struct as defined in HISTO2 or HISTO3.
     If q is a vector, the each row of Q returns the q(i)-th percentile 

 see also: HISTO2, HISTO3, QUANTILE



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 PRCTILE calculates the percentiles of histograms and sample arrays.



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quantile


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 QUANTILE calculates the quantiles of histograms and sample arrays.  

  Q = quantile(Y,q)
  Q = quantile(Y,q,DIM)
     returns the q-th quantile along dimension DIM of sample array Y.
     size(Q) is equal size(Y) except for dimension DIM which is size(Q,DIM)=length(Q)

  Q = quantile(HIS,q)
     returns the q-th quantile from the histogram HIS. 
     HIS must be a HISTOGRAM struct as defined in HISTO2 or HISTO3.
     If q is a vector, the each row of Q returns the q(i)-th quantile 

 see also: HISTO2, HISTO3, PERCENTILE



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 QUANTILE calculates the quantiles of histograms and sample arrays.



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range


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 RANGE calculates the range of Y 
  Missing values (encoded as NaN) are ignored. 

  Q = range(Y)
  Q = range(Y,DIM)
     returns the range along dimension DIM of sample array Y.

  Q = range(HIS)
     returns the RANGE from the histogram HIS.
     HIS must be a HISTOGRAM struct as defined in HISTO2 or HISTO3.

 see also: IQR, MAD, HISTO2, HISTO3, PERCENTILE, QUANTILE



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 RANGE calculates the range of Y 
  Missing values (encoded as NaN) are ignored.



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rankcorr


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 RANKCORR calculated the rank correlation coefficient.
 This function is replaced by CORRCOEF. 
 Significance test and confidence intervals can be obtained from CORRCOEF, too. 

 R = CORRCOEF(X, [Y, ] 'Rank');

 The rank correlation   r = corrcoef(ranks(x)). 
 is often confused with Spearman's rank correlation.  
 Spearman's correlation is defined as 
   r(x,y) = 1-6*sum((ranks(x)-ranks(y)).^2)/(N*(N*N-1))
 The results are different. Here, the former version is implemented. 

 see also: CORRCOEF, SPEARMAN, RANKS

 REFERENCES:
 [1] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
 [2] http://mathworld.wolfram.com/CorrelationCoefficient.html



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 RANKCORR calculated the rank correlation coefficient.



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ranks


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 RANKS gives the rank of each element in a vector.
 This program uses an advanced algorithm with averge effort O(m.n.log(n)) 
 NaN in the input yields NaN in the output.
 
 r = ranks(X[,DIM])
   if X is a vector, return the vector of ranks of X adjusted for ties.
   if X is matrix, the rank is calculated along dimension DIM. 
   if DIM is zero or empty, the lowest dimension with more then 1 element is used. 
 r = ranks(X,DIM,'traditional')
   implements the traditional algorithm with O(n^2) computational 
   and O(n^2) memory effort
 r = ranks(X,DIM,'mtraditional')
   implements the traditional algorithm with O(n^2) computational 
   and O(n) memory effort
 r = ranks(X,DIM,'advanced   ')
   implements an advanced algorithm with O(n*log(n)) computational 
   and O(n.log(n)) memory effort
 r = ranks(X,DIM,'advanced-ties')
   implements an advanced algorithm with O(n*log(n)) computational 
   and O(n.log(n)) memory effort
   but without correction for ties 
   This is the fastest algorithm 

 see also: CORRCOEF, SPEARMAN, RANKCORR

 REFERENCES:
 --



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 RANKS gives the rank of each element in a vector.



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rms


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 RMS calculates the root mean square
   can deal with complex data. 

 y = rms(x,DIM,W)

 DIM	dimension
	1 STD of columns
	2 STD of rows
 	N STD of  N-th dimension 
	default or []: first DIMENSION, with more than 1 element
 W	weights to compute weighted s.d. (default: [])
	if W=[], all weights are 1. 
	number of elements in W must match size(x,DIM) 

 y	estimated standard deviation

 features:
 - can deal with NaN's (missing values)
 - weighting of data 
 - dimension argument also in Octave
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, MEAN



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 RMS calculates the root mean square
   can deal with complex data.



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row_col_deletion


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 ROW_COL_DELETION selects the rows and columns for removing any missing values. 
    A heuristic based on maximizing the number of remaining sample values
    is used. In other words, if there are more rows than columns, it is 
    more likely that a row-wise deletion will be applied and vice versa. 
 
    [rix,cix] = row_col_deletion(d)
    [rix,cix] = row_col_deletion(d,c,w)
 
 Input: 
    d        data (each row is a sample, each column a feature)
    c        classlabels (not really used) [OPTIONAL]
    w        weight for each sample vector [OPTIONAL]   
 Output:
    rix      selected samples
    cix      selected columns    
 
   d(rix,cix) does not contain any NaN's i.e. missing values      
 
 see also: TRAIN_SC, TEST_SC



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 ROW_COL_DELETION selects the rows and columns for removing any missing values.



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sem


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 SEM calculates the standard error of the mean
 
 [SE,M] = SEM(x [, DIM [,W]])
   calculates the standard error (SE) in dimension DIM
   the default DIM is the first non-single dimension
   M returns the mean. 
   Can deal with complex data, too. 

 DIM	dimension
	1: SEM of columns
	2: SEM of rows
 	N: SEM of  N-th dimension 
	default or []: first DIMENSION, with more than 1 element
 W	weights to compute weighted mean and s.d. (default: [])
	if W=[], all weights are 1. 
	number of elements in W must match size(x,DIM) 

 features:
 - can deal with NaN's (missing values)
 - weighting of data 
 - dimension argument 
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, MEAN, VAR, STD



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 SEM calculates the standard error of the mean
 
 [SE,M] = SEM(x [, DIM [,W]])
 



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skewness


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 SKEWNESS estimates the skewness 

 y = skewness(x,DIM)
   calculates skewness of x in dimension DIM

 DIM	dimension
	1: STATS of columns
	2: STATS of rows
	default or []: first DIMENSION, with more than 1 element

 features:
 - can deal with NaN's (missing values)
 - dimension argument 
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN, STATISTIC

 REFERENCE(S):
 http://mathworld.wolfram.com/



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 SKEWNESS estimates the skewness 



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spearman


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 SPEARMAN Spearman's rank correlation coefficient.
 This function is replaced by CORRCOEF. 
 Significance test and confidence intervals can be obtained from CORRCOEF. 

 [R,p,ci1,ci2] = CORRCOEF(x, [y, ] 'Rank');

 For some (unknown) reason, in previous versions Spearman's rank correlation  
   r = corrcoef(ranks(x)). 
 But according to [1], Spearman's correlation is defined as 
   r = 1-6*sum((ranks(x)-ranks(y)).^2)/(N*(N*N-1))
 The results are different. Here, the later version is implemented. 

 see also: CORRCOEF, RANKCORR

 REFERENCES:
 [1] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
 [2] http://mathworld.wolfram.com/CorrelationCoefficient.html



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 SPEARMAN Spearman's rank correlation coefficient.



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statistic


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 STATISTIC estimates various statistics at once.
 
 R = STATISTIC(x,DIM)
   calculates all statistic (see list of fun) in dimension DIM
   R is a struct with all statistics 

 y = STATISTIC(x,fun)
   estimate of fun on dimension DIM
   y gives the statistic of fun	

 DIM	dimension
	1: STATS of columns
	2: STATS of rows
 	N: STATS of  N-th dimension 
	default or []: first DIMENSION, with more than 1 element

 fun	'mean'	mean
	'std'	standard deviation
	'var'	variance
	'sem'	standard error of the mean
	'rms'	root mean square
	'meansq' mean of squares
	'sum'	sum
	'sumsq'	sum of squares
	'CM#'	central moment of order #
	'skewness' skewness 
	'kurtosis' excess coefficient (Fisher kurtosis)
	'mad'	mean absolute deviation

 features:
 - can deal with NaN's (missing values)
 - dimension argument 
 - compatible to Matlab and Octave

 see also: SUMSKIPNAN

 REFERENCE(S):
 [1] http://www.itl.nist.gov/
 [2] http://mathworld.wolfram.com/



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 STATISTIC estimates various statistics at once.



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std


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 STD calculates the standard deviation.
 
 [y,v] = std(x [, opt[, DIM [, W]]])
 
 opt   option 
	0:  normalizes with N-1 [default]
		provides the square root of best unbiased estimator of the variance
	1:  normalizes with N, 
		this provides the square root of the second moment around the mean
 	otherwise: 
               best unbiased estimator of the standard deviation (see [1])      

 DIM	dimension
 	N STD of  N-th dimension 
	default or []: first DIMENSION, with more than 1 element
 W	weights to compute weighted s.d. (default: [])
	if W=[], all weights are 1. 
	number of elements in W must match size(x,DIM) 

 y	estimated standard deviation

 features:
 - provides an unbiased estimation of the S.D. 
 - can deal with NaN's (missing values)
 - weighting of data 
 - dimension argument also in Octave
 - compatible to Matlab and Octave

 see also: RMS, SUMSKIPNAN, MEAN, VAR, MEANSQ,


 References(s):
 [1] http://mathworld.wolfram.com/StandardDeviationDistribution.html



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 STD calculates the standard deviation.



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sumskipnan


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 SUMSKIPNAN adds all non-NaN values. 

 All NaN's are skipped; NaN's are considered as missing values. 
 SUMSKIPNAN of NaN's only  gives O; and the number of valid elements is return. 
 SUMSKIPNAN is also the elementary function for calculating 
 various statistics (e.g. MEAN, STD, VAR, RMS, MEANSQ, SKEWNESS, 
 KURTOSIS, MOMENT, STATISTIC etc.) from data with missing values.  
 SUMSKIPNAN implements the DIMENSION-argument for data with missing values.
 Also the second output argument return the number of valid elements (not NaNs) 
 
 Y = sumskipnan(x [,DIM])
 [Y,N,SSQ] = sumskipnan(x [,DIM])
 [...] = sumskipnan(x, DIM, W)
 
 x	input data 	
 DIM	dimension (default: [])
	empty DIM sets DIM to first non singleton dimension	
 W	weight vector for weighted sum, numel(W) must fit size(x,DIM)
 Y	resulting sum
 N	number of valid (not missing) elements
 SSQ	sum of squares

 the function FLAG_NANS_OCCURED() returns whether any value in x
  is a not-a-number (NaN)

 features:
 - can deal with NaN's (missing values)
 - implements dimension argument. 
 - computes weighted sum 
 - compatible with Matlab and Octave

 see also: FLAG_NANS_OCCURED, SUM, NANSUM, MEAN, STD, VAR, RMS, MEANSQ, 
      SSQ, MOMENT, SKEWNESS, KURTOSIS, SEM



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 SUMSKIPNAN adds all non-NaN values.



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sumsq


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 SUMSQ calculates the sum of squares.
 
 [y] = sumsq(x [,  DIM])
 
 DIM	dimension
 	N STD of  N-th dimension 
	default or []: first DIMENSION, with more than 1 element

 y	estimated standard deviation

 features:
 - can deal with NaN's (missing values)
 - dimension argument also in Octave
 - compatible to Matlab and Octave

 see also: RMS, SUMSKIPNAN, MEAN, VAR, MEANSQ,


 References(s):



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 SUMSQ calculates the sum of squares.



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tcdf


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 TCDF returns student cumulative distribtion function

 cdf = tcdf(x,DF);

 Computes the CDF of the students distribution 
    with DF degrees of freedom 
 x,DF must be matrices of same size, or any one can be a scalar. 

 see also: NORMCDF, TPDF, TINV 



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 TCDF returns student cumulative distribtion function



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test_sc


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 TEST_SC: apply statistical and SVM classifier to test data 

  R = test_sc(CC,D,TYPE [,target_Classlabel]) 
       R.output     	output: "signed" distance for each class. 
		This represents the distances between sample D and the separating hyperplane
		The "signed distance" is possitive if it matches the target class, and 
		and negative if it lays on the opposite side of the separating hyperplane. 
       R.classlabel 	class for output data
  The target class is optional. If it is provided, the following values are returned. 
       R.kappa 	Cohen's kappa coefficient
       R.ACC   	Classification accuracy 
       R.H     	Confusion matrix 

 The classifier CC is typically obtained by TRAIN_SC. If a statistical 
 classifier is used, TYPE can be used to modify the classifier. 
    TYPE = 'MDA'    mahalanobis distance based classifier
    TYPE = 'MD2'    mahalanobis distance based classifier
    TYPE = 'MD3'    mahalanobis distance based classifier
    TYPE = 'GRB'    Gaussian radial basis function 
    TYPE = 'QDA'    quadratic discriminant analysis
    TYPE = 'LD2'    linear discriminant analysis
    TYPE = 'LD3', 'LDA', 'FDA, 'FLDA'   (Fisher's) linear discriminant analysis
    TYPE = 'LD4'    linear discriminant analysis
    TYPE = 'GDBC'   general distance based classifier
 
 see also: TRAIN_SC

 References: 
 [1] R. Duda, P. Hart, and D. Stork, Pattern Classification, second ed. 
       John Wiley & Sons, 2001.



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 TEST_SC: apply statistical and SVM classifier to test data 



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tiedrank


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 TIEDRANK compute rank of samples, the mean value is used in case of ties
  this function is just a wrapper for RANKS, and provided for compatibility 
  with the statistics toolbox of matlab(tm)
 
    R = tiedrank(X)
	computes the rank R of vector X
    
 see also: RANKS



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 TIEDRANK compute rank of samples, the mean value is used in case of ties
  this



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tinv


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 TINV returns inverse cumulative function of the student distribution

 x = tinv(p,v);

 Computes the quantile (inverse of the CDF) of a the student
    cumulative distribution with mean m and standard deviation s
 p,v must be matrices of same size, or any one can be a scalar. 

 see also: TPDF, TCDF, NORMPDF, NORMCDF, NORMINV 



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 TINV returns inverse cumulative function of the student distribution



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tpdf


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 TPDF returns student probability density 

 pdf = tpdf(x,DF);

 Computes the PDF of a the student distribution 
    with DF degreas of freedom
 x,DF must be matrices of same size, or any one can be a scalar. 

 see also: TINV, TCDF, NORMPDF, NORMCDF, NORMINV 



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 TPDF returns student probability density 



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train_lda_sparse


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 Linear Discriminant Analysis for the Small Sample Size Problem as described in
 Algorithm 1 of J. Duintjer Tebbens, P. Schlesinger: 'Improving
 Implementation of Linear Discriminant Analysis for the High Dimension/Small Sample Size
 Problem', Computational Statistics and Data Analysis, vol. 52, no. 1, pp. 423-437, 2007.  
 Input:
               X                 ......       (sparse) training data matrix
               G                 ......       group coding matrix of the training data
               test              ......       (sparse) test data matrix
               Gtest             ......       group coding matrix of the test data
               par               ......       if par = 0 then classification exploits sparsity too
               tol               ......       tolerance to distinguish zero eigenvalues
 Output:
               err               ......       Wrong classification rate (in %)
               trafo             ......       LDA transformation vectors

 Reference(s): 
 J. Duintjer Tebbens, P. Schlesinger: 'Improving
 Implementation of Linear Discriminant Analysis for the High Dimension/Small Sample Size
 Problem', Computational Statistics and Data Analysis, vol. 52, no. 1, 
 pp. 423-437, 2007.

 Copyright (C) by J. Duintjer Tebbens, Institute of Computer Science of the Academy of Sciences of the Czech Republic,
 Pod Vodarenskou vezi 2, 182 07 Praha 8 Liben, 18.July.2006. 
 This work was supported by the Program Information Society under project
 1ET400300415.


 Modified for the use with Matlab6.5 by A. Schloegl, 22.Aug.2006

	$Id$
       This function is part of the NaN-toolbox
       http://pub.ist.ac.at/~schloegl/matlab/NaN/



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 Linear Discriminant Analysis for the Small Sample Size Problem as described in




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train_sc


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 Train a (statistical) classifier
 
  CC = train_sc(D,classlabel)
  CC = train_sc(D,classlabel,MODE)
  CC = train_sc(D,classlabel,MODE, W)
	weighting D(k,:) with weight W(k) (not all classifiers supported weighting)

 CC contains the model parameters of a classifier which can be applied 
   to test data using test_sc. 
   R = test_sc(CC,D,...) 

   D		training samples (each row is a sample, each column is a feature)	
   classlabel	labels of each sample, must have the same number of rows as D. 
 		Two different encodings are supported: 
		{-1,1}-encoding (multiple classes with separate columns for each class) or
		1..M encoding. 
 		So [1;2;3;1;4] is equivalent to 
			[+1,-1,-1,-1;
			[-1,+1,-1,-1;
			[-1,-1,+1,-1;
			[+1,-1,-1,-1]
			[-1,-1,-1,+1]
		Note, samples with classlabel=0 are ignored. 

  The following classifier types are supported MODE.TYPE
    'MDA'      mahalanobis distance based classifier [1]
    'MD2'      mahalanobis distance based classifier [1]
    'MD3'      mahalanobis distance based classifier [1]
    'GRB'      Gaussian radial basis function     [1]
    'QDA'      quadratic discriminant analysis    [1]
    'LD2'      linear discriminant analysis (see LDBC2) [1]
		MODE.hyperparameter.gamma: regularization parameter [default 0] 
    'LD3', 'FDA', 'LDA', 'FLDA'
               linear discriminant analysis (see LDBC3) [1]
		MODE.hyperparameter.gamma: regularization parameter [default 0] 
    'LD4'      linear discriminant analysis (see LDBC4) [1]
		MODE.hyperparameter.gamma: regularization parameter [default 0] 
    'LD5'      another LDA (motivated by CSP)
		MODE.hyperparameter.gamma: regularization parameter [default 0] 
    'RDA'      regularized discriminant analysis [7]
		MODE.hyperparameter.gamma: regularization parameter 
		MODE.hyperparameter.lambda =
		gamma = 0, lambda = 0 : MDA
		gamma = 0, lambda = 1 : LDA [default]
		Hint: hyperparameter are used only in test_sc.m, testing different 
		the hyperparameters do not need repetitive calls to train_sc, 
		it is sufficient to modify CC.hyperparameter before calling test_sc. 	
    'GDBC'     general distance based classifier  [1]
    ''         statistical classifier, requires Mode argument in TEST_SC	
    '###/DELETION'  if the data contains missing values (encoded as NaNs), 
		a row-wise or column-wise deletion (depending on which method 
		removes less data values) is applied;  
    '###/GSVD'	GSVD and statistical classifier [2,3], 
    '###/sparse'  sparse  [5] 
		'###' must be 'LDA' or any other classifier 
    'PLS'	(linear) partial least squares regression 
    'REG'      regression analysis;
    'WienerHopf'	Wiener-Hopf equation  
    'NBC'	Naive Bayesian Classifier [6]     
    'aNBC'	Augmented Naive Bayesian Classifier [6]
    'NBPW'	Naive Bayesian Parzen Window [9]     

    'PLA'	Perceptron Learning Algorithm [11]
		MODE.hyperparameter.alpha = alpha [default: 1]
		 w = w + alpha * e'*x
    'LMS', 'AdaLine'  Least mean squares, adaptive line element, Widrow-Hoff, delta rule 
		MODE.hyperparameter.alpha = alpha [default: 1]
    'Winnow2'  Winnow2 algorithm [12]

    'PSVM'	Proximal SVM [8] 
		MODE.hyperparameter.nu  (default: 1.0)
    'LPM'      Linear Programming Machine
                 uses and requires train_LPM of the iLog CPLEX optimizer 
		MODE.hyperparameter.c_value = 
    'CSP'	CommonSpatialPattern is very experimental and just a hack
		uses a smoothing window of 50 samples.
    'SVM','SVM1r'  support vector machines, one-vs-rest
		MODE.hyperparameter.c_value = 
    'SVM11'    support vector machines, one-vs-one + voting
		MODE.hyperparameter.c_value = 
    'RBF'      Support Vector Machines with RBF Kernel
		MODE.hyperparameter.c_value = 
		MODE.hyperparameter.gamma = 
    'SVM:LIB'    libSVM [default SVM algorithm)
    'SVM:bioinfo' uses and requires svmtrain from the bioinfo toolbox        
    'SVM:OSU'   uses and requires mexSVMTrain from the OSU-SVM toolbox 
    'SVM:LOO'   uses and requires svcm_train from the LOO-SVM toolbox 
    'SVM:Gunn'  uses and requires svc-functios from the Gunn-SVM toolbox 
    'SVM:KM'    uses and requires svmclass-function from the KM-SVM toolbox 
    'SVM:LINz'  LibLinear [10] (requires train.mex from LibLinear somewhere in the path)
            z=0 (default) LibLinear with -- L2-regularized logistic regression
            z=1 LibLinear with -- L2-loss support vector machines (dual)
            z=2 LibLinear with -- L2-loss support vector machines (primal)
            z=3 LibLinear with -- L1-loss support vector machines (dual)
    'SVM:LIN4'  LibLinear with -- multi-class support vector machines by Crammer and Singer
    'DT'	decision tree - not implemented yet.  

 {'REG','MDA','MD2','QDA','QDA2','LD2','LD3','LD4','LD5','LD6','NBC','aNBC','WienerHopf','LDA/GSVD','MDA/GSVD', 'LDA/sparse','MDA/sparse', 'PLA', 'LMS','LDA/DELETION','MDA/DELETION','NBC/DELETION','RDA/DELETION','REG/DELETION','RDA','GDBC','SVM','RBF','PSVM','SVM11','SVM:LIN4','SVM:LIN0','SVM:LIN1','SVM:LIN2','SVM:LIN3','WINNOW', 'DT'};

 CC contains the model parameters of a classifier. Some time ago,     
 CC was a statistical classifier containing the mean 
 and the covariance of the data of each class (encoded in the 
  so-called "extended covariance matrices". Nowadays, also other 
 classifiers are supported. 

 see also: TEST_SC, COVM, ROW_COL_DELETION

 References: 
 [1] R. Duda, P. Hart, and D. Stork, Pattern Classification, second ed. 
       John Wiley & Sons, 2001. 
 [2] Peg Howland and Haesun Park,
       Generalizing Discriminant Analysis Using the Generalized Singular Value Decomposition
       IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(8), 2004.
       dx.doi.org/10.1109/TPAMI.2004.46
 [3] http://www-static.cc.gatech.edu/~kihwan23/face_recog_gsvd.htm
 [4] Jieping Ye, Ravi Janardan, Cheong Hee Park, Haesun Park
       A new optimization criterion for generalized discriminant analysis on undersampled problems.
       The Third IEEE International Conference on Data Mining, Melbourne, Florida, USA
       November 19 - 22, 2003
 [5] J.D. Tebbens and P. Schlesinger (2006), 
       Improving Implementation of Linear Discriminant Analysis for the Small Sample Size Problem
	Computational Statistics & Data Analysis, vol 52(1): 423-437, 2007
       http://www.cs.cas.cz/mweb/download/publi/JdtSchl2006.pdf
 [6] H. Zhang, The optimality of Naive Bayes, 
	 http://www.cs.unb.ca/profs/hzhang/publications/FLAIRS04ZhangH.pdf
 [7] J.H. Friedman. Regularized discriminant analysis. 
	Journal of the American Statistical Association, 84:165–175, 1989.
 [8] G. Fung and O.L. Mangasarian, Proximal Support Vector Machine Classifiers, KDD 2001.
        Eds. F. Provost and R. Srikant, Proc. KDD-2001: Knowledge Discovery and Data Mining, August 26-29, 2001, San Francisco, CA.
 	p. 77-86.
 [9] Kai Keng Ang, Zhang Yang Chin, Haihong Zhang, Cuntai Guan.
	Filter Bank Common Spatial Pattern (FBCSP) in Brain-Computer Interface.
	IEEE International Joint Conference on Neural Networks, 2008. IJCNN 2008. (IEEE World Congress on Computational Intelligence). 
	1-8 June 2008 Page(s):2390 - 2397
 [10] R.-E. Fan, K.-W. Chang, C.-J. Hsieh, X.-R. Wang, and C.-J. Lin. 
       LIBLINEAR: A Library for Large Linear Classification, Journal of Machine Learning Research 9(2008), 1871-1874. 
       Software available at http://www.csie.ntu.edu.tw/~cjlin/liblinear 
 [11] http://en.wikipedia.org/wiki/Perceptron#Learning_algorithm
 [12] Littlestone, N. (1988) 
       "Learning Quickly When Irrelevant Attributes Abound: A New Linear-threshold Algorithm" 
       Machine Learning 285-318(2)
 	http://en.wikipedia.org/wiki/Winnow_(algorithm)



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 Train a (statistical) classifier
 
  CC = train_sc(D,classlabel)
  CC = train_s



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trimean


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 TRIMEAN yields the weighted mean of the median and the quartiles
    m = TRIMEAN(y).

 The trimean is  m = (Q1+2*MED+Q3)/4
    with quartile Q1 and Q3 and median MED   

 N-dimensional data is supported
 
 REFERENCES:
 [1] http://mathworld.wolfram.com/Trimean.html



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 TRIMEAN yields the weighted mean of the median and the quartiles
    m = TRIME



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trimmean


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 TRIMMEAN calculates the trimmed mean by removing the fraction of p/2 upper and 
 p/2 lower samples. Missing values (encoded as NaN) are ignored and not taken into account. 
 The same number from the upper and lower values are removed, and is compatible to various
 spreadsheet programs including GNumeric [1], LibreOffice, OpenOffice and MS Excel.

  Q = trimmean(Y,p)
  Q = trimmean(Y,p,DIM)
     returns the TRIMMEAN along dimension DIM of sample array Y.
  If p is a vector, the TRIMMEAN for each p is computed. 

 see also: MAD, RANGE, HISTO2, HISTO3, PERCENTILE, QUANTILE

 References:
 [1] http://www.fifi.org/doc/gnumeric-doc/html/C/gnumeric-trimmean.html



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 TRIMMEAN calculates the trimmed mean by removing the fraction of p/2 upper and 



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ttest


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 TTEST (paired) t-test
     For a sample X from a normal distribution with unknown mean and
     variance, perform a t-test of the null hypothesis `mean (X) == M'.
     Under the null, the test statistic T follows a Student
     distribution with `DF = length (X) - 1' degrees of freedom.

     TTEST treads NaNs as "Missing values" and ignores these. 

 H = ttest(x,m)
	tests Null-hypothesis that mean of x is m. 		
 H = ttest(x,y)
 	size of x and size of y must match, it is tested whether the 
	difference x-y is significantly different to m=0; 
 H = ttest(x,y,alpha)
 H = ttest(x,y,alpha,tail)
 H = ttest(x,y,alpha,tail,DIM)
 [H,PVAL] = ttest(...)

     H=1 indicates a rejection of the Null-hypothesis at a significance 
     level of alpha (default alpha = 0.05).	 
 
     With the optional argument string TAIL, the alternative of interest
     can be selected.  If TAIL is '!=' or '<>' or 'both', the null is tested
     against the two-sided Alternative `mean (X) ~= mean (Y)'.  If TAIL
     is '>' or 'right', the one-sided Alternative `mean (X) > mean (Y)' is used.
     Similarly for '<' or 'left', the one-sided Alternative `mean (X) < mean
     (Y)' is used.  The default is the two-sided case.
 
     H returns whether the Null-Hypotheses must be rejected. 
     The p-value of the test is returned in PVAL. 
 
     TTEST works on the first non-singleton dimension or on DIM. 

     If no output argument is given, the p-value of the test is
     displayed.




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 TTEST (paired) t-test
     For a sample X from a normal distribution with unkno



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ttest2


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 TTEST2 (unpaired) t-test
     For two samples x and y from normal distributions with unknown
     means and unknown equal variances, perform a two-sample t-test of
     the null hypothesis of equal means.  Under the null, the test
     statistic T follows a Student distribution with DF degrees of
     freedom.

     TTEST2 treads NaNs as "Missing values" and ignores these. 

 H = ttest2(x,y)
 H = ttest2(x,y,alpha)
 H = ttest2(x,y,alpha,tail)
 H = ttest2(x,y,alpha,tail,vartype)
 H = ttest2(x,y,alpha,tail,vartype,DIM)
 [H,PVAL] = ttest2(...)
 [h,p,ci,stats] = ttest2(...)

     H=1 indicates a rejection of the Null-hypothesis at a significance 
     level of alpha (default alpha = 0.05).	 
 
     With the optional argument string TAIL, the Alternative of interest
     can be selected.  If TAIL is '!=' or '<>' or 'both', the null is tested
     against the two-sided Alternative `mean (X) ~= mean (Y)'.  If TAIL
     is '>' or 'right', the one-sided Alternative `mean (X) > mean (Y)' is used.
     Similarly for '<' or 'left', the one-sided Alternative `mean (X) < mean
     (Y)' is used.  The default is the two-sided case.
 
     vartype support only 'equal' (default value); the value 'unequal' is not supported. 

     H returns whether the Null-Hypotheses must be rejected. 
     The p-value of the test is returned in PVAL. 
 
     TTEST2 works on the first non-singleton dimension or on DIM. 

     If no output argument is given, the p-value of the test is
     displayed.




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 TTEST2 (unpaired) t-test
     For two samples x and y from normal distributions



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var


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 VAR calculates the variance.
 
 y = var(x [, opt[, DIM]])
   calculates the variance in dimension DIM
   the default DIM is the first non-single dimension

 opt   0: normalizes with N-1 [default]
	1: normalizes with N 
 DIM	dimension
	1: VAR of columns
	2: VAR of rows
 	N: VAR of  N-th dimension 
	default or []: first DIMENSION, with more than 1 element
 W	weights to compute weighted variance (default: [])
	if W=[], all weights are 1. 
	number of elements in W must match size(x,DIM) 
 
 usage: 
	var(x)	
	var(x, opt, DIM)	
	var(x, [], DIM)	
	var(x, W, DIM)
	var(x, opt, DIM, W)	

 features:
 - can deal with NaN's (missing values)
 - weighting of data 
 - dimension argument 
 - compatible to Matlab and Octave

 see also: MEANSQ, SUMSQ, SUMSKIPNAN, MEAN, RMS, STD,



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 VAR calculates the variance.



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xcovf


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 XCOVF generates cross-covariance function. 
 XCOVF is the same as XCORR except 
   X and Y can contain missing values encoded with NaN.
   NaN's are skipped, NaN do not result in a NaN output. 
   The output gives NaN only if there are insufficient input data

 [C,N,LAGS] = xcovf(X,MAXLAG,SCALEOPT);
      calculates the (auto-)correlation function of X
 [C,N,LAGS] = xcovf(X,Y,MAXLAG,SCALEOPT);
      calculates the crosscorrelation function between X and Y

  SCALEOPT   [character string] specifies the type of scaling applied
          to the correlation vector (or matrix). is one of:
    'none'      return the unscaled correlation, R,
    'biased'    return the biased average, R/N, 
    'unbiased'  return the unbiassed average, R(k)/(N-|k|), 
    'coeff'     return the correlation coefficient, R/(rms(x).rms(y)),
          where "k" is the lag, and "N" is the length of X.
          If omitted, the default value is "none".
          If Y is supplied but does not have the ame length as X,
          scale must be "none".


 see also: COVM, XCORR



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 XCOVF generates cross-covariance function.



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xptopen


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 XPTOPEN read of several file formats and writing of the SAS Transport Format (*.xpt)

        X = xptopen(filename)
        X = xptopen(filename,'r')
                read filename and return variables in struct X
        Supported are ARFF, SAS-XPT and STATA files.

        X = xptopen(filename,'w',X)
                save fields of struct X in filename.

        The fields of X must be column vectors of equal length.
        Each vector is either a numeric vector or a cell array of strings.

 The SAS-XPT format stores Date/Time as numeric value counting the number of days since 1960-01-01.



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 XPTOPEN read of several file formats and writing of the SAS Transport Format (*



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xval


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 XVAL is used for crossvalidation 

  [R,CC] = xval(D,classlabel)
  .. = xval(D,classlabel,CLASSIFIER)
  .. = xval(D,classlabel,CLASSIFIER,type)
  .. = xval(D,{classlabel,W},CLASSIFIER)
  .. = xval(D,{classlabel,W,NG},CLASSIFIER)
 
  example: 
      load_fisheriris;    %builtin iris dataset      
      C = species;
      K = 5; NG = [1:length(C)]'*K/length(C);
      [R,CC] = xval(meas,{C,[],NG},'NBC');            

 Input:
    D:	data features (one feature per column, one sample per row)
    classlabel	labels of each sample, must have the same number of rows as D. 
 		Two different encodings are supported: 
		{-1,1}-encoding (multiple classes with separate columns for each class) or
		1..M encoding. 
 		So [1;2;3;1;4] is equivalent to 
			[+1,-1,-1,-1;
			[-1,+1,-1,-1;
			[-1,-1,+1,-1;
			[+1,-1,-1,-1]
			[-1,-1,-1,+1]
		Note, samples with classlabel=0 are ignored. 

    CLASSIFIER can be any classifier supported by train_sc (default='LDA')
       {'REG','MDA','MD2','QDA','QDA2','LD2','LD3','LD4','LD5','LD6','NBC','aNBC','WienerHopf', 'RDA','GDBC',
	 'SVM','RBF','PSVM','SVM11','SVM:LIN4','SVM:LIN0','SVM:LIN1','SVM:LIN2','SVM:LIN3','WINNOW'}
       these can be modified by ###/GSVD, ###/sparse and ###/DELETION. 
	   /DELETION removes in case of NaN's either the rows or the columns (which removes less data values) with any NaN
	   /sparse and /GSVD preprocess the data an reduce it to some lower-dimensional space. 
       Hyperparameters (like alpha for PLA, gamma/lambda for RDA, c_value for SVM, etc) can be defined as 
 	CLASSIFIER.hyperparameter.alpha, etc. and 
 	CLASSIFIER.TYPE = 'PLA' (as listed above). 
       See train_sc for details.
    W:	weights for each sample (row) in D. 
	default: [] (i.e. all weights are 1)
	number of elements in W must match the number of rows of D 
    NG: used to define the type of cross-valdiation
 	Leave-One-Out-Method (LOOM): NG = [1:length(classlabel)]' (default)
 	Leave-K-Out-Method: NG = ceil([1:length(classlabel)]'/K)
	K-fold XV:  NG = ceil([1:length(classlabel)]'*K/length(classlabel))
	group-wise XV (if samples are not indepentent) can be also defined here
	samples from the same group (dependent samples) get the same identifier
	samples from different groups get different classifiers
    TYPE:  defines the type of cross-validation procedure if NG is not specified 
	'LOOM'  leave-one-out-method
       k	k-fold crossvalidation

 OUTPUT: 
    R contains the resulting performance metric
    CC contains the classifier  

    plota(R) shows the confusion matrix of the results

 see also: TRAIN_SC, TEST_SC, CLASSIFY, PLOTA

 References: 
 [1] R. Duda, P. Hart, and D. Stork, Pattern Classification, second ed. 
       John Wiley & Sons, 2001. 
 [2] A. Schlögl, J. Kronegg, J.E. Huggins, S. G. Mason;
       Evaluation criteria in BCI research.
       (Eds.) G. Dornhege, J.R. Millan, T. Hinterberger, D.J. McFarland, K.-R.Müller;
       Towards Brain-Computer Interfacing, MIT Press, 2007, p.327-342



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 XVAL is used for crossvalidation 



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zScoreMedian


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 zScoreMedian removes the median and standardizes by the 1.483*median absolute deviation

 Usage:  Z = zScoreMedian(X, DIM)
 Input:  X  : data
         DIM: dimension along which z-score should be calculated (1=columns, 2=rows) 
              (optional, default=first dimension with more than 1 element
 Output: Z  : z-scores



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 zScoreMedian removes the median and standardizes by the 1.483*median absolute d



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zscore


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 ZSCORE removes the mean and normalizes data 
 to a variance of 1. Can be used for pre-whitening of data, too. 

 [z,mu, sigma] = zscore(x [,OPT [, DIM])
   z   z-score of x along dimension DIM
   sigma is the inverse of the standard deviation
   mu   is the mean of x

 The data x can be reconstucted with 
     x = z*diag(sigma) + repmat(m, size(z)./size(m))  
     z = x*diag(1./sigma) - repmat(m.*v, size(z)./size(m))  

 DIM	dimension
	1: STATS of columns
	2: STATS of rows
	default or []: first DIMENSION, with more than 1 element

 see also: SUMSKIPNAN, MEAN, STD, DETREND

 REFERENCE(S):
 [1] http://mathworld.wolfram.com/z-Score.html



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 ZSCORE removes the mean and normalizes data 
 to a variance of 1.