######################################################################## ## ## Copyright (C) 2017-2020 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or . ## ## This file is part of Octave. ## ## Octave 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. ## ## Octave 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 Octave; see the file COPYING. If not, see ## . ## ######################################################################## ## -*- texinfo -*- ## @deftypefn {} {@var{Z} =} normalize (@var{X}) ## @deftypefnx {} {@var{Z} =} normalize (@var{X}, @var{dim}) ## @deftypefnx {} {@var{Z} =} normalize (@dots{}, @var{method}) ## @deftypefnx {} {@var{Z} =} normalize (@dots{}, @var{method}, @var{Option}) ## @deftypefnx {} {@var{Z} =} normalize (@dots{}, @var{scale}, @var{ScaleOption}, @var{center}, @var{CenterOption}) ## @deftypefnx {} {[@var{Z}, @var{C}, @var{S}] =} normalize (@dots{}) ## ## Returns the normalized version of @var{X}. The default normalization is the ## @code{zscore} of @var{X}, defined as the number of standard deviations each ## element is from the mean of @var{X}. The returned value @var{Z} will have ## the same size as @var{X}. The optional return variables @var{C} and @var{S} ## are the centering and scaling factors used in the normalization such that: ## ## @example ## @group ## @var{z} = (@var{x} - @var{c}) ./ @var{s} ## @end group ## @end example ## ## If @var{X} is a vector, @code{normalize} will operate on the data in @var{X}. ## ## If @var{X} is a matrix, @code{normalize} will operate independently on ## each column in @var{X}. ## ## If @var{X} is a n-dimensional array, @code{normalize} will operate ## independently on the first non-singleton dimension in @var{X}. ## ## The optional second input variable @var{DIM} can be used to force @code{normalize} ## to operate over the specified dimension. @var{DIM} must be an integer scalar. ## ## The optional input variables @var{Method} and @var{Option} can be used to ## specify the type of normalization performed on @var{X}. Note that only the ## @option{scale} and @option{center} options may be specified together using ## any of the methods defined below. Valid normalization methods are: ## ## @table @code ## @item zscore ## (Default) Normalizes the elements in @var{X} to the scaled distance from a ## central value. Valid Options: ## ## @table @code ## @item std ## (Default) Data is centered at @code{mean(@var{X})} and scaled by the # standard deviation. ## @item robust ## Data is centered at @code{median(@var{X})} and scaled by the median ## absolute deviation. ## @end table ## ## @item norm ## @var{Z} is the general vector norm of @var{X}, with @var{OPTION} being the ## normalization factor @var{p} that determines the vector norm type according ## to: ## @tex ## $$z = \left (\sum_k \left | x_k \right |^p \right )^{1/p}$$ ## @end tex ## @ifnottex ## ## @example ## @group ## @var{z} = [sum (abs(@var{x})^@var{p})]^(1/@var{p}) ## @end group ## @end example ## ## @end ifnottex ## @var{P} can be any positive scalar, specific values being: ## ## @table @code ## @item @var{p} = 1 ## @var{x} is normalized by sum (abs (@var{x})) ## @item @var{p} = 2 ## (Default) @var{x} is normalized by the Euclidian norm, or vector magnitude ## of the elements. ## @item @var{p} = Inf ## @var{x} is normalized by @code{max (abs (@var{x}))} ## @end table ## ## @item scale ## @var{x} is scaled by the value specified by either the numerical value ## @var{Option} or one of the following: ## ## @table @code ## @item std ## (Default) @var{x} is scaled by its standard deviation. ## @item mad ## @var{x} is scaled by its median absolute deviation. ## @item first ## @var{x} is scaled by the first element in the vector. ## @item iqr ## @var{x} is scaled by the vector's interquartile range. ## @end table ## ## @item range ## @var{x} is scaled to fit the range specified by @var{Option} as a two element ## scalar row vector. The default range is [0 1]. ## ## @item center ## @var{X} is shifted by an amount determined by @var{Option}, which can be a ## numeric scalar or one of the following: ## ## @table @code ## @item mean ## (Default) @var{x} is shifted such that @code{mean (@var{X}) = 0}. ## @item median ## @var{x} is shifted such that @code{median (@var{X}) = 0}. ## @end table ## ## @item medianiqr ## @var{X} is shifted such that the median equals zero and is scaled by the ## vector's interquartile range. ## @end table ## ## Known @sc{matlab} incompatibilities: ## ## @enumerate ## @item ## The DataVariables option is not yet implemented in @code{normalize} for ## Table class @var{X} inputs. ## ## @item ## Certain arrays containing NaN elements may not return Matlab compatible ## output. ## @end enumerate ## ## @seealso{zscore, iqr, norm, rescale, std, median, mean, mad} ## @end deftypefn ## Author: Nicholas Jankowski ## Created: 2020-11-25 function [z, centervalues, scalevalues] = normalize (x, varargin) ## TODO: until "NANFLAG" is implemented in std, mean, median, etc, normalize ## cannot efficiently reproduce some behavior with NaN's in x. xtests have ## been added to BISTs to capture this. ## TODO: when table class is implemented, remove DataVariables error line ## and DataVariables switch statement. Other cases should not require ## modification. ## input checks if ((nargin < 1) || (nargin > 8)) print_usage (); endif if (~ (isnumeric (x) || islogical (x))) error ("normalize: X must be a numeric vector, matrix, or array"); endif if (nargin == 1) ## directly handle simple 1 input case. centervalues = mean (x); if (x == Inf) ##FIXME: hack for std(Inf) not currently being matlab compatible scalevalues = NaN; else scalevalues = std (x); endif z = (x - centervalues)./ scalevalues; else ## parse inputs to determine options dim = []; %default = vector dimension, matrix column, or 1st n-dim >1 method = []; %default = zscore methodoption = []; %default = std datavariables_flag = false; datavar = []; scale_and_center_flag = false; input_idx = 1; ## Default scaling factors: centervalues = 0; scalevalues = 1; ##only second optional input can be numeric without following a method if isnumeric (varargin{1}) dim = varargin{1}; ##check for valid dimensions if !((isscalar (dim)) && (dim == fix (dim)) && (dim > 0)) error ("normalize: DIM must be a positive scalar integer"); endif input_idx++; endif ## parse varargin to determine methods then options while (input_idx <= (nargin - 1)) ##only second optional input can be numeric without following a method if isnumeric (varargin{input_idx}) print_usage (); endif %keyboard prop = tolower (varargin{input_idx}); if (strcmp (prop,"datavariables")) ##TODO: remove error on next line when Tables implemented error ("normalize: DataVariables method not yet implemented."); if (input_idx == (nargin - 1)) error ("normalize: DataVariables requires a table variable be specified"); elseif (datavariables_flag == true) error ("normalize: DataVariables may only be specified once"); else datavariables_flag = true; datavar = varargin{input_idx+1}; ##if Tables case sensitive, cannot tolower input_idx++; endif else if (~isempty (method)) ## catch if a second method is passed if (scale_and_center_flag) ## if true, already specified two methods, throw error error ("normalize: too many methods specified"); elseif ((strcmp ({method, prop}, {"center", "scale"})) ... || strcmp ({method, prop}, {"scale", "center"})) ## only scale and center can be combined scale_and_center_flag = true; ##scale/center order doesn't matter, preserve first one from being overwritten stored_method = method; method = []; stored_methodoption = methodoption; methodoption = []; else error ("normalize: methods `%s` and `%s` may not be combined", prop, method); endif endif ##determine method and whether there's an appropriate option specified switch (prop) case "zscore" method = "zscore"; if (input_idx < (nargin - 1)) nextprop = tolower (varargin{input_idx+1}); if (strcmp (nextprop, "std")|| strcmp (nextprop, "robust")) if (~isempty (methodoption)) error ("normalize: only one method option may be specified"); endif methodoption = nextprop; input_idx++; endif endif if (isempty (methodoption)) methodoption = "std"; endif case "norm" method = "norm"; if ((input_idx < (nargin - 1)) && isnumeric (varargin{input_idx+1})) nextprop = varargin{input_idx+1}; if (isscalar(nextprop) && (nextprop > 0)) if (~isempty (methodoption)) error ("normalize: only one method option may be specified"); endif methodoption = nextprop; input_idx++; else error ("normalize: 'norm' option must be a positive scalar or Inf") endif endif if (isempty (methodoption)) methodoption = 2; endif case "range" method = "range"; if (input_idx < (nargin - 1) && isnumeric (varargin{input_idx+1})) nextprop = varargin{input_idx+1}; if (size (nextprop) == [1 2]) if (~isempty (methodoption)) error ("normalize: only one method option may be specified"); endif methodoption = nextprop; input_idx++; else error ("normalize: 'range' must be specified as a 2-element row vector [a b]"); endif endif if (isempty (methodoption)) methodoption = [0 1]; endif case "scale" method = "scale"; if (input_idx < (nargin - 1)) nextprop = tolower (varargin{input_idx+1}); if (isnumeric (nextprop)) if (~isscalar (nextprop)) error ("normalize: scale value must be a scalar"); else methodoption = nextprop; input_idx++; endif elseif (any (strcmp (nextprop, {"std", "mad", "first", "iqr"}))) if (~isempty (methodoption)) error ("normalize: only one method option may be specified"); endif methodoption = nextprop; input_idx++; endif endif if (isempty (methodoption)) methodoption = 'std'; endif case "center" method = "center"; if (input_idx < (nargin - 1)) nextprop = tolower (varargin{input_idx+1}); if (isscalar (nextprop) || any (strcmp (nextprop, {"mean", "median"}))) if (~isempty (methodoption)) error ("normalize: only one method option may be specified"); endif methodoption = nextprop; input_idx++; elseif (isnumeric (nextprop)) error ("normalize: center shift must be a scalar value"); endif endif if (isempty (methodoption)) methodoption = 'mean'; endif case "medianiqr" method = "medianiqr"; otherwise error ("normalize: unknown method '%s'", prop); endswitch endif input_idx++; endwhile if isempty (method) method = 'zscore'; methodoption = 'std'; endif if isempty (dim) if isvector (x) if iscolumn (x) dim = 1; else dim = 2; endif elseif (ismatrix(x)) ## matrix x, operate independently along columns dim = 1; else ## operate on first non-singleton dimension. dim = find ((size (x) > 1), 1); endif endif ## TODO: DataTables option not handled below. Fix after Table Class implemented. if scale_and_center_flag method = "scaleandcenter"; endif switch (method) case "zscore" switch (methodoption) case "std" centervalues = mean (x, dim); scalevalues = std (x, [], dim); case "robust" ##center/median to zero and MAD = 1 centervalues = median (x, dim); scalevalues = median (abs (x - centervalues), dim); endswitch case "norm" switch methodoption case 1 scalevalues = sum (abs (x), dim); case Inf scalevalues = max (abs (x), [], dim); otherwise scalevalues = sum ((abs (x) .^ methodoption) , dim) .^ (1/methodoption); endswitch case "range" ## if any range element = 0, avoid divide by zero by replacing that range ## element with 1. output will be zero+min due to x-min(x)=0. ## can't use rescale(), as it applies full scaling to entire array x_range = range (x, dim); x_range(x_range == 0) = 1; z_range = methodoption(2)-methodoption(1); #scalar scalevalues = x_range ./ z_range; centervalues = min (x, [], dim) - (methodoption(1) .* scalevalues); case "scale" scalevalues = process_scale_option (x, dim, methodoption); case "center" centervalues = process_center_option (x, dim, methodoption); case "scaleandcenter" ## repeats scale and center using appropriate order and info switch (stored_method) case "scale" ## stored info is scale, latest info is center center_option = methodoption; scale_option = stored_methodoption; case "center" ## stored info is center, latest info is scale center_option = stored_methodoption; scale_option = methodoption; endswitch scalevalues = process_scale_option (x, dim, scale_option); centervalues = process_center_option (x, dim, center_option); case "medianiqr" ##try with new iqr if this fails for some x's and dims centervalues = median (x, dim); scalevalues = iqr (x, dim); endswitch endif ## div by scale factor. if scale = 0, div by zero = Inf is ok. z = (x - centervalues) ./ scalevalues; endfunction function c = process_center_option (x, dim, center_option) if (isnumeric (center_option)) c = center_option; else switch (center_option) case "mean" c = mean (x, dim); case "median" c = median (x, dim); endswitch endif endfunction function s = process_scale_option (x, dim, scale_option) warning("off", "Octave:divide-by-zero"); if (isnumeric (scale_option)) s = scale_option; else switch (scale_option) case "std" s = std (x, [], dim); case "mad" s = mad (x, 1, dim); case "first" dim_vector = repmat ({':'}, ndims(x), 1); dim_vector{dim} = 1; s = x(dim_vector{:}); case "iqr" s = iqr (x, dim); endswitch endif warning("on", "Octave:divide-by-zero") endfunction ## TODO - add the following two once xtests can pass ##(normalize ([1,2,3],1), [NaN,NaN,NaN]) ##(normalize (ones (3,2,2,2)), NaN (3,2,2,2)) ## no method specified, using zscore & std %!assert (normalize ([1,2,3]), [-1,0,1]) %!assert (normalize ([1,2,3], 2), [-1,0,1]) %!assert (normalize (single ([1,2,3])), single ([-1,0,1])) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2]), [1,0,-1;0,1,0;-1,-1,1]) %!assert (normalize (magic (3)), [[3;-2;-1]/sqrt(7),[-1;0;1],[1;2;-3]/sqrt(7)]) %!assert (normalize (magic (3), 2), [[3 -4 1]/sqrt(13);[-1 0 1];[-1 4 -3]/sqrt(13)]) ## Method: zscore, [std, robust] %!assert (normalize ([1,2,3],"zscore","std"), [-1,0,1]) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"zscore","std"), [1,0,-1;0,1,0;-1,-1,1]) %!assert (normalize (magic (3),"zscore","std"), [[3;-2;-1]/sqrt(7),[-1;0;1],[1;2;-3]/sqrt(7)]) %!assert (normalize ([1,2,3],"zscore","robust"), [-1,0,1]) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"zscore","robust"), [1,0,-1;0,1,0;-1,-1,1]) %!assert (normalize (magic (3),"zscore","robust"), [4 -1 0; -1 0 1; 0 1 -4]) ## Method: norm [1, 2, inf] %!assert (normalize ([1,2,3],"norm",1), [1/6 1/3 1/2]) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"norm",1), [1,0,-1;0,1,0;-1,-1,1]/2) %!assert (normalize (magic (3),"norm",1), magic(3)/15) %!assert (normalize ([1,2,3],"norm",2), [1 2 3]./3.741657386773941, eps) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"norm",2), [1,0,-1;0,1,0;-1,-1,1]*(sqrt(2)/2), eps) %!assert (normalize ([1,2,3],"norm",Inf), [1/3 2/3 1]) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"norm",Inf), [1,0,-1;0,1,0;-1,-1,1]) %!assert (normalize (magic (3),"norm",Inf), [[8;3;4]/8,[1;5;9]/9,[6;7;2]/7]) ## Method: range %!assert (normalize ([1,2,3],"range"), [0 0.5 1]) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"range",[0 1]), [1,0.5,0;0.5,1,0.5;0,0,1]) %!assert (normalize (magic (3),"range",[-1 1]), [1 -1 0.6; -1 0 1; -0.6 1 -1], eps) ## Method: scale [mad first iqr number] %!assert (normalize ([1,2,3],"scale"), [1 2 3]) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"scale","std"), [1 0 -1; 0 1 0; -1 -1 1]) %!assert (normalize (magic (3),"scale",2), (magic(3)/2)) %!assert (normalize ([1,2,3],"scale", "mad"), [1 2 3]) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"scale","mad"), [1 0 -1; 0 1 0; -1 -1 1]) %!assert (normalize (magic (3),"scale","mad"), [8 0.25 6; 3 1.25 7; 4 2.25 2]) %!assert (normalize ([1,2,3],"scale", "first"), [1 2 3]) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"scale","first"), [1 NaN 1; 0 Inf 0; -1 -Inf -1]) %!assert (normalize (magic (3),"scale","first"), [1 1 1; 3/8 5 7/6; 0.5 9 1/3]) %!assert (normalize (magic (3),2,"scale","first"), [1 1/8 3/4;1 5/3 7/3;1 9/4 0.5]) %!test %! x = reshape (magic (4),2,2,2,2); %! y3 = cat (4, cat (3,ones(2),[1/8 7/9;11/5 7/2]), cat (3,ones(2),[13/3 2; 4/5 1/15])); %! y4 = cat (4, ones (2,2,2), cat (3,[3/16 2/3; 2 15/4],[6.5 12/7; 8/11 1/14] )); %! assert (normalize (x, 3, "scale", "first"), y3); %! assert (normalize (x, 4, "scale", "first"), y4); %!assert (normalize ([1,2,3], "scale", "iqr"), [1 2 3]*2/3) %!assert (normalize ([1,2,3]', "scale", "iqr"), ([1 2 3]')*2/3) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2],"scale","iqr"), [1 0 -1; 0 1 0; -1 -1 1]* 2/3, eps) %!assert (normalize (magic (3),"scale","iqr"), [[8;3;4]/3.75,[1;5;9]/6,[6;7;2]/3.75],eps) ## Method: center [mean median number] %!assert (normalize ([1,2,3], "center"), [-1 0 1]) %!assert (normalize ([1,2,3], 1, "center"), [0 0 0]) %!assert (normalize ([1,2,3], "center", 10), [-9 -8 -7]) %!assert (normalize ([1 2 3 10], "center", "mean"), [-3 -2 -1 6]) %!assert (normalize ([1 2 3 10], "center", "median"), [-1.5 -0.5 0.5 7.5]) ## Method: medianiqr %!assert (normalize ([1,2,3], "medianiqr"), [-1 0 1]*2/3) %!assert (normalize ([1,2,3]', "medianiqr"), ([-1 0 1]')*2/3) %!assert (normalize ([2,0,-2;0,2,0;-2,-2,2], "medianiqr"), [1 0 -1; 0 1 0; -1 -1 1]*2/3) %!assert (normalize (magic (3), "medianiqr"), [8/5 -1 0; -2/5 0 2/5; 0 1 -8/5]*2/3) ##test nan and inf %!assert (normalize ([1 2 Inf], 2), [NaN, NaN, NaN]) %!assert (normalize ([1 2 3], 1), [NaN, NaN, NaN]) %!assert (normalize ([1 2 3], 3), [NaN, NaN, NaN]) %!assert (normalize (Inf), NaN) %!assert (normalize (NaN), NaN) %!assert (normalize ([Inf, NaN]), [NaN, NaN]) %!assert (normalize ([Inf, NaN]'), [NaN, NaN]') %!assert (normalize ([Inf, Inf], 1), [NaN, NaN]) %!assert (normalize ([Inf, Inf], 2), [NaN, NaN]) %!assert (normalize ([Inf, Inf]', 1), [NaN, NaN]') %!assert (normalize ([Inf, Inf]', 2), [NaN, NaN]') %!assert (normalize ([1 2 NaN; NaN 3 4], 1), [NaN -1 NaN; NaN 1 NaN]*sqrt(2)/2, eps) ##Two input methods, must be scale and center %!assert (normalize (magic(3), "scale", "center"), normalize (magic(3), "zscore"), eps) %!assert (normalize (magic(3), "center", "scale"), normalize (magic(3), "zscore"), eps) ##Test additional outputs %!test %! [z, c, s] = normalize ([1, 2, 3], 2); %! assert ({z, c, s}, {[-1 0 1], [2], [1]}); %! [z, c, s] = normalize (magic (3), "zscore", "std"); %! assert ({z, c, s}, {[[3;-2;-1]/sqrt(7),[-1;0;1],[1;2;-3]/sqrt(7)], [5 5 5], [sqrt(7) 4 sqrt(7)]}); %! [z, c, s] = normalize (magic (3), "zscore", "robust"); %! assert ({z, c, s}, {[4 -1 0; -1 0 1; 0 1 -4], [4 5 6], [1 4 1]}); %! [z, c, s] = normalize (magic (3), "norm", 1); %! assert ({z, c, s}, {magic(3)/15 , 0, [15 15 15]}); %! [z, c, s] = normalize ([2,0,-2;0,2,0;-2,-2,2],"norm",2); %! assert ({z, c, s}, {[1,0,-1;0,1,0;-1,-1,1]*(sqrt(2)/2), 0, [1 1 1]*2*sqrt(2)}, eps) %! [z, c, s] = normalize ([1 2 3], "norm", Inf); %! assert ({z, c, s}, {[1 2 3]/3, 0, 3}, eps); %! [z, c, s] = normalize (magic (3),"range",[-1 1]); %! assert ({z, c, s}, {[1 -1 0.6; -1 0 1; -0.6 1 -1], [5.5 5 4.5], [2.5 4 2.5]}, eps) %! [z, c, s] = normalize (magic (3),"scale","mad"); %! assert ({z, c, s}, {[8 0.25 6; 3 1.25 7; 4 2.25 2], 0, [1 4 1]}); %! [z, c, s] = normalize (magic (3),"scale","first"); %! assert ({z, c, s}, {[1 1 1; 3/8 5 7/6; 0.5 9 1/3],0, [8 1 6]}, eps); %! [z, c, s] = normalize ([1,2,3]', "scale", "iqr"); %! assert ({z, c, s}, {([1 2 3]')*2/3, 0, 1.5}); %! [z, c, s] = normalize ([1,2,3], "center", 10); %! assert ({z, c, s}, {[-9 -8 -7], 10, 1}); %! [z, c, s] = normalize ([1 2 3 10], "center", "mean"); %! assert ({z, c, s}, {[-3 -2 -1 6], 4, 1}) %! [z, c, s] = normalize ([1 2 3 10], "center", "median"); %! assert ({z, c, s}, {[-1.5 -0.5 0.5 7.5], 2.5, 1}); %! [z, c, s] = normalize (magic (3), "medianiqr"); %! assert ({z, c, s}, {[8/5 -1 0; -2/5 0 2/5; 0 1 -8/5]*2/3, [4 5 6], [3.75 6 3.75]}, eps) %! [z, c, s] = normalize ([1 2 Inf], 2); %! assert ({z, c, s}, {[NaN, NaN, NaN], Inf, NaN}); %! [z, c, s] = normalize (Inf); %! assert ({z, c, s}, {NaN, Inf, NaN}); ## Matlab ignores NaNs, operating as if the vector had one less element, then ## returns the result retaining the NaN in the solution. %!xtest assert (normalize ([1 2 NaN], 2), [-1, 1, NaN]*sqrt(2)/2) %!xtest assert (normalize ([1 2 NaN; 1 2 3], 2), [[-1 1 NaN]*sqrt(2)/2; -1 0 1], eps) ## Test input validation %!error normalize () %!error normalize (1, 2, 3) %!error normalize (['A'; 'B']) %!error normalize (1, ones (2,2)) %!error normalize (1, 1.5) %!error normalize (1, -1) %!error normalize (1, [1 2]) %!error normalize (1, 0) %!error normalize ([1 2 3], "norm", "zscore") %!error normalize ([1 2 3], "norm", "foo") %!error normalize ([1 2 3], "scale", "center", "norm") %!error <'norm' option must be a positive scalar or Inf> normalize ([1 2 3], "norm", -1) %!error <'norm' option must be a positive scalar or Inf> normalize ([1 2 3], "norm", -Inf) %!error <'norm' option must be a positive scalar or Inf> normalize ([1 2 3], "norm", [1 2]) %!error <'range' must be specified as> normalize ([1 2 3], "range", [1 2]') %!error <'range' must be specified as> normalize ([1 2 3], "range", [1 2 3]) %!error <'range' must be specified as> normalize ([1 2 3], "range", 1) %!error normalize ([1 2 3], "scale", [1 2 3]) %!error normalize ([1 2 3], "scale", [1 2 3]') %!error normalize ([1 2 3], "scale", [1 2; 3 4]) %!error
normalize ([1 2 3], "center", [1 2]) %!error
normalize ([1 2 3], "center", [1 2]') %!error
normalize ([1 2 3], "center", [1 2; 3 4]) %!error normalize ([1 2 3], "foo")