rescale#

Purpose#

Scales the columns of a matrix

Format#

{ x_s, location, scale_factor } = rescale(x, method)#

x_s = rescale(x, location, scale_factor)

Parameters:

x (NxK matrix or NxK array) – data to be rescaled
location (1xK vector) – used for column centering
scale_factor (1xK vector) – used for column scaling

method (string) –

name of scaling and centering method:

Method	Location	scale_factor
”euclidean”	0	Euclidean length: \(\sqrt{\sum_{i=1}^nx_i^2}\)
”mad”	Median	Absolute deviation from the median
”maxabs”	0	Maximum absolute value
”midrange”	\(\frac{(Max+Min)}{2}\)	\(\frac{Range}{2}\)
”range”	Minimum	Range
”standardize”	Mean	Standard deviation
”sum”	0	Sum
”ustd”	0	Standard deviation about the origin

Returns:

x_s (matrix or multi-dimensional array) – containing the scaled columns of x
location (1xK vector) – containing the values used to center the columns of the input matrix x
scale_factor (1xK vector) – containing the values used to scale the columns of the input matrix x

Examples#

Specifying a scaling method#

// Create a column vector
x = {   12.5,
        18.2,
        10.8,
         8.3,
        15.4,
        21.5,
        14.6,
        16.7 };

// Standardize 'x' and return the location and scaling factors
{ x_s, location, scale_factor } = rescale(x, "standardize");

print "x_s = " x_s;
print "location = " location;
print "scale_factor = " scale_factor;

After the code above:

x_s =
    -0.53463295
     0.81977052
    -0.93857785
     -1.5326145
     0.15444952
      1.6038989
   -0.035642197
     0.46334856

location =        14.750000
scale_factor =    4.2084948

Specifying a scaling method for multiple columns#

// Create a matrix with 2 columns
x = {   12.5 1088.5,
        18.2  879.3,
        10.8 1232.0,
         8.3 1189.8,
        15.4  932.1,
        21.5 1009.2,
        14.6  656.7,
        16.7 1251.5 };

// Standardize 'x' and return the location and scaling factors
{ x_s, location, scale_factor } = rescale(x, "standardize");

print "x_s = " x_s;
print "location = " location;
print "scale_factor = " scale_factor;

After the code above:

x_s =
    -0.53463295       0.28751716
     0.81977052      -0.73869039
    -0.93857785       0.99144060
     -1.5326145       0.78443315
     0.15444952      -0.47968581
      1.6038989      -0.10148025
   -0.035642197       -1.8306302
     0.46334856        1.0870957

location =        14.750000        1029.8875
scale_factor =    4.2084948        203.85740

Applying previously created location and scaling factors#

Continuing with the variables used in the previous example, we can apply the returned location and scaling factors to standardize additional observations of our two variables.

// Additional observations
x_new = { 9.3  964.1,
         10.9 1173.7,
         11.1 1232.0,
          9.1 1051.2,
         14.6 1124.1,
         18.4  815.3,
         20.2 1292.6,
         18.5  833.1 };

// Standardize 'x' using the location and scaling factors
print "location = " location;
print "scale_factor = " scale_factor;

// returned in the previous example
x_new_s = rescale(x_new, location, scale_factor);

print "x_new_s = " x_new_s;
print "x_new = " x_new;

After the code above:

location =        14.750000        1029.8875
scale_factor =    4.2084948        203.85740

x_new_s =
     -1.2949998      -0.32271333
    -0.91481638       0.70545637
    -0.86729345       0.99144060
     -1.3425227       0.10454612
   -0.035642197       0.46214904
     0.86729345       -1.0526353
      1.2949998        1.2887072
     0.89105492      -0.96531940

x_new =
      9.3000000        964.10000
      10.900000        1173.7000
      11.100000        1232.0000
      9.1000000        1051.2000
      14.600000        1124.1000
      18.400000        815.30000
      20.200000        1292.6000
      18.500000        833.10000