# Data Transformations¶

## Normalizing and scaling data¶

The rescale() function provides 8 different scaling options and returns the rescaled data along with the location and scale factors.

Method

Location

Scale Factor

“euclidean”

0

Euclidean length

median

Absolute deviation from median

“maxabs”

0

Maximum absolute value

“midrange”

(Max+Min)/2

Range/2

“range”

Minimum

Range

“standardize”

Mean

Standard deviation

“sum”

0

Sum

“ustd”

0

Standard deviation around origin

### Example: Rescaling with a specified 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_s, scale_factor_s } = rescale(x, "standardize");

// Rescale the x using the median
{ x_m, location_m, scale_factor_m } = rescale(x, "mad");

print "Standardized rescaling:";
print "x_s = " x_s;
print "location_s = " location_s;
print "scale_factor_s = " scale_factor_s;

print "Median rescaling:";
print "x_m = " x_m;
print "location_m = " location_m;
print "scale_factor_m = " scale_factor_m;


After the code above:

Standardized rescaling:
x_s =
-0.53463295
0.81977052
-0.93857785
-1.5326145
0.15444952
1.6038989
-0.035642197
0.46334856

location_s =           14.75
scale_factor_s =        4.21

Median rescaling:
x_m =
-0.87719298
1.1228070
-1.4736842
-2.3508772
0.14035088
2.2807018
-0.14035088
0.59649123

location_m =           15.00
scale_factor_m =        2.85


The rescale() function can also be used with a known location and scale factor to rescale data.

### Example: Rescaling using known location and scaling factors¶

// Additional observations
x2    = { 9.3,
10.9,
11.1,
9.1,
14.6,
18.4,
20.2,
18.5 };

// Rescale matrix above using
// location and scale matrix
// from above
x_s2 = rescale(x2, location_s, scale_factor_s);


After the code above x_s2 is equal to:

  -1.2949998
-0.91481638
-0.86729345
-1.3425227
-0.035642197
0.86729345
1.2949998
0.89105492


The rescale() function can also be used to rescale multiple columns at time.

### Example: Rescaling 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;

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


## Recoding and reclassifying¶

GAUSS provides a variety of tools for recoding and reclassifying data. These functions can be divided into functions for numeric data and functions for categorical data.

Numeric Functions

Description

Recoding specifier

reclassify()

Replaces specified values of a matrix, array or string array.

User-specified values.

reclassifycuts()

Replaces values of a matrix or array within specified ranges.

User-specified values.

code()

Creates a new matrix based on recoding of an existing numeric vector.

Based on logical expression.

recode()

Recodes the values of an existing vector of numeric data.

Based on logical expression.

substute()

Substitutes new values for old values in a matrix, depending on the outcome of a logical expression.

Based on logical expression.

Categorical Functions

reorderCatLabels()

Changes relative order of categorical variable. This changes the key values associated with the categorical labels.

recodeCatLabels()

Replaces the labels of categorical variables with new labels.

Recoding and reclassifying non-categorical data

Both the code() and recode() procedures can be used to recode data using conditional expressions.

The code() procedure:

• Creates a new matrix which splits existing data into classes.

• Uses N logical expressions to determine N+1 classes.

• Works for vectors only.

### Example: Coding blood pressure data to create a new (binary) class variable¶

// Blood pressure data
x = { 91,
121,
99,
135,
110,
155 };

// Values for the classes
new_val = { 1, 2 };

/*
** Create a vector containing a 1 for every element
** which is less than 120, or a 0 otherwise
*/
logical = x .<  120;

/*
** Create a new vector which contains the class
** assignment for each element in 'x'
*/
x_class = code(logical, new_val);


The code above generates a new vector x_class which splits the original data into two classes based on whether x is less than 120.

x = 91   logical =  1   x_class = 1
121              0             2
99              1             1
135              0             2
110              1             1
155              0             2


### Example: Coding blood pressure data to create a new multi-class variable¶

// Blood pressure data
x = { 91,
121,
99,
135,
110,
155 };

// Values for the classes
new_val = { 1,
2,
3 };

// Create a vector containing a 1 for every element
// which is less than 100, or a 0 otherwise
logical_1 = x .<= 100;

// Create a vector containing a 1 for every element
// which is between 100 and 120, or a 0 otherwise
logical_2 = x .> 100 .and x .<=  120;

// Form a 2 column logical vector using
// horizontal concatenation
logical = logical_1 ~ logical_2;

// Create a new vector which contains the class
// assignment for each element in 'x'
x_class = code(logical, new_val);


Now x_class splits the original data into three classes based on whether x is less than or equal to 100, falls between 100 and 120, or is greater 120.

x =  91    logical = 1 0     x_class = 1
121              0 0               3
99              1 0               1
135              0 0               3
110              0 1               2
155              0 0               3


Note

The setColLabels() function can be used to specify x_class as a categorical variable and to assign labels to the classes.

Recoding values of an existing vector

The recode() procedure :

• Replaces specific values of an existing vector with new values.

• Uses a logical expression to determine where and how to replace values.

• Is valid for vectors.

Some notes to remember about recode():

• There should be no more than a single 1 in any row of logical expression matrix.

• For any given row of a data matrix and logical expression matrix, if a column of the logical expression is 1, the corresponding replacement values with replace the original element of the data matrix.

• If every column of logical expression matrix contains a 0, the original value of the data matrix will be unchanged.

### Example: Recoding numeric values based on ranges¶

x = { 20,
45,
32,
63,
29 };

// Create 4 column vectors with a 1 where the statement
// evaluates as 'true'

// Check if 20 < x <= 30
e1 = (20 .lt x) .and (x .le 30);

// Check if 30 < x <= 40
e2 = (30 .lt x) .and (x .le 40);

// Check if 40 < x <= 50
e3 = (40 .lt x) .and (x .le 50);

// Check if 50 < x <= 60
e4 = (50 .lt x) .and (x .le 60);

// Horizontally concatenate the column vectors into a 5x4
// matrix
logical = e1~e2~e3~e4;

v = { 1.2,
2.4,
3.1,
4.6 };

// Replace elements of 'x' with elements from 'v' based upon
// the 0's and 1's in 'e'
x_new = recode(x, logical, v);


Note that in this example x_new is as follows:

          0   0   0   0
0   0   1   0
logical = 0   1   0   0
0   0   0   0
1   0   0   0

// Since the third column of the second row of 'e' is equal
// to 1, the second row of 'y' is set equal to the third
// element of 'v', etc.
20.000000
3.1000000
x_new = 2.4000000
63.000000
1.2000000


Reclassifying data

The reclassify() and reclassifyCuts() procedures can be used to reclassify existing values to new values.

The reclassify() procedure:

• Replaces values in a from input with values specified in a to input.

• Works for matrices, arrays, and string arrays.

• Can be used to reclassify matrices to string arrays and vice versa.

Note

The reclassify() function can reclassify matrices to string arrays but does not create a dataframe. To create a dataframe with a string labels from an existing matrix see asDF().

### Example: Change instances of 1, 2 and 3 to ‘low’, ‘medium’ and ‘high’.¶

// Vector to be changed
x = { 2,
3,
2,
1,
2,
3 };

from = { 1,
2,
3 };

// Create a 3x1 string array using
// string vertical concatenation operator
to = "low" $| "medium"$| "high";

x_new = reclassify(x, from, to);
print x_new;


After the code above, x_new is equal to:

medium
high
medium
low
medium
high


In this case, if the number of specified strings in to is less than the number of unique values in x, the unmapped values will be converted directly into strings.

// Vector to be changed
x = { 2,
3,
2,
1,
2,
3 };

from = { 1,
2};

// Create a 3x1 string array using
// string vertical concatenation operator
to = "low" $| "medium"; x_new = reclassify(x, from, to); print x_new;  Now x_new is medium 3 medium low medium 3  ### Example: Change instances of tea types: ‘black’, ‘green’, ‘oolong’ to 9.95, 11.95 and 10.50, respectively.¶ string orders = { "green", "green", "oolong", "green", "green", "green", "black" }; string tea_types = { "black", "green", "oolong" }; price = { 9.95, 11.95, 10.50 }; order_prices = reclassify(orders, tea_types, price); print order_prices;  The vector order_prices is equal to: 11.95 11.95 10.50 11.95 11.95 11.95 9.95  In this case, if the number of specified values in to is less than the number of unique strings in x, unmapped strings will be reclassified as missings: string orders = { "green", "green", "oolong", "green", "green", "green", "black" }; string tea_types = { "black", "green" }; price = { 9.95, 11.95 }; order_prices = reclassify(orders, tea_types, price); print order_prices;  Now order_prices is: 11.950000 11.950000 . 11.950000 11.950000 11.950000 9.9500000  The reclassifyCuts() procedure: • Splits the data in x into classes based on specified cutoff values. • Works for matrices and arrays. • Cutoff points can be used to define the right endpoint of an interval or the starting points of the next interval. The default is to use the cutoff points as starting points of the next interval. ### Example: Basic sequence¶ // Create column vector to place in categories x = { 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 }; // Cut points for data in 'x' cut_pts = { 0.2, 0.5 }; // Class 0: x <= 0.2 // Class 1: 0.2 < x <= 0.5 // Class 2: 0.5 < x r_open = reclassifyCuts(x, cut_pts); // Class 0: x < 0.2 // Class 1: 0.2 <= x < 0.5 // Class 2: 0.5 <= x r_closed = reclassifyCuts(x, cut_pts, 1); print "x = " x; print; print "r_open = " r_open; print; print "r_closed = " r_closed; print; print "cut_pts = " cut_pts;  This results in: x = 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 r_open = 0.00 0.00 0.00 1.0 1.0 1.0 2.0 2.0 r_closed = 0.00 0.00 1.0 1.0 1.0 2.0 2.0 2.0 cut_pts = 0.20 0.50  ### Example: Classifying blood pressure data¶ // Create a column of blood pressure data bp = { 87, 154, 127, 112, 159, 90, 151, 109, 125, 107 }; // Assign cut points cut_pts = { 120, 140 }; // Create categorical variable bp_category = reclassifyCuts(bp, cut_pts); print "bp = " bp; print; print "bp_category = " bp_category; print; print "cut_pts = " cut_pts;  This splits the data in bp into three categories: those that fall below 120, those that greater than or equal to 120 but less than 140, and those that are greater than or equal to 140:  87 154 127 112 bp = 159 90 151 109 125 107 0 2 1 0 bp_category = 2 0 2 0 1 0 cut_pts = 120 140  ## Substituting values¶ The substute() function replaces values in a matrix based on the outcome of a logical expression. ### Example: Setting very small values to zero¶ // Create example vector x = { 3.8e-21, 1.0, 3.5, 2.7e-18, 0.5, 3.0, 1.1e-16, 0.5, 2.2, 4.0 }; // Substitute all values less than 2.2e-16 with a zero x_new = substute(x, x .< 2.25e-16, 0);  This results in x_new equal to: 0.00000000 1.0000000 3.5000000 0.00000000 0.50000000 3.0000000 0.00000000 0.50000000 2.2000000 4.0000000  Recoding categorical data The recodeCatLabels() can be use to change the labels on categorical variables in a dataframe. ### Example: Recoding categories in yarn dataset¶ // Load data fname = getGAUSSHome$+ "examples/yarn.xlsx";

// Get column labels for yarn_length
{ labels, keyvalues } = getColLabels(yarn, "yarn_length");

// Print results
sprintf("%11s", "Key"$~"Labels"); sprintf("%10.0f %10s", keyvalues, labels); // Recode yarn_length variable from // 'low', 'medium', and 'high' // to 'sm', 'md', 'lg' yarn_recoded = recodecatlabels(yarn, "low"$|"med"$|"high", "sm"$|"md"$|"lg", "yarn_length"); // Get column labels for yarn_length { labels, keyvalues } = getColLabels(yarn_recoded, "yarn_length"); // Print results print "Yarn recoded labels"; sprintf("%11s", "Key"$~"Labels");
sprintf("%10.0f %10s", keyvalues, labels);


This prints the following:

Yarn labels
Key     Labels

0       high
1        low
2        med

Yarn recoded labels
Key     Labels

0         lg
1         sm
2         md


Reordering categorical data

The reorderCatLabels() can be use to change the key values associated with categorical labels.

// Load data
fname = getGAUSSHome $+ "examples/yarn.xlsx"; yarn = loadd(fname, "cat(yarn_length) + cat(amplitude) + cat(load) + cycles"); // Get column labels for yarn_length { labels, keyvalues } = getColLabels(yarn, "yarn_length"); // Print results print "Yarn labels"; sprintf("%11s", "Key"$~"Labels");
sprintf("%10.0f %10s", keyvalues, labels);

// Order labels
yarn_reordered = reordercatlabels(yarn, "med"$|"high"$|"low", "yarn_length");

// Get column labels for yarn_length
{ labels, keyvalues } = getColLabels(yarn_reordered, "yarn_length");

// Print results
print "Reordered yarn labels";

sprintf("%11s", "Key"$~"Labels"); sprintf("%10.0f %10s", keyvalues, labels);  This prints the following: Yarn labels Key Labels 0 high 1 low 2 med Reordered yarn labels Key Labels 0 med 1 high 2 low  ## Dummy variables¶ Categorical variables in dataframes will automatically be treated as dummy variables in GAUSS estimation routines. This means no extra steps are necessary to include categorical variables in regression. ### Example: Include a categorical variable in OLS¶ // Load data fname = getGAUSSHome$+ "examples/auto2.dta";

// Include the rep78
// categorical variable in
// ols estimation
call olsmt(fname, "price ~ mpg + rep78");


The categorical variable rep78 will automatically be included in the OLS regression as a dummy variable with the base case excluded from the regression. In addition, the category labels will be displayed in the printed output table.

Standard                                                  Prob   Standardized  Cor with
Variable             Estimate      Error      t-value     >|t|     Estimate    Dep Var
---------------------------------------------------------------------------------------

CONSTANT                10450     2251.04     4.64229     0.000       ---         ---
mpg                  -280.261     61.5767    -4.55142     0.000   -0.564519   -0.455949
rep78: Fair           877.635     2063.28    0.425358     0.672   0.0971824  -0.0223477
rep78: Average        1425.66     1905.44    0.748204     0.457     0.24444   0.0859051
rep78: Good           1693.84     1942.67    0.871914     0.387    0.257252   -0.015317
rep78: Excellent      3131.98     2041.05      1.5345     0.130    0.396546   -0.035102


The categories of rep78, Fair, Average, Good, Excellent are included as dummy variables in the regression. The Poor category is excluded from the regression, as it is the base case.

## Example: Including a categorical variable in GLM estimation¶

// Load data
fname = getGAUSSHome \$+ "examples/auto2.dta";

// Loadd data and remove missing values
data = packr(loadd(fname, "price + mpg + rep78"));

// Include the rep78
// categorical variable in
// linear regression using glm
call glm(data, "price ~ mpg + rep78", "normal");

Standard                                                                        Prob
Variable                 Estimate            Error          t-value             >|t|
----------------     ------------     ------------     ------------     ------------
CONSTANT                    10450             2251           4.6423         < 0.0001
mpg                       -280.26           61.577          -4.5514         < 0.0001
rep78: Fair                877.63           2063.3          0.42536         0.672025
rep78: Average             1425.7           1905.4           0.7482         0.457121
rep78: Good                1693.8           1942.7          0.87191         0.386566
rep78: Excellent             3132             2041           1.5345         0.129915


Outside of estimation, dummy variables can be created using a number of procedures:

Functions

Description

design()

Creates dummy variables from discrete data that is split into classes.

dummybr()

Creates dummy variables from continuous data based on break points. The highest (rightmost) category is bounded on the right.

dummydn()

Creates dummy variables from continuous data based on break points. The highest (rightmost) category is unbounded on the right, and a specified column of dummies is dropped.

dummy()

Creates dummy variables from continuous data based on break points. The highest (rightmost) category is unbounded on the right.

### Example: Create dummy variables based on BP classes¶

This example builds on an earlier example, in which BP data was split into 3 classes using reclassify().

// Classified BP data
bp_class = { 1,
3,
1,
3,
2,
3 };

// Create matrix of dummy
// variables using design
dv_bp_classes = design(bp_class);


After this code dv_bp_classes is equal to:

dv_bp_classes;

1      0      0
0      0      1
1      0      0
0      0      1
0      1      0
0      0      1


## Example: Create dummy variables from continuous BP data¶

The dummybr() variable can be used to generate dummy variables from the ranges of original BP data.

// Create a column of blood pressure data
bp = { 91,
121,
99,
135,
110,
155 };

// Create breakpoints
v = { 100, 120 };

// Create dummy variables
dv_bp = dummy(bp, v);


Note that dv_bp is the same as dv_bp_classes from the first example:

1      0      0
0      0      1
1      0      0
0      0      1
0      1      0
0      0      1


## Example: Create dummy variables from continuous BP data and drop first column¶

The dummydn() variable can be used to generate dummy variables from the ranges of original BP data.

// Create a column of blood pressure data
bp = { 91,
121,
99,
135,
110,
155 };

// Create breakpoints
v = { 100, 120 };

// Create dummy variables
dv_bp_drop = dummydn(bp, v, 1);


Now the dv_bp_drop matrix is the same as the second and third columns of dv_bp and dv_bp_classes:

0      0
0      1
0      0
0      1
1      0
0      1