normalize

Normalize data

Syntax

N = normalize(A)

N = normalize(A,dim)

N = normalize(___,method)

N = normalize(___,method,methodtype)

N = normalize(___,"center",centertype,"scale",scaletype)

N = normalize(___,Name,Value)

[N,C,S] = normalize(___)

Description

N = normalize(A) returns the vectorwise z-score of the data in A with center 0 and standard deviation 1.

• If A is a vector, then normalize operates on the entire vector A.

• If A is a matrix, then normalize operates on each column of A separately.

• If A is a multidimensional array, then normalize operates along the first dimension of A whose size does not equal 1.

• If A is a table or timetable, then normalize operates on each variable of A separately.

N = normalize(A,dim) specifies the dimension of A to operate along. For example, normalize(A,2) normalizes each row.

N = normalize(___,method) specifies a normalization method with any of the previous syntaxes. For example, normalize(A,"norm") normalizes the data in A by the Euclidean norm (2-norm).

N = normalize(___,method,methodtype) specifies the type of normalization for the given method. For example, normalize(A,"norm",Inf) normalizes the data in A using the infinity norm.

N = normalize(___,"center",centertype,"scale",scaletype) uses the "center" and "scale" methods at the same time. These are the only methods you can use together. If you do not specify centertype or scaletype, then normalize uses the default method type for that method (centering to have a mean of 0 and scaling by the standard deviation).

Use this syntax with any center and scale type to perform both methods together. For instance, N = normalize(A,"center","median","scale","mad"). You can also use this syntax to specify centering and scaling values C and S from a previously computed normalization. For instance, normalize one data set and save the parameters with [N1,C,S] = normalize(A1). Then, reuse those parameters on a different data set with N2 = normalize(A2,"center",C,"scale",S).

N = normalize(___,Name,Value) specifies additional parameters for normalizing using one or more name-value arguments. For example, normalize(A,"DataVariables",datavars) normalizes the variables specified by datavars when A is a table or timetable.

[N,C,S] = normalize(___) additionally returns the centering and scaling values C and S used to perform the normalization. Then, you can normalize different input data using the values in C and S with N = normalize(A2,"center",C,"scale",S).

Alternative

You can use normalize functionality interactively by adding the Normalize Data task to a live script.

Examples

Vector and Matrix Data

Normalize data in a vector and matrix by computing the z-score.

Create a vector v and compute the z-score, normalizing the data to have mean 0 and standard deviation 1.

v = 1:5;
N = normalize(v)

N = 1×5

   -1.2649   -0.6325         0    0.6325    1.2649

Create a matrix B and compute the z-score for each column. Then, normalize each row.

B = magic(3)

B = 3×3

     8     1     6
     3     5     7
     4     9     2

N1 = normalize(B)

N1 = 3×3

    1.1339   -1.0000    0.3780
   -0.7559         0    0.7559
   -0.3780    1.0000   -1.1339

N2 = normalize(B,2)

N2 = 3×3

    0.8321   -1.1094    0.2774
   -1.0000         0    1.0000
   -0.2774    1.1094   -0.8321

Scale Data

Scale a vector A by its standard deviation.

A = 1:5;
Ns = normalize(A,"scale")

Ns = 1×5

    0.6325    1.2649    1.8974    2.5298    3.1623

Scale A so that its range is in the interval [0, 1].

Nr = normalize(A,"range")

Nr = 1×5

         0    0.2500    0.5000    0.7500    1.0000

Specify Method Type

Create a vector A and normalize it by its 1-norm.

A = 1:5;
Np = normalize(A,"norm",1)

Np = 1×5

    0.0667    0.1333    0.2000    0.2667    0.3333

Center the data in A so that it has mean 0.

Nc = normalize(A,"center","mean")

Nc = 1×5

    -2    -1     0     1     2

Table Variables

Create a table containing height information for five people.

LastName = ["Sanchez";"Johnson";"Lee";"Diaz";"Brown"];
Height = [71;69;64;67;64];
T = table(LastName,Height)

T=5×2 table
    LastName     Height
    _________    ______

    "Sanchez"      71  
    "Johnson"      69  
    "Lee"          64  
    "Diaz"         67  
    "Brown"        64  

Normalize the height data by the maximum height.

N = normalize(T,"norm",Inf,"DataVariables","Height")

N=5×2 table
    LastName     Height 
    _________    _______

    "Sanchez"          1
    "Johnson"    0.97183
    "Lee"        0.90141
    "Diaz"       0.94366
    "Brown"      0.90141

Complex Vector

Create a vector containing real and imaginary components.

a = [1; 2; 3; 4];
b = [2; -2; 7; -7];
z = complex(a,b)

z = 4×1 complex

   1.0000 + 2.0000i
   2.0000 - 2.0000i
   3.0000 + 7.0000i
   4.0000 - 7.0000i

Normalize the complex vector. To scale the magnitude while maintaining the phase, scale by the infinity norm, or the largest magnitude. Specify the Inf option with the norm method. The function returns a complex unit vector.

N = normalize(z,"norm",Inf)

N = 4×1 complex

   0.1240 + 0.2481i
   0.2481 - 0.2481i
   0.3721 + 0.8682i
   0.4961 - 0.8682i

Verify that the normalized vector is within the complex unit circle.

Nmag = max(abs(N))

Nmag = 
1

Verify that the ratios between the corresponding elements of the normalized and original vectors are the same.

r = N ./ z

r = 4×1

    0.1240
    0.1240
    0.1240
    0.1240

Verify that the phase angle of the normalized vector is the same as the phase angle of the original vector.

ztheta = angle(z)

ztheta = 4×1

    1.1071
   -0.7854
    1.1659
   -1.0517

Ntheta = angle(N)

Ntheta = 4×1

    1.1071
   -0.7854
    1.1659
   -1.0517

Normalize Multiple Data Sets with Same Parameters

Normalize a data set, return the computed parameter values, and reuse the parameters to apply the same normalization to another data set.

Create a timetable with two variables: Temperature and WindSpeed. Then create a second timetable with the same variables, but with the samples taken a year later.

rng default 
Time1 = (datetime(2019,1,1):days(1):datetime(2019,1,10))';
Temperature = randi([10 40],10,1);
WindSpeed = randi([0 20],10,1);
T1 = timetable(Temperature,WindSpeed,'RowTimes',Time1)

T1=10×2 timetable
       Time        Temperature    WindSpeed
    ___________    ___________    _________

    01-Jan-2019        35             3    
    02-Jan-2019        38            20    
    03-Jan-2019        13            20    
    04-Jan-2019        38            10    
    05-Jan-2019        29            16    
    06-Jan-2019        13             2    
    07-Jan-2019        18             8    
    08-Jan-2019        26            19    
    09-Jan-2019        39            16    
    10-Jan-2019        39            20    

Time2 = (datetime(2020,1,1):days(1):datetime(2020,1,10))';
Temperature = randi([10 40],10,1);
WindSpeed = randi([0 20],10,1);
T2 = timetable(Temperature,WindSpeed,'RowTimes',Time2)

T2=10×2 timetable
       Time        Temperature    WindSpeed
    ___________    ___________    _________

    01-Jan-2020        30            14    
    02-Jan-2020        11             0    
    03-Jan-2020        36             5    
    04-Jan-2020        38             0    
    05-Jan-2020        31             2    
    06-Jan-2020        33            17    
    07-Jan-2020        33            14    
    08-Jan-2020        22             6    
    09-Jan-2020        30            19    
    10-Jan-2020        15             0    

Normalize the first timetable. Specify three outputs: the normalized table, and also the centering and scaling parameter values C and S that the function uses to perform the normalization.

[T1_norm,C,S] = normalize(T1)

T1_norm=10×2 timetable
       Time        Temperature    WindSpeed
    ___________    ___________    _________

    01-Jan-2019      0.57687       -1.4636 
    02-Jan-2019        0.856       0.92885 
    03-Jan-2019      -1.4701       0.92885 
    04-Jan-2019        0.856       -0.4785 
    05-Jan-2019     0.018609       0.36591 
    06-Jan-2019      -1.4701       -1.6044 
    07-Jan-2019      -1.0049      -0.75997 
    08-Jan-2019     -0.26052       0.78812 
    09-Jan-2019      0.94905       0.36591 
    10-Jan-2019      0.94905       0.92885 

C=1×2 table
    Temperature    WindSpeed
    ___________    _________

       28.8          13.4   

S=1×2 table
    Temperature    WindSpeed
    ___________    _________

      10.748        7.1056  

Now normalize the second timetable T2 using the parameter values from the first normalization. This technique ensures that the data in T2 is centered and scaled in the same manner as T1.

T2_norm = normalize(T2,"center",C,"scale",S)

T2_norm=10×2 timetable
       Time        Temperature    WindSpeed
    ___________    ___________    _________

    01-Jan-2020      0.11165      0.084441 
    02-Jan-2020      -1.6562       -1.8858 
    03-Jan-2020      0.66992       -1.1822 
    04-Jan-2020        0.856       -1.8858 
    05-Jan-2020       0.2047       -1.6044 
    06-Jan-2020      0.39078       0.50665 
    07-Jan-2020      0.39078      0.084441 
    08-Jan-2020      -0.6327       -1.0414 
    09-Jan-2020      0.11165       0.78812 
    10-Jan-2020       -1.284       -1.8858 

By default, normalize operates on any variables in T2 that are also present in C and S. To normalize a subset of the variables in T2, specify the variables to operate on with the DataVariables name-value argument. The subset of variables you specify must be present in C and S.

Specify WindSpeed as the data variable to operate on. normalize operates on that variable and returns Temperature unchanged.

T2_partial = normalize(T2,"center",C,"scale",S,"DataVariables","WindSpeed")

T2_partial=10×2 timetable
       Time        Temperature    WindSpeed
    ___________    ___________    _________

    01-Jan-2020        30         0.084441 
    02-Jan-2020        11          -1.8858 
    03-Jan-2020        36          -1.1822 
    04-Jan-2020        38          -1.8858 
    05-Jan-2020        31          -1.6044 
    06-Jan-2020        33          0.50665 
    07-Jan-2020        33         0.084441 
    08-Jan-2020        22          -1.0414 
    09-Jan-2020        30          0.78812 
    10-Jan-2020        15          -1.8858 

Input Arguments

A — Input data

scalar | vector | matrix | multidimensional array | table | timetable

Input data, specified as a scalar, vector, matrix, multidimensional array, table, or timetable.

If A is a numeric array and has type single, then the output also has type single. Otherwise, the output has type double.

normalize ignores NaN values in A.

Data Types: double | single | table | timetable
Complex Number Support: Yes

dim — Operating dimension

positive integer scalar

Operating dimension, specified as a positive integer scalar. If no value is specified, then the default is the first array dimension whose size does not equal 1.

For table or timetable input data, dim is not supported and operation is along each table or timetable variable separately.

method — Normalization method

"zscore" (default) | "norm" | "scale" | "range" | "center" | "medianiqr"

Normalization method, specified as one of the options in this table.

Method	Description
"zscore"	Compute the z-score. Center data to have mean 0, and scale data to have standard deviation 1.
"norm"	Scale data by the 2-norm, also known as the Euclidean norm.
"scale"	Scale data to have standard deviation 1.
"range"	Rescale range of data to [0, 1].
"center"	Center data to have mean 0.
"medianiqr"	Center data to have median 0, and scale data to have interquartile range 1.

To return the parameters the function uses to normalize the data, specify the C and S output arguments.

methodtype — Method type

array | table | two-element row vector | type name

Method type, specified as an array, table, two-element row vector, or type name, depending on the specified method.

Method	Method Type Options	Description
"zscore"	"std" (default)	Compute the z-score. Center data to have mean 0, and scale data to have standard deviation 1.
	"robust"	Compute the z-score. Center data to have mean 0, and scale data to have median absolute deviation 1.
"norm"	Positive numeric scalar (default is 2)	Scale data by the p-norm, where p is a positive numeric scalar.
	Inf	Scale data by the p-norm, where p is Inf. The infinity norm, or maximum norm, is the same as the largest magnitude of the elements in the data.
"scale"	"std" (default)	Scale data to have standard deviation 1.
	"mad"	Scale data to have median absolute deviation 1.
	"first"	Scale data by the first element of the data.
	"iqr"	Scale data to have interquartile range 1.
	Numeric array	Scale data by an array of numeric values. The array must have a compatible size with input A.
	Table	Scale data by variables in a table. Each table variable in the input data A is scaled using the value in the similarly named variable in the scaling table.
"range"	2-element row vector (default is [0 1])	Rescale range of data to [a b], where a < b.
"center"	"mean" (default)	Center data to have mean 0.
	"median"	Center data to have median 0.
	Numeric array	Shift center by an array of numeric values. The array must have a compatible size with input A.
	Table	Shift center by variables in a table. Each table variable in the input data A is centered using the value in the similarly named variable in the centering table.

To return the parameters the function uses to normalize the data, specify the C and S output arguments.

centertype, scaletype — Center and scale method types

array | table | type name

Center and scale method types, specified as any valid methodtype option for the "center" or "scale" methods, respectively. See the methodtype argument description for a list of available options for each of the methods.

Example: N = normalize(A,"center",C,"scale",S)

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: normalize(T,ReplaceValues=false)

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: normalize(T,"ReplaceValues",false)

DataVariables — Table variables to operate on

table variable name | scalar | vector | cell array | pattern | function handle | table vartype subscript

Table variables to operate on, specified as one of the options in this table. The DataVariables value indicates which variables of the input table to fill.

Other variables in the table not specified by DataVariables pass through to the output without being normalized.

Indexing Scheme	Values to Specify	Examples
Variable names	A string scalar or character vector A string array or cell array of character vectors A pattern object	"A" or 'A' — A variable named A ["A" "B"] or {'A','B'} — Two variables named A and B "Var"+digitsPattern(1) — Variables named "Var" followed by a single digit
Variable index	An index number that refers to the location of a variable in the table A vector of numbers A logical vector. Typically, this vector is the same length as the number of variables, but you can omit trailing 0 (false) values.	3 — The third variable from the table [2 3] — The second and third variables from the table [false false true] — The third variable
Function handle	A function handle that takes a table variable as input and returns a logical scalar	@isnumeric — All the variables containing numeric values
Variable type	A vartype subscript that selects variables of a specified type	vartype("numeric") — All the variables containing numeric values

Example: normalize(T,"DataVariables",["Var1" "Var2" "Var4"])

ReplaceValues — Replace values indicator

true or 1 (default) | false or 0

Replace values indicator, specified as one of these values when A is a table or timetable:

• true or 1 — Replace input table variables with table variables containing normalized data.

• false or 0 — Append input table variables with table variables containing normalized data.

For vector, matrix, or multidimensional array input data, ReplaceValues is not supported.

Example: normalize(T,"ReplaceValues",false)

Output Arguments

N — Normalized values

array | table | timetable

Normalized values, returned as an array, table, or timetable.

N is the same size as A unless the value of ReplaceValues is false. If the value of ReplaceValues is false, then the width of N is the sum of the input data width and the number of data variables specified.

normalize generally operates on all variables of input tables and timetables, except in these cases:

• If you specify DataVariables, then normalize operates on only the specified variables.

• If you use the syntax normalize(T,"center",C,"scale",S) to normalize a table or timetable T using previously computed parameters C and S, then normalize automatically uses the variable names in C and S to determine the data variables in T to operate on.

C — Centering values

array | table

Centering values, returned as an array or table.

When A is an array, normalize returns C and S as arrays such that N = (A - C) ./ S. Each value in C is the centering value used to perform the normalization along the specified dimension. For example, if A is a 10-by-10 matrix of data and normalize operates along the first dimension, then C is a 1-by-10 vector containing the centering value for each column in A.

When A is a table or timetable, normalize returns C and S as tables containing the centers and scales for each table variable that was normalized, N.Var = (A.Var - C.Var) ./ S.Var. The table variable names of C and S match corresponding table variables in the input. Each variable in C contains the centering value used to normalize the similarly named variable in A.

S — Scaling values

array | table

Scaling values, returned as an array or table.

When A is an array, normalize returns C and S as arrays such that N = (A - C) ./ S. Each value in S is the scaling value used to perform the normalization along the specified dimension. For example, if A is a 10-by-10 matrix of data and normalize operates along the first dimension, then S is a 1-by-10 vector containing the scaling value for each column in A.

When A is a table or timetable, normalize returns C and S as tables containing the centers and scales for each table variable that was normalized, N.Var = (A.Var - C.Var) ./ S.Var. The table variable names of C and S match corresponding table variables in the input. Each variable in S contains the scaling value used to normalize the similarly named variable in A.

More About

Z-Score

z-scores measure the distance of a data point from the mean in terms of the standard deviation. The standardized data set has mean 0 and standard deviation 1, and retains the shape properties of the original data set (same skewness and kurtosis).

For a random variable X with mean μ and standard deviation σ, the z-score of a value x is $$z=\frac{\left(x-\mu \right)}{\sigma }.$$ For sample data with mean $$\overline{X}$$ and standard deviation S, the z-score of a data point x is $$z=\frac{\left(x-\overline{X}\right)}{S}.$$

P-Norm

The general definition for the p-norm of a vector v that has N elements is

$${\Vert v\Vert }_p={\left[{\displaystyle \sum _{k=1}^N{\left|v_k\right|}^p}\right]}^{1/p},$$

where p is any positive real value, Inf, or -Inf. Some common values of p are 1, 2, and Inf.

• If p is 1, then the resulting 1-norm is the sum of the absolute values of the vector elements.

• If p is 2, then the resulting 2-norm gives the vector magnitude or Euclidean length of the vector.

• If p is Inf, then $${\Vert v\Vert }_{\infty }={\max }_i\left(\left|v\left(i\right)\right|\right)$$.

Rescaling

Rescaling changes the distance between the min and max values in a data set by stretching or squeezing the points along the number line. The z-scores of the data are preserved, so the shape of the distribution remains the same.

The equation for rescaling data X to an arbitrary interval [a b] is

$$X_{rescaled}=a+\left[\frac{X-{\min }_X}{{\max }_X-{\min }_X}\right]\left(b-a\right).$$

If A is constant, then normalize returns the lower bound of the interval (0 by default) or NaN (when the specified interval contains Inf).

While the normalize and rescale functions can both rescale data to any arbitrary interval, rescale also permits clipping the input data to specified minimum and maximum values.

Interquartile Range

The interquartile range (IQR) of a data set describes the range of the middle 50% of values when the values are sorted. If Q1 is the 25th percentile of the data and Q3 is the 75th percentile of the data, then $$\text{IQR = Q3 - Q1}$$.

If A is constant, then the interquartile range of A is 0, but if the values are missing or infinite, then the interquartile range of A is NaN.

The IQR is generally preferred over looking at the full range of the data when the data contains outliers (very large or very small values) because the IQR excludes the largest 25% and smallest 25% of values in the data.

Median Absolute Deviation

The median absolute deviation (MAD) of a data set is the median value of the absolute deviations from the median $$\tilde{X}$$ of the data: $$\text{MAD}=\text{median}\left(\left|x-\tilde{X}\right|\right)$$. Therefore, the MAD describes the variability of the data in relation to the median.

The MAD is generally preferred over using the standard deviation of the data when the data contains outliers (very large or very small values) because the standard deviation squares deviations from the mean, giving outliers an unduly large impact. Conversely, the deviations of a small number of outliers do not affect the value of the MAD.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The normalize function supports tall arrays with the following usage notes and limitations:

• The outputs C and S are not supported.

• The "center" and "scale" methods cannot be specified at the same time.

• The supported method types for "center" are: "mean", "median", or a numeric scalar.

• The supported method types for "scale" are: "std", "mad", "first", or a numeric scalar.

• The DataVariables name-value argument cannot specify a function handle.

• Normalization methods that require calculation of the median or interquartile range along the first dimension only support tall column vector data. This includes the methods normalize(___,"zscore","robust"), normalize(___,"scale","mad"), normalize(___,"scale","iqr"), normalize(___,"center","median"), and normalize(___,"medianiqr").

For more information, see Tall Arrays.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• When the method types for "center" and "scale" are both tables and DataVariables is not provided, the method types must have table variable names in the same order.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The normalize function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

Usage notes and limitations:

• The syntax normalize(___,"medianiqr") is not supported.

• The syntax normalize(___,"scale","iqr") is not supported.

For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced in R2018a

R2022a: Append normalized values

You can now append, instead of replace, input table variables with table variables containing normalized data by setting the ReplaceValues name-value argument to false.

The ReplaceValues name-value argument is supported only for table and timetable input data.

R2021a: Normalize multiple data sets with same parameters

Return and reuse the centering and scaling normalization parameter values to normalize subsequent data sets. For example, normalize array A and then normalize array B with the same parameters.

[Anorm,C,S] = normalize(A);
Bnorm = normalize(B,"center",C,"scale",S);

The new outputs, centering value C and scaling parameter S, allow for reuse in a later normalization step. Specify the "center" and "scale" normalization methods at the same time. These are the only two normalization methods that you can specify together.

When method is "center" or "scale", the possible values of methodtype include arrays and tables. While these methodtype values are intended to work with the new outputs C and S, you also can compute your own normalization parameters to specify.