normalize - Normalize data - MATLAB (original) (raw)

Syntax

Description

[N](#mw%5F9405678c-6c76-491a-83e4-b928ee0b5cee) = normalize([A](#d126e1172741)) 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, thennormalize operates along the first dimension ofA whose size does not equal 1.
• If A is a table or timetable, thennormalize operates on each variable of A separately.

example

[N](#mw%5F9405678c-6c76-491a-83e4-b928ee0b5cee) = normalize([A](#d126e1172741),[dim](#mw%5Ff104f5ca-9a1b-4bb7-ad9d-6452bd07c519)) specifies the dimension of A to operate along. For example,normalize(A,2) normalizes each row.

example

[N](#mw%5F9405678c-6c76-491a-83e4-b928ee0b5cee) = normalize(___,[method](#mw%5F78930ffd-d2e4-41a6-9ec7-a6a00749332d)) specifies a normalization method with any of the previous syntaxes. For example,normalize(A,"norm") normalizes the data inA by the Euclidean norm (2-norm).

example

[N](#mw%5F9405678c-6c76-491a-83e4-b928ee0b5cee) = normalize(___,[method](#mw%5F78930ffd-d2e4-41a6-9ec7-a6a00749332d),[methodtype](#mw%5Fc53d6eba-62ae-44c6-a262-cc16fe9b5655)) specifies the type of normalization for the given method. For example,normalize(A,"norm",Inf) normalizes the data inA using the infinity norm.

example

[N](#mw%5F9405678c-6c76-491a-83e4-b928ee0b5cee) = normalize(___,"center",[centertype](#mw%5F5af7d9db-9955-4e5a-9cb8-9a019a1624c6),"scale",[scaletype](#mw%5F5af7d9db-9955-4e5a-9cb8-9a019a1624c6)) uses the "center" and "scale" methods at the same time. These are the only methods you can use together. If you do not specifycentertype or scaletype, thennormalize 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 valuesC 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).

example

[N](#mw%5F9405678c-6c76-491a-83e4-b928ee0b5cee) = normalize(___,[Name,Value](#namevaluepairarguments)) specifies additional parameters for normalizing using one or more name-value arguments. For example, normalize(A,"DataVariables",datavars) normalizes the variables specified by datavars whenA is a table or timetable.

example

[[N](#mw%5F9405678c-6c76-491a-83e4-b928ee0b5cee),[C](#mw%5F5eed55d6-0907-415f-8d20-33b5c8682305),[S](#mw%5Fadd83049-8c13-478f-a91c-7ff738c5967d)] = normalize(___) additionally returns the centering and scaling values C andS used to perform the normalization. Then, you can normalize different input data using the values in C andS 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.

example

Examples

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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 = 3×3

 8     1     6
 3     5     7
 4     9     2

N1 = 3×3

1.1339   -1.0000    0.3780

-0.7559 0 0.7559 -0.3780 1.0000 -1.1339

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")

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.

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

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 = 4×1

1.1071

-0.7854 1.1659 -1.0517

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

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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 typesingle, then the output also has typesingle. Otherwise, the output has typedouble.

normalize ignores NaN values inA.

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.

To return the parameters the function uses to normalize the data, specify theC 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.

To return the parameters the function uses to normalize the data, specify theC 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 asName1=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 byDataVariables pass through to the output without being normalized.

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

ReplaceValues — Replace values indicator

true or1 (default) | false or 0

Replace values indicator, specified as one of these values whenA 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

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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, thennormalize operates on only the specified variables.
• If you use the syntaxnormalize(T,"center",C,"scale",S) to normalize a table or timetable T using previously computed parameters C andS, then normalize automatically uses the variable names in C and S to determine the data variables inT 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 thatN = (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, thenC 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 andS 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 andS 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 thatN = (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, thenS 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 andS 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 andS 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

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_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=(x−μ)σ. For sample data with mean X¯ and standard deviation S, the_z_-score of a data point x is z=(x−X¯)S.

_P_-Norm

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

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 ‖v‖∞=maxi(|v(i)|).

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

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 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 ofA 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 X˜ of the data: MAD=median(|x−X˜|). 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.

Thenormalize 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 methodsnormalize(___,"zscore","robust"),normalize(___,"scale","mad"),normalize(___,"scale","iqr"),normalize(___,"center","median"), andnormalize(___,"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 andDataVariables 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

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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 parameterS, 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.