normalize
Normalize data
Syntax
Description
returns the vectorwise zscore of the data in N
= normalize(A
)A
with
center 0 and standard deviation 1.
If
A
is a vector, thennormalize
operates on the entire vectorA
.If
A
is a matrix, thennormalize
operates on each column ofA
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 ofA
separately.
specifies the type of normalization for the given method. For example,
N
= normalize(___,method
,methodtype
)normalize(A,"norm",Inf)
normalizes the data in
A
using the infinity norm.
uses the N
= normalize(___,"center",centertype
,"scale",scaletype
)"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)
.
specifies additional parameters for normalizing using one or more namevalue
arguments. For example, N
= normalize(___,Name,Value
)normalize(A,"DataVariables",datavars)
normalizes the variables specified by datavars
when
A
is a table or timetable.
Examples
Vector and Matrix Data
Normalize data in a vector and matrix by computing the zscore.
Create a vector v
and compute the zscore, 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 zscore 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 1norm.
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
___________ ___________ _________
01Jan2019 35 3
02Jan2019 38 20
03Jan2019 13 20
04Jan2019 38 10
05Jan2019 29 16
06Jan2019 13 2
07Jan2019 18 8
08Jan2019 26 19
09Jan2019 39 16
10Jan2019 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
___________ ___________ _________
01Jan2020 30 14
02Jan2020 11 0
03Jan2020 36 5
04Jan2020 38 0
05Jan2020 31 2
06Jan2020 33 17
07Jan2020 33 14
08Jan2020 22 6
09Jan2020 30 19
10Jan2020 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
___________ ___________ _________
01Jan2019 0.57687 1.4636
02Jan2019 0.856 0.92885
03Jan2019 1.4701 0.92885
04Jan2019 0.856 0.4785
05Jan2019 0.018609 0.36591
06Jan2019 1.4701 1.6044
07Jan2019 1.0049 0.75997
08Jan2019 0.26052 0.78812
09Jan2019 0.94905 0.36591
10Jan2019 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
___________ ___________ _________
01Jan2020 0.11165 0.084441
02Jan2020 1.6562 1.8858
03Jan2020 0.66992 1.1822
04Jan2020 0.856 1.8858
05Jan2020 0.2047 1.6044
06Jan2020 0.39078 0.50665
07Jan2020 0.39078 0.084441
08Jan2020 0.6327 1.0414
09Jan2020 0.11165 0.78812
10Jan2020 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
namevalue 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
___________ ___________ _________
01Jan2020 30 0.084441
02Jan2020 11 1.8858
03Jan2020 36 1.1822
04Jan2020 38 1.8858
05Jan2020 31 1.6044
06Jan2020 33 0.50665
07Jan2020 33 0.084441
08Jan2020 22 1.0414
09Jan2020 30 0.78812
10Jan2020 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 

 Compute the zscore. Center data to have mean 0, and scale data to have standard deviation 1. 
 Scale data by the 2norm, also known as the Euclidean norm. 
 Scale data to have standard deviation 1. 
 Rescale range of data to [0, 1]. 
 Center data to have mean 0. 
 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  twoelement row vector  type name
Method type, specified as an array, table, twoelement row vector, or type name, depending on the specified method.
Method  Method Type Options  Description 


 Compute the zscore. Center data to have mean 0, and scale data to have standard deviation 1. 
 Compute the zscore. Center data to have mean 0, and scale data to have median absolute deviation 1.  
 Positive numeric scalar (default is 2)  Scale data by the pnorm, where p is a positive numeric scalar. 
 Scale data by the pnorm, where
p is Inf . The
infinity norm, or maximum norm, is the same as the
largest magnitude of the elements in the data.  

 Scale data to have standard deviation 1. 
 Scale data to have median absolute deviation 1.  
 Scale data by the first element of the data.  
 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.  
 2element row vector (default is [0 1])  Rescale range of data to [a
b] , where a <
b . 

 Center data to have mean 0. 
 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)
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue 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  Examples 

Variable names:


Variable index:


Function handle:


Variable type:


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
or1
— Replace input table variables with table variables containing normalized data.false
or0
— 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
, thennormalize
operates on only the specified variables.If you use the syntax
normalize(T,"center",C,"scale",S)
to normalize a table or timetableT
using previously computed parametersC
andS
, thennormalize
automatically uses the variable names inC
andS
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 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 10by10 matrix of data
and normalize
operates along the first dimension, then
C
is a 1by10 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 10by10 matrix of data
and normalize
operates along the first dimension, then
S
is a 1by10 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
ZScore
zscores 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 zscore of a value x is $$z=\frac{\left(x\mu \right)}{\sigma}.$$ For sample data with mean $$\overline{X}$$ and standard deviation S, the zscore of a data point x is $$z=\frac{\left(x\overline{X}\right)}{S}.$$
PNorm
The general definition for the pnorm of a vector v that has N elements is
$${\Vert v\Vert}_{p}={\left[{\displaystyle \sum _{k=1}^{N}{\left{v}_{k}\right}^{p}}\right]}^{\text{\hspace{0.17em}}1/p}\text{\hspace{0.17em}},$$
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 1norm is the sum of the absolute values of the vector elements.
If p is 2, then the resulting 2norm gives the vector magnitude or Euclidean length of the vector.
If p is
Inf
, then $${\Vert v\Vert}_{\infty}={\mathrm{max}}_{i}\left(\leftv\left(i\right)\right\right)$$.
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 zscores 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
$${X}_{rescaled}=a+\left[\frac{X{\mathrm{min}}_{X}}{{\mathrm{max}}_{X}{\mathrm{min}}_{X}}\right]\left(ba\right)\text{\hspace{0.17em}}.$$
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 the median of the data is Q2, the median of the lower half of the data is Q1, and the median of the upper half of the data is Q3, then $$\text{IQR=Q3Q1}$$.
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: $$\text{MAD}=\text{median}\left(\leftx\tilde{X}\right\right)$$. 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.
Usage notes and limitations:
The outputs
C
andS
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
namevalue 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")
, andnormalize(___,"medianiqr")
.
For more information, see Tall Arrays.
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
The
ReplaceValues
namevalue argument is not supported.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.
ThreadBased Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports threadbased environments. For more information, see Run MATLAB Functions in ThreadBased Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. 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 R2018aR2022a: Append normalized values
You can now append, instead of replace, input table variables with table variables
containing normalized data by setting the ReplaceValues
namevalue argument to false
.
The ReplaceValues
namevalue 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.
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