# uniquetol

Unique values within tolerance

## Description

example

C = uniquetol(A,tol) returns the unique elements in A using tolerance tol. Two values, u and v, are within tolerance if

abs(u-v) <= tol*max(abs(A(:)))

That is, uniquetol scales the tol input based on the magnitude of the data.

uniquetol is similar to unique. Whereas unique performs exact comparisons, uniquetol performs comparisons using a tolerance.

example

C = uniquetol(A) uses a default tolerance of 1e-6 for single-precision inputs and 1e-12 for double-precision inputs.

C = uniquetol(A,tol,occurrence), where occurrence is 'highest', specifies that when several values are within tolerance of each other, the highest value should be selected as being unique. The default for occurrence is 'lowest', which selects the lowest value as being unique.

example

[C,IA,IC] = uniquetol(___) returns index vectors IA and IC, such that C = A(IA) and A~C(IC) (or A(:)~C(IC) if A is a matrix), where ~ means the values are within tolerance of each other. You can use any of the input arguments in previous syntaxes.

example

[___] = uniquetol(___,Name,Value) uses additional options specified by one or more Name-Value pair arguments using any of the input or output argument combinations in previous syntaxes. For example, uniquetol(A,'ByRows',true) determines the unique rows in A.

## Examples

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Create a vector x. Obtain a second vector y by transforming and untransforming x. This transformation introduces round-off differences in y.

x = (1:6)'*pi;
y = 10.^log10(x);

Verify that x and y are not identical by taking the difference.

x-y
ans = 6×1
10-14 ×

0.0444
0
0
0
0
-0.3553

Use unique to find the unique elements in the concatenated vector [x;y]. The unique function performs exact comparisons and determines that some values in x are not exactly equal to values in y. These are the same elements that have a nonzero difference in x-y. Thus, c contains values that appear to be duplicates.

c = unique([x;y])
c = 8×1

3.1416
3.1416
6.2832
9.4248
12.5664
15.7080
18.8496
18.8496

Use uniquetol to perform the comparison using a small tolerance. uniquetol treats elements that are within tolerance as equal.

C = uniquetol([x;y])
C = 6×1

3.1416
6.2832
9.4248
12.5664
15.7080
18.8496

By default, uniquetol looks for unique elements that are within tolerance, but it also can find unique rows of a matrix that are within tolerance.

Create a numeric matrix, A. Obtain a second matrix, B, by transforming and untransforming A. This transformation introduces round-off differences to B.

A = [0.05 0.11 0.18; 0.18 0.21 0.29; 0.34 0.36 0.41; 0.46 0.52 0.76];
B = log10(10.^A);

Use unique to find the unique rows in A and B. The unique function performs exact comparisons and determines that all of the rows in the concatenated matrix [A;B] are unique, even though some of the rows differ by only a small amount.

unique([A;B],'rows')
ans = 8×3

0.0500    0.1100    0.1800
0.0500    0.1100    0.1800
0.1800    0.2100    0.2900
0.1800    0.2100    0.2900
0.3400    0.3600    0.4100
0.3400    0.3600    0.4100
0.4600    0.5200    0.7600
0.4600    0.5200    0.7600

Use uniquetol to find the unique rows. uniquetol treats rows that are within tolerance as equal.

uniquetol([A;B],'ByRows',true)
ans = 4×3

0.0500    0.1100    0.1800
0.1800    0.2100    0.2900
0.3400    0.3600    0.4100
0.4600    0.5200    0.7600

Create a vector, x. Obtain a second vector, y, by transforming and untransforming x. This transformation introduces round-off differences to some elements in y.

x = (1:5)'*pi;
y = 10.^log10(x);

Combine x and y into a single vector, A. Use uniquetol to reconstruct A, treating the values that are within tolerance as equal.

A = [x;y]
A = 10×1

3.1416
6.2832
9.4248
12.5664
15.7080
3.1416
6.2832
9.4248
12.5664
15.7080

[C,IA,IC] = uniquetol(A);
newA = C(IC)
newA = 10×1

3.1416
6.2832
9.4248
12.5664
15.7080
3.1416
6.2832
9.4248
12.5664
15.7080

You can use newA with == or functions that use exact equality like isequal or unique in subsequent code.

D1 = unique(A)
D1 = 6×1

3.1416
3.1416
6.2832
9.4248
12.5664
15.7080

D2 = unique(newA)
D2 = 5×1

3.1416
6.2832
9.4248
12.5664
15.7080

Use the occurrence option to control which elements uniquetol selects as being unique.

Create a vector and find which elements are unique within a tolerance of 1e-1.

a = [1 1.1 1.11 1.12 1.13 2];
c = uniquetol(a,1e-1)
c = 1×2

1     2

Since the first five elements in A all have similar values with respect to the tolerance of 1e-1, only the lowest value among them is selected as being unique. This is because uniquetol begins with the lowest value in a and does not find a new element that is not within tolerance until the 2 at the end of the vector.

Use the 'highest' option to specify that uniquetol should begin with the highest value in a. Now, the 1.13 element is selected as being unique since uniquetol works down from the highest values.

d = uniquetol(a,1e-1,'highest')
d = 1×2

1.1300    2.0000

Create a cloud of 2-D sample points constrained to be inside a circle of radius 0.5 centered at the point $\left(\frac{1}{2},\frac{1}{2}\right)$.

x = rand(10000,2);
insideCircle = sqrt((x(:,1)-.5).^2+(x(:,2)-.5).^2)<0.5;
y = x(insideCircle,:);

Find a reduced set of points, such that each point of the original dataset is within tolerance of a point.

tol = 0.05;
C = uniquetol(y,tol,'ByRows',true);

Plot the reduced set of points as red dots on top of the original data set. The red dots are all members of the original data set. All the red dots are at least a distance tol apart.

plot(y(:,1),y(:,2),'.')
hold on
axis equal
plot(C(:,1), C(:,2), '.r', 'MarkerSize', 10)

Create a vector of random numbers and determine the unique elements using a tolerance. Specify OutputAllIndices as true to return all of the indices for the elements that are within tolerance of the unique values.

A = rand(100,1);
[C,IA] = uniquetol(A,1e-2,'OutputAllIndices',true);

Find the average value of the elements that are within tolerance of the value C(2).

C(2)
ans = 0.0318
allA = A(IA{2})
allA = 3×1

0.0357
0.0318
0.0344

aveA = mean(allA)
aveA = 0.0340

By default, uniquetol uses a tolerance test of the form abs(u-v) <= tol*DS, where DS automatically scales based on the magnitude of the input data. You can specify a different DS value to use with the DataScale option. However, absolute tolerances (where DS is a scalar) do not scale based on the magnitude of the input data.

First, compare two small values that are a distance eps apart. Specify tol and DS to make the within tolerance equation: abs(u-v) <= 10^-6.

x = 0.1;
uniquetol([x, exp(log(x))], 10^-6, 'DataScale', 1)
ans = 0.1000

Next, increase the magnitude of the values. The round-off error in the calculation exp(log(x)) is proportional to the magnitude of the values, specifically to eps(x). Even though the two large values are a distance eps from one another, eps(x) is now much larger. Therefore, 10^-6 is no longer a suitable tolerance.

x = 10^10;
uniquetol([x, exp(log(x))], 10^-6, 'DataScale', 1)
ans = 1×2
1010 ×

1.0000    1.0000

Correct this issue by using the default (scaled) value of DS.

format long
Y = [0.1 10^10];
uniquetol([Y, exp(log(Y))])
ans = 1×2
1010 ×

0.000000000010000   1.000000000000000

Create a set of random 2-D points, then use uniquetol to group the points into vertical bands that have a similar (within tolerance) x-coordinate. Use these options with uniquetol:

• Specify ByRows as true since the point coordinates are in the rows of A.

• Specify OutputAllIndices as true to return the indices for all points that have an x-coordinate within tolerance of each other.

• Specify DataScale as [1 Inf] to use an absolute tolerance for the x-coordinate while ignoring the y-coordinate.

A = rand(1000,2);
DS = [1 Inf];
[C,IA] = uniquetol(A, 0.1, 'ByRows', true, ...
'OutputAllIndices', true, 'DataScale', DS);

Plot the points and average value for each band.

hold on
for k = 1:length(IA)
plot(A(IA{k},1), A(IA{k},2), '.')
meanAi = mean(A(IA{k},:));
plot(meanAi(1), meanAi(2), 'xr')
end

## Input Arguments

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Query array, specified as a scalar, vector, matrix, or multidimensional array. A must be full.

Data Types: single | double

Comparison tolerance, specified as a positive, real scalar. uniquetol scales the tol input using the maximum absolute value in input array A. Then uniquetol uses the resulting scaled comparison tolerance to determine which elements in A are unique. If two elements in A are within tolerance of each other, then uniquetol considers them to be equal.

Two values, u and v, are within tolerance if abs(u-v) <= tol*max(abs(A)).

To specify an absolute tolerance, specify both tol and the 'DataScale' Name-Value pair.

Example: tol = 0.05

Example: tol = 1e-8

Example: tol = eps

Data Types: single | double

Occurrence of unique values, specified as one of the options in this table. The value of occurrence determines which elements uniquetol selects as being unique.

OptionDescription
'lowest'

uniquetol starts with the lowest value in A, then repeatedly finds the next lowest value in A that is not within tolerance of any other value in C. The effect of this is that when several input values are within tolerance of each other, the lowest one is selected as being unique.

'highest'

uniquetol starts with the highest value in A, then repeatedly finds the next highest value in A that is not within tolerance of any other value in C. The effect of this is that when several input values are within tolerance of each other, the highest one is selected as being unique.

Example: C = uniquetol(A,tol,'highest')

Example: C = uniquetol([1 2 3],2,'highest') returns 3, since uniquetol starts with the highest value in the input and all other values are within tolerance.

Data Types: char | string

### 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.

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

Example: C = uniquetol(A,'ByRows',true)

Output index type, specified as the comma-separated pair consisting of 'OutputAllIndices' and either false (default), true, 0, or 1. uniquetol interprets numeric 0 as false and numeric 1 as true.

When OutputAllIndices is true, the uniquetol function returns the second output, IA, as a cell array. The cell array contains the indices for all elements in A that are within tolerance of a value in C. That is, each cell in IA corresponds to a value in C, and the values in each cell correspond to locations in A.

Example: [C,IA] = uniquetol(A,tol,'OutputAllIndices',true)

Row comparison toggle, specified as the comma-separated pair consisting of 'ByRows' and either false (default), true, 0, or 1. uniquetol interprets numeric 0 as false and numeric 1 as true. Use this option to find rows in A that are unique, within tolerance.

When ByRows is true:

• A must be a 2-D array.

• uniquetol compares the rows of A by considering each column separately. For two rows to be within tolerance of one another, each column has to be in tolerance.

• Each row in A is within tolerance of a row in C. However, no two rows in C are within tolerance of each other.

Two rows, u and v, are within tolerance if all(abs(u-v) <= tol*max(abs(A),[],1)).

Example: C = uniquetol(A,tol,'ByRows',true)

Scale of data, specified as the comma-separated pair consisting of 'DataScale' and either a scalar or vector. Specify DataScale as a numeric scalar, DS, to change the tolerance test to be abs(u-v) <= tol*DS.

When used together with the ByRows option, the DataScale value also can be a vector. In this case, each element of the vector specifies DS for a corresponding column in A. If a value in the DataScale vector is Inf, then uniquetol ignores the corresponding column in A.

Example: C = uniquetol(A,'DataScale',1)

Example: [C,IA,IC] = uniquetol(A,'ByRows',true,'DataScale',[eps(1) eps(10) eps(100)])

Data Types: single | double

Range preservation toggle, specified as the comma-separated pair consisting of 'PreserveRange' and either false (default), true, 0, or 1. uniquetol interprets numeric 0 as false and numeric 1 as true. Use this option to specify that the minimum and maximum values in the output C should be the same as those in A.

If the minimum and maximum values in A are within the tolerance tol of each other, then uniquetol returns only one of the values.

Example: C = uniquetol(A,tol,'PreserveRange',true)

## Output Arguments

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Unique elements in A (within tolerance), returned as a vector or matrix. If A is a row vector, then C is also a row vector. Otherwise, C is a column vector. The elements in C are sorted in ascending order. Each element in A is within tolerance of an element in C, but no two elements in C are within tolerance of each other.

If the ByRows option is true, then C is a matrix containing the unique rows in A. In this case, the rows in C are sorted in ascending order by the first column. Each row in A is within tolerance of a row in C, but no two rows in C are within tolerance of each other.

Index to A, returned as a column vector of indices to the first occurrence of repeated elements, or as a cell array. IA generally satisfies C = A(IA), with the following exceptions:

• If the ByRows option is true, then C = A(IA,:).

• If the OutputAllIndices option is true, then IA is a cell array and C(i)~A(IA{i}) where ~ means the values are within tolerance of each other.

Index to C, returned as a column vector of indices. IC satisfies the following properties, where ~ means the values are within tolerance of each other.

• If A is a vector, then A~C(IC).

• If A is a matrix, then A(:)~C(IC).

• If the ByRows option is true, then A~C(IC,:).

## Algorithms

uniquetol sorts the input lexicographically, and then starts at the lowest or highest value to find unique values within tolerance. As a result, changing the sorting of the input could change the output. For example, uniquetol(-A) might not give the same results as -uniquetol(A).

There can be multiple valid C outputs that satisfy the condition, no two elements in C are within tolerance of each other. The uniquetol function can return several of the valid outputs, depending on whether the value of occurrence is 'highest' or 'lowest' and whether the PreserveRange option is specified.

## Version History

Introduced in R2015a

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