A Julia package for Cluster Validity Indices (CVIs).
Author AP6YC
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March 2021


A Julia package for Cluster Validity Indices (CVI) algorithms.

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This package is developed and maintained by Sasha Petrenko with sponsorship by the Applied Computational Intelligence Laboratory (ACIL). This project is supported by grants from the Night Vision Electronic Sensors Directorate, the DARPA Lifelong Learning Machines (L2M) program, Teledyne Technologies, and the National Science Foundation. The material, findings, and conclusions here do not necessarily reflect the views of these entities.

Please read the documentation for detailed usage and tutorials.

Table of Contents


This Julia project contains an outline of the conceptual usage of CVIs along with many example scripts. Quickstart provides an overview of how to use this project, while Structure outlines the project file structure, giving context to the locations of every component of the project. Usage outlines the general syntax and workflow of the ICVIs, while Authors gives credit to the author(s).


This section provides a quick overview of how to use the project. For more detailed code usage, please see Usage.

This project has several example scripts to demonstrate the functionality of CVIs in the ClusterValidityIndices.jl package. In ICVI-Examples/src/examples/, the scripts db.jl, ps.jl, and xb.jl demonstrate usage of the Davies-Boudin (DB), Partition Separation (PS), and Xie-Beni (XB) metrics, respectively.

NOTE Each of these scripts must be run at the top level of the project to correctly point to the datasets. For example, they can be run in the shell with

julia src/examples/db.jl

or in a Julia REPL session with


Three preprocessed datasets are provided under data/ to demonstrate the correct partitioning, over partitioning, and under partitioning of samples by a clustering algorithm to illustrate how the CVIs behave in each case. The data consists of 2000 samples of 2-element features with the clustering label appended in the third column. You can change which dataset is used in each script above.

Lastly, there is a large experiment script src/examples/combined.jl that runs every CVI with all three datasets. The common code for all scripts is contained under src/common.jl, while the experiment subroutines referenced in these scripts are under src/experiments.jl, so feel free to modify them to further explore the behavior and usage of these CVIs.


├── .github/workflows       // GitHub: workflows for testing and documentation.
├── data                    // Data: CI and example data location.
├── src                     // Source: scripts and common helper functions.
│   └───examples            //      Example scripts for CVI usage.
├── test                    // Test: unit, integration, and environment tests.
├── .gitignore              // Git: .gitignore for the whole project.
├── LICENSE                 // Doc: the license to the project.
├── Manifest.toml           // Julia: the explicit package versions used.
├── Project.toml            // Julia: the Pkg.jl dependencies of the project.
└──               // Doc: this document.


The usage of these CVIs requires an understanding of:


Because Julia is programmed in a column-major fashion, all CVIs make the assumption that the first dimension (columns) contains features, while the second dimension (rows) contains samples. This is more important for batch operation, as incremental operation accepts 1-D sample of features at each time step by definition.

For example,

# Load data from somewhere
data = load_data()
# The data shape is dimsion x samples
dim, n_samples = size(data)

NOTE: As of ClusterValidityIndices.jl v0.1.3, all the CVIs assume that the labels are presented sequentially initially, starting with index 1 (e.g., 1, 1, 2, 2, 3, 2, 2, 1, 3, 4, 4 ...). You may repeat previously seen label indices, but skipping label indices (e.g., 1, 2, 4) results in undefined behavior. In this project, this is ameliorated with the function

relabel_cvi_data(labels::Array{M, 1}) where {M<:Int}

For example,

data_file = "path/to/data.csv"
data, labels = get_cvi_data(data_file)
labels = relabel_cvi_data(labels)

Alternatively, you may pairwise sort the entirety of the data with

sort_cvi_data(data::Array{N, 2}, labels::Array{M, 1}) where {N<:Real, M<:Int}

NOTE* sort_cvi_data reorders the input data as well, which will lead to different ICVI results than with relabel_cvi_data.


The names of each CVI are capital abbreviations of their literature names, often based upon the surname of the principal authors of the papers that introduce the metrics. All CVIs are implemented with the default constructor, such as

cvi = DB()

Incremental vs. Batch

The CVIs in this project all contain incremental and batch implementations. When evaluated in incremental mode, they are often called ICVIs (incremental cluster validity indices). In documentation, CVI refers to both modalities (as in the literature), but in code, CVI means batch and ICVI means incremental.

The funtions that differ between the two modes are how they are updated

# Incremental
# Batch

and their respective porcelain functions

# Incremental
# Batch

They both compute their most recent criterion values with


NOTE: Any CVI can switch to be updated incrementally or in batch, as the CVI data structs are update mode agnostic.


The CVIs in this project all contain internal parameters that must be updated. Each update function modifies the CVI, so they use the Julia nomenclature convention of appending an exclamation point to indicate as much.

In both incremental and batch modes, the parameter update requires:

  • The CVI being updates
  • The sample (or array of samples)
  • The label(s) that was/were prescribed by the clustering algorithm to the sample(s)

More concretely, they are

# Incremental updating
param_inc!(cvi::C, sample::Array{T, 1}, label::I) where {C<:AbstractCVI, T<:Real, I<:Int}
# Batch updating
param_batch!(cvi::C, data::Array{T, 2}, labels::Array{I, 1}) where {C<:AbstractCVI, T<:Real, I<:Int}

Every CVI is a subtype of the abstract type AbstractCVI. For example, we may instantiate and load our data

cvi = DB()
data = load_data()
labels = get_cluster_labels(data)
dim, n_samples = size(data)

then update the parameters incrementally with

# Iterate over all samples
for ix = 1:n_samples
    sample = data[:, ix]
    label = labels[ix]
    param_inc!(cvi, sample, labels)

or in batch with

param_batch!(cvi, data, labels)

Furthermore, any CVI can alternate between being updated in incremental or batch modes, such as

# Create a new CVI
cvi_mixed = DB()

# Update on half of the data incrementally
i_split = n_samples/2
for ix = 1:i_split
    param_inc!(cvi, data[:, ix], labels[ix])

# Update on the other half all at once
param_batch!(cvi, data[:, (i_split+1):end])

Criterion Values

The CVI parameters are separate from the criterion values that they produce. This is partly because in batch mode computing the criterion value is only relevant at the last step, which eliminates unnecessarily computing it at every step. This is also provide granularity to the user that may only which to extract the criterion value occasionally during incremental mode.

Because the criterion values only depend on the internal CVI parameters, they are computed (and internally stored) with

evaluate!(cvi::C) where {C<:AbstractCVI}

To extract them, you must then simply grab the criterion value from the CVI struct with

criterion_value = cvi.criterion_value

For example, after loading the data

cvi = DB()
data = load_data()
labels = get_cluster_labels(data)
dim, n_samples = size(data)

we may extract and return the criterion value at every step with

criterion_values = zeros(n_samples)
for ix = 1:n_samples
    param_inc!(cvi, data[:, ix], labels[ix])
    criterion_values[ix] = cvi.criterion_value

or we may get it at the end in batch mode with

param_batch!(cvi, data, labels)
criterion_value = cvi.criterion_value


Taken from the git convention of calling low-level operations plumbing and high-level user-land functions porcelain, the package comes with a small set of porcelain function that do common operations all at once for the user.

For example, you may compute, evalute, and return the criterion value all at once with the functions

# Incremental
# Batch

Exactly as in the usage for updating the parameters, the functions take the cvi, sample(s), and clustered label(s) as input:

# Incremental
get_icvi!(cvi::C, x::Array{N, 1}, y::M) where {C<:AbstractCVI, N<:Real, M<:Int}
# Batch
get_cvi!(cvi::C, x::Array{N, 2}, y::Array{M, 1}) where {C<:AbstractCVI, N<:Real, M<:Int}

For example, after loading the data you may get the criterion value at each step with

criterion_values = zeros(n_samples)
for ix = 1:n_samples
    criterion_values[ix] = get_icvi!(cvi, data[:, ix], labels[ix])

or you may get the final criterion value in batch mode with

criterion_value = get_cvi!(cvi, data, labels)

Required Packages

Used By Packages