Need help to understand LAMDA clustering algorithm

Need help to understand LAMDA clustering algorithm, taken from some article.

Here is the algorithm from the article

LAMDA is a conceptual clustering and classification methodology that computes the degree of adequation of an object to a class with all the partial or marginal information available . The difference between this algorithm and the classical clustering and classification approaches is that LAMDA models the total indistinguishability or homogeneity inside the context or universe from which the information is extracted. This is done by means of a special class, the so called non-informative class (NIC), which accepts all objects under the same status. Therefore, the adequation degree of the objects of NIC acts as a minimum threshold to assign an element to a significant class. Hence, the minimum threshold is not fixed arbitrarily but is auto- matically determined by the proper context. Algorithm 1 summarizes the main steps of LAMDA clustering. Given a set of objects X = { $\vec{x_1}$, $\vec{x_2}$,..., $\vec{x_n}$}, where an object is represented by an m-dimensional vector $\vec{x_k}$ = {$\vec{x_1^k}$, $\vec{x_2^k}$, ..., $\vec{x_m^k}$}, the algorithm starts by creating an initial class consisting of one of the objects selected at random.

For each remaining object $\vec{x_k}$ and for each existing class $C_j$, LAMDA computes for every descriptor the so-called marginal adequacy degree $MAD_{ij}$ ($x^k_i$ ) between the values that the ith descriptor takes over $\vec{x_k}$ and the class $C_j$ . Thus, a vector $MAD_j$($\vec{x_k}$) can be associated with object $\vec{x_k}$ for each class $C_j$ . $MAD_j$($\vec{x_k}$) is a membership function derived from a fuzzy generalization of a binomial probability law, as expressed in the algorithm. In the expression, ν ($x^k_i$ , $c_ij$ ) is a distance function between the descriptor $x^k_i$ and the attribute $c_{ij}$ of the center of the class $C_j$ ; $ρ_{ij}$ is the possibility of the descriptor $x^k_i$ to belong to class $C_j$ .

Algorithm: Data clustering with LAMDA

Input: A set of data objects X = { $\vec{x_1}$, $\vec{x_2}$,..., $\vec{x_n}$}, where $\vec{x_k}$ = {$\vec{x_1^k}$, $\vec{x_2^k}$, ..., $\vec{x_m^k}$}

Output: Γ = {$C_j$} set of classes where each class $C_j$ is represented by the parameters $c_{ij}$ and $ρ_{ij}$

Initialization: ν($x_i^k$) = 1 - $||$$x_i^k - c_{ij}$$||$ and $ρ_{init}$ = 0.5, α ∈ ]0,1[, $T_{norm}$, $T_{conorm}$ and $C$ = 1

begin

for $k$ ← 1 to $n$ do

for $j$ ← 1 to $C$ do

for $i$ ← 1 to $m$ do

$MAD_{ij}$($x_i^k$) = $ρ_{ij}^{ν(x_i^k,c_{ij})}*(1 - ρ_{ij})^{1 - ν(x_i^k, c_{ij})}$

end

$MAD_j$($\vec{x_k}$) = {$MAD_{ij}$($\vec{x_i^k}$) | 1 ≤ i ≤ m}

$GAD_j$($\vec{x_k}$) = $L_α$($MAD_j$($\vec{x_k}$)) = α × $T_{norm}$($MAD_j$($\vec{x_k}$))+(1 − α) × $T_{conorm}$($MAD_j$($\vec{x_k}$))

end

j ← arg $max_{1≤l≤C}$($GAD_l$ ($\vec{x_k}$))

if ($GAD_j$($\vec{x_k}$) > 0.5) then

//1 − Affect object $\vec{x_k}$ to class j

$\vec{x_k}$ → $C_j$

//2 − Update parameters ρ and c for class $C_j$

for $i$ ← 1 to $m$ do

$\sum_{i=0}^m$(δ/δ$c_{ij}$ )ν($\vec{x_k}$,$\hat{c}_{ij}$ )=0

$\hat{ρ}_{ij}$ =1/n $\sum_{i=0}^n$ν($x^k_i$,$\hat{c}_{ij}$)

end

else

//Create a new class

Γ ← Γ ∪ {$C_j$}

C ← C + 1

//Initialize the new class parameters

$ρ_{ij}$ = $ρ_{init}$

$c_{ij}$=$x^k_i$

end

end

return Γ

end

When computing $GAD_j$ ($\vec{x_k}$ ), if the value is smaller or equal to 0.5, the object is considered as part of NIC and automatically assigned as the first element of the new class as a result. Otherwise, after comput- ing the GAD values corresponding to all the classes, the object will be assigned to the class with the greatest GAD value. Using sample data, the $ρ_{ij}$ and $c_{ij}$ values for each class are estimated by minimizing a maximum likelihood criterion as expressed in the algorithm.

Question 1:

What is ${c}_{ij}$ in this article? Is it center of the cluster?

Question 2:

What is $x_i^k$? Is it i-th component of k-th vector (scalar value)? If so, what does norm operation mean in the following statement:

ν($x_i^k$) = 1 - $||$$x_i^k - c_{ij}$$||$

Question 3:

There is a statement in article:

Using sample data, the $ρ_{ij}$ and $c_{ij}$ values for each class are estimated by minimizing a maximum likelihood criterion as expressed in the algorithm.

It points out on the following statements:

for $i$ ← 1 to $m$ do

$\sum_{i=0}^m$(δ/δ$c_{ij}$ )ν($\vec{x_k}$,$\hat{c}_{ij}$ )=0

$\hat{ρ}_{ij}$ =1/n $\sum_{i=0}^n$ν($x^k_i$,$\hat{c}_{ij}$)

3.1 What does hat symbol in these statements mean?

3.2 There is cycle by i and sum by i (in article). How could it be? Is it a typo?

3.3 How would I calculate ${c}_{ij}$ and ${ρ}_{ij}$? Need some help with turning these statements into pseudocode.

It would be useful to know your reference. This algorithm is not popular and there is a lack of details on its original creation (J Aguilar-Martin and R De Mantaras. 1982. The process of classification and learning the meaning of linguistic descriptors of concepts. Approximate Reasoning in Decision Analysis, 1982:165–175) and on modifications to it. I've implemented LAMDA years ago following recommendations from Tatiana Kempowsky.

The literals you have questions are diferent from the original paper, but I'll try to say something.

1. $c_{ij}$ is the $i-$th component of the centroid of the $j-$th class/cluster.

2. $x_{i}^k\in\mathbb{R}$ is the $i-$th component of the $k-$th input sample. Thus, the norm operation is over real values (component-wise), so it is the absolute value actually.

3. Here comes a different thing w.r.t the original paper: $v(x_{i}^k)$ is a modification, so $MAD_{ij}$ is also different. The original version is: $$MAD_{ij}=\rho_{ij}^{x_{i}^k}(1-\rho_{ij})^{1-x_{i}^k},$$
which is a fuzzy membership function, rather than a probability density function. There are different membership functions, not only the binomial. The notation also changes. In the version you are showing, $c_{ij}\ne \rho_{ij}$, however, in the original version $\rho_{ij}$ is the $i-$th component of thecentroid of the $j-$th class/cluster. $c_{ij}$ does not exist.

I'm not sure what does $\delta$ mean. It probably is used in the supervised learning version of the algorithm, which I don't know. Also I think there is not likelihood criterion. The original version of the algorithm uses an online mean for updating the centroid of each cluster: $$\hat{\rho}_j=\rho_j+\frac{\rho_j-x^k}{1+N_j}$$ where $\hat{\rho}_j\in\mathbb{R}^n$ is the new (updated) centroid, $x^k\in\mathbb{R}^n$ is the $k-$th (current) sample that the $GAD$ has decided to belong to the $j-$ class. $N_j$ is the number of samples that already are members of this class.

At the beginning of the online processing of samples, only the NIC class does exist. Its centroid is $\rho_{ij}=\rho_{i,NIC}=0.5$ for $i=1,\dots,n$. The fisrt sample ($k=1$) belongs to it, so the algorithm creates a new centroid by means of the online mean (last equation above), where the resulting $\hat{\rho}_j$ becomes the centroid of a new cluster, and so on for the next samples, but they are now compared with the existent $\rho_j$, inlcuding $\rho_{i,NIC}$, as they are already created.

I've worked the LAMDA algorithm since 2009 and I recommend reading the following paper where my brother and me propose a novel LAMDA algorithm:

Javier Fernando Botía Valderrama, Diego José Luis Botía Valderrama, On LAMDA clustering method based on typicality degree and intuitionistic fuzzy sets, Expert Systems with Applications, Volume 107, 2018, Pages 196-221.