Dynamic clustering of interval data based on hybrid $L_q$ distance

Abstract

Dynamic clustering defines partitions within data and prototypes to each partition. Distance metrics are responsible for checking the closeness between instances and prototypes. Considering the literature about interval data, distances depend on interval bounds and the information inside the intervals is ignored. This paper proposes new distances, which explore the information inside of intervals. It also presents a mapping of intervals to points, which preserves their spatial location and internal variation. We formulate a new hybrid distance for interval data based on the well-known $L_q$ distance for point data. This new distance allows for a weighted formulation of the hybridism. Hence, we propose a Hybrid $L_q$ distance, a Weighted Hybrid $L_q$ distance, as well as the adaptive version of the Hybrid $L_q$ distance for interval data. Experiments with synthetic and real interval data sets illustrate the usefulness of the hybrid approach to improve dynamic clustering for interval data.

Appendices

A Proof of Proposition 1

Fixing cluster k and dimension j, the hybrid weights of the $WHL_q$ distance are obtained using Lagrange multipliers under the restrictions: $w_{k,1}^j + w_{k,2}^j = 1$; $w_{k,1}^j \ge 0$; $w_{k,2}^j \ge 0$; and $t > 1$. Let

$$\begin{aligned} \xi _{k,1}^j = \sum _{\gamma _n \in C_k} | \underline{\gamma }_n^j -\underline{g}_k^j |^q \quad \text{ and } \quad \,\, \xi _{k,2}^j =\sum _{\gamma _n \in C_k} | \breve{\gamma }_n^j - \breve{g}_k^j |^q. \end{aligned}$$

The Hybrid weight values are computed by:

$$\begin{aligned} w_{k,1}^j = \left\{ 1 + \left( \frac{\xi _{k,1}^j}{\xi _{k,2}^j} \right) ^{\frac{1}{t-1}} \right\} ^{-1} \text{ and } \,\,\, w_{k,2}^j = \left\{ 1 + \left( \frac{\xi _{k,2}^j}{\xi _{k,1}^j} \right) ^{\frac{1}{t-1}} \right\} ^{-1}. \end{aligned}$$

Proof

The partitional dynamic clustering criterion for the $WHL_q$ distance is given by

$$\begin{aligned} J_{d_{WHLq}} = \sum _{k=1}^K \sum _{\gamma _n \in C_k} \sum _{j=1}^p \left\{ (w_{k,1}^j)^t \, |\underline{\gamma }_n^j -\underline{g}_k^j|^q + (w_{k,2}^j)^t \, |\breve{\gamma }_n^j-\breve{g}_k^j|^q \right\} , \end{aligned}$$

(28)

under the restrictions: $w_{k,1}^j + w_{k,2}^j = 1$, $w_{k,1}^j \ge 0$, $w_{k,2}^j \ge 0$ and $t > 1$. The solution can be found using Lagrange multipliers. Let $J_{d_{WHLq}}(\varLambda _1^1,\ldots , \varLambda _K^p)$ be the version of Eq. (28) with the Lagrange multipliers ($\varLambda _k^j$) associated restrictions. Thus, it becomes

$$\begin{aligned} J_{d_{WHLq}}(\varLambda _1^1,\ldots , \varLambda _K^p)= & {} \sum _{k=1}^K \sum _{\gamma _n \in C_k} \sum _{j=1}^p \left\{ (w_{k,1}^j)^t \, |\underline{\gamma }_n^j-\underline{g}_k^j|^q + (w_{k,2}^j)^t \, |\breve{\gamma }_n^j-\breve{g}_k^j|^q \right\} \nonumber \\&- \sum _{k=1}^K \sum _{j=1}^p \left\{ \varLambda _k^j \, (w_{k,1}^j + w_{k,2}^j - 1) \right\} . \end{aligned}$$

(29)

Weights can be found when the partial derivatives of $J_{d_{WHLq}}$ are equal to 0. Fixing cluster k and dimension j, deriving $J_{d_{WHLq}}$ according to the first weight component ($w_{k,1}^j$), we get

$$\begin{aligned}&\frac{\partial J_{d_{WHLq}}(\varLambda _1^1,\ldots , \varLambda _K^p)}{\partial w_{k,1}^j} = \sum _{\gamma _n \in C_k} \left\{ t \, (w_{k,1}^j)^{t-1} \, |\underline{\gamma }_n^j -\underline{g}_k^j|^q \right\} - \varLambda _k^j = 0 \end{aligned}$$

(30)

$$\begin{aligned}&t \, (w_{k,1}^j)^{t-1} \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j-\underline{g}_k^j|^q - \varLambda _k^j = 0. \end{aligned}$$

(31)

Defining

$$\begin{aligned} \xi _{k,1}^j = \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j-\underline{g}_k^j|^q, \end{aligned}$$

(32)

and isolating the $w_{k,1}^j$ term, we get

$$\begin{aligned} t \, (w_{k,1}^j)^{t-1} \xi _{k,1}^j - \varLambda _k^j = 0 \Longrightarrow w_{k,1}^j = \left( \frac{\varLambda _k^j}{t \, \xi _{k,1}^j } \right) ^{\frac{1}{t-1}}. \end{aligned}$$

(33)

Now, deriving $J_{d_{WHLq}}$ according to the second weight component ($w_{k,2}^j$), we obtain

$$\begin{aligned}&\displaystyle \frac{\partial J_{d_{WHLq}}(\varLambda _1^1,\ldots , \varLambda _K^p)}{\partial w_{k,2}^j} = \sum _{\gamma _n \in C_k} \left\{ t \, (w_{k,2}^j)^{t-1} \, |\breve{\gamma }_n^j -\breve{g}_k^j|^q \right\} - \varLambda _k^j = 0 \end{aligned}$$

(34)

$$\begin{aligned}&\displaystyle t \, (w_{k,2}^j)^{t-1} \sum _{\gamma _n \in C_k} |\breve{\gamma }_n^j-\breve{g}_k^j|^q - \varLambda _k^j = 0. \end{aligned}$$

(35)

Defining

$$\begin{aligned} \xi _{k,2}^j = \sum _{\gamma _n \in C_k} |\breve{\gamma }_n^j-\breve{g}_k^j|^q, \end{aligned}$$

(36)

we get $w_{k,2}^j$, as follows:

$$\begin{aligned} t \, (w_{k,2}^j)^{t-1} \xi _{k,2}^j - \varLambda _k^j = 0 \Longrightarrow w_{k,2}^j = \left( \frac{\varLambda _k^j}{t \, \xi _{k,2}^j } \right) ^{\frac{1}{t-1}}. \end{aligned}$$

(37)

With the expressions above, used to find weights, we compute the Lagrange multiplier ($\varLambda _k^j$) using the restriction $w_{k,1}^j + w_{k,2}^j =1$. Then,

$$\begin{aligned}&\displaystyle w_{k,1}^j + w_{k,2}^j = 1 \end{aligned}$$

(38)

$$\begin{aligned}&\displaystyle \left( \frac{\varLambda _k^j}{t \, \xi _{k,1}^j } \right) ^{\frac{1}{t-1}} + \left( \frac{\varLambda _k^j}{t \, \xi _{k,2}^j } \right) ^{\frac{1}{t-1}} = 1 \end{aligned}$$

(39)

$$\begin{aligned}&\displaystyle \varLambda _k^j = \left\{ \left( \frac{1}{t \, \xi _{k,1}^j } \right) ^{\frac{1}{t-1}} + \left( \frac{1}{t \, \xi _{k,2}^j } \right) ^{\frac{1}{t-1}}\right\} ^{-(t-1)} \end{aligned}$$

(40)

Replacing Eq. (40)) in Eq. (33)), we get

$$\begin{aligned} w_{k,1}^j = \left( \frac{\left\{ \left( \frac{1}{t \, \xi _{k,1}^j } \right) ^{\frac{1}{t-1}} + \left( \frac{1}{t \, \xi _{k,2}^j } \right) ^{\frac{1}{t-1}}\right\} ^{-(t-1)} }{(t \, \xi _{k,1}^j) }\right) ^{\frac{1}{t-1}} = \left\{ 1+ \left( \frac{\xi _{k,1}^j}{\xi _{k,2}^j} \right) ^{\frac{1}{t-1}}\right\} ^{-1}. \end{aligned}$$

(41)

Now, replacing Eq. (40)) in Eq. (37)), we get

$$\begin{aligned} w_{k,2}^j = \left( \frac{\left\{ \left( \frac{1}{t \, \xi _{k,1}^j } \right) ^{\frac{1}{t-1}} + \left( \frac{1}{t \, \xi _{k,2}^j } \right) ^{\frac{1}{t-1}}\right\} ^{-(t-1)} }{(t \, \xi _{k,2}^j) }\right) ^{\frac{1}{t-1}} = \left\{ 1+ \left( \frac{\xi _{k,2}^j}{\xi _{k,1}^j} \right) ^{\frac{1}{t-1}}\right\} ^{-1}. \end{aligned}$$

(42)

So, the hybrid weights can be computed using Eqs. (41) and (42). $\square$

B Proof of Proposition 2

Fixing cluster k, the hybrid weights of the $WHL_{\infty }$ distance are obtained using Lagrange multipliers under the following restrictions: $w_{k,1} + w_{k,2} = 1$; $w_{k,1} \ge 0$; $w_{k,2} \ge 0$; and $t > 1$. Let

$$\begin{aligned} \xi _{k,1} = \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ | \underline{\gamma }_n^j - \underline{g}_k^j |\right\} \, \text{ and } \, \xi _{k,2} = \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ | \breve{\gamma }_n^j - \breve{g}_k^j |\right\} . \end{aligned}$$

Then,

$$\begin{aligned} w_{k,1} = \left\{ 1 + \left( \frac{\xi _{k,1}}{\xi _{k,2}} \right) ^{\frac{1}{t-1}} \right\} ^{-1} \, \text{ and } \, \, w_{k,2} = \left\{ 1 + \left( \frac{\xi _{k,2}}{\xi _{k,1}} \right) ^{\frac{1}{t-1}} \right\} ^{-1}. \end{aligned}$$

Proof

The partitional dynamic clustering criterion for the $WHL_{\infty }$ distance is given by

$$\begin{aligned} J_{d_{WHLq}} = \sum _{k=1}^K \sum _{\gamma _n \in C_k} \left\{ (w_{k,1})^t \max _{j=1}^p \left\{ |\underline{\gamma }_n^j-\underline{g}_k^j|\right\} +(w_{k,2})^t\max _{j=1}^p \left\{ |\breve{\gamma }_n^j -\breve{g}_k^j| \right\} \right\} \end{aligned}$$

(43)

under the restrictions: $w_{k,1} + w_{k,2} = 1$; $w_{k,1} \ge 0$; $w_{k,2} \ge 0$; and $t > 1$. This solution can be computed using Lagrange multipliers. Eq. (43) is rewritten to incorporate Lagrange multipliers ($\varLambda _k$) and its respective restrictions. It then becomes

$$\begin{aligned} J_{d_{WHLq}}(\varLambda _1, \ldots , \varLambda _K)= & {} \sum _{k=1}^K \sum _{\gamma _n \in C_k} \left( (w_{k,1})^t \max _{j=1}^p \left\{ |\underline{\gamma }_n^j-\underline{g}_k^j| \right\} \right. \nonumber \\&+\left. (w_{k,2})^t \max _{j=1}^p \left\{ |\breve{\gamma }_n^j-\breve{g}_k^j| \right\} \right) -\sum _{k=1}^K \varLambda _k (w_{k,1} + w_{k,2} - 1)\qquad \end{aligned}$$

(44)

Weights can be found when partial derivatives of $J_{d_{WHLq}}$, with respect to the weights, are equal to 0. For fixed cluster k and dimension j, deriving $J_{d_{WHLq}}$ according to the first weight component ($w_{k,1}$), we get

$$\begin{aligned}&\displaystyle \frac{\partial J_{d_{WHLq}}(\varLambda _1, \ldots , \varLambda _K)}{\partial w_{k,1}} = \sum _{\gamma _n \in C_k} t \, (w_{k,1})^{t-1} \, \max _{j=1}^p \left\{ |\underline{\gamma }_n^j-\underline{g}_k^j| \right\} - \varLambda _k = 0 \end{aligned}$$

(45)

$$\begin{aligned}&\displaystyle t \, (w_{k,1})^{t-1} \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\underline{\gamma }_n^j-\underline{g}_k^j| \right\} - \varLambda _k = 0. \end{aligned}$$

(46)

Defining

$$\begin{aligned} \xi _{k,1} = \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\underline{\gamma }_n^j-\underline{g}_k^j| \right\} , \end{aligned}$$

(47)

we get

$$\begin{aligned} t \, (w_{k,1})^{t-1} \xi _{k,1} - \varLambda _k = 0 \Longrightarrow w_{k,1} = \left( \frac{\varLambda _k}{t \, \xi _{k,1}^j } \right) ^{\frac{1}{t-1}}. \end{aligned}$$

(48)

Now, deriving $J_{d_{WHLq}}$ according to the second weight component ($w_{k,2}$), we get

$$\begin{aligned}&\displaystyle \frac{\partial J_{d_{WHLq}}(\varLambda _1, \ldots , \varLambda _K)}{\partial w_{k,2}} = \sum _{\gamma _n \in C_k} t \, (w_{k,2})^{t-1} \max _{j=1}^p \left\{ |\breve{\gamma }_n^j-\breve{g}_k^j| \right\} - \varLambda _k = 0 \end{aligned}$$

(49)

$$\begin{aligned}&\displaystyle t \, (w_{k,2})^{t-1} \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\breve{\gamma }_n^j-\breve{g}_k^j| \right\} - \varLambda _k = 0. \end{aligned}$$

(50)

Defining

$$\begin{aligned} \xi _{k,2} = \sum _{\gamma _n \in C_k}\max _{j=1}^p \left\{ |\breve{\gamma }_n^j-\breve{g}_k^j| \right\} , \end{aligned}$$

(51)

it becomes

$$\begin{aligned} t \, (w_{k,2})^{t-1} \xi _{k,2} - \varLambda _k = 0 \Longrightarrow w_{k,2} = \left( \frac{\varLambda _k}{t \, \xi _{k,2}} \right) ^{\frac{1}{t-1}}. \end{aligned}$$

(52)

The Lagrange multiplier ($\varLambda _k$) is computed based on restriction $w_{k,1}+ w_{k,2} =1$. Then,

$$\begin{aligned}&\displaystyle w_{k,1} + w_{k,2} = 1 \end{aligned}$$

(53)

$$\begin{aligned}&\displaystyle \left( \frac{\varLambda _k}{t \, \xi _{k,1} } \right) ^{\frac{1}{t-1}} + \left( \frac{\varLambda _k}{t \, \xi _{k,2} } \right) ^{\frac{1}{t-1}} = 1 \end{aligned}$$

(54)

$$\begin{aligned}&\displaystyle \varLambda _k = \left\{ \left( \frac{1}{t \, \xi _{k,1} } \right) ^{\frac{1}{t-1}} + \left( \frac{1}{t \, \xi _{k,2}} \right) ^{\frac{1}{t-1}}\right\} ^{-(t-1)} \end{aligned}$$

(55)

Replacing Eq. (55) in Eq. (48), we get

$$\begin{aligned} w_{k,1} = \left( \frac{\left\{ \left( \frac{1}{t \, \xi _{k,1} } \right) ^{\frac{1}{t-1}} + \left( \frac{1}{t \, \xi _{k,2}} \right) ^{\frac{1}{t-1}}\right\} ^{-(t-1)} }{(t \, \xi _{k,1}) }\right) ^{\frac{1}{t-1}} = \left\{ 1+\left( \frac{\xi _{k,1}}{\xi _{k,2}} \right) ^{\frac{1}{t-1}}\right\} ^{-1}. \end{aligned}$$

(56)

Now, replacing Eq. (55) in Eq. (52), we get

$$\begin{aligned} w_{k,2} = \left( \frac{\left\{ \left( \frac{1}{t \, \xi _{k,1} } \right) ^{\frac{1}{t-1}} + \left( \frac{1}{t \, \xi _{k,2} } \right) ^{\frac{1}{t-1}}\right\} ^{-(t-1)} }{(t \, \xi _{k,2}) } \right) ^{\frac{1}{t-1}} =\left\{ 1+ \left( \frac{\xi _{k,2}}{\xi _{k,1}} \right) ^{\frac{1}{t-1}}\right\} ^{-1}. \end{aligned}$$

(57)

So, the hybrid weights are computed by Eqs. (56) and (57). $\square$

C Proof of Proposition 3

Fixing cluster k and dimension j, the prototype for the $HL_1$ and $HL_{\infty }$ distances has an analytic solution, given by Eq. (58),

$$\begin{aligned} \underline{g}_k^j = \underset{\gamma _n \in C_k }{Me } \left\{ \underline{\gamma }_{n}^j \right\} \, \text{ and } \, \bar{g}_k^j = \underline{g}_k^j + \underset{\gamma _n \in C_k }{Me } \left\{ \breve{\gamma }_{n}^j \right\} . \end{aligned}$$

(58)

Proof

The criterion to be minimized for the $HL_1$ distance is

$$\begin{aligned} J_{d_{HL_1}} = \sum _{k=1}^K \sum _{\gamma _n \in C_k} \sum _{j=1}^p \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j | + |\breve{\gamma }_k^j - \breve{g}_k^j | \right\} . \end{aligned}$$

(59)

Fixing cluster k and dimension j, it is possible to reduce the optimization complexity to

$$\begin{aligned} \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j -\underline{g}_k^j | + \sum _{\gamma _n \in C_k} |\breve{\gamma }_k^j -\breve{g}_k^j | . \end{aligned}$$

(60)

The problem is resumed to optimize the two sums

$$\begin{aligned} \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j - \underline{g}_k^j | \, \text{ and } \, \sum _{\gamma _n \in C_k} |\breve{\gamma }_k^j -\breve{g}_k^j | . \end{aligned}$$

(61)

Each sum is minimized by the median of the respective set [27]. Then,

$$\begin{aligned} \underline{g}_k^j = \underset{\gamma _n \in C_k }{Me } \left\{ \underline{\gamma }_n^j \right\} \, \text{ and } \, \breve{g}_k^j =\underset{\gamma _n \in C_k }{Me } \left\{ \breve{\gamma }_n^j\right\} . \end{aligned}$$

(62)

The criterion to be minimized for the $HL_{\infty }$ distance is

$$\begin{aligned} J_{d_{HL_{\infty }}} = \sum _{k=1}^K \sum _{\gamma _n \in C_k} \left( \max _{j=1}^p \{ |\underline{\gamma }_n^j - \underline{g}_k^j | \} +\max _{j=1}^p \{ |\breve{\gamma }_k^j - \breve{g}_k^j | \} \right) . \end{aligned}$$

(63)

Fixing the cluster k, it is possible to reduce the optimization complexity to

$$\begin{aligned} \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j |\right\} + \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\breve{\gamma }_k^j - \breve{g}_k^j | \right\} . \end{aligned}$$

(64)

The problem is reduced to optimizing the two sums independently,

$$\begin{aligned} \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j | \right\} \, \text{ and } \, \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\breve{\gamma }_k^j - \breve{g}_k^j|\right\} . \end{aligned}$$

(65)

The $\max$ function can be rewritten as a limit of the $HL_q$ distance when $q \rightarrow \infty$, so

$$\begin{aligned} \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j | \right\} = \sum _{\gamma _n \in C_k} \lim _{q \rightarrow \infty } \left\{ \sum _{j=1}^p |\underline{\gamma }_n^j -\underline{g}_k^j |^q \right\} ^{\frac{1}{q}} \end{aligned}$$

(66)

and

$$\begin{aligned} \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\breve{\gamma }_k^j -\breve{g}_k^j | \right\} = \sum _{\gamma _n \in C_k} \lim _{q \rightarrow \infty } \left\{ \sum _{j=1}^p |\breve{\gamma }_k^j -\breve{g}_k^j |^q \right\} ^{\frac{1}{q}}. \end{aligned}$$

(67)

As the terms of the sums are positive, their minimization entails the minimization of all sums. Fixing dimension j, the problem is reduced to

$$\begin{aligned} \sum _{\gamma _n \in C_k} \lim _{q \rightarrow \infty } \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j |^q\right\} ^{\frac{1}{q}} = \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j - \underline{g}_k^j | \end{aligned}$$

(68)

and

$$\begin{aligned} \sum _{\gamma _n \in C_k} \lim _{q \rightarrow \infty } \left\{ |\breve{\gamma }_k^j - \breve{g}_k^j |^q\right\} ^{\frac{1}{q}} =\sum _{\gamma _n \in C_k} |\breve{\gamma }_k^j - \breve{g}_k^j |. \end{aligned}$$

(69)

This follows the $HL_1$ optimization problem, whose solution is the medians of lower bounds and ranges. Then,

$$\begin{aligned} \underline{g}_k^j = \underset{\gamma _n \in C_k }{Me } \left\{ \underline{\gamma }_n^j \right\} \, \text{ and } \, \breve{g}_k^j =\underset{\gamma _n \in C_k }{Me } \left\{ \breve{\gamma }_n^j\right\} . \end{aligned}$$

(70)

Using the inverse mapping [see Eq. (15)], we compute the upper bounds as

$$\begin{aligned} b_{g_k}^j = \underline{g}_k^j + \breve{g}_k^j = \underline{g}_k^j + \underset{\gamma _n \in C_k }{Me } \left\{ \breve{\gamma }_n^j\right\} . \end{aligned}$$

(71)

$\square$

D Proof of Proposition 4

Fixing cluster k and dimension j, the prototype for the $HL_2$ distance has an analytic solution, which is the mean of the interval bounds. It is computed by Eq. (72),

$$\begin{aligned} \underline{g}_k^j = \frac{1}{|C_k|} \sum _{j=1}^p \underline{\gamma }_n^j\, \text{ and } \, b_{g_k}^j = \underline{g}_k^j + \frac{1}{|C_k|} \sum _{j=1}^p \breve{\gamma }_{n}^j, \end{aligned}$$

(72)

where $|C_k|$ is the number of instances allocated in the cluster $C_k$.

Proof

The criterion to be minimized for the $HL_2$ distance is

$$\begin{aligned} J_{d_{HL_2}} = \sum _{k=1}^K \sum _{\gamma _n \in C_k} \sum _{j=1}^p \left\{ (\underline{\gamma }_n^j - \underline{g}_k^j)^2| + (\breve{\gamma }_k^j - \breve{g}_k^j)^2 \right\} . \end{aligned}$$

(73)

Fixing cluster k and dimension j, it is possible to reduce the optimization complexity to

$$\begin{aligned} J_k^j=\sum _{\gamma _n \in C_k} (\underline{\gamma }_n^j -\underline{g}_k^j)^2+ \sum _{\gamma _n \in C_k} (\breve{\gamma }_k^j -\breve{g}_k^j)^2 . \end{aligned}$$

(74)

The solution is found using minimum squares. Partial derivatives of $J_k^j$ with respect to $\underline{g}_k^j$ and $\breve{g}_k^j$ should be null. So,

$$\begin{aligned} \frac{\partial J_k^j}{\partial \underline{g}_k^j} = \sum _{\gamma _n \in C_k} 2 \cdot (\underline{\gamma }_n^j - \underline{g}_k^j) = 0 \Longrightarrow \underline{g}_k^j = \frac{1}{|C_k|} \sum _{\gamma _n \in C_k} \underline{\gamma }_n^j \end{aligned}$$

(75)

and

$$\begin{aligned} \frac{\partial J_k^j}{\partial \underline{g}_k^j} = \sum _{\gamma _n \in C_k} 2 \cdot (\breve{\gamma }_k^j - \breve{g}_k^j) = 0 \Longrightarrow \breve{g}_k^j = \frac{1}{|C_k|} \sum _{\gamma _n \in C_k} \breve{\gamma }_k^j \end{aligned}$$

(76)

where $|C_k|$ is the number of instances allocated in cluster k. The $HL_2$ prototypes are computed by:

$$\begin{aligned} \underline{g}_k^j = \frac{1}{|C_k|} \sum _ {\gamma _n \in C_k } \underline{\gamma }_n^j \, \text{ and } \, \breve{g}_k^j = \frac{1}{|C_k|} \sum _ {\gamma _n \in C_k } \breve{\gamma }_n^j. \end{aligned}$$

(77)

Using the inverse mapping (see Eq. (15)), we compute the upper bounds as

$$\begin{aligned} b_{g_k}^j = \underline{g}_k^j + \breve{g}_k^j = \underline{g}_k^j + \frac{1}{|C_k|} \sum _ {\gamma _n \in C_k } \breve{\gamma }_{n}^j. \end{aligned}$$

(78)

$\square$

E Proof of Proposition 5

Fixing cluster k and dimension j, the prototype for the $HL_q$ distance (when $q > 1$) can be found using the Newton–Raphson numeric method. Let the sets $L_k^j =\left\{ \underline{\gamma }_n^j | \gamma _n \in C_k \right\}$ and $R_k^j = \left\{ \breve{\gamma }_k^j | \gamma _n \in C_k \right\}$. Algorithm 2 shows how to compute the prototype components $\underline{g}_k^j$ and $\breve{g}_k^j$, respectively.

Proof

Let an ascending ordered set $X = \left\{ x_1, x_2, \ldots , x_N\right\}$, i.e., $x_i \le x_{i+1}$, and the function $f:\mathfrak {R}\leftarrow \mathfrak {R}$,

$$\begin{aligned} f(v) = \sum _{i=1}^N |x_i- v|^q \end{aligned}$$

(79)

with $q > 1$. We are interested on the value which minimizes f(v). This function can be rewritten as follows:

$$\begin{aligned} f(v) = \sum _{i=1}^N (x_i- v)^q \cdot \left\{ sgn(x_i-v) \right\} ^q, \end{aligned}$$

(80)

where $sgn(\cdot )$ is a constant, defined as

$$\begin{aligned} sgn(x)= {\left\{ \begin{array}{ll} \,\,\, 1 &{} \text{ if } x \ge 0 \\ -1 &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$

(81)

The first derivative of $f(\cdot )$ is given by:

$$\begin{aligned} f'(v)&= -q \sum _{i=1}^N (x_i- v)^{q-1} \cdot \left\{ sgn(x_i-v) \right\} ^{q} \end{aligned}$$

(82)

$$\begin{aligned} f'(v)&= -q \sum _{i=1}^N |x_i- v|^{q-1} \cdot sgn(x_i-v), \end{aligned}$$

(83)

and the second derivative of $f(\cdot )$ is given by:

$$\begin{aligned} f''(v)&= q (q-1) \sum _{i=1}^N (x_i- v)^{q-2} \cdot \left\{ sgn(x_i-v) \right\} ^{q} \end{aligned}$$

(84)

$$\begin{aligned} f''(v)&= q (q-1) \sum _{i=1}^N |x_i- v|^{q-2}. \end{aligned}$$

(85)

When $q > 1$ the second derivative is always positive. We conclude that the first derivative is monotonically increasing for any v.

The value $v_*$ which minimizes f(v) must satisfy $f'(v_*)=0$. Suppose a value $v_{-}$ with $v_{-} < x_1$. Then, $v_{-} < x_i, \forall x_i$. So, $x_i -v_{-} > 0$, which implies that $sgn(x_i-v_{-})=1, \forall x_i$. Then, the first derivative becomes

$$\begin{aligned} f'(v_{-}) = -q \sum _{i=1}^N |x_i- v_{-}|^{q-1} \end{aligned}$$

(86)

which always assumes a negative value. So, $f'(v_{-}) < 0$.

Now, suppose that $v_+ > x_N$. Then, $v_+ > x_i, \forall x_i$. So, $x_i-v_+<0$, implying $sgn(x_i-v_+)=-1$. Then, the first derivative becomes

$$\begin{aligned} f'(v_+) = q \sum _{i=1}^N |x_i- v_+|^{q-1} \end{aligned}$$

(87)

which assumes positive values. So, $f'(v_+) > 0$.

When $v < x_1$, $f'(v) < 0$ and when $v > x_N$, $f'(v) > 0$, so, the function $f'(v)$ changes its signal on the interval $[x_1, x_N]$, then $\exists \, v_* \in [x_1, x_N]$ such that $f'(v_*)=0$. As $f'(v)$ is monotonically increasing, this solution is unique. Unfortunately, the $f'(v)$ expression is too complex and a general analytic solution cannot be computed. We propose the use of Newton−Raphson numeric method to find $v_*$. In this case, a initial value $v_0$ is chosen randomly on interval $[x_1,x_N]$. Iterative values $\left\{ v_i \right\}$ are computed as follows:

$$\begin{aligned} v_i = v_{i-1} - \frac{f'(v_{i-1})}{f''(v_{i-1})}. \end{aligned}$$

(88)

Convergence occurs when $|v_i - v_{i-1}| < \epsilon$, with $\epsilon > 0$.

The criterion to be minimized for the $HL_q$ distance is given by

$$\begin{aligned} J_{d_{HL_q}} = \sum _{k=1}^K \sum _{\gamma _n \in C_k} \sum _{j=1}^p \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j +|\breve{\gamma }_k^j - \breve{g}_k^j| \right\} . \end{aligned}$$

(89)

Fixing the kth cluster and jth dimension results in

$$\begin{aligned} \sum _{\gamma _n \in C_k} \left\{ |\underline{\gamma }_n^j -\underline{g}_k^j|^q + |\breve{\gamma }_k^j - \breve{g}_k^j|^q\right\} , \end{aligned}$$

(90)

and the two sums must be minimized:

$$\begin{aligned} \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j -\underline{g}_k^j|^q \quad \text{ and } \quad \sum _{\gamma _n \in C_k} |\breve{\gamma }_k^j - \breve{g}_k^j|^q. \end{aligned}$$

(91)

The steps described above (for the f function) can be applied for each sum independently. The sets $L_k^j =\left\{ \underline{\gamma }_n^j | \gamma _n \in C_k \right\}$ and $R_k^j = \left\{ \breve{\gamma }_k^j | \gamma _n \in C_k \right\}$ can be replaced by set X, and components $\underline{g}_k^j$ and $\breve{g}_k^j$ are determined. Algorithm 2 shows the steps to compute them using the Newton–Raphson numeric method. $\square$

F Proof of Proposition 6

Fixing cluster k, dimension j and parameter q ($q \ge 1$), the prototypes for $WHL_q$ and $AHL_q$ distances are computed according to one of three cases:

• 1. If $(q=1$ or $q=\infty )$, prototypes have an analytic solution, given by:

  $$\begin{aligned} \underline{g}_k^j = \underset{\gamma _n \in C_k }{Me } \left\{ \underline{\gamma }_n^j \right\} \, \quad \text{ and } \quad \, b_{g_k}^j = \underline{g}_k^j + \underset{\gamma _n \in C_k }{Me } \left\{ \breve{\gamma }_n^j \right\} . \end{aligned}$$

• 2. If $(q=2)$, prototypes have an analytic solution, given by:

  $$\begin{aligned} \underline{g}_k^j = \frac{1}{|C_k|} \sum _{j=1}^p \underline{\gamma }_n^j \quad \text{ and } \quad \,\, b_{g_k}^j =\underline{g}_k^j + \frac{1}{|C_k|} \sum _{j=1}^p \breve{\gamma }_n^j, \end{aligned}$$

  where $|C_k|$ is the number of instances in cluster k.

• 3. If ($q\ne 1$ and $q \ne 2$ and $q \ne \infty$), the Newton–Raphson numeric method is used, as described by Algorithm 2. The sets $L_k^j = \left\{ \underline{\gamma }_n^j | \gamma _n \in C_k \right\}$ and $R_k^j = \left\{ \breve{\gamma }_k^j | \gamma _n \in C_k \right\}$ are manipulated by it, resulting on the values of $\underline{g}_k^j$ and $\breve{g}_k^j$, respectively. Prototype upper bounds are found by $b_{g_k}^j=\underline{g}_k^j +\breve{g}_k^j$, for $j=1\ldots ,p$.

Proof

The optimization criterion for the $AHL_q$ distance is given by:

$$\begin{aligned} J_{d_{AHLq}} = \sum _{k=1}^K \sum _{\gamma _n \in C_k} \sum _{j=1}^p \lambda _k^j \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j |^q +|\breve{\gamma }_k^j - \breve{g}_k^j |^q \right\} . \end{aligned}$$

(92)

Fixing cluster k and dimension j, the optimization problem can be reduced to

$$\begin{aligned} \lambda _k^j \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j -\underline{g}_k^j |^q + \lambda _k^j \sum _{\gamma _n \in C_k} |\breve{\gamma }_k^j - \breve{g}_k^j |^q . \end{aligned}$$

(93)

The optimization criterion for the $WHL_q$ distance is given by:

$$\begin{aligned} J_{d_{WHLq}} = \sum _{k=1}^K \sum _{\gamma _n \in C_k} \sum _{j=1}^p (w_{k,1}^j)^t \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j |^q + (w_{k,2}^j)^t |\breve{\gamma }_k^j - \breve{g}_k^j |^q \right\} . \end{aligned}$$

(94)

Fixing cluster k and dimension j, the optimization problem can be reduced to

$$\begin{aligned} (w_{k,1}^j)^t \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j -\underline{g}_k^j |^q + (w_{k,2}^j)^t \sum _{\gamma _n \in C_k} |\breve{\gamma }_k^j - \breve{g}_k^j |^q . \end{aligned}$$

(95)

Adaptive and Hybrid weights become constants when classes and dimensions are fixed. The problem is reduced to optimize the following two sums:

$$\begin{aligned} \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j - \underline{g}_k^j |^q \, \quad \text{ and } \quad \, \sum _{\gamma _n \in C_k} |\breve{\gamma }_k^j - \breve{g}_k^j |^q . \end{aligned}$$

(96)

Solutions are proposed according to the value of the q parameter. If $q=1$, the optimization becomes

$$\begin{aligned} \sum _{\gamma _n \in C_k} |\underline{\gamma }_n^j - \underline{g}_k^j | \, \quad \text{ and } \quad \, \sum _{\gamma _n \in C_k} |\breve{\gamma }_k^j -\breve{g}_k^j |, \end{aligned}$$

(97)

whose solution, according to Proposition 3, is given by:

$$\begin{aligned} \underline{g}_k^j = \underset{\gamma _n \in C_k }{Me } \left\{ \underline{\gamma }_n^j \right\} \, \quad \text{ and } \quad \, \breve{g}_k^j =\underset{\gamma _n \in C_k }{Me } \left\{ \breve{\gamma }_n^j\right\} . \end{aligned}$$

(98)

If $q=2$, the optimization becomes

$$\begin{aligned} \sum _{\gamma _n \in C_k} (\underline{\gamma }_n^j -\underline{g}_k^j)^2 \, \quad \text{ and } \quad \, \sum _{\gamma _n \in C_k} (\breve{\gamma }_k^j - \breve{g}_k^j)^2. \end{aligned}$$

(99)

The solution, according to Proposition 4, is given by:

$$\begin{aligned} \underline{g}_k^j = \frac{1}{|C_k|} \sum _{j=1}^p \underline{\gamma }_n^j \quad \text{ and } \quad \,\, \breve{g}_k^j =\frac{1}{|C_k|} \sum _{j=1}^p \breve{\gamma }_{n}^j, \end{aligned}$$

(100)

where $|C_k|$ is the number of instances in cluster k.

The optimization criterion for the $WHL_{\infty }$ distance is given by:

$$\begin{aligned} J_{d_{HL_{\infty }}} = \sum _{k=1}^K \sum _{\gamma _n \in C_k} \left( (w_{k,1} )^t \max _{j=1}^p \left\{ |\underline{\gamma }_n^j -\underline{g}_k^j | \right\} + (w_{k,2})^t \max _{j=1}^p \left\{ |\breve{\gamma }_k^j - \breve{g}_k^j | \right\} \right) . \end{aligned}$$

(101)

Fixing cluster k, the optimization problem can be reduced to

$$\begin{aligned} (w_{k,1} )^t \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j | \right\} + (w_{k,2} )^t \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\breve{\gamma }_k^j- \breve{g}_k^j | \right\} . \end{aligned}$$

(102)

When the cluster is fixed, the hybrid weights become constants; then, the problem is reduced to optimizing the two sums independently:

$$\begin{aligned} \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\underline{\gamma }_n^j - \underline{g}_k^j | \right\} \, \quad \text{ and } \quad \, \sum _{\gamma _n \in C_k} \max _{j=1}^p \left\{ |\breve{\gamma }_k^j - \breve{g}_k^j | \right\} . \end{aligned}$$

(103)

The solution are the medians of lower bounds and ranges (as showed on Proposition 3). Then,

$$\begin{aligned} \underline{g}_k^j = \underset{\gamma _n \in C_k }{Me } \left\{ \underline{\gamma }_n^j \right\} \, \quad \text{ and } \quad \, \breve{g}_k^j =\underset{\gamma _n \in C_k }{Me } \left\{ \breve{\gamma }_n^j \right\} . \end{aligned}$$

(104)

If $q>1$, $q \ne 2$ or $q\ne \infty$, it is not possible to express an analytic solution for Eq. in (96). So the solution is computed as discussed on Proposition 5. The Newton–Raphson numeric method is used, as described by Algorithm 2. The sets $L_k^j = \left\{ \underline{\gamma }_n^j | \gamma _n \in C_k \right\}$ and $R_k^j = \left\{ \breve{\gamma }_k^j | \gamma _n \in C_k \right\}$ are parameters for this algorithm, resulting on the values of $\underline{g}_k^j$ and $\breve{g}_k^j$, respectively.

Using the inverse mapping (see Eq. (15)), we compute the upper bounds as

$$\begin{aligned} b_{g_k}^j&= \underline{g}_k^j + \breve{g}_k^j. \end{aligned}$$

(105)

$\square$