Distance-based Learning for Mixed Data:
The manydist Package

MuViSU Quarterly Check-in

Alfonso Iodice D’Enza, Angelos Markos,
Michel van de Velden and Carlo Cavicchia

2026-09-30

Outline

01 · Setting the scene
Methodological background

02 · Building distances with manydist
Variable-specific dissimilarities, scaling, and aggregation

03 · Distance diagnostics
Understanding how variables shape the resulting distances

04 · Distance-based learning pipelines
Integrating distance construction into modelling workflows

05 · Examples and practical insights
Connecting distance choices to downstream results

01 · Setting the scene

Learning from mixed-type data

Many statistical learning methods rely, explicitly or implicitly, on comparing observations.

Terminology alert!

I will often use distance as shorthand for pairwise dissimilarity.

Some measures discussed here are not metrics in the strict mathematical sense,
but they all quantify how different two observations are.

If the data are mixed-type, then comparison is not straightforward:

Many learning methods can operate directly on a dissimilarity matrix.

Dimension reduction

Multidimensional scaling (MDS)¹

Clustering methods

Hierarchical clustering (HC), PAM², and spectral clustering³

Nearest-neighbour prediction

Nearest-neighbour classification and nearest-neighbour averaging for regression

Desirable properties¹

Multivariate Additivity

Let $\mathbf{x}_i=\left(x_{i1}, \dots, x_{iQ}\right)$ denote a $Q-$dimensional vector. A distance function $d\left(\mathbf{x}_i,\mathbf{x}_\ell\right)$ between observations $i$ and $\ell$ is multivariate additive if

\[ d\left(\mathbf{x}_i,\mathbf{x}_\ell\right)=\sum_{j=1}^{Q} d_j\left(\mathbf{x}_i,\mathbf{x}_\ell\right), \]

where $d_j\left(\mathbf{x}_i,\mathbf{x}_\ell\right)$ denotes the $j-$th variable specific distance.

Manhattan distance satisfies the additivity property; the Euclidean distance does not

Desirable properties¹

If additivity holds, by-variable distances are added together: they should be on equivalent scales

Commensurability

Let ${\boldsymbol X}_i =\left(X_{i1}, \dots, X_{iQ}\right)$ denote a $Q-$dimensional random variable corresponding to an observation $i$. Furthermore, let $d_{j}$ denote the distance function corresponding to the $j-$th variable.

We have commensurability if, for all $j$, and $i \neq \ell$,

\[ E[d_{j}({ X}_{ij}, {X}_{\ell j})] = c, \]

where $c$ is some constant.

Mixed-data setup

a mixed data set

$n$ observations described by $Q$ variables, $Q_{n}$ numerical and $Q_{c}$ categorical
the $n\times Q$ data matrix ${\bf X}=\left[{\bf X}_{n},{\bf X}_{c}\right]$ is column-wise partitioned

A formulation for mixed distance between observations $i$ and $\ell$:

\[\begin{eqnarray}\label{genmixeddist_formula} d\left(\mathbf{x}_i,\mathbf{x}_\ell\right)&=& \sum_{j_n=1}^{Q_n} d_{j_n}\left(\mathbf{x}_i^{(n)},\mathbf{x}_\ell^{(n)}\right)+ \sum_{j_c=1}^{Q_c} d_{j_c}\left(\mathbf{x}_i^{(c)},\mathbf{x}_\ell^{(c)}\right)=\\ &=& \sum_{j_n=1}^{Q_n} w_{j_n} \delta^{(n)}_{j_n}\left(\mathbf{x}_i^{(n)},\mathbf{x}_\ell^{(n)}\right)+ \sum_{j_c=1}^{Q_c} w_{j_c}\delta^{(c)}_{j_c}\left(\mathbf{x}_i^{(c)},\mathbf{x}_\ell^{(c)}\right) \end{eqnarray}\]

numeric case

$\delta^n_{j_n}$ is a function quantifying the dissimilarity between observations on the $j_n-$th numerical variable
$w_{j_n}$ is a weight for the $j_n-$th variable.

categorical case

dissimilarity between the categories chosen by subjects $i$ and $\ell$ for categorical variable $j_c$

$w_{j_c}$ is a weight for the $j_c-$th variable

Distributions, scaling and bias: the numeric case

Distributions, scaling and bias: the categorical case

$q\in \{2,3,5,10\}$
$p_1 \in \{0.05,0.1,0.2,0.33, 0.5,0.66, 0.8,0.9,0.95\}$
$p_j = (1-p_1)/(q-1)$, with $j=2,\dots,q$,

The expected distance increases with the heterogeneity of the distribution and with the number of categories

Independence-based distances

Independence-based pairwise distance

No inter-variable relations are considered.

in the continuous case: Euclidean or Manhattan distances
in the categorical case: Hamming / matching distance, among many others
in the mixed-data case: Gower dissimilarity index

variable contributions may be balanced, but still treated as separate sources of information

Beyond commensurability

commensurability makes variable contributions comparable across scales and data types.

If variables are correlated or associated, the same information may contribute repeatedly to the distance: redundancy

by variable differences: independence-based

When variables are correlated or associated, shared information is effectively counted multiple times
inflated dissimilarities may cause potential distortions in downstream unsupervised learning tasks.

by variable differences: independence-based

When variables are correlated or associated, shared information is effectively counted multiple times
inflated dissimilarities may cause potential distortions in downstream unsupervised learning tasks.

by variable differences: independence-based

The Euclidean distance $\longrightarrow$ shared information is over-counted

accounting for inter-variable relations: association-based

The Mahalanobis distance $\longrightarrow$ shared information is not over-counted

this is an association-based distance for continuous data

association-based distance

Association-based for continuous: Mahalanobis distance

Let ${\bf X}_{con}$ be $n\times Q_{d}$ a data matrix of $n$ observations described by $Q_{d}$ continuous variables, and let $\bf S$ the sample covariance matrix, the Mahalanobis distance matrix is

\[ {\bf D}_{mah} = \left[\operatorname{diag}({\bf G})\,{\bf 1}_{n}^{\sf T} + {\bf 1}_{n}\,\operatorname{diag}({\bf G})^{\sf T} - 2{\bf G}\right]^{\odot 1/2} \] where

$[\cdot]^{\odot 1/2}$ denotes the element-wise square root
${\bf G}=({\bf C}{\bf X}_{con}){\bf S}^{-1}({\bf C}{\bf X}_{con})^{\sf T}$ is the Mahalanobis Gram matrix
${\bf C}={\bf I}_{n}-\tfrac{1}{n}{\bf 1}_{n}{\bf 1}_{n}^{\sf T}$ is the centering operator

association-based distance

Association-based for categorical: total variation distance (TVD)(Le & Ho, 2005)

The distance matrix ${\bf D}_{tvd}$ can be defined via the delta framework¹ upon properly defining the block-diagonal matrix ${\bf \Delta}$

Let ${\bf X}_{cat}$ be $n\times Q_{c}$ a data matrix of $n$ observations described by $Q_{c}$ categorical variables.

\[ {\bf D} = {\bf Z}{\Delta}{\bf Z}^{\sf T} = \left[\begin{array}{ccc} {\bf Z}_{1} & \dots & {\bf Z}_{Q_{c}} \end{array} \right]\left[\begin{array}{ccc} {\bf\Delta}_1 & & \\ & \ddots &\\ & & {\bf\Delta}_{Q_{c}} \end{array} \right] \left[ \begin{array}{c} {\bf Z}_{1}^{\sf T}\\ \vdots \\ {\bf Z}_{Q_{c}}^{\sf T} \end{array} \right] \]

in the framework, setting ${\Delta}_j$ determines the categorical distance measure of choice (independent- or association-based)

association-based distance

Association-based for categorical: total variation distance (TVD) (Le & Ho, 2005) (2)

Consider the empirical joint probability distributions stored in the off-diagonal blocks of ${\bf P}$:

\[ {\bf P} = \frac{1}{n} \begin{bmatrix} {\bf Z}_1^{\sf T}{\bf Z}_1 & {\bf Z}_1^{\sf T}{\bf Z}_2 & \cdots & {\bf Z}_1^{\sf T}{\bf Z}_{Q_c} \\ \vdots & \ddots & \vdots & \vdots \\ {\bf Z}_{Q_c}^{\sf T}{\bf Z}_1 & {\bf Z}_{Q_c}^{\sf T}{\bf Z}_2 & \cdots & {\bf Z}_{Q_c}^{\sf T}{\bf Z}_{Q_c} \end{bmatrix}. \]

The block matrix $\bf R$ refer to the conditional probability distributions for each variable $j$ given each variable $i$ ($i,j=1,\ldots,Q_c$, $i\neq j$), stored in the block matrix

\[ {\bf R} = {\bf P}_z^{-1}({\bf P} - {\bf P}_z). \]

where ${\bf P}_z = {\bf P} \odot {\bf I}_{Q^*}$, and ${\bf I}_{Q^*}$ is the $Q^*\times Q^*$ identity matrix.

association-based distance: a small example

Data

Consider two categorical variables:

$X_1$ with categories $A,B,C$
$X_2$ with categories $u,v$

# A tibble: 10 × 3
      id X1    X2   
   <int> <fct> <fct>
 1     1 A     u    
 2     2 A     u    
 3     3 A     v    
 4     4 B     u    
 5     5 B     u    
 6     6 B     v    
 7     7 C     u    
 8     8 C     v    
 9     9 C     v    
10    10 C     v

Indicator matrices

\[ {\bf Z}_1 = \begin{pmatrix} 1&0&0\\ 1&0&0\\ 1&0&0\\ 0&1&0\\ 0&1&0\\ 0&1&0\\ 0&0&1\\ 0&0&1\\ 0&0&1\\ 0&0&1 \end{pmatrix}, \qquad {\bf Z}_2 = \begin{pmatrix} 1&0\\ 1&0\\ 0&1\\ 1&0\\ 1&0\\ 0&1\\ 1&0\\ 0&1\\ 0&1\\ 0&1 \end{pmatrix}. \]

association-based distance: from ${\bf Z}$ to ${\bf P}$

Let

\[ {\bf Z} = [{\bf Z}_1,{\bf Z}_2]. \]

The empirical co-occurrence matrix is

\[ {\bf P} = \frac{1}{10}{\bf Z}^{\sf T}{\bf Z}. \]

For this example,

\[ {\bf P} = \begin{pmatrix} \color{#2A9D8F}{0.30} & \color{#2A9D8F}{0} & \color{#2A9D8F}{0} & \color{#E76F51}{0.20} & \color{#E76F51}{0.10}\\ \color{#2A9D8F}{0} & \color{#2A9D8F}{0.30} & \color{#2A9D8F}{0} & \color{#E76F51}{0.20} & \color{#E76F51}{0.10}\\ \color{#2A9D8F}{0} & \color{#2A9D8F}{0} & \color{#2A9D8F}{0.40} & \color{#E76F51}{0.10} & \color{#E76F51}{0.30}\\ \color{#E76F51}{0.20} & \color{#E76F51}{0.20} & \color{#E76F51}{0.10} & \color{#2A9D8F}{0.50} & \color{#2A9D8F}{0}\\ \color{#E76F51}{0.10} & \color{#E76F51}{0.10} & \color{#E76F51}{0.30} & \color{#2A9D8F}{0} & \color{#2A9D8F}{0.50} \end{pmatrix}. \]

diagonal blocks contain marginal information; off-diagonal blocks contain joint proportions

association-based distance: from ${\bf P}$ to ${\bf R}$

The diagonal part of ${\bf P}$ is

\[ {\bf P}_z = {\bf P} \odot {\bf I}_{Q^*} = \operatorname{diag}(0.30,0.30,0.40,0.50,0.50). \]

The block matrix of conditional profiles is

\[ {\bf R} = {\bf P}_z^{-1}({\bf P}-{\bf P}_z). \]

For this example,

\[ {\bf R} = \begin{pmatrix} \color{#2A9D8F}{0} & \color{#2A9D8F}{0} & \color{#2A9D8F}{0} & \color{#E76F51}{0.67} & \color{#E76F51}{0.33}\\ \color{#2A9D8F}{0} & \color{#2A9D8F}{0} & \color{#2A9D8F}{0} & \color{#E76F51}{0.67} & \color{#E76F51}{0.33}\\ \color{#2A9D8F}{0} & \color{#2A9D8F}{0} & \color{#2A9D8F}{0} & \color{#E76F51}{0.25} & \color{#E76F51}{0.75}\\ \color{#E76F51}{0.40} & \color{#E76F51}{0.40} & \color{#E76F51}{0.20} & \color{#2A9D8F}{0} & \color{#2A9D8F}{0}\\ \color{#E76F51}{0.20} & \color{#E76F51}{0.20} & \color{#E76F51}{0.60} & \color{#2A9D8F}{0} & \color{#2A9D8F}{0} \end{pmatrix}. \]

each off-diagonal block contains conditional profiles across variables

association-based distance

Association-based for categorical: total variation distance (TVD)(Le & Ho, 2005) (3)

Let ${\bf r}^{ji}_a$ and ${\bf r}^{ji}_b$ be the rows of ${\bf R}_{ji}$, the $(j,i)$th off-diagonal block of ${\bf R}$.

The category dissimilarity between $a$ and $b$ for variable $j$ based on the total variation distance (TVD) is defined as

\[ \delta^{j}_{tvd}(a,b) = \sum_{i\neq j}^{Q_c} w_{ji} \Phi^{ji}({\bf r}^{ji}_{a},{\bf r}^{ji}_{b}) = \sum_{i\neq j}^{Q_c} w_{ji} \left[\frac{1}{2}\sum_{\ell=1}^{q_i} |{\bf r}^{ji}_{a\ell}-{\bf r}^{ji}_{b\ell}|\right], \label{ab_delta} \]

where $w_{ji}=1/(Q_c-1)$ for equal weighting (can be user-defined).

TVD-based dissimilarity matrix is, therefore,

\[ {\bf D}_{tvd}= {\bf Z}{\Delta}^{(tvd)}{\bf Z}^{\sf T}. \]

association-based distance: reading ${\bf R}_{12}$

For the categories of $X_1$, the relevant block is

\[ {\bf R}_{12} = \begin{pmatrix} 0.67 & 0.33\\ 0.67 & 0.33\\ 0.25 & 0.75 \end{pmatrix}. \]

Interpretation

Rows of ${\bf R}_{12}$ describe the distribution of $X_2$ within each category of $X_1$:

$P(X_2=u \mid X_1=A)=\color{#E76F51}{\mathbf{0.67}}$, $P(X_2=v \mid X_1=A)=\color{#E76F51}{\mathbf{0.33}}$
$P(X_2=u \mid X_1=B)=\color{#E76F51}{\mathbf{0.67}}$, $P(X_2=v \mid X_1=B)=\color{#E76F51}{\mathbf{0.33}}$
$P(X_2=u \mid X_1=C)=\color{#E76F51}{\mathbf{0.25}}$, $P(X_2=v \mid X_1=C)=\color{#E76F51}{\mathbf{0.75}}$

categories are compared through their association profiles

association-based distance: from ${\bf R}$ to $\Delta_1^{(tvd)}$

Compare the rows of ${\bf R}_{12}$ using TVD.

\[ \delta^{1}_{tvd}(A,B) = \frac{1}{2} \left( |0.67-0.67| + |0.33-0.33| \right) = {\color{coral}0}. \]

\[ \delta^{1}_{tvd}(A,C) = \frac{1}{2} \left( |0.67-0.25| + |0.33-0.75| \right) = {\color{coral}0.42}. \]

\[ \delta^{tvd}_{1}(B,C) = {\color{coral}0.42}. \]

Therefore,

\[ \Delta^{(tvd)}_1 = \begin{pmatrix} 0 & 0 & {\color{coral}0.42}\\ 0 & 0 & {\color{coral}0.42}\\ {\color{coral}0.42} & {\color{coral}0.42} & 0 \end{pmatrix}. \]

$A$ and $B$ are close because they have the same profile with respect to $X_2$

association-based distance: reading ${\bf R}_{21}$

For the categories of $X_2$, the relevant block is

\[ {\bf R}_{21} = \begin{pmatrix} 0.40 & 0.40 & 0.20\\ 0.20 & 0.20 & 0.60 \end{pmatrix}. \]

Interpretation

Rows of ${\bf R}_{21}$ describe the distribution of $X_1$ within each category of $X_2$:

$P(X_1=A \mid X_2=u)=\color{#E76F51}{\mathbf{0.40}}$, $P(X_1=B \mid X_2=u)=\color{#E76F51}{\mathbf{0.40}}$, $P(X_1=C \mid X_2=u)=\color{#E76F51}{\mathbf{0.20}}$
$P(X_1=A \mid X_2=v)=\color{#E76F51}{\mathbf{0.20}}$, $P(X_1=B \mid X_2=v)=\color{#E76F51}{\mathbf{0.20}}$, $P(X_1=C \mid X_2=v)=\color{#E76F51}{\mathbf{0.60}}$

categories of $X_2$ are compared through their association profiles with respect to $X_1$

association-based distance: from ${\bf R}_{21}$ to $\Delta_2^{(tvd)}$

Compare the rows of ${\bf R}_{21}$ using TVD.

\[ \delta^{tvd}_{1}(u,v) = \frac{1}{2} \left( |0.40-0.20| + |0.40-0.20| + |0.20-0.60| \right) = {\color{coral}0.40}. \]

Therefore,

\[ \Delta^{(tvd)}_2 = \begin{pmatrix} 0 & {\color{coral}0.40}\\ {\color{coral}0.40} & 0 \end{pmatrix}. \]

$u$ and $v$ are different because they imply different profiles over $X_1$

association-based distance: from $\Delta$ to ${\bf D}$

We collect the category dissimilarity matrices in a block-diagonal matrix:

\[ \Delta^{(tvd)} = \begin{pmatrix} \color{#2A9D8F}{\Delta^{(tvd)}_1} & \color{#E76F51}{0}\\ \color{#E76F51}{0} & \color{#2A9D8F}{\Delta^{(tvd)}_2} \end{pmatrix}. \]

The observation-level categorical distance matrix is then

\[ {\bf D}_{tvd} = {\bf Z}\Delta^{(tvd)}{\bf Z}^{\sf T} = \begin{bmatrix} {\bf Z}_1 & {\bf Z}_2 \end{bmatrix} \begin{pmatrix} \Delta^{(tvd)}_1 & 0\\ 0 & \Delta^{(tvd)}_2 \end{pmatrix} \begin{bmatrix} {\bf Z}_1^{\sf T}\\ {\bf Z}_2^{\sf T} \end{bmatrix}. \]

Equivalently,

\[ {\bf D}_{tvd} = {\bf Z}_1\Delta^{(tvd)}_1{\bf Z}_1^{\sf T} + {\bf Z}_2\Delta^{(tvd)}_2{\bf Z}_2^{\sf T}. \]

category-level dissimilarities are translated into observation-level distances

02 · Building distances with `manydist`

Building distance-based pipelines

manydist

A package to construct, diagnose, and use distances for continuous, categorical, and mixed-type data.

Distance construction

mdist()
- presets and custom specifications
- response-aware distances
step_mdist()
- integrates distance construction into tidymodels workflows

Diagnostics

lovo_mdist()
compare_lovo_mdist()
benchmark_mdist()

Learning: model specs

Unsupervised learning

pam_dist()
spectral_dist()

Supervised learning

nearest_neighbor_dist()

the WDI data

A 2022 World Bank / WDI snapshot of country-level socio-economic indicators.

Country	Region	Income group	World Bank lending category	GDP per capita (k USD)	Life expectancy (years)	Unemployment (%)	Urban population (% total)	Population growth (%)
Tajikistan	Europe & Central Asia	Lower middle income	IDA	1.1	71.6	7.1	26.2	2.14
West Bank and Gaza	Middle East, North Africa, Afghanistan & Pakistan	Lower middle income	Not classified	3.8	76.7	24.4	86.6	2.43
Belarus	Europe & Central Asia	Upper middle income	IBRD	8.0	74.1	3.6	78.5	-0.80
United Arab Emirates	Middle East, North Africa, Afghanistan & Pakistan	High income	Not classified	50.8	80.5	2.9	85.5	5.09
El Salvador	Latin America & Caribbean	Upper middle income	IBRD	5.1	72.0	3.0	74.1	0.39
New Zealand	East Asia & Pacific	High income	Not classified	49.1	82.0	3.3	83.9	-0.06
Cyprus	Europe & Central Asia	High income	Not classified	33.2	80.4	6.8	66.7	1.06
Zambia	Sub-Saharan Africa	Lower middle income	IDA	1.4	65.3	6.0	44.6	2.76

`mdist()`: the main arguments

Data

x: training data—numeric, categorical, or mixed
new_data: optional observations to compare with x
response: optional outcome for response-aware distances

Choose a construction

preset: a ready-made distance specification
Use preset = "custom" for full control

Custom distance controls

method_cat: categorical dissimilarity (matching, tvd and MANY more)
method_num: numerical preprocessing (std, robust, pc_scores)
commensurable: balance variable contributions

Specialized controls

ncomp, threshold: component retention
gower_average: average or sum Gower contributions

`mdist()` output: an `MDist` R6 object

The object

[1] "MDist" "R6"

The result keeps the distance and its specification together:

d$distance: pairwise or new-to-training dissimilarities
d$preset: selected construction
d$data: training data
d$params: fitted parameters and metadata
d$to_dist(): convert to a standard dist object

Printing the object

MDist object
  preset : u_mix 
  number of observations : 181 
  number of continuous variables   : 5 
  number of categorical variables   : 3 
  parameters:
    - categorical method: tvd
    - numerical preprocessing: std
    - commensurability adjustment: TRUE

Reports the preset, number of observations and variables, and the active distance parameters.

Summarising the distance

MDist summary
  Preset : u_mix 
  Observations : 181 
  Continuous vars : 5 
  Categorical vars: 3 

Distance matrix statistics:
  Mean : 8.0444 
  SD   : 3.5231 
  Min  : 0.3832 
  Max  : 24.0576 
  Diagonal all zeros (block): YES 
  Symmetric (block check)   : YES 
  Triangle viol. rate (block): 0.000 
  NA count:0
  Inf count:0

Commensurable mixed-type distance using std numerical preprocessing and tvd categorical component.

Adds:

mean, SD, minimum, and maximum dissimilarity
zero-diagonal and symmetry checks
an approximate triangle-inequality check
counts of missing and infinite values

Constructing a mixed-type distance using a preset

Directly with mdist()

d_preset <- mdist(
  wdi_x,
  preset = "u_dep"
)

Inside a recipe with step_mdist()

rec_preset <- recipe(~ ., data = wdi_x) |>
  step_mdist(
    all_predictors(),
    preset = "u_dep"
  )

d_preset_step <- rec_preset |>
  prep(training = wdi_x) |>
  bake(new_data = NULL)

all.equal(
  unname(as.matrix(d_preset$distance)),
  unname(as.matrix(d_preset_step))
)

[1] TRUE

`step_mdist()` embeds the same distance specification into a modelling workflow

Constructing a custom mixed-type distance

Directly with mdist()

wdi_x <- wdi_data |> 
  dplyr::select(-country)

d_custom <- mdist(
  wdi_x,
  method_cat  = "eskin",
  method_num  = "std",
  commensurable = FALSE
)

Inside a recipe with step_mdist()

rec_custom <- recipe(~ ., data = wdi_x) |>
  step_mdist(
    all_predictors(),
    method_cat  = "eskin",
    method_num  = "std",
    commensurable = FALSE
  )

d_custom_step <- rec_custom |>
  prep(training = wdi_x) |>
  bake(new_data = NULL)

all.equal(
  unname(as.matrix(d_custom$distance)),
  unname(as.matrix(d_custom_step))
)

[1] TRUE

03 · Distance diagnostics

No one-size-fits-all distance

Different distance definitions induce different distance-based representations of the same data.

Same data, different representation

Changing the distance changes the global dissimilarity structure on which downstream learning methods rely.

Leave-one-variable-out diagnostics

How can we measure the contribution of each variable to this structure?

compare the dissimilarity matrix computed with and without the variable in question

LOVO-based diagnostics: what is evaluated?

For each variable $X_j$, compare the full distance ${\bf D}$ with the distance obtained after removing it, ${\bf D}_{-j}$.

Distance structure

Always computed

mean absolute difference
relative distance change

Outputs include mad_importance and relative_distance.

MDS geometry

Optional: mds = TRUE

Compare MDS configurations using congruence and the alienation coefficient.

dims controls the number of MDS dimensions.

Cluster stability

Optional: supply cluster_k

Compare full and LOVO partitions using the adjusted Rand index (ARI).

Importance is reported as $1-\mathrm{ARI}$.

LOVO can stop at the distance matrix—or continue into a chosen downstream representation.

`lovo_mdist()`: controlling the diagnostics

Data and distance

x: data used to construct the distance
response, response_used: optional response-aware construction
...: arguments passed to mdist(), such as preset, method_cat, and method_num
keep_dist: retain the full and LOVO distance matrices

Optional MDS diagnostics

mds = TRUE: enable MDS comparisons
dims: number of retained MDS dimensions

Optional clustering diagnostics

cluster_k: number of clusters; NULL disables clustering
cluster_methods: "pam", "hclust", and/or "spectral"
hclust_method: linkage for hierarchical clustering
spectral_sigma, spectral_nstart: spectral-clustering controls

Output

An MDistLOVO R6 object. Its $results tibble has one row per left-out variable and supports print(), summary(), and autoplot().

`compare_lovo_mdist()`: WDI example

compare_lovo_mdist() applies the same LOVO diagnostics to a named list of distance specifications.

mds and cluster_k make the optional diagnostics available; autoplot() selects the metric to display.

WDI: MDS geometry

autoplot(lovo_wdi_compare, metric = "ac_importance")

Larger alienation means that leaving out a variable produces a stronger change in the MDS configuration.

WDI: PAM clustering stability

autoplot(lovo_wdi_compare, metric = "pam_importance")

Larger $1-\mathrm{ARI}$ means that leaving out a variable produces a stronger change in the PAM partition.

`benchmark_mdist()`: comparing distance choices

While LOVO asks which variables matter within a distance, benchmark_mdist() asks how much candidate distance constructions differ from one another.

Main features

one mdist() run for each row of specs
failures are recorded without stopping the benchmark
successful distances are compared pairwise
distance and MDS diagnostics are computed; clustering ARI is optional
returns an MDistBenchmark tibble

Inspecting the result

summary(wdi_benchmark)
autoplot(wdi_benchmark, metric = "relative_distance")

`benchmark_mdist()`: WDI summary

summary(wdi_benchmark)

Summary of MDistBenchmark
  specifications : 4
  successful     : 4
  failed         : 0
  comparisons    : 6

Pairwise diagnostic summary:
# A tibble: 5 × 4
  metric              min median   max
  <chr>             <dbl>  <dbl> <dbl>
1 mad               1.06   4.66  7.64 
2 relative_distance 0.132  1.20  1.81 
3 mds_congruence    0.947  0.972 0.989
4 alienation        0.150  0.235 0.320
5 ari_pam           0.637  0.663 0.763

The summary reports the range of pairwise differences across successful distance specifications: raw and relative distance changes, MDS congruence and alienation, and—when requested—clustering ARI.

`benchmark_mdist()`: WDI pairwise comparison

autoplot(wdi_benchmark, metric = "relative_distance")

Each cell compares two complete distance constructions; larger values indicate more substantial disagreement between their pairwise dissimilarities.

04 · Distance-based learning pipelines

Distance-based classification pipeline

set.seed(123)
wdi_region <- wdi_data |>
  dplyr::filter(region != "North America") |>
  dplyr::mutate(
    region = droplevels(region)
  )
wdi_split <- initial_split(
  wdi_region,
  strata = region
)
wdi_train <- training(wdi_split)
wdi_test  <- testing(wdi_split)
wdi_rec <- recipe(region ~ ., data = wdi_train) |>
  update_role(country, new_role = "id") |>
  step_mdist(
    all_predictors(),
    preset = "u_dep"
  )
knn_spec <- nearest_neighbor_dist(
  mode = "classification",
  neighbors = tune()
)
wdi_wf <- workflow() |>
  add_recipe(wdi_rec) |>
  add_model(knn_spec)

Data

Prepare the classification task.

Distance-based classification pipeline

set.seed(123)
wdi_region <- wdi_data |>
  dplyr::filter(region != "North America") |>
  dplyr::mutate(
    region = droplevels(region)
  )
wdi_split <- initial_split(
  wdi_region,
  strata = region
)
wdi_train <- training(wdi_split)
wdi_test  <- testing(wdi_split)
wdi_rec <- recipe(region ~ ., data = wdi_train) |>
  update_role(country, new_role = "id") |>
  step_mdist(
    all_predictors(),
    preset = "u_dep"
  )
knn_spec <- nearest_neighbor_dist(
  mode = "classification",
  neighbors = tune()
)
wdi_wf <- workflow() |>
  add_recipe(wdi_rec) |>
  add_model(knn_spec)

Data

Prepare the classification task.

Split

Create training and test sets.

Distance-based classification pipeline

set.seed(123)
wdi_region <- wdi_data |>
  dplyr::filter(region != "North America") |>
  dplyr::mutate(
    region = droplevels(region)
  )
wdi_split <- initial_split(
  wdi_region,
  strata = region
)
wdi_train <- training(wdi_split)
wdi_test  <- testing(wdi_split)
wdi_rec <- recipe(region ~ ., data = wdi_train) |>
  update_role(country, new_role = "id") |>
  step_mdist(
    all_predictors(),
    preset = "u_dep"
  )
knn_spec <- nearest_neighbor_dist(
  mode = "classification",
  neighbors = tune()
)
wdi_wf <- workflow() |>
  add_recipe(wdi_rec) |>
  add_model(knn_spec)

Data

Prepare the classification task.

Split

Create training and test sets.

Recipe

Use step_mdist() to construct the distance representation.

Distance-based classification pipeline

set.seed(123)
wdi_region <- wdi_data |>
  dplyr::filter(region != "North America") |>
  dplyr::mutate(
    region = droplevels(region)
  )
wdi_split <- initial_split(
  wdi_region,
  strata = region
)
wdi_train <- training(wdi_split)
wdi_test  <- testing(wdi_split)
wdi_rec <- recipe(region ~ ., data = wdi_train) |>
  update_role(country, new_role = "id") |>
  step_mdist(
    all_predictors(),
    preset = "u_dep"
  )
knn_spec <- nearest_neighbor_dist(
  mode = "classification",
  neighbors = tune()
)
wdi_wf <- workflow() |>
  add_recipe(wdi_rec) |>
  add_model(knn_spec)

Data

Prepare the classification task.

Split

Create training and test sets.

Recipe

Use step_mdist() to construct the distance representation.

Model

Specify a distance-based KNN classifier.

Distance-based classification pipeline

set.seed(123)
wdi_region <- wdi_data |>
  dplyr::filter(region != "North America") |>
  dplyr::mutate(
    region = droplevels(region)
  )
wdi_split <- initial_split(
  wdi_region,
  strata = region
)
wdi_train <- training(wdi_split)
wdi_test  <- testing(wdi_split)
wdi_rec <- recipe(region ~ ., data = wdi_train) |>
  update_role(country, new_role = "id") |>
  step_mdist(
    all_predictors(),
    preset = "u_dep"
  )
knn_spec <- nearest_neighbor_dist(
  mode = "classification",
  neighbors = tune()
)
wdi_wf <- workflow() |>
  add_recipe(wdi_rec) |>
  add_model(knn_spec)

Data

Prepare the classification task.

Split

Create training and test sets.

Recipe

Use step_mdist() to construct the distance representation.

Model

Specify a distance-based KNN classifier.

Workflow

Combine preprocessing and model specification.

Tuning a distance-based KNN pipeline

set.seed(123)
wdi_folds <- vfold_cv(
  wdi_train,
  v = 5,
  strata = region
)
knn_grid <- tibble(
  neighbors = c(1, 3, 5, 7, 9, 11, 15)
)
knn_tuned <- tune_grid(
  wdi_wf,
  resamples = wdi_folds,
  grid = knn_grid,
  metrics = metric_set(accuracy)
)
best_k <- select_best(
  knn_tuned,
  metric = "accuracy"
)
final_wf <- finalize_workflow(
  wdi_wf,
  best_k
)
final_res <- last_fit(
  final_wf,
  split = wdi_split,
  metrics = metric_set(accuracy)
)

Resample

Create cross-validation folds on the training set.

Tuning a distance-based KNN pipeline

set.seed(123)
wdi_folds <- vfold_cv(
  wdi_train,
  v = 5,
  strata = region
)
knn_grid <- tibble(
  neighbors = c(1, 3, 5, 7, 9, 11, 15)
)
knn_tuned <- tune_grid(
  wdi_wf,
  resamples = wdi_folds,
  grid = knn_grid,
  metrics = metric_set(accuracy)
)
best_k <- select_best(
  knn_tuned,
  metric = "accuracy"
)
final_wf <- finalize_workflow(
  wdi_wf,
  best_k
)
final_res <- last_fit(
  final_wf,
  split = wdi_split,
  metrics = metric_set(accuracy)
)

Resample

Create cross-validation folds on the training set.

Grid

Define candidate values for the number of neighbours.

Tuning a distance-based KNN pipeline

set.seed(123)
wdi_folds <- vfold_cv(
  wdi_train,
  v = 5,
  strata = region
)
knn_grid <- tibble(
  neighbors = c(1, 3, 5, 7, 9, 11, 15)
)
knn_tuned <- tune_grid(
  wdi_wf,
  resamples = wdi_folds,
  grid = knn_grid,
  metrics = metric_set(accuracy)
)
best_k <- select_best(
  knn_tuned,
  metric = "accuracy"
)
final_wf <- finalize_workflow(
  wdi_wf,
  best_k
)
final_res <- last_fit(
  final_wf,
  split = wdi_split,
  metrics = metric_set(accuracy)
)

Resample

Create cross-validation folds on the training set.

Grid

Define candidate values for the number of neighbours.

Tune

Evaluate each candidate value by cross-validation.

Tuning a distance-based KNN pipeline

set.seed(123)
wdi_folds <- vfold_cv(
  wdi_train,
  v = 5,
  strata = region
)
knn_grid <- tibble(
  neighbors = c(1, 3, 5, 7, 9, 11, 15)
)
knn_tuned <- tune_grid(
  wdi_wf,
  resamples = wdi_folds,
  grid = knn_grid,
  metrics = metric_set(accuracy)
)
best_k <- select_best(
  knn_tuned,
  metric = "accuracy"
)
final_wf <- finalize_workflow(
  wdi_wf,
  best_k
)
final_res <- last_fit(
  final_wf,
  split = wdi_split,
  metrics = metric_set(accuracy)
)

Resample

Create cross-validation folds on the training set.

Grid

Define candidate values for the number of neighbours.

Tune

Evaluate each candidate value by cross-validation.

Select

Choose the best-performing number of neighbours.

Tuning a distance-based KNN pipeline

set.seed(123)
wdi_folds <- vfold_cv(
  wdi_train,
  v = 5,
  strata = region
)
knn_grid <- tibble(
  neighbors = c(1, 3, 5, 7, 9, 11, 15)
)
knn_tuned <- tune_grid(
  wdi_wf,
  resamples = wdi_folds,
  grid = knn_grid,
  metrics = metric_set(accuracy)
)
best_k <- select_best(
  knn_tuned,
  metric = "accuracy"
)
final_wf <- finalize_workflow(
  wdi_wf,
  best_k
)
final_res <- last_fit(
  final_wf,
  split = wdi_split,
  metrics = metric_set(accuracy)
)

Resample

Create cross-validation folds on the training set.

Grid

Define candidate values for the number of neighbours.

Tune

Evaluate each candidate value by cross-validation.

Select

Choose the best-performing number of neighbours.

Finalize

Insert the selected value into the workflow.

Tuning a distance-based KNN pipeline

set.seed(123)
wdi_folds <- vfold_cv(
  wdi_train,
  v = 5,
  strata = region
)
knn_grid <- tibble(
  neighbors = c(1, 3, 5, 7, 9, 11, 15)
)
knn_tuned <- tune_grid(
  wdi_wf,
  resamples = wdi_folds,
  grid = knn_grid,
  metrics = metric_set(accuracy)
)
best_k <- select_best(
  knn_tuned,
  metric = "accuracy"
)
final_wf <- finalize_workflow(
  wdi_wf,
  best_k
)
final_res <- last_fit(
  final_wf,
  split = wdi_split,
  metrics = metric_set(accuracy)
)

Resample

Create cross-validation folds on the training set.

Grid

Define candidate values for the number of neighbours.

Tune

Evaluate each candidate value by cross-validation.

Select

Choose the best-performing number of neighbours.

Finalize

Insert the selected value into the workflow.

Test

Fit the finalized workflow on the training set and evaluate it on the test set.

Tuning results

Test-set performance

Metric	Accuracy
accuracy	0.553

Workflow

last_fit() fits the finalized workflow on the full training set and evaluates it once on the held-out test set.

05 · Examples and practical insights

UCI Bank Marketing Dataset

Applied question

Can we identify distinct customer profiles from stable customer characteristics?

Variable	Type	Description
`age`	Numeric	Age in years
`job`	Categorical	Occupation
`marital`	Categorical	Marital status
`education`	Categorical	Education level
`default`	Binary	Credit currently in default
`balance`	Numeric	Average yearly account balance (€)
`housing`	Binary	Has a housing/mortgage loan
`loan`	Binary	Has a personal loan
`y`	Binary	Subscribed to a term deposit; external only

Choice of candidate distances

Objective

Are conclusions stable across defensible definitions of similarity?

Distance	Preset	Interpretation
Gower	gower	range-normalized Manhattan + simple matching
u_indep	u_indep	standard-deviation scaling + simple matching (commensurable)
Hennig-Liao	hl	standard-deviation scaling + Hennig--Liao indicator scaling
u_mix	u_mix	standard-deviation scaling + total variation distance (commensurable)

First check: what cluster structure is suggested?

Distance	Best k	ASW
u_mix	4	0.331
u_indep	5	0.284
Gower	7	0.263
Hennig-Liao	8	0.165

Within (k=2,,8), different distance constructions may favour different values of (k).

For the remainder of the comparison, however, we deliberately fix (k=6) for every method.

Controlled comparison: hold (k) fixed

What stays constant?

same 1,500 customers + same variables + same PAM algorithm + (k=6)

What changes?

Only the definition of dissimilarity.

This isolates the consequences of distance construction for the resulting geometry and partition.

LOVO across distances

If a variable is removed, how much does the distance representation change?

LOVO across distances

This asks a different question: if a variable is removed, how much does the PAM partition change?

Benchmarking the distance constructions

All solutions use PAM with (k=6).

Therefore, disagreement in ARI reflects changes in cluster membership induced by the distance construction.

One solution in detail: ordinary Gower + PAM

Cluster	n	Age	Balance	Housing	Marital	Education	Main jobs	Subscribed (%)
1	221	33.5	978	yes (100%)	single (82%)	secondary (75%)	technician (39%); blue-collar (20%)	10.9%
2	204	39.0	1994	yes (100%)	married (67%)	tertiary (92%)	management (69%); technician (9%)	11.3%
3	257	51.3	1756	no (100%)	married (88%)	secondary (67%)	blue-collar (29%); retired (17%)	14.0%
4	433	42.0	1095	yes (100%)	married (88%)	secondary (75%)	blue-collar (43%); admin. (19%)	6.9%
5	187	33.3	1092	no (100%)	single (83%)	secondary (71%)	technician (33%); blue-collar (13%)	19.3%
6	198	41.4	1948	no (100%)	married (66%)	tertiary (89%)	management (67%); technician (9%)	15.2%

This table gives a substantive interpretation of one of the four (k=6) solutions.

The previous analyses tell us which parts of that segmentation are sensitive—or robust—to changing the distance definition.

External profiling: term-deposit uptake

Distance	k	ASW	Cramér's V	Min subscribed	Max subscribed
Gower	6	0.262	0.124	6.9%	19.3%
u_indep	6	0.258	0.138	4.3%	16.3%
Hennig-Liao	6	0.136	0.150	5.1%	21.4%
u_mix	6	0.293	0.135	4.4%	16.0%

y was not used to construct any distance or clustering solution.

All methods again use (k=6), so these are comparable external descriptions of the four partitions.

MCA of the clustering solutions

Clusters that lie close in the MCA representation tend to have similar patterns of customer membership.

Which customer groups persist?

Which clusters obtained under different distance definitions actually contain the same customers?

Strongest cross-method cluster correspondences

Cluster 1	Cluster 2	n1	n2	Shared	Jaccard	% of cluster 1	% of cluster 2
u_indep: C6	u_mix: C6	27	27	27	1.00	100%	100%
u_indep: C3	u_mix: C3	184	183	183	0.99	99%	100%
u_indep: C2	u_mix: C2	408	419	403	0.95	99%	96%
u_indep: C5	u_mix: C5	301	286	277	0.89	92%	97%
u_indep: C1	u_mix: C1	295	298	278	0.88	94%	93%
u_indep: C4	u_mix: C4	285	287	264	0.86	93%	92%
Hennig-Liao: C3	u_indep: C3	178	184	167	0.86	94%	91%
Hennig-Liao: C3	u_mix: C3	178	183	166	0.85	93%	91%

Jaccard measures symmetric overlap.

The final two columns show how much of each cluster is recovered by its corresponding cluster under the alternative distance.

What do the recurring customer groups look like?

Cluster 1	Cluster 2	Shared n	Mean age	Mean balance	Housing	Marital	Education	Main job	Term-deposit uptake
u_indep: C6	u_mix: C6	27	38.6	-129	yes (67%)	married (52%)	secondary (70%)	blue-collar (33%)	11.1%
u_indep: C3	u_mix: C3	183	39.7	1030	yes (73%)	married (67%)	secondary (67%)	blue-collar (27%)	4.4%
u_indep: C2	u_mix: C2	403	41.8	1217	yes (100%)	married (90%)	secondary (67%)	blue-collar (36%)	7.7%
u_indep: C5	u_mix: C5	277	38.4	1936	no (73%)	married (63%)	tertiary (96%)	management (73%)	15.2%
u_indep: C1	u_mix: C1	278	33.0	1489	yes (75%)	single (98%)	secondary (64%)	technician (26%)	16.5%
u_indep: C4	u_mix: C4	264	50.3	1713	no (100%)	married (81%)	secondary (68%)	blue-collar (24%)	15.5%

These profiles describe the shared core of customers assigned together under two different distance constructions.

They reveal substantive customer groups that persist when the definition of similarity changes.

Practical insight from the application

Not: default distance → clustering → interpretation

A clustering result inherits the assumptions of the distance construction.

Instead: distance-aware experimentation

define plausible distances → inspect variable behaviour → compare representations → compare partitions → identify recurring groups → interpret

With manydist, distance choice becomes something we can investigate rather than something we silently inherit from a default.

Main references

van de Velden, M., Iodice D’Enza, A., Markos, A., & Cavicchia, C. (2026). Unbiased mixed-variable distance. Journal of Computational and Graphical Statistics. https://doi.org/10.1080/10618600.2026.2680181

van de Velden, M., Iodice D’Enza, A., Markos, A., & Cavicchia, C. (2024). A general framework for implementing distances for categorical variables. Pattern Recognition, 153, 110547.

Kaufman, L., & Rousseeuw, P. J. (2009). Finding groups in data: An introduction to cluster analysis. John Wiley & Sons.

Borg, I., & Groenen, P. J. (2005). Modern multidimensional scaling: Theory and applications. Springer.

Le, S. Q., & Ho, T. B. (2005). An association-based dissimilarity measure for categorical data. Pattern Recognition Letters, 26(16), 2549–2557.

Ng, A. Y., Jordan, M. I., & Weiss, Y. (2002). On spectral clustering: Analysis and an algorithm. Advances in Neural Information Processing Systems, 14, 849–856. MIT Press.

CARME 2027 in Corfu

June 22–25, 2027

Ionian University
Corfu, Greece

The tenth international CARME conference brings together work on correspondence analysis, categorical data, visualization, clustering, and related methods.

Invited speakers

Mikael Börjesson
Uppsala University

Paul Eilers
Erasmus University Medical Center

Patrick J. F. Groenen
Erasmus School of Economics

Trevor Hastie
Stanford University

Heungsun Hwang
McGill University

Ndèye Niang
CNAM Paris

carme2027corfu.wixsite.com/conference

Contact and slides

Contact

Alfonso Iodice D’Enza
iodicede@unina.it

Angelos Markos
amarkos@eled.duth.gr

Distance-based Learning for Mixed Data: The manydist Package

Outline

01 · Setting the scene

Learning from mixed-type data

Desirable properties1

Desirable properties1

Mixed-data setup

Distributions, scaling and bias: the numeric case

Distributions, scaling and bias: the categorical case

Independence-based distances

by variable differences: independence-based

by variable differences: independence-based

by variable differences: independence-based

accounting for inter-variable relations: association-based

association-based distance

association-based distance

association-based distance

association-based distance: a small example

association-based distance: from \({\bf Z}\) to \({\bf P}\)

diagonal blocks contain marginal information; off-diagonal blocks contain joint proportions

association-based distance: from \({\bf P}\) to \({\bf R}\)

each off-diagonal block contains conditional profiles across variables

association-based distance

association-based distance: reading \({\bf R}_{12}\)

categories are compared through their association profiles

association-based distance: from \({\bf R}\) to \(\Delta_1^{(tvd)}\)

\(A\) and \(B\) are close because they have the same profile with respect to \(X_2\)

association-based distance: reading \({\bf R}_{21}\)

categories of \(X_2\) are compared through their association profiles with respect to \(X_1\)

association-based distance: from \({\bf R}_{21}\) to \(\Delta_2^{(tvd)}\)

\(u\) and \(v\) are different because they imply different profiles over \(X_1\)

association-based distance: from \(\Delta\) to \({\bf D}\)

category-level dissimilarities are translated into observation-level distances

02 · Building distances with manydist

Building distance-based pipelines

the WDI data

mdist(): the main arguments

mdist() output: an MDist R6 object

Constructing a mixed-type distance using a preset

step_mdist() embeds the same distance specification into a modelling workflow

Constructing a custom mixed-type distance

03 · Distance diagnostics

No one-size-fits-all distance

LOVO-based diagnostics: what is evaluated?

lovo_mdist(): controlling the diagnostics

compare_lovo_mdist(): WDI example

WDI: MDS geometry

WDI: PAM clustering stability

benchmark_mdist(): comparing distance choices

benchmark_mdist(): WDI summary

benchmark_mdist(): WDI pairwise comparison

04 · Distance-based learning pipelines

Distance-based classification pipeline

Distance-based classification pipeline

Distance-based classification pipeline

Distance-based classification pipeline

Distance-based classification pipeline

Tuning a distance-based KNN pipeline

Tuning a distance-based KNN pipeline

Tuning a distance-based KNN pipeline

Tuning a distance-based KNN pipeline

Tuning a distance-based KNN pipeline

Tuning a distance-based KNN pipeline

Tuning results

05 · Examples and practical insights

UCI Bank Marketing Dataset

Choice of candidate distances

First check: what cluster structure is suggested?

Controlled comparison: hold (k) fixed

LOVO across distances

LOVO across distances

Benchmarking the distance constructions

One solution in detail: ordinary Gower + PAM

External profiling: term-deposit uptake

MCA of the clustering solutions

Which customer groups persist?

Strongest cross-method cluster correspondences

What do the recurring customer groups look like?

Practical insight from the application

Main references

Distance-based Learning for Mixed Data:
The manydist Package

Desirable properties¹

Desirable properties¹

02 · Building distances with `manydist`

`mdist()`: the main arguments

`mdist()` output: an `MDist` R6 object

`step_mdist()` embeds the same distance specification into a modelling workflow

`lovo_mdist()`: controlling the diagnostics

`compare_lovo_mdist()`: WDI example

`benchmark_mdist()`: comparing distance choices

`benchmark_mdist()`: WDI summary

`benchmark_mdist()`: WDI pairwise comparison