# DimReduction - Interactive Graphic Environment for Dimensionality Reduction

###### Abstract

Feature selection is a pattern recognition approach to choose important
variables according to some criteria to distinguish or explain certain
phenomena. There are many genomic and proteomic applications which rely on
feature selection to answer questions such as: selecting signature genes which
are informative about some biological state, e.g. normal tissues and several
types of cancer; or defining a network of prediction or inference among
elements such as genes, proteins, external stimuli and other elements of
interest. In these applications, a recurrent problem is the lack of samples to
perform an adequate estimate of the joint probabilities between element
states. A myriad of feature selection algorithms and criterion functions are
proposed, although it is difficult to point the best solution in general. The
intent of this work is to provide an open-source multiplataform graphical
environment to apply, test and compare many feature selection approaches
suitable to be used in bioinformatics problems.

Fabrício Martins Lopes(1,2), David Correa Martins-Jr(1) and Roberto M. Cesar-
Jr(1)

(1)Instituto de Matemática e Estatística da Universidade de São Paulo, Brazil.

(2)Universidade Tecnológica Federal do Paraná, Brazil.

##  1 Introduction

The pattern recognition methods allow the classification of objects or
patterns in a number of classes [1]. Specifically in statistical pattern
recognition, given a set $Y=\\{y_{1},...,y_{c}\\}$ of classes and an unknown
pattern $\mathbf{X}=\\{X_{1},X_{2},...,X_{n}\\}$, a pattern recognition system
associates $\mathbf{x}$ to a class $y_{i}$ based on defined measures in a
feature space. In many applications, especially in bioinformatics, the feature
space dimension tends to be very large, making difficult the classification
task. In order to overcome this inconvenient situation, the study of
dimensionality reduction problem in pattern recognition becomes imperative.

The so called “curse of dimensionality” [2] is a phenomenon in which the
number of training samples required to a satisfactory classifier performance
is given by an exponential function of the feature space. This is the main
motivation by which performing of dimensionality reduction is important in
problems with large number of features and small number of training samples.
Many bioinformatics applications are perfectly inserted in this context. Data
sets containing mRNA transcription expressions from microarray or SAGE, for
example, possess thousands of genes (features) and only some dozens of samples
that may be cell states or types of tissues. If time is a factor involved, the
samples are called dynamical states, otherwise they are called steady states.

There are basically two dimensionality reduction approaches: feature
extraction and feature selection [1, 3, 4]. The feature extraction methods
create new features from transformations or combinations of the original
feature set. On the other hand, feature selection algorithms just search for
the optimal feature subset according to some criterion function. The software
proposed in this paper is initially focused on feature selection methods.

A feature selection method is composed by two main parts: a search algorithm
and a criterion function. As far as the search algorithms, there are two main
categories: the optimal and sub-optimal algorithms. The optimal algorithms
(including exhaustive and branch-and-bound searches) return the best feature
subspace, but their computational costs are very high to be applied in
general. The sub-optimal algorithms do not guarantee that the solution is
optimal, but some of them present a reasonable cost-benefit between
computational cost and quality of the solution. Up to now, we have implemented
in the software the exhaustive search (optimal), the Sequential Forward
Selection (SFS - sub-optimal) and the Sequential Forward Floating Selection
(SFFS - sub-optimal with excellent cost-benefit) [5].

There is a large number of criterion functions proposed in the literature. The
most common functions are based on the classifier error and distances between
patterns. There are also criterion functions based on information theory. They
are closely related to the classifier error, but instead of using the error,
it is based on the conditional entropy of the class probabilities
distributions given the observed pattern.

Due to the curse of dimensionality phenomenon, error estimation is a crucial
issue. We have developed some ways to embed error estimation in the criterion
functions based on classifier error or conditional entropy. The main idea is
based on penalization of non-oberved or rarely observed instances. A good
advantage in doing this is that the right dimension of the feature subset
solution is also estimated (the dimension parameter is not required). After
the feature selection, it is possible to apply classical error estimation
techniques like resubstitution, leave-one-out, cross validation or bootstrap.

The software is implemented in Java, so it can be executed in many operational
systems. It is open source and intended to be continuously developed in a
world-wide collaboration. The software is available at
http://dimreduction.incubadora.fapesp.br/.

Following this introduction, Section 2 and 3 will describe the feature
selection algorithms and criterion functions implemented so far. Section 4
discusses the implemented software. Section 5 will shows some preliminary
results obtained on gene regulation networks and classification of breast
cancer cells. This paper is finalized with some conclusions in Section 6.

##  2 Implemented feature selection algorithms

The first and simpler feature selection algorithm implemented in this work is
the exhaustive search. This algorithm searches the whole search space, and as
a result, the selected features are optimal. However in bioinformatics
context, normally the computational cost makes this approach inadequate. Then,
it is clear the existence of a trade-off between optimality and computational
cost.

An alternative way is to adopt sub-optimal search methods. In this work we
have implemented two sub-optimal approaches with unique solution, which are
known as top down and bottom up. In the first one, the selection subset starts
empty and features are inserted by optimizing a criterion function until a
stop condition is satisfied, which is often based on the subset size or a
threshold. In the second algorithm, the subset starts full and features are
removed, trying to optimize the criterion function until a stop condition is
reached. Methods that implement these approaches are known as SFS (Sequential
Forward Search) and SBS (Sequential Backward Search), respectively.
Considering the context of this work, our choice was to implement the SFS
approach.

However, these suboptimal search methods present an undesirable drawback known
as nesting effect. This effect happens because the discarded features in the
top-down approach are not inserted anymore, or the inserted features in the
bottom-up approach are never discarded.

In order to circumvent this problem, the Sequential Forward Floating Selection
(SFFS) [5] was also implemented. The SFFS algorithm tries to avoid the nesting
effect allowing to insert and exclude features on subset in a floating way,
i.e. without defining the number of insertions or exclusions.

The SFFS may be formalized as in [5]. Let $\mathbf{X_{k}}=\\{x_{i}:1\leq i\leq
k,x_{i}\in\mathbf{X}\\}$ be the subset with $k$ features of the complete set
$\mathbf{X}=\\{x_{i}:1\leq i\leq n\\}$ with $n$ features available. Let
$E(\mathbf{X_{k}})$ the criterion function value for the subset
$\mathbf{X_{k}}$. The algorithm initializes with $k=0$, therefore the subset
$\mathbf{X_{k}}$ is empty.

First Step (insert): using the SFS method, select the feature $x_{k+1}$ of the
set $\mathbf{X}-\mathbf{X_{k}}$ to form the set $\mathbf{X_{k+1}}$, such that
$x_{k+1}$ be the most relevant feature of the subset $\mathbf{X_{k}}$. The new
subset is $\mathbf{X_{k+1}}=\mathbf{X_{k}}\cup x_{k+1}$.

Second Step (conditional exclusion): Find the least relevant feature in the
set $\mathbf{X_{k+1}}$. If $x_{k+1}$ is the least relevant feature in the
subset $\mathbf{X_{k+1}}$, then $k\leftarrow k+1$,
$\mathbf{X_{k}}\leftarrow\mathbf{X_{k+1}}$ and back to the first step. If
$x_{r},1\leq r\leq k$ is the least relevant feature in the subset
$\mathbf{X_{k+1}}$, then exclude $x_{r}$ from $\mathbf{X_{k+1}}$ to form a new
subset $\mathbf{X^{\prime}_{k}}=\mathbf{X_{k+1}}-x_{r}$ and $k\leftarrow k-1$.
If $k=2$, then $\mathbf{X_{k}}=\mathbf{X^{\prime}_{k}}$, and return to the
first step, else execute the third step.

Third Step (continuation of conditional exclusion): Find the least relevant
feature $x_{s}$ in the set $\mathbf{X^{\prime}_{k}}$. If
$E(\mathbf{X^{\prime}_{k}}-x_{s})\leq E(\mathbf{X_{k-1}})$, then
$\mathbf{X_{k}}\rightarrow\mathbf{X^{\prime}_{k}}$ and return to first step.
If $E(\mathbf{X^{\prime}_{k}}-x_{s})>E(\mathbf{X_{k-1}})$ then exclude $x_{s}$
from $\mathbf{X^{\prime}_{k}}$ to form a new reduced subset
$\mathbf{X^{\prime}_{k-1}}=\mathbf{X^{\prime}_{k}}-x_{s}$ and $k\rightarrow
k-1$. If $k=2$, then $\mathbf{X_{k}}=\mathbf{X^{\prime}_{k}}$ and return to
first step, else repeat the third step.

The SFFS algorithm starts by setting $k=0$ e $\mathbf{X_{k}}=0$, and the SFS
method is used until the subset size $k=2$.Then the SBS is performed in order
to exclude bad features. SFFS proceeds by alternating between SFS and SBS
until a stop criteria is reached. The best result set for each cardinality is
stored in a list. The best set among them is selected as algorithm result, and
tie occurs, the set with lower cardinality is selected.

##  3 Implemented criterion functions

We implemented criterion functions based on classifier information (mean
conditional entropy) and classifier error (Coefficient of Determination [6]),
introducing some penalization on poorly or non-observed patterns.

###  3.1 Mean conditional entropy

The information theory was originated by Shannon [7] and can be employed on
feature selection problems [3]. The Shannon’s entropy $H$ is a measure of
randomness of a variable $Y$ given by:

$H(Y)=-\sum_{y\in Y}P(y)logP(y)\textrm{,}$ (1)

where $P$ is the probability distribution function. By convention $0\cdot
log0=0$.

The conditional entropy is a fundamental concept related to the mutual
information. It is given by the following equation:

$H(Y|\mathbf{X}=\mathbf{x})=-\sum_{y\in
Y}P(y|\mathbf{X}=\mathbf{x})logP(y|\mathbf{X}=\mathbf{x})$ (2)

where $\mathbf{X}$ is a feature vector and $P(Y|\mathbf{X}=\mathbf{x})$ is the
conditional probability of $Y$ given the observation of an instance
$\mathbf{x}\in\mathbf{X}$. And finally, the mean conditional entropy of $Y$
given all the possible instances $\mathbf{x}\in\mathbf{X}$ is given by:

$H(Y|\mathbf{X})=\sum_{\mathbf{x}\in\mathbf{X}}P(\mathbf{x})H(Y|\mathbf{x})$
(3)

Lower values of $H$ yield better feature subspaces (the lower $H$, the larger
is the information gained about $Y$ by observing $\mathbf{X}$).

###  3.2 Coefficient of Determination

The Coefficient of Determinstion (CoD) [6], like the conditional entropy, is a
non-linear criterion useful for feature selection problems [8]. It is given
by:

$CoD_{Y}(\mathbf{X})=\frac{1-\max_{y\in
Y}P(y)-(1-\sum_{\mathbf{x}\in\mathbf{X}}P(\mathbf{x})\max_{y\in
Y}P(y|\mathbf{x}))}{1-\max_{y\in Y}P(y)}$ (4)

where $1-\max_{y\in Y}P(y)$ is the error of predicting $Y$ in the absence of
other observations (let us denote it by $\varepsilon_{Y}$) and
$1-\sum_{\mathbf{x}\in\mathbf{X}}\max_{y\in Y}P(\mathbf{x},y)$ is the error of
predicting $Y$ based on the observation of $\mathbf{X}$ (let us denote it by
$\varepsilon_{Y}(\mathbf{X})$). Larger values of $CoD$ yield better feature
subspaces ($CoD=0$ means that the feature subspace does not improve the priori
error and $CoD=1$ means that the error was fully eliminated).

###  3.3 Penalization of non-observed instances

A way to embed the error estimation caused by using feature vectors with large
dimensions and insufficient number of samples is to involve non-observed
instances in the criterion value calculus [9]. A positive probability mass is
attributed to the non-observed instances and their contribution is the same as
observing only the $Y$ values with no other observations.

In the case of mean conditional entropy, the non-observed instances get the
entropy equal to $H(Y)$ and, for the $CoD$, they get the prior error
$\varepsilon_{Y}$ value. The probability mass for the non-observed instances
is parametrized by $\alpha$. This parameter is added to the relative frequency
(number of occurrences) of all possible instances. So, the mean conditional
entropy with this type of penalization becomes:

$H(Y|\mathbf{X})=\frac{1}{\alpha
M+s}\left[\alpha(M-N)H(Y)+\sum_{i=1}^{N}(f_{i}+\alpha)H(Y|\mathbf{X}=\mathbf{x_{i}})\right]$
(5)

where $M$ is the number of possible instances of the feature vector
$\mathbf{X}$, $N$ is the number of observed instances (so, the number of non-
observed instances is given by $M-N$), $f_{i}$ is the relative frequence
(number of observations) of the instance $\mathbf{x_{i}}$ and $s$ is the
number of samples.

And $CoD$ becomes:

$CoD_{Y}(\mathbf{X})=\frac{\varepsilon_{Y}-\left[\frac{\alpha(M-N)\varepsilon_{Y}}{\alpha
M+s}+1-\sum_{i=1}^{N}\frac{(f_{i}+\alpha)}{\alpha M+s}\max_{y\in
Y}P(y|\mathbf{x_{i}})\right]}{\varepsilon_{Y}}$ (6)

###  3.4 Penalization of rarely observed instances

In this penalization, the non-observed instances are not taken into account.
This penalization consists in changing the conditional probability
distribution of the instances that have just a unique observation [10]. It
makes sense because if an instance $\mathbf{x}$ has only 1 observation, the
value of $Y$ is fully determined ($H(Y|\mathbf{X}=\mathbf{x})=0$ and
$CoD_{Y}(\mathbf{X})=1$), but the confidence about the real distribution of
$P(Y|\mathbf{X}=\mathbf{x})$ is very low. A parameter $\beta$ gives a
confidence value that $Y=y$. The main idea is to distrubute $1-\beta$ equally
over all $P(Y\neq y|\mathbf{X}=\mathbf{x})$ and to attribute $\beta$ to
$P(Y=y|\mathbf{X}=\mathbf{x})$. In Barrera et al [10], the $\beta$ value is
$\frac{1}{|Y|}$ where $|Y|$ is the number of classes (cardinality of $Y$),
becoming the uniform distribution (strongest penalization).

Adapting this penalization to the Equation 3, the mean conditional entropy
becomes:

$H(Y|\mathbf{x})=\frac{M-N}{s}H(F(Y))+\sum_{\mathbf{x}\in\mathbf{X}:P(\mathbf{x})>\frac{1}{s}}P(\mathbf{x})H(Y|\mathbf{x})\textrm{,}$
(7)

where $F(Y)$ is the probability distribution given by

$F(i)\>=\>\left\\{\begin{array}[]{ll}\beta,&\textrm{if }i=1\\\
\frac{1-\beta}{c-1},&\textrm{if }i=2,3...,c\end{array}\right.\textrm{,}$

and $N$ in this case is the number of instances $\mathbf{x}$ with
$P(\mathbf{x})>\frac{1}{s}$ (more than one observation).

Since $\varepsilon_{Y}(\mathbf{x})=1-\beta$ when
$P(Y|\mathbf{x})=\frac{1}{t}$, the $CoD$ with this penalization is given by:

$CoD_{Y}(\mathbf{X})=\frac{\varepsilon_{Y}-(1-\frac{(M-N)}{s}\beta-\sum_{\mathbf{x}\in\mathbf{X}:P(\mathbf{x})>\frac{1}{s}}P(\mathbf{x})max_{y\in
Y}P(y|\mathbf{x}))}{\varepsilon_{Y}}$ (8)

###  3.5 Classifier design and generalization

After the feature selection using $H$ or $CoD$, the classifier is designed
from the table of conditional probabilities where each row is a possible
instance $\mathbf{x}\in\mathbf{X}$, each column is a possible class $Y=y$ and
each cell of this table represents $P(Y|\mathbf{X}=\mathbf{x})$. This table is
used as a Bayesian classifier where, for each given instance, the chosen label
$Y=y$ is the one with maximum conditional probability for the considered
instance. In case of instances that have two or more labels of maximum
probability (including non-observed instances), it is possible to generalize
these instances according to some criterion. A commonly used criterion is the
nearest neighbors with some distance metric [1]. We implemented the nearest
neighbors using Euclidean distance. In this implementation, the nearest
neighbors are taken successively. The occurrences of each label are summed
until only one of such labels has the maximum number of occurrences and may be
chosen as the class to which the considered instance belongs. This featured
can be turned off. In this case, the label is guessed, i.e., chosen randomly
from the labels with maximum number of occurrences (including non-observed
instances).

##  4 Software description

(a) Upload the biological data

(b) Quantization process

(c) Single execution

(d) Cross-validation

Figure 1: Application panels.

(a) Scatterplot

(b) Parallel coordinates

Figure 2: Examples of scatterplot and parallel coordinates generated by the
software.

The software is implemented in Java in order to be executable in different
platforms. It is open source and intended to be continuously developed in a
world-wide collaboration. The software is available at
http://dimreduction.incubadora.fapesp.br/.

There are four main panels: the first panel allows the user to load the data
set (Figure 1-a). The second is optional for the user to define a quantization
degree to the data set. The quantized data may be visualized (Figure 1-b). It
is worth noting that some feature selection criteria like mean conditional
entropy or CoD require data quantization to discrete values. This fact
explains the quantization step available in the software. The data
quantization is based on a common rule, searching for the extreme values
(positive and negative) and dividing equally the negative and positive space
considering the number of divisions specified by the quantization degree
parameter.

The next step can be the single execution or cross-validation. The first one
is dedicated to perform single tests (Figure 1-c). It is represented by a
panel where the user is able to enter input parameters such as the feature
selection algorithm (see Section 2 for the algorithms implemented) and the
criterion function (see Section 3 for the criteria implemented). Other
implemented utilities, including the visualization results of the feature
selection, area found in the middle of the panel. There are three forms to
visualize the results: graphs (Figure 4), scatterplot (Figure 2-a) and
parallel coordinates (Figure 2-b). The graphs show the connections among
different classes, chosen in feature selection execution, as directed edges
between selected vertices. The parallel coordinates proposed by [11] allows to
visualize in adjacent axes (selected features) similar patterns of behavior in
data, visually indicating how separated are the classes, considering the
adjacent features. In the software application, the features and it and its
order to build he parallel coordinates chart are defined by the user.

The cross-validation panel (Figure 1-d) is very similar to the prior. Cross-
validation [12] consists in to divide the whole data set in two subsets:
training and test, mutually exclusive, and the user can define the size of
both sets. The training set is entered as input to the feature selection
algorithm. The classifier designed from the feature selection and the joint
probability distributions table labels the test set samples. At the end of the
cross-validation process, it is plotted a chart with the results of each
execution, and it is possible to visualize the rate of hits and its variation
along the executions.

Another available option is the generalization of non-observed instances. With
this option selected, the instances of the selected feature set not present in
the training samples are generalized by a nearest neighbors method [1] with
Euclidean distance (see Section 3.5 for more details). This method is also
applied to take a decision among classes with tied maximum conditional
probability distributions given a certain instance.

##  5 Illustrative Results

This section presents the results in two main aspects. Initially the software
was applied as feature selection in a biological classification problem to
classify breast cancer cells in two possible classes: benign and malignant.
The biological data used here was obtained from [13] which has 589 instances
and 32 features. The results shown figure 3, presents very low variations and
high accurate classification achieving 99.96% of accuracy on average.

Figure 3: Cross-validation results using 10 executions, 80% of data as
training set and 20% as test set. Figure 4: Identified network: dashed lines
represent the false positives and solid lines the positives. There are no
false negatives.

The second computational biology problem addressed was gene network recovery.
In this case we used an artificial gene network generated by the approach
presented in [14]. The parameters used were: 10 nodes, binary quantization, 20
observations (timestamps), 1 average of edges per vertex and Random graphs of
Erdös-Rényi as network architecture. In figure 4, it is presented the network
recovered. This result did not present false negatives and just few false
positives.

##  6 Conclusion

The proposed feature selection environment allows data analysis using several
algorithms, criterion functions and graphic visualization tools. Since it is
an open-source and multi-platform software, it is suitable for the user that
wants to analyze data and draw some conclusions about it, as well as for the
specialist that has as objective to compare several combinations of approaches
and parameters for each specific data set or to include more features in the
software such as a new algorithm or a new criterion function. This system can
evolve and include feature extraction methods as well, not limited only to
feature selection methods.

The environment can be used in many pattern recognition applications, although
the main concern is with Bioinformatics tasks, especially those involving
high-dimensional data (large number of genes, for example) with small number
of samples. Even users not familiar with programming are allowed to manipulate
the software in an easy way, just by clicking to select file inputs,
quantization, algorithms, criterion functions, error estimation methods and
visualization of the results. The environment is implemented as “wizard
style”, i.e., it has tabs delimiting each procedure.

This software opens a great space for future works. The next step consists in
the implementation of other classical feature selection algorithms (e.g. GSFS
and PTA [1, 15]), criterion functions (e.g. based on distances between classes
[1]), error estimation methods (e.g. Leave-one-out and Bootstrap) and then the
inclusion of classical methods of feature extraction (e.g. PCA [16]).

## Acknowledgement

This work was supported by FAPESP, CNPq and CAPES.

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# Electronic stopping power: the IAEA database and state of the art
experimental knowledge

C. C. Montanari  F. Bivort-Haiek  P. Dimitriou  L. Marian  A. M. P. Mendez  J.
P. Peralta  Instituto de Astronomía y Física del Espacio, CONICET –
Universidad de Buenos Aires,
Buenos Aires, Argentina. [label1] organization=Instituto de Astronomía y
Física del Espacio. CONICET and Universidad de Buenos Aires, addressline=IAFE,
Ciudad Universitaria, postcode=1428, city=Buenos Aires, country=Argentina

###### Abstract

In this contribution, we review the state of the art of the experimental
electronic stopping power data available in the IAEA database. We examine
thoroughly the abundance or lack of data, quality of recent measurements, and
trends. We analyze the evolution in time of the experimental stopping power
measurements, characteristics, achievements, and failures. We also discuss
comparisons with SRIM-2013 calculations for many interesting cases. Sparsely
measured systems and energy regions are observed, stressing the need for new
and reliable data and independent theoretical predictions.

###### keywords:

stopping-power , ion , electrons , IAEA

††journal: Nuclear Instruments and Methods in Physics Research B

[label2] organization=Division of Physical and Chemical Sciences,
International Atomic Energy Agency, addressline=Vienna International Centre,
P.O. Box 100, postcode=A-1400, city=Vienna, country=Austria

##  1 Introduction

The electronic stopping power cross-sections are relevant to a wide range of
applications, from ion beam analysis, deposition ranges, ion implantation, and
radiation damage to medical studies and treatments. Reliable stopping powers
are also needed in isotope production for medical applications, fusion
technologies, non-destructive assay for nonproliferation, and detector
developments. By the end of 2015, the International Atomic Energy Agency
(IAEA) became responsible for hosting, maintaining, updating, and improving
the electronic stopping power database [1, 2]. This database is the legacy of
Helmut Paul [3, 4, 5], pioneer of this field, who in the 1990s had the vision
of making all stopping power data available to the whole scientific community.
The database includes almost a century of experimental measurements in
different laboratories worldwide, including the early measurements by
Rosenblum in 1928 [6] for alpha particles in Li, Cs, Zn, Pd, and Pb, and by
Bätzner in 1936 [7] for hydrogen in Al, Cu, Sn, Ag, and Au. Since 2015, the
database has been periodically updated with newly published data [1].

The IAEA Stopping Power database currently contains experimental values for
1526 collisional systems, of which around 60% (913) have only one dataset,
while 233 have two datasets, which do not necessarily overlap in energies.
This implies that less than $30\%$ of the systems in the database can be
considered reasonably well-known from an experimental point of view. Most of
the compiled experimental measurements have been performed on solids, but
there are also data for gases and liquids, single atomic targets, and
compounds.

A large number of systems with a single dataset is well described by semi-
empirical methods, such as SRIM-2013 [8]. This agreement is expected but does
not represent an actual independent comparison. SRIM-2013 calculations agree
with data published before 2013 within 5-10% for all ions [9, 10]. The
agreement worsens to an average of 11.4% (19.2% for H ion, 10.6% for He ion,
6.6% for the rest) for measurements performed after 2013, which were not
considered in the empirical fit [11]. On the other hand, determining the
target thickness or normalizing data with SRIM-2013 at high energies (far from
the stopping maximum) are always conflicting points of discussion.

The stopping of slow ions in matter is a major topic of research in
fundamental physics. It depends on the electronic configuration and the target
type (metal or insulator, solid or gas); for compounds, the low energy region
coincides with the region where the Bragg rule is unreliable. Experimentally,
the determination of the electronic stopping power of slow ions is complex (it
requires separation of nuclear and electronic stopping), resulting in sparse
and dispersed values, as we will show in this work.

This contribution aims to review the state of the art of the electronic
stopping power from the point of view of the experimental data by considering
the abundance or lack of data, preferred ion-target systems in recent
measurements, trends, and reliability of data. We highlight the need for new
measurements (ion-target systems, energy regions) and models with descriptive
and predictive power but also recognize the importance of assessing the
existing data and making recommendations to the user community.

The measured systems and data acquisition over time will be shown in Section
2. Some examples of interest are analyzed in Section 3. Particularly, we
examine water, due to its germaneness in medical applications, and oxides with
many recent measurements, i.e., Ta2O5 and TiO2. Challenges, limitations,
differences among data sets, and energy regions without values are presented
and commented on in Section 4. Finally, we summarize our conclusions in
Section 5.

##  2 Data dissemination and exploration

Figure 1: Datasets in the database.

The IAEA Stopping Power database (https://nds.iaea.org/stopping/) was recently
modernized. The data files and metadata were sorted, restructured and coupled
to an API leading to a more versatile and dynamic exploration of the content
of the database (including data and references) and flexible retrieval of the
data. The data content was kept practically the same, although several
improvements were made to complete relevant descriptors. The layout of the
webpage is now more user-friendly, with the possibility of selecting different
ion-target systems. All the systems can be inspected on-demand by ion-target
combination or author. The data is displayed with interactive plots, extensive
data tables, and hyperlinks to the online publications. The retrieval of the
data and publications can be done via the user-friendly interface as well as
with direct use of the API.

Figure 1 shows a picture of the existing stopping power experimental data.
Each bubble represents one ion whose dimension is proportional to the number
of datasets. It is evident from this figure that most of the measurements
correspond to H and He ions, followed by O, Li, and N. The distribution of
targets per ion is also illustrated in the figure, with $58\%$ of the systems
composed of atomic targets and $42\%$ of compounds. We draw attention to these
proportions, stressing the importance of measuring stopping in compounds to
extend the knowledge beyond the Bragg additive rule. In Figure 1, we highlight
with dark grey the single dataset per ion-target combinations. As mentioned
before, the collisional systems with single ones add up to 60%; however, we
can see from the figure that there are certain ions, such as W, Mo, and Ag,
for which the large majority of measurements are single datasets. Among all
the targets, some atomic ones have no experimental stopping values by any
impinging ion, namely, Na, P, K, S, I, Ba, Os, As, Pm, Eu, and Tl.

Figure 2: Distribution of total publications, datasets, and measurements over
the years.

A representation of the evolution of the experimental data over the years is
displayed in Figure 2, where we considered the publication date, the number of
datasets measured, and the number of data points. The number of publications
on stopping power per year is larger than the one shown in the left plot of
Figure 2. This plot counts only for publications of the experimental data
featured in the database, which only considers absolute values as a function
of the impact energies. Publications on other related topics, such as relative
measurements, differential stopping cross-sections, comparisons, simulations,
extrapolations, applications, and theoretical models, are not considered. The
middle plot shown in Figure 2 separates the number of datasets by ion: H, He,
and heavier ones. It can be noted that most of the measurements since 1980
correspond to heavy ions. The number of measurements over the years displayed
in the right plot of Figure 2 shows an interesting peak around 1980 and 2015.
In the first case, it corresponded to an increment in the number of datasets
published, while in the second one, this feature may be related to the
improvement of experimental techniques and systematization of data
acquisition.

As a general comment on Figure 2, despite the lack of stopping data already
mentioned and the partial energy range coverage, the number of publications,
measured datasets, and data points has decreased since 2015. It is likely that
the overconfidence in the stopping power values provided by SRIM-2013 [8, 9,
10] could explain this phenomenon. It is a fact that SRIM-2013 [8] can predict
the stopping power of any ion-target system, even of compounds not measured
yet, as long as the Bragg rule is valid.

##  3 Examples of interest

In this section, we present an in-depth analysis of three compounds of
interest: water, titanium oxide, and tantalum oxide.

###  3.1 Stopping power of water

The case of water is interesting for biological targets and medical
applications, mainly hadron-therapy [12, 13]. Only four ions in water feature
energy loss experimental data are available: H, He, Li, and C, as shown in
Figure 3. The stopping power has been scaled with $Z^{2}$, which provides an
interesting overlapping of the different ion data at high energies (per
nucleon). In this figure, we include the experimental data for gas (g), solid
(s), and liquid (l) water with markers of different colors per ion and type
per dataset. The dashed curves correspond to SRIM-2013 [8] for gas water. The
measurements of protons in water cover an extended energy range and different
water phases. However, the other cases are sparsely measured. Carbon in water
is the system less examined, which is surprising considering its importance in
carbon ion therapy [14, 15, 16]. From Figure 3, one can make the following
remarks: solid-gas-liquid differences are important below the stopping
maximum, while they are negligible for energies above 1 MeV/amu. In the low
energy region, the measurements for solid water date from the 50s and 70s [2].
New measurements for protons in ice with up-to-date techniques would be very
interesting. The scaled stopping cross-sections allow us to inter-compare
different ions and phases, which is an extra test for the experimental data.
The data for He and Li ions is forty years old and has already been considered
in the semi-empirical SRIM-2013 fit. On the other hand, the recent data for C
ions for liquid water [17] suggests a different stopping maximum compared to
SRIM-2013. The importance of C makes this ion deserving of being examined more
extensively from an experimental perspective at higher and lower energies.
Independent theoretical calculations should accompany these studies, and
ideally, all the available data should be critically assessed to reveal
possible deficiencies in the measurements and lead to recommended experimental
values.

Figure 3: Scaled stopping power cross-section of H, He, Li, and C ions on H2O.
Experimental data of target in solid (s), liquid (l), and gaseous (g) states
[1] (symbols) and semi-empirical SRIM-2013 values [8](dashed curves).

###  3.2 Stopping power of oxides

Figure 4: Scaled stopping power cross-section of H, He, B, C, O, Ti, Cr, and
Ni ions on TiO2. Experimental data [1] (symbols) and semi-empirical SRIM-2013
values [8] (dashed curves). Figure 5: Scaled stopping power cross-section of
H, He, Li, Be, C, O, F, Si, Cl, Ti, Ni, Br, I, and Au ions on Ta2O5.
Experimental data [1] (symbols) and semi-empirical SRIM-2013 values [8]
(dashed curves).

The stopping in solid oxides has received both experimental and theoretical
attention, with measurements in Al2O3 (17 different ions), SiO2 (14 ions), and
Ta2O5 (14 ions), but also for ZrO2 (9 ions), Nb2O5 (8 ions), TiO2 (5 ions),
HfO2 (5 ions), Er2O3 (H and He ions), VO2 (H ion), WO3, UO2, and SO2 (only
measurements for He ion), and SnO2 (only for F ion). The case of the different
ions in aluminum oxide is shown in Figure 4 of [2]. In the present work, the
scaled stopping data of all ions in TiO2 and Ta2O5 are displayed in Figures 4
and 5. We choose these oxides following the recent surge in measurements of
these oxides as seen in the literature [18, 19, 20, 21, 22, 23]. As can be
observed in these two figures, the energy-range coverage in these measurements
is uneven. In Figure 4, we note the differences between the experimental data
and SRIM-2013 for H ion [20] (Bro2017), C and O ions [24] (BA2012c), and Ni
ion [18] (Dib2023). Remarkably, the new data for Ti and Cr ions in TiO2 in the
very recent work of Dib and co-workers [18] agree very well with the SRIM-2013
predictions, even though the latter did not consider this data as it dates 10
years prior to these measurements.

The number of ions in Ta2O5 makes it difficult to see details in a single
plot. The low-energy regions with and without data are very clear, and the
agreement of the values by Roth et al. [25] (Roth2017b) with previous
SRIM-2013 predictions is not bad, considering that the latter uses the Bragg
rule in the low-energy region. The agreement with the data around the stopping
maximum is very good, except for C and O ions by Mizohata et al. [23]
(Miz2013).

##  4 Challenges

There are many open challenges in the field of stopping power. We draw
attention to some of the issues that, to our understanding, require attention:
the lack of data for many systems; the extended use of the Bragg rule to
describe molecules combining the constituent atoms; the use of stopping values
of sparsely-measured (or even not measured) atomic systems to predict stopping
in molecules, resulting in the combination of two possible factors of
uncertainty; the large number of single-dataset systems that agree with
SRIM-2013 but without any independent confirmation; the small fraction of
systems with stopping values measurements at low energies ; the differences
between new measurements, previous data, and SRIM-2013 predictions around the
stopping maximum, which calls for a critical assessment of the data. In what
follows, we will analyze the last three of these open subjects.

###  4.1 Single dataset systems and SRIM

As mentioned before, 60% of the measured systems contain only one dataset.
This number reflects our poor knowledge of the stopping power since we do not
have an independent confirmation of the measure stopping power for 60% of the
systems in the database. In the top row of Figure 6, we illustrate this
situation with subplots for three of them: protons in lanthanides Lu, Nd, and
Tb. The measurements by Krist and Mertens from 1983 [26] (Kt1983) are
displayed with red bullets. We also included the SRIM-2013 results with dashed
curves. The agreement of SRIM-2013 with the experimental data is almost
perfect. However, this publication [26] also provides measurements of the same
ion on many other targets. Three of them are the most measured collisional
systems in the history of stopping power: protons in Au, Ag, and Al. The
bottom row of Figure 6 presents with red bullets the results in Ref. [26] for
protons in Al, Ag, and Au. This data disagrees with most of the experimental
values for the three targets. Moreover, the available experimental stopping of
Lu, Nd, and Tb cannot be described thus far by theoretical models such as CASP
[27, 28], DPASS [29, 30], and SLPA [31]. These circumstances make it
imperative to carry out new measurements in lanthanides and perhaps in many
more single-set systems.

Figure 6: Stopping power cross-section of H on Lu, Nd, Tb, Al, Ag, and Au.
Experimental data [1] (symbols) and semi-empirical SRIM-2013 values [8]
(dashed curves). Measurements by Krist and Mertens from 1983 [26] are
illustrated in red.

###  4.2 New data versus the status quo

In some cases, the recent stopping power measurements differ from the present
status of knowledge represented by the previous measurements and SRIM-2013. In
Figure 7, we display the cases of protons in Pt, Ta, and Gd. The data
published from 1967 to 1996 is shown in grey symbols, while we illustrate in
red recent measurements, from 2012 to date, by the groups led by Primetzhofer
(Uppsala) [32, 33, 34, 35], Arista (Bariloche) [36], Grande (Porto Alegre)
[37], and Valdés (Valparaíso) [38]. The disagreement around the maximum with
the older data and the discrepancies with the SRIM-2013 curves are clear.
These differences are reasonable since SRIM-2013 is a model with fitted
parameters that efficiently describe the data published up to its latest
version, which dates over a decade ago. On the other hand, new full
theoretical works describe quite well these recent measurements [31, 37, 39,
40, 41]. It is clear that in these cases, both the old and new data need to be
critically assessed, outliers identified and recommended values offered to the
model developers and end users.

Figure 7: Stopping power cross-section for H on Pt, Ta, and Gd. Experimental
data [1] (symbols) and semi-empirical SRIM-2013 values [8] (dashed curves).
Recent measurements [32, 33, 34, 35, 36, 37, 38] are illustrated in red.

###  4.3 Stopping power of slow ions

Figure 8: Stopping power cross-section of CH4 for 16 ions. Experimental data
[1] (symbols) and semi-empirical SRIM-2013 values [8] (dashed curves). Slow
ion datasets are illustrated in red (see text).

The electronic stopping power for low-velocity impact projectiles is complex,
mainly because the nuclear stopping is not negligible and must be subtracted
from the total measured values. Recent theoretical works cast doubts on the
separation of nuclear and electronic stopping based on the Coulomb deflection
of the ion at very low energies and the impact-parameter dependence of the
electronic stopping [42, 43]. This phenomenon may lead to an overestimation of
the inelastic part of the total stopping and deviations from velocity-
proportionality and isotope effects for very slow ions, which could be
noticeable for the lighter projectiles (protons and helium). Differences
between measurements by transmission and backscattering techniques and also
among recent values and previous ones reinforce the importance of having more
sets of stopping cross-sections by slow ions.

At present, the available data for low-energy stopping (per nucleon) is
scarce; however, there have been many experimental efforts in the last ten
years by different scientific groups and laboratories around the world:
Uppsala University (Sweden) [25, 34, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54], Universidad Técnica Federico Santa María (Chile) [38, 55, 56], iThemba
LABS (South Africa) [57, 58, 59], Johannes Kepler University Linz (Austria)
[33, 60, 61], University of Jyväskylä (Finland) [62, 63], Centro Atómico
Bariloche (Argentina) [36, 64], University of Tennessee (USA) [21, 65],
University of São Paulo (Brazil) [66], and Colorado School of Mines (USA)
[67].

Around 52% of the collisional systems in the database lack stopping power
measurements for slow ions. Out of this 52%, when considering the targets, 47%
correspond to atomic systems and 53% to compounds; with regards to the
projectiles, 25% are for light ($Z\leq 4$), 58% for medium ($5\leq Z\leq 54$),
and 13% for heavy ($Z>54$) ions. To obtain these percentages, we analyzed the
entire database, and for each collisional system, we found the stopping power
maximum (experimental or the one given by SRIM). Then, we regarded all the
datasets with at least one value in the energy region below half the stopping
maximum as slow ion measurement. In Figure 8, we present an example of the
present analysis for all the available ions on CH4. The markers colored with
red are the ones considered as slow ion values. Note that some sets in red
include measurements for low energies but also near the stopping maximum (He,
Li, and N), and they have been included together since they belong to the same
dataset. In the case of CH4, only 25% of the systems have measurements below
half the maxima. However, other targets –such as Au (45 ions) or Ag (36 ions)–
have around 80% of their collisional systems featuring slow ion measurements.
These statistics, together with the discrepancies among recent values (last
ten years) and older ones reinforce the importance of having more sets of
experimental stopping cross-sections by slowing ions and conducting a proper
assessment of the techniques and quality of the various data.

##  5 Summary and conclusions

The modernization of the stopping database created by Helmut Paul in the 1990s
makes it possible to mine its rich content efficiently and analyze various
aspects of the experimental data, from the large number of data available to
the gaps existing for certain collisional systems. Although experimental
techniques have improved significantly along with our knowledge of the
stopping power, the field still faces challenges, and new measurements are
needed in our continued effort to understand the physics underlying the energy
loss of ions in matter. In this work, we focused our attention on some cases
of interest. The behavior of ions in water and the differences observed in
solid, gas, and liquid phases remain open questions, with the case of carbon
in water standing out because of it’s importance for applications. The studies
on oxides also show the contrast between the number of measurements and ions
versus the not-yet-covered energy regions. The unexpected validity of the
Bragg rule around the stopping maximum and below in most cases is noteworthy.

We mentioned, at least briefly, some of the open challenges in the field. We
presented the statistics of systems with single datasets and slow velocity
ions, which show the limitations and gaps in the available experimental
values. The agreement of SRIM-2013 in systems with few datasets is expected.
By examining similar measurements in other cases with many independent
measurements (like Au, Al, or Ag), we cast doubts on some of the single
datasets. We consider this not a problem of the SRIM-2013 predictions but of
the lack of data. To elucidate these differences, new measurements are
required. Discrepancies among recent and old measurements around the stopping
maximum and below are also noted. They are reasonable because they reflect the
results of new techniques and continuous efforts. On the other hand, they
mandate a critical assessment of all the techniques and a careful evaluation
of the data with uncertainty quantification before the data are used to
develop models or implemented directly in applications.

## Acknowledgements

This work was partially supported by the following projects:
PIP11220200102421CO by CONICET Argentina, PICT-2020-SERIE A-01931 by the
ANPCyT Argentina, and by the IAEA Coordinated Research Project F41035,
Contract No. 28067. FBH acknowledges support from the IAEA in the form of two
Special Service Agreements awarded in the March - July 2022 period.

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