# Meta-learning for unsupervised outlier detection with optimal transport

Prabhant Singh & Joaquin Vanschoren
Eindhoven University of Technology
Eindhoven 5600 MB, Netherlands
<EMAIL_ADDRESS>

###### Abstract

Automated machine learning has been widely researched and adopted in the field
of supervised classification and regression, but progress in unsupervised
settings has been limited. We propose a novel approach to automate outlier
detection based on meta-learning from previous datasets with outliers. Our
premise is that the selection of the optimal outlier detection technique
depends on the inherent properties of the data distribution. We leverage
optimal transport in particular, to find the dataset with the most similar
underlying distribution, and then apply the outlier detection techniques that
proved to work best for that data distribution. We evaluate the robustness of
our approach and find that it outperforms the state of the art methods in
unsupervised outlier detection. This approach can also be easily generalized
to automate other unsupervised settings.

##  1 Introduction

Outlier detection(OD) is the process of identifying data points that are
significantly different from the rest of the data. These data points can be
caused by errors in the data collection process, incorrect values, or unusual
events. Outlier detection can be used to improve the quality of the data or to
find unusual events that could be interesting to different business and
scientific domains . The term ”outlier detection” can be interchangeably used
with ”anomaly detection”. For consistency, we will use the term ”outlier
detection” in this paper. Outlier detection has multiple applications such as
medicine (Chow & keung Tse, 1990; Ma et al., 2021b), chemistry (Egan & Morgan,
1998) and molecular biology (Cho & Eo, 2016). Outlier detection has been a
particularly hard problem. A number of Outlier detection algorithms have been
introduced in the last two decades (Aggarwal, 2013). Unsupervised outlier
detection is a very challenging task with no universally good model which
works optimally on every task (Campos et al., 2015).

AutoML (Hutter et al., 2019) has shown reliable performance and benefits in
model selection and hyperparameter optimization (Hutter et al., 2019; Feurer
et al., 2015; Thornton et al., 2013). The research in Automated machine
learning has been highly focused on supervised machine learning where we can
focus on the performance on the hold-out dataset to define an optimization
metric for the search algorithm which finds the optimal algorithms by
iterating over the search space. This setting is very reliable (Feurer et al.,
2015) but the research on unsupervised setting is rather limited. In recent
years frameworks like MetaOD (Zhao et al., 2021) have appeared which attempt
to solve automated outlier detection via meta-learning (Vanschoren, 2018).

In this work we propose an automated framework for unsupervised machine
learning tasks LOTUS(Learning to learn with Optimal Transport for Unsupervised
Scenarios), which leverages meta-learning (Vanschoren, 2018) and optimal
transport distances (Peyré & Cuturi, 2019; Scetbon & Cuturi, 2022). In this
work we use LOTUS to perform model selection on a given unsupervised outlier
detection task. We also develop GAMAOD, an extension to the popular AutoML
framework GAMA (Gijsbers & Vanschoren, 2021) for supervised outlier detection
problems. In summary, we make the following 4 contributions:

  * •

A Meta-learner for outlier detection: We propose LOTUS: Learning to learn with
Optimal Transport for Unsupervised Scenarios, an optimal transport based meta-
learner which recommends an optimal outlier detection algorithm based on a
historical collection of datasets and models in a zero-shot learning scenario.
Our solution can be used in cold start settings for model selection on
unsupervised outlier detection.

  * •

A supervised automated framework for unsupervised outlier detection: We
develop GAMAOD which is an extension of an AutoML tool GAMA. GAMAOD can use
different search algorithms to find optimal unsupervised outlier detection
algorithms for a given dataset when labels are available. We develop GAMAOD to
populate the meta-data for model selection.

  * •

Experiments and results: We empirically evaluate LOTUS in combination with
existing state of the art methods. We demonstrate the robustness of our
approach against existing state of the art meta-learners and learners.

  * •

Open source: We open-source the code for LOTUS and GAMAOD for researchers to
use and reproduce our experiments. Our tools can be extended with new datasets
and algorithms.

##  2 Background

This section describes related work regarding Automated Machine learning for
unsupervised outlier detection, optimal transport and meta-learning.

###  2.1 AutoML for Outlier detection

AutoML (Hutter et al., 2019) for unsupervised outlier detection is an
extremely hard problem due lack of an optimization metric to perform algorithm
selection. One can argue that the use of internal metrics like Excess-Mass
(Goix, 2016), Mass-Volume (Goix, 2016) and IREOS (Marques et al., 2015) can
make algorithm selection possible. Ma et al. (2021a) shows in their
experiments that these internal metrics are computationally very expensive and
do not scale well for large datasets. This makes it unfeasible to use these
metrics in AutoML tools for most real world scenarios.

There has been recent research on AutoML for outlier detection. PyODDS and
MetaOD (Li et al., 2020; Zhao et al., 2021) are among the few tools which have
been shown to automate outlier detection.

To the best of our knowledge MetaOD (Zhao et al., 2021) is the current state
of the art meta-learner for model selection on outlier detection tasks for
tabular data. MetaOD uses meta-learning as a recommendation engine using
landmark meta-features and model based meta-features with collaborative
filtering (Stern et al., 2010) to perform model selection for a given task.

###  2.2 Meta learning

Meta-learning or ”learning to learn” in AutoML (Vanschoren, 2019) is the study
of learning from historical performances of machine learning models on a
variety of tasks and using this knowledge to find better models for new tasks.
Meta-learning can help to speed up the model selection process and find better
architectures. Meta-learning is often proposed as a solution to cold start
problem, by initializing the hyperparameters or search space for the AutoML
algorithm. This is often called warm-starting for AutoML.

Meta-learning in existing AutoML tools: Different AutoML tools use different
meta-learning schemes to solve this cold start problem. AutoSklearn-2.0
(Feurer et al., 2020) learns pipeline portfolios, MetaOD (Zhao et al., 2021)
trains a collaborative filtering based algorithm (Stern et al., 2010) with
landmark-based and model-based metafeatures (Castiello et al., 2005), FLAML
uses in-built meta-learned defaults for warm starting. MetaBu (Rakotoarison et
al., 2022) uses Fused Gromov Wasserstein with proximal gradient method on
landmark meta-features for warm-starting AutoSklearn(More discussion about
LOTUS vs MetaBu is provided in the section 5.3.2).

###  2.3 Optimal transport and dataset distances

Optimal transport(OT) theory deals with the problem of finding an optimal
transport map between two probability measures, often on different metric
spaces. It is closely related to Monge’s problem (Villani, 2008), in which one
searches for the optimal transport map between two given measures.

An Optimal transport problem consists of minimizing the cost of transporting
mass from one distribution to another. For cost function(ground metric)
between pair of points, we calculate the cost matrix $C$ with dimensionality
$n\times m$, the OT problem minimizes the loss function $L_{c}(P):=\langle
C,P\rangle$ w.r.t a coupling matrix $P$. Most common approach with
practitioners is to use a regularized approach which is computationally more
efficient $L_{c}^{\epsilon}(P):=\langle C,P\rangle+\epsilon r(P)$ where r is
negative entropy sinkhorn algorithm (Cuturi, 2013) which is computationally
more efficient. A discrete OT problem can be defined with two finite
pointclouds, $\\{x^{(i)}\\}^{n}_{i=1}$
,$\\{y^{(j)}\\}^{m}_{j=1},x^{(i)},y^{(j)}\in\mathbb{R}^{d}$, which can be
described as two empirical distributions:
$\mu:=\sum^{n}_{i=1}a_{i}\delta_{x^{(i)}},\nu:=\sum^{m}_{j=1}b_{j}\delta_{y^{(j)}}$.
Here $a$ and $b$ are the probability vectors of size $n$ and $m$. In this work
we are interested in the the Gromov Wasserstein(GW) distance between these two
discrete probability distributions. Gromomv Wasserstein allows us to match
points taken within different metric spaces. This problem can be written as a
function of $(a,A),(b,B)$ (Villani, 2008; Scetbon et al., 2022):

$GW((a,A),(b,B))=\min_{P\in\Pi_{a,b}}\mathcal{Q}_{A,B}(P)$ (1)

Where $\Pi_{a,b}:=\\{P\in\mathbb{R}^{n\times
m}_{+}|P\mathbf{1}_{m}=a,P^{T}\mathbf{1}_{n}=b\\}$

the energy $\mathcal{Q}_{A,B}$ is a quadratic function of $P$ which can be
described as

$\mathcal{Q}_{A,B}(P):=\sum_{i,j,i^{{}^{\prime}},j^{{}^{\prime}}}(A_{i,i^{\prime}}-B_{j,j^{\prime}})^{2}P_{i,j}P_{i^{\prime},j^{\prime}}$
(2)

In this work we are interested in the Entropic Gromov Wasserstein cost (Peyré
et al., 2016):

$GW_{\varepsilon}((a,A),(b,B))=\min_{P\in\Pi_{a,b}}\mathcal{Q}_{A,B}(P)-_{\varepsilon}H(P)$
(3)

where $GW_{\epsilon}$ is the Entropic Gromov Wasserstein cost between our
distributions $A$ and $B$, and $\varepsilon H(P)$ is the shannon entropy. The
problem with Gromov Wasserstein is that it is NP-hard and the entropic
approximation of GW still has cubic complexity. To speed up the computations
and use it in a realistic AutoML settings we use the Low-Rank Gromov
Wasserstein (GW-LR) approximation (Scetbon et al., 2021; Scetbon & Cuturi,
2022; Scetbon et al., 2022), which reduces the computational cost from cubic
to linear time. Scetbon et al. (2022) consider the GW problem with low-rank
couplings, linked by a common marginal $g$. Therefore, the set of possible
transport plans is restricted to those adopting the factorization of the form
$P_{r}=Qdiag(1/g)R^{T}$. In this form $Q$ and $R$ are thin matrices with
dimensionality of $n\times r$, $r\times m$ respectively and $g$ is a $r-$
dimensional probability vector. The GW-LR distance is be described as:

$\text{GW-
LR}^{(r)}((a,A),(b,B)):=\min_{(Q,R,g)\in\mathcal{C}_{a,b,r}}\mathcal{Q}_{A,B}(Qdiag(1/g)R^{T})$
(4)

Our primary inspiration for LOTUS comes from two different works.

  1. 1.

Alvarez-Melis & Fusi (2020) proposes optimal transport dataset distance(OTDD)
which uses optimal transport to learn a mapping over the joint feature and
label spaces. Alvarez-Melis & Fusi (2020) proposed that optimal transport
distances can be used as a similarity metric between different datasets from
different domains and subdomains.

  2. 2.

Work of Nies et al. (2021) argues that optimal transport measures can be used
as a correlation measure between two random variables via transport
dependency.

There have been other studies exploring the space of dataset and task
similarity with distance measures. Gao & Chaudhari (2021) proposes “coupled
transfer distance” which utilises optimal transport distances as a transfer
learning distance metric. Achille et al. (2021) explores connections between
Deep Learning, Complexity Theory, and Information Theory through their
proposed asymmetric distance on tasks.

##  3 Methodology (Meta learning for unsupervised outlier detection)

###  3.1 problem statement

In this section, we formally describe the problem of model selection for
unsupervised outlier detection.
Problem Statement: Given a new dataset without any labels, our meta-learner
needs to selects an optimal algorithm with associated hyperparameters from a
collection of previously evaluated pipeline. In this setting, we cannot
optimize the given model for the dataset as there are no given labels. This
problem becomes from a Combined model selection and hyperparameter
optimization problem to a zero-shot model reccomendation problem.
Given a new input dataset (i.e., detection task) $D_{new}=(X_{new})$ without
any labels, Select a model $A^{*}_{\lambda^{*}}\in\mathcal{A}$ to employ on
$X_{new}$. Where $A^{*}_{\lambda^{*}}$ is a tuned model for a similar dataset
to $X_{new}$.

###  3.2 LOTUS+GAMAOD: A combined framework for automated unsupervised outlier
detection

In this work, we present a meta-learner for outlier detection (LOTUS) and a
tool to collect meta-data for automated supervised outlier detection (GAMAOD).
We call the training phase of GAMAOD meta-training, where different datasets
are supplied to GAMAOD with labels. This meta-data contains two major
attributes:

  * •

A collection of $n$ meta datasets $\mathcal{D}_{meta}=\\{D_{1},...,D_{n}\\}$
with labels, test and train splits such that
$D_{i}=(X^{train}_{i},y^{train}),(X^{test},y^{test})$

  * •

A collection of $n$ optimized algorithm(s) with associated hyperparamters for
every dataset in $\mathcal{D}_{meta}$;
$\mathcal{A}=\\{A^{*}_{\lambda^{*}_{1}},...,A^{*}_{\lambda^{*}_{n}}\\}$

An overview of our system can be found in Figure 1. We will discuss the
building of our meta-data before describing our meta-learning approach.

Figure 1: An overview of LOTUS

####  3.2.1 GAMAOD: Automated supervised learning for outlier detection

Problem Formulation: A Combined model selection and hyperparameter
optimization problem (Thornton et al., 2013) for a supervised learning task is
as follows:

$\begin{split}A^{*}_{\lambda^{*}}=\operatorname*{argmin}_{\begin{subarray}{c}\forall
A^{j}\in\bm{A}\\\
\forall\lambda\in\bm{\Lambda_{A}}\end{subarray}}\frac{1}{k}\sum_{f=1}^{k}L\left(A^{j}_{\lambda},\big{\\{}\bm{X}^{f}_{train},\bm{y}^{f}_{train}\big{\\}},\big{\\{}\bm{X}^{f}_{val},\bm{y}^{f}_{val}\big{\\}}\right)\end{split}$
(5)

in equation 5, $A_{\lambda^{*}}^{*}$ is an optimal combination of learning
algorithm from search space $A$ with associated hyperparameter space
$\Lambda_{A}$ over $k$ cross validation folds of dataset $D$ where
$D=\\{X,y\\}$ with training and validation splits. $L$ is our evaluation
measure.

The CASH problem from equation 5 relies on the validation split to optimise
for the optimal configuration. However, in unsupervised outlier detection
scenario the learning algorithm does not have access to labels but the AutoML
framework does. We cannot do cross validation folds in the unsupervised
outlier model selection setting as the learning algorithms are trained without
labels and evaluated with labels. Hence performing k-fold CV is not useful in
this setting. Our modified CASH formulation to select the optimal unsupervised
algorithm with access to labels is as follows:

$\begin{split}A^{*}_{\lambda^{*}}=\operatorname*{argmin}_{\begin{subarray}{c}\forall
A^{j}\in\bm{A}\\\
\forall\lambda\in\bm{\Lambda_{A}}\end{subarray}}L\left(A^{j}_{\lambda},\big{\\{}\bm{X}_{train}\\}\big{\\{}\bm{y}_{train}\big{\\}}\right)\end{split}$
(6)

Figure 2: An overview of GAMAOD

GAMAOD: To populate our meta-data we develop an extension on top of GAMA
(Gijsbers & Vanschoren, 2021), we call this extension GAMAOD. GAMAOD’s search
space consists of outlier detectors from PyOD (Zhao et al., 2019), PyOD is a
Python library for detecting outlying objects in multivariate data. GAMAOD can
use different metrics to optimise for the given task like AUC score and PR.

####  3.2.2 LOTUS: Learning to learn with optimal transport for unsupervised
scenarios

We describe LOTUS approach in algorithm 1. Our meta-learning approach has two
components. First, we have a transformation function that is applied to the
dataset. Let’s call this function $\phi$, which is applied on given datasets
$D_{a}$ and $D_{b}$. In the second phase, we calculate the dataset similarity
$\mathcal{O}$ based on some distance metric $\psi$ in equation 7. Because our
distributions are lie on different metric spaces we calculate the Low Rank
Gromov Wasserstein distance from equation 4 on these transformed distributions
in equation 8.

$\mathcal{O}=\psi(\phi(D_{a})\phi(D_{b}))$ (7) $\mathcal{O}=\text{GW-
LR}^{(r)}(\phi(D_{a})\phi(D_{b}))$ (8)

To find the most similar dataset $\Gamma$, we take the smallest distance from
datasets from our distance matrix $\\{\mathcal{O}_{1},...,\mathcal{O}_{n}\\}$
between a new dataset $D_{new}$ and existing datasets from
$\mathcal{D}_{meta}$. which results in:

$\Gamma=argmin\\{\mathcal{O}_{1},...,\mathcal{O}_{n}\\}$ (9)

Our meta-approach for capturing the similarity metric between datasets can be
described as:

$\Gamma_{\Gamma\in\mathcal{D}_{meta}}=argmin(\text{GW-
LR}^{(r)}(\phi(D_{new})\iota\\{(\phi(D_{D\in\mathcal{D}_{meta}}))\\}))$ (10)

where $\iota$ is the iterator over $\mathcal{D}_{meta}$.
LOTUS then accesses the optimal pipeline from $\mathcal{A}$:
$A^{*}_{\lambda^{*}new}=A^{*}_{\lambda^{*}\Gamma}$ Where
$A^{*}_{\lambda^{*}new}$ is the optimal pipeline for $D_{new}$.

Algorithm 1 Pseudocode for LOTUS

Inputs: $D_{new},\mathcal{D}_{meta},\mathcal{A}$

while $D_{i}\in\mathcal{D}_{meta}$ do

$\mathcal{O}_{i}\leftarrow\psi(\phi(D_{new},D_{i}))$ $\triangleright$ Distance
calculation

$\Gamma\leftarrow\mathcal{D}_{meta}[argmin\\{\mathcal{O}_{1},...,\mathcal{O}_{n}\\}]$
$\triangleright$ Retrieval of most similar dataset

$A^{*}_{\lambda^{*}new}\leftarrow A^{*}_{\lambda^{*}\Gamma}$ $\triangleright$
Model Selection

##  4 experiments on ADBench

For our experiments, we use ADBench (Han et al., 2022) and retrieve all
tabular datasets. This collection consists of 46 datasets. As we do not have
access to multiple benchmarks we use the N-1 strategy for the evaluation of
our system, i.e., we take one dataset at a time from ADBench and use the other
datasets in the meta-data. This ensures independent meta-training on the
following datasets. We compare our approach with 7 outlier detection
algorithms available in PyOD (Zhao et al., 2019) and the current state of the
art meta-learner for outlier detection MetaOD (Zhao et al., 2021). From PyOD
we compare our approach with the following algorithms: IForest (Liu et al.,
2008), ABOD (Kriegel et al., 2008), OCSVM (Schölkopf et al., 1999), LODA
(Pevný, 2015), KNN (Angiulli & Pizzuti, 2002; Ramaswamy et al., 2000), HBOS
(Goldstein & Dengel, 2012).

For experimental consistency, we use the same search space for GAMAOD
pretraining as MetaOD (A.3) to ensure a fair comparison. To populate our meta-
data, we run GAMAOD for 2 hours on every ADBench dataset to find the optimal
pipeline for a given dataset. We use an asynchronous evolutionary algorithm to
iterate over the search space and return the optimal pipeline. GAMAOD uses AUC
score as internal metric for model selection.

Implementation details: We use Independent Component Analysis(ICA) (Hyvärinen
& Oja, 2000) from scikit-learn (Pedregosa et al., 2011) as our transformation
function $\phi$. We use OTT-JAX library (Cuturi et al., 2022) library to
implement Low Rank Gromov Wassersstein distance. For this experiment, we set
the rank parameter of Low Rank Gromov Wasserstein to 6. The model selection
phase of LOTUS in our experiments is as follows: First the datasets are
transformed via ICA and then converted into JAX (Bradbury et al., 2018)
pointclouds geometry objects 111https://ott-
jax.readthedocs.io/en/latest/_autosummary/ott.geometry.pointcloud.PointCloud.html
and then we turn these distributions into a quadratic regularized optimal
transport problem (Peyré et al., 2016). We input this quadratic problem to our
Gromov Wasserstein Low Rank solver which returns us the distance(cost) between
two datasets. When a new dataset is given to LOTUS, the pipeline corresponding
to the dataset with the lowest distance(except the new dataset itself) is
chosen from the optimal pipeline database.

##  5 results and discussion

###  5.1 Experimental results

We use the Bayesian Wilcoxon signed-rank test (or ROPE test, Benavoli et al.
(2017; 2014)) to analyze the results of our experiments. ROPE defines an
interval wherein the differences in model performance are considered
equivalent to the null value. Using this test allows us to compare model
performances in a more practical sense. We set the ROPE value to 1% for our
experiments. We use the baycomp library (Benavoli et al., 2017) to run and
visualize the analyses.

LOTUS vs MetaOD: The ROPE test on AUC scores comparing LOTUS and MetaOD are
shown in Figure 3. There is a 74.2 % probability that LOTUS is better than
MetaOD. LOTUS proves to be more robust than MetaOD, since
$p(LOTUS)>p(MetaOD)$. We show the per-dataset performances in Appendix A.1.

Figure 3: ROPE test LOTUS vs MetaOD.

Results against other estimators: ROPE testing on the results of ROCAUC
performance of LOTUS vs other estimators are shown in Table 1. LOTUS proves to
be significantly better than other techniques, with default parameters, in
PyOD. In this case $P(LOTUS)>>P(PyOD_{estimators})$, Figure 5 shows the
simplex plots of MetaOD vs other estimators. We also include a Critical
Difference Plot of performances between LOTUS and PyOD estimators (lower is
better) in Figure 4. The detailed experimental results are reported in
appendix A.1 table 3.

Figure 4: Comparison of average rank (lower is better) of methods w.r.t. performance across datasets in ADBench. Estimator name | p(LOTUS) | p(rope) | p(Estimator)
---|---|---|---
IForest | 0.99954 | 0.0 | 0.00046
ABOD | 1.0 | 0.0 | 0.0
OCSVM | 1.0 | 0.0 | 0.0
LODA | 1.0 | 0.0 | 0.0
KNN | 1.0 | 0.0 | 0.0
HBOS | 0.99982 | 0.0 | 0.00018
COF | 1.0 | 0.0 | 0.0
Table 1: Rope testing results with LOTUS vs PyOD estimators with rope=1%

(a) LOTUS vs ABOD

(b) LOTUS vs HBOS

(c) LOTUS vs COF

(d) LOTUS vs IForest

(e) LOTUS vs LODA

(f) LOTUS vs KNN

(g) LOTUS vs OCSVM

Figure 5: ROPE test result of LOTUS vs (a) ABOD (b) HBOS (c) COF (d) IForest
(e) LODA (f) KNN (g) OCSVM

###  5.2 Using optimal transport distances as a similarity measure

In our experiments, we show that LOTUS is more robust and better than current
state of the art meta-learner MetaOD for unsupervised outlier detection tasks
and other outlier detection algorithms in default configuration.

In our method we experimentally show that using optimal transport distances
like GW-LR is a feasible approach for dataset similarity and meta-learning. We
would like to emphasize that this similarity measure should only be used as a
relative similarity measure, for e.g. in our case where we use this similarity
measure to find the most similar dataset from a collection of datasets in
$\mathcal{D}_{meta}$. To estimate to what degree datasets are similar Nies et
al. (2021) proposes optimal transport based correlation measures that can be
leveraged. Our approach assumes that Wasserstein distances can capture
intrinsic properties of datasets and can capture the similarity between them,
Alvarez-Melis & Fusi (2020) also proposes their approach with optimal
transport distances to provide some sort of distance between dataset.

###  5.3 Related Works

In this section we will discuss the difference between closest approaches to
LOTUS which are MetaOD and MetaBu.

####  5.3.1 LOTUS vs MetaOD

LOTUS and MetaOD solve the same problem of model selection problem for
unsupervised outlier detection. The major difference in LOTUS and MetaOD is
meta-feature generation. LOTUS aims to capture the similarity of the given
source and target representations via optimal transport. MetaOD captures
similarity with a combination of landmark-features and model-based features
and uses a rank-based criteria called discounted cumulative gain for model
selection. MetaOD also uses stochastic algorithms such as Isolation Forest and
LODA for model-based meta-feature generation which means that the absolute
dataset similarity and ranking can differ based on the number of runs. LOTUS
on the other hand uses non stochastic methods for distance calculation. Our
approach is generalises more than MetaOD as well for different unsupervised
tasks as it aims to find similar dataset independent of task, whereas MetaOD’s
similarity is highly coupled with the task of outlier detection.

####  5.3.2 LOTUS vs MetaBu

MetaBu (Rakotoarison et al., 2022) was proposed as a solution to cold start
problem in supervised learning scenario. Rakotoarison et al. (2022) uses
Fused-Gromov-Wasserstein distance with multi dimensional scaling (Cox & Cox,
2008) by first extracting meta-features from the target representation and
source representation and proximal gradient method (Xu et al., 2020). LOTUS on
the other hand uses independent component analysis and Low Rank Gromov
Wasserstein. LOTUS is much faster than MetaBu simply because of computational
complexity differences between Low Rank Gromov Wasserstein($O(n)$) and Fused
Gromov Wasserstein($O(n^{3})$). LOTUS is a simpler approach as well as
compared to MetaBu(2 phases vs 5 phases). Lastly, LOTUS is a solution for
unsupervised setting whereas MetaBu relies on landmark features from PyMFE
(Alcobaça et al., 2020) which are more reliable for datasets with labels.
Similar to MetaOD, MetaBu setting is limited to only one task (supervised
classification) as it relies on landmark-features which require labels. LOTUS
therefore can generalize to more tasks than MetaBu.

###  5.4 limitations

  1. 1.

LOTUS: LOTUS depends on the quality of meta-data, i.e. range of datasets and
algorithms in our case. In the worst case scenario, if there are no similar
datasets in the $\mathcal{D}_{meta}$, LOTUS can recommend a dataset which is
not sufficiently similar to new dataset. On the other hand, it is expected to
improve as more benchmarks and datasets with different properties become
available.

  2. 2.

OT distance: The computation cost of GW-LR on really large datasets can still
be very high. In these cases we recommend using stratified sampling or random
sampling depending on the nature of dataset and problem.

  3. 3.

Parameter tuning: Tuning rank of GW-LR can be tricky. Low rank can result in
faster computation but high loss and high rank can result in less efficient
algorithm. Scetbon et al. (2022) states an experiment where they study the
affect of rank of GW-LR. This rank can also be tuned by minimizing the loss
between GW and GW-LR.

##  6 conclusion and future work

Model selection for unsupervised outlier detection is a challenging task. We
do not have efficient internal metrics for evaluating an algorithm without
ground truth. In this work, we proposed a new meta-learner: LOTUS, which uses
optimal transport distances to capture the similarity between datasets and
uses that similarity measure to recommend pipelines from a meta-data. To run
these experiments, we also developed another tool GAMAOD which is an extension
of GAMA and allows users to find optimal outlier detection algorithms in a
supervised setting. Through our experiments, we demonstrate that LOTUS
outperforms MetaOD and other built-in estimators in PyOD. The LOTUS approach
also enables researchers to use a simplified meta-learning framework as
compared to other landmark and model-based meta-features methods where meta-
features are highly specialized according to the domain.

We believe that our approach can be easily extended as a meta-learner to
perform model selection in other unsupervised machine learning tasks as well.
These include clustering, distance metric learning, density estimation and
covariance estimation. This approach can also be used as a meta-learner to
warm-start neural architecture search(NAS) problems.

##  7 reproducibilty statement

We opensource both LOTUS and GAMAOD with hyperparameters used for this
experiment. We also provide scripts which can be used to perform these
experiment for just one dataset without making the meta-data first(not
reccomended). We aim to provide modularity to researchers therefore we users
them to save and retrieve meta-data in whatever format they want. More
information about reproducing our experiments can be found in the README.md of
the supplementary code repository. To reproduce LOTUS for other tasks and
dataset, users are simply required to change the datasets and algorithms in
meta-data. The approach works out of the box for other scenarios. While
reproducing the experiments, the results can differ due to stochasticity of
few algorithms.

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##  Appendix A Appendix

###  A.1 Performances

Table 2 contains the performances of LOTUS and MetaOD on 42 datasets, We had
to eliminate 4 datasets from this experiment because MetaOD returned invalid
models for these datasets(i.e. models with invalid values). Scores are in bold
where AUC of LOTUS $>$ MetaOD or differ by less than a %. The dataset names
are as they were in ADBench (Han et al., 2022).

Dataset | LOTUS | MetaOD
---|---|---
19_landsat | 0.7902 | 0.5931
25_musk | 0.9895 | 0.9655
24_mnist | 1.0000 | 1.0000
32_shuttle | 0.9216 | 0.9163
23_mammography | 0.6434 | 0.6477
42_WBC | 0.8521 | 0.8655
15_Hepatitis | 0.9353 | 0.9353
43_WDBC | 0.8548 | 0.9671
12_fault | 0.9246 | 0.9043
10_cover | 0.9463 | 0.9436
34_smtp | 0.2744 | 0.5212
11_donors | 0.8064 | 0.8049
29_Pima | 0.8804 | 0.7197
37_Stamps | 0.9275 | 0.9339
44_Wilt | 0.7765 | 0.5327
40_vowels | 0.8491 | 0.9355
8_celeba | 0.9908 | 0.9906
1_ALOI | 0.8954 | 0.8957
30_satellite | 0.8913 | 0.7890
26_optdigits | 0.9996 | 0.9997
2_annthyroid | 0.8472 | 0.8445
41_Waveform | 0.9758 | 0.9413
28_pendigits | 0.8597 | 0.9265
4_breastw | 0.7466 | 0.7438
21_Lymphography | 0.9441 | 0.9861
20_letter | 0.9701 | 0.9891
39_vertebral | 0.7634 | 0.8424
47_yeast | 0.9089 | 0.9097
3_backdoor | 1.0000 | 1.0000
13_fraud | 0.9646 | 0.8904
45_wine | 0.9841 | 0.9481
22_magic.gamma | 0.9322 | 0.8122
9_census | 0.9819 | 1.0000
7_Cardiotocography | 0.9392 | 0.9378
35_SpamBase | 0.9446 | 0.9015
46_WPBC | 0.7811 | 0.8088
36_speech | 1.0000 | 0.4344
6_cardio | 0.9794 | 0.9793
31_satimage-2 | 0.9552 | 0.8100
18_Ionosphere | 0.8072 | 0.8338
27_PageBlocks | 0.7164 | 0.7668
5_campaign | 0.9922 | 0.9996
Table 2: AUC scores of MetaOD vs LOTUS on ADBench

Table 3 reports the auc scores over datasets from ADBench. The bold number
shows scores where LOTUS is better than all other estimators in PyOD.

Dataset | IForest | ABOD | OCSVM | LODA | KNN | HBOS | COF | LOTUS
---|---|---|---|---|---|---|---|---
44_Wilt | 0.471963 | 0.568222 | 0.301310 | 0.408280 | 0.472095 | 0.281412 | 0.544269 | 0.7765
6_cardio | 0.943738 | 0.498576 | 0.939676 | 0.892753 | 0.741544 | 0.865343 | 0.544550 | 0.9794
43_WDBC | 0.987241 | 0.987241 | 0.989655 | 0.987586 | 0.960345 | 0.998966 | 0.771034 | 0.8548
4_breastw | 0.976321 | 0.976321 | 0.778694 | 0.981964 | 0.947386 | 0.969329 | 0.381366 | 0.7466
42_WBC | 0.993567 | 0.993567 | 0.994103 | 0.995980 | 0.911954 | 0.991691 | 0.754757 | 0.8521
47_yeast | 0.431011 | 0.417114 | 0.448353 | 0.492504 | 0.413668 | 0.410032 | 0.428639 | 0.9089
45_wine | 0.735205 | 0.735205 | 0.681612 | 0.923158 | 0.471241 | 0.891757 | 0.412289 | 0.9841
5_campaign | 0.692549 | 0.642977 | 0.645556 | 0.566477 | 0.696817 | 0.771387 | 0.564588 | 0.9922
46_WPBC | 0.522489 | 0.522489 | 0.475911 | 0.562133 | 0.419170 | 0.555259 | 0.495170 | 0.7811
7_Cardiotocography | 0.752439 | 0.539423 | 0.810433 | 0.785916 | 0.582569 | 0.623355 | 0.572511 | 0.9392
8_celeba | 0.757810 | 0.757810 | 0.761861 | 0.718291 | 0.632204 | 0.805965 | 0.393545 | 0.9908
9_census | 0.598140 | 0.598140 | 0.523211 | 0.325589 | 0.650628 | 0.633393 | 0.413254 | 0.9819
39_vertebral | 0.377788 | 0.377788 | 0.427308 | 0.284423 | 0.417163 | 0.282356 | 0.321923 | 0.7634
41_Waveform | 0.669757 | 0.698172 | 0.474443 | 0.611266 | 0.782120 | 0.639714 | 0.804121 | 0.9758
38_thyroid | 0.979620 | 0.979620 | 0.867786 | 0.699534 | 0.951152 | 0.952834 | 0.871991 | 0.7910
40_vowels | 0.708373 | 0.956714 | 0.532701 | 0.655924 | 0.971722 | 0.646130 | 0.849763 | 0.8491
3_backdoor | 0.734361 | 0.734361 | 0.802264 | 0.708914 | 0.738679 | 0.665487 | 0.728995 | 1.0000
32_shuttle | 0.996250 | 0.618768 | 0.987461 | 0.951075 | 0.678578 | 0.994925 | 0.557606 | 0.9216
31_satimage-2 | 0.996844 | 0.762625 | 0.983527 | 0.987126 | 0.909884 | 0.985936 | 0.451384 | 0.9552
26_optdigits | 0.771433 | 0.525541 | 0.527237 | 0.623480 | 0.398194 | 0.852822 | 0.423611 | 0.9996
1_ALOI | 0.501898 | 0.609567 | 0.532848 | 0.549594 | 0.555634 | 0.478001 | 0.635583 | 0.8954
35_SpamBase | 0.657074 | 0.390792 | 0.520510 | 0.273952 | 0.515358 | 0.651507 | 0.416468 | 0.9446
36_speech | 0.469975 | 0.729473 | 0.462061 | 0.448529 | 0.473192 | 0.476358 | 0.553156 | 1.0000
34_smtp | 0.696899 | 0.670223 | 0.018006 | 0.372124 | 0.744582 | 0.878626 | 0.890630 | 0.2744
22_magic.gamma | 0.704407 | 0.799144 | 0.594241 | 0.635940 | 0.823228 | 0.681717 | 0.663549 | 0.9322
23_mammography | 0.859409 | 0.859409 | 0.854704 | 0.814810 | 0.859614 | 0.871755 | 0.792004 | 0.6434
24_mnist | 0.794443 | 0.750330 | 0.834765 | 0.743575 | 0.828259 | 0.619057 | 0.733384 | 1.0000
20_letter | 0.581556 | 0.880889 | 0.485185 | 0.627407 | 0.867111 | 0.540593 | 0.829704 | 0.9701
30_satellite | 0.707795 | 0.538013 | 0.605468 | 0.609243 | 0.646056 | 0.768130 | 0.556999 | 0.8913
19_landsat | 0.495534 | 0.500057 | 0.374050 | 0.382382 | 0.577134 | 0.556768 | 0.542057 | 0.7902
37_Stamps | 0.909527 | 0.909527 | 0.878255 | 0.944582 | 0.746473 | 0.928582 | 0.636364 | 0.9275
18_Ionosphere | 0.867847 | 0.867847 | 0.765359 | 0.858325 | 0.862297 | 0.667416 | 0.850478 | 0.8072
21_Lymphography | 0.997003 | 0.997003 | 0.993506 | 0.667582 | 0.512862 | 0.995005 | 0.934316 | 0.9441
25_musk | 0.999923 | 0.085936 | 0.818675 | 0.959047 | 0.701124 | 1.000000 | 0.400387 | 0.9895
17_InternetAds | 0.700473 | 0.673305 | 0.710028 | 0.580881 | 0.712320 | 0.704318 | 0.693902 | 1.0000
16_http | 1.000000 | 1.000000 | 0.995308 | 0.000000 | 0.001340 | 0.994638 | 0.583110 | 0.7106
15_Hepatitis | 0.742736 | 0.742736 | 0.722262 | 0.772817 | 0.467871 | 0.813292 | 0.425388 | 0.9353
14_glass | 0.818496 | 0.818496 | 0.459264 | 0.632274 | 0.740799 | 0.791758 | 0.882668 | 0.8374
13_fraud | 0.934023 | 0.941569 | 0.914391 | 0.751185 | 0.916394 | 0.941169 | 0.914591 | 0.9646
11_donors | 0.794215 | 0.794215 | 0.723436 | 0.260784 | 0.829936 | 0.763981 | 0.720262 | 0.8064
12_fault | 0.571477 | 0.676490 | 0.494426 | 0.436072 | 0.713079 | 0.479224 | 0.612146 | 0.9246
2_annthyroid | 0.824922 | 0.824922 | 0.606069 | 0.305845 | 0.730291 | 0.691522 | 0.704828 | 0.8472
27_PageBlocks | 0.889696 | 0.684494 | 0.892650 | 0.753280 | 0.769997 | 0.788657 | 0.673234 | 0.7164
28_pendigits | 0.949714 | 0.673023 | 0.938642 | 0.951140 | 0.705836 | 0.921169 | 0.475639 | 0.8597
29_Pima | 0.660016 | 0.660016 | 0.580166 | 0.606169 | 0.685681 | 0.713573 | 0.566752 | 0.8804
10_cover | 0.914310 | 0.767605 | 0.886407 | 0.866889 | 0.899776 | 0.795243 | 0.870260 | 0.9463
Table 3: AUC Scores: LOTUS vs PyOD estimators with default configuration

###  A.2 Baslines

The 8 baslines estimators and frameworks are listed below with brief
description from PyOD’s (Zhao et al., 2019) documentation for reference here:

  1. 1.

MetaOD: MetaOD is the first automated tool for outlier detection. MetaOD use
collaborative filtering, landmark and model based meta-features to recommend
the model for given task.

  2. 2.

IForest: IsolationForest ‘isolates’ observations by randomly selecting a
feature and then randomly selecting a split value between the maximum and
minimum values of the selected feature.

  3. 3.

LOF:The anomaly score of each sample is called Local Outlier Factor. It
measures the local deviation of density of a given sample with respect to its
neighbors. It is local in that the anomaly score depends on how isolated the
object is with respect to the surrounding neighborhood. More precisely,
locality is given by k-nearest neighbors, whose distance is used to estimate
the local density. By comparing the local density of a sample to the local
densities of its neighbors, one can identify samples that have a substantially
lower density than their neighbors. These are considered outliers.

  4. 4.

ABOD:For an observation, the variance of its weighted cosine scores to all
neighbors could be viewed as the outlying score.

  5. 5.

HBOS: Histogram- based outlier detection assumes the feature independence and
calculates the degree of outlier by building histograms.

  6. 6.

KNN: kNN class for outlier detection. For an observation, its distance to its
kth nearest neighbor could be viewed as the outlying score.

  7. 7.

COF: Connectivity-Based Outlier Factor uses the ratio of average chaining
distance of data point and the average of average chaining distance of k
nearest neighbor of the data point, as the outlier score for observations.

  8. 8.

LDOA: Lightweight on-line detector of anomalies detects anomalies in a dataset
by computing the likelihood of data points using an ensemble of one-
dimensional histograms.

  9. 9.

OCSVM: One class support vector machines unsupervised outlier Detection.
Estimate the support of a high-dimensional distribution.

###  A.3 LOTUS+GAMAOD search space and MetaOD reproducibility

We implement the same searchspace as MetaOD github repository for a fair
comparison.
222https://github.com/yzhao062/MetaOD/blob/master/metaod/models/base_detectors.py,
MetaOD also uses all the existing datasets from ADbench. We believe that we
have fairly evaluated MetaOD against out baseline. We believe that our
Benchmark setting was more challenging than the one evaluated in Zhao et al.
(2021) where it take child and parent datasets.
333https://github.com/yzhao062/MetaOD/blob/2a8ed2761468d2f8ee2cd8194ce36b0f817576d1/metaod/models/train_metaod.py


# New data acquisition system of the Lintott magnetic spectrometer at the
S-DALINAC

M. Singer  U. Bonnes  A. D’Alessio  M. Hilcker  P. von Neumann-Cosel
<EMAIL_ADDRESS>N. Pietralla  Institut für Kernphysik, Technische
Universität Darmstadt, 64289 Darmstdt, Germany

###### Abstract

A new data acquisition system for the high resolution magnetic spectrometer
Lintott at the superconducting Darmstadt electron linear accelerator S-DALINAC
was developed. It allows inclusive and coincidence electron scattering
experiments with event rates up to 10 kHz.

###### keywords:

data acquisition , nuclear science , electron scattering , magnetic
spectrometer

††journal: Nuclear Instruments and Methods in Physics Research Section A

##  1 Introduction

The Institute for Nuclear Physics at the Technical University Darmstadt has a
long-standing history in nuclear structure studies using electron scattering
at low momentum transfers [1] with a focus on the magnetic dipole response of
nuclei [2]. A central research instrument has been the Lintott spectrometer
[3] with a beam line designed for dispersion matching [4], thus allowing high-
resolution measurements.

In 2006, the original focal plane detector consisting of overlapping
scintillators was replaced by a system based on silicon strip detectors [5],
which considerably improved the data taking because the long and cumbersome
measurements of the varying scintillator efficiencies due to radiation damage
could be avoided. For maximum data rates no list mode data were recorded but
the position information from the triggered strip was directly converted into
an energy histogram. In recent years, a number of current nuclear structure
questions have been successfully addressed with this system including the
astrophysically important transitions to the Hoyle state state in 12C [6, 7]
and to the first excited state in 9Be [8], investigations of mixed-symmetry
states [9, 10, 11, 12], of shape coexistence [13] or a precision measurement
of the first excited $2^{+}$ state in 12C testing novel ab initio-based shell-
model approaches [14].

However, the recent implementation of a high-energy scraper in the beam line
[15], which allows in principle extremely high-resolution measurements opens a
new opportunity to increase the energy resolution even further by offline
analysis requiring list mode data. This has led to the need of a redesign and
further development of the data acquisition presented here.

##  2 Lintott Spectrometer

The Lintott spectrometer is a vertically placed magnetic spectrometer built in
the 1970s [3]. A cross section is presented in Fig. 1. The most important
characteristics of the spectrometer are summarized in Tab. 1. A special
operational mode of the Lintott spectrometer the so-called dispersion
matching, where the dispersion of the extracted beam is matched to the
dispersion of the spectrometer. Basically, the momentum spread of the electron
beam is transformed into a spatial distribution on the target. The width of
that distribution matches the different path lengths of the electrons inside
the dipole magnet due to the momentum spread, such that they focus to a point
at the focal plane where the detector is placed [4]. In this way the energy
spread of the electron beam can be eliminated.

Electrons scattered into the spectrometer port with a solid angle of 6 msr are
collimated by vertical and horizontal slits to improve the energy
resolution.The dipole magnet separates the electrons by their momentum with a
dispersion of 3.72 cm/% in the focal plane. The electron position in the focal
plane is measured by a silicon strip detector located inside the vacuum system
of the spectrometer.

Figure 1: Cross section of the Lintott magnetic spectrometer. Table 1: Spectrometer characteristics defined by design. The energy resolution is limited by the position resolution of the silicon strip detector.  Characteristic | Value
---|---
Momentum acceptance |  $\pm 2$%
Max. solid angle |  $6$ msr
Max. momentum |  $120$ MeV/c
Dispersion |  $3.72$ cm/%
Energy resolution |  $<1\cdot 10^{-4}$ (FWHM)
Spectrometer angles |  $69$∘-$165$∘, $12$∘ steps
Max. event rate (DAQ) | 10 kHz

After passing the position detector, the electrons leave the spectrometer
through a mylar foil and hit the trigger scintillator and the Cherenkov
detector. The latter serves for the suppression of the background due to
gamma-rays mainly originating from the target. The read-out electronics for
the detector system and the supply voltage for the photo-multipliers, the
silicon strips and other electronics are placed on the top of the spectrometer
shielded from radiation during the experiment.

###  2.1 Silicon strip detector

The focal plane detector consists of four silicon strip modules with 96 strips
each. One strip has a width of 650 $\mu$m and a thickness of 500 $\mu$m. For
the lowest energy of 20 MeV at the Lintott spectrometer, electrons deposite
290 keV in a silicon strip, which corresponds to a charge of about 13 fC. For
amplifying such small charges and for further processing, each 16 silicon
strips are connected to a Gassiplex chip [16]. Three Gassiplex chips are
connected in a serial way, so that they act as a unit with 48 inputs and one
output. Two such 48-channel Gassiplex units are connected by a ribbon cable to
one 96-channel silicon strip detector. Because of the special configuration
only 8 analog multiplexed outputs are needed to transmit channel level
information from all 384 strips. Figure 2 displays a photograph of the system
inside the vacuum chamber.

Figure 2: Focal plane detector system inside the vacuum chamber of the 169∘
spectrometer. In the center 4 silicon strip detectors, each 96 channel wide,
are visible. Around them are Gassiplex chips on the PCB. The 8 multiplexed
output signals and the control signals for the Gassiplex chips are connected
through vacuum feedthroughs to the readout electronics.

The Gassiplex chips require a special read-out scheme shown in Fig. 3. After a
charge peaking time of 520 ns a Hold signal freezes all voltage levels at the
inputs connected to the silicon strips. Next a Clock signal provided to the
serializer produces a multiplexed signal with voltage levels at the output.
Finally, after 2 $\mu$s needed for the baseline recovery, a Clear pulse sets
the Gassiplex chip back into the operational mode.

Figure 3: Read-out scheme of a Gassiplex chip. A Hold signal let the Gassiplex
chips keep the voltage levels depending on the collected charge inside the
silicon strips. At every new Clock pulse they are successively put to the
output driver. Finally, a Clear signal sets the electronics back to the live
state (adapted from Ref. [5]).

###  2.2 Previous Data Acquisition

In the previous data acquisition setup, the 8 cables with the analog
multiplexed signals were connected to a special FPGA-based module described in
Ref. [5]. It processes all the steps from the digitization of the multiplexed
signal to the ready-to-use histogram containing detected electron positions.
The communication with the module was realized via Telnet. The module also
produced control signals for the Gassiplex read-out logic, triggered by the
scintillator detector. The complete setup is shown schematically in the left
part of Fig. 4.

##  3 New read-out process

Figure 4: Scheme of the old (left) and new (right) data acquisition. The
trigger logic of the new system is realized with NIM logic modules. Data are
extracted with VME modules. For details see text.

The data acquisition setup described in this work is schematically shown in
the r.h.s. of Fig. 4. It starts with the analog multiplex signal coming from
the Gassiplex chips, hence it replaces the FPGA-based module. The trigger
coincidence logic and the dead-time logic are realized with NIM modules. To
create the necessary control signals for the Gassiplex electronics a VME
module Wiener MDGG8 [17] was used. It provides a number of built-in logic
gates that can be connected to inputs and outputs by setting corresponding VME
registers. Access to the configuration registers and the read-out process of
the VME ADC and TDC modules is realized with the Multi Branch System (MBS)
[18]. The data is written to a file server and analyzed by a custom software.

###  3.1 Gassiplex control process

The control scheme for the Gassiplex chips shown in Fig. 3 is realized with a
multi-purpose Wiener MDGG8 module. First, a coincidence signal between the
scintillator and the Cherenkov detector is passed through the dead-time logic
to an input of the MDGG8 module. After a peaking time of 520 ns required by
the Gassiplex chips, the MDGG8 module produces a Hold signal to freeze all
voltage levels of the silicon strips. At the same time a Clock signal
consisting of 48 pulses with a period length of 144 ns is generated to produce
a multiplexed signal of 48 channel levels at all 8 analog outputs. Finally,
after a 2 $\mu$s long baseline recovery pause a 80 ns long Clear pulse sets
the Gassiplex chip back to live mode.

###  3.2 Data read-out

The differential multiplexed signals from the Gassiplex electronics are
connected to a 16-bit Flash-ADC VME module Struck SIS3302 [19] with 100
$\Omega$ differential inputs. The internal clock frequency of the ADC was set
to 25 MHz to reduce the number of sampling points. The number of samples to be
digitized was set to 184 corresponding to a trace length of 7360 ns and 368
Byte per channel and trigger.

The trigger signal starts the read-out process by activating the GSI TRIVA7
module [20] whose status is polled by MBS running on a CES RIO4-8072 VME
master controller [21]. First, the scaler and TDC data are read and then the
digitized traces. The Flash-ADC provides the possibility to write and read the
data to two independent memory pages for every input channel. A memory page is
8 MByte deep sufficient to store $>$20k events. With every read-out event the
memory page is switched and the data from the last active memory page can be
read using multi-block-transfer mode. The binary data stream from the VME
modules is transported over an Ethernet connection to a local data acquisition
server, where the data is written to so-called list mode data (LMD) files.

###  3.3 Live time

The live time of the system is defined by two factors. The first one is the
time needed by the Gassiplex electronics to serialize the analog channel
levels to a multiplex signal, whose length is about 9.5 $\mu$s including 144
ns/ch $\times$ 48 ch for the desired analog signal, gaps after 16 and 32
channels and the baseline recovery time. To cover this period of time a fixed,
non-extendable dead time of 10 $\mu$s length was introduced by a back-looped
logic directly after the trigger detector coincidence unit.

The second factor is the transfer rate of the VME bus between the digitizer
and the VME controller measured to be about 30 MByte/s. It introduces a limit
for the maximum event rate as illustrated in Fig. 5 showing the event rate as
a function of the trigger rate for the old and the new system. For a dead time
fraction of 10%, the dead time of 10 $\mu$s corresponds to a maximum event
rate of about 10 kHz, which is more than sufficient for most electron-
scattering experiments at the S-DALINAC.

Figure 5: Measured event rate for the new data acquisition. The maximum event
rate is limited by the read-out data rate.

###  3.4 Beam current measurement

After passing the scattering target the electron beam is collected in a
Faraday cup to measure the beam current. The Faraday cup is connected via an
BNC cable to an in-house made linear current-to-frequency converter that
produces square pulses with a frequency depending on the measured beam
current. The conversion factor, selected by a resistor on the converter PCB,
was chosen to 101.6 $\mu$A / 32.768 MHz to cover a range from 0.1 nA to 10
$\mu$A. The output of this converter is connected to a VME scaler that counts
the number of pulses since the last read-out, which corresponds to the
collected charge. Different to the old data acquisition system the measurement
of the beam current was integrated into the new data acquisition to have all
relevant values in one data file.

###  3.5 Expandability for (e,e′x) Coincidence Experiments

To achieve the best signal-to-noise ratio and hence reduce the beam time in a
coincidence (e,e′x) experiment it is most important to optimize the resolution
of time coincidence peak between the detected electron and the coincidence
detectors. A TDC module with internal ring buffer and 30 ps (RMS) time
resolution was implemented into the current setup. The purpose of this module
is to measure the arriving times of the trigger detector signals with respect
to the trigger signal after the dead-time logic, which serves as a reference
signal and is provided to the read-out system of coincidence detectors. By
comparing the electron arriving times and the coincidence detector times to
the reference signal a coincidence histogram can be built.

The current trigger detector setup consists of a 0.5 cm thin plastic
scintillator detector and a Cherenkov detector with a thickness of 5 cm. Each
detector is read out by one photo-multiplier. Because of this configuration
the time resolution of the trigger signal is limited by the size of the
scintillator to $\approx 8$ ns. To overcome this limitation a new setup for
the trigger detector is presently under development. It will consists of two 1
cm thick scintillators, each with two read-out detectors that will improve the
time resolution to about $1.3$ ns (FWHM). The properties of the TDC module
take this development into account.

##  4 Data processing

###  4.1 Data extraction

The binary data from the VME master controller is transferred over an Ethernet
connection to a data acquisition server, where it is written to LMD files. For
further processing an application written in C++ with a graphical user
interface realized with Qt [22]. It covers all necessary steps from the binary
LMD files to the summarized experimental data presented to an experimenter. A
graphical user interface allows to perform basic data analysis like fitting a
model function to excited states in the energy spectrum.

Figure 6: Simplified scheme of the processing of the data stream from the data
acquisition.

The most important steps in data processing built into the application are
schematically shown in Fig. 6. In the first step, the ADC data is extracted
from MBS binary files to a data structure with eight lists of digitized
multiplexed signals, representing the analog voltage levels of the silicon
strips. Figure 7 presents an example of the digitized multiplexed signal from
the Gassiplex electronics. To extract the actual voltage levels, a software
leading edge discriminator with a threshold set to half of the typical signal
height was applied to find the starting point of the first level in the data.

In the next step, an iterative routine assigns sampling points to the actual
silicon strips. In this procedure only the central part of a voltage level
region with a length of 64 ns is used to avoid problems with rising and
falling edges due to the limited bandwidth of the signal driver of the
Gassiplex chip electronics. Considering the Gassiplex clock period length of
144 ns and the sampling point distance of 40 ns, there is always at least one
sampling point for each silicon strip. When two sampling points are available,
then the average value of both is assigned to a channel. For each channel a
histogram is produced to monitor the voltage levels. It mostly contains values
without an electron signal and some increased voltage levels for situations
when electrons have hit the region close to this particular channel of the
detector. After recording several thousand events, the maximum value recorded
is used as the default voltage level for that specific channel. A list of
default voltage levels is written into a text file and can be used for
upcoming experiments.

Figure 7: Example of two digitized multiplexed signals from the Gassiplex
electronics without and with a hit (strip 5). The hit signal amplitude is
about 250 mV in this case. Because of the low bandwidth the channel edges are
fuzzy. To eliminate the uncertainties due to the edges only data points in the
red marked areas are used for further processing.

###  4.2 Estimation of the hit position

For the identification of an electron hit on the focal plane the default level
values are subtracted from the measured ones (cf. Fig. 6). Next, every channel
is compared to a threshold set to ten times the standard deviation of the
channel level distribution. This value is high enough to avoid an erroneous
triggering of the algorithm but low enough not to exclude any true events as
demonstrated in Fig. 8.

Figure 8: Blue: Histogram for ADC values of a single silicon strip. The
maximum of the Gaussian peak provides the default voltage level. Because an
electrons can deposit energy in two or three silicon strips, the charge
measured for one strip is continuous, causing a flat plateau in the histogram.
Orange: Distribution for strips with the largest charge in a detected cluster.

Because electrons impinge on the detector under an average angle of
$34^{\circ}$, usually two or three silicon strips show a signal. This provides
additional information and allows to calculate a continuous value for the
electron hit position, causing a flat plateau in Fig. 6. Its value is
determined in the following way. First the algorithm is looking for a silicon
strip with a signal above threshold. For a true signal an additional cluster
search in the neighboring $\pm 2$ strips around is performed to find the strip
with the largest value in the cluster ($\textmd{ch}_{\textmd{l}}$). In the
next step we assume that $\textmd{val}(\textmd{ch})$ – the charge deposited in
one silicon strip – depends linearly on the distance to that strip. Then, the
actual electron position can be calculated by linear interpolation normalized
to the total charge of the two involved silicon strips

$\textmd{ch}=\textmd{ch}_{\textmd{l}}+\textmd{p}$ (1)

with

$\textmd{p}=\frac{-\textmd{val}(\textmd{ch}_{\textmd{l}}-1)}{\textmd{val}(\textmd{ch}_{\textmd{l}}-1)+\textmd{val}(\textmd{ch}_{\textmd{l}})}$
(2)

for
$\textmd{val}(\textmd{ch}_{\textmd{l}}-1)>\textmd{val}(\textmd{ch}_{\textmd{l}}+1)$
and

$\textmd{p}=\frac{\textmd{val}(\textmd{ch}_{\textmd{l}}+1)}{\textmd{val}(\textmd{ch}_{\textmd{l}}+1)+\textmd{val}(\textmd{ch}_{\textmd{l}})}$
(3)

else.

The upper part of Fig. 9 shows the resulting density histogram for inter-strip
positions $P(x)$ versus the highest strip amplitude inside a cluster. The
distribution $P(x)$ is expected to be flat but exhibits a dent around zero.
This is the region where most of the total charge is deposited in a single
strip only. Nevertheless, the neighbor strips have non-zero values due to the
statistical variations in the level voltages, shifting the estimated position
away from zero. To correct this problem an nonlinear correction function is
applied

$\mathrm{p}_{\mathrm{corr}}(p)=\frac{1}{\sum_{x=0.0}^{0.5}P(x)}\sum_{x=0.0}^{p}P(x)$
(4)

if the position is $>0$ and

$\mathrm{p}_{\mathrm{corr}}(p)=\frac{1}{\sum_{x=0.0}^{-0.5}P(x)}\sum_{x=0.0}^{p}P(-x)$
(5)

else. The result of the correction is visible in the bottom part of Fig. 9.
The inter strip positions are now flatly distributed. By this technique the
achieved position resolution can be improved from 0.65 $\mu$m defined by the
size of the silicon strips to 0.36 $\mu$m around zero and 0.18 $\mu$m towards
$P(x)=\pm 0.5$.

Figure 9: Density plot of the calculated continuous electron hit position
around a discrete silicon strip on the x-axis before (top figure) and after
the correction by Eq. (missing) 4 or (5). The y axis shows the measured
amplitude of the highest silicon strip inside a cluster. The uncorrected case
shows a dent around zero in the histogram subfigure that disappears after the
correction. The width of the distribution between $\pm 0.5$ and 0.0
corresponds to the achieved position resolution.

The improved position resolution can be utilized to achieve a better energy
resolution because the bin size of the position histogram can be reduced. The
shape of the elastic line and transitions to excited states in electron
scattering is usually described by an analytical model with five fit
parameters [23]. Depending on the kinematics of the experiment, the width of
the lines may correspond to $3-4$ channels only, creating problems with over-
fitting and uncertainty estimations in the past. The interpolated data with
more channels per line shown in the bottom part makes the model fit algorithm
more robust.

###  4.3 Automated beam energy shift correction

The electron beam energies and thus the position of the elastic line in the
detector plane shows fluctuations with time scales from minutes to sub-
seconds. The reasons for that are various, partially due to the complexity of
the accelerator. Some important contributions are initial energy variations
caused by the thermionic gun or even caused by long-time drifts due to thermal
variations of the accelerator RF components. Figure 10 demonstrates the
phenomenon by a continuous measurement of the position of the elastic line
during the commissioning experiment described below. The channel position was
estimated from histograms of the electron hit position using 200 events and
taking the mean position of a Gaussian model function fitted to the data. The
number of 200 events was chosen because it is large enough for stable fit
results and small enough to catch variations on time levels down to several
hundreds of milliseconds (depending on the actual event rate).

Figure 10: Estimated position of the elastic line in the focal plane detector
during a commissioning run. The time between two data points corresponds to
approximately 0.5 seconds. In the shown experiment one channel corresponds to
4.4 keV, so the position variations are of the order of the energy resolution.
Figure 11: Exited $\frac{7}{2}^{+}$ state of the ${}^{\text{197}}$Au(e,e′)
reaction before (orange) and after the automated position walk correction
(red) for the run shown in Fig. 10. The blue peak represents the old data
acquisition for comparison.

With the new data acquisition an automated shift correction was introduced.
The position of the elastic line extracted from the first histogram with 200
events is set as the initial position. The position for the subsequent
electron hits is shifted by the difference between the initial position and
the mean position of the 200 event packages. The benefit of this simple method
is shown in Fig. 11 for the first exited state in 197Au observed in the
commissioning run. The line width deduced from the corrected histogram (red)
is improved by 22 % compared to the uncorrected spectrum (orange) and a
spectrum corresponding to the old data acquisition (blue).

##  5 Commissioning

Figure 12: Excitation spectrum of the ${}^{\text{197}}$Au(e,e′) reaction used
for commissioning after an automated peak shift correction. The gaps in the
spectrum arise from the physical gaps between the four silicon strip
detectorsvisible in Fig. 2.

Commissioning of the new data acquisition was conducted with a $(42.5\pm 0.5)$
MeV electron beam impinging on a ${}^{\text{197}}$Au target with an areal
density $\rho=1$ mg/cm2. The Lintott spectrometer was placed at 90∘. The
measured excitation spectrum is shown in Fig. 12. Peaks corresponding to
elastic scattering and due to the first excited $\frac{7}{2}^{+}$ state at
547.5 keV in 197Au are visible. The measured energy resolution of the elastic
line was 11 keV (FWHM), a value regularly achieved at the Lintott spectrometer
since the installation of the new high energy scraper system [15].

##  6 Summary

A new event-based data acquisition for the existing silicon strip focal plane
detector at the Lintott magnetic spectrometer at the S-DALINAC was developed.
The system allows to conduct high resolution electron scattering experiments
with event rates up to 10 kHz. Because of the event-based design and time
information available from the trigger detectors, coincidence experiments are
possible in future. The event-based design provides also the possibility to
detect and correct shifts in the energy spectrum, leading to significant
improvements of the energy resolution.

## Acknowledgments

We thank the accelerator crew at the S-DALINAC for providing excellent beams.
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Grant No. SFB 1245 (project ID 279384907) and GRK
2128 (project ID 264883531).

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