# COSTI: a New Classifier for Sequences
of Temporal Intervals ††thanks: The project was funded by POB Research Centre
for Artificial Intelligence and Robotics of Warsaw University of Technology
within the Excellence Initiative Program – Research University (ID-UB).

Jakub Michał Bilski  Faculty of Mathematics and Information Science
Warsaw University of Technology
Warsaw, Poland
<EMAIL_ADDRESS>Agnieszka Jastrzębska  Faculty of Mathematics
and Information Science
Warsaw University of Technology
Warsaw, Poland
<EMAIL_ADDRESS>
###### Abstract

Classification of sequences of temporal intervals is a part of time series
analysis which concerns series of events. We propose a new method of
transforming the problem to a task of multivariate series classification. We
use one of the state-of-the-art algorithms from the latter domain on the new
representation to obtain significantly better accuracy than the state-of-the-
art methods from the former field. We discuss limitations of this workflow and
address them by developing a novel method for classification termed COSTI
(short for Classification of Sequences of Temporal Intervals) operating
directly on sequences of temporal intervals. The proposed method remains at a
high level of accuracy and obtains better performance while avoiding
shortcomings connected to operating on transformed data. We propose a
generalized version of the problem of classification of temporal intervals,
where each event is supplemented with information about its intensity. We also
provide two new data sets where this information is of substantial value.

###### Index Terms:

classification, sequences of temporal events, multivariate temporal datasets

##  I Introduction

This paper is concerned with classifying sequences of temporal intervals, a
type of data that consists of sequences of different intervals limited by
start and end timestamps.

The contribution of this paper is threefold. Firstly, we introduce a
methodology for the generalization of classification of sequences of temporal
intervals by adding information about the event’s intensity. This
generalization widens the scope of practical applications of algorithms from
this domain. We share two new data sets concerning music analysis and
meteorology where intensity values are of substantial, informative importance.
The proposed scheme can be applied to a plethora of real-world domains.
Example novel areas of application include:

  * •

healthcare (one sequence corresponds to one patient, events correspond to
symptoms or reactions to drugs, each symptom/reaction can be described with
its intensity that can vary in time),

  * •

business process mining (one sequence corresponds to one process, intensity
depicts the number of people participating in a given stage/event in the
process),

  * •

smart homes (one sequence corresponds to one house, intensity values can be
used to describe number of persons in a room, energy consumption, etc.).

The list above is not exclusive. The new approach opens the room for novel
studies on temporal events analysis where it is essential to measure the time
of a change with high accuracy.

Secondly, we show how to convert data available in a generalized
representation format to multivariate time series, a widely known time-varying
phenomena representation form. This transformation allows using algorithms
developed for the latter data representation format to be applied to the
former. We show that using MiniRocketMultivariate [1], one of the multivariate
time series classification methods, on the converted data gives considerably
better results than dedicated state-of-the-art methods in the classification
of sequences of temporal intervals (SMILE [2] and Stife [3]). However, we can
only apply the described transformation between data representation formats
after establishing a sampling step, which is a challenging problem in itself.
Moreover, utilization of sampling introduces time complexity depending on the
duration of the sequences divided by the sampling step. When high accuracy is
needed, and the duration of sequences is very high, this leads to a very high
execution time. Based on the observed facts, we postulate that there is a need
for an algorithm that would work directly on interval data (without sampling),
not depend on the magnitude of timestamps and perform at a high level of
accuracy.

The culminating aspect of the contribution is a new algorithm termed COSTI
(short for Classification of Sequences of Temporal Intervals) for generalized
classification of sequences of temporal intervals, adapting the core ideas of
the MiniRocketMultivariate algorithm in a new method that works very well on a
different type of input data, with some necessary conceptual changes and a
completely new implementation. While retaining a high level of accuracy, the
presented algorithm does not depend on the duration of sequences in time
complexity but rather on the number of intervals present in the data. As a
result, the new algorithm is faster than MiniRocketMultivariate for each
considered real-world dataset; it is over 50 times faster in two cases. We
show that the new algorithm is well-fitted for sequences of temporal
intervals. Not only is the new method applicable to the generalized version of
the problem, but it also performs on a level of accuracy higher than state-of-
the-art methods when the constrained version of the problem is concerned.

An indispensable part of the study presented here is the implementation of the
algorithm. It was written in Python and compiled via Numba [4]. Our code was
made available at https://bit.ly//3sTiHup. In the paper, we address an
empirical study concerning eight datasets, which are at the moment deemed as
benchmark datasets in this domain [2].

The remainder of the paper is structured as follows. Section II introduces
preliminary notions and relevant literature positions. Section III
demonstrates the proposed data format transformation procedure and discusses
its limitations. Section IV describes the proposed new method of
classification of sequences of temporal intervals. Section V introduces two
new data sets and identifies assumptions made in the experiments part. Section
VI examines results of the conducted experiments. Section VII is devoted to
conclusions and critical discussion.

##  II Existing solutions

###  II-A Problem statement: multivariate time series classification – even
steps in time

A common assumption in data analysis of time-varying phenomena is that the
data are present as a sequence of real numbers, and successive values were
collected with some fixed, evenly spaced interval. We were not given an
analytic form of the relation describing the phenomenon in time. Still, we
imagine it as some continuous real-valued function that we can approximate by
analyzing data points collected in a time series. Technically, we can
represent such data as a vector containing real numbers indexed with a
timestamp. We can say that this vector includes data from a single channel of
information. In a situation where a given time-varying phenomenon is observed
using more than one channel, we can store such data in a matrix, where
successive columns correspond to consecutive information channels, and a
timestamp is used to index the rows. An important assumption is to maintain
the same sampling rate in each channel.

###  II-B Multivariate time series classification – even steps in time

As Pasos Ruiz et al. [5] suggest, Dynamic Time Warping (DTW)-based approaches
are deemed to be the ‘gold standard’ solution to time series classification.
It is often used as a bottom-line method with which more sophisticated
approaches are compared. DTW, in its simple form, is suitable for univariate
time series. However, the literature proposes a couple of generalizations that
allow to apply it in a scheme aiming at multivariate time series
classification. As Shokoohi-Yekta et al. [6] outline, the most straightforward
adaptation is to run the procedure independently for all variables in the time
series and then combine the results. The second strategy assumes that warping
is the same across all dimensions [7]. Some approaches adaptively incorporate
independent and dependent warping strategies [8].

When it comes to algorithms dedicated to multivariate data, we shall mention
Multiple Representation Sequence Learner, MrSEQL [9] which is an ensemble
operating on symbolic time series representation and deep neural architectures
addressed by Fawaz et al. [10].

Finally, we shall mention that univariate time series classifiers can be
adapted to deal with multivariate time series. We can list several studies
addressing this line of research, where for example, HIVE-COTE [11] or RISE
[12] algorithms were applied in a new role. The problems arise less in terms
of classification quality but in classifiers’ performance. In some cases,
including the case of a classifier called WEASEL [13] or CIF [14], one can
find several ways to enhance the algorithm’s efficiency to process
multivariate data faster.

###  II-C Univariate time series classification using Rocket

Rocket [15], introduced in 2019, achieved a state-of-the-art accuracy while
requiring only a fraction of the training time required by the existing
methods of univariate time series classification. Rocket uses random
convolutional kernels to create a set of features. Convolutional kernel is a
concept that was first utilized in convolutional neural networks. A
convolutional kernel $f$ as used in Rocket can be expressed as (version with
no padding):

$f(X)=\bigg{\\{}(\sum^{L-1}_{k=0}w_{k}\cdot
X[t+dk])-b\bigg{\\}}_{t=1,...,T-(L-1)d}$ (1)

where $X\in R^{T}$ is the input vector, $L\in N$ is the length of the kernel,
$w_{k}\in R$ is its k-th weight, $b\in R$ is bias and $d\in N$ is dilation.
When padding is enabled, a number of zeros is appended to $X$ at its beginning
and end, so that the iteration of $t$ can start with the middle weight getting
multiplied by $X[1]$ and end by the middle weight getting multiplied by
$X[T]$. Please note that $f:R^{T}\rightarrow R^{T-dL}$ or $f:R^{T}\rightarrow
R^{T}$, depending on whether padding is used, so one convolutional kernel
applied to one sequence produces many real numbers.

Rocket generates a high number of random kernels, i.e., kernels with random
length, weights, bias, dilation, and padding. After a kernel is applied on a
time series, two statistics are calculated based on the output – maximum value
and proportion of positive values. These two statistics become new features
describing the sequence. Normally, a very high number of kernels is used (10
000), generating a set of 20 000 features for each sequence. Then, a
classifier is used to perform the final classification. Most often, Ridge
Classifier is used, sometimes replaced by a single softmax layer when number
of examples is very big.

###  II-D Univariate time series classification – even steps in time:
MiniRocket

Even though Rocket was extremely fast comparing to other algorithms in the
field, its authors were able to come up with a version up to 75 times faster
on larger data sets. MiniRocket [1], introduced in 2020, was able to retain
the same level of accuracy as Rocket, while greatly reducing the time of
execution and making the whole process almost deterministic (or fully
deterministic, at an additional computational expense). This was done mainly
by reducing to a bare minimum the random space from which the convolutional
kernels were generated. Length of the kernel in MiniRocket is fixed to 9 and
all random kernels must contain three weights equal to $-2$ and six weights
equal to 1, creating a set of 84 possible kernels. Maximal value statistic is
discarded, making proportion of positive values the only method used for
feature generation. Possible dilation values now consist of 32 integer values
evenly distributed over an exponential distribution over interval $[1,T/8]$.
Also, the choice of biases now relies on creating a histogram of a
convolutional output for a random training example, from which random
quantiles are extracted and become biases. Many other small improvements were
also introduced. These all changes made it possible to introduce many
computational shortcuts contributing to the performance of the method. Authors
declare that from now on, MiniRocket should be used as a default version of
the algorithm.

###  II-E Multivariate time series classification – even steps in time:
MiniRocketMultivariate

Although MiniRocket was developed as an algorithm for univariate time series,
a basic extension for multivariate case was also created by the same authors.
The basic premise of this algorithm is treating data in the same way as in
MiniRocket, only now instead of multiplying the weight by one element, it gets
multiplied by sum of elements from a subset of channels. This subset is chosen
at random individually for each feature.

###  II-F Problem statement: classification of sequences of temporal intervals

The subsequent generalization considered in this paper is to move to the case
where the classification task concerns a sequence of temporal events with
assigned duration. Each event is assigned a type, a start time, and an end
time. The end time is always greater than or equal to the start time. When it
is equal, according to Kotsifakos et al. [16], we say that an event is
instantaneous. We assume that no events of the same type can overlap, while
events of different types can.

It is possible to extend this description by adding a weight to an event,
representing its intensity. Additionally, we can describe the absence of an
event as an event with a weight value equal to zero. With this encoding, it is
possible to represent this type of phenomena in a manner analogous to the case
of multivariate time series–the proposed generalization results in an event
being interpreted as an information channel.

###  II-G Classification of sequences of temporal intervals

Classification of sequences of temporal intervals has received substantially
less attention than the main stream of time series classification. To our best
knowledge, there are only two algorithms that constitute the current state of
the art. Those are Stife and SMILE, the latter being an improved version of
the former. SMILE works by creating a vector of features for each sample.
Extracted features are then used to perform classification using Random Forest
[17]. Features are generated in four ways:

  1. 1.

Static features: a set of 14 predefined statistics. These range from very
simple, like total number of events in a sequence, to more complex, like
summed duration of time intervals when there was a maximal number of events
active at the same time.

  2. 2.

Class-based medoid distance features: distances to medoids representing each
class. Medoids are chosen from training data based on IBSM, a distance metric
for sequences of temporal intervals, introduced by Kotsifakos et al. [16].

  3. 3.

Interval relation-pair features: numbers of occurrences of seven Allen’s
temporal relations, introduced by Allen [18]. These relations can occur
between pairs of events and the number of their occurrences is calculated for
each configuration of different types of events (e.g. relation “X precedes Y”
between event of type 2 and event of type 4 occurs 19 times in the sequence,
19 is the feature). Only 75 features from this group are chosen to be included
in the final set of features. This choice is made based on information gain
calculated on the training set.

  4. 4.

Interval segment features: a set of distances between the sequence and 75
subsequences randomly extracted from the training set. The function used to
measure distance is called ABIDE [19].

Features generated by these four methods are concatenated and passed to a
Random Forest classifier. Workflow of Stife is very similar, only it does not
use Interval segment features.

##  III Classification through transformation between data representation
formats

The introduced data transformation method opens the possibility to apply
methods of classification operating on the latter type of data, which can be
very beneficial given a high development of the field of multivariate series
classification. In addition, we show that using MiniRocketMultivariate on the
transformed data gives considerably better accuracy than the current state-of-
the-art described in Section II-G.

###  III-A Transforming sequences of temporal intervals into multivariate time
series

Figure 1: Left: data structure suitable to store multivariate time series.
Right: example sequence of events considered in a problem of sequences of
temporal intervals classification.

Figure 1 illustrates an example of multivariate time series data and an event
sequence to indicate how conceptually transparent the transition between one
format and the other is. The general idea is to sample channels (events) at a
chosen frequency. We can only do this after choosing a step in time. Using
that step, consecutive rows in the matrix can be populated with values of
channels, according to the formula: $M[r,c]=X_{c}[r\cdot s]$, where $r$ is
row, $c$ is column and $s$ is the chosen step. In a particular case of
MiniRocketMultivariate, we want all matrices to be of the same height, as the
algorithm operates on sequences of equal length. However, when it is implicit
that time in $X$ starts from 0, it is not always explicitly stated what the
end of the sequences is. In that case, we can only depend on the last
timestamp of a sequence, i.e., the end of the last interval. Thus, the number
of rows of the matrix is set to the number of steps resulting from parsing the
sequence that contains the highest end timestamp. All other sequences are
effectively padded with zeros. Other techniques of ensuring the same size of
input matrices are also possible and we describe one such technique in Section
VI-B.

A number of difficulties can arise as a consequence of operating on the data
created with the described transformation. Too high a sampling rate
unnecessarily increases the volume of data. Too low a frequency, in turn,
results in too much information being lost. Selecting the right frequency is
difficult and demands domain knowledge from the user. Sometimes even the best
rate will give unsatisfactory results because the first variant is not
suitable for storing the second type of information efficiently. These not-
yet-solved issues motivated us to develop a new method for classification of
sequences of temporal intervals, operating directly on the interval data. We
compare the performance of our solution with the results obtained using
transformation between formats and show that not only it allows to omit the
problematic sampling stage, it also benefits from operating on the original
data in terms of time efficiency.

##  IV Method

In this section, we present our new approach to the task of sequences of
temporal intervals classification. The new algorithm can be conceptually
summarized as a processing pipeline consisting of the following steps:

  1. 1.

Transforming interval-based data set to a custom data structure.

  2. 2.

Fitting model parameters using a training set.

  3. 3.

Calculating features for training and test sets.

  4. 4.

Performing data classification using Ridge Classifier.

The proposed solution can be seen as a generalization of the time series
processing steps present in MiniRocketMultivariate algorithm.

###  IV-A Introductory notes on the new approach

Calculation of a single feature is performed using a set of randomly chosen
parameters: a subset of channels $C$, bias $b$, dilation $d$ and a
convolutional kernel with weights $\\{w_{0},…,w_{8}\\}$. Now, let us consider
a case when observations $X$ are defined on a continuous range
$<t_{s},t_{e}>$. This would allow us to introduce the following way of
computing features:

$a~{}=\frac{1}{t_{e}-t_{s}}\int_{t_{s}}^{t_{e}}H\bigg{(}(\sum^{8}_{k=0}\sum_{c\in
C}w_{k}\cdot X_{c}[t+dk])-b\bigg{)}dt$ (2)

where $H(\cdot)$ is the Heaviside step function given as:

$H(x)=\left\\{\begin{array}[]{ll}1,&\textrm{ $x>0$}\\\ 0,&\textrm{ $x\leq
0$}\\\ \end{array}\right.$ (3)

Please note that the value of the integral is effectively the sum of lengths
of all intervals where the expression inside $H(\cdot)$ is larger than zero.

From now on, let us assume that each channel of $X$ is defined by a number of
intervals of constant values, as described in Section II-F. We can utilize
this characteristic to create an efficient method of computing the integral in
(2). The key approach here will be aligning sequences corresponding to each k
in $X_{c}[t+dk]$, and then creating a sorted concatenation using those. This
will allow us to calculate the integral by just iterating over a sequence of
timestamps between which the value of the expression inside $H(\cdot)$ does
not change. Below the algorithm description, you may find a toy example
illustrating this approach.

###  IV-B The new algorithm – steps

  1. 1.

Create a list $L_{0}$ containing all timestamps where the value of all channel
changes is sorted according to the timestamp in ascending order. Store
information about the difference between the new and the old value and the
channel’s id where the change happened.

  2. 2.

$L=L_{0}\bigcup L_{1}\bigcup...\bigcup L_{8}$, where $L_{k}$ is created by
adding $dk$ to each element of $L_{0}$. Save the information about $k$ which
the element originated from. This information will be used to choose the
weight of the convolutional kernel.

  3. 3.

Sort all elements of $L$ by their timestamps in ascending order. Please note
that the individual lists $L_{k}$ are already sorted, which can be used to
greatly speed up the procedure.

  4. 4.

Set $psum=-b$, $a=0$.

  5. 5.

Traverse $L$, adding to $psum$ the changes of values of intervals multiplied
by the weight corresponding to the $k$ from which the change point originated
from. Take into account only changes in the channels belonging to the subset
$C$. Whenever $psum$ over the current interval is larger than zero, add the
length of the interval to $a$.

  6. 6.

Feature = $a/(t_{e}-t_{s})$.

Please note that the first three points of this algorithm demand only the
value of the dilation to be known. As the original MiniRocketMultivariate
algorithm uses a maximum of 32 different dilation values, we suggest to
analogously compute the lists in points 1-3 for these 32 dilations, and for
each feature calculate only points 4-6.

###  IV-C Illustrative Example

Figure 2: Example sequence consisting of two events. Timestamp | Value | Channel | Weight index
---|---|---|---
10.33 | 0.73 | 1 | 1
30.33 | 0.73 | 1 | 2
803.88 | 1.58 | 2 | 1
815.40 | -0.73 | 1 | 1
823.88 | 1.58 | 2 | 2
835.40 | -0.73 | 1 | 2
1000.00 | -1.58 | 2 | 1
1020.00 | -1.58 | 2 | 2
Figure 3: Toy case study data structure containing $L$.

Fig. 3 presents $L$, a data structure created based on the sequence visualized
in Fig. 2. Please note that the consecutive rows in the $L$ data structure
correspond to changes in value of the expression psum
$=\sum^{8}_{k=0}\sum_{c\in C}w^{c}_{k}\cdot X_{c}[t+dk]-b$, only in this toy
example 8 is replaced by 1. This can be visualized as a convolutional kernel
of length equal to 2 moving above the sequence along the time axis. When one
of its inputs approaches the next change point, a new value of the expression
is computed by adding $w_{k}\cdot v$ to psum, where $v$ is “value” and $k$ is
“weight index”. Before we update psum, we check if the previous value was
greater than 0. If it was, the $H(\cdot)$ function would return 1, so we can
add the length of the interval that just ended to the final integral value
$a$.

Data structure created this way can be used to calculate the integral for any
given kernel and bias. Moreover, if we want to take a smaller combination, say
$C={1}$, we can just ignore timestamps that have “channel” field equal to $2$
during the structure iteration.

###  IV-D Selecting final timestamp

We denote the highest timestamp found in the data by $t_{max}$ and set
$t_{min}$ always equal to 0. These values are then used to determine $t_{s}$
and $t_{e}$, depending on the usage of padding (see Section IV-G). This
approach can be seen as padding all sequences with events with intensity equal
to zero to meet duration of the longest sequence.

###  IV-E Fitting biases

Choice of bias depends on a histogram of $\sum^{8}_{k=0}\sum_{c\in
C}w^{c}_{k}\cdot X_{c}[t+dk]$, from which random quantiles are extracted and
become biases. Here, $X$ is an element from the training set, chosen at random
for each combination of convolutional kernels and dilation values. Extraction
of a random quantile can be performed using a cumulative sum of duration over
intervals of occurring values sorted by values. This technique does not create
any serious computational overhead.

###  IV-F Fitting dilations

The fact that the number of used dilations is limited to 32 directly
contributes to achieving high performance using this workflow. The new
algorithm can use floating-point numbers for dilation values as observations
are no longer defined only in fixed timestamps. Please note that
MiniRocketMultivariate dilation distribution starts at 1, as no smaller
dilation can be used. No such lower boundary can be established in the
presented solution, as timestamps can be of any magnitude. Thus, dilations are
taken from evenly divided exponential distribution over interval
$[d_{min},(t_{max}-t_{min})/8]$, where $d_{min}$ is the smallest difference
between different timestamps found in the training data. Using continuous
values means also avoiding duplicates (in MiniRocketMultivariate there can be
multiple equal dilations, if $T$ is small enough). This could, in theory,
convey to a higher modelling power. However, experiments on data containing
only integer timestamps show that the solution performs better when dilations
are rounded to the nearest integers. This may suggest that when there is a
strong sampling frequency present in the data, it is beneficial to use only
multiples of this frequency as dilations. However, the experimental data was
too limited to decide on the significance of this relationship. Also, we did
not want to permit the usage of floating-point timestamps. As a compromise,
dilation values are rounded to integers only when it is detected that all
timestamps in train data are integers.

###  IV-G Padding

Similarly to MiniRocketMultivariate, we use padding in half of the features.
The way features are computed when padding is on corresponds to
$t_{s}=t_{min}+4d$, $t_{e}=t_{max}+4d$, so the interval depends on the middle
weight remaining in the range of the original data (taking into consideration
positive shifts in dilation introduced in (2)). The case with no padding is
$t_{s}=t_{min}+8d$, $t_{e}=t_{max}$. No padding is used when biases are
generated.

###  IV-H Time complexity

Let us address time complexity of the proposed approach and compare it with
MiniRocketMultivariate, which was the starting point for our research.

MiniRocketMultivariate has linear scalability in the number of features ($m$),
number of examples ($n$) and time series length ($l_{input}$). The number of
channels chosen for computation of a single feature is taken from a
logarithmic distribution, which makes the overall complexity equal to
$O(m\cdot n\cdot l_{input}\cdot log(C))$, where $C$ is the number of channels.
When used to classify sequences of intervals, MiniRocketMultivariate takes a
set of matrices as an input, which normally must be generated from the
interval data. The complexity of this operation is proportional to the number
of cells in all matrices, which is $O(n\cdot l_{input}\cdot C)$.

Creating the data structure and calculating features in COSTI comes with time
complexity $O(m\cdot n\cdot e\cdot log(C))$, where $e$ is the number of all
events present in the data. Additionally, sorting the input sequences $L_{0}$
at the beginning is $O(n\cdot e\cdot log(e))$. Please note that step 3 of the
algorithm described in Section IV-B involves concatenating and sorting 9
sorted lists. This is implemented as iterating over 9 sequences using a
priority queue, achieving linear time complexity.

MiniRocketMultivariate
---
$O(m\cdot n\cdot l_{input}\cdot log(C)+n\cdot l_{input}\cdot C)$
COSTI
$O(m\cdot n\cdot e\cdot log(C)+n\cdot e\cdot log(e))$
$m$ \- number of features
$n$ \- number of samples
$e$ \- number of events
$C$ \- number of channels
$l_{input}$ \- number of rows in a matrix

##  V Data sets and basic experiment assumptions

The proposed methodology of transitions and generalizations between time
series data sets and sequences data sets is a novel contribution of this
paper. Furthermore, the literature does not offer any solutions designed to
solve the problem of classification of sequences of temporal intervals after
generalization we outlined in Section II-F (by adding values corresponding to
events).

In Section V-B and Section V-C, we present two new data sets that we
contribute alongside this paper. They concern real-world phenomena that are
best described as sequences of events with meaningful weights (which is an
extended temporal data representation format, more versatile than sequences of
events with no weights).

###  V-A Existing data sets

Nonetheless, to compare the new solution with other algorithms, we also had to
perform tests on a constrained version of the classification of event
intervals problem, with no values, just events happening or not. There exists
a small benchmark data collection for this problem, containing six data sets.
Recent algorithms in this field, Stife [3] and SMILE [2], used this data for
accuracy evaluation.

Please note that we did not prepare a specialized version of the algorithm
despite using the algorithm for a constrained problem, where timestamps are
integers and there are no values associated with intervals. For the use of
comparison with other algorithms, all values were set to 1 when an event
occurs (and to 0 when it does not, but this is also true for an unconstrained
problem). This was also true for the two new data sets.

###  V-B New dataset for musical key recognition

The data set contains time series extracted from 240 MIDI files taken from the
MASETRO Dataset [20] and from the public domain. The files contained
recordings of piano pieces, either created manually or recorded during live
performance. We processed each MIDI file to create a series of events spanning
across 12 channels. Each channel corresponds to one of the notes (C, C#$=$Db,
D, D#$=$Eb, E, F, F#$=$Gb, G, G#$=$Ab, A, A#$=$Bb, B). An event in a channel
is present when at least one key corresponding to the channel’s note is being
pressed. Timestamps in the time series were taken from the MIDI files, as
these also operate on timestamps. However, timestamps in MIDI are purely
relative – they cannot be easily converted to time, and it is impossible to
compare them across different files. Considering this, we decided to resize
all timestamps, so the end of the last interval in each recording would be
equal to 24000. This value should be big enough to ensure that all rounding
errors are negligible (for a ten minutes-long piece, this is 40 Hz, and in a
normal tempo like moderato, one thirty-second note lasts 1/12 of a second). In
a version of the data set with values associated with intervals, value of an
interval is set to the velocity the key was pressed with. The classification
task is to predict which of the 24 key signatures the piece was written in.
There are 12 classes with 20 samples each and 12 classes with ten samples
each.

###  V-C New dataset concerning weather events

The data set was extracted from a weather events data set, US Weather Events
(2016 - 2020), a countrywide data set of 6.3 million weather events. We pulled
only the events recorded in 2017 by weather stations in the state of
Louisiana. Each channel corresponds to the presence of some specific type of
weather conditions. Individual series were made from the recordings for every
single weather station and each single month, summing up to 12 series for a
single station. Each series contains a month of weather events, with
timestamps rounded with 10 minutes step, counting from the month’s start
(please note, that even using a one second step would not hinder the
performance of the presented algorithm. However, this would be hardly
reasonable given the nature of the data). In a version of the data set with
values associated with intervals, the value of an interval is equal to the
intensity of the weather condition. The classification task is to predict
which month the series was recorded in. For a detailed description of the data
and levels of intensity, please refer to [21].

TABLE I: Summary of the eight data sets used in experimentation. Musekey and Weather are the new datasets contributed with this paper. Data set | Classes | Samples | Channels | Max duration | Events
---|---|---|---|---|---
Auslan2 | 10 | 200 | 12 | 30 | 12.24
Blocks | 8 | 210 | 8 | 123 | 5.75
Context | 5 | 240 | 54 | 284 | 80.65
Hepatitis | 2 | 498 | 63 | 7555 | 108.28
Pioneer | 3 | 160 | 92 | 80 | 55.93
Skating | 6 | 530 | 41 | 6829 | 43.78
Musekey | 24 | 360 | 12 | 24000 | 2914.61
Weather | 12 | 540 | 7 | 4465 | 63.16

##  VI Results

###  VI-A Comparison with state-of-the-art

There are two state-of-the-art algorithms in the field of classification of
sequences of temporal intervals: Stife and SMILE. No implementation of SMILE
was made available to the public up to the time this paper was submitted, so
we followed [2] and recreated SMILE using python and Numba. We decided to use
our own implementation of Stife, even though its implementation, written in
Java, is available in the Internet. Both hyperparameters of SMILE, that is
number of 2-lets and number of e-lets, were set to 75, as suggested in the
paper. We used 75 as the number of 2-lets in Stife. As for the accuracy
comparison, we followed the example of [2] and included two extracted parts of
SMILE, namely Static features (STAT) and Class-based medoid distance features
(MEDOID), described in Section II-G. These were used as individual methods of
feature extraction based on which classification was performed. Default
version of a Random Forest classifier from scikit-learn package [22] was used
as the final step in all four above methods. Additionally, we included IBSM
method, which SMILE was also compared to. IBSM method works by 1-nearest-
neighbour classification using the IBSM distance. As for the presented
algorithm, we used default values of 10 000 features and 32 dilations. All
experiments were performed using 10-fold cross-validation. Every experiment
was repeated 10 times and mean values of accuracy were calculated.

TABLE II: Accuracy comparison with state-of-the-art. Data set | STAT | MEDOID | IBSM | STIFE | SMILE | COSTI
---|---|---|---|---|---|---
Auslan2 | 0.331 | 0.437 | 0.415 | 0.503 | 0.509 | 0.579
Blocks | 0.840 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
Context | 0.947 | 0.961 | 0.967 | 0.991 | 0.991 | 0.996
Hepatitis | 0.661 | 0.697 | 0.603 | 0.799 | 0.795 | 0.841
Pioneer | 0.822 | 0.823 | 0.949 | 0.977 | 0.975 | 1.000
Skating | 0.599 | 0.902 | 0.969 | 0.954 | 0.970 | 0.994
Musekey | 0.143 | 0.280 | 0.039 | 0.473 | 0.473 | 0.768
Weather | 0.611 | 0.495 | 0.603 | 0.685 | 0.680 | 0.909

Results of experiments are presented in Table II. The new method achieved a
significant improvement over the current state-of-the-art in the field of
classification of sequences of temporal intervals, reaching the highest scores
for all eight data sets.

###  VI-B Comparison with MiniRocketMultivariate

In all the experiments, step during matrices creation $s=1$, so we simulate a
situation when no information is discarded. Matrices creation for sequences
that have different highest timestamps can be performed by padding shorter
sequences with zeros, as explained in Section III-A. An alternative approach
would be to perform bilinear interpolation on each matrix that has a number of
rows smaller than maximal, stretching it to meet the number of rows of the
longest matrix. A similar technique is used in IBSM distance [16] when two
sequences of different length are compared. Decision on the matrix creation
technique could be made with regard to the domain knowledge. We decided to
test both approaches on each data set. Please note that this dilemma concerns
only MiniRocketMultivariate. COSTIoperates directly on the interval data and
does not use matrix representation.

In all tests, we used the implementation of MiniRocketMultivariate available
through the sktime package. In all algorithms, default numbers of features (10
000) and dilations (32) were used. Time of execution of MiniRocketMultivariate
did not include creating a matrix representation of the data. Time of
execution of the new algorithm did include sorting the data at the beginning.
All experiments were performed using 10-fold cross-validation. Every
experiment was repeated 10 times and mean values of accuracy and time were
calculated.

TABLE III: Comparison with MiniRocketMultivariate used on a matrix representation. | accuracy
---|---
| MiniRocketMultivariate | COSTI
Data set | Scaling | Padding |
Auslan2 | 0.557 | 0.570 | 0.579
Blocks | 1.000 | 1.000 | 1.000
Context | 0.996 | 0.996 | 0.996
Hepatitis | 0.829 | 0.824 | 0.841
Pioneer | 1.000 | 1.000 | 1.000
Skating | 0.987 | 0.992 | 0.994
Musekey | 0.755 | 0.752 | 0.768
Weather | 0.881 | 0.910 | 0.909
| time [s]
| MiniRocketMultivariate | COSTI
Data set | Scaling | Padding |
Auslan2 | 0.59 | 0.59 | 0.52
Blocks | 1.00 | 1.08 | 0.55
Context | 2.18 | 2.25 | 1.18
Hepatitis | 134.44 | 133.39 | 2.80
Pioneer | 0.76 | 0.80 | 0.67
Skating | 108.05 | 102.73 | 1.69
Musekey | 257.54 | 258.27 | 184.47
Weather | 31.04 | 31.55 | 5.57

Table III presents accuracy and time of execution for all eight data sets.
Although the presented solution achieved slightly better results in accuracy
than the two variants of MiniRocketMultivariate, the difference is too little
to consider it statistically significant. A much bigger difference in favour
of the presented solution was observed for the time of execution. Please note
that compared algorithms differ significantly in terms of important factors
affecting the time complexity. For example, the higher the maximal timestamp
is, the longer it takes to execute both versions of MiniRocketMultivariate. On
the other hand, dealing with a very high number of events makes COSTIslower.
Experimental results confirm this – the presented solution easily beats
MiniRocketMultivariate in efficiency on data sets with a high maximal
timestamp (Hepatitis, Skating) and performs at a similar level when the
maximal timestamp is very low (Auslan2, Pioneer) or the number of events is
very high (Musekey). This is not surprising. The real conclusion resulting
from this experiment is different: the new algorithm was successfully
implemented with a level of efficiency that makes using it on real data
profitable when compared to MiniRocketMultivariate with a matrix
representation. Not only faster, using it should also be easier, as it allows
to use real values of timestamps, and no domain knowledge to establish a
sampling step is needed.

##  VII Conclusion and Critical Discussion

We believe that the main contribution presented in COSTI does not come from
its speed advantage over MiniRocketMultivariate on matrix representation, but
rather from a new approach to processing information, more fitting to the
problem of classification of interval sequences. Possibility of using real
values of timestamps and avoiding any type of sampling is important not only
when the modelling power is concerned. It can also prove to be advantageous
from a practical perspective, making the framework easier to use for users
with various levels of experience. We strongly believe there are many ways to
improve accuracy of the presented method. One of possible research directions
would be to examine the impact of different distributions of floating-point
dilations. Other ways of analysing the convolution output sequences, more
elaborate than the proportion of positive values, can also be considered. A
weak spot of the proposed approach is its memory complexity. When we consider
input data in a typical format (each interval is represented by three real
values: start, end, and intensity, and one integer value describing the
channel), the size of the data structure $L$ used in the algorithm is about
$18$ times the size of the input. For the biggest data set we use (Musekey)
this is roughly equal to 400 megabytes. When much bigger data sets emerge in
the field, this trait of the method can become a significant downside, making
it unsuitable for severely resource-constraint domains, like wearable
electronics. However, we believe that mainstream domains of application of the
developed approach are not falling into this area.

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# A note on meromorphic functions on a compact Riemann surface having poles at
a single point

Gollakota V V Hemasundar  Department of Mathematics
SIWS College, affiliated to Mumbai University

Wadala, Mumbai-400031

###### Abstract.

On a compact Riemann surface $X$ of genus $g$, one of the questions is the
existence of meromorphic functions having poles at a point $P$ on $X$. One of
the theorems is the Weierstrass gap theorem that determines a sequence of $g$
numbers $1<n_{k}<2g$, $1\leq k\leq g$ for which a meromorphic function with
the order with $n_{k}$ fails to exist at $P$. In this note, we give proof of
the Weierstrass gap theorem in cohomology terminology. We see that an
interesting combinatorial problem may be formed as a byproduct from the
statement of the Weierstrass gap theorem.

###### Key words and phrases:

Compact Riemann surfaces, Weierstrass points, Meromorphic functions

###### 2020 Mathematics Subject Classification:

Primary 30F10

##  1\. Introduction

Let $X$ be a compact Riemann surface of genus $g$. One of the important
questions is the existence of a meromorphic function having a pole at a point
$P$, and is holomorphic in $X\setminus\\{P\\}$. The following theorem provides
a bound on the order of a pole at $P$, and is a simple consequence of the
Riemann Roch theorem:

######  Theorem 1.1.

Suppose $X$ is a Riemann surface of genus $g$ and $P$ is a point on $X$. Then
there is a non-constant meromorphic function $f\in\mathcal{M}(X)$ which has a
pole of order $\leq g+1$ at $P$ and is holomorphic in $X\setminus\\{P\\}$.

There are some exceptional points, known as Weierstrass points that play a
role in the existence of meromorphic functions with lower order. The following
theorem clarifies that a lower order pole exists at Weierstrass points:

######  Theorem 1.2.

Suppose $X$ is a Compact Riemann surface of genus $g$ and $P$ is a point on
$X$. Then there exists a non-constant meromorphic function on $X$ of genus $g$
which has a pole of order $\leq g$ at $P$ and is holomorphic in
$X\setminus\\{P\\}$ if and only if $P$ is a Weierstrass point.

In this note we prove the following theorem which provides more information on
the order of a pole at a point:

######  Main Theorem 1.3 (Weierstrass gap theorem).

Let X be a compact Riemnn surface, of genus $g\geq 1$. Suppose $P$ is a point
on $X$. Then there are precisely $g$ integers, $n_{k}$ such that

(1.1) $1=n_{1}<n_{2}<\dots<n_{g}<2g$

such that there does not exist a meromorphic function on $X$ with a pole of
order $n_{k}$ at $P$.

The numbers $n_{k}$, for $k=1,\dots,g$ are called gaps at $P$ and their
complement in $\mathbb{N}$ are called non-gaps. There are precisely $g$ non-
gaps in $\\{2,\dots,2g\\}$. Further, the gap sequence is uniquely determined
by the point $P$.

The case $g=0$ is trivial as there is always a function on the sphere with a
simple pole. In the case of $g=1$, there is no meromorphic function on $X$
with a simple pole at $P$ and is holomorphic in $X\setminus\\{P\\}$ since
there is no doubly periodic meromorphic function having precisely a simple
pole at the single point in any period parallelogram.

In this note, Theorem 1.3 will be proved using the language of sheaf
cohomology. A proof of the Weierstrass-gap theorem can be derived as a special
case of Noether theorem. See Farkas and Kra [2]. The proofs are generally
given as an application of the Riemann-Roch Theorem [6].

In section 2, we mention some of the consequences of Riemann-Roch and Serre
Duality theorems [3] that are useful for comparing the dimensions of
cohomology groups. The Proof of the Weierstrass-gap theorem is given in
Section 3. In Section 4, we form a combinatorial problem that appears to be a
byproduct of the statement of the Weierstrass gap theorem. We conclude with
some remarks in section 5 on Weierstrass points.

##  2\. Some Consequences of Riemann-Roch and Serre Duality Theorems

######  Definition 2.1.

Let $D$ be a divisor on $X$. For any open set $U\subset X$, we define
$\mathcal{O}_{D}(U)$ the set of all those meromorphic functions which are
multiples of the divisor $-D$, i.e.,

$\mathcal{O}_{D}(U):=\\{f\in\mathcal{M}(U):ord_{x}(f)\geq-D(x)\mbox{ for every
}x\in U\\}$

One can verify with the natural restriction mappings, $\mathcal{O}_{D}$ is a
sheaf. See [3]. If $D=0$, one has $\mathcal{O}_{0}=\mathcal{O}$ which is the
set of holomorphic functions on $X$.

######  Theorem 2.2.

Let $X$ be a compact Riemann surface and $D\in Div(X)$. If $degD<0$ then
$H^{0}\left(X,\mathcal{O}_{D}\right)=0$.

###### Proof.

Suppose $f\in H^{0}\left(X,\mathcal{O}_{D}\right)$ with $f\neq 0$. Then
$(f)\geq-D$ and thus

$deg(f)\geq-degD>0$

This contradicts the fact that $deg(f)=0$. ∎

We can use Serre-Duality theorem to obtain equality of dimensions:

(2.1) $\dim H^{0}\left(X,\Omega_{-D}\right)=\dim
H^{1}\left(X,\mathcal{O}_{D}\right)$

For $D=0$ we obtain

(2.2) $\dim H^{0}\left(X,\Omega\right)=g=\dim H^{1}\left(X,\mathcal{O}\right)$

Here $H^{0}\left(X,\Omega\right)=\Omega(X)$ which denotes the sheaf of
holomorphic 1-forms on $X$. The following equation can also be obtained as an
application of Seree Duality Theorem:

(2.3) $\dim H^{1}\left(X,\Omega\right)=\dim H^{0}\left(X,\mathcal{O}\right)=1$

######  Remark 2.3.

This is a known result that there are no non-constant holomorphic maps on a
compact Riemann surface.

Suppose $X$ is a compact Riemann surface of genus $g$. Let $K$ be the
canonical divisor on $X$. Then deg $K=2g-2$.

Our proof of Theorem 1.3 derived as an application of the following theorem
which is central in the theory of compact Riemann surfaces:

######  Theorem 2.4.

The Riemann-Roch Theorem Suppose $X$ is a compact Riemann surface of genus $g$
and $D\in Div(X)$. Then $H^{0}\left(X,\mathcal{O}_{D}\right)$ and
$H^{1}\left(X,\mathcal{O}_{D}\right)$ are finite dimensional vector spaces and

(2.4) $\dim H^{0}\left(X,\mathcal{O}_{D}\right)-\dim
H^{1}\left(X,\mathcal{O}_{D}\right)=1-g+degD$

By using Equation(2.1), we can also formulate in the following way:

(2.5) $\dim H^{0}\left(X,\mathcal{O}_{D}\right)-\dim
H^{0}\left(X,\Omega_{-D}\right)=1-g+degD$

##  3\. Proof of Theorem 1.3 (Weierstrass Gap Theorem)

###### Proof.

Suppose $P\in X$. If $D$ is a zero divisor, then $\dim H^{1}(X,\mathcal{O})=g$
and $deg(D)=0$.

By the Riemann Roch theorem $\dim H^{0}(X,\mathcal{O})=1$. Therefore, there
are no non-constant holomorphic functions on $X$.
Define the divisor $D_{P}$ such that

$D_{P}(P)=\begin{cases}1\mbox{ if }x=P\\\ 0\mbox{ if }x\neq P\end{cases}$

$Deg(D_{P})=1$. Once again by the Riemann -Roch theorem,

$\dim H^{0}(X,\mathcal{O}_{D_{P}})=2-g+\dim H^{0}(X,\Omega_{-D_{P}})$

If $\dim H^{0}(X,\Omega_{-D_{P}})=g$, then $\dim
H^{0}(X,\mathcal{O}_{D_{P}})=2$, hence there exists $f\in\mathcal{M}(X)$ which
has a simple pole at $P$ and is holomorphic in $X\setminus\\{P\\}$.

If $\dim H^{0}(X,\Omega_{-D_{P}})=g-1$, then $\dim
H^{0}(X,\mathcal{O}_{D_{P}})=1$, hence there is no meromorphic function which
has a simple pole at $P$ and is holomorphic in $X\setminus\\{P\\}$.

Now we want to see the effect of changing $D_{P}$ from $(n-1)D_{P}$ (denoted
by $D_{(n-1)P}$) to $nD_{P}$ (denoted by $D_{nP}$). By the Riemann - Roch
Theorem

$\dim H^{0}(X,\mathcal{O}_{D_{(n-1)P}})=n-g+\dim
H^{0}(X,\Omega_{-D_{(n-1)P}})$

and

$\dim H^{0}(X,\mathcal{O}_{D_{nP}})=n+1-g+\dim H^{0}(X,\Omega_{-D_{nP}})$

If

$\dim H^{0}(X,\Omega_{-D_{(n-1)P}})=\dim H^{0}(X,\Omega_{-D_{nP}})$

then

$\dim H^{0}(X,\mathcal{O}_{D_{(n-1)P}})+1=\dim H^{0}(X,\mathcal{O}_{D_{nP}}).$

So there exists a meromorphic function $f\in\mathcal{M}(X)$ with a pole of
order $n$ at $P$ and is holomorphic in $X\setminus\\{P\\}$.

If

$\dim H^{0}(X,\Omega_{-D_{(n-1)P}})=\dim H^{0}(X,\Omega_{-D_{nP}})+1$

then

$\dim H^{0}(X,\mathcal{O}_{{D_{nP}}})=\dim H^{0}(X,\mathcal{O}_{D_{(n-1)P}}).$

So there will not exist a function with a pole of order $n$ at $P$ and is
holomorphic in $X\setminus\\{P\\}$.

So if $\dim H^{0}(X,\Omega_{-D_{nP}})$ remains the same as $n$ increases by
$1$, a new linearly independent function is added in going from the sheaf
$\mathcal{O}_{D_{(n-1)P}}$ to $\mathcal{O}_{D_{(n)P}}$.

From the Equation (2.2), we have $\dim H^{0}(X,\Omega)=g$. Suppose $\omega$ is
a non-vanishing meromorphc 1-form on a compact Riemann surface of genus $g$,
and $K$ is its divisor, then deg $(\omega)=2g-2$.

we can deduce that

$\dim H^{0}(X,\Omega_{-D_{(2g-1)P}})=0.$

Therefore, the number of times $\dim H^{0}(X,\Omega_{-D_{nP}})$ does not
remain the same must be $g$ times and at each change it decreases by 1.
This completes the proof. ∎

##  4\. Analyzing gaps and non-gaps

Let $d$ be the least non-gap value at the point $P$, if $n>d$ is a gap then
$n-d$ is again a gap value. Therefore, all the gaps occur in finite
arithmetical sequences of the form

$j,j+d,j+2d,\dots,j+\lambda_{j}d,$

where $j=1,2,\dots,d-1$ and $\lambda_{j}=0,1,2,\dots.$ See pp. 124 [4]. For
more details on gap sequence see [2].

######  Example 4.1.

Suppose $X$ is a compact Riemann surface fo $g=3$. We see the possible gap
sequences are
$\\{1,3,5\\}$, $\\{1,2,3\\}$, $\\{1,2,4\\}$, $\\{1,2,5\\}$
and corresponding non-gap sequences in $\\{2,\dots,2g\\}$ are:
$\\{2,4,6\\}$, $\\{4,5,6\\}$, $\\{3,5,6\\}$, $\\{3,4,6\\}$.
Note that $2g$ is always a non-gap.

Suppose $P\in X$. If $f$ has a pole of order $s$ at $P$, and $g$ has a pole of
order $t$ at $P$, then $fg$ has a pole of order $s+t$ at $P$. The following
problem may be of some interest to see the number of possible gaps and non-
gaps $\leq 2g$ on $X$. The reader must note that this problem is nothing to do
with the proof of the theorem.

######  Problem 4.2.

Write the numbers $2$ to $2g-1$ into two (disjoint) parts
$G=\\{n_{1},n_{2},\dots,n_{g-1}\\}$ and
$G^{\prime}=\\{m_{1},m_{2},\dots,m_{g-1}\\}$ such that no number in $G$ is a
sum of any combination of numbers in $G^{\prime}$. How many pairs of such $G$
and $G^{\prime}$ exist?

Clearly then $\\{1\\}\cup G$ gives possible gap sequence and
$G^{\prime}\cup\\{2g\\}$ gives corresponding non-gap sequence in
$\\{2,\dots,2g\\}$ at a point $P$ on $X$.

##  5\. Concluding Remarks

We define a Weierstrass point by using gap sequence as follows:

######  Definition 5.1.

Suppose $P\in X$ and

$0<n_{1}<n_{2}<\dots n_{g}<2g$

be the gap sequence at $P$. In terms of the gap sequence we define the weight
of the point $P$, denoted by $\omega(P)$ and defined by

$\omega(P)=\sum_{i=1}^{g}(n_{i}-i)$

Note that $w(P)\geq 0$ for all $P\in X$.

######  Definition 5.2.

A point $P\in X$ is called a Weierstrass point if $\omega(P)>0$.

These points provide an important information about the possible order of a
pole of meromorphic function at a point $P$ on $X$.

Hyperelliptic Riemann surfaces are characterized by the following gap sequence
at the Weierstrass points:

(5.1) $G=\\{1,3,\dots,2g-1\\}$

hence the non-gaps $(\leq 2g)$ are $G^{\prime}=\\{2,4,\dots,2g\\}$

Jenkins proved that if $h$ is a first non-gap sequence at $P$ and $(h,k)=1$,
then $k$ is a gap if

$g>\frac{(h-1)(k-1)}{2}.$

Further he observes, hyperelliptic Riemann surfaces are a special case whose
first non-gap sequence starts with a prime number. See [5].

For the Exceptional Riemann surfaces the gap sequence $G$ and the non-gap
sequence $G^{\prime}$ at each Weierstrass point is given by

(5.2) $G=\\{1,2,3,\dots,g-1,g+1\\}\mbox{ and
}G^{\prime}=\\{g,g+2,\dots,2g-1\\}.$

It is clear that the Theorems 1.1, 1.2 and 1.3 will give information on the
order of a pole of a meromorphic function which has a pole at a point $P$ and
holomorphic in $X\setminus\\{P\\}$. The following example illustrates this:

######  Example 5.3.

Suppose $X$ is a hyperelliptic Riemann surface of genus 6 and $P$ is a
Weierstrass point. Then there exist meromorphic functions with order 2, 4 and
6 at $P$. But at the same time if $P$ is not a Weierstrass point, there is no
meromorphic function with a pole of order $<7$ at $P$, and holomorphic in
$X\setminus\\{P\\}$.

### Acknowledgements

This work was supported by The National Board for Higher Mathematics (NBHM)
Govt. of India. Grant No. 02011/6/2023 NBHM(R.P)/4301.

## References

  * [1]
  * [2] H.M. Farkas and I. Kra, Riemann surfaces, GTM, Springer, Second Edition, 1991.
  * [3] O. Forster, Lectures on Riemann surfaces, GTM 81, Springer-Verlag, New York, 1993.
  * [4] R.C. Gunning, Lectures on Riemann surfaces, Princeton University Press, second printing 1968.
  * [5] James A Jenkins, Some Remarks on the Weierstrass points, Proceedings of the American Mathematical Society , Volume 44, Number 1, May 1974.
  * [6] George Springer, Introduction to Riemann surfaces, Addison Wesley Publishing Company, Inc, 1957.


# Quantum logic control and precision measurements of molecular ions in a ring
trap - a new approach for testing fundamental symmetries

Trevor N. Taylor  Joshua O. Island  Yan Zhou<EMAIL_ADDRESS>Department of
Physics and Astronomy, University of Nevada, Las Vegas, Las Vegas, Nevada
89154, USA

###### Abstract

This letter describes an experimental platform that implements quantum logic
control of polar molecular ions in a ring surface ion trap for precision
measurements. This approach targets unity state preparation and detection and
long spin precession coherence. A unique feature is bridging the spin-
polarization in a static frame and the spin-precession in a rotating frame.
The new method accommodates temporally and spatially localized multiplexing
measurements and can be ubiquitously applied to generic ion species. This
platform may be applied to the electron’s Electric Dipole Moment search using
232ThF+ and could be extended to the Nuclear Magnetic Quadrupole Momentum
search using 181TaO+.

††preprint: APS/123-QED

The Standard Model of particle physics serves as a cornerstone in describing
the fundamental constituents of matter and their interactions[1, 2]. However,
it is an incomplete theory raising many unanswered questions, such as what
dark matter and dark energy are, what mechanism induces matter and antimatter
imbalance, and why the strong interactions do not exhibit significant Charge
and Parity (CP) violation[3, 4, 5, 6, 7, 8]. One of the ways to test the
Standard Model is to probe and explore new particles created in a high-energy
collider[9]. A more accessible, table-top alternative is to sense the forces
and interactions generated by the new particles using quantum sensors, such as
polar molecules or highly charged ions[9, 10, 11]. The quantum sensor method
is conceptually akin to electron-neutron scattering experiments but at a much
lower energy scale and in tightly bound atomic or molecular systems. The probe
electron in $s$-type orbitals spends significant time close to and interacting
with a heavy nucleus with relativistic energy. Such interactions will result
in a subtle but detectable frequency shift of the electron spin resonance by
precision spectroscopy[12, 13, 14, 15].

This letter proposes a new approach for measurements of the electron’s
Electric Dipole Moment (eEDM) using quantum logically controlled molecular
ions in a segmented ring ion trap. The quantum logic spectroscopy (QLS) method
enables unity state preparation and detection[16, 17, 18, 19, 20]. The
circular motions of ions in a micro-fabricated ring trap generate an up to 32
V/cm rotating biased electric field and support extensive scalability and
parallel measurements. To adapt the QLS method and precision metrology in a
rotating frame, we propose and numerically investigate spin-polarization
protection during a transition from a static trap site with a DC magnetic
field, to a rotating ring trap with synchronously rotating electric and
magnetic fields. This new scheme completely decouples the active quantum
control of molecular ions with lasers and the free evolution of spin-
precession and facilitating high efficiencies of state preparation and
detection and long coherence simultaneously. Furthermore, we implement
temporally and spatially localized multiplexing measurements that would
suppress dominant systematics from inhomogeneous and time-variant magnetic
fields. The proposed scheme inherits many successful features from a well-
established experimental platform from the JILA eEDM group[21, 22, 23, 24, 25,
26]. The newly designed experimental platform is expected to achieve a $\mu$Hz
accuracy equivalent to new physics at a mass scale above 50 TeV.

Figure 1:  Experimental overview of the quantum logic control and spin-
precession measurements in a ring ion trap. (a) An energy diagram of the state
preparation of ThF+ using optical-microwave pumping (OMP) and quantum logic
spectroscopy (QLS) with Yb+. OMP concentrates $\sim$30$\%$ population of ThF+
in ${}^{1}\Sigma^{+},v=0,J=0,m_{F}=-1/2$. QLS using a pair of referenced
optical frequency combs (OFCs) transfers the population to one of the eEDM
sensitive states, ${}^{3}\Delta_{1},v=0,J=1,m_{F}=-3/2$, + parity. Motional
coupling induced by Coulomb force transports the QLS information to the Yb+
detected by electron shelving and fluorescence. (b) Schematic diagram of the
segmented ring ion trap for implementing the QLS scheme in the precision
measurements with biased electric and magnetic fields. The circular motion of
ions generates the rotating electric field and synchronous magnetic in a field
gradient, which is created by a pair of anti-Helmholtz coils (not shown). (c)
Time sequence of the precision measurements. ThF+, Yb+: creating and trapping
ThF+ and Yb+. OP: optical pumping. GS: ground-state cooling. QL: quantum logic
readout. TP: transport the ions from the state preparation site (No. 1 in the
red box in (b)) to another empty site. ROT: ramp up the rotation from 0 to 100
kHz. $\sim$ROT: ramp down the rotation from 100 kHz to 0. The dashed red arrow
indicates the starting point after the first measurement.

State preparation and detection by QLS Two valence electrons and closely
spaced states with opposite parities in ThF+ provide a significant eEDM
enhancement in a weak bias electric field. However, simultaneously, the large
number of nearly degenerate quantum states substantially dilutes the desired
coherent signal. The JILA eEDM group has demonstrated a $\sim$50$\%$ state
preparation of ThF+ in an eEDM-sensitive state (${}^{3}\Delta_{1}$, $v$=0,
$J$=1, single $m_{F}$ and $\Omega$) by applying optical and microwave pumping
(OMP)[27]. Furthermore, they have also developed a resonant enhanced
multiphoton dissociation (REMPD) detection scheme with $\sim$30$\%$
efficiency[28]. The impurity of the state preparation comes from population
leakage to unknown dark states (from 3000 to 10000 cm-1) and vibration excited
states (v$>$1) during the OMP. The non-ideal efficiency of the state detection
is determined by the complexity of multiple dissociative and radiative
channels of the highly excited ThF+. Improvements in state preparation and
detection could require substantial spectroscopic work and more repumping
lasers. In this letter, we develop a universal method by integrating OMP and
QLS for single quantum state preparation and detection with unity efficiency,
as shown in Figure 1(a). This method could be directly applied to other
molecular ions, even polyatomic molecules, without substantial strategy
modifications or spectroscopic work.

Summarizing the procedure, First, ThF+ ions are generated by resonance-
enhanced multi-photon ionization (REMPI) and distributed in a single vibronic
state but four rotational states. Second, one ThF+ and one Yb+ are loaded into
a radiofrequency (RF) ion trap and Doppler cooled. Third, OMP lasers and
microwaves transfer $\sim$30$\%$ ThF+ to ${}^{1}\Sigma^{+}$, $v=0$, $J=0$,
$m_{F}=-1/2$ state. Spin polarization is generated by a static magnetic field
and a circularly polarized pumping laser. Fourth, a ground state cooling and
electron shelving are applied to Yb+ for preparing the ${}^{2}D_{3/2}$, $F=2$,
$n=0$ state, where $n$ labels the normal mode of Yb+ and ThF+ in the harmonic
trap. Fifth, a pair of Raman pulses from referenced optical frequency combs
coherently transfer ThF+ to the eEDM sensitive state, ${}^{3}\Delta_{1}$,
$v=0$, $J=1$, $m_{F}=-3/2$, $+$ parity, $n=1$ with motional excitation that
impacts on Yb+ as well. The $m_{F}$ selectivity comes from the polarization of
the Raman beams and 100 kHz Zeeman shift (about 10 Gauss B field). After OMP
in step 3, if ThF+ is not in ${}^{1}\Sigma^{+}$, $v=0$, $J=0$, $m_{F}=-1/2$
state, the motional excitation will not happen, and the normal mode of the ion
pair is still $n=0$. Sixth, a blue sideband $\pi$ pulse is applied to
resonantly de-excite the ground electronic and motional state, Yb+ to
${}^{2}S_{1/2}$, $F=1$, $n=0$, which can be optically detected by a cycling
transition. If the number of collected photons is above a detection threshold,
according to the motional entanglement between Yb+ and ThF+, it is 100$\%$
that the ThF+ is at the ${}^{3}\Delta_{1}$, $v=0$, $J=1$, $m_{F}=-3/2$, $+$
parity state. Otherwise, the above process from 1 to 6 needs to be processed
again. A numerical model using typical experimental parameters shows a unity
eEDM sensitive state preparation of ThF+ could be achieved after $\sim$30 ms
OMP and QLS. A similar model has also been applied to more complicated
molecules, such as 229ThF+ with more hyperfine states. Numerical simulations
show a low state preparation efficiency using OMP only but a unity efficiency
if QLS is implemented. The state detection is quantum logically transferring
ThF+ from ${}^{3}\Delta_{1}$ to ${}^{1}\Sigma^{+}$ with a similar state
readout method.

Segmented ring ion trap and rotating electric and magnetic fields. To probe
fundamental symmetry breaks using molecular ions, a biased electric field must
be applied to mix states with opposite parities. However, this biased electric
field is typically incompatible with trapped ions because even a weak DC
electric field ($\sim$10 V/cm) will eject the ions out of the trap. The JILA
eEDM group first demonstrated polarization of the molecular ions using a
rapidly rotating electric field[23, 24]. However, implementing QLS in a
rotating frame is challenging because of difficulties in light delivery and
excess heating. In this letter, we solve this conflict by completely
decoupling the QLS and spin-precession measurements. An essential step of our
solution is to design and fabricate a segmented ring trap, as shown in Figure
1(b), which inherits many features from a trap designed by Sandia National Lab
for scalable quantum computation[29].

The fabrication of the proposed trap can be realized using existing
microfabrication techniques. The most critical elements of the fabrication are
creating a planarized surface for the trap electrodes and keeping obstructions
that protrude from the trap surface away from the trap. To achieve a flat
surface that is free of bonding wires that may interfere with trapping lasers,
several studies have investigated the use of silicon plug/via technology[29,
30, 31, 32]. Harnessing CMOS technology in this way allows the freedom to
fabricate multilayer structures with both RF and DC electrical access[33]. The
tight radial confinements (2.5 MHz) are from three concentric RF (20 MHz, 200
Vpp) and ground electrodes. The loose axial confinement (1 MHz) is from the
segmented DC electrodes inside and outside the RF ring electrodes. The radius
of the ring trap is 3 mm, and the ions are located about 100 $\mu$m above the
surface. The ring trap consists of 96 segmented electrodes divided into 12
groups (+V, 0, -V, 0) that form 12 trapping sites. The trapping sites and ions
inside can be rotated clockwise or counterclockwise by modulating the
segmented electrodes sinusoidally. Centripetal acceleration provides a
rotating electric field that polarizes polar molecules. The trap radius and
maximum rotating frequency determine the amplitude of the electric field (32
V/cm at 100 kHz). A radial magnetic field gradient is applied by a pair of
anti-Helmholtz coils concentric to the ring trap. When the ions are rotating,
they will experience a rotating magnetic field that is parallel or anti-
parallel to the electric field in the rotating frame. Reversing the coil
current can reverse the magnetic field direction. Conversely, when the ions
are standstill, there is still a non-zero static inward/outward magnetic field
that defines the quantization axis for the QLS scheme of the state preparation
and detection.

Figure 2:  Spin-polarization during the rotating ramp-up procedure. Panels (a)
and (b) show two types of rotational coupling at low and intermediate electric
fields. The blue arrows mark Stark-induced frequency shifts, and the red
arrows mark rotation-induced couplings (darker and thicker arrows represent
stronger couplings). Panel (c) shows the evolution of quantum states with a
fast rotation ramp (3.3 MHz/ms). Panel (d) shows the evolution of quantum
states with slow and intermediate rotation ramps (10 kHz/ms first, and 300
kHz/ms after). Panel (e) shows numerical evaluations of spin-polarization
dynamics with a 1 ms linear ramp. The red arrow marks the population transfer
described in Panel (a), and the blue arrow marks the population transfer
described in Panel (b). Panel (f) compares different rotating ramp-ups for
four spin-polarized states. The color scale indicates the percentage of the
population in which the spin polarization is preserved.

Spin-polarization protection.  By integrating QLS state preparation, our
segmented ring trap, and static quadrupole magnetic potential, we now show how
to prepare ThF+ in one spin-polarized eEDM sensitive state (A, B, C, or D in
Figure 2). The standstill ions will be accelerated and transferred to a fast-
rotating frame experiencing a $\sim$32 V/cm biased electric field. However,
during the rotating ramp-up, the rotation-induced coupling, $H_{rot}$, could
mix magnetic sub-levels ($\Delta m_{F}$ = $\pm$1) to depopulate the spin-
polarized states[21]:

$H_{rot}=-\omega_{rot}(F_{z}\cos{\theta}-F_{x}\sin{\theta}),$ (1)

where $\theta$ is the angle between the rotating and quantization axes and
equal to 90∘ in this case, $\omega_{rot}$ is the rotational angular frequency.
Such interaction could be strong when the perturbed states ($m_{F}$=$\pm$1/2)
are degenerate to the spin-polarized states ($m_{F}$=$\pm$3/2) due to a
cancellation of Zeeman and Stark shifts at $\sim$1.5 V/cm and a cancellation
of Stark and hyperfine shifts at $\sim$12 V/cm electric fields respectively,
thick red arrows in Figure 2 (a) and (b). Moreover, nearly-degenerate
interactions that could contribute to about $0.1\%$ of population transfer are
marked in Figure 2 using light or thin (dashed) pink arrows. We numerically
simulate the dynamics during the rotating ramp-up to evaluate such degenerate
interaction induced de-polarization. Figure 2 (e) shows that the rotational
coupling completely depopulates the A and D states and partially depopulates
the C state with a 1 ms linear ramp. Only the population of state B is
completely preserved.

One solution to suppress spin-depolarization induced by the degenerate
interactions is increasing the ramp rate of the rotation, as shown in Figure 2
(c). The numerical model shows a 30 $\mu$s ramp time (3.3 MHz/ms ramp rate)
can preserve the spin-polarization by $\sim$93$\%$, as shown in Figure 2 (f).
However, by applying a rapidly accelerating field, the ions could not evolve
adiabatically which leads to excessive heating or even detrapping of ions. An
alternative method is first selectively preparing the population in A’, B, C,
D’ in Figure 2 (a) and (d). Then, the frequency of rotation $f_{rot}$ is
ramped up slowly (10 kHz/ms). Thus, the degenerate couplings adiabatically
transfer the population from A’ to A and D’ to D. Due to the lack of
degenerate interactions, populations of B and C are conserved. However, with
such a slow frequency ramp, populations of A and C could leak to $F$ = 1/2
states adiabatically, as shown in Figure 2 (d). To solve this issue, when
$f_{rot}$ = 30 kHz, we increase the ramp rate to 300 kHz/ms until $f_{rot}$ =
100 kHz. Thus, the adiabatic interactions between A, C, and the $F$ = 1/2
states are suppressed. The maximum ramp rate here is still ten times lower
than the first scheme’s. Therefore, the excess heating or de-trapping can be
largely reduced. Figure 2 (f) summarizes the spin-polarization efficiency
after rotation ramp-up using different methods.

Coherent superpositions A coherent superposition of states A and C, or B and D
must be initiated before and after the spin precession measurements. Following
the JILA eEDM group, we will generate such coherence using a fourth-order
interaction from combined perturbations of rotational coupling and Stark
mixing[24]. An effective two-level Hamiltonian (ignoring the EDM contribution,
different $g_{F}$-factors of A/C and B/D) of states A and C, or B and D, is :

H = [-3gFμBBp(ωrotErot)3p(ωrotErot)3+3gFμBB]
=[-3gFμBBp′(1ωrot)3p′(1ωrot)3+3gFμBB]

(2)

where $g_{F}$ is the $g$ factor of the $F=3/2$ state, $\mu_{B}$ is the Bohr
magneton, $B$ is the amplitude of the effective rotating magnetic field
(${\bf{r}}\cdot\nabla{\bf{B}}$), and all other parameters are grouped in $p$
and $p^{\prime}$. The electric field and rotating frequency are connected to
be $E_{rot}=m\omega_{rot}^{2}r/e$, where $r$ is the trap radius, and $e$ is a
unity charge. Therefore, the diagonal matrix elements can be tuned by changing
the applied magnetic field gradient, and the off-diagonal matrix elements can
be adjusted by changing the rotating frequency. Thus, generating off-resonant
$\pi/2$ pulses is possible by either modulating the rotating frequency or the
magnetic field gradient. The latter method is preferred because of the non-
degradation of spin-polarization.

Experimental sequence Integrating all procedures described above, Figure 1 (c)
shows a complete time sequence of the measurements. First, ThF+ and Yb+ are
created and trapped. Second, the QLS state preparation scheme will be applied
to create a single spin-polarized state of ThF+ with a static magnetic field.
To guarantee the unity success of a single target state preparation, multi-
cycle of optical pumping (OP), ground state cooling (GS), and quantum logic
readout (QL) could be required. Third, the single state-prepared ThF+ and Yb+
are transported to other trap nodes (TP). All twelve spin-polarized ThF+ ions
are expected to be prepared in a second. The twelve ions could be prepared in
the same quantum state or interleaved in different states ($\Omega=\pm 1$) for
systematics suppression. Fourth, a programmed rotation ramp transfers all ions
from a static frame to a 100 kHz rotating frame with a 32 V/cm bias electric
field. Fifth, the magnetic field amplitude is ramped down from 10 Gauss to 10
milli-Gauss. Immediately after that, the magnetic field is abruptly turned off
and on for $\sim$100 ms to induce coherence (effective $\pi$/2 pulse) between
the spin-polarized states. Sixth, a phase accumulation during a long spin-
precession happens in the rotating frame. Then, another $\pi$/2 pulse is
applied to map the phase evolution to the population. Seventh, reversing steps
2 to 5 – the magnetic field is ramped up, the rotation frequency is ramped
down, and the QLS method is applied to readout for all twelve ions. Lastly, if
the spin-depolarization is negligible after quantum state readout, ThF+ should
have already been in a pure spin-polarization state. Therefore, a subsequent
spin-precession measurement can start immediately without further state
preparation.

Systematics Following the JILA eEDM experiments, three binary experimental
switches – the direction of magnetic field ($B$ switch), the direction of
rotation ($R$ switch), and the molecular orientation ($D$ switch), will be
implemented[24]. A linear combination of eight independent measurements will
be applied to eliminate leading terms of systematics. Differently, in this
work, twelve ions in the ring trap will be prepared in $\Omega$ = $\pm$1
(interleaved with the binary switch $D$). The spin precession measurements
with opposite effective electric fields happen simultaneously in both time and
space. The spatially and temporally parallel measurements would reduce
systematics from drifting and inhomogeneous electric and magnetic fields. The
simultaneous measurements will be essential for reducing systematics when the
coherence time is increased substantially.

Scalability and sensitivity Compared to the precision measurements using
ensembles of molecules, the scheme of a single molecule per trap site
compromises the signal readout. To rectify the low molecular number, the
scalability of the ring trap with many trap sites is essential. Furthermore,
integrated photonic technology developed by MIT Lincoln lab could be applied
to scale light delivery to the trap as well[34]. Therefore, a large ring trap
with hundreds of trap sites is practical. Moreover, reliable long-term data
acquisition could be easily accessed with an autopilot control system because
of no high-power pulsed lasers, unstable molecular sources, or mechanically
moving parts.

Table 1 describes estimations of statistical sensitivities of the proposed
measurements. Generation I could be performed at room temperature aiming at
(1) demonstrating near-unit state preparation and readout schemes using QLS,
(2) demonstrating the validity of the proposed precision metrology, and (3)
preliminary investigation of systematics. Generation II could be conducted at
cryogenic temperature for (1) maximum QLS efficiency with low surface noise,
(2) highly multiplexed measurements, and (3) long coherence time.

In this letter, we describe a design of a novel experimental platform for
precision measurements using quantum logically controlled molecular ions. In
addition to the eEDM measurements using 232ThF+, this scheme is ubiquitous to
generic ion species, such as 181TaO+ for probing Nuclear Magnetic Quadrupole
Momentum[35] and HfO+ for detecting Nuclear-spin dependent parity
violation[36].

Parameters | Gen I (300 K) | Gen II (4 K)
---|---|---
No. of trap sites | 12 | 400
State preparation | 90$\%$ | 99$\%$
State readout | 90$\%$ | 99$\%$
Contrast | 0.9 | 0.99
Coherent time | 4 s | 100 s
Duty cycle | 90$\%$ | 99$\%$
Data collection | 300 h | 1000 h
Frequency accuracy | 15 $\mu$Hz | 0.2 $\mu$Hz
Table 1:  Target sensitivities of the Phase I and Phase II measurements.

###### Acknowledgements.

We thank B. Zygelman, A. Jayich, J. Chou, and R. W. Field for helpful
discussions. This work is financially supported by UNLV Faculty Opportunity
Awards, UNLV Undergraduate Research Mentor-Mentee Program (MMP), and NSF MRI
grant (2117253).

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