# Improving Time Series Classification Algorithms Using Octave-Convolutional
Layers

Samuel Harford
Department of Mechanical & Industrial Engineering
University of Illinois at Chicago
<EMAIL_ADDRESS>
&Fazle Karim
Department of Mechanical & Industrial Engineering
University of Illinois at Chicago
<EMAIL_ADDRESS>
&Houshang Darabi
Department of Mechanical & Industrial Engineering
University of Illinois at Chicago
<EMAIL_ADDRESS>
###### Abstract

Deep learning models utilizing convolution layers have achieved state-of-the-
art performance on univariate time series classification tasks. In this work,
we propose improving CNN based time series classifiers by utilizing Octave
Convolutions (OctConv) to outperform themselves. These network architectures
include Fully Convolutional Networks (FCN), Residual Neural Networks
(ResNets), LSTM-Fully Convolutional Networks (LSTM-FCN), and Attention LSTM-
Fully Convolutional Networks (ALSTM-FCN). The proposed layers significantly
improve each of these models with minimally increased network parameters. In
this paper, we experimentally show that by substituting convolutions with
OctConv, we significantly improve accuracy for time series classification
tasks for most of the benchmark datasets. In addition, the updated ALSTM-
OctFCN performs statistically the same as the top two time series classifers,
TS-CHIEF and HIVE-COTE (both ensemble models). To further explore the impact
of the OctConv layers, we perform ablation tests of the augmented model
compared to their base model.

_Keywords_ Time Series $\cdot$ Classification $\cdot$ Octave Convolution

##  1 Introduction

Time series data is ubiquitous. It exists in weather readings Taylor et al.
(2009), financial recordings Tsay (2005), industrial observations Alwan and
Roberts (1988), and psychological signals Jebb et al. (2015); Kadous et al.
(2002). Over the past two decades, classifying these time series data has
received enormous attention, and their models are constantly improving. The
main objectives of these time series classification algorithms are to perform
accurately and efficiently Sharabiani et al. (2018).

Several models have been proposed to classify time series data accurately.
Typically, these models are feature-based, ensembles, or deep learning models.
Some promising feature-based univariate time series classification models are
Bag-of-Words (BoW) Lin et al. (2007), Bag-of-features (TSBF) Baydogan et al.
(2013), Bag-of-SFA-Symbols (BOSS) Schäfer (2015), BOSSVS Schäfer (2016), and
Word ExtrAction for time Series cLassification (WEASEL) Schäfer and Leser
(2017). In 2007, Lin et al. propose representing time series data using a
symbolic representation, which is fed into a BoW classifier Lin et al. (2007).
Subsequently, Baydogan et al. propose utilizing multiple subsequences of
random lengths to capture local information of time series data (TSBF)
Baydogan et al. (2013). These subsequences act as a cookbook to classify time
series data. In 2015 and 2016, Schafer et al. proposed BOSS Schäfer (2015) and
BOSSVS Schäfer (2016) representing time series data symbolically utilizing
histograms and a symbolic Fourier approximation. BOSSVS is an extension of
BOSS that reduces the time complexity by utilizing a vector space model. In
2017, Schafer and Leser proposed WEASEL Schäfer and Leser (2017), an accurate
and scalable time series classification model. WEASEL extracts important
features by applying a chi-square test on discriminative words that are
obtained through a symbolic Fourier approximation. Fast logistic regression
utilizes these features to classify time series data. Around the same time,
Flynn et al. present RISE, which extracts features of a time series in the
time domain (using autocorrelation function, partial autocorrelation function,
and an autoregressive model) and features from the frequency domain using
power spectrum Flynn et al. (2019). A random forest is trained on these
extracted features to classify the time series data. Nguyen et al. propose
utilizing multiple resolutions of symbolic representations, multiple domains
representations, and efficient navigation of a huge symbolic word space via a
modified symbolic sequence classifier to achieve state-of-the-art results.

Several ensemble models, such as Elastic Ensemble (EE) Lines and Bagnall
(2015), Shapelet Ensembles (SE) Bagnall et al. (2015), Flat Collective of
Transformation-Based Ensembles (FLAT-COTE) Bagnall et al. (2015), Hierarchical
Vote COTE (HIVE-COTE) Lines et al. (2016), and Time Series Combination of
Heterogeneous and Integrated Embeddings Shifaz et al. (2020) and have yielded
state-of-the-art performance. EE integrates 11 1-NN algorithms via a weighted
ensemble method. SE integrates a heterogeneous ensemble on a transformed
shapelet. FLAT-COTE. FLAT-COTE combines 35 various time series classifiers
into a single classifier by applying a weighted majority vote. HIVE-COTE
extends FLAT-COTE by integrating the BOSS and RISE algorithms. More recently,
TS-CHIEF Shifaz et al. (2020) utilizes tree-structured classifiers to
efficiently ensemble classifiers yielding results similar to HIVE-COTE for a
fraction of the time.

Deep learning models have received much attention. Fully Convolutional Network
(FCN) Wang et al. (2017), Residual Neural Network (ResNet) Wang et al. (2017),
LSTM-FCN, and ALSTM-FCN are currently the state-of-the-art deep learning
algorithms for time series classification Karim et al. (2017). All these
algorithms take advantage of convolutional neural network (CNN) layers to
extract features to classify univariate time series data. This paper proposes
improving all CNN based time series classifiers by substituting the CNN layers
with a modified Octave-Convolutional layer. An OctConv layer factorizes the
mixed feature maps of a convolutional layer by their frequencies. OctConv has
never been applied to time series data or on 1D vectors. Results indicate
OctConv to statistically improve the classification accuracy of the various
time series deep learning architectures.

The remainder of this paper is structured as follows: Section 2 provides
background on univariate time series classifiers and neural network
architectures. Section 3 details the proposed methods for this work. Section 4
presents the experiments and results of our proposed methodology on the
benchmark univariate time series classifiers. Section 5 concludes the paper
and discusses future work.

##  2 Literature Review

###  2.1 Fully Convolutional Network

FCNs have proven to be effective deep learning classifiers in many domains
including computer vision Wang et al. (2015) and natural language processing
Tachibana et al. (2018). Wang et al. introduced the use of FCNs as a good base
for a deep learning based time series classifier Wang et al. (2017). The input
block accepts a univariate time series. This is then passed to three
convolutional blocks where each block has a 1-D convolutional layer, followed
by batch normalization, and finally a ReLU activation. The convolutional
layers have filter sizes of {128, 256, 128} and kernel sizes of {8, 5, 3},
respectively. The convolutional blocks are followed by a global average
pooling layer and finally a softmax layer to output a class probability
vector.

###  2.2 Residual Neural Network

ResNets utilize deeper structures of networks and employ skip connections to
utilize information from bottom level gradients. ResNets are often utilized in
object detection Jung et al. (2017). Wang et al. explore the use of ResNets
for time series classification Wang et al. (2017). The input block accepts a
univariate time series. This is then passed to three residual blocks. Each
residual block goes through a 1-D convolutional layer, batch normalization,
and ReLU layer three times. The final ReLU layer of each residual block has a
skip connection to the input of the block. The filter size for convolutional
layers in the same block is the same where the sizes are {64, 128, 128} for
the respective residual blocks. The three residual blocks are then followed by
a global average pooling layer and a softmax output.

###  2.3 LSTM-FCN & ALSTM-FCN

The LSTM-FCN architecture aims to extract feature information by branching the
input Karim et al. (2017). The input block accepts a univariate time series.
The input is then fed to two branches. One branch is an FCN with the same
parameters defined in Section 2.1. The other branch performs a dimension
shuffle on the input. This step transposes the input shape from a univariate
time series to a multivariate time series where each channel is a single time
step. Without the dimension shuffle the model training results in severe
overfitting. This branch then passes to an LSTM layer or Attention LSTM layer
Bahdanau et al. (2016). The final step of this branch is a dropout layer. The
result of the two branches is then concatenated and fed to a softmax output
layer.

###  2.4 Octave Convolution Layer

The Octave Convolution is formulated as a replacement for general
convolutional layers that reduces the spatial redundancy and computational
cost Chen et al. (2019). An OctConv operation consists of decomposing the
input into a high and low frequency. Let the high and low frequency output of
the OctConv be represented by $Y=\\{Y^{H},Y^{L}\\}$ where the frequency
components are calculated as:

$\displaystyle Y^{H}=Y^{H\rightarrow H}+Y^{L\rightarrow H}$ $\displaystyle
Y^{L}=Y^{L\rightarrow L}+Y^{H\rightarrow L}$

where $Y^{H\rightarrow H}$ and $Y^{L\rightarrow L}$ represent the intra-
frequency updates, and $Y^{L\rightarrow H}$ and $Y^{H\rightarrow L}$ are the
inter-frequency communications. The intra-frequency component is used to
update the information within each frequency, while inter-frequency
communication further enables information exchange between frequencies.

These intra and inter frequencies can are calculated using the convolutional
kernel $W$ and input tensor $X$. The representation of $W$ is separated into
high and low components $W=\\{W^{H},W^{L}\\}$ responsible for convolving with
the respective input component $X=\\{X^{H},X^{L}\\}$. Each of these components
can further be divided into the intra and inter frequencies
$W^{H}=\\{W^{H\rightarrow H},W^{H\rightarrow L}\\}$ and
$W^{L}=\\{W^{L\rightarrow H},W^{L\rightarrow L}\\}$. This breakdown of
components formalize the output tensor calculations as:

$\displaystyle Y_{p,q}^{H}=Y^{H\rightarrow H}+Y^{L\rightarrow H}$
$\displaystyle=\sum_{i,j\in\mathcal{N}_{k}}W^{H\rightarrow
H}_{i+\tfrac{k-1}{2},j+\tfrac{k-1}{2}}\top X^{H}_{p+i,q+i}$
$\displaystyle+\sum_{i,j\in\mathcal{N}_{k}}W^{L\rightarrow
H}_{i+\tfrac{k-1}{2},j+\tfrac{k-1}{2}}\top
X^{L}_{\lfloor\frac{p}{2}\rfloor+i,\lfloor\frac{q}{2}\rfloor+i}$

$\displaystyle Y_{p,q}^{L}=Y^{L\rightarrow L}+Y^{H\rightarrow L}$
$\displaystyle=\sum_{i,j\in\mathcal{N}_{k}}W^{L\rightarrow
L}_{i+\tfrac{k-1}{2},j+\tfrac{k-1}{2}}\top X^{L}_{p+i,q+i}$
$\displaystyle+\sum_{i,j\in\mathcal{N}_{k}}W^{H\rightarrow
L}_{i+\tfrac{k-1}{2},j+\tfrac{k-1}{2}}\top X^{H}_{2*p+0.5+i,2*q+0.5+i}$

where $(p,q)$ is a location coordinate,
$\mathcal{N}_{k}=\\{(i,j):i=\\{\frac{-k-1}{2},\dots,\frac{k-1}{2}\\},j=\\{\frac{-k-1}{2},\dots,\frac{k-1}{2}\\}\\}$
is a neighboring coordinate, and $\lfloor\rfloor$ represents the floor
operation. Figure 1 illustrates the interactions of the Octave Convolution.

Figure 1: Octave Convolution.

##  3 Methodology

###  3.1 1D Octave Convolutional Layer

Deep learning models have proven to be successful models to classify
univariate time series Wang et al. (2017). The majority of deep learning
models for time series classification utilize a series of CNN layers as the
primary feature extractor of the neural network architecture. While CNNs have
proven to be effective for a variety of deep learning classification tasks,
there remains room for improvement in the feature extraction of these layers.
In this work, we propose extensions of deep learning models for time series
classification with the use of OctConv layers instead of CNN layers. We
replace the convolutional layers for Octave Convolutions, OctConv Fully
Convolutional Network (OctFCN), OctConv Residual Neural Network (OctResNet),
LSTM-OctConv Fully Convolutional Network (LSTM-OctFCN), Attention LSTM-OctConv
Fully Convolutional Network and (ALSTM-OctFCN) we use are for the OctConv
layer to the case of 1D sequence models to enhance classification accuracies
for each model.

Figure 2: Neural Network Augmentations and OctConv Blocks

The OctConv augmentation to the models in Figure 2D-F aims to enhance the
feature extraction at each layer. The initial OctConv layer takes the same
input shape as the original convolutional layer. The filter size $f_{init}$
for the initial layer is split based on a parameter $\alpha$ which is set to
0.5. The high filter is achieved by feeding the input to a 1-D convolutional
layer with filter size of $\alpha*f_{init}$. The low filter is achieved by
first downsampling the input through a 1D average pool and then fed to a 1-D
convolutional layer with filter size of $(1-\alpha)*f_{init}$. The initial
layer then outputs a high and low filter component. The intermediate layers of
the OctConv blocks take as input a high and low component and again split each
component by the factor $\alpha$ and the filter size $f_{inte}$. The
high$\rightarrow$high component is obtained through a 1-D convolution of the
high input with a filter size of $(1-\alpha)*f_{inte}$. The
low$\rightarrow$high component is obtained through a 1-D convolution of the
low input with a filter size of $(1-\alpha)*f_{inte}$ and then performing a
1-D upsampling. The final high component is calculated by an elementwise
addition of the high$\rightarrow$high and low$\rightarrow$high components. The
high$\rightarrow$low component is obtained through a 1-D average pooling of
the high input and then fed to a 1-D convolution with a filter size of
$\alpha*f_{inte}$. The low$\rightarrow$low component is obtained through a 1-D
convolution of the low input with a filter size of $\alpha*f_{inte}$. The
final low component is the elementwise addition of the high$\rightarrow$low
and low$\rightarrow$low components. The final layer of the OctConv block takes
as input a high and low component and the filter size $f_{fin}$. Both the high
and low inputs are fed to 1-D convolutions with filter size $f_{fin}$. The low
component is then upsampled to match shapes. The high and low components are
finally added together to can the final OctConv output.

OctConv layers aim to save on computational processing and memory through the
use of the high and low filter Chen et al. (2019). In addition, the processing
for features in the high and low filters results in better model performance
overall. In this work, we aim to understand the impact of OctConv’s ability to
capture global information through the task of time series classification. The
Octave Convolution function aims to improve performance over transitional
convolutions by exploiting the low- and high-frequency information and
increasing the field size to learn better global information.

###  3.2 Network Architectures

In this section, we describe the model architectures for the four most common
CNN based time series classifiers.

####  3.2.1 OctConv Fully Convolutional Network

The OctFCN architecture is an extension of the FCN introduced in Section 2.1.
Figure 2A illustrates the OctFCN architecture. The input block accepts a
univariate time series. This is then passed to three OctConv blocks where each
block has a 1-D Octave Convolutional layer, followed by batch normalization,
and finally a ReLU activation. The 1-D Octave Convolutional layers have filter
sizes of {128, 256, 128} and kernel sizes of {8, 5, 3}, respectively. The
three OctConv blocks are followed by a global average pooling layer and
finally a softmax layer to output a class probability vector.

####  3.2.2 OctConv Residual Neural Network

The OctResNet architecture is an extension of the ResNet introduced in Section
2.2. Figure 2B illustrates the OctResNet architecture. The input block accepts
a univariate time series. This is then passed to three modified residual
blocks. Each of these blocks goes through a 1-D Octave Convolutional layer,
batch normalization, and ReLU layer three times. The final ReLU layer of each
residual block has a skip connection to the input of the block. The filter
size for 1-D Octave Convolutional layers in the same block is the same where
the sizes are {64, 128, 128} for the respective blocks. The three modified
residual blocks are then followed by a global average pooling layer and a
softmax output.

####  3.2.3 A/LSTM-OctConv Fully Convolutional Network

The LSTM-OctFCN architecture is an extension of the LSTM-FCN introduced in
Section 2.3. Figure 2C illustrates the LSTM-OctFCN architecture. The input
block accepts a univariate time series. The input is then fed to two branches.
One branch is an OctFCN with three OctConv blocks where each block has a 1-D
Octave Convolutional layer, followed by batch normalization, and finally a
ReLU activation. The 1-D Octave Convolutional layers have filter sizes of
{128, 256, 128} and kernel sizes of {8, 5, 3}, respectively. The other branch
performs a dimension shuffle on the input. This step transposes the input
shape from a univariate time series to a multivariate time series where each
channel is a single time step. The final step of this branch is a dropout
layer. This branch is not modified because there are no CNN layers. The result
of the two branches is then concatenated and fed to a softmax output layer.

###  3.3 Training & Evaluation Methodology

Time series datasets are defined as a tensor of shape (N, Q, M), where N is
the number of samples, Q is the maximum length of a time series, and M is the
number of dimensions. In the case of univariate time series M is 1 for all
datasets. While this study focuses on univariate time series classification,
the models can easily be extended to multivariate applications.

The primary modeling parameters for each network architecture are defined in
Section 3.1. The majority of these parameters are constant for all datasets
that are modeling. During the training of the neural networks and their
OctConv variants, a gridsearch of optimal parameters is not conducted. The
objective of these experiments is to compare the augmentation of the
convolution layers, not to obtain the optimal model. The only exception is the
number of units in the LSTM for the LSTM-FCN and LSTM-OctFCN models. These
units have been optimized in previous works and are utilized in our study
Karim et al. (2017, 2019).

In this paper, various models are evaluated using accuracy and rank. To fairly
compare these models, we use an average of 20 runs for each model.

##  4 Experiments & Results

###  4.1 Univariate Time Series Benchmarks

The current University of California-Riverside (UCR) time series
classification benchmark consists of 128 univariate datasets Dau et al.
(2019). The first 85 datasets were introduced in the 2015 version of the
benchmarks. These datasets consist of univariate time series with uniform
length and z-normalization preprocessing. The additional 43 datasets in the
current version aim to address the issues of the data and task the user with
more realistic classification benchmarks. This includes the addition of
varying length time series and removes the preprocessing of time series
benchmarks.

###  4.2 VS Deep Learning Classifiers

This study aims to show the impact of the OctConv layer as a replacement for a
traditional convolutional layer. This replacement is evaluated by comparing
neural networks trained using the convolutional layer compared to the same
architecture with OctConv in place of the convolutional layers. These neural
networks include the FCN, ResNet and LSTM-FCN introduced in Figure 2. In
addition, a modified LSTM-FCN for an attention lstm (ALSTM-FCN) is evaluated
during the experiments.

To compare a neural network to its OctConv version, the average accuracy of 20
trained networks is used as the final metric for each classifier. The Wilcoxon
signed-rank test (WSRT) is a non-parametric statistical test that hypothesizes
the median of the rank between the compared models is similar. The alternative
hypothesis of the WSRT is that the median of the rank is not similar for
compared models. The WSRT is used to evaluate the performance of base neural
networks with their OctConv versions. Table 1 shows the results of the WSRT
for the four neural networks explored in this study. At a p-value of 0.05 all
of the evaluated networks show a significant improvement in accuracy across
the benchmark datasets. This shows the impact of the OctConv in place of the
traditional convolution results in significantly better models in a variety of
neural networks for the task of time series classification.

Table 1: Wilcoxon Signed-Rank Test comparing AVERAGE testing accuracy of Models developed with traditional convolutions vs OctConvs (p-value shown) Model | Oct P-Value
---|---
FCN | 4.60E-07
Resnet | 3.19E-13
LSTM-FCN | 3.51E-10
ALSTM-FCN | 1.63E-09

In addition to the average accuracy, the maximum accuracy of 20 trained
networks is compared. Table 2 shows the results of the WSRT for the four
neural networks based on the maximum accuracy. At a p-value of 0.05 all
networks except ALSTM-FCN show to have a significant improvement in maximum
accuracy on the benchmark datasets.

Table 2: Wilcoxon Signed-Rank Test comparing MAXIMUM testing accuracy of Models developed with traditional convolutions vs OctConvs (p-value shown) Model | Oct P-Value
---|---
FCN | 7.44E-05
Resnet | 1.33E-03
LSTM-FCN | 9.21E-04
ALSTM-FCN | 5.22E-02

Figure 3 illustrates a critical difference diagram for the models evaluated in
this study. The critical difference diagram is conducted using a Wilcoxon-Holm
post-hoc analysis to evaluate the arithmetic means of the ranks Fawaz et al.
(2019). This figure confirms that each neural network has a lower rank than
that of its original version.

Figure 3: Critical difference diagram of the arithmetic means of the ranks for Deep Learning Models Figure 4: Ablation visual representation of the input signal after transformation in FCN and OctConv through randomly selected filters from each layer Table 3: Wilcoxon Signed-Rank Test comparing AVERAGE testing accuracy of Models developed with OctConvs to some of the SOTA models (p-value shown) Model | HIVE-COTE | TS-CHIEF | ROCKET
---|---|---|---
OctFCN | 4.15E-02 | 9.78E-02 | 1.06E-02
OctResnet | 2.56E-03 | 1.96E-02 | 7.37E-04
LSTM-OctFCN | 6.31E-04 | 4.65E-03 | 9.81E-05
ALSTM-OctFCN | 7.20E-02 | 1.12E-01 | 1.33E-02
Figure 5: Ablation visual representation of the input signal after
transformation in FCN and OctConv through randomly selected filters from each
layer

###  4.3 VS State-of-the-Art Classifiers

As an additional evaluation, we compare our results to those reported as
existing non-deep learning state-of-the-art (SOTA) models. These include
models such as TS-CHIEF Shifaz et al. (2020), ROCKET Dempster et al. (2020),
HIVE-COTE Lines et al. (2018) and more. As stated in Section 3.3, our model
accuracies are evaluated on an average of 20 runs. For existing models, we
utilize the metrics reported in the literature.

Table LABEL:results_long shows the results for the models developed in this
study compared to the existing SOTA models for time series classification on
the 128 benchmark datasets. The first 85 datasets are from the original
achieve. These datasets have been available for research and compared more
rigorously than the new datasets. For this reason, the SOTA models are more
easily available for comparison through the existing literature. The remainder
shows the SOTA for the remaining 43 benchmark datasets. Bold cells in these
tables indicate the developed models that achieve higher accuracy than the
existing SOTA.

Figure 3 illustrates the critical difference diagram of our models and the top
SOTA models. These SOTA models often rely on ensemble techniques to boost
optimal performance. The purpose of this study is not to statistically
outperform SOTA models on the benchmark, rather to illustrate the
effectiveness of the OctConv layer in the place of convolutional layers. Table
3 shows the results of the WSRT for the four OctConv neural networks compared
to TS-CHIEF, ROCKET, and HIVE-COTE. At a significance level of 0.05, both
OctResNet and LSTM-OctFCN is outperformed by HIVE-COTE, TS-CHIEF, and ROCKET.
Oct-FCN performs statistically the same as TS-CHIEF. Interestingly, ALSTM-FCN
performs statistically the same as the top 2 time series classifiers, HIVE-
COTE and TS-CHIEF. This result shows that a simple update of a convolution
layer to an octave convolution layer, not only improves its own performance,
but leads ALSTM-OctFCN to perform similarly to top two SOTA TSC classifiers
(HIVE-COTE and TS-CHIEF).

###  4.4 Ablation

As an additional evaluation for the impact of the OctConv layer, an ablation
study is conducted to show the impact of the replacement for feature
extraction. Figure 5 shows the differences in input transformation of FCN and
OctFCN for a single input of the CBF dataset. These visuals depict randomly
selected filters for each of the models. Both models show clear learned
feature extraction in the series of layers, however the two models result in
very different noise reduction steps. While these visuals clearly show the
superior feature extraction of the OctConv layers, we further study the
outcomes through the use of a linear kernel classifier.

In order to explicitly determine the impact of these different methods, we
evaluate the features from the developed models when applied to an SVM
classifier with a linear kernel. For each of the developed models (FCN,
ResNet, LSTM-FCN, and ALSTM-FCN) the features are fed to SVM classifiers to
get an explicit evaluation of the extracted features. For the LSTM-FCN and
ALSTM-FCN models, only the FCN component of the network is analyzed in this
ablation study.

Table 4 shows the results of the WSRT comparing the accuracy for an SVM
developed with convolutional features vs an SVM developed with OctConv
features. These results show that (at a p-value of 0.05) there is a
significant improvement of the SVM developed with OctConv features than that
of the convolutional features. Based on this analysis, we can conclude that
the OctConv is doing a better job creating linearly separable classes than the
convolutional layer. We postulate that the OctConv layers are better because
the high and low filters do a better job of capturing the global information
at each layer.

Table 4: Wilcoxon Signed-Rank Test comparing the SVM models developed from traditional convolutions vs OctConvs information (p-value shown) Model | P-Value
---|---
FCN | 6.06E-05
Resnet | 8.67E-08
LSTM-FCN | 1.70E-08
ALSTM-FCN | 8.35E-06

##  5 Conclusion

In this work, we propose the replacement of traditional convolutional layers
with Octave Convolutional layers. We evaluate this replacement on four state-
of-the-art neural network architectures including FCN, ResNet, LSTM-FCN, and
ALSTM-FCN. The experiments for this study utilize the 128 datasets provided by
the University of California-Riverside time series classification archive.
These experiments show that on an average of 20 runs on each dataset the
tested neural networks perform significantly better when the convolutional
layers are replaced with Octave Convolutional layers. Based on the average
testing accuracy of 20 runs, ALSTM-OctFCN performs statistically the same as
the top two time series classifiers, TS-CHIEF and HIVE-COTE. To further
evaluate the replaced layers, we provide an ablation study of the features
extracted by each neural network classifier. This ablation study establishes
that the features extracted by the Octave Convolutional layers provide better
information than that of the traditional convolutional layers. Future research
in this area can explore the use of other alternatives to convolutional layers
that may outperform the Octave Convolution. In addition, the generated models
may be considered in an ensemble model to achieve state-of-the-art performance
on the time series classification benchmarks.

Table 5:  Classification Accuracy for the Benchmark Datasets Dataset | FCN | Oct | ResNet | Oct | LSTM | LSTM | ALSTM | ALSTM | SOTA
---|---|---|---|---|---|---|---|---|---
. | . | FCN | . | ResNet | -FCN | -OctFCN | -FCN | -OctFCN | .
Adiac | 0.768 | 0.801 | 0.833 | 0.828 | 0.826 | 0.823 | 0.811 | 0.821 | 0.857
ArrowHead | 0.816 | 0.855 | 0.813 | 0.846 | 0.869 | 0.884 | 0.813 | 0.868 | 0.880
ChlorineConcentration | 0.800 | 0.800 | 0.817 | 0.813 | 0.800 | 0.811 | 0.798 | 0.807 | 0.875
InsectWingbeatSound | 0.382 | 0.451 | 0.448 | 0.541 | 0.649 | 0.654 | 0.385 | 0.454 | 0.653
Lighting7 | 0.740 | 0.814 | 0.703 | 0.784 | 0.745 | 0.775 | 0.749 | 0.814 | 0.863
Wine | 0.596 | 0.778 | 0.759 | 0.681 | 0.672 | 0.770 | 0.717 | 0.756 | 0.889
WordsSynonyms | 0.532 | 0.607 | 0.609 | 0.655 | 0.657 | 0.668 | 0.544 | 0.608 | 0.779
50words | 0.656 | 0.694 | 0.683 | 0.722 | 0.781 | 0.801 | 0.659 | 0.696 | 0.846
Beef | 0.747 | 0.680 | 0.773 | 0.757 | 0.803 | 0.757 | 0.763 | 0.720 | 0.933
DistPhalOutAgeGroup | 0.799 | 0.799 | 0.805 | 0.785 | 0.823 | 0.807 | 0.825 | 0.816 | 0.835
DistPhalOutCorrect | 0.803 | 0.800 | 0.798 | 0.790 | 0.803 | 0.798 | 0.801 | 0.794 | 0.820
DistalPhalanxTW | 0.776 | 0.764 | 0.766 | 0.762 | 0.779 | 0.767 | 0.783 | 0.771 | 0.790
ECG200 | 0.876 | 0.900 | 0.892 | 0.907 | 0.884 | 0.899 | 0.891 | 0.898 | 0.920
ECGFiveDays | 0.989 | 0.993 | 0.980 | 0.991 | 0.985 | 0.990 | 0.986 | 0.993 | 1.000
BeetleFly | 0.810 | 0.855 | 0.850 | 0.850 | 0.860 | 0.900 | 0.785 | 0.850 | 0.950
BirdChicken | 0.935 | 0.915 | 0.925 | 0.910 | 0.900 | 0.900 | 0.935 | 0.910 | 0.950
ItalyPowerDemand | 0.959 | 0.956 | 0.961 | 0.958 | 0.959 | 0.956 | 0.959 | 0.956 | 0.970
SonyAIBORobotSur | 0.964 | 0.957 | 0.916 | 0.928 | 0.938 | 0.934 | 0.961 | 0.961 | 0.985
SonyAIBORobotSurII | 0.972 | 0.965 | 0.980 | 0.970 | 0.972 | 0.962 | 0.976 | 0.966 | 0.978
MidPhalOutAgeGroup | 0.749 | 0.736 | 0.751 | 0.733 | 0.742 | 0.737 | 0.746 | 0.742 | 0.814
MidPhalOutCor | 0.817 | 0.798 | 0.816 | 0.791 | 0.814 | 0.806 | 0.816 | 0.800 | 0.808
MiddlePhalanxTW | 0.585 | 0.605 | 0.595 | 0.593 | 0.595 | 0.599 | 0.594 | 0.593 | 0.612
ProxPhalOutAgeGroup | 0.833 | 0.835 | 0.853 | 0.844 | 0.845 | 0.837 | 0.845 | 0.835 | 0.883
ProxPhalOutCorrect | 0.902 | 0.910 | 0.914 | 0.908 | 0.906 | 0.910 | 0.906 | 0.910 | 0.918
ProximalPhalanxTW | 0.811 | 0.802 | 0.816 | 0.791 | 0.814 | 0.799 | 0.810 | 0.805 | 0.815
MoteStrain | 0.918 | 0.906 | 0.909 | 0.901 | 0.927 | 0.908 | 0.920 | 0.903 | 0.950
MedicalImages | 0.768 | 0.753 | 0.756 | 0.755 | 0.768 | 0.751 | 0.771 | 0.760 | 0.792
Strawberry | 0.964 | 0.965 | 0.964 | 0.962 | 0.963 | 0.965 | 0.966 | 0.966 | 0.980
ToeSegmentation1 | 0.975 | 0.967 | 0.965 | 0.964 | 0.971 | 0.965 | 0.970 | 0.964 | 0.974
Coffee | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
Cricket_X | 0.773 | 0.791 | 0.788 | 0.814 | 0.746 | 0.762 | 0.782 | 0.791 | 0.821
Cricket_Y | 0.764 | 0.798 | 0.798 | 0.819 | 0.759 | 0.798 | 0.770 | 0.795 | 0.826
Cricket_Z | 0.774 | 0.812 | 0.804 | 0.823 | 0.786 | 0.809 | 0.783 | 0.812 | 0.836
uWaveGestLib_X | 0.757 | 0.773 | 0.763 | 0.782 | 0.832 | 0.834 | 0.766 | 0.774 | 0.855
uWaveGestLib_Y | 0.649 | 0.674 | 0.652 | 0.682 | 0.727 | 0.745 | 0.668 | 0.676 | 0.759
uWaveGestLib_Z | 0.737 | 0.745 | 0.736 | 0.747 | 0.777 | 0.773 | 0.740 | 0.745 | 0.792
ToeSegmentation2 | 0.903 | 0.929 | 0.899 | 0.928 | 0.905 | 0.933 | 0.905 | 0.927 | 0.962
DiatomSizeReduction | 0.940 | 0.930 | 0.952 | 0.933 | 0.937 | 0.937 | 0.937 | 0.936 | 0.977
Car | 0.913 | 0.932 | 0.888 | 0.907 | 0.917 | 0.902 | 0.915 | 0.927 | 0.933
CBF | 0.994 | 0.996 | 0.995 | 0.998 | 0.994 | 0.996 | 0.993 | 0.996 | 1.000
CinC_ECG_torso | 0.832 | 0.847 | 0.809 | 0.843 | 0.890 | 0.890 | 0.830 | 0.848 | 0.996
Computers | 0.844 | 0.832 | 0.814 | 0.814 | 0.775 | 0.734 | 0.836 | 0.828 | 0.848
Earthquakes | 0.763 | 0.793 | 0.760 | 0.794 | 0.794 | 0.801 | 0.783 | 0.797 | 0.801
ECG5000 | 0.939 | 0.941 | 0.940 | 0.940 | 0.942 | 0.942 | 0.940 | 0.941 | 0.948
ElectricDevices | 0.714 | 0.705 | 0.722 | 0.705 | 0.742 | 0.732 | 0.717 | 0.719 | 0.799
FaceAll | 0.922 | 0.864 | 0.848 | 0.813 | 0.910 | 0.875 | 0.914 | 0.868 | 0.929
FaceFour | 0.934 | 0.953 | 0.950 | 0.953 | 0.874 | 0.876 | 0.934 | 0.955 | 1.000
FacesUCR | 0.933 | 0.952 | 0.942 | 0.946 | 0.917 | 0.932 | 0.934 | 0.951 | 0.965
Fish | 0.937 | 0.969 | 0.971 | 0.983 | 0.958 | 0.967 | 0.953 | 0.972 | 0.994
FordA | 0.919 | 0.924 | 0.921 | 0.928 | 0.918 | 0.931 | 0.915 | 0.924 | 0.973
FordB | 0.880 | 0.902 | 0.916 | 0.918 | 0.883 | 0.904 | 0.881 | 0.902 | 0.917
Gun_Point | 1.000 | 1.000 | 0.997 | 1.000 | 0.995 | 0.998 | 1.000 | 1.000 | 1.000
Ham | 0.715 | 0.750 | 0.712 | 0.730 | 0.717 | 0.727 | 0.720 | 0.750 | 0.781
HandOutlines | 0.521 | 0.755 | 0.880 | 0.871 | 0.842 | 0.875 | 0.623 | 0.806 | 0.960
Haptics | 0.426 | 0.494 | 0.517 | 0.553 | 0.489 | 0.514 | 0.465 | 0.510 | 0.551
Herring | 0.634 | 0.703 | 0.614 | 0.661 | 0.641 | 0.698 | 0.647 | 0.727 | 0.703
InlineSkate | 0.374 | 0.423 | 0.482 | 0.432 | 0.465 | 0.446 | 0.424 | 0.427 | 0.613
LargeKitApp | 0.890 | 0.895 | 0.903 | 0.897 | 0.685 | 0.744 | 0.897 | 0.897 | 0.896
Lighting2 | 0.708 | 0.711 | 0.730 | 0.726 | 0.752 | 0.769 | 0.718 | 0.728 | 0.885
MALLAT | 0.963 | 0.966 | 0.938 | 0.961 | 0.963 | 0.961 | 0.966 | 0.966 | 0.980
Meat | 0.863 | 0.875 | 0.938 | 0.942 | 0.895 | 0.922 | 0.885 | 0.910 | 1.000
NonInvFatECGThor1 | 0.949 | 0.958 | 0.954 | 0.955 | 0.962 | 0.962 | 0.955 | 0.956 | 0.962
NonInvFatECGThor2 | 0.929 | 0.947 | 0.949 | 0.956 | 0.954 | 0.957 | 0.948 | 0.951 | 0.969
OliveOil | 0.533 | 0.720 | 0.827 | 0.847 | 0.593 | 0.770 | 0.587 | 0.767 | 0.933
OSULeaf | 0.979 | 0.970 | 0.974 | 0.965 | 0.895 | 0.907 | 0.979 | 0.971 | 0.988
PhalOutlCor | 0.813 | 0.821 | 0.825 | 0.821 | 0.816 | 0.817 | 0.812 | 0.817 | 0.854
Phoneme | 0.336 | 0.345 | 0.354 | 0.345 | 0.252 | 0.265 | 0.340 | 0.343 | 0.349
Plane | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
RefrigerationDevices | 0.510 | 0.528 | 0.555 | 0.534 | 0.466 | 0.480 | 0.507 | 0.525 | 0.581
ScreenType | 0.637 | 0.637 | 0.648 | 0.627 | 0.545 | 0.510 | 0.639 | 0.634 | 0.707
ShapeletSim | 0.956 | 0.926 | 0.948 | 0.874 | 0.966 | 0.953 | 0.927 | 0.921 | 1.000
ShapesAll | 0.885 | 0.903 | 0.912 | 0.917 | 0.890 | 0.904 | 0.884 | 0.902 | 0.930
SmallKitApp | 0.734 | 0.759 | 0.769 | 0.735 | 0.610 | 0.599 | 0.766 | 0.757 | 0.853
StarlightCurves | 0.964 | 0.975 | 0.976 | 0.977 | 0.965 | 0.974 | 0.964 | 0.975 | 0.982
SwedishLeaf | 0.969 | 0.966 | 0.960 | 0.960 | 0.971 | 0.968 | 0.968 | 0.968 | 0.966
Symbols | 0.973 | 0.983 | 0.972 | 0.980 | 0.963 | 0.963 | 0.976 | 0.984 | 0.982
synthetic_control | 0.985 | 0.999 | 0.994 | 0.998 | 0.978 | 0.999 | 0.980 | 0.999 | 1.000
Trace | 0.250 | 0.250 | 0.250 | 0.250 | 0.250 | 0.250 | 0.250 | 0.250 | 1.000
Two_Patterns | 0.863 | 0.942 | 0.963 | 0.991 | 0.994 | 0.984 | 0.873 | 0.943 | 1.000
TwoLeadECG | 0.999 | 0.998 | 0.999 | 0.996 | 0.999 | 0.998 | 0.999 | 0.998 | 1.000
UWaveGestLibAll | 0.822 | 0.864 | 0.833 | 0.874 | 0.955 | 0.959 | 0.824 | 0.861 | 0.976
Wafer | 0.997 | 0.999 | 0.995 | 0.997 | 0.998 | 0.999 | 0.997 | 0.998 | 1.000
Worms | 0.612 | 0.650 | 0.598 | 0.626 | 0.494 | 0.534 | 0.606 | 0.651 | 0.805
WormsTwoClass | 0.773 | 0.774 | 0.771 | 0.762 | 0.683 | 0.699 | 0.781 | 0.773 | 0.831
yoga | 0.860 | 0.879 | 0.878 | 0.888 | 0.903 | 0.904 | 0.879 | 0.883 | 0.918
ACSF1 | 0.927 | 0.935 | 0.917 | 0.932 | 0.903 | 0.920 | 0.913 | 0.940 | 0.850
AllGestWiX | 0.615 | 0.646 | 0.707 | 0.718 | 0.693 | 0.724 | 0.718 | 0.722 | 0.717
AllGestWiY | 0.746 | 0.778 | 0.764 | 0.773 | 0.753 | 0.781 | 0.788 | 0.856 | 0.730
AllGestWiZ | 0.694 | 0.725 | 0.715 | 0.720 | 0.704 | 0.758 | 0.733 | 0.746 | 0.651
BME | 0.947 | 0.980 | 0.953 | 0.968 | 0.952 | 0.996 | 0.833 | 0.900 | 0.999
Chinatown | 0.938 | 0.956 | 0.964 | 0.973 | 0.988 | 0.988 | 0.985 | 0.995 | 0.970
Crop | 0.764 | 0.783 | 0.789 | 0.793 | 0.764 | 0.875 | 0.762 | 0.799 | 0.793
DodgLpDay | 0.614 | 0.647 | 0.647 | 0.648 | 0.634 | 0.652 | 0.486 | 0.550 | 0.588
DodgLpGm | 0.826 | 0.833 | 0.846 | 0.856 | 0.893 | 0.907 | 0.779 | 0.823 | 0.928
DodgLpWnd | 0.938 | 0.967 | 0.949 | 0.949 | 0.927 | 0.933 | 0.970 | 0.973 | 0.986
EOGHzSgn | 0.651 | 0.656 | 0.652 | 0.663 | 0.617 | 0.632 | 0.686 | 0.680 | 0.884
EOGVtSgn | 0.473 | 0.490 | 0.465 | 0.472 | 0.504 | 0.529 | 0.514 | 0.518 | 0.815
EthLevel | 0.737 | 0.743 | 0.753 | 0.737 | 0.725 | 0.763 | 0.738 | 0.727 | 0.875
FrzRegTr | 0.957 | 0.991 | 0.970 | 0.965 | 0.998 | 1.000 | 1.000 | 1.000 | 0.999
FRSmlTr | 0.829 | 0.842 | 0.816 | 0.822 | 0.807 | 0.836 | 0.878 | 0.897 | 0.995
Fungi | 0.983 | 0.998 | 0.994 | 0.996 | 0.984 | 1.000 | 0.995 | 1.000 | 0.839
GestMidAirD1 | 0.736 | 0.742 | 0.719 | 0.746 | 0.703 | 0.751 | 0.714 | 0.751 | 0.639
GestMidAirD2 | 0.672 | 0.677 | 0.657 | 0.669 | 0.694 | 0.681 | 0.640 | 0.628 | 0.608
GestMidAirD3 | 0.421 | 0.440 | 0.390 | 0.433 | 0.452 | 0.433 | 0.444 | 0.435 | 0.377
GestPebZ1 | 0.944 | 0.973 | 0.941 | 0.949 | 0.947 | 0.933 | 0.921 | 0.921 | 0.826
GestPebZ2 | 0.857 | 0.883 | 0.892 | 0.864 | 0.873 | 0.913 | 0.832 | 0.818 | 0.779
GunPtAgeSp | 0.982 | 0.973 | 0.998 | 0.995 | 0.986 | 0.988 | 1.000 | 0.970 | 1.000
GunPontMVsF | 0.962 | 0.987 | 0.976 | 0.957 | 0.980 | 0.989 | 1.000 | 1.000 | 1.000
GunPointOVsY | 0.942 | 0.949 | 0.995 | 0.957 | 0.974 | 0.983 | 1.000 | 0.986 | 1.000
HouseTwenty | 0.945 | 0.955 | 0.908 | 0.961 | 0.970 | 0.962 | 0.948 | 0.970 | 0.979
InsEPGRegTr | 1.000 | 1.000 | 1.000 | 0.995 | 1.000 | 0.994 | 1.000 | 0.991 | 1.000
InsEPGSmlTr | 0.928 | 0.958 | 1.000 | 0.920 | 1.000 | 0.981 | 1.000 | 0.976 | 1.000
MelbPed | 0.972 | 0.993 | 0.983 | 0.988 | 0.973 | 0.951 | 0.976 | 0.979 | 0.848
MxShpRegTr | 0.928 | 0.942 | 0.975 | 0.984 | 0.963 | 0.991 | 0.968 | 0.981 | 0.971
MxShpSmlTr | 0.918 | 0.941 | 0.928 | 0.936 | 0.922 | 0.927 | 0.924 | 0.915 | 0.947
PickGestWiZ | 0.853 | 0.878 | 0.864 | 0.851 | 0.897 | 0.895 | 0.899 | 0.900 | 0.660
PigAryPress | 0.750 | 0.760 | 0.753 | 0.740 | 0.793 | 0.824 | 0.756 | 0.826 | 0.977
PigArtPress | 0.964 | 0.974 | 0.974 | 0.977 | 0.985 | 1.000 | 0.995 | 1.000 | 0.975
PigCVP | 0.864 | 0.869 | 0.871 | 0.937 | 0.922 | 0.941 | 0.928 | 0.956 | 0.961
PLAID | 0.481 | 0.489 | 0.482 | 0.494 | 0.282 | 0.483 | 0.506 | 0.515 | 0.840
PowerCons | 0.973 | 0.984 | 0.973 | 0.981 | 1.000 | 1.000 | 0.984 | 1.000 | 0.993
Rock | 0.718 | 0.727 | 0.676 | 0.729 | 0.764 | 0.782 | 0.795 | 0.805 | 0.867
SgHdGendCh2 | 0.943 | 0.969 | 0.937 | 0.959 | 0.945 | 0.986 | 0.852 | 0.879 | 0.969
SgHdMovCh2 | 0.612 | 0.626 | 0.698 | 0.630 | 0.703 | 0.714 | 0.653 | 0.716 | 0.891
SgHdSubCh2 | 0.919 | 0.927 | 0.925 | 0.932 | 0.917 | 0.917 | 0.902 | 0.917 | 0.951
ShkGestWiZ | 0.961 | 0.981 | 0.973 | 0.966 | 0.946 | 0.992 | 0.975 | 0.974 | 0.860
SmthSub | 0.986 | 0.983 | 0.993 | 0.988 | 1.000 | 0.988 | 1.000 | 1.000 | 0.998
UMB | 0.989 | 1.000 | 1.000 | 0.982 | 1.000 | 0.966 | 0.989 | 1.000 | 0.983

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