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http://www.aimspress.com/journal/Math
AIMS Mathematics, 9(1): 792–817.
DOI: 10.3934/math.2024041
Received: 19 August 2023
Revised: 06 November 2023
Accepted: 15 November 2023
Published: 04 December 2023
Research article
Features gradient-based signals selection algorithm of linear complexity for
convolutional neural networks
Yuto Omae1,*, Yusuke Sakai2and Hirotaka Takahashi2
1College of Industrial Technology, Nihon University, 1-2-1, Izumi, Narashino, Chiba 275-8575,
Japan
2Research Center for Space Science, Advanced Research Laboratories and Department of Design
and Data Science, Tokyo City University, Kanagawa 224-8551, Japan
*Correspondence: Email: oomae.yuuto@nihon-u.ac.jp.
Abstract: Recently, convolutional neural networks (CNNs) for classification by time domain data
of multi-signals have been developed. Although some signals are important for correct classification,
others are not. The calculation, memory, and data collection costs increase when data that include
unimportant signals for classification are taken as the CNN input layer. Therefore, identifying and
eliminating non-important signals from the input layer are important. In this study, we proposed a
features gradient-based signals selection algorithm (FG-SSA), which can be used for finding and re-
moving non-important signals for classification by utilizing features gradient obtained by the process of
gradient-weighted class activation mapping (grad-CAM). When we defined nsas the number of signals,
the computational complexity of FG-SSA is the linear time O(ns) (i.e., it has a low calculation cost).
We verified the effectiveness of the algorithm using the OPPORTUNITY dataset, which is an open
dataset comprising of acceleration signals of human activities. In addition, we checked the average of
6.55 signals from a total of 15 signals (five triaxial sensors) that were removed by FG-SSA while main-
taining high generalization scores of classification. Therefore, FG-SSA can find and remove signals
that are not important for CNN-based classification. In the process of FG-SSA, the degree of influence
of each signal on each class estimation is quantified. Therefore, it is possible to visually determine
which signal is effective and which is not for class estimation. FG-SSA is a white-box signal selection
algorithm because it can understand why the signal was selected. The existing method, Bayesian opti-
mization, was also able to find superior signal sets, but the computational cost was approximately three
times greater than that of FG-SSA. We consider FG-SSA to be a low-computational-cost algorithm.
Keywords: convolutional neural network; signal importance; signal selection algorithm;
time-directional Grad-CAM
Mathematics Subject Classification: 68T01, 68T07, 68T20
793
1. Introduction
Recently, many convolutional neural networks (CNNs) have been developed for classification in the
time domain of signals, e.g., recognizing movement-intention using electroencephalography (EEG)
signals [1], human activity recognition using inertial signals [2, 3], and swimming stroke detection
using acceleration signals [4]. Although these studies used all signals, signals that are not important
for correct classification may be included in the CNN input layer. These signals worsen the accuracy of
the classification and increase calculation costs, required memory, and data collection costs. Therefore,
finding and removing non-important signals from the CNN input layer and creating a classification
model for the minimum signals possible are crucial. However, many CNNs use all signals [4–6] or
manually select signals [7, 8].
We can visually find unimportant regions using gradient-weighted class activation mapping (grad-
CAM) embedded in CNNs. Grad-CAM [9] uses the gradient of the feature maps output from the
convolutional layer immediately before the fully connected layer to visualize which regions on the
image influenced class estimation in the form of a heat map. For example, Figure 1 (A) is the generation
process of grad-CAM when the image of 2 is input to CNN. First, the gradient of the output vector
with respect to the feature maps is computed. Then, the feature maps are weighted by the sum of the
gradients and they are stacked. A heat map is generated by passing the weighted feature map through
the ReLu function. Considering this, we can understand that the CNN used the upper side of the image
to estimate “2”. Grad-CAM was proposed by improving CAM [10]. CAM is the white-box method for
only CNNs, which do not have fully connected layers. In contrast, Grad-CAM is applicable to various
CNN model-families [9]. The method is primarily used to input image data, e.g., chest CT images [11],
3D positron emission tomography images [12] and chest X-ray images [13–15].
Cases of applying grad-CAM to CNNs that input time domain data of signals, as shown in Figure 1
(B), are continuously increasing. For example, the classification of sleep stages [16], prognostication of
comatose patients after cardiac arrest [17], classification of schizophrenia and healthy participants [18],
and classification of motor imagery [19] are performed using EEG signal(s). Electrocardiogram (ECG)
signals are used to detect nocturnal hypoglycemia [20] and predict 1-year mortality [21]. Acceleration
signals have been used to detect hemiplegic gait [22] and human activity recognition [23].
From these studies, we visually find unimportant signals for correct classification by applying grad-
CAM to CNN, inputting the time domain of the signals. In the example shown in Figure 1 (B), signals
s1and snare strongly activated, but signal s2is not. Therefore, s2can be interpreted as not important for
class estimation. However, observing all of them and finding unimportant signals is extremely difficult
because one grad-CAM for one input data is generated. Therefore, we propose features gradient-based
signals selection algorithm (FG-SSA), which can be used to find and remove unimportant signals from
a CNN input layer by utilizing features gradient obtained by the calculation process of grad-CAM. The
algorithm provides a signal subset consisting of only important signals. The computational complexity
of the proposed algorithm is linear order O(ns) (i.e., FG-SSA has a low calculation cost) when we
define nsas the number of all signals:
The academic contributions of this study are twofold, which are specified as follows.
•We propose a method for quantifying the effect of signals on classes estimation.
•We propose an algorithm to remove signals of low importance.
This algorithm can remove unnecessary signals and enable CNNs and datasets to be smaller in size.
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Figure 1. Schematic diagram of (A) image data-based grad-CAM and (B) multi signals-
based grad-CAM. “C” and “FC” mean “convolution” and “fully-connected”, respectively.
The MNIST dataset is used for the figure (A). The red color represents high activation and
the dark blue color represents low activation.
2. Related works
The proposed algorithm, FG-SSA, is applied to CNNs that estimate class labels. It selects signals
to be used in the input layer. This is similar to the feature selection problem of which features to
use among all the features. The large number of patterns in feature combinations makes it difficult to
find a globally optimal solution that maximizes verification performance for a subset of features. For
example, a combination of 10 out of 100 features exceeds 1014 patterns. Therefore, the feature selection
problem is known to be NP-hard [24, 25]. For NP-hard problems, finding a good feasible solution is
better than finding a global optimal solution [26]. Therefore, many methods for finding approximate
solutions to the feature selection problem have been proposed, e.g., within-class variance between-
class variance ratio [27, 28], classification and regression tree [29], Lasso [30], ReliefF [31, 32], out-
of-bag error [33], and others. Recently, some studies have used Bayesian optimization for feature
selection [34–36], which expresses whether to use a feature as a binary variable and evaluates the
combination of features. These feature selection methods reduce the computation time of machine
learning and improve the prediction accuracy [37,38].
The solution to the signal selection problem in CNNs also has a large number of patterns, which
is obtained through several signal combinations. However, the feature selection algorithm described
above cannot be used for signal selection problems and is not suitable for CNNs. Compared to other
machine learning methods (support vector machine, decision tree, k-nearest neighbor, and others),
CNNs require more computation time to construct a model. In addition, CNNs have many hyperpa-
rameters (number of layers, size of kernel filters, etc.), which also require computational resources
for tuning. Therefore, instead of searching for an appropriate signals set, all of the prepared signals
are often used such as [39, 40]. Hence, a method that can find an approximate solution at a small
computational cost is needed for the feature selection and signal selection problems.
Several studies have attempted to address the signal selection problem. Some studies applied
Bayesian optimization to signal selection [41, 42]. Because Bayesian optimization is a method for
finding optimal solutions in black box functions, it can be applied to both feature selection and sig-
nal selection. However, it is not suitable for CNNs. Another approach is signal selection by group
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Lasso [43]. These signal selection algorithms are important research, but they are difficult to visually
interpret because they do not utilize Grad-CAM. To the best of the authors’ knowledge, only these pre-
vious studies involved signal selection. We feel that developing signal selection algorithms for CNNs
is an important issue but has not been sufficiently studied. Therefore, we propose a signal selection
algorithm, FG-SSA, that provides a visual interpretation of which signals contribute to which class
estimation and is computationally inexpensive.
3. Proposed method
3.1. Initial signals set
We consider a situation in which the task is to estimate a class cbelonging to class set Cdefined by
C={ci|i=1,· · · ,ncm},(3.1)
where ncm is the number of classes. The measurement data of the initial signal set is defined by
S={si|i=1,· · · ,ns},(3.2)
where nsdenotes the number of signals and sidenotes the signal identification names. Some of the ns
signals are important for classification, whereas others are not. Therefore, finding a signal subset Suse
that has removed the unimportant signals from all signal sets Sis crucial. In this study, we provide
an algorithm for finding such a subset of signals Suse ⊆S. The applicable targets of the proposed
method are all tasks of solving classification problems using CNNs by inputting the time domain of
multisignals.
3.2. Class estimation
An example of CNN with multiple signals as input is shown in Figure 2. This figure shows the CNN
solving the problem of classifying 10 human motion labels from 15 acceleration signals mentioned in
Subsection 4.2; however, the proposed algorithm can be applied to various problems. The data for the
input layer are in the form of a matrix w×ns.wis the window length of the sliding window method [44]
and ns=|S|is the number of signals.
Subsequently, the input data are convoluted using kernel filters of the time-directional convolution
size sc. The number of generated feature maps is nfbecause the number of filters is nf. The reason for
adopting only time-directional convolution is to avoid mixing a signal and others in the convolution
process. The input data are convoluted by these ncconvolution layers, and the CNN generates the
feature maps shown in “extracted feature maps” in Figure 2. Subsequently, the CNN generates the
output vector using fully connected layers and the SoftMax function. We define the output vector yas
y=[yc]∈R|C|,c∈C,X
c∈C
yc=1.(3.3)
The estimation class c′is
c′=argmax
c∈C
{yc|c∈C}.(3.4)
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Herein, we only define the output vector ybecause it appears in the grad-CAM definition. We explain
grad-CAM for time domain signals in the next subsection.
Figure 2. An example of CNN with multiple signals as input. This is the CNN using the
initial signals set Sand classes set Cdefined by Equations (4.1) and (4.2) described in
Subsection 4.2. In other words, the CNN structure for 10 human activities recognition from
15 acceleration signals.
3.3. Time-directional grad-CAM
We defined the vertical size of the feature map before the fully connected layer as sf, as shown in
Figure 2. Let us denote the kth feature map of signal sas
fs,k=[fs,k
1fs,k
2· · · fs,k
sf]⊤∈Rsf,s∈S,k∈ {1,· · · ,nf}.(3.5)
In the case of CNNs for image-data-based classification, the form of feature maps is a matrix. However,
in the case of CNNs consisting of time-directional convolution layers, the form of the feature maps is
a vector. We defined the effect of the feature map fs,kon the estimation class c′as
αs,k
c′=1
sf
sf
X
j=1
∂yc′
∂fs,k
j
.(3.6)
Then, we applied the grad-CAM of signal sto estimation class c′in the vector form of
Zs
c′=ReLU1
nf
nf
X
k=1
αs,k
c′fs,k,Zs
c′∈Rsf
≥0.(3.7)
The activated region of each signal can be understood by calculating Zs
c′for all signals ∀s∈S.
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In this study, we refer to Zs
c′as “time-directional grad-CAM” because it is calculated by the sum-
ming partial differentiations of the time direction. This was defined by a minor change in the ba-
sic grad-CAM for image data [9]. In the case of the basic grad-CAM [9], the CNN generates one
grad-CAM for one input data. In contrast, in the case of time-directional grad-CAM, CNN generates
grad-CAMs as many as the number of signals nsfor one input data.
Figure 3 shows the examples of Zs
c′calculated using the CNN presented in Figure 2 and
the dataset described in Subsection 4.2. From top to bottom, these results correspond to c′=
Stand N,Stand L,Stand R, and Walk N. Notable, an estimation class c′is an element of C. We can
determine the signals that are important by viewing the time-directional grad-CAM shown in Fig-
ure 3. For example, signals from the left arm and left shoe are not used for estimating the Stand N
class. Moreover, left arm signals are not used for the estimation of the Stand R class. Therefore, the
time-directional grad-CAM is effective for finding unimportant signals.
Figure 3. Examples of time-directional grad-CAMs of 15 acceleration signals for four esti-
mation classes (Stand N, Stand L, Stand R, and Walk N) calculated by the CNN illustrated
in Figure 2 using the dataset described in Subsection 4.2. The red color represents high acti-
vation and the dark blue color represents low activation.
3.4. Signals importance index
Although we can find unimportant signals by viewing time-directional grad-CAM, the result varies
for each input data. Therefore, viewing all the grad-CAMs is difficult when the data size is large.
Herein, we quantify the importance of signal sfor classification based on αs,k
c′defined in Eq (3.6).
Here, we denote the input dataset of size ndat as
X={Xi|i=1,· · · ,ndat},Xi∈Rw×ns.(3.8)
The size of ith input data Xiis w×ns, which is the window length wand number of signals ns. When
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we define the input dataset of estimation class c′∈Cas Xc′, we can represent the input dataset Xas
X=[
c′∈C
Xc′.(3.9)
Using the set Xc′, we define the importance of the signal s∈Sto the estimation class c′∈Cas
Ls(Xc′)=1
|Xc′|X
Xi∈Xc′
gs
c′(Xi),(3.10)
where
gs
c′(Xi)=1
nf
nf
X
k=1
βs,k
c′(Xi), βs,k
c′(Xi)=
αs,k
c′(Xi),if αs,k
c′(Xi)≥0
0,otherwise ,Xi∈Xc′.(3.11)
In addition, αs,k
c′(Xi) is αs,k
c′of the input data Xito a CNN, and c′is the estimated class. Therefore,
Ls(Xc′) represents the importance of signal sto class c′based on the grad-CAM. Notably, we ignore
terms with negative partial derivatives to extract the positive effect on classification, as shown in Eq
(3.11). Moreover, using Ls(Xc′), we can obtain the matrix
Imat(Xc1,Xc2,· · · ,Xcncm )=
Ls1(Xc1)Ls1(Xc2)· · · Ls1(Xcncm )
.
.
..
.
.....
.
.
Lsns(Xc1)Lsns(Xc2)· · · Lsns(Xcncm )
=[Ls(Xc′)] ∈R|S|×|C|
≥0,s∈S,c′∈C.(3.12)
As shown in Eq (3.9), since X=Xc1∪Xc2∪ · · · ∪ Xcncm is satisfied, we define
Imat(X)def
=Imat(Xc1,Xc2,· · · ,Xcncm ).(3.13)
We refer to Imat(X) as the “signals importance matrix (SIM)” because the matrix comprises the im-
portance of all signals and classes using the input dataset X. The effect of each signal to all classes
can be understood by calculating and viewing SIM Imat(X).
Although Imat(X) includes important information, summarizing the SIM is necessary to find signals
that are unimportant to all classes. Therefore, by calculating the summation of the row values of SIM,
we denote the importance of signal sas
Is(X)=1
|C|X
c′∈C
Is,c′
mat(X),(3.14)
where Is,c′
mat(X) is the value of row sand column c′of Imat(X). By calculating Is(X) for all signals s,
we obtain the vector as
Ivec(X)=[Is(X)] ∈R|S|
≥0.(3.15)
We refer to Ivec(X) as the “signals importance vector (SIV)” because it is the vector that consists of
the signal importance.
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The calculated examples of SIM and SIV are shown in Figure 4. This is the effect of the each
individual signal on the class estimation by the CNN. These values were calculated using the CNN for
Condition A described in Section 5. The result of SIM (columns 1 to 10 in Figure 4) includes various
important data. First, we can confirm that the Right arm X signal is used for right hand movement
(Stand R, Walk R, and Sit R) estimation. In the case of the left hand movement (Stand L, Walk L)
estimation, the signals from the Back sensor responds strongly. Since the Back sensor is attached to
the body trunk, it is assumed that the body trunk movement is used to estimate the movement of the left
hand. The Left arm X responds to another left hand movement (Sit L) estimation. In addition, the Left
shoe Y and Right shoe Y respond strongly to the motion estimation of Walk N. Based on these results,
the SIM is considered reasonable. However, since some results are not comfortable, it is important to
remove unnecessary signals by FG-SSA.
Moreover, we can identify the signals that are important for all classes by viewing SIV (the column
11 in Figure 4). From SIV, we denote the minimum importance signal smin and maximum signal smax
as
(smin,smax )=argmin
s∈S
Is(X),argmax
s∈S
Is(X).(3.16)
We expect to maintain the estimation accuracy even when removing smin because it is a minimum
importance signal. In contrast, we expect the accuracy to decrease when we remove smax and re-learn
the CNN because it is the most important signal.
Figure 4. Example of signals importance to all classes expressed by SIM Imat(X) and SIV
Ivec(X) calculated to CNNs of Condition A described in Section 5. Columns 1 to 10 are SIM
and column 11 is SIV (“All classes”). The values are standardized from zero to one in each
column. The greater the cell’s value, the greater is the relevance of signal son estimation
class c′.
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Algorithm 1 Features gradient-based signals selection algorithm (FG-SSA)
Input:
Training dataset Xtrain
Validation dataset Xvalid
Initial signals set S
Maximum number of using signals γ
Output: Using signals set Suse
1: Initialization of a signals set S0←S
2: for t=0 to |S| − 1do
3: Learning a CNN by the training dataset XSt
train
4: Calculating validation accuracy A(XSt
valid)
5: Finding a minimum importance signal smin ←argmin
s∈St
Is(XSt
valid), see Equation (3.16)
6: Removing a signal: St+1←St\smin
7: end for
8: Suse ←argmax
S′∈{S0,··· ,S|S|−1}
A(XS′
valid),s.t., |Suse| ≤ γ
9: return Suse
Notably, XSt
train and XSt
valid are all the input data belonging to the training and validation datasets using
the signal set St, respectively.
3.5. Signals selection algorithm
In this study, we proposed an algorithm to find a desirable signal subset Suse ⊆Sby removing
unimportant signals. The proposed method is presented in Algorithm 1. The main inputs to the al-
gorithm are the training dataset Xtrain, validation dataset Xvalid, and input initial signals set S. To
avoid data leakage, the test dataset was not used in the algorithm. The elements of set Sare the signal
identification names defined in Eq (3.2). The first procedure is to create the initial signal set S0, which
consists of all signals (line 1). The next step is to develop a CNN using the training dataset XS0
train, con-
sisting of S0(line 3). Subsequently, we measure the validation accuracy A(XS0
valid) using the validation
dataset XS0
valid (line 4). Next, we determine the most unimportant signal smin ∈S0based on Eq (3.16)
(line 5). Finally, we obtain the next signal set S1by removing smin from S0(line 6).
This procedure is repeated until the number of signals reaches one (i.e., t=|S| − 1), and the
validation accuracies are recorded. The algorithm returns the signal subset Suse leading to maximum
validation accuracy (lines 8 and 9). Notably, the case wherein the signal subset leading to the maximum
accuracy is the initial signal set can occur (i.e., Suse =S). In cases wherein the main purpose is to
achieve maximum accuracy, adopting the initial signal set as an optimal subset is a better option, if
Suse =Soccurs. However, some cases exist that require a decrease in the number of signals. Therefore,
we prepared constraints wherein the number of adopted signal sizes is γor less (i.e., |Suse| ≤ γ). This
is a hyperparameter of the proposed algorithm.
When the size of the initial signal set Sis ns, the number of developing CNNs in Algorithm 1 is as
follows:
T(ns)=ns−1
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∼ns.(3.17)
That is, the computational complexity of Algorithm 1 is O(ns). This means that even if the signal size
nsincreases, the signal subset is returned in realistic time.
Generally, the total pattern for choosing mfrom nsignals is nCm. Because m, which is the number
of signals leading to the maximum validation accuracy, is not known, the total number of combinations
of input layers Tbs(ns) is similar to
Tbs(ns)=
ns
X
m=1
nsCm
∼max
mnsCm
=nsC⌊ns/2⌋.(3.18)
Therefore, finding the optimal signal subset using a brute-force search is difficult. The computation
time tends to be large because the CNN includes other hyperparameters. From this viewpoint, a fast
algorithm such as the proposed method is important.
4. Experiment 1: Relationship between validation accuracy and the number of deleted signals
4.1. Objective and outline
Herein, we examine the reliability of smin,max defined in Eq (3.16). We expect to maintain estimation
accuracy even if the signal smin is removed because it is the most unimportant signal. In contrast,
because signal smax is the most important signal, the estimation accuracy may decrease by removing it.
We gradually removed signal smin,max using the processes of lines 2 to 7 of Algorithm 1 to verify the
aforementioned hypothesis. In addition, we repeatedly developed CNNs and recorded their validation
accuracies. We performed verification using a total of 100 seeds because the training process of CNN
depends on randomness. The adopted layer structure of the CNN is explained in the Appendix section.
The number of epochs was 300.
4.2. Dataset
Herein, we define the initial signals set Sand the classes set Cdescribed in Subsection 3.1 con-
cretely with real data. We used the dataset “OPPORTUNITY Activity Recognition”. This is the
dataset for activity recognition, and it is used in “Activity Recognition Challenge” held by IEEE in
2011 [45, 46]. The dataset is regarded as reliable because it is used for the performance evaluation of
machine learning in some studies [47, 48]. The dataset contains data on multiple inertial measurement
units, 3D acceleration sensors, ambient sensors, and 3D localization information. Four participants
performed a natural execution of daily activities. The activity annotations include locomotions (e.g.,
sitting, standing, and walking) and left- and right-hand actions (e.g., reach, grasp, release, and etc.).
Details of the dataset are described in [45, 46].
In this study, we used five triaxial acceleration sensors from all the measurement devices (sampling
frequency: 32 [Hz]). Attachment points of sensors on the human body are “back,” “right arm,” “left
arm,” “right shoe,” and “left shoe.” The total number of signals was 15 because we adopted five triaxial
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sensors (5 ×3=15). We adopted data splitting using the sliding window method with window length
w=60 and sliding length 60. One signal length was approximately 2 s since the sampling frequency
was 32 [Hz]. Searching for the optimal window length size is important because it is a hyperparameter
that affects the estimation accuracy [44, 49, 50]. However, we did not tune the window length size
because the aim of this study was to provide a signal-selection algorithm.
Subsequently, we labeled the motion class label for each dataset based on human activity. The class
labels are combinations of locomotions (“Stand,” “Walk,” “Sit,” and “Lie”) and three hands activity
(“R”: moving right hand, “L”: moving left hand, and “N”: not moving hands). For example, the class
label “Sit R” refers to the sitting and right-hand motions.
Table 1 lists the results of applying the described procedures to all the data. This indicates that the
data size of the left-hand motion is small. Nearly all participants were right-handed (notably, we could
not find a description of the participants’ dominant arms in the explanation of the OPPORTUNITY
dataset). Moreover, data belonging to Lie R and L are absent. Therefore, these classes were removed
from the estimation task (i.e., the total number of classes was 10).
Table 1. Samples size of each class label.
Labels Samples Rates [%]
Stand N 1070 20.68
Stand L 193 3.73
Stand R 642 12.41
Walk N 1405 27.16
Walk L 63 1.22
Walk R 150 2.90
Sit N 571 11.04
Sit L 130 2.51
Sit R 536 10.36
Lie N 414 8.00
Lie L 0 0.00
Lie R 0 0.00
Total 5174 100.00
Subsequently, we randomly split all the data into a training dataset (80%) and a test dataset (20%).
Moreover, we assigned 20% of the training dataset to the validation dataset. The training, validation,
and test datasets were independent because we adopted the sliding window method for the same win-
dow and slide length (60 steps). We defined the signal set S, which has 15 elements, and the class set
C, which has 10 elements, as follows:
S={Back X,Back Y,Back Z,Right arm X,Right arm Y,Right arm Z,Left arm X,Left arm Y,
Left arm Z,Right shoe X,Right shoe Y,Right shoe Z,Left shoe X,Left shoe Y,Left shoe Z},
(4.1)
C={Stand N,Stand L,Stand R,Walk N,Walk L,Walk R,Sit N,Sit L,Sit R,Lie N}.(4.2)
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In other words, CNNs solve 10 classification problems from 15 multisignals. Moreover, an algorithm
for removing unimportant signals was provided while maintaining the estimation accuracy. Although
the sets Sand Crepresent acceleration signals and human activities, respectively, the proposed algo-
rithm can be used for other diverse signals, such as EEG and ECG.
4.3. Result and discussion
First, we indicate the validation accuracies when the most unimportant signal, smin, is gradually
removed, as shown in Figure 5 (A). From left to right in Figure 5 (A), the number of deleted signals
increases (i.e., the number of signals in the CNN input layer decreases). The leftmost and rightmost
results are obtained using all signals and only one signal as the CNN input layer, respectively. The
results show that even if six signals were deleted, the validation accuracy did not decrease. Moreover,
the accuracy significantly decreases by removing seven or more signals. In this case, although CNNs
estimate class labels from 15 signals in the set S, six signals are unnecessary.
We calculated the average removed timings of the 15 signals to determine the unnecessary signals.
Figure 5 (B) shows the results. This means that the lower the value, the earlier the signal is removed
in the procedure of Algorithm 1. The results indicate that the signals of the right and left shoes are
removed earlier than those of the other sensors. In contrast, some signals from the back, right arm, and
left arm did not disappear early. Therefore, we can regard shoe signals as unimportant for classification.
In addition, shoe sensors are important for walk motion classification. However, even if shoe sensors
disappear, we consider that the back sensor attached to centroids of the human body contributes to the
classification of walk motions because these motions are periodic.
Figure 5. (A) Maximum validation accuracy in 300 epochs when the most unimportant
signal smin is gradually removed and CNNs are re-learned. (B) Average removed timings of
each signal. It means the smaller the timing, the earlier the signal is removed. The results of
both (A) and (B) are average values of 100 seeds, and the error bars are standard deviations.
The pvalues in (A) represent the results of two-sided ttest, and the dashed line represents
the timing of statistically decreasing validation accuracy.
Then, we indicate the validation accuracies when the most important signal, smax , is gradually re-
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moved in Figure 6 (A). We can verify that the validation accuracy statistically decreases by removing
one of the most important signals. Therefore, removing the most important signal, smax, leads to a
worse accuracy. The average timing of the signal removal is shown in Figure 6 (B). From this figure,
we confirm that the signals of back X, right arm X, and left arm Y are removed early. Moreover, we
confirm that shoe sensor signals are not removed early. These tendencies are in contrast compared with
the case of removing the most unimportant signal smin .
Figure 6. (A) Maximum validation accuracy in 300 epochs when the most important signal
smax is gradually removed and CNNs are re-learned. (B) Average removed timings of each
signal. In contrast to that shown in Figure 5, the most important signals are removed.
Clearly, (1) even if we remove the most unimportant signal smin, the estimation performance re-
mains, and (2) the performance decreases when we remove the most important signal smax. Therefore,
the signal importance Is(X) defined in Eq (3.14) is reliable.
5. Experiment 2: Effectiveness of FG-SSA on generalization scores
5.1. Objective and outline
Herein, we confirmed the effect of FG-SSA indicated in Algorithm 1 on the generalization perfor-
mance of the classification. Therefore, we developed the following three conditions for CNNs:
•Condition A: CNN using all signals (i.e., FG-SSA is not used).
•Condition B: CNN for applied FG-SSA of γ=ns.
•Condition C: CNN for applied FG-SSA of γ=9.
Condition A means that the CNN does not remove signals but uses all signal sets S. Condition B refers
to the CNN using the signal subset Suse obtained by the FG-SSA, given nsas the constraint parameter
γ. Under this condition, we allow the algorithm to return Sas Suse when the maximum validation
accuracy is achieved using all signal sets S. In other words, a case wherein no signals are removed can
occur. Condition C implies that the number of adopted signals is nine or less for a CNN input layer
(i.e., |Suse| ≤ 9). In other words, six or more signals were deleted because the initial number of signals
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nswas 15. The value of hyper-parameter γ=9 was determined by referring to the result shown in
Figure 5 (A).
We developed CNNs using the signal set, which was determined by FG-SSA. Moreover, the optimal
epochs leading to a maximum validation accuracy were adopted. The search range of epochs was from
1 to 300. Subsequently, the generalization performance is measured using the test dataset. The test
dataset was used only at this time (i.e., it was not used for the parameter search). We developed CNNs
for Conditions A, B, and C with a total of 100 seeds to remove the randomness effect.
5.2. Result and discussion
By developing CNNs under Conditions A, B, and C on a total of 100 seeds, the average number
of signals used was 15.00, 11.94, and 8.35, respectively. In other words, an average of 3.06 and 6.65
signals were removed by FG-SSA in the cases of Conditions B and C, respectively. The generalization
performance (F score, precision, and recall) measured by the test dataset for each condition is shown
in Figure 7. These are histograms of 100 CNNs developed using 100 random seeds.
Figure 7. Histograms of estimation score using the test dataset (total 100 seeds). The macro
averages of 10 classes of F score, precision, and recall from left to right, and Conditions A,
B, and C from top to bottom.
The results indicated that the generalization scores were nearly the same for Conditions A, B, and
C. Moreover, pvalues of the two-sided t-test were not statistically significant. The confusion matrices
obtained by CNNs for each condition are shown in Figure 8. The values were averaged by 100 seed
results and standardized, where the summation in each row was 1. Consequently, the confusion ma-
trices were nearly the same. Although the proposed algorithm removed some signals, generalization
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errors did not increase. Therefore, we consider that FG-SSA can find and remove signals that are not
important for CNN-based classification.
Figure 8. Confusion matrices of the Conditions A, B, and C using the test dataset. The
values were averaged by 100 seed results and standardized, where the summation in each
row was 1.
From another viewpoint, the number of correct classifications of left hand motions (Stand L, Walk
L, and Sit L) was small in all conditions. We adopted weighted cross-entropy-based learning for CNNs
because the original data size of the left hand motions is small, as shown in Table 1. However, we
consider that an appropriate classification cannot be performed because the data diversity of these
motions is low. Although we believe that nearly all participants are right-handed, the explanation of
participant’s dominant arm in the OPPORTUNITY dataset is insufficient to the best of our knowledge.
Note that this study also validates FG-SSA on a dataset other than OPPORTUNITY. This dataset is
for an experiment that classifies swimming styles with signals measured from a single inertial sensor.
The details of this experiment and the results obtained are presented in Appendix B. The experimental
results also indicate that FG-SSA is effective at signal selection.
6. Comparison of FG-SSA and other methods
6.1. Objective and outline
From the described results, it was confirmed that signals can be removed without degrading the
generalization performance using FG-SSA. In this section, we compare the generalization performance
obtained by signal subsets selected by Condition C mentioned in Section 5 and signal subsets obtained
by other signal selection methods. Then, we evaluate the effectiveness of the proposed method. Since
Condition C has a problem of selecting a maximum of nine signals from 15 signals, this condition was
also adopted in existing methods.
Bayesian optimization (BO) and random search (RS) were adopted as existing methods. BO is a
search algorithm that solves combinatorial optimization problems and is sometimes applied to hyper-
parameter tuning in machine learning and deep learning [51, 52]. In this approach, the next hyperpa-
rameters to be verified are determined based on validation accuracies obtained with the previously set
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hyperparameters. This can be applied to feature selection problems in classification problems [34–36].
Feature selection is a method of selecting features to be used for classification from all features pre-
pared in advance. This is similar to the method of choosing signal subset from all signals. Therefore,
BO can also be applied to signal selection and has been applied in several studies [41,42]. Therefore,
in this study, we adopted BO as an existing method for comparison with FG-SSA.
As shown in Eq (3.18), when selecting some signals from nssignals, the number of selectable
combinations becomes enormous. In contrast, as shown in Algorithm 1, FG-SSA can obtain the signal
subset to be used only by constructing nsCNNs. On the other hand, the greater the number of iterations
for BO, the higher the possibility of discovering a desirable signal subset. However, in comparing FG-
SSA and BO, it is fairer to set the number of CNN constructions to be the same. Therefore, we set the
number of iterations of BO to ns. Note that nsis 15 in this experiment, as mentioned in Section 5. We
adopted tree-structured Parzen estimator (TPE) [53] as a specific method of BO and Optuna [54] as a
framework for implementation. TPE has the property of trying to obtain the desired solution with a
small number of iterations [55].
Moreover, RS was adopted as a baseline. This is a method that repeats the process of randomly
selecting signals and constructing a CNN for a specified number of times and uses the signal subset
with the highest validation accuracy. This time, we decided to repeat the process of selecting up to
nine signals from 15 signals nstimes. In other words, the numbers of constructing CNNs for RS, BO,
and FG-SSA are the same.
The details of the process of existing methods are described in Algorithm 2. The inputs for this
algorithm are the training dataset Xtrain, validation dataset Xvalid, initial signals set Sdefined in Eq
(4.1), maximum number of selecting signals Smax , number of iterations T, and searching method
M∈ {BO,RS}. BO means the TPE-based Bayesian optimization and RS means random search. In
Algorithm 2, the signal subset Stconsisting of Smax signals is extracted by “SignalsSelection” function
(line 2 in Algorithm 2, BO or RS method). We developed a CNN using the training dataset XSt
train
consisting of the signals subset Stand measured the validation accuracy A(XSt
valid) using the validation
dataset XSt
valid. We repeated it from t=1 to t=T(i.e., Ttimes) and selected the signal subset leading
to the maximum validation accuracy. This subset is Suse ⊆Sand it is returned by Algorithm 2 to
users. Then, we developed a CNN using the signal subset Suse and measured the generalization scores
(F score, precision, and recall) using the test dataset. The layer structure and learning condition of the
CNN are the same as the previous experiment described in Section 5.
In general, for both BO and RS, better signal subsets can be found as the number of iterations T
increases. However, CNN requires a long time to build; thus, it is not desirable to increase T. As
shown in Eq (3.17), the strength of FG-SSA is that it is a fast algorithm. Therefore, we decided to
make the number of iterations Tof BO and RS and the maximum number of selected signals Smax the
same as those of FG-SSA (Section 5, Condition C). In other words, we set T=15,Smax =9. We ran
Algorithm 2 100 times while changing the random seed to remove the effect of randomness (FG-SSA
in Condition C is also the same).
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Algorithm 2 Existing signals selection algorithm
Input:
Training dataset Xtrain
Validation dataset Xvalid
Initial signals set S
Maximum number of using signals Smax
Number of iterations T
Search method M∈ {BO,RS}
Output: Using signals set Suse
1: for t=1 to Tdo
2: St←SignalsSelection(S,Smax,M)
3: Learning a CNN by the training dataset XSt
train
4: Calculating validation accuracy A(XSt
valid)
5: end for
6: Suse ←argmax
S′∈{S1,··· ,ST}
A(XS′
valid)
7: return Suse
Note: “SignalsSelection(S,Smax,M)” selects and returns subset St⊆Sconsisting of Smax signals
from the initial signals set Sbased on method M. When Mis “Bayesian optimization (BO)”, tree-
structured Parzen estimator is used for signals selection. When Mis “random search (RS)”, signals are
randomly selected.
6.2. Result and discussion
The generalization scores of FG-SSA, BO, and RS measured by the test dataset are shown in Table
2. The results consist of F score, precision, and recall of 10 classes belonging to the class set Cdefined
by Eq (4.2). The bold and underlined numbers indicate the highest and lowest scores, respectively. The
values in the cells indicate the average scores of 100 runs, and pvalues are the result of the two-sided
ttest between FG-SSA and BO.
Based on Table 2, the baseline RS has the lowest score. A possible reason for this is that random
signal selection is not strategic in terms of improving the generalization performance. In addition, FG-
SSA has obtained better scores in more classes than BO. Although some results have better scores with
BO, there is no significant difference in comparison with FG-SSA. Additionally, looking at the macro
averages, we confirmed that FG-SSA scores higher than BO and RS in all three indicators. Consider-
ing this, it can be interpreted that FG-SSA can discover signal subsets leading to higher generalization
performance than BO and RS. Therefore, FG-SSA is an excellent method from the viewpoint of per-
forming good signal selection with a small number of iterations.
Since the number of CNN constructions in FG-SSA is 15, the number of Bayesian optimization
iterations was set to 15 in the previous analyses. As shown in Table 2, FG-SSA demonstrated higher
estimation performance than BO. However, by increasing the number of iterations of BO, it may be
possible to find a signal set that yields scores similar to those of FG-SSA. To test this hypothesis,
the generalization performance was measured when the number of BO iterations was 15, 30, and 45
(BO15, BO30, and BO45). Table 3 shows the obtained results. It can be seen that the generalization
performance increases slightly as the number of BO iterations increases. When comparing FG-SSA
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and BO15, FG-SSA performed significantly higher; however, when comparing FG-SSA and BO45,
no significant differences were observed for all indices. Therefore, the two signal sets obtained with
BO45 and FG-SSA could be similar. Conversely, the computation time for BO45 was approximately
three times longer than that for FG-SSA. Therefore, FG-SSA can find a good signals set at a lower
computational cost than BO, which is an existing method. Note that the computational environment
used in this experiment was OS: Ubuntu 20.04 LTS, CPU: Xeon W-2225 (4.1 GHz), RAM: 32 GB
DDR4-2933, GPU: NVIDIA RTX A6000 (48 GB).
Table 2. Estimation scores of FG-SSA, BO, and RS obtained by the test dataset (average
results of 100 seeds). pvalues mean the two-sided ttest between FG-SSA and BO. The bold
numbers mean the highest values and the underlined numbers mean the lowest values.
F score Precision Recall
Class FG-SSA BO RS pFG-SSA BO RS pFG-SSA BO RS p
Stand N .671 .668 .642 n.s. .721 .702 .701 * .635 .643 .600 n.s.
Stand L .281 .260 .235 * .282 .280 .244 n.s. .299 .259 .252 **
Stand R .597 .585 .530 * .560 .549 .503 n.s. .653 .638 .577 n.s.
Walk N .859 .852 .838 ** .882 .880 .847 n.s. .840 .828 .832 *
Walk L .247 .233 .184 n.s. .259 .234 .192 n.s. .254 .244 .195 n.s.
Walk R .402 .380 .289 * .345 .322 .257 ** .501 .483 .357 n.s.
Sit N .586 .594 .534 n.s. .679 .686 .644 n.s. .531 .536 .476 n.s.
Sit L .349 .352 .312 n.s. .426 .427 .373 n.s. .322 .319 .302 n.s.
Sit R .686 .688 .657 n.s. .627 .619 .593 n.s. .772 .785 .756 n.s.
Lie N .989 .983 .975 ** .988 .981 .973 ** .990 .986 .976 **
Macro ave. .567 .560 .520 * .577 .568 .533 ** .580 .572 .532 *
**: p< .01, *: p< .05, n.s.: otherwise (FG-SSA vs. BO)
Table 3. Estimation scores of FG-SSA, BO15, BO30, and BO45 obtained by the test dataset
and the corresponding calculation times (average and standard deviation of 100 seeds). The
bold numbers indicate the highest values, and the underlined numbers indicate the lowest
values. pvalues are obtained from the two-sided ttest between FG-SSA and BO.
Macro F score Macro precision Macro recall Calculation
Ave. pAve. pAve. ptime [sec]
FG-SSA .567 ±.021 -.577 ±.022 -.580 ±.022 - 2179.53 ±8.37
BO15 .560 ±.023 * .568 ±.024 ** .572 ±.025 * 2024.54 ±9.27
BO30 .563 ±.020 n.s. .572 ±.022 n.s. .573 ±.022 * 4101.60 ±29.97
BO45 .564 ±.020 n.s. .572 ±.022 n.s. .574 ±.021 n.s. 6174.26 ±41.47
**: p< .01, *: p< .05, n.s.: otherwise (FG-SSA vs. other methods)
7. Conclusion, limitations, and future works
In this study, we described the following topics to find and remove unimportant signals for CNN-
based classification:
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•(1) The signal importance indices SIM Imat(X) and SIV Ivec(X) are explained in Subsection 3.4.
•(2) The algorithm of the linear complexity O(ns) for obtaining the signals subset Suse from the
initial signals set Sby finding and removing unimportant signals (see Algorithm 1).
Although the proposed algorithm performed well in the case of the OPPORTUNITY dataset, it had
some limitations. This is explained in the following sections. Future work will confirm these findings.
•(A) The results described in this paper were obtained from limited cases. Although we assume
that the algorithm can be applied to data other than acceleration signals (e.g., EEG and ECG), it
is not validated.
•(B) In this experiment, the initial signals set size was 15. The number of signals may be much
higher depending on some situations. It is important to determine the relationship between the
number of unimportant signals and the effectiveness of FG-SSA.
•(C) We assume that the algorithm may extract the signals’ subset weighted specific target class
by providing class weights to the signal importance Is(X) defined in Eq (3.14). In other words,
we denote the importance of signal sweighted classes as
Is(X;w)=1
|C|X
c′∈C
wc′Is,c′
mat(X),(7.1)
where wc′is the weight of class c′and wis the class weight vector, which is defined as
w=[wc′]∈R|C|,s.t., X
c′∈C
wc′=1.(7.2)
We note that the algorithm based on Is(X;w) can return the signals’ subset weighted specific
classes. However, this effect is not confirmed in this study. This will be studied in the future.
A. Appendix: Layers structure of CNNs
The CNN’s layer structure is illustrated in Figure 2. First, an input layer exists for inputting w×ns
size data (w: window length and ns: the number of signals). Then, nc=3 convolution layers exist to
generate nffeature maps using kernel filters of convolution size sc. The convolutions are only in the
direction of time. Subsequently, the generated feature maps are transformed into a vector form by the
flattened layer. Vector dimensions are gradually reduced to 200, 100, and 50. Finally, an output vector
yof 10 dimensions is generated, and classification is performed. The total number of CNN layers is
nine (one input layer, three convolution layers, four dense layers, and one output layer). The activation
function of each layer is ReLU.
We performed a hyperparameter search for convolution size sc, the number of kernel filters nf, and
learning rate r. Particularly, CNNs were trained using the training and validation datasets described in
Subsection 3.1 and 300 epochs. We adopted a weighted cross-entropy-based loss function based on the
inverse values of class sample sizes because the sample sizes of each class are imbalanced, as shown
in Table 1.
The parameter candidates are as follows:
sc∈ {5,10,15},nf∈ {5,10},r∈ {10−3,10−4}.(A.1)
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The combination of hyperparameters leading to the maximum validation accuracy was (sc,nf,r)=
(10,10,10−4) (accuracy: 0.726). Therefore, the adopted CNN architecture is as follows:
•Layer 1: Conv. layer, the kernel filter: (sc,nf)=(10,10)
•Layer 2: Conv. layer, the kernel filter: (sc,nf)=(10,10)
•Layer 3: Conv. layer, the kernel filter: (sc,nf)=(10,10)
•Layer 4: Flatten layer based on the final conv. layer’s output size
•Layer 5: Dense layer of 200 dimensions
•Layer 6: Dense layer of 100 dimensions
•Layer 7: Dense layer of 50 dimensions
•Layer 8: Classification layer
•Note: the activation function for all layers except the classification layer is ReLU.
We adopted this architecture for all CNNs used in this study. Although CNNs have many other
hyperparameters, we avoided excessive tuning because the aim of this study was to provide a solution
to the signal selection problem.
B. Appendix: Effectiveness of FG-SSA on swimming style classification
B.1. Outline
The effectiveness of the proposed method should be validated by multiple datasets, not only by a
single dataset. Therefore, in this study, FG-SSA was applied to the task of automatically classifying
swimming style (backstroke, breaststroke, butterfly, and front crawl) based on signals measured by
an inertial sensor and CNN. Swimming style classification with inertial sensors is a typical research
topic in sports engineering (e.g., [56, 57]). The subjects were 16 Japanese university students who
selected two swimming styles from four swimming styles. The pool had a length of 25 m, and each
swimmer made one round trip of swimming (total 50 m). A sensor was attached to the lower back of
the swimmers, and it measured the tri-axial acceleration and angular velocity (six signals in total). The
sampling frequency of the inertial sensor was 100 Hz, the acceleration range was ±5 G, and the angular
velocity range was ±1500 dps. All signals were standardized to a minimum of 0 and a maximum of
1. To verify whether FG-SSA can eliminate unnecessary signals and select only important signals,
nine meaningless signals generated by uniform random numbers in the 0 to 1 range were prepared.
Considering the nine meaningless signals, the total number of signals was 3 +3+9=15.
B.2. Preprocessing
Of the 32 swims (16 subjects ×2 swims), there was a measurement error (sensor fell offwhile
swims) in 5 swims. Therefore, usable data were available for 27 swimmings. The sliding window
method was applied to the data with the window length and sliding length set to 1 sec. Since window
length and sliding length are of the same size, the cut waveforms did not overlap. This process resulted
in sample sizes of 91, 145, 332, and 205 for backstroke, breaststroke, butterfly, and front crawl, respec-
tively. Of the available data, 80 % was used as training data and 20 % as test data. In addition, 20 % of
the training data was used as validation data. Since the data splitting is random, the performance may
be higher (or lower) than the average tendency due to random effects. To obtain reliable results, the
experiment was run with a total of 100 random seeds.
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B.3. Result and discussion
Figure 9 (a) shows the maximum validation accuracies in 300 epochs when FG-SSA is applied to
the aforementioned task and unnecessary signals are removed one by one. In the initial half of FG-SSA,
the estimation performance improves as more unnecessary signals are deleted, reaching a maximum
when the number of deletions is 9. Then, in the later half of FG-SSA, the estimation performance
degrades. This indicates that the estimation performance is worse when there are unnecessary signals
in the input layer or when important signals are removed. Figure 9 (b) shows the deletion timing of
each signal. Randomly generated meaningless signals are removed early in the algorithm, and the
tri-axial acceleration and angular velocity signals are removed later. This result indicates that FG-SSA
can efficiently find and remove only meaningless signals while keeping important signals.
The number of signals deleted by processing line 8 of Algorithm 1 was 9.66 ±1.88 signals (average
and standard deviation of the results of 100 seeds). In other words, the number of signals selected as
the input layer of the CNN by FG-SSA was 15 −9.66 =5.34. Next, the generalization performance
was measured by the test dataset for CNNs using all signals (Condition A, 15 signals) and for CNNs
using the signal set selected by FG-SSA (Condition B, average 5.34 signals). The result is shown in
Figure 10. When all signals were used (Condition A), confusion in the estimation of “breaststroke”
and “butterfly” was observed. On the other hand, when unnecessary signals were removed by FG-
SSA (Condition B), misclassification was greatly reduced. Therefore, it can be concluded that signal
reduction by FG-SSA is beneficial for improving the generalization performance.
Figure 9. (A) Maximum validation accuracy in 300 epochs when the most unimportant
signal smin is gradually removed by FG-SSA and CNNs are re-learned. (B) Average removed
timings of each signal. The results of both (A) and (B) are average values of 100 seeds, and
the error bars present standard deviations. The pvalues in (A) represent the results of two-
sided ttest.
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Figure 10. Confusion matrices of the Conditions A and B using the test dataset (“Ba”:
backstroke, “Br”: breaststroke, “Bu”: butterfly, and “Fr”: front crawl). The values were
averaged over 100 seed results and standardized, where the summation in each row was 1.
Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
This work was supported in part by JSPS Grant-in-Aid for Scientific Research (C) (Grant Nos.
21K04535 and 23K11310), and JSPS Grant-in-Aid for Young Scientists (Grant No. 19K20062). This
work was also supported by the Tokyo City University Prioritized Studies.
Conflict of interest
The authors declare no conflicts of interest.
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