# ACCTS: AN ADAPTIVE MODEL TRAINING POLICY FOR CONTINUOUS CLASSIFICATION OF TIME SERIES

**Anonymous authors**
Paper under double-blind review

ABSTRACT

More and more real-world applications require to classify time series at every time.
For example, critical patients should be detected for vital signs and diagnosed at all
times to facilitate timely life-saving. For this demand, we propose a new concept,
Continuous Classification of Time Series (CCTS), to achieve the high-accuracy
classification at every time. Time series always evolves dynamically, changing
features introducing the multi-distribution form. Thus, different from the existing
one-shot classification, the key of CCTS is to model multiple distributions simultaneously. However, most models are hard to achieve it due to their independent
identically distributed premise. If a model learns a new distribution, it will likely
forget old ones. And if a model repeatedly learns similar data, it will likely be
overfitted. Thus, two main problems are the catastrophic forgetting and the over
fitting. In this work, we define CCTS as a continual learning task with the unclear
distribution division. But different divisions differently affect two problems and a
fixed division rule may become invalid as time series evolves. In order to overcome
two main problems and finally achieve CCTS, we propose a novel Adaptive model
training policy - ACCTS. Its adaptability represents in two aspects: (1) Adaptive multi-distribution extraction policy. Instead of the fixed rules and the prior
knowledge, ACCTS extracts data distributions adaptive to the time series evolution
and the model change; (2) Adaptive importance-based replay policy. Instead of
reviewing all old distributions, ACCTS only replays the important samples adaptive
to the contribution of data to the model. Experiments on four real-world datasets
show that our method can classify more accurately than all baselines at every time.

1 INTRODUCTION

In the real world, many applications need to classify time series data at every time (Gupta et al.
(2020)). For example, in the Intensive Care Unit (ICU), most detected vital signs of patients change
dynamically according to the development of decease. The status perception is needed at any time as
the real-time diagnosis provides more opportunities for doctors to rescue lives (Chen et al. (2014)).
In response to the current demand, we propose a new concept – Continuous Classification of Time
Series (CCTS). It aims to classify as accurately as possible st every time with the data evolvement.

For most time series extracted from practical applications, the development, caused by their changed
data characteristics, leads to the evolved data distribution, and finally produces the multi-distribution
form. For example, in Figure 1, the data distributions of blood pressure of 2,000 sepsis patients vary
among early, middle and late time stages during hospitalization, bring a triple-distribution.

In the background of CCTS, modeling multiple data distributions simultaneously is the requirement:
when the data distribution changes, the model performance cannot decrease. However, a single
model, like deep neural network, is lack of ability to learn all distributions simultaneously as they are
restricted by the premise of independent identically distributed (i.i.d) data (Shim et al. (2021)). If a
model learns a new distribution, it will negatively affect its performance on old ones.

Therefore, when learning the multi-distribution, a model will face two problems:

-  Catastrophic forgetting. A time series usually has a large number of time points. For example,
the blood pressure of a critical patient could be sampled hundreds of times. If a model is trained
at all points, it will learn hundreds of data distributions continuously. Frequent learning of new
knowledge will inevitably lead to the forgetting of old ones (Parisi et al. (2019));


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Figure 1: Multi-distribution in Time Series.
The statistics of mean blood pressure of 2,000 sepsis
represent the distributions (Kiyasseh et al. (2021)).


Figure 2: Continuous Classification Mode.
Different from the existing time series classification task,
CCTS need the continuous classification.



-  Over fitting. As the development of time series needs a process, the data distributions of adjacent
time are always similar as shown in Figure 1. And in order to alleviate the catastrophic forgetting,
many methods replay old distributions to enhance memory (Kiyasseh et al. (2021)). Over-learning
of similar data will cause the strict function and poor generalization (Saha et al. (2021)).

Two issues are intertwined. The old knowledge review could avoid the forgetting but worsen the
overfitting, while the iteration avoidance could alleviate the overfitting but worsen the forgetting. The
additional policy like the selective review, could help to balance the above problems. However, the
policy formulation is based on the known data distribution, yet the distribution in CCTS is not clear
as its data form is sequential and can be divided by different rules. Therefore, before planning how to
learn the multi-distribution, we should also determine the well-divided multi-distribution as:

-  Less divided distributions may worsen the catastrophic forgetting problem and omit important data
features. If dividing time series less, distributions will be more different. It will make the model
change greatly after learning the new distribution and seriously affect the model performance on
old distributions. Besides, the key distribution may not be included due to the incomplete sampling;

-  More divided distributions may worsen the over fitting problem and have the low training efficiency.
If dividing time series more, distributions will have many overlaps. It will make some similar
features to be learned repeatedly and over fit the trained model to the recurring dataset. Besides, in
the model training phrase, learning more distributions will spend more time.

The optimal multi-distribution is hard to obtain. Unlike images, the time series is more abstract and
its characteristics are not explicit (Xing et al. (2011)). Although some methods can describe time
series like Shaplets (Liang & Wang (2021)), they still need the prior knowledge. Most importantly,
the artificial rule needs to be determined before training the model and remains the same over time.
But because the time series has been evolving dynamically, a fixed rule is likely to be outdated.

In this work, instead of the fixed methods, we design an Adaptive model training policy for Continuous
Classification of Time Series (ACCTS). It has two data-based adaptive policy:

-  Adaptive multi-distribution extraction policy. It explores the policy space according to the reward
of distribution difference and classification accuracy based on the reinforcement learning strategy,
and finally extracts data distributions adaptive to the time series evaluation and the model change;

-  Adaptive importance-based replay policy. It leans the impact of each sample on the model, applying
partial replay to balance the result accuracy and the time efficiency. The important samples in each
distributions are obtained adaptive to the time series evaluation and the model change.
Both polices are dynamic rather than static to trade off the forgetting and the overfitting. Experimental
results on real-world datasets show that ACCTS is more accurate than all baselines at every time.

2 RELATED WORK

The popularity of time series classification has attracted increasing attention in many practical
fields (Santos & Kern (2016)). The foundation is Classification of Time Series (CTS), making
classification based on the full-length data (Fawaz et al. (2019)). But in time-sensitive applications,
Early Classification of Time Series (ECTS) is more critical, making classification at an early time
(Gupta et al. (2020)). For example, early diagnosis helps for sepsis outcomes (Liu et al. (2018a)).
Both of them give one classification result, while CCTS needs continuous multiple results. Thus,
based on the classification mode, the existing work can be summarized into two categories.


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2.1 ONE-SHOT CLASSIFICATION

This mode classifies time series at a fixed time: as shown in Figure 2, once the classification is
complete, the action will not continue. Both CTS and ECTS are belong to this and always use Deep
Learning (DL) models. RNNs-based methods recur in evolution direction to learn the sequential
dependency (Fawaz et al. (2019)). CNN-based methods use kernels to extract local features (Huang
et al. (2017)). Choi et al. (2017) models the long-term dependencies; Tan et al. (2020) and Sun et al.
(2021) address the time irregularity; Hsu et al. (2019) learns frequency features; Lai et al. (2015)
designs Temporal Convolutional Network (TCN) to integrate RNN and CNN. However, the above
DL-based methods just perform well on i.i.d data at a fixed time, like early 6 hours sepsis diagnosis
(Reyna et al. (2019a)), but fail for classifying old distributions after learning new distributions.

2.2 CONTINUOUS CLASSIFICATION

This mode classifies time series at every time as shown in Figure 2. Most of the existing methods
apply multi-models to model multi-distribution, like SR (Mori et al. (2018)) and ECEC (Lv et al.
(2019)). They divide data according to time stages and design different classifiers for different
distributions. But they only consider the data division, ignoring the strategic training method. Besides,
the operation of classifier selection in multi-models framework will result in additional losses.

CCTS has the multi-distributed data, leading to the catastrophic forgetting problem. Recently,
Continual Learning (CL) (Delange et al. (2021)) aims to address the issue of static models incapable
of adapting their behavior for new knowledge. It learns a new task at every new moment and each
new task corresponds to a new data distribution. Replay-based methods re-train the model by old data
to consolidate memory (Rolnick et al. (2019); Kiyasseh et al. (2021); Isele & Cosgun (2018); Rebuffi
et al. (2017)); Regularization-based methods restrain parameter update of neural networks to limit
forgetting (Kirkpatrick et al. (2016); Lopez-Paz & Ranzato (2017); Liu et al. (2018b); Zhang et al.
(2020)); Model-based methods change network structure or apply multiple models to response to
different tasks (Fernando et al. (2017); Mallya & Lazebnik (2018)). But most of the above methods
have the problems of storage limitation, distributions drifts and model overfitting. Most importantly,
in CL, the definition of old and new tasks is clear and the division of distribution is fixed. But in
CCTS, the distributions, that is, the tasks in CL, is not determined and need to be defined.

Thus, CCTS is a new proposed concept, there is no systematic analysis and corresponding solutions.

3 PROBLEM FORMULATION

**Definition 1 Continuous Classification.labeled with a class C ∈C. Continuous classification tasks aim to classify the time series A time series X = {x1, ...xT } having T time points is X at every**
_time t, t = 1, ..., T with the minimum additive loss_ _t=1_

_[L][(][f]_ [(][X][1:][t][)][, C][)][.]

Note that most of the existing time series classification tasks are belong to the one-shot classification
mode, where they optimize the objective with a single minimum loss[P][T] (f (X1:t), C). In continuous
_L_
classification, the model need to learn multiple distributions, which is firstly proposed in CL field. A
CL issue T = {T [1], T [2], ..., T _[N]_ _} has a sequence of N tasks, learning a new task T = (X, C) at every_
time. The goal is to control the statistical risk of all seen tasks _n=1_ [E][(][X] _[n][,C][n][)][[][L][(][f][ n][((][X]_ _[n][;][ θ][)][, C]_ _[n][)]][.]_
Based on this settings, we give the definition of CCTS:

**Definition 2 Continuous Classification of Time Series (CCTS).has T, time points and is labeled with a class, ...,** _. Each distribution M_ _[n] Cis represented by the subsequence set ∈C. CCTS has a sequence of[P][N]_ _A time series N distributions X X =1:t {n_ _. The goalx1, ...x MT = }_
_{M[1]_ _M[2]_ _M[N]_ _}_
_is the statistical risk control of classification with all distributions:_ _n=1_ [E][M][n] [[][L][(][f][ n][(][M][t][;][ θ][)][, C][)]]
_with the loss L, the network function f and parameters θ of the classification model. When the model_
_f_ _[n]_ _is trained by the current distribution_ _, its performance on all observed data cannot degrade:_
_M[n]_ [P][N]


minL(f _[n], M[n])_


_t[n]_


_t[n]_ (1)

(f _[n][−][1](X1:t; θ[n][−][1]), C)_
_L_
_t=1_

X


_subject to_ [1]

_t[n]_


(f _[n](X1:t; θ[n]),C)_
_L_ _≤_
_t=1_

X


_t[n][−][1]_


-----

Figure 3: Adaptive Model Training Process for Continuous Classification of Time Series

4 ADAPTIVE MODEL TRAINING POLICY

Figure 3 shows the model training process by the policy of ACCTS. When a model is trained by time
series from the initial to the final time, ACCTS gives two decisions:

-  Whether the current time series segment forms a new distribution to be learned. If yes, train the
model by the current time series; Otherwise, do not train and continue to get new data points;

-  Which old samples need to be replayed and learned again. If the previous decision is yes, train the
model by the obtained old samples again after train it by the current time series.

4.1 ADAPTIVE MULTI-DISTRIBUTION EXTRACTION

The first decision is got by the adaptive multi-distribution extraction polity. It is an agent that decides
whether to extract the current time series sequence to train the model. It solves a 3-triple partiallyobservable Markov decision process {S, A, R} (Kaelbling et al. (1995)), where the observation
arrive from a state s at each time, an action a is sampled using a learned policy, and a reward r is
observed according to the selected action’s quality. The objective is to optimize long-term rewards.

**State S. It is represented by the characteristics of the currently data and the adaptability of the old**
model to the current data. It is intuitive: First, the model need to be trained by the dataset with
different features from the previous data for the comprehensive modeling; Second, the model must be
trained again when it performs poorly on the current data for the overall accuracy.

At the current time t, we use the Long Short-Term Memory (LSTM) network as the base model to
learn the hidden characteristics of a time series X1:t, generating low-dimensional vector representation
_ht. We also propose the model gradients gt to evaluate the adaptability of the model to the current_
time series. The model gradient can help for the interpretation of DL model by explaining the
response of the neural network to input data (Srinivas & Fleuret (2019)). Large gradient fluctuation
reflects the low adaptability of the model to the input data. Thus, the state st of current time series is:

_st = concatenate(LSTM(xt), MG(X1:t))_ (2)

_ft = σ(Wf_ [ht 1, xt] + bf ), it = σ(Wi[ht 1, xt] + bi),
_−_ _−_
_ct = ft_ _ct_ 1 + it _η(Wc[ht_ 1, xt] + bc), LSTM
_·_ _−_ _·_ _−_
_ot = σ(Wo[ht_ 1, xt]+bo), ht = ot _η(ct)._
_−_ _·_


_∂_

MG
_∂θf n_

_[L][(][f][ n][(][X][1:][t][, θ][f]_ [)][, C][)]


_gt =_


**Action A. At the current time t, the action at dictates the decisions of ACCTS agent: If at = 0,**
continue to accept the value point of time series and let LSTM move forward one time step; If at = 1,
extract the current time series X1:t as a new distribution to be learned. For the action selection, we
use ε-greedy selection to avoid abundant exploitation. at is replaced with a random action with the
probability ε of exponentially decreasing from 1 to 0 during the training process.

_at,_ with probability 1 _ε_
_at =_ _−_ _,_ _at_ 0, 1 (3)
_random, with probability ε_ _∈{_ _}_


**Reward R. The agent observes the return which can qualify the parameters of the current policy. The**
goal of CCTS is the high accurate classification by solving the problems of catastrophic forgetting


-----

**Algorithm 1 Adaptive Multi-distribution Extraction Policy**


**Input: Data D = {(X, C)};**
Classifier net f .

**Output: Extraction policy (µ, Q).**

1: for t = 1 to T do
2: _st_ Equation 2

3: _at ←_ _µ(st, θActor)_

4: **if a ←t = 1 then**

5: _rt_ Equation 4

6: Update ← _Q, µ by:_


_∇θQ_ _Ocritic =_

_∇θµ_ _Oactor =_


(y −∇θQ _Q (st, at|θQ))_

_µ(st)Q (st, µ (st))_ _θµ_ _µ (st_ _θµ)_
_∇_ _∇_ _|_


7: Soft update Q[′], µ[′] by:

_θQ′ ←_ _τθQ + (1 −_ _τ_ )θQ′
_θµ′_ _τθ[µ]_ + (1 _τ_ )θµ′
_←_ _−_

8: Train f by (X1:t, C)

9: **end if**

10: end for


and over fitting as we analysed in Section 1. Thus, we pursue the higher accuracy of the current
classifier on all potential data distributions to control the catastrophic forgetting, and we limit the
number of extracted distributions by the time span between distributions to control the over-fitting.
Thus, at the current time t, after applying the action at, the reward rt is consisted of two components.
The first term is for the high accuracy of the current model f _[n]_ on all data, the second term is for less
divisions by using the time length between the current time t[n] and the last data extraction time t[n][−][1].


_rt =_
_−_ _T[α]_


_T_

(f _[n], X_ [1:][t]) + [(1][ −] _[α][)]_
_L_ _T_
_t=1_

X


(|t[n] _−_ _t[n][−][1]|)_ (4)


When using the transition probability P (st+1 _st, at), the total reward of the trajectory is the is sum_
_|_
of the reward in each time. Thus, the objective is to maximize the total reward R = _t=1_ _[r][t][. The]_
policy gradient method (Sutton et al. (1999)) learns the policy πθ(st, at) = P (at _st) for the larger_
_|_
return. The objective is J(θ) = E[r(s, a)πθ(s, a)]. For ACCTS, we apply Actor-Critic (Zhang et al.

[P][T]
(2019)) structure with two components of the main net and the target net. The main net of Actor µ
use the state s to generate the action a; The main net of Critic Q judges the action a through reward r
by Q-function (Watkins & Dayan (1992)). The target nets of Actor and Critic µ[′], Q[′] put the target Q
value stable for a period of time, making the algorithm performance more stable.

_OActor (θµ) = Est_ _S [_ (st, µ (st _θµ)_ _θ_ )]
_∈_ _Q_ _|_ _|_ _Q_ (5)

_OCritic(θQ) =Est∈S[(rt + γQ[′](st+1, µ[′](st+1|θµ′_ )|θQ′ ) −Q(st, a|θQ))[2]]

4.2 ADAPTIVE IMPORTANCE-BASED REPLAY

The replay mechanism can help to alleviate the catastrophic forgetting (Borsos et al. (2020)). However,
the operation of repeated replay easily causes the over-fitting problem, especially for time series with
small difference between the two adjacent times. In CL, many methods only replay the representative
data, such as the class means (Rebuffi et al. (2017)) and the class prototype (Mazumder et al. (2021)),
each representative is fixed to its distribution. But in CCTS, we still need to consider whether all the
representative need to be learned again and whether the representative will change over time.

Thus, we focus on the adaptive method to explore a wider space, where the replayed data is dynamic
and determined according to the current state. We introduce an importance-based replay method. In
each round, it only re-trained the model by some important samples to the model. The importance of
each sample is learned from the objective of an additive loss function.

We incorporate the importance parameter βi of a time series Xi in the replay buffer B[n] as a coefficient
of its loss _n,i. The overall loss at the current time t[n]_ is the sum of loss of each sample’s loss:
_L_

1 _|B[n]|_
_n =_ (βn,i[2] _[L][n,i]_ [+][ λ][(][β][n,i]
_L_ _|B[n]|_ _i=1_ _[−]_ [1)][2][)] (6)

X

= _X1:t[n]_ _,X[˜]i_ _βn_ 1,i < ϵ
_B[n]_ _{_ _|_ _−_ _}_

_β is learned by the gradient descent βn,i ←_ _βn,i −_ _∂βn,i∂Ln_ [. Thus, if a sample][ X][i][ is hard to classify, its]

loss _,i will be larger. In order to minimize the loss, its β_ _,i will be smaller. Based on this, in each_
_L∗_ _∗_


-----

**Algorithm 2 The Model Training Process by ACCTS Policy**


**Input: Data D = {(X, C)}; ACCTS Actor µ.**
**Output: Final Classifier net f** _[N]_ .

1: Initialize buffer 1 _X1:1_
2: Initialize DL Classifier net B _←{_ _f_ [1]}.
//TRAVERSE EVERY TIME POINT

3: for t = 2 to T do
4: _st_ _ht, gt_ from Eq.2

5: _at ←{µ(st)_ _}_
_←_


//ADAPTIVE EXTRACTION

6: **if at = 1 then**

8:7: _fB[n]n ←B←_ Trainn + f {[n]X[−]1:[1]tby} _B[n]_ with Eq.6
//IMPORTANCE STORAGE

10:9: **end ifB[n][+1]** _←{Xi|Xi ∈B[n], βi < ϵ}_

11: end for


learning phrase, the buffer contains the current time series X1:tn and the important old time series
_X˜_, who are the first few difficult learning samples ( B[n] _βn_ 1,i < ϵ) in the last buffer . Meanwhile,
_−_ _B[n][−][1]_
as β is the confidence of loss, if β = 0, the loss are hard to be optimized. Thus, inspired by Kiyasseh
et al. (2021), we introduce a regularization term (β − 1)[2] and initialize β = 1 to penalize it when
rapidly decaying toward 0. As β is re-obtained after each model training process, the important
samples _X[˜] are changed adaptively and the buffer B is updated iteratively._

4.3 OVERALL TRAINING PROCESS FOR CLASSIFIER MODEL

The adaptive multi-distribution extraction policy, which is achieved by the Actor net µ, is trained
before the classifier training process, as shown in Algorithm 1. First, LSTM calculates the current
sate st (Line 2) and gives the action at (Line 3). Then, the reward rt is obtained by the long-term
accuracy to update the net (Line 6), where Actor and Critic are updated alternately. Main Critic net is
updated by Q value, calculating from both two Critic. Main Actor is updated by the back-propagation
gradient of main Critic. Target Actor and Critic are learned by the soft update (Line 7).

The adaptive importance-based replay policy is trained along with the classifier training process, as
shown in Algorithm 2. First, in each time step, the Actor of ACCTS determines if a new distribution
appears (Line 4,5). If yes, train the classifier from f _[n]_ to f _[n][−][1]_ by datasets in the buffer B[n] (Line 7,8),
and get the important samples according to β to form a new buffer B[n][+1] (Line 9); Else, continue to
get new values in next time point t + 1. At the final time, we can get the well-trained classifier f _[N]_ .

Note that the two processes of the adaptive multi-distribution extraction and the adaptive importancebased replay are relevant rather than independent. The extraction policy is based on the feature of the
buffer data, and the replay policy selects the important samples based on the extracted data. Both of
them are data-based, which helps for adaptive combination. That’s why we design the replay-based
policy rather than the regularization-based policy after the distribution extraction.

5 EXPERIMENTS

5.1 DATASETS

-  UCR-EQ dataset (Chen et al. (2015)) has 471 earthquake records from UCR time series database
archive. It is the univariate time series of seismic feature value. Natural disaster early warning,
like earthquake warning, helps to reduce casualties and property losses (Ammon et al. (2021)).

-  USHCN dataset (Menne & R. (2016)) has the daily meteorological data of 48 states in U.S. from
1887 to 2014. It is the multivariate time series of 5 weather features. Rainfall warning is not only
the demand of daily life, but also can help prevent natural disasters (Lee et al. (2021)).

-  COVID-19 dataset (Yan L (2020)) has 6,877 blood samples of 485 COVID-19 patients from Tongji
Hospital, Wuhan, China. It is the multivariate time series of 74 laboratory test features. Mortality
prediction helps for the personalized treatment and resource allocation (Sun et al. (2020)).

-  SEPSIS dataset (Reyna et al. (2019b)) has 30,336 patients’ records, including 2,359 diagnosed
sepsis. It is the multivariate time series of 40 related patient features. Early diagnose of sepsis is
critical to improve the outcome of ICU patients (Seymour et al. (2017)).
Not that for each time series in the above four datasets, every time point is tagged with a class label,
which is the same as its outcome label, such as ‘mortality’, ‘sepsis’, ‘earthquake’ and ‘rain’.


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5.2 BASELINES

CCTS is related to ECTS and CL, we use baselines in these fields. The first is ECTS-based methods.
All methods use the same base model of LSTM and the same structure of fully connected layers.

-  LSTM (Wiens et al. (2012); Choi et al. (2017)). It contains a single classifier model. For one time
series, the classifier model is trained by all subsequences from time 1 to time t, where t = 2, ..., T .

-  SR (Mori et al. (2018)). It has multiple base models. All models are trained by the full-length time
series. The final classification is the fusion result. It also has a stop rule of classification stop time.

-  ECEC (Lv et al. (2019)). It has a set of base models. Each model is trained by time series in
different time stages. When classifying, the data selects the classifier based on its time stages.
The second type is CL-based methods, including regularization and replay methods:

-  EWC (Kirkpatrick et al. (2016)). It is a regularization-based method, training a model to remember
the old tasks by constraining important parameters to stay close to their old values.

-  GEM (Lopez-Paz & Ranzato (2017)). It is a regularization-based method, training a model to
remember the old tasks by finding the new gradients which are at acute angles to the old gradients.

-  CLEAR (Rolnick et al. (2019)). It is a replay-based method, using the reservoir sampling to limit
the number of stored samples to a fixed budget assuming an i.i.d. data stream.

-  CLOPS (Kiyasseh et al. (2021)). It is a replay-based method, training a base model by replaying
old tasks with importance-guided buffer storage and uncertainty-based buffer acquisition.

5.3 EVALUATION METRICS
The accuracy is evaluated by Area Under Curve of Receiver Operating Characteristic (AUC-ROC).
The performance of continuous mode is evaluated by Backward Transfer (BWT) and Forward Transfer
(FWT), the influence that learning a current has on the old/future. R ∈ R[|M|×|M|] is an accuracy
matrix, Ri,j is the accuracy on M[j] after learning M[i]. b is the accuracy with random initialization.

1 _|M|−1_ 1 _|M|_
BWT = _|M| −_ 1 _i=1_ _R|M|,i −_ _Ri,i,_ FWT = _|M| −_ 1 _i=2_ _Ri−1,i −_ _bi,i_ (7)

X X

5.4 RESULTS AND ANALYSIS
We test the baselines from the classification accuracy, analyze our ACCTS methods from ablation
study and coefficient test, and show the representation of time series in continuous classification.

5.4.1 CLASSIFICATION ACCURACY
ACCTS has the best performance on classification accuracy. As shown in Table 1, it can classify time
series more accurately than all baselines at every time. The average accuracy is about 2% higher.

5.4.2 CATASTROPHIC FORGETTING AND OVER FITTING
ACCTS is the best when solving these two problems with the highest BWT and FWT as shown in
Table 2. In Table 3, for CL-based methods (EWC, GEM, CLEAR and CLOPS), the accuracy on
validation set is much lower than that on training set, shown as ↓. And in Table 1, in late stages,
the accuracy of CL-based methods is also lower than that of ECTS-based methods. They all show
the over-fitting of CL-based methods caused by training and reviewing at all time. But ACCTS can
alleviate this problem by training and reviewing at the selected time so that it accuracy is higher.

5.4.3 ABLATION STUDY
Both policies of ACCTS are necessary. As shown in Figure 4, compared with the extraction at all
times and random extraction, the adaptive extraction performs best in both overall data and the early
distribution; Compared with no-replay and all-replay, the adaptive replay has the best performance.
Besides, the accuracy of importance-based replay is higher than regularization. It demonstrate a good
fit between two data-based polices of ACCTS: Importance-based replay and distribution extraction
fit better than regularization and distribution extraction. This confirms the reason why we refer to
replay-based instead of regularization-based CL methods.

5.4.4 COEFFICIENT TEST
ACCTS has two definable coefficients α and ϵ, belong to two policies separately. Larger α review
more distribution to learn. Larger ϵ causes more samples to review. As shown in Figure 5, the practice
is to set them in the direct ratio: Within a reasonable range, more distributions need more review.


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*20% means the current classification time is 20% of the total time of the full-length time series;Table 1: Baselines Classification Accuracy (AUC-ROC↑) for 4 Real-world Datasets at 5 Time Steps.
Bold font indicates the highest accuracy;
More detailed results are in Appendix.

Dataset Method 20%* 40% 60% 80% 100%


LSTM 0.711±0.038 0.843±0.019 0.874±0.012 0.909±0.014 0.924±0.012
SR 0.736±0.014 0.863±0.015 0.888±0.017 0.928±0.105 0.941±0.104
ECEC 0.738±0.018 0.865±0.014 0.890±0.015 0.929±0.107 0.940±0.009
EWC 0.768±0.018 0.874±0.016 0.895±0.014 0.923±0.102 0.933±0.003
GEM 0.767±0.017 0.876±0.016 .0900±0.015 0.929±0.008 0.934±0.004
CLEAR 0.770±0.015 0.880±0.013 0.904±0.012 0.923±0.004 0.932±0.005
CLOPS 0.773±0.016 0.878±0.016 0.902±0.015 0.917±0.006 0.925±0.005
**ACCTS** **0.774±0.023** **0.882±0.022** **0.906±0.005** **0.933±0.010** **0.946±0.003**

LSTM 0.700±0.028 0.745±0.028 0.820±0.015 0.852±0.014 0.891±0.002
SR 0.730±0.022 0.761±0.023 0.836±0.016 0.902±0.013 0.933±0.009
ECEC 0.736±0.024 0.760±0.025 0.837±0.016 0.906±0.017 0.931±0.009
EWC 0.736±0.025 0.798±0.024 0.834±0.016 0.896±0.017 0.926±0.007
GEM 0.728±0.026 0.781±0.023 0.838±0.013 0.899±0.010 0.928±0.005
CLEAR 0.738±0.025 0.784±0.024 0.837±0.010 0.879±0.012 0.921±0.004
CLOPS 0.740±0.024 0.781±0.025 0.835±0.016 0.877±0.011 0.919±0.013
**ACCTS** **0.742±0.017** **0.791±0.021** **0.841±0.012** **0.910±0.015** **0.939±0.013**

LSTM 0.701±0.033 0.833±0.015 0.888±0.013 0.925±0.014 0.944±0.015
SR 0.730±0.024 0.867±0.016 0.900±0.018 0.946±0.006 0.962±0.005
ECEC 0.732±0.028 0.870±0.016 0.904±0.014 0.948±0.015 0.963±0.017
EWC 0.769±0.015 0.888±0.028 0.923±0.014 0.940±0.013 0.954±0.008
GEM 0.779±0.017 0.885±0.022 0.924±0.018 0.939±0.010 0.953±0.005
CLEAR 0.785±0.019 0.879±0.016 0.926±0.014 0.941±0.007 0.952±0.008
CLOPS 0.775±0.013 0.900±0.017 0.925±0.015 0.940±0.007 0.954±0.006
**ACCTS** **0.790±0.023** **0.901±0.022** **0.927±0.006** **0.960±0.011** **0.967±0.008**

LSTM 0.629±0.035 0.736±0.064 0.748±0.043 0.795±0.027 0.827±0.039
SR 0.659±0.015 0.791±0.026 0.827±0.037 0.845±0.014 0.866±0.023
ECEC 0.669±0.019 0.793±0.016 0.815±0.014 0.849±0.016 0.863±0.014
EWC 0.733±0.023 0.827±0.036 0.838±0.024 0.848±0.015 0.854±0.016
GEM 0.730±0.024 0.826±0.033 0.836±0.028 0.849±0.014 0.853±0.012
CLEAR 0.732±0.024 0.825±0.035 0.839±0.028 0.847±0.010 0.848±0.016
CLOPS 0.733±0.025 0.824±0.036 0.838±0.026 0.850±0.017 0.857±0.018
**ACCTS** **0.734±0.038** **0.828±0.030** **0.842±0.034** **0.857±0.012** **0.872±0.012**


UCR-EQ

USHCN

COVID-19

SEPSIS


Table 2: Continual Learning Performance of Baselines.
The left table is BWT↑ results, the right table is FWT↑ results.

Dataset Method EWC GEM CLEAR CLOPS ACCTS Dataset Method EWC GEM CLEAR CLOPS ACCTS


UCR-EQ +0.039 +0.041 +0.053 +0.052 +0.058
USHCN +0.058 +0.054 +0.063 +0.074 +0.084
COVID-19 +0.011 +0.012 +0.009 +0.014 +0.020
SEPSIS +0.019 +0.017 +0.030 +0.032 +0.035


UCR-EQ +0.321 +0.329 +0.312 +0.301 +0.345
USHCN +0.312 +0.328 +0.335 +0.301 +0.342
COVID-19 +0.426 +0.421 +0.427 +0.439 +0.455
SEPSIS +0.295 +0.265 +0.401 +0.397 +0.410


Table 3: COVID-19 Classification Accuracy with Non-uniform Training Sets and Validation Sets.
_↓_ means the accuracy is greatly reduced; More detailed results are in the appendix.

Subset SR ECEC EWC GEM CLEAR CLOPS ACCTS

Male 0.968±0.014 0.969±0.016 0.965±0.012 0.965±0.004 0.978±0.009 0.978±0.014 0.971±0.010
Female 0.945±0.004 0.947±0.015 0.939±0.018 0.938±0.003 0.919±0.008↓ 0.921±0.009↓ 0.947±0.002

Age 30- 0.965±0.014 0.967±0.015 0.967±0.013 0.964±0.009 0.977±0.008 0.979±0.012 0.972±0.010
Age 30+ 0.941±0.007 0.943±0.018 0.931±0.008↓ 0.923±0.040↓ 0.902±0.006↓ 0.914±0.007↓ 0.945±0.006

Test 0.964±0.013 0.968±0.015 0.966±0.012 0.962±0.006 0.979±0.009 0.978±0.010 0.970±0.007
Valid. 0.962±0.006 0.963±0.014 0.954±0.003 0.953±0.005 0.952±0.009↓ 0.954±0.004↓ 0.967±0.006


-----

Figure 4: Ablation Study of Two Policies of ACCTS with the Case Study of COVID-19


Figure 6: Extracted Six Distributions from
Figure 7: Impact of Data
SEPSIS Dataset by ACCTS.


Figure 5: Parameter Test
of α, ϵ in ACCTS.


Size to Distributions.

10

98 UCRE-EQUSHCNCOVID-19

7 SEPSIS

6

5

4

Ditribution number

3

2

1 20 40 60 80 100

Data size (%)


Figure 8: The Important Samples in Four SEPSIS Distribution Buffers (2,3,4,5 in Figure 6)

5.4.5 ANALYSIS OF MULTI-DISTRIBUTION
The case study of SEPSIS in Figure 6 shows that ACCTS only extracts six distributions and the
difference among distributions is relatively large. The extraction is concentrated in 85%-length late
stage, which may be because the patient’s vital signs change significantly near the outcome time.

5.4.6 ANALYSIS OF IMPORTANT SAMPLES
The important samples includes not only the data hard to learn, but also the representative data as
shown in Figure 8. It might because that, the representative data is similar to the most common data,
resulting in a greater additive loss, therefore leading to smaller coefficients in Equation 6. Through
the experimental results, we can redefine the important samples: Important samples are samples that
can represent most data of a class and samples that are difficult to distinguish by the model.

5.4.7 IMPACT OF DATA SIZE
ACCTS prefers to divide more distributions for large data size as shown in Figure 7. Besides, ACCTS
has greater advantages over baselines in larger data size, detailed results are in Appendix.

6 CONCLUSION

In this paper, we propose a new concept of Continuous Classification of Time Series (CCTS) to meet
the real needs. It has two major difficulties of catastrophic forgetting and over fitting. In CCTS, the
multi-distribution of time series is not clearly defined, and the distribution division directly affects
the above two difficulties. Thus, we design an Adaptive model training policy named ACCTS. It
contains a multi-distribution extraction policy adaptive to the time series evaluation and the model
change, and an importance-based replay policy adaptive to the data features and final accuracy. We
test the methods on four real-world datasets and analyze the method from perspectives of accuracy,
continual learning, ablation study and parameter setting. Future work could refer to the distribution
extraction policy that fits to regularization-based methods.


-----

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A DETAILED EXPERIMENTAL RESULTS

A.1 BASIC CLASSIFICATION ACCURACY

*k% means the current classification time is k% of the total time of the full-length time series;Table 4: Classification Accuracy (AUC-ROC↑) for 4 Real-world Datasets at the First 5 Time Steps.
Bold font indicates the highest accuracy.

Dataset Method 10% 20% 30% 40% 50%


LSTM 0.695±0.044 0.711±0.038 0.803±0.024 0.843±0.019 0.854±0.017
SR 0.700±0.015 0.736±0.014 0.830±0.016 0.863±0.015 0.871±0.024
ECEC 0.703±0.013 0.738±0.018 0.828±0.017 0.865±0.014 0.873±0.026
EWC 0.724±0.015 0.768±0.018 0.848±0.014 0.874±0.016 0.883±0.025
GEM 0.723±0.014 0.767±0.017 0.850±0.015 0.876±0.016 0.890±0.024
CLEAR 0.729±0.015 0.770±0.015 0.852±0.019 0.880±0.013 0.899±0.026
CLOPS 0.728±0.016 0.773±0.016 0.855±0.015 0.878±0.016 0.896±0.028
**ACCTS** **0.730±0.022** **0.774±0.023** **0.856±0.015** **0.882±0.022** **0.900±.0017**

LSTM 0.682±0.014 0.700±0.028 0.721±0.013 0.745±0.028 0.784±0.023
SR 0.702±0.014 0.730±0.022 0.745±0.016 0.761±0.023 0.809±0.024
ECEC 0.707±0.017 0.736±0.024 0.748±0.015 0.760±0.025 0.806±0.025
EWC 0.727±0.018 0.736±0.025 0.768±0.017 0.798±0.024 0.805±0.022
GEM 0.720±0.019 0.728±0.026 0.772±0.015 0.781±0.023 0.801±0.026
CLEAR 0.728±0.016 0.738±0.025 0.773±0.018 0.784±0.024 0.802±0.027
CLOPS 0.728±0.012 0.740±0.024 0.769±0.019 0.781±0.025 0.800±0.024
**ACCTS** **0.730±0.018** **0.742±0.017** **0.775±0.016** **0.791±0.021** **0.810±.0133**

LSTM 0.605±0.044 0.701±0.033 0.793±0.022 0.833±0.015 0.844±0.013
SR 0.636±0.014 0.730±0.024 0.810±0.013 0.867±0.016 0.901±0.013
ECEC 0.639±0.013 0.732±0.028 0.829±0.013 0.870±0.016 0.901±0.026
EWC 0.703±0.022 0.769±0.015 0.870±0.014 0.888±0.028 0.915±0.017
GEM 0.699±0.025 0.779±0.017 0.871±0.015 0.885±0.022 0.914±0.019
CLEAR 0.710±0.013 0.785±0.019 0.870±0.016 0.879±0.016 0.916±0.024
CLOPS 0.709±0.017 0.775±0.013 0.869±0.012 0.900±0.017 0.918±0.026
**ACCTS** **0.712±0.021** **0.790±0.023** **0.872±0.013** **0.901±0.022** **0.919±0.016**

LSTM 0.576±0.063 0.629±0.035 0.735±0.064 0.736±0.064 0.745±0.056
SR 0.626±0.035 0.659±0.015 0.768±0.013 0.791±0.026 0.803±0.018
ECEC 0.623±0.024 0.669±0.019 0.761±0.016 0.793±0.016 0.811±0.015
EWC 0.671±0.027 0.733±0.023 0.799±0.015 0.827±0.036 0.832±0.028
GEM 0.670±0.026 0.730±0.024 0.802±0.018 0.826±0.033 0.834±0.026
CLEAR 0.680±0.028 0.732±0.024 0.801±0.015 0.825±0.035 0.833±0.025
CLOPS 0.684±0.025 0.733±0.025 0.802±0.017 0.824±0.036 0.830±0.023
**ACCTS** **0.690±0.032** **0.734±0.038** **0.812±0.022** **0.828±0.036** **0.835±0.024**


UCR-EQ

USHCN

COVID-19

SEPSIS


-----

*k% means the current classification time is k% of the total time of the full-length time series;Table 5: Classification Accuracy (AUC-ROC↑) for 4 Real-world Datasets at the Last 5 Time Steps.
Bold font indicates the highest accuracy.

Dataset Method 60% 70% 80% 90% 100%


LSTM 0.874±0.012 0.913±0.034 0.909±0.014 0.919±0.008 0.924±0.012
SR 0.888±0.017 0.924±0.010 0.928±0.105 0.936±0.103 0.941±0.104
ECEC 0.890±0.015 0.923±0.013 0.929±0.107 0.936±0.006 0.940±0.009
EWC 0.895±0.014 0.910±0.017 0.923±0.102 0.930±0.005 0.933±0.003
GEM 0.900±0.015 0.920±0.015 0.929±0.008 0.935±0.003 0.934±0.004
CLEAR 0.904±0.012 0.918±0.019 0.923±0.004 0.928±0.007 0.932±0.005
CLOPS 0.902±0.015 0.915±0.010 0.917±0.006 0.921±0.009 0.925±0.005
**ACCTS** **0.906±0.005** **0.928±0.007** **0.933±0.010** **0.940±0.005** **0.946±0.003**

LSTM 0.820±0.015 0.837±0.024 0.852±0.014 0.869±0.025 0.891±0.002
SR 0.836±0.016 0.886±0.023 0.902±0.013 0.921±0.026 0.933±0.009
ECEC 0.837±0.016 0.887±0.027 0.906±0.017 0.920±0.028 0.931±0.009
EWC 0.834±0.016 0.867±0.026 0.896±0.017 0.906±0.020 0.926±0.007
GEM 0.838±0.013 0.868±0.029 0.899±0.010 0.910±0.021 0.928±0.005
CLEAR 0.837±0.010 0.867±0.023 0.879±0.012 0.899±0.027 0.921±0.004
CLOPS 0.835±0.016 0.861±0.024 0.877±0.011 0.895±0.016 0.919±0.013
**ACCTS** **0.841±0.012** **0.898±0.022** **0.910±0.015** **0.928±0.013** **0.939±0.013**

LSTM 0.888±0.013 0.918±0.033 0.925±0.014 0.939±0.005 0.944±0.015
SR 0.900±0.018 0.935±0.010 0.946±0.006 0.952±0.017 0.962±0.005
ECEC 0.904±0.014 0.937±0.008 0.948±0.015 0.952±0.008 0.963±0.017
EWC 0.923±0.014 0.935±0.007 0.940±0.013 0.950±0.013 0.954±0.008
GEM 0.924±0.018 0.936±0.009 0.939±0.010 0.949±0.017 0.953±0.005
CLEAR 0.926±0.014 0.933±0.011 0.941±0.007 0.948±0.009 0.952±0.008
CLOPS 0.925±0.015 0.935±0.013 0.940±0.007 0.947±0.006 0.954±0.006
**ACCTS** **0.927±0.006** **0.955±0.008** **0.960±0.011** **0.963±0.009** **0.967±0.008**

LSTM 0.748±0.043 0.773±0.032 0.795±0.027 0.813±0.025 0.827±0.039
SR 0.827±0.037 0.835±0.013 0.845±0.014 0.859±0.022 0.866±0.023
ECEC 0.815±0.014 0.827±0.016 0.849±0.016 0.859±0.017 0.863±0.014
EWC 0.838±0.024 0.842±0.030 0.848±0.017 0.850±0.014 0.854±0.016
GEM 0.836±0.028 0.841±0.034 0.849±0.014 0.851±0.016 0.853±0.012
CLEAR 0.839±0.028 0.842±0.031 0.847±0.010 0.850±0.019 0.848±0.016
CLOPS 0.838±0.026 0.842±0.030 0.850±0.017 0.853±0.010 0.857±0.018
**ACCTS** **0.842±0.034** **0.852±0.023** **0.857±0.012** **0.866±0.014** **0.872±0.012**


UCR-EQ

USHCN

COVID-19

SEPSIS


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A.2 CLASSIFICATION ACCURACY ON SUBSETS WITH DIFFERENT SEMANTICS

Table 6: COVID-19 Classification Accuracy with Non-uniform Training Sets and Validation Sets.
_↓_ means the accuracy is greatly reduced.

Subset LSTM SR ECEC EWC

Male 0.955±0.013 0.968±0.014 0.969±0.016 0.965±0.012
Female 0.924±0.013 0.945±0.004 0.947±0.015 0.939±0.018

Age 30- 0.954±0.013 0.965±0.014 0.967±0.015 0.967±0.013
Age 30+ 0.923±0.014 0.941±0.007 0.943±0.018 0.931±0.008↓

Test 0.950±0.011 0.964±0.013 0.968±0.015 0.966±0.012
Valid. 0.944±0.014 0.962±0.006 0.963±0.014 0.954±0.003

Subset GEM CLEAR CLOPS ACCTS


Male 0.965±0.004 0.978±0.009 0.978±0.014 0.971±0.010
Female 0.938±0.003 0.919±0.008↓ 0.921±0.009↓ 0.947±0.002

Age 30- 0.964±0.009 0.977±0.008 0.979±0.012 0.972±0.010
Age 30+ 0.923±0.040↓ 0.902±0.006↓ 0.914±0.007↓ 0.945±0.006

Test 0.962±0.006 0.979±0.009 0.978±0.010 0.970±0.007
Valid. 0.953±0.005 0.952±0.009↓ 0.954±0.004↓ 0.967±0.006


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A.3 CLASSIFICATION ACCURACY ON SUBSETS WITH DIFFERENT DATA SIZE

Table 7: Classification Accuracy (AUC-ROCk% means the volume of sub dataset is k% of the corresponding original dataset;↑) for Subsets with Different Data Size.
Bold font indicates the highest accuracy;

-  means that the accuracy of ACCTS is higher 2% than this method.

Dataset Method 20% 40% 60% 80% 100%


LSTM 0.724* 0.765* 0.804* 0.809* 0.813*
SR 0.758* 0.784* 0.828* 0.813* 0.831*
ECEC 0.790 0.770* 0.815* 0.827* 0.838*
EWC 0.785 0.791* 0.833* 0.855* 0.862*
GEM 0.780 0.775* 0.840* 0.857* 0.863*
CLEAR 0.784 0.808 0.859 0.864* 0.870*
CLOPS 0.792 0.809 0.864 0.871 0.875*
**ACCTS** **0.797** **0.817** **0.872** **0.886** **0.896**

LSTM 0.701* 0.730* 0.732* 0.760* 0.763*
SR 0.731* 0.769* 0.782* 0.801* 0.805*
ECEC 0.747* 0.774 0.800* 0.807* 0.816*
EWC 0.739* 0.768* 0.810 0.817 0.826*
GEM 0.737* 0.772 0.809 0.811* 0.818*
CLEAR 0.757 0.780 0.812 0.819 0.823*
CLOPS 0.775 0.785 0.817 0.825 0.839
**ACCTS** **0.776** **0.790** **0.821** **0.835** **0.843**

LSTM 0.713* 0.730* 0.765* 0.819* 0.834*
SR 0.751* 0.767* 0.806 0.822* 0.842*
ECEC 0.755* 0.770* 0.796* 0.829* 0.856*
EWC 0.763 0.785 0.794* 0.835* 0.849*
GEM 0.769 0.772* 0.793* 0.849 0.856*
CLEAR 0.776 0.791 0.810 0.856 0.866*
CLOPS 0.775 0.789 0.809 0.848 0.874
**ACCTS** **0.781** **0.800** **0.821** **0.863** **0.888**

LSTM 0.658* 0.669* 0.691* 0.733* 0.747
SR 0.682 0.700 0.725* 0.759* 0.768
ECEC 0.679* 0.702 0.719* 0.755* 0.770
EWC 0.685 0.708 0.729* 0.768* 0.772*
GEM 0.693 0.704 0.740* 0.771* 0.781*
CLEAR 0.687 0.705 0.741 0.776 0.789
CLOPS 0.698 0.710 0.745 0.779 0.783*
**ACCTS** **0.701** **0.712** **0.760** **0.794** **0.803**


UCR-EQ

USHCN

COVID-19

SEPSIS


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