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Research Article

An Empirical Study of Machine Learning Algorithms for

Stock Daily Trading Strategy

Dongdong Lv ,1Shuhan Yuan,2Meizi Li,1,3 and Yang Xiang 1

1College of Electronics and Information Engineering, Tongji University, Shanghai 201804, China

2University of Arkansas, Fayetteville, AR 72701, USA

3College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, Shanghai 200234, China

Correspondence should be addressed to Yang Xiang; shxiangyang@tongji.edu.cn

Received 17 October 2018; Revised 3 March 2019; Accepted 19 March 2019; Published 14 April 2019

Academic Editor: Kemal Polat

Copyright © Dongdong Lv et al. is is an open access article distributed under the Creative Commons Attribution License,

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

According to the forecast of stock price trends, investors trade stocks. In recent years, many researchers focus on adopting machine

learning (ML) algorithms to predict stock price trends. However, their studies were carried out on small stock datasets with

limited features, short backtesting period, and no consideration of transaction cost. And their experimental results lack statistical

signicance test. In this paper, on large-scale stock datasets, we synthetically evaluate various ML algorithms and observe the

daily trading performance of stocks under transaction cost and no transaction cost. Particularly, we use two large datasets of

S&P index component stocks (SPICS) and CSI index component stocks (CSICS) from to and compare

six traditional ML algorithms and six advanced deep neural network (DNN) models on these two datasets, respectively. e

experimental results demonstrate that traditional ML algorithms have a better performance in most of the directional evaluation

indicators. Unexpectedly, the performance of some traditional ML algorithms is not much worse than that of the best DNN models

without considering the transaction cost. Moreover, the trading performance of all ML algorithms is sensitive to the changes of

transaction cost. Compared with the traditional ML algorithms, DNN models have better performance considering transaction

cost. Meanwhile, the impact of transparent transaction cost and implicit transaction cost on trading performance are dierent.Our

conclusions are signicant to choose the best algorithm for stock trading in dierent markets.

1. Introduction

e stock market plays a very important role in modern

economic and social life. Investors want to maintain or

increase the value of their assets by investing in the stock

of the listed company with higher expected earnings. As a

listed company, issuing stocks is an important tool to raise

funds from the public and expand the scale of the industry.

In general, investors make stock investment decisions by

predicting the future direction of stocks’ ups and downs. In

modern nancial market, successful investors are good at

making use of high-quality information to make investment

decisions, and, more importantly, they can make quick and

eective decisions based on th e information they have already

had. erefore, the eld of stock investment attracts the

attention not only of nancial practitioner and ordinary

investors but also of researchers in academic [].

In the past many years, researchers mainly constructed

statistical models to describe the time s eries of stock price and

trading volume to forecast the trends of future stock returns

[–]. It is worth noting that the intelligent computing meth-

ods represented by ML algorithms also present a vigorous

development momentum in stock market prediction with

the development of articial intelligence technology. e

main reasons are as follows. () Multisource heterogeneous

nancial data are easy to obtain, including high-frequency

trading data, rich and diverse technical indicators data,

macroeconomic data, industry policy and regulation data,

market news, and even social network data. () e research

of intelligent algorithms has been deepened. From the early

linear model, support vector machine, and shallow neural

network to DNN models and reinforcement learning algo-

rithms, intelligent computing methods have made signicant

improvement. ey have been eectively applied to the elds

Hindawi

Mathematical Problems in Engineering

Volume 2019, Article ID 7816154, 30 pages

https://doi.org/10.1155/2019/7816154

Mathematical Problems in Engineering

of image recognition and text analysis. In some papers, the

authors think that these advanced algorithms can capture the

dynamic changes of the nancial market, simulate the trading

process of stock, and make automatic investment decisions.

() e rapid development of high-performance computing

hardware, such as Graphics Processing Units (GPUs), large

servers, and other devices, can provide powerful storage

space and computing power for the use of nancial big data.

High-performance computer equipment, accurate and fast

intelligent algorithms, and nancial big data together can

provide decision-making support for programmed and auto-

mated trading of stocks, which has gradually been accepted

by industry practitioners. erefore, the power of nancial

technology is reshaping the nancial market and changing

the format of nance.

Over the years, traditional ML methods have shown

strong ability in trend prediction of stock prices [–].

In recent years, articial intelligence computing methods

represented by DNN have made a series of major break-

throughs in the elds of Natural Language Processing, image

classication, voice translation, and so on. It is noteworthy

that some DNN algorithms have been applied for time series

prediction and quantitative trading [–]. However, most

of the previous studies focused on the prediction of the

stock index of major economies in the world ([, , , ,

–,,,,],etc.)orselectingafewstockswith

limited features according to their own preferences ([–, ,

, , , , ], etc.) or not considering transaction cost

([,,,],etc.),ortheperiodofbacktestingisvery

short([,,,,,,,],etc.).Meanwhile,thereis

no statistical signicance test between dierent algorithms

which were used in stock trading ([–, ], etc.). at is, the

comparison and evaluation of the various trading algorithms

lack large-scale stocks datasets, considering transaction cost

and statistical signicance test. erefore, the performance of

backtesting may tend to be overly optimistic. In this regard,

we need to clarify two concerns based on a large-scale stock

dataset: () whether the trading strategies based on the DNN

models can achieve statistically signicant results compared

with the traditional ML algorithms without transaction cost;

() how do transaction costs aect trading performance

of the ML algorithm? ese problems constitute the main

motivation of this research and they are very important

for quantitative investment practitioners and portfolio man-

agers. ese solutions of these problems are of great value for

practitioners to do stock trading.

In this paper, we select SPICS and CSICS from

to as research objects. e SPICS and CSICS

represent the industry development of the world’s top two

economies and are attractive to investors around the world.

e stock symbols are shown in the “Data Availability”. For

each stock in SPICS and CSICS, we construct technical

indicators as shown in the “Data Availability”. e label

on the -th trading day is the symbol for the yield of

the +1-th trading day relative to the -th trading day.

at is, if the yield is positive, the label value is set to ,

otherwise . For each stock, we choose tec hnical indicators

of trading days before December , , to build

a stock dataset. Aer the dataset of a stock is built, we

choose the walk-forward analysis (WFA) method to train

the ML models step by step. In each step of training, we

use traditional ML methods which are support vector

machine (SVM), random forest (RF), logistic regression (LR),

na¨

ıve Bayes model (NB), classication and regression tree

(CART), and eXtreme Gradient Boosting algorithm (XGB)

and DNN models which are widely in the eld of text

analysis and voice translation such as Multilayer Perceptron

(MLP), Deep Belief Network (DBN), Stacked Autoencoders

(SAE), Recurrent Neural Network (RNN), Long Short-Term

Memory (LSTM), and Gated Recurrent Unit (GRU) to

train and forecast the trends of stock price based on the

technical indicators. Finally, we use the directional evaluation

indicators such as accuracy rate (AR), precision rate (PR),

recall rate (RR),F-Score (F), Area Under Curve (AUC), and

the performance evaluation indicators such as winning rate

(WR), annualized return rate (ARR), annualized Sharpe ratio

(ASR), and maximum drawdown (MDD)) to evaluate the

trading performance of these various algorithms or strategies.

From the experiments, we can nd that the traditional ML

algorithms have a better performance than DNN algorithms

in all directional evaluation indicators except for PR in

SPICS; in CSICS, DNN algorithms have a better performance

in AR, PR, and F expert for RR and AUC. () Trading

performance without transaction cost is as follows: the WR

of traditional ML algorithms have a better performance than

those of DNN algorithms in both SPICS and CSICS. e

ARR and ASR of all ML algorithms are signicantly greater

than those of the benchmark index (S&P index and

CSI index) and BAH strategy; the MDD of all ML

algorithms are signicantly greater than that of BAH strategy

and are signicantly less than that of the benchmark index.

In all ML algorithms, there are always some traditional ML

algorithms whose trading performance (ARR, ASR, MDD)

can be comparable to the best DNN algorithms. erefore,

DNN algorithms are not always the best choice, and the

performance of some traditional ML algorithms has no

signicant dierence from that of DNN algorithms; even

those traditional ML algorithms can perform well in ARR

and ASR. () Trading performance with transaction cost

is as follows: the trading performance (WR, ARR, ASR,

and MDD) of all machine learning algorithms is decreasing

with the increase of transaction cost as in actual trading

situation. Under the same transaction cost structure, the

performance reductions of DNN algorithms, especially MLP,

DBN, and SAE, are smaller than those of traditional ML

algorithms, which shows that DNN algorithms have stronger

tolerance and risk control ability to the changes of transaction

cost. Moreover, the impact of transparent transaction cost

on SPICS is greater than slippage, while the opposite is

true on CSICS. rough multiple comparative analysis of

the dierent transaction cost structures, the performance of

trading algorithms is signicantly smaller than that without

transaction cost, which shows that trading performance is

sensitive to transaction cost. e contribution of this paper

is that we use nonparametric statistical test methods to

compare dierences in trading performance for dierent

ML algorithms in both cases of transaction cost and no

transaction cost. erefore, it is helpful for us to select the

Mathematical Problems in Engineering

1. Data Acquisition

Data Source

Soware

2. Data Preparation

EX Right/Dividend

Feature Generation

Data Normalization

3. Learning

Algorithm

Machine Learning

Algorithms

Walk-Forward

Training/Prediction

Algorithm Design of

Trading Signals

4. Performance

Calculation

Directional

Evaluation Indicators

Performance

Evaluation Indicators

Back-testing

Algorithms

5. e Experimental

Results

Statistical Testing

Method

Trading Evaluation

without Transaction

Cost

Trading Evaluation

with Transaction Cost

F : e framework for predicting stock price trends based on ML algorithms.

most suitable algorithm from these ML algorithms for stock

trading both in the US stock market and the Chinese A-share

market.

e remainder of this paper is organized as follows:

Section describes the architecture of this work. Section

gives the parameter settings of these ML models and the

algorithm for generating trading signals based on the ML

models mentioned in this paper. Section gives the direc-

tional evaluation indicators, performance evaluation indi-

cators, and backtesting algorithms. Section uses nonpa-

rameter statistical test methods to analyze and evaluate the

performance of these dierent algorithms in the two markets.

Section gives the analysis of impact of transaction cost

on performance of ML algorithms for trading. Section

gives some discussions of dierences in trading performance

among dierent algorithms from the perspective of data,

algorithms, transaction cost, and suggestions for algorithmic

trading. Section provides a comprehensive conclusion and

future research directions.

2. Architecture of the Work

e general framework of predicting the future price trends

of stocks, trading process, and backtesting based on ML

algorithms is shown in Figure . is article is organized

from data acquisition, data preparation, intelligent learning

algorithm, and trading performance evaluation. In this study,

data acquisition is the rst step. Where should we get data

and what soware should we use to get data quickly and

accurately are something that we need to consider. In this

paper, we use R language to do all computational procedures.

Meanwhile, we obtain SPICS and CSICS from Yahoo nance

and Netease Finance, respectively. Secondly, the task of

data preparation includes ex-dividend/rights for the acquired

data, generating a large number of well-recognized technical

indicators as features, and using max-min normalization to

deal with the features, so that the preprocessed data can

be used as the input of ML algorithms []. irdly, the

trading signals of stocks are generated by the ML algorithms.

In this part, we train the DNN models and the traditional

ML algorithms by a WFA method; then the trained ML

models will predict the direction of the stocks in a future

time which is considered as the trading signal. Fourthly, we

give some widely used directional evaluation indicators and

performance evaluation indicators and adopt a backtesting

algorithm for calculating the indicators. Finally, we use the

trading signal to implement the backtesting algorithm of

stock daily trading strategy and then apply statistical test

method to evaluate whether there are statistical signicant

dierences among the performance of these trading algo-

rithms in both cases of transaction cost and no transaction

cost.

3. ML Algorithms

3.1. ML Algorithms and eir Parameter Settings. Given a

training dataset D, the task of ML algorithm is to classify

classlabelscorrectly.Inthispaper,wewillusesixtraditional

ML models (LR, SVM, CART, RF, BN, and XGB) and six

DNN models (MLP, DBN, SAE, RNN, LSTM, and GRU) as

classiers to predict the ups and downs of the stock prices

[]. e main model parameters and training parameters of

these ML learning algorithms are shown in Tables and .

In Tables and , features and class labels are set according

to the input format of various ML algorithms in R language.

Matrix (m, n) represents a matrix with mrows and ncolumns;

Array (p,m,n) represents a tensor and each layer of the

tensor is Matrix (m,n) and the height of the tensor is p.c

(h1,h2,h3,...) represents a vector, where the length of the

vector is the number of hidden layers and the -th element

of cis the number of neurons of the -th hidden layer. In

the experiment, = represents that we use the data of

the past trading days as training samples in each round

of WFA; = represents that the data of each day has

features. In Table , the parameters of DNN models such as

activation function, learning rate, batch size, and epoch are

all default values in the algorithms of R programs.

3.2. WFA Method. WFA [] isa rolling training method. We

use the latest data instead of all past data to train the model

Mathematical Problems in Engineering

T : Main parameter settings of traditional ML algorithms.

Input Features Label Main parameters

LR Matrix(,) Matrix(,) A specication for the model link function is logit.

SVM Matrix(,) Matrix(,) e kernel function used is Radial Basis kernel; Cost of constraints violation is .

CART Matrix(,) Matrix(,) e maximum depth of any node of the nal tree is ; e splitting index can be Gini coecient.

RF Matrix(,) Matrix(,) e Number of trees is ; Number of variables randomly sampled as candidates at each split is .

BN Matrix(,) Matrix(,) the prior probabilities of class membership is the class proportions for the training set.

XGB Matrix(,) Matrix(,) e maximum depth of a tree is ; the max number of iterations is ; the learning rate is ..

T : Main parameter settings of DNN algorithms.

Input Features Label Learning rate Dimensions of hidden layers Activation function Batch size Epoch

MLP Matrix(,) Matrix(,) . c(,,,) sigmoid

DBN Matrix(,) Matrix(,) . c(,,,) sigmoid

SAE Matrix(,) Matrix(,) . c(,,) sigmoid

RNN Array(,,) Array(,,) . c(,) sigmoid

LSTM Array(,,) Array(,,) . c(,) sigmoid

GRU Array(,,) Array(,,) . c(,) sigmoid

and then apply the trained model to implement the prediction

for the out-of-sample data (testing dataset) of the future time

period. Aer that, a new training set, which is the previous

training set walk one step forward, is carried out the training

of the next round. WFA can improve the robustness and the

condence of the trading strategy in real-time trading.

In this paper, we use ML algorithms and the WFA method

to do stock price trend predictions as trading signals. In each

step,weusethedatafromthepastdays(oneyear)asthe

training set and the data for the next days (one week) as

the test set. Each stock contains data of , trading days,

so it takes (-)/ = training sessions to produce a

total of , predictions which are the trading signals of daily

trading strategy. e WFA method is as shown in Figure .

3.3. e Algorithm Design of Trading Signal. In this part, we

use ML algorithms as classiers to predict the ups and downs

of the stock in SPICS and CSICS and then use the prediction

resultsastradingsignalsofdailytrading.WeusetheWFA

method to train each ML algorithm. We give the generating

algorithm of trading signals according to Figure , which is

shown in Algorithm .

4. Evaluation Indicators and

Backtesting Algorithm

4.1. Directional Evaluation Indicators. In this paper, we use

ML algorithms to predict the direction of stock price, so

the main task of the ML algorithms is to classify returns.

erefore, it is necessary for us to use directional evaluation

indicators to evaluate the classication ability of these algo-

rithms.

e actual label values of the dataset are sequences of

sets {DOWN,UP}. erefore, there are four categories of

predicted label values and actual label values, which are

expressed as TU, FU, FD, and TD. TU denotes the number of

UP that the actual label values are UP and the predicted label

T : Confusion matrix of two classication results of ML

algorithm.

Predicted label values

UP DOWN

Actual label values UP TU FD

DOWN FU TD

values are also UP; FU denotes the number of UP that the

actual label values are DOWN but the predicted label values

are UP; TD denotes the number of DOWN that the actual

label values are DOWN and the predicted label values are

DOWN; FD denotes the number of DOWN that the actual

label values are UP but the predicted label values are DOWN,

as shown in Table . Table is a two-dimensional table called

confusion matrix. It classies predicted label values according

to whether predicted label values match real label values. e

rst dimension of the table represents all possible predicted

label values and the second dimension represents all real label

values. When predicted label values equal real label values,

they are correct classications. e correct prediction label

values lie on the diagonal line of the confusion matrix. In

this paper, what we are concerned about is that when the

direction of stock price is predicted to be UP tomorrow, we

buy the stock at today’s closing price and sell it at tomorrow’s

closing price; when we predict the direction of stock price to

beDOWNtomorrow,wedonothing.SoUPisa“positive”

label of our concern.

In most of classication tasks, AR is generally used

to evaluate performance of classiers. AR is the ratio of

the number of correct predictions to the total number of

predictions.atisasfollows.

= ( + )

(+++

)()

Mathematical Problems in Engineering

ML

Algorithm

44-dim

2000-dim

44-dim

44-dim

44-dim

44-dim

44-dim

44-dim

250-dim

5-dim250-dim

5-dim

250-dim

5-dim

1-dim

1-dim

1-dim

1-dim

5-dim5-dim5-dim

1750-dim

...

...

Concatenate

ML

Algorithm

ML

Algorithm

Raw Input Data

F : e schematic diagram of WFA (training and testing).

Input:StockSymbols

Output:TradingSignals

() N=Length of Stock Symbols

() L=Length of Trading Days

() P=Length of Features

() k= Length of Training Dataset for WFA

() n= Length of Sliding Window for WFA

() for (i in : N) {

() Stock=Stock Symbols[i]

() M=(L-k)/n

() Trading Signal=NULL

() for (j in :M) {

() Dataset= Stock[(k+n∗(j-)):(k+n+n∗(j-)), :(P+)]

() Train=Dataset[:k,:(+P)]

() Test= Dataset[(k+):(k+n),:P]

() Model=ML Algorithm(Train)

() Probability=Model(Test)

() if (Probability>=.) {

() Trading Signal=

() }else {

() Trading Signal=

() }

() }

() Trading Signal=c (Trading Signal, Trading Signal)

() }

() return (Trading Signal)

A : Generating trading signal in R language.

Mathematical Problems in Engineering

In this paper, “UP” is the prot source of our trading

strategies. e classication ability of ML algorithm is to eval-

uate whether the algorithms can recognize “UP”. erefore,

it is necessary to use PR and RR to evaluate classication

results. ese two evaluation indicators are initially applied

in the eld of information retrieval to evaluate the relevance

of retrieval results.

PR is a ratio of the number of correctly predicted UP to

all predicted UP. at is as follows.

=

( + )()

High PR means that ML algorithms can focus on “UP”

rather than “DOWN”.

RR is the ratio of the number of correctly predicted “UP”

to the number of actually labeled “UP”. at is as follows.

=

( + )()

High RR can capture a large number of “UP” and be

eectively identied. In fact, it is very dicult to present an

algorithm with high PR and RR at the same time. erefore,

it is necessary to measure the classication ability of the

ML algorithm by using some evaluation indicators which

combine PR with RR. F-Score is the harmonic average of

PR and AR. F is a more comprehensive evaluation indicator.

at is as follows.

1=2∗∗

( + )()

Here,itisassumedthattheweightsofPRandRRareequal

when calculating F, but this assumption is not always correct.

It is feasible to calculate F with dierent weights for PR and

RR, but determining weights is a very dicult challenge.

AUC is the area under ROC (Receiver Operating Charac-

teristic) curve. ROC curve is oen used to check the tradeo

between nding TU and avoiding FU. Its horizontal axis

is FU rate and its vertical axis is TU rate. Each point on

the curve represents the proportion of TU under dierent

FU thresholds []. AUC reects the classication ability of

classier. e larger the value, the better the classication

ability. It is worth noting that two dierent ROC curves may

lead to the same AUC value, so qualitative analysis should be

carried out in combination with the ROC curve when using

AUC value. In this paper, we use R language package “ROCR”

to calculate AUC.

4.2. Performance Evaluation Indicator. Performance evalua-

tion indicator is used for evaluating the protability and risk

control ability of trading algorithms. In this paper, we use

trading signals generated by ML algorithms to conduct the

backtestingandapplytheWR,ARR,ASR,andMDDtodo

the trading performance evaluation []. WR is a measure

of the accuracy of trading signals; ARR is a theoretical rate

of return of a trading strategy; ASR is a risk-adjusted return

which represents return from taking a unit risk [] and the

risk-free return or benchmark is set to in this paper; MDD

is the largest decline in the price or value of the investment

period, which is an important risk assessment indicator.

4.3. Backtesting Algorithm. Using historical data to imple-

ment trading strategy is called backtesting. In research and

the development phase of trading model, the researchers

usually use a new set of historical data to do backtesting. Fur-

thermore, the backtesting period should be long enough,

because a large number of historical data can ensure that the

trading model can minimize the sampling bias of data. We

can get statistical performance of trading models theoretically

by backtesting. In this paper, we get trading signals for

each stock. If tomorrow’s trading signal is , we will buy the

stock at today’s closing price and then sell it at tomorrow’s

closing price; otherwise, we will not do stock trading. Finally,

we get AR, PR, RR, F, AUC, WR, ARR, ASR, and MDD by

implementing backtesting algorithm based on these trading

signals.

5. Comparative Analysis of

Different Trading Algorithms

5.1. Nonparametric Statistical Test Method. In this part, we

use the backtesting algorithm(Algorithm ) to calculate the

evaluation indicators of dierent trading algorithms. In order

to test whether there are signicant dierences between

the evaluation indicators of dierent ML algorithms, the

benchmark indexes, and the BAH strategies, it is necessary

to use analysis of variance and multiple comparisons to give

the answers. erefore, we propose the following nine basic

hypotheses for signicance test in which Hja (=,,,,

, , , , ) are the null hypothesis, and the corresponding

alternative assumptions are Hjb (=,,,,,,,,).e

level of signicance is ..

For any evaluation indicator ∈ {,,, 1,,

,,,}and any trading strategy ∈{,

,,,, ,,, ,,,

,, }, the null hypothesis a is Hja,

alternative hypotheses b is Hjb (=,,,,,,,,

represent AR, PR, RR, F, AUC, WR, ARR, ASR, MDD,

respectively.).

Hja: the evaluation indicator jof all strategies are the

same

Hjb: the evaluation indicator jof all strategies are not

the same

It is worth noting that any evaluation indicator of all

trading algorithm or strategy does not conform to the basic

hypothesis of variance analysis. at is, it violates the assump-

tion that the variances of any two groups of samples are the

same and each group of samples obeys normal distribution.

erefore, it is not appropriate to use t-test in the analysis

of variance, and we should take the nonparametric statistical

test method instead. In this paper, we use the Kruskal-Wallis

rank sum test [] to carry out the analysis of variance. If the

alternative hypothesis is established, we will need to further

applytheNemenyitest[]todothemultiplecomparisons

between trading strategies.

5.2. Comparative Analysis of Performance of Dierent Trading

Strategies in SPICS. Table shows the average value of

Mathematical Problems in Engineering

Input: TS TS is trading signals of a stock.

Output: AR, PR, RR, F, AUC, WR, ARR, ASR, MDD

() N=length of Stock Code List SPICS, and CSICS.

() Bt=Benchmark Index [“Closing Price”] B is the closing price of benchmark index.

() WR=NULL; ARR=NULL; ASR=NULL; MDD=NULL

() for (i in : N) {

() Stock Data=Stock Code List[i]

() Pt=Stock Data [“Closing Price”]

() Labelt=Stock Data [“Label”]

() BDRRt=(Bt-Bt-1)/ Bt-1 BDRR is the daily return rate of benchmark index.

() DRRt=(P

t-Pt-1)/Pt-1DRR is daily return rate. at is daily return rate of BAH strategy.

() TDRRt=lag (TSt)∗DRRtTDRR is the daily return through trading.

() Table=Confusion Matrix(TS, Label)

() AR[i]=sum(adj(Table))/sum(Table)

() PR[i]=Table[2,2]/sum(Table[,2])

() RR[i]=Table[2, 2]/sum(Table[2,])

() F=∗PR[i]∗RR[i]/(PR[i]+RR[i])

() Pred=prediction (TS, Label)

() AUC[i]=performance (Pred, measure=“auc”)@y.values[[1]]

() WR[i]=sum (TDRR>)/sum(TDRR =)

() ARR[i]=Return.annualized (TDRR) TDRR, BDRR, or DRR can be used.

() ASR[i]=SharpeRatio.annualized (TDRR) TDRR, BDRR, or DRR can be used.

() MDD[i]=maxDrawDown (TDRR) TDRR, BDRR, or DRR can be used.

() AR=c (AR, AR[i])

() PR=c (PR, PR[i])

() RR=c (RR, RR[i])

() F=c (F, F[i])

() AUC=c (AUC, AUC[i])

() WR=c (WR, WR[i])

() ARR=c (ARR, ARR[i])

() ASR=c (ASR, ASR[i])

() MDD=c (MDD, MDD[i])

() }

() Performance=cbind(AR,PR,RR,F,AUC,WR,ARR,ASR,MDD)

() return (Performance)

A : Backtesting algorithm of daily trading strategy in R language.

T : Trading performance of dierent trading strategies in the SPICS. Best performance of all trading strategies is in boldface.

Index BAH MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM XGB

AR — — . . . . . . . . . . . 0.6600

PR — — 0.7861 . . . . . . . . . . .

RR — — . . . . . . . . . . . 0.6767

F — — . . . . . . . . . . . 0.6751

AUC — — . . . . . . . . . . . 0.6590

WR . . . . . . . . . 0.5930 . . . .

ARR . . 0.3333 . . . . . . . . . . .

ASR . . . . . . . . . . 1.6768 . . .

MDD 0.1939 . . . . . . . . . . . . .

various trading algorithms in AR, PR, RR, F, AUC, WR,

ARR, ASR, and MDD. We can see that the AR, RR, F, and

AUC of XGB are the greatest in all trading algorithms. e

WR of NB is the greatest in all trading strategies. e ARR

of MLP is the greatest in all trading strategies including the

benchmark index (S&P index) and BAH strategy. e

ASR of RF is the greatest in all trading strategies. e MDD of

the benchmark index is the smallest in all trading strategies.

It is worth noting that the ARR and ASR of all ML algorithms

are greater than those of BAH strategy and the benchmark

index.

() rough the hypothesis test analysis of Ha and Hb,

we can obtain p value<.e-.

erefore, there are statistically signicant dierences

between the AR of all trading algorithms. erefore, we need

to make multiple comparative analysis further, as shown in

Mathematical Problems in Engineering

T : Multiple comparison analysis between the AR of any two trading algorithms. e p value of the two trading strategies with signicant

dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN 0.0000 0.0000 0.0000

LSTM 0.0000 0.0000 0.0000 .

GRU 0.0000 0.0000 0.0000 . .

CART 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

NB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

RF 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0232 0.0000

LR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 .

SVM 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 . 0.0000 0.0000 0.0000

XGB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 . 0.0000

T : Multiple comparison analysis between the PR of any two trading algorithms. e p value of the two trading strategies with signicant

dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN 0.0000 0.0000 0.0000

LSTM 0.0000 0.0000 0.0000 0.0034

GRU 0.0000 0.0000 0.0000 0.0000 .

CART 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

NB 0.0000 0.0000 0.0000 . . 0.0000 0.0000

RF 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 . 0.0000

LR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 . 0.0000 .

SVM 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0008 0.0000 . 0.0000

XGB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0491 0.0000 .

Table . e number in the table is a p value of any two algo-

rithms of Nemenyi test. When p value<., we think that

the two trading algorithms have a signicant dierence,

otherwise we cannot deny the null assumption that the mean

values of AR of the two algorithms are equal. From Tables

and , we can see that the AR of all DNN models are signif-

icantly lower than those of all traditional ML models. e AR

of MLP, DBN, and SAE are signicantly greater than those of

RNN, LSTM, and GRU. ere are no signicant dierences

among the AR of MLP, DBN, and SAE. ere are no sig-

nificant dierences among the AR of RNN, LSTM, and GRU.

() rough the hypothesis test analysis of Ha and Hb,

we can obtain p value<.e-. So, there are statistically sig-

nicant dierences between the PR of all trading algorithms.

erefore, we need to make multiple comparative analysis

further, as shown in Table . e number in the table is a p

valueofanytwoalgorithmsofNemenyitest.FromTables

and , we can see that the PR of MLP, DBN, and SAE are

signicantly greater than that of other trading algorithms.

e PR of LSTM is not signicantly dierent from that of

GRU and NB. e PR of GRU is signicantly lower than that

of all traditional ML algorithms. e PR of NB is signicantly

lower than that of other traditional ML algorithms.

() rough the hypothesis test analysis of Ha and Hb,

we can obtain p value<.e-. So, there are statistically

signicant dierences between the RR of all trading algo-

rithms erefore, we need to make multiple comparative

analysis further, as shown in Table . e number in the

table is a p value of any two algorithms of Nemenyi test.

From Tables and , we can see that there is no signicant

dierence among the RR of all DNN models, but the RR

of any DNN model is signicantly lower than that of all

traditional ML models. e RR of NB is signicantly lower

than that of other traditional ML algorithms. e RR of

CART is signicantly lower than that of other traditional ML

algorithms except for NB.

() rough the hypothesis test analysis of Ha and Hb,

we can obtain p value<.e-. So, there are statistically sig-

nicant dierences between the F of all trading algorithms.

erefore, we need to make multiple comparative analysis

further, as shown in Table . e number in the table is a p

valueofanytwoalgorithmsofNemenyitest.FromTables

and , we can see that there is no signicant dierence

among the F of MLP, DBN, and SAE. e F of MLP, DBN,

and SAE are signicantly greater than that of RNN, LSTM,

GRU, and NB, but are signicantly smaller than that of RF, LR,

SVM, and XGB. e F of GRU and LSTM have no signicant

dierence, but they are signicantly smaller than that of all

traditional ML algorithms. e F of XGB is signicantly

greater than that of all other trading algorithms.

Mathematical Problems in Engineering

T : Multiple comparison analysis between the RR of any two trading algorithms. e p value of the two trading strategies with signicant

dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN . . .

LSTM . . . .

GRU . . . . .

CART 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

NB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

RF 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0485 0.0000

LR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 .

SVM 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0197 0.0000 0.0000 0.0000

XGB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0010 . 0.0000

T : Multiple comparison analysis between the F of any two trading algorithms. e p value of the two trading strategies with signicant

dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN 0.0000 0.0000 0.0000

LSTM 0.0000 0.0000 0.0000 .

GRU 0.0000 0.0000 0.0000 0.0000 .

CART . 0.0061 0.0117 0.0000 0.0000 0.0000

NB 0.0000 0.0000 0.0000 . 0.0000 0.0000 0.0000

RF 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0078 0.0000

LR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0173 0.0000 .

SVM 0.0007 0.0000 0.0000 0.0000 0.0000 0.0000 . 0.0000 . .

XGB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

T : Multiple comparison analysis between the AUC of any two trading algorithms. e p value of the two trading strategies with

signicant dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN . . .

LSTM . . . .

GRU . . . . .

CART 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

NB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

RF 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0270 0.0000

LR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 .

SVM 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 . 0.0000 0.0000 0.0000

XGB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 . 0.0000

() rough the hypothesis test analysis of Ha and Hb,

we can obtain p value<.e-. So, there are statistically

signicant dierences between the AUC of all trading algo-

rithms. erefore, we need to make multiple comparative

analysis further, as shown in Table . e number in the

table is a p value of any two algorithms of Nemenyi test.

From Tables and , we can see that there is no signicant

dierence among the AUC of all DNN models. e AUC of

all DNN models are signicantly smaller than that of any

traditional ML model.

() rough the hypothesis test analysis of Ha and Hb,

we can obtain p value<.e-. So, there are statistically sig-

nicant dierences between the WR of all trading algorithms.

erefore, we need to make multiple comparative analysis

further,asshowninTable.enumberinthetableisp

value of any two algorithms of Nemenyi test. From Tables

Mathematical Problems in Engineering

T : Multiple comparison analysis between the WR of any two trading algorithms. e p value of the two trading strategies with signicant dierenceisinboldface.

Index BAH MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

BAH 0.0000

MLP 0.0000 0.0000

DBN 0.0000 0.0000 .

SAE 0.0000 0.0000 . .

RNN 0.0000 0.0000 0.0000 0.0000 0.0000

LSTM 0.0000 0.0000 0.0000 0.0000 0.0000 .

GRU 0.0011 0.0000 0.0001 0.0000 0.0000 . .

CART 0.0000 . 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

NB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0031 0.0000 0.0038 0.0000

RF 0.0000 0.0000 0.0000 0.0000 0.0000 0.0118 0.0001 0.0140 0.0000 .

LR 0.0000 0.0000 0.0000 0.0000 0.0000 . . . 0.0000 0.0432 .

SVM 0.0000 0.0000 0.0000 0.0000 0.0000 . . . 0.0000 0.0001 0.0006 .

XGB 0.0000 0.0000 0.0000 0.0000 0.0000 . 0.0084 . 0.0000 . . . 0.0376

Mathematical Problems in Engineering

T : Multiplecomparison analysis between the ARR of any two trading strategies.e p value of the two trading strategies with signicant

dierence is in boldface.

Index BAH MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

BAH 0.0000

MLP 0.0000 0.0000

DBN 0.0000 0.0000 .

SAE 0.0000 0.0000 . .

RNN 0.0000 0.0000 0.0001 0.0006 0.0001

LSTM 0.0000 0.0000 0.0000 0.0002 0.0000 .

GRU 0.0000 0.0000 0.0001 0.0008 0.0001 . .

CART 0.0000 0.0000 . . . 0.0001 0.0000 0.0001

NB 0.0000 0.0000 0.0021 0.0094 0.0022 . . . 0.0018

RF 0.0000 0.0000 . . . . . . . .

LR 0.0000 0.0000 0.0002 0.0012 0.0002 . . . 0.0002 . .

SVM 0.0000 0.0000 . . . . . . . . . .

XGB 0.0000 0.0000 . . . . . . . . . . .

T : Multiple comparison analysis betweenthe ASR of any two trading strategies. e p value of the two trading strategieswith signicant

dierence is in boldface.

Index BAH MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

BAH .

MLP 0.0000 0.0000

DBN 0.0000 0.0000 .

SAE 0.0000 0.0000 . .

RNN 0.0000 0.0000 . . .

LSTM 0.0000 0.0000 . . . .

GRU 0.0000 0.0000 . . . . .

CART 0.0000 0.0000 0.0002 0.0005 0.0002 0.0000 0.0000 0.0000

NB 0.0000 0.0000 0.0467 0.0233 . . . . 0.0000

RF 0.0000 0.0000 0.0000 0.0000 0.0000 0.0291 0.0042 . 0.0000 .

LR 0.0000 0.0000 . . . . . . 0.0000 . .

SVM 0.0000 0.0000 . . . . . . 0.0000 . . .

XGB 0.0000 0.0000 0.0099 0.0044 0.0122 . . . 0.0000 . . . .

and , we can see that the WR of MLP, DBN, and SAE have

no signicant dierence, but they are signicantly higher

than that of BAH and benchmark index, and signicantly

lower than that of other trading algorithms. e WR of RNN,

LSTM, and GRU have no signicant dierence, but they are

signicantly higher than that of CART and signicantly lower

than that of NB and RF. e WR of LR is not signicantly

dierent from that of RF, SVM, and XGB.

() rough the analysis of the hypothesis test of Ha

and Hb, we obtain p value<.e-. erefore, there are

signicant dierences between the ARR of all trading strate-

gies including the benchmark index and BAH. We need

to do further multiple comparative analysis, as shown in

Table . F r o m Ta b l e s and , w e c a n s ee t h at t h e A RR of

the benchmark index and BAH are signicantly lower than

that of all ML algorithms. e ARR of MLP, DBN, and SAE

are signicantly greater than that of RNN, LSTM, GRU, NB,

and LR, but not signicantly dierent from that of CART,

RF, SVM, and XGB; there is no signicant dierence between

the ARR of MLP, DBN, and SAE. e ARR of RNN, LSTM,

and GRU are signicantly less than that of CART, but they

are not signicantly dierent from that of other traditional

ML algorithms. In all traditional ML algorithms, the ARR of

CART is signicantly greater than that of NB and LR, but,

otherwise, there is no signicant dierence between ARR of

any other two algorithms.

() rough the hypothesis test analysis of Ha and Hb,

we obtain p value<.e-. erefore, there are signicant

dierences between ASR of all trading strategies including

the benchmark index and BAH. e results of our multiple

comparative analysis are shown in Table . From Tables

and , we can see that the ASR of the benchmark index and

BAH are signicantly smaller than that of all ML algorithms.

e ASR of MLP and DBN are signicantly greater than that

of CART and are signicantly smaller than that of NB, RF,

and XGB, but there is no signicant dierence between MLP,

DBN, and other algorithms. e ASR of SAE is signicantly

greater than that of CART and signicantly less than that of

RF and XGB, but there is no signicant dierence between

SAE and other algorithms. e ASR of RNN and LSTM

Mathematical Problems in Engineering

T : Multiple comparison analysis between the MDD of any two trading strategies. e p value of the two trading strategies with

signicant dierence is in boldface.

Index BAH MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

BAH 0.0000

MLP 0.0000 0.0052

DBN 0.0000 0.0031 .

SAE 0.0000 0.0012 . .

RNN 0.0000 0.0000 . . .

LSTM 0.0000 0.0000 . . . .

GRU 0.0000 0.0000 0.0245 0.0381 . . .

CART 0.0000 0.0000 . . . . . .

NB 0.0000 0.0000 . . . . . . .

RF 0.0000 0.0000 0.0002 0.0004 0.0012 . . . . .

LR 0.0000 0.0000 . . . . . . . . .

SVM 0.0000 0.0000 . . . . . . . . . .

XGB 0.0000 0.0000 0.0103 0.0167 0.0360 . . . . . . . .

T : Trading performance of dierent trading strategies in CSICS. Best performance of all trading strategies is in boldface.

Index BAH MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM XGB

AR — — 0.5175 . . . . . . . . . . .

PR — — 0.7548 . . . . . . . . . . .

RR — — . . . . . . . . . 0.5318 . .

F — — 0.6150 . . . . . . . . . . .

AUC — — . . . . . . . . . 0.5086 . .

WR . . . . . . . . . . . 0.5809 . .

ARR . . . . . . . . . 0.6125 . . . .

ASR . . . . . . . . . . . 1.5582 . .

MDD 0.4808 . . . . . . . . . . . . .

are signicantly greater than that of CART and signicantly

less than that of RF, but there is no signicant dierence

between RNN, LSTM, and other algorithms. e ASR of GRU

is signicantly greater than that of CART, but there is no

signicant dierence between GRU and other traditional ML

algorithms. In all traditional ML algorithms, the ASR of all

algorithms are signicantly greater than that of CART, but

otherwise, there is no signicant dierence between ASR of

any other two algorithms.

() rough the hypothesis test analysis of Ha and Hb,

we obtain p value<.e-. erefore, there are signicant

dierences between MDD of trading strategies including

the benchmark index and the BAH. e results of multiple

comparative analysis are shown in Table . From Tables

and,wecanseethatMDDofanyMLalgorithmis

signicantly greater than that of the benchmark index but

signicantly smaller than that of BAH strategy. e MDD

of MLP and DBN are signicantly smaller than those of

GRU, RF, and XGB, but there is no signicant dierence

between MLP, DBN, and other algorithms. e MDD of

SAE is signicantly smaller than that of XGB, but there is

no signicant dierence between SAE and other algorithms.

Otherwise, there is no signicant dierence between MDD of

any other two algorithms.

In a word, the traditional ML algorithms such as NB,

RF, and XGB have good performance in most directional

evaluation indicators such as AR, PR, and F. e DNN

algorithms such as MLP have good performance in PR and

ARR. In traditional ML algorithms, the ARR of CART, RF,

SVM, and XGB are not signicantly dierent from that of

MLP, DBN, and SAE; the ARR of CART is signicantly

greater than that of LSTM, GRU, and RNN, but otherwise

the ARR of all traditional ML algorithms are not signicantly

worse than that of LSTM, GRU, and RNN. e ASR of all

traditional ML algorithms except CART are not signicantly

worse than that of the six DNN models; even the ASR of NB,

RF, and XGB are signicantly greater than that of some DNN

algorithms. e MDD of RF and XGB are signicantly less

that of MLP, DBN, and SAE; the MDD of all traditional ML

algorithms are not signicantly dierent from that of LSTM,

GRU, and RNN. e ARR and ASR of all ML algorithms are

signicantly greater than that of BAH and the benchmark

index; the MDD of any ML algorithm is signicantly greater

than that of the benchmark index, but signicantly less than

that of BAH strategy.

5.3. Comparative Analysis of Performance of Dierent Trading

Strategies in CSICS. eanalysismethodsofthispartare

similar to Section .. From Table , we can see that the AR,

PR, and F of MLP are the greatest in all trading algorithms.

e RR, AUC, WR, and ASR of LR are the greatest in

all trading algorithms, respectively. e ARR of NB is the

Mathematical Problems in Engineering

T : Multiplecomparison analysis between the AR of any two trading algorithms. e p value of the twotrading strategies with signicant

dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN 0.0000 0.0000 0.0000

LSTM 0.0000 0.0000 0.0000 .

GRU 0.0000 0.0000 0.0000 . .

CART 0.0000 0.0000 0.0000 . 0.0024 0.0131

NB 0.0000 0.0001 0.0002 0.0022 0.0000 0.0000 .

RF 0.0000 0.0002 0.0005 0.0007 0.0000 0.0000 . .

LR 0.0000 0.0000 0.0000 0.0076 0.0000 0.0000 . . .

SVM 0.0217 . . 0.0000 0.0000 0.0000 0.0003 . . .

XGB 0.0000 0.0001 0.0001 0.0025 0.0000 0.0000 . . . . .

T : Multiplecomparison analysis between the PR ofany two trading algorithms. e p value of the two trading strategies with signicant

dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN 0.0000 0.0000 0.0000

LSTM 0.0000 0.0000 0.0000 0.0000

GRU 0.0000 0.0000 0.0000 0.0000 .

CART 0.0000 0.0000 0.0000 0.0000 . .

NB 0.0000 0.0000 0.0000 0.0000 . . .

RF 0.0000 0.0000 0.0000 0.0000 0.0319 0.0205 . .

LR 0.0000 0.0000 0.0000 0.0000 . . . . .

SVM 0.0000 0.0000 0.0000 . 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

XGB 0.0000 0.0000 0.0000 0.0000 . . . . . . 0.0000

highest in all trading strategies. e MDD of CSI index

(benchmark index) is the smallest in all trading strategies.

e WR, ARR, and ASR of all ML algorithms are greater than

those of the benchmark index and BAH strategy.

() rough the hypothesis test analysis of Ha and Hb,

wecanobtainpvalue<.e-. erefore, there are signicant

dierences between the AR of all trading algorithms. ere-

fore, we need to do further multiple comparative analysis and

the results are shown in Table . e number in the table is a

p value of any two algorithms of Nemenyi test. From Tables

and , we can see that the AR of MLP, DBN, and SAE have no

signicant dierence, but they are signicantly greater than

that of all other trading algorithms except for SVM. e AR

of GRU is signicantly smaller than that of all traditional ML

algorithms. ere is no signicant dierence between the AR

of any two traditional ML algorithms except for CART and

SVM.

() rough the hypothesis test analysis of Ha and Hb,

wecanobtainpvalue<.e-. erefore, there are signicant

dierences between the PR of all trading algorithms. ere-

fore, we need to do further multiple comparative analysis and

the results are shown in Table . e number in the table

is a p value of any two algorithms of Nemenyi test. From

Tables and , we can see that the PR of MLP, DBN, and

SAE are signicantly greater than that of all other trading

algorithms, and the PR of MLP, DBN, and SAE have no

signicant dierence. e PR of SVM is signicantly greater

than that of all other traditional ML algorithms which have

no signicant dierence between any two algorithms except

for SVM. e PR of RNN is signicantly greater than that

of all traditional ML algorithms except for SVM. e PR of

GRU and LSTM are not signicantly dierent from that of all

traditional ML algorithms except for SVM and LR.

() rough the hypothesis test analysis of Ha and Hb,

we can obtain p value<.e-. erefore, there are signicant

dierences between the RR of all trading algorithms. ere-

fore, we need to do further multiple comparative analysis and

the results are shown in Table . e number in the table is

apvalueofanytwoalgorithmsofNemenyitest.FromTables

and , we can see that the RR of all DNN models are

not signicantly dierent. ere is no signicant dierence

among the RR of all traditional ML algorithms. e RR of

RNN, GRU, and LSTM are signicantly smaller than that of

any traditional ML algorithm except for CART.

() rough the hypothesis test analysis of Ha and Hb,

we can obtain p value<.e-. erefore, there are signicant

dierences between the F of all trading algorithms. ere-

fore, we need to do further multiple comparative analysis and

Mathematical Problems in Engineering

T : Multiplecomparison analysis between the RR of any two trading algorithms. e p value of the two trading strategies with signicant

dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN . . .

LSTM . . . .

GRU . . . . .

CART . . . . . .

NB . . . 0.0075 0.0004 0.0007 .

RF . . 0.0260 0.0028 0.0001 0.0002 . .

LR 0.0330 0.0328 0.0152 0.0015 0.0001 0.0001 . . .

SVM . . . 0.0434 0.0033 0.0059 . . . .

XGB 0.0193 0.0192 0.0085 0.0007 0.0000 0.0000 . . . . .

T : Multiple comparison analysis between the F of any two trading algorithms. e p value of the two trading strategies with signicant

dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN 0.0000 0.0000 0.0000

LSTM 0.0000 0.0000 0.0000 0.0000

GRU 0.0000 0.0000 0.0000 0.0000 .

CART 0.0000 0.0000 0.0000 0.0000 . .

NB 0.0000 0.0000 0.0000 0.0136 0.0132 0.0099 .

RF 0.0000 0.0000 0.0000 . 0.0016 0.0011 . .

LR 0.0000 0.0000 0.0000 0.0000 . . . . .

SVM 0.0000 0.0000 0.0000 0.0178 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

XGB 0.0000 0.0000 0.0000 0.0001 . . . . . . 0.0000

T : Multiple comparison analysis between the AUC of any two trading algorithms. e p value of the two trading strategies with

signicant dierence is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

DBN .

SAE . .

RNN . . .

LSTM . . . .

GRU . . . . .

CART . . . . 0.0014 0.0096

NB 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 .

RF 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 . .

LR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 . . .

SVM 0.0027 0.0014 0.0001 0.0000 0.0000 0.0000 . . . .

XGB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 . . . . .

the results are shown in Table . e number in the table is a

p value of any two algorithms of Nemenyi test. From Tables

and , we can see that the F of MLP, DBN, and SAE have no

signicant dierence, but they are signicantly greater than

that of all other trading algorithms. ere is no signicant

dierence among traditional ML algorithms except SVM, and

the F of SVM is signicantly greater than that of all other

traditional ML algorithms.

() rough the hypothesis test analysis of Ha and Hb,

we can obtain p value<.e-. erefore, there are signicant

dierences between the AUC of all trading algorithms. ere-

fore, we need to do further multiple comparative analysis and

the results are shown in Table . e number in the table is

apvalueofanytwoalgorithmsofNemenyitest.FromTables

and , we can see that the AUC of all DNN models have

no signicant dierence. ere is no signicant dierence

Mathematical Problems in Engineering

T : Multiple comparison analysis between the WR of any two trading algorithms. e p value of the two trading strategies with

signicant dierence is in boldface.

Index BAH MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

BAH .

MLP 0.0000 0.0000

DBN 0.0000 0.0000 .

SAE 0.0000 0.0000 . .

RNN 0.0000 0.0000 0.0002 0.0006 0.0000

LSTM 0.0000 0.0000 0.0000 0.0000 0.0000 .

GRU 0.0000 0.0000 0.0000 0.0000 0.0000 . .

CART . 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

NB 0.0031 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

RF 0.0000 0.0000 0.0000 0.0000 0.0000 0.0205 . . 0.0000 0.0000

LR 0.0000 0.0000 0.0000 0.0000 0.0000 0.0010 . . 0.0000 0.0000 .

SVM 0.0000 0.0000 0.0000 0.0000 0.0000 . . . 0.0000 0.0000 . .

XGB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

T : Multiple comparison analysis between the ARR of any two trading strategies. e p value of the two trading strategies withsig nicant

dierence is in boldface.

Index BA H MLP DBN SAE R NN LSTM GRU CART NB RF LR SVM

BAH 0.0007

MLP 0.0000 0.0000

DBN 0.0000 0.0000 .

SAE 0.0000 0.0000 . .

RNN 0.0000 0.0000 . . .

LSTM 0.0000 0.0000 . . . .

GRU 0.0000 0.0000 . . . . .

CART 0.0000 0.0000 . . . . . .

NB 0.0000 0.0000 . . . . . . .

RF 0.0000 0.0000 0.0020 0.0048 0.0076 . . . . 0.0006

LR 0.0000 0.0000 . . . . . . . . .

SVM 0.0000 0.0000 . . . . . . . 0.0165 . .

XGB 0.0000 0.0000 . 0.0333 0.0484 . . . . 0.0057 . . .

between the AUC of all traditional ML algorithms. e

AUC of all traditional ML algorithms except for CART are

signicantly greater than that of any DNN model. ere is

no signicant dierence among the AUC of MLP, SAE, DBN,

RNN, and CART.

() rough the hypothesis test analysis of Ha and Hb,

wecanobtainpvalue<.e-. erefore, there are signicant

dierences between the WR of all trading algorithms. ere-

fore, we need to do further multiple comparative analysis and

theresultsareshowninTable.enumberinthetableis

apvalueofanytwoalgorithmsofNemenyitest.FromTables

and , we can see that the WR of BAH and benchmark

index have no signicant dierence, but they are signicantly

smaller than that of any ML algorithm. e WR of MLP, DBN,

and SAE are signicantly smaller than that of the other trad-

ing algorithms, but there is no signicant dierence between

the WR of MLP, DBN, and SAE. e WR of LSTM and

GRU have no signicant dierence, but they are signicantly

smaller than that of XGB and signicantly greater than that of

CART and NB. In traditional ML models, the WR of NB and

CART are signicantly smaller than that of other algorithms.

e WR of XGB is signicantly greater than that of all other

ML algorithms.

() rough the analysis of the hypothesis test of Ha and

Hb, we obtain p value<.e-.

erefore, there are signicant dierences between the

ARR of all trading strategies including the benchmark index

and BAH strategy. erefore, we need to do further multiple

comparative analysis and the results are shown in Table .

From Tables and , we can see that ARR of the benchmark

index and BAH are signicantly smaller than that of all

trading algorithms. e ARR of MLP is signicantly higher

than that of RF, but there is no signicant dierence between

MLP and other algorithms. e ARR of SAE and DBN are

signicantly higher than that of RF and XGB, but they are

not signicantly dierent from ARR of other algorithms. e

ARR of NB is signicantly higher than that of RF, SVM,

andXGB.But,otherwise,thereisnosignicantdierence

Mathematical Problems in Engineering

T : Multiple comparison analysis between the ASR of any two trading strategies. e pv alue of the two trading strategies with signicant

dierence is in boldface.

Index BAH MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

BAH .

MLP 0.0000 0.0000

DBN 0.0000 0.0000 .

SAE 0.0000 0.0000 . .

RNN 0.0000 0.0000 . . .

LSTM 0.0000 0.0000 . . . .

GRU 0.0000 0.0000 . . . . .

CART 0.0000 0.0000 0.0158 0.0195 0.0327 0.0000 0.0000 0.0000

NB 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 .

RF 0.0000 0.0000 . . . . . . 0.0018 0.0000

LR 0.0000 0.0000 . . . . . . 0.0000 0.0000 .

SVM 0.0000 0.0000 . . . . . . 0.0042 0.0000 . .

XGB 0.0000 0.0000 . . . . . . 0.0001 0.0000 . . .

T : Multiple comparison analysis between the MDD of any two trading strategies. e p value of the two trading strategies with

signicant dierence is in boldface.

Index BAH MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM

BAH 0.0000

MLP 0.0000 0.0006

DBN 0.0000 0.0004 .

SAE 0.0000 0.0023 . .

RNN 0.0000 0.0000 0.0320 0.0421 0.0111

LSTM 0.0000 0.0000 0.0002 0.0003 0.0000 .

GRU 0.0000 0.0000 0.0001 0.0001 0.0000 . .

CART 0.0000 0.0000 . . . . . .

NB 0.0000 . 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

RF 0.0000 0.0000 . . . . . . . 0.0000

LR 0.0000 0.0000 0.0001 0.0002 0.0000 . . . . 0.0000 .

SVM 0.0000 0.0000 . . . . . . . 0.0000 . .

XGB 0.0000 0.0000 0.0308 0.0405 0.0106 . . . . 0.0000 . . .

between any other two algorithms. erefore, the ARR of

most traditional ML models are not signicantly worse than

that of the best DNN model.

() rough the hypothesis test analysis of Ha and Hb,

we obtain p value<.e-. erefore, ere are signicant

dierences between ASR of all trading strategies including the

benchmark index and BAH strategy. e results of multiple

comparative analysis are shown in Table . From Tables

and , we can see that the ASR of the benchmark index

and BAH are signicantly smaller than that of all trading

algorithms. e ASR of all ML algorithms are signicantly

higher than that of CART and NB, but there is no signicant

dierence between the ASR of CART and NB. Beyond that,

there is no signicant dierence between any other two

algorithms. erefore, the ASR of all traditional ML models

except NB and CART are not signicantly worse than that of

any DNN model.

() rough the hypothesis test analysis of Ha and Hb,

we obtain p value<.e-. erefore, there are signicant

dierences between the MDD of these trading strategies

including the benchmark index and the BAH strategy.

e results of multiple comparative analysis are shown in

Table . F r om Tables a n d , w e c a n see that t h e M D D

of the benchmark index is signicantly smaller than that of

other trading strategies including BAH strategy. e MDD of

BAH is signicantly greater than that of all trading algorithms

except NB. e MDD of MLP, DBN, and SAE are signicantly

lower than that of NB, but signicantly higher than that

ofRNN,LSTM,GRU,LR,andXGB.eMDDofNBis

signicantly greater than that of all other trading algorithms.

Beyond that, there is no signicant dierence between any

other two algorithms. erefore, all ML algorithms expect

NB, especially LSTM, RNN, GRU, LR, and XGB, can play a

role in controlling trading risk.

Inaword,someDNNmodelssuchasMLP,DBN,and

SAE have good performance in AR, PR, and F; traditional

ML algorithms such as LR and XGB have good performance

in AUC and WR. e ARR of some traditional ML algorithms

such as CART, NB, LR, and SVM are not signicantly

dierent from that of the six DNN models. e ASR of the

Mathematical Problems in Engineering

six DNN algorithms are not signicantly dierent from all

traditional ML models except NB and CART. e MDD of

LR and XGB are signicantly smaller than those of MLP,

DBN, and SAE, and are not signicantly dierent from

that of LSTM, GRU, and RNN. e ARR and ASR of all

ML algorithms are signicantly greater than those of BAH

and benchmark index; the MDD of all ML algorithms are

signicantly smaller than that of the benchmark index but

signicantly greater than that of BAH strategy.

From the above analysis and evaluation, we can see that

the directional evaluation indicators of some DNN models

are very competitive in CSICS, while the indicators of some

traditional ML algorithms have excellent performance in

SPICS. Whether in SPICS or CSICS, the ARR and ASR of

all ML algorithms are signicantly greater than that of the

benchmark index and BAH strategy, respectively. In all ML

algorithms, there are always some traditional ML algorithms

which are not signicantly worse than the best DNN model

for any performance evaluation indicator (ARR, ASR, and

MDD). erefore, if we do not consider transaction cost and

other factors aecting trading, performance of DNN models

are alternative but not the best choice when they are applied

to stock trading.

Inthesameperiod,theARRofanyMLalgorithmin

CSICS is signicantly greater than that of the same algorithm

in SPICS (p value <. in the Nemenyi test). Meanwhile, the

MDD of any ML algorithm in CSICS is signicantly greater

than that of the same algorithm in SPICS (p value <.

in the Nemenyi test). e results show that the quantitative

trading algorithms can more easily obtain excess returns in

the Chinese A-share market, but the volatility risk of trading

in Chinese A-share market is signicantly higher than that of

the US stock market in the past years.

6. The Impact of Transaction Cost on

Performance of ML Algorithms

Trading cost can aect the protability of a stock trading

strategy. Transaction cost that can be ignored in long-term

strategies is signicantly magnied in daily trading. However,

many algorithmic trading studies assume that transaction

cost does not exist ([, ], etc.). In practice, frictions such

as transaction cost can distort the market from the perfect

model in textbooks. e cost known prior to trading activity

is referred to as transparent such as commissions, exchange

fees, and taxes. e costs that has to be estimated are known

as implicit, including comprise bid-ask spread, latency or

slippage, and related market impact. is section focuses on

the transparent and implicit cost and how do they aect

trading performance in daily trading.

6.1. Experimental Settings and Backtesting Algorithm. In this

part, the transparent transaction cost is calculated by a certain

percentage of transaction turnover for convenience; the

implicit transaction cost is very complicated in calculation,

and it is necessary to make a reasonable estimate for the

random changes of market environment and stock prices.

erefore,weonlydiscusstheimpactofslippageontrading

performance.

e transaction cost structures of American stocks are

similar to that of Chinese A-shares. We assume that transpar-

ent transaction cost is calculated by a percentage of turnover

such as less than .% [, ] and .% and .% in the

literature []. It is dierent for the estimation of slippage.

In some quantitative trading simulation soware such as

JoinQuant [] and Abuquant [], the slippage is set to ..

e transparent transaction cost and implicit transaction cost

are charged in both directions when buying and selling. It

is worth noting that the transparent transaction cost varies

with the dierent brokers, while the implicit transaction cost

is related to market liquidity, market information, network

status, trading soware, etc.

We set slippages s = {s=, s=., s=., s=.,

s=.4}; the transparent transaction cost c = {c=, c=.,

c=., c=., c=., c=.5}.Fordierent{s,c}

combinations, we study the impact of dierent transaction

cost structures on trading performance. We assume that

buying and selling positions are one unit, so the turnover is

the corresponding stock price. When buying stocks, we not

only need to pay a certain percentage cost of the purchase

price, but also need to pay an uncertain slippage cost. at

is, we need to pay a higher price than the real-time price

𝑡−1 when we are buying. But, when selling stocks, we not

only need to pay a certain percentage cost of the selling

price, but also to pay an uncertain slippage cost. Generally

speaking, we need to sell at a price lower than the real-time

price 𝑡. It is worth noting that our trading strategy is self-

nancing. If ML algorithms predict the continuous occur-

rence of buying signals or selling signals, i.e., |𝑡−

𝑡−1|=0, we will continue to hold or do nothing,

so the transaction cost at this time is . when |𝑡−

𝑡−1|=1, it is indicated that the position may

be changed from holding to selling or from empty position

to buying. At this time, we would pay transaction cost

due to the trading operation. Finally, we get a real yield

is

𝑡≤𝑡−

𝑡−1

𝑡−1 .

𝑡=

𝑡

∗1−∗

𝑡−

𝑡−1

−∗

𝑡−

𝑡−1

𝑡−1 =

𝑡−1

∗1+∗

𝑡−1 −

𝑡−2

+∗

𝑡−1 −

𝑡−2

𝑡=𝑡−

𝑡−1

𝑡−1

()

where 𝑡denotes the -th closing price,

𝑡denotes the -th trading signal, 𝑡denotes

the -th executing price, and 𝑡denotes the -th return rate.

We propose a backtesting algorithm with transaction cost

based on the above analysis, as is shown in Algorithm .

Mathematical Problems in Engineering

Input: TS TS is trading signals of a stock.

ssisslippage.

c c is transparent transaction cost.

Output: WR, ARR, ASR, MDD

() N=length of Stock Code List SPICS, and CSICS.

() WR=NULL; ARR=NULL; ASR=NULL; MDD=NULL

() for (i in : N) {

() Stock Data=Stock Code List[i]

() ClosePricet=Stock Data [“Closing Price”]

() Pt=ClosePricet∗(-c∗abs(TSt-TSt-1)) - s∗abs(TSt-TSt-1)

() Pt-1=ClosePricet∗(+c∗abs(TSt-TSt-1)) + s∗abs(TSt-T St-1)

() Rett=(P

t-P

t-1)/ Pt Ret is the return rate series.

() TDRR=lag (TS)∗Ret TDRR is the daily return through trading.

() WR[i]=sum (TDRR>)/sum(TDRR =)

() ARR[i]=Return.annualized (TDRR)

() ASR[i]=SharpeRatio.annualized (TDRR)

() MDD[i]=maxDrawDown (TDRR)

() WR=c (WR, WR[i]);

() ARR=c (ARR, ARR[i]);

() ASR=c (ASR, ASR[i]);

() MDD=c (MDD, MDD[i])

() }

() return (WR, ARR, ASR, MDD)

A : Backtesting algorithm with transaction cost in R language.

6.2. Analysis of Impact of Transaction Cost on the Trading

Performance of SPICS. Transaction cost is one of the most

important factors aecting trading performance. In US stock

trading, transparent transaction cost can be charged accord-

ing to a xed fee per order or month, or a oating fee based

on the volume and turnover of each transaction. Sometimes,

customers can also negotiate with broker to determine

transaction cost. e transaction cost charged by dierent

brokers varies greatly. Meanwhile, implicit transaction cost

is not known beforehand and the estimations of them are

very complex. erefore, we assume that the percentage

of turnover is the transparent transaction cost for ease of

calculation. In the aspect of implicit transaction cost, we only

consider the impact of slippage on trading performance.

(1)AnalysisofImpactofTransactionCostonWR.Ascanbe

seen from Table , WR is decreasing with the increase of

transaction cost for any trading algorithm, which is intuitive.

When the transaction cost is set to (s, c) = (., .), the

WR of each algorithm is the lowest. Compared with setting

(s, c) = (, ), the WR of MLP, DBN, SAE, RNN, LSTM,

GRU, CART, NB, RF, LR, and SVM to XGB are reduced by

.%, .%, .%, .%, .%, .%, .%, .%,

.%, .%, .%, and .%, respectively. erefore,

MLP, DBN, and SAE are more tolerant to transaction cost.

Generally speaking, the DNN models have stronger capacity

to accommodate transaction cost than the traditional ML

models. From the single trading algorithm such as MLP, if

we do not consider slippage, i.e., s=, the average WR of

MLP is . under transaction cost structures {(s, c),

(s, c), (s, c), (s, c), (s, c) };ifwedonotconsider

transparent transaction cost, i.e., c=, the average WR of MLP

is . under transaction cost structures {(s, c), (s, c),

(s, c), (s, c) }; so transparent transaction cost has greater

impact than slippage. rough multiple comparative analysis,

the WR under the transaction cost structure (s, c) is not

signicantly dierent from the WR without transaction cost

forMLP,DBN,andSAE.eWRunderallothertransaction

cost structures are signicantly smaller than the WR without

transaction cost. For all trading algorithms except for MLP,

DBN, and SAE, the WR under the transaction cost structure

{(s, c), (s, c) }are not signicantly dierent from the WR

without transaction cost; the WR under all other transaction

cost structures are signicantly smaller than the WR without

transaction cost.

(2) Analysis of Impact of Transaction Cost on ARR.Ascan

be seen from Table , ARR is decreasing with the increase

of transaction cost for any trading algorithm. Undoubtedly,

when the transaction cost is set to (s, c) = (., .), the

ARR of each algorithm is the lowest. Compared with the

settings without transaction cost, the ARR of MLP, DBN,

and SAE reduce by .%, .%, and .%, respectively,

while the ARR of other trading algorithms decrease by more

than % compared with those without transaction cost.

erefore, excessive transaction cost can lead to serious losses

in accounts. For a general setting of s and c, i.e., (s, c) = (.,

.), ARR of MLP, DBN, and SAE decrease by .%,

.%, and .%, respectively, while the ARR of other

algorithms decrease by more than % and that of CART

and XGB decrease by more than %. erefore, MLP, DBN,

and SAE are more tolerant to high transaction cost. From

single trading algorithm such as RNN, if we do not consider

slippage, i.e., s=, the average ARR of RNN is . under

Mathematical Problems in Engineering

T : e WR of SPICS for daily trading with dierent transaction cost. e result that there is no signicant dierence between performance without transaction cost and that with

transactioncostisinboldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM XGB

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) 0.5649 0.5653 0.5656 0.5778 0.5751 0.5785 0.5190 0.5861 0.5824 0.5782 0.5759 0.5788

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . 0.5729 0.5697 0.5741 0.5134 0.5811 0.5758 0.5726 0.5704 0.5713

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

Mathematical Problems in Engineering

T : e ARR of SPICS for daily trading with dierent transaction cost. e result that there is no signicant dierence between performance withouttransactioncostandthatwith

transactioncostisinboldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM XGB

(s, c) . . . . . . . . . . . .

(s, c) 0.3128 0.3087 0.3118 . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . -.

(s, c) . . . . -. . -. . -. -. -. -.

(s, c) 0.3249 0.3212 0.3242 0.2778 0.2729 0.2794 0.2953 0.2818 0.2898 0.2747 0.2855 0.2762

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . .

(s, c) . . . . . . -. . -. . . -.

(s, c) . . . . -. . -. . -. -. -. -.

(s, c) 0.3167 0.3129 0.3159 0.2613 0.2538 0.2654 0.2589 0.2661 0.2663 0.2551 0.2643 0.2484

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . -.

(s, c) . . . . . . -. . -. . -. -.

(s, c) . . . . -. . -. . -. -. -. -.

(s, c) 0.3085 0.3046 0.3076 . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . -.

(s, c) . . . . -. . -. . -. -. -. -.

(s, c) . . . -. -. . -. . -. -. -. -.

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . .

(s, c) . . . . . . -. . -. . . -.

(s, c) . . . . -. . -. . -. -. -. -.

(s, c) . . . -. -. . -. -. -. -. -. -.

Mathematical Problems in Engineering

the transaction cost structures {(s, c), (s, c), (s, c), (s,

c), (s, c) }; if we do not consider transparent transaction

cost, i.e., c=, the average ARR of RNN is . under the

transaction cost structure {(s, c), (s, c), (s, c), (s,

c) }; so transparent transaction cost has greater impact than

slippage. rough multiple comparative analysis, the ARR

under the transaction cost structures {(s, c), (s, c), (s,

c), (s, c) }are not signicantly dierent from the ARR

without transaction cost for MLP, DBN, and SAE; the ARR

under all other transaction cost structures are signicantly

smaller than the ARR without transaction cost. For all trading

algorithms except for MLP, DBN, and SAE, the ARR under

the transaction cost structures {(s, c), (s, c) }are not

signicantly dierent from the ARR without transaction

cost; the ARR under all other transaction cost structures are

signicantly smaller than the ARR without transaction cost.

(3)AnalysisofImpactofTransactionCostonASR.Ascan

be seen from Table , ASR is decreasing with the increase

of transaction cost for any trading algorithm. Undoubtedly,

when the transaction cost is set to (s, c) = (., .), the

ASR of each algorithm is the lowest. Compared with setting

without transaction cost, the ASR of MLP, DBN, and SAE

reduce by .%, .%, and .%, respectively, while the

ASR of other trading algorithms reduce by more than %

compared with the case of no transaction cost. erefore,

excessive transaction cost will signicantly reduce ASR. For

a general setting of s and c, i.e., (s, c) = (., .), the

ASRofMLP,DBN,andSAEdecreaseby.%,.%

and .% respectively. while the ASR of other algorithms

decrease by more than %; the ASR of CART and XGB

decrease by more than %. erefore, MLP, DBN, and SAE

are more tolerant to transaction cost. From single trading

algorithm such as NB, if we do not consider slippage, i.e.,

s=, the average ASR of NB is . under the transaction

cost structure {(s0, c1),(s,c),(s,c),(s,c),(s0, c5)};

if we do not consider transparent transaction cost, i.e., c=,

the average ASR of NB is . under the transaction cost

structures {(s1, c0),(s,c),(s,c),(s,c)};sotransparent

transaction cost has greater impact than slippage. rough

multiple comparative analysis, the ASR under the transaction

cost structures {(s1, c0),(s,c),(s,c),(s0,c1)} are not

signicantly dierent from the ASR without transaction cost

forMLP,DBN,andSAE;theASRunderallothertransaction

cost structures are signicantly smaller than the ASR without

transaction cost. For all trading algorithms except for MLP,

DBN, and SAE, the ASR under the transaction cost structures

{(s1,c0),(s2, c0)} are not signicantly dierent from the ASR

without transaction cost; the ASR under all other transaction

cost structures are signicantly smaller than the ASR without

transaction cost.

(4)AnalysisofImpactofTransactionCostonMDD.Ascan

be seen from Table , MDD increases with the increase

of transaction cost for any trading algorithm. Undoubtedly,

when the transaction cost is set to (s, c) = (., .), the

MDD of each algorithm increases to the highest level. In this

case, compared with the settings without transaction cost, the

MDD of MLP, DBN, and SAE increase by .%, .%, and

.%, respectively. e MDD of other trading algorithms

increase by more than % compared with those without

considering transaction cost. erefore, excessive transaction

cost can cause serious potential losses to the account. For a

general setting of s and c, i.e., (s, c) = (., .), the MDD

of MLP, DBN, and SAE increase by .%, .%, and .%,

respectively, while the MDD of other algorithms increase by

more than %, and the MDD of CART, RF, and XGB increase

by more than %. erefore, MLP, DBN, and SAE are more

tolerant to transaction cost. As a whole, the DNN models have

stronger capacity to accommodate transaction cost than the

traditional ML models. From single trading algorithm such

as GRU, if we do not consider slippage, i.e., s=, the average

MDD of GRU is . under the transaction cost structures

{(s0,c1),(s,c),(s,c),(s,c),(s0,c5)};ifwedonot

consider transparent transaction cost, i.e., c=, the average

MDD of GRU is . under the transaction cost structures

{(s1,c0),(s,c),(s,c),(s4, c0)};sotransparenttransaction

cost has greater impact than slippage. rough multiple

comparative analysis, the MDD under any the transaction

cost structure is not signicantly dierent from the MDD

without transaction cost for MLP, DBN, and SAE. For all

trading algorithms except for MLP, DBN, and SAE such as

LR, the MDD under the transaction cost structures {(s0,c1),

(s, c), (s, c), (s3,c0)} are not signicantly dierent from

the MDD without transaction cost; the MDD under all other

transaction cost structures are signicantly greater than the

MDD without transaction cost.

rough the analysis of the Table performance eval-

uation indicators, we nd that trading performance aer

considering transaction cost will be worse than that without

considering transaction cost as is in actual trading situation.

It is noteworthy that the performance changes of DNN

algorithms, especially MLP, DBN, and SAE, are very small

aer considering transaction cost. is shows that the three

algorithms have good tolerance to changes of transaction

cost. Especially for the MDD of the three algorithms, there

is no signicant dierence with that with no transaction

cost. So, we can consider applying them in actual trading.

Meanwhile, we conclude that the transparent transaction

cost has greater impact on the trading performances than

the slippage for SPICS. is is because the prices of SPICS

are too high when the transparent transaction cost is set

to a certain percentage of turnover. In actual transactions,

special attention needs to be paid to the fact that the trans-

action performance under most transaction cost structures

is signicantly lower than the trading performance without

considering transaction cost. It is worth noting that the

performance of traditional ML algorithm is not worse than

that of DNN algorithms without considering transa ction cost,

while the performance of DNN algorithms is better than that

of traditional ML algorithms aer considering transaction

cost.

6.3. Analysis of Impact of Transaction Cost on the Trading

Performance of CSICS. Similar to Section ., we will discuss

the impact of transaction cost on trading performance of

CSICS in the followings. In the Chinese A-share market,

the transparent transaction cost is usually set to a certain

Mathematical Problems in Engineering

T : e ASR of SPICS for daily trading with dierent transaction cost. e result that there is no signicant dierence between performance withouttransactioncostandthatwith

transactioncostisinboldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM XGB

(s, c) . . . . . . . . . . . .

(s, c) 1.4562 1.4478 1.4577 . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . .

(s, c) . . . . . . -. . . . . -.

(s, c) . . . . -. . -. . -. -. -. -.

(s, c) 1.5121 1.5057 1.5149 1.4927 1.4606 1.5119 1.2449 1.5424 1.5582 1.4825 1.4974 1.4886

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . -.

(s, c) . . . . . . -. . -. . . -.

(s, c) . . . . -. . -. . -. -. -. -.

(s, c) 1.477 1.4699 1.4792 1.4081 1.3632 1.4403 1.0946 1.4601 1.4390 1.3822 1.3922 1.3462

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . -.

(s, c) . . . . . . -. . -. . -. -.

(s, c) . . . -. -. . -. . -. -. -. -.

(s, c) 1.4415 1.4337 1.4432 . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . .

(s, c) . . . . . . -. . . . . -.

(s, c) . . . . -. . -. . -. -. -. -.

(s, c) . . . -. -. . -. . -. -. -. -.

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . -. . . . . .

(s, c) . . . . . . -. . -. . . -.

(s, c) . . . . -. . -. . -. -. -. -.

(s, c) . . . -. -. . -. -. -. -. -. -.

Mathematical Problems in Engineering

T : e MDD of SPICS for daily trading with dierent transaction cost. e result that there is no signicant dierence between

performance without transaction cost and that with transaction cost is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM XGB

(s, c) . . . . . . . . . . . .

(s, c) 0.3629 0.3638 0.3594 0.3779 0.3986 0.3636 0.4072 0.3712 0.3843 0.3963 0.3972 0.4203

(s, c) 0.3677 0.3695 0.3647 . . . . . . . . .

(s, c) 0.3727 0.3756 0.3703 . . . . . . . . .

(s, c) 0.3781 0.3821 0.3764 . . . . . . . . .

(s, c) 0.3839 0.3890 0.3828 . . . . . . . . .

(s, c) 0.3596 0.3600 0.3560 0.3500 0.36130 0.3446 0.3574 0.3502 0.3414 0.3585 0.3569 0.3540

(s, c) 0.3642 0.3655 0.3609 0.3907 . 0.3717 . 0.3814 . . . .

(s, c) 0.3691 0.3712 0.3662 . . . . . . . . .

(s, c) 0.3742 0.3774 0.3720 . . . . . . . . .

(s, c) 0.3796 0.3839 0.3781 . . . . . . . . .

(s, c) 0.3856 0.3909 0.3847 . . . . . . . . .

(s, c) 0.3609 0.3615 0.3573 0.3607 0.3756 0.3517 0.3770 0.3586 0.3586 0.3739 0.3727 0.3787

(s, c) 0.3656 0.3671 0.3623 . . . . 0.3929 . . . .

(s, c) 0.3705 0.3729 0.3678 . . . . . . . . .

(s, c) 0.3756 0.3792 0.3736 . . . . . . . . .

(s, c) 0.3812 0.3859 0.3799 . . . . . . . . .

(s, c) 0.3873 0.3930 0.3866 . . . . . . . . .

(s, c) 0.3622 0.3631 0.3588 0.3729 0.3912 0.3594 0.4004 0.3685 0.3795 0.3909 0.3909 0.4081

(s, c) 0.3669 0.3687 0.3639 . . . . . . . . .

(s, c) 0.3719 0.3746 0.3694 . . . . . . . . .

(s, c) 0.3772 0.3811 0.3754 . . . . . . . . .

(s, c) 0.3829 0.3879 0.3818 . . . . . . . . .

(s, c) 0.3894 0.3954 0.3888 . . . . . . . . .

(s, c) 0.3635 0.3647 0.3602 0.3861 0.4082 0.3678 . 0.3798 . . . .

(s, c) 0.3683 0.3704 0.3654 . . . . . . . . .

(s, c) 0.3734 0.3765 0.3712 . . . . . . . . .

(s, c) 0.3790 0.3833 0.3775 . . . . . . . . .

(s, c) 0.3851 0.3904 0.3841 . . . . . . . . .

(s, c) 0.3917 0.3981 0.3913 . . . . . . . . .

percentage of turnover, and it is the same as the assumption

in the experimental settings. As in the US stock market, the

smallest unit of price change is . (one tick). It is reasonable

to set slippage to be .-.. Of course, it should be noted

that the prices uctuation may be more intense when closing

than that in the middle of a trading day.

(1)AnalysisofImpactofTransactionCostonWR.Ascanbe

seen from Table , the WR is decreasing with the increase

of transaction cost for any trading algorithm. When the

transaction cost is set to (s, c) = (., .), the WR of each

algorithm is the smallest. Compared with the settings without

transaction cost, the WR of MLP, DBN, SAE, RNN, LSTM,

GRU,CART,NB,RF,LR,SVM,andXGBarereducedby

.%, .%, .%, .%, .%, .%, .%, .%,

.%, .%, .%, and .%, respectively. For a general

setting of s and c, i.e., (s, c) = (., .), the WR of

MLP, DBN, and SAE decrease by .%, .%, and .%,

respectively, while the WR of other algorithms decrease by

more than %; the WR of CART, RF, and XGB decrease by

more than %. erefore, MLP, DBN, and SAE are more

tolerant to transaction cost. From single trading algorithm

such as LSTM, if we do not consider slippage, i.e., s=, the

average WR of DBN is . under the transaction cost

structures {(s, c), (s, c), (s, c), (s, c), (s, c) };ifwe

do not consider transparent transaction cost, i.e., c=, the

average WR of LSTM is . under the transaction cost

structures {(s, c), (s, c), (s, c), (s, c)};sotransparent

transaction cost has smaller impact than slippage. rough

multiple comparative analysis, the WR under the transaction

cost structures {(s, c), (s, c), (s, c) }are not signicantly

dierent from the WR without transaction cost for MLP,

DBN, SAE, and NB; the WR under all other transaction

cost structures are signicantly smaller than the WR without

transaction cost. For all trading algorithms except for MLP,

DBN, SAE, and NB, the WR under the transaction cost

structure (s, c) is not signicantly dierent from the WR

without transaction cost; the WR under all other transaction

cost structures are signicantly smaller than the WR without

transaction cost.

Mathematical Problems in Engineering

T : e WR of CSICS for daily trading with dierent transaction cost. e result that there is no signicant dierence between

performance without transaction cost and that with transaction cost is in boldface.

MLP DBN SAE RNN LSTM GRU CART NB RF LR SVM XGB

(s, c) . . . . . . . . . . . .

(s, c) 0.5523 0.5527 0.5525 0.5525 0.5608 0.5620 0.5009 0.5227 0.5612 0.5665 0.5571 0.5595

(s, c) 0.5488 0.5492 0.5489 . . . . 0.5149 . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) 0.5494 0.5499 0.5497 . . . . 0.5170 . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(s, c) . . . . . . . . . . . .

(2)AnalysisofImpactofTransactionCostonARR.Ascan

be seen from Table , ARR is decreasing with the increase

of transaction cost for any trading algorithm. Undoubtedly,

when the transaction cost is set to (s, c) = (., .), the

ARR of each algorithm is the smallest. Compared with the

settings without transaction cost, the ARR of MLP, DBN,

and SAE reduce by .%, .%, and .%, respectively.

While the ARR of other trading algorithms decrease by

more than % compared with those algorithms without

transaction cost. erefore, excessive transaction cost can

lead to serious losses in the accounts. For a general setting

of s and c, i.e., (s, c) = (., .), ARR of MLP, DBN,

and SAE decrease by .%, .%, and .% respectively,

while the ARR other algorithms decrease by more than %

and that of CART, NB, RF, and XGB decrease by more than

%. erefore, MLP, DBN, and SAE are more tolerant to

transaction cost. From single trading algorithm such as SAE,

if we do not consider slippage, i.e., s=, the average ARR of

SAE is . under the transaction cost structure {(s, c),

(s, c), (s, c), (s, c), (s, c) };ifwedonotconsider

transparent transaction cost, i.e., c=, the average ARR of SAE

is . under the transaction cost structures {(s, c), (s,

c), (s, c), (s, c) };sotransparenttransactioncosthas

smaller impact than slippage. rough multiple comparative

analysis, the ARR under the transaction cost structures {(s,

c), (s, c), (s, c), (s, c), (s, c) }are not signicantly

dierent from the ARR without transaction cost for MLP,

DBN, and SAE; the ARR under all other transaction cost

structures are signicantly smaller than the ARR without

transaction cost. For RNN, LSTM, GRU, CART, RF, LR, and