L/ng dung
mang noi-ron nhan tao (ANNs) trong du bao gia dong cufa
cac ma co phieu niem yet tren san chiTne khoan
* - ' LE THI THG GIANG"
VG THj HUYEM TRANG"
Tom tat
Do tinh bie'n dgng khdng ngdng cua thi trudng tdi chinh, du bdo gid co phii'u Id mgt nhiim vu day thdch thiie. Nhieu nhd nghien cdu da cd gang dif bdo gid co phieu bang cdch sd dung cdc logi hinh thd'ng ki, cdc md hinh kinh ti'lugng hay mgng nef-ron nhdn tgo (ANNs) khdc nhau. Trong nghien cdu ndy, chung toi sd dung mo hinh ANNs de du bdo gid ddng ciia cua cdc md eo phii'u duac niem yet cdng khai trin sdn chdng khodn vd thuc nghiim trin 60 sd lieu cua 2 md cd phii'u Id VCB vd VIB cua 2 ngdn hdng: Ngogi thuang Viet Nam (VCB) vd Qudc ti' Viet Nam (VIB). Ke't qud cho thdy, cdc md hinh hogt ddng khd hieu qua, dgc bift Id khi sd lieu khong cd bie'n dgng qud Idn vdi trung binh tuyet dd'i cua sai sd tuang ddi ehi khodng 20%.
Ttf khoa: mang na-ron nhdn tao, md hinh dU bdo, du bdo gid cd phii'u, thudt todn truyen nguge Summary
Due lo the ever-changing volatility of the financial markets, stock price prediction is a challenging task. Many researchers have tried to predict stock price by using different types of statistics, eeanometric models, or artificial neural networks. In this study, we employ artificial neural network model lo forecast the closing prices of slocks listed publicly on the slock market and experimentally on the dalaset of two stocks which are VCB and VIB (respectively stand for Joint Stock Commercial Bank for Foreign Trade of Vietnam and Vietnam International Commercial Joint Slock Bank). The results show that the models work quite well, especially when the data are not very volatile with the absolute average of the relative error of only about 20%.
Keywords: artificial neural network, forecasting model, slock price prediction, baekpropagalion algorithm
GIOI THIEU tam cua nhieu nha dau ttf, chuyen gia, nha khoa hgc.
Tuy nhien, do tinh bien dgng nhanh theo su" tac d6ng O bat ky qu6'c gia nao, thi tri/cJng cua thi tru'dng, di/bao gia co phieu la mot trong nhu'ng chiJng khoan cung la m6t trong nhu'ng bai toan day thach thiJc.
thanh phan quan trgng trong nin kinh te. Nhieu phu'dng phap du' bao da va dang difdc Bdi vay, viec hieu difdc xu hu'dng cua thj phat tri6'n de dif bao xu hu'dng bien dgng ve gia co tru'dng nay la rat can thiet. Do do, cung phie'u nham tim kie'm cac co phie'u tiim nang de vdi sif phat trien cua thi trifdng chifng dau tu'. Trong dd, nhiing phu'dng phap phan tich va khoan, dif bao gid co phieu da trd thanh dy bao bSng dinh lu'dng thdng qua cac mo hinh toan mot chu de thii vi, thu hut du'dc .sif quan hgc da difdc quan tam do tinh khach quan va cd sd 'ThS., " T h S . , Trifctng Dai hoc Thifdng mai
Ngdy nhan bdi: 15/01/2020; Ngdy iham dinh- 05/02/2020. Ngdy duyel ddng: 12/02/2020
khoa hgc ciia chiing. Ddn gian nha't la cac mo hinh hdi quy tuye'n tinh, h i i quy da thifc... de dy bao xu hifdng thi tnfdng. Tie'p de'n la nhu'ng mo hinh thd'ng ke CO dien, bao gom cac phifOng phap: trung binh tru'dt, ARIMA.
Ngay nay, ANNs dang difdc siJ dung rdng rai trong rat nhieu ITnh vurc. Difa vao cac kha nang hgc, kha nang khdi quat hda va kha nang xu* ly thong tin song song, ANNs da cho tha'y tinh hieu qua cua no trong viec giai quyS't bai toan dif bao. Tuy nhien, tai Viet Nam, cac cdng tnnh nghien ciJu ve van de dif bao gia CO phi6'u ANNs khdng nhieu.
Trong bai bao nay, nhdm tac gia siV dung ANNs trong dif bao gia mdt so' ma cd phid'u cua c^c ngan hang niem yet cong khai tren san chd'ng khoan. Cu the la cd phid'u VCB, VIB. Day la mgt kenh tham khao cho cac nha dau tif va cac nha phan tich thi trifdng.
CO s d LY THUYET VA P H U O N G PHAP NGHIEN Cl?U
Mang ntf-ron nhan tao
Mang nd-ron nhan lao (Artificial neural networks - ANNs) la mot md hinh hgc may difdc la'y cam hifng tlif ca'u triic mang nd-ron ty nhien. Ve cd ban, chiing bao gom: cac nd-ron dau vao (nhu cac khdp than kinh), difdc nhan vdi cac trgng so', chinh la cifdng do cua cac tin hieu tu'dng ifng va sau dd du'dc tinh bang mgt ham toan hgc xac dinh sy kich hoat cua nd-ron (activation function). Mot ham khac (cd Xhi la ham dong nha't) se tinh toan dau ra cua te bao than kinh nhan tao.
Trong so' cua mgt nd-ron nhan tao cang cao, thi dau vao difdc nhan len cang manh. Trgng so nay co the am hoac du'dng. Bang each dieu chinh trgng so' cda mot nd-ron nhan tao, chung ta cd the cd difdc dau ra ma chiing ta mud'n vdi cac dau vao cu the. Tuy nhien, khi mang nd-ron gom hang tram hoac hang ngan te bao than kinh, viec tinh toan ra cac trgng so' nay se rat phiJc tap. Liic nay, ta can siJ dung cac thuat toan de dieu chinh trgng so' cua ANNs nham co dUdc d^u ra mong mud'n mdt each nhanh nha't. Qua trinh dieu chinh nay du'dc ggi la "hoc" (learning) hoac "dao tao"
(training). Trong mot mang nd-ron nhdn tao co the cd mdt hoac nhieu Idp an trung gian (hidden layers), nhifng mang no-ron nhu" vay difdc ggi la mang nd-ron da tang. Khi dd, mdi d^u ra cua Idp trddc chinh la dau vao cua Idp lien sau no. Vecto gia tri cua Idp thiJ difdc xac dmh bdi cdng thu'c:
a^ =f(wla^ J + b^)
Trong do, w, la ma tran trong so' tai Idp thiJ 1, f la ham kich hoat (activation function) tai Idp dd. Thong thifdng, ta dijng mgt trong so' cdc ham phi tuygn sau dSy lam ham kich hoat
- Ham Sigmoid: f(x) - Ham Tank. / W = ; - Ham ReLU'.f(x) = max(x, 0)
Ham Sigmoid va ham Tanh deu la cac ham dd'i xiJng, dOn dieu tang vd cd dau ra tie'n tdi I khi d^u vao rat Idn, Neu nhif ham Sigmoid chi nhan gia tri difdng, thi ham Tanh lay^cac gid tri trong khoang [-1,1]. Uu diem ciia cac ham nay Id cho dao ham ra't dep vd md ta kha td't nhu'ng md hinh vdi bien dau ra cd dang nhi phan hoac cdc bieu diin xac sua't. Tuy nhien, mgt nhifdc didm rat dd nhan thd'y Id khi dau vdo cd gid trj tuyet do'i Idn, dao ham cua 2 loai hdm ndy rat gan vdi 0. Dieu nay cd nghia la trong tru'dng hdp do, cac trgng so' gin nhif khdng difdc cap nhat, khi ta diing cdng thifc gradient desent.
Can day, hdm ReLV (Rectified Linear Unit) du'dc sxt dung rgng rdi vi tinh ddn gian va hieu qua cua nd. Khi sif dung hdm nay trong cac mang nd-ron da tang, tdc do hdi tu cao hdn nhieu so vdi viec diing ham Tanh hay Sigmoid. Mac dii ham ReLU CO nhifdc diem Id cd dao ham bang 0 vdi cac gid tri dm, nhtfng thyc nghiem cho tha'y rang, nhifdc diem nay cd the du'dc khac phuc bdng each tdng sd'niit an.
Ke ty md hinh than kinh dau tien ciia McCulloch vd Pitts (1943), da cd hdng tram md hinh ANNs khdc nhau du'dc difa ra. Sif khdc bi6t giifa cac m6 hinhdd cd the la cac ham, ca'u triic lien ke't, thuat toan hgc tap... Trong nghien cdu ndy, nhdm tac gia se sii dung mdt mang nd-ron du'Oc dya tren thuSt toan backpropagation (Rumelhart va cdng sy, 1986), mot trong nhffng mo hinh phd bie'n nha't ddOc s^ dung trong ANNs de tim ra cac trgng so' phii hdp trong viec dy bdo gia cd phieu ty cdc djl lidu qua khy.
Thuat toan truyin ngifdc (Backpropagation)
^ Khi dy lieu truyen qua he't cdc idp de cd dddc dau ra, ta se Unh du'Oc sai sd' giya ke't qua cua phifdng phap'vdi ket qua thyc te. Sai so nay Id mdt hdm phi tuyen vdi dd'i so' la cac ma tran trgng so' cua md hinh. Ham sd dd du'Oc "oi la
"hdm ma't mat" (loss function). Cd nhiiu each dinh nghTa ham nuVt n nghien cyu ndy, chung tdi s binh binh phu'dng sai sd dinh nghia, nhu'sau:
MSE = -LY^(y,-y)' 6 day, >',^ la gia tri thyc te ket qua thu'c nghiem tii phu'Oi^
dy bdo >'^.
trong 'i-ung
• f c
„ la p'dd
Thuat toan backpropagation (Rumelhart va cdng sy, 1986) du'dc dung trong feed- forward ANNs. Cdc te bao than kinh nhan tao du:dc to chtfc thanh cdc Idp va gu'i tin hieu cua chiing ve phia trifdc, sau do cac l5i du'dc truyen ngUi^c lai.
Y tu'dng ctia thuat toan truyen ngu'dc (backpropagation) Id giam sai so' nay bang each thay ddi cac trgng s6' cho de'n khi ANNs hoc du'dc tii dfl lieu ddo tao. Cdch td't nha't de ldm giam sai sd la hua'n luyen mo hinh sao cho sai so' cang gan vdi gid tri be nha't cua ham ma't mat cang tdt. Do dd, de tim ra bg trong sd' phu hdp, ta se giai gan diing bai toan to'i Ifu do. Gradient Descent la mgt thuat toan ra't phd bie'n de xac dinh gan diing nghiem td'i u\i. Cu the:
Budc 1: Dy dodn mgt ma tran trgng sd khdi tao W = W^
Budc 2: Tinh sai so' dau ra vdi trgng so' W|| dd
Budc 3: Cap nhat bd trgng sd mdi W^
= W - TiV L(W )
Budc i: Cdp nhdt den khi sai sd dat mtfc cha'p nhan difdc: W^^, = W^
Trong do, v^L(W^) Id dao hdm (Gradient) ciia ham ma't mat L(W) tai W^.
De viec cap nhdt trgng so' dat du'dc ket qua mong mud'n, ddng thdi ciing tranh hien tu'dng sd' bifdc lap can sii dung cho qud trinh hgc la qua Idn, chiing ta thifdng difa ra mdc sai sd' va gidi han so' bifdc lap cua mdi thuat toan. Cdch ldm nay cung giiip phifOng phdp khong bi rdi vdo hien tifdng
"over fitting", nghia la md hinh tim difdc "qua khdp" vdi dd lieu, ddn de'n cd thd k6't qua se ra't khdng chinh xac, khi mgt sd' dif lieu dddc sii dung cd nhieu Idn.
PhUdng phap nghien ci?u Budc 1: Thu thap vd ldm sach dfl lieu Mdt trong nhffng bifdc quan trgng nhd't trong vi6c xii ly so' lieu Id thu thap difdc dfl heu td't, thyc hiSn cdc bfldc lam sach dfl lieu bang phifdng phap phd hdp. Trong nghien cflu nay, cac dfl lieu chdng khoan du'dc download tfl trang https://www.vndirect.com.vn dffdi dang mdt bang excel vdi cdc thdng tin, bao gom: ngdy thdng (Dates), gia md ciia (Open), gia cao nha't (High), gid tha'p nha't (Low), gid ddng ciia (Close), gia ddng cu'a dieu chinh (Adj Close) va khd'i Iffdng giao dich (Volumn). TS't ca
cdc dfl lieu dd deu la cac sd' thyc ngoai trfl ngdy, thdng. Vi vdy, cdch ddn gian nhd't de mdy tinh cd the hieu dffdc va bieu thi thdng so' ndy tren dd thi la chuyen ngay, thdng sang dang so' bang each tao them mot cot ben canh cot "Dates" va dat ten la
"Date values", sau do copy toan bg thdng so' d cot
"Dates" sang va siidiinglenh "Text." "Date values"
vd "Close" sau do se dffdc chuyen thdnh cdc gia trj nhd hdn nhd lenh "Scale".
Trong nghien cffu nay, chiing tdi Iffa chgn MinMaxScaler d l dffa cac gid tri tren ve doan nhd cdng thffc:
= y ' ym,n
•^'"•'"' y,„^ - y.u„
Vdi y^^^^, y^^^ lan IffOt Id gia ddng ciia cao nha't va thd'p nhat ciia mgt ma cd phie'u trong tap dff lieu.
Elide 2: Lya chon md hinh
Chiing tdi sti dung mang nd-ron de dff bao gia ddng ciia Ciia mdt ma cd phie'u vao mot ngay bd't ky trong tffdng lai khi da cd cac dff kien trong qua khff vdi thff vien Tensor Flow. Mang nd-ron gom mgt gia tri dau vdo la ngay can dff bao (dang "Date value" sau khi da scale) vd mot dau ra Id gid dff bdo ciia mgt ma cd phie'u vao ngay hdm do. Mang gdm nhieu Idp an.
Chung toi se thii nghiem ke't qua khi thay ddi sd' Idp an, sd' nO-ron trong moi Idp an do vd thay ddi cac hdm kich hoat d mdi Idp. O bdi bao nay, chiing tdi sfl dung 3 md hinh ANNs:
- Md hinh 1: Cd 3 Idp an. Vdi moi Idp an, ham kich hoatmd ta sii dung la hdm "7a«/]".SdnO-ron tren mdi Idp an la 50.
- Mo hinh 2: Cd 5 Idp d'n. Hai Idp d^u va 2 Idp cud'i diing ham '"Tanh" de kich hoat, Idp thff 3 ta sii dung ham '"ReLu". So'nd-ron cua mdi Idp van la 50.
Cac thdng sd tren cd thd thay ddi de Iffa^chgn ra md hinh phii hdp hdn. Cu thd, chung ta cd the tang so' Idp d'n, tang so nO-ron trong mdi Idp, ddng thdi^cd the sii dung cdc ham kich hoat khdc trong cdc Idp an do.
Budc 3: Kiem nghi6m ke't qua
Vdi mdi md hinh, khi sff dung bg so' lieu cu thd cdc tdc gia dd tinh toan ra cac ket qua dy bao tifdng ffng, ddng thdi kiem tra do chinh xdc bang each tinh sai sd tffdng dd'i gifla dfl lieu chinh xac vdi ke't qua dff bao vdi cdng thffc sau:
err = ^ ^ * 100%
d ddy, yl y, lan lfft?t la gid ddng cffa thifc te' va gia dff bao sau khi da scale ciia mgt ma cd phieu tai thdi diem t. Trong trffdng hdp y, = 0, tac gia da thay gid tri ndy bdi trung binh cdng ciia y^_, vd y^^^.
Ngoai ra, cdc tac gia cung tinh trung binh binh phffdng sai sd (MSE) va trung binh tuyet dd'i sai sd tffdng dd'i (MAPE) gifla ke't qua dff bdo va gia tri thffc te cua hai md hinh tran de ddnh gid dd chinh xac cua cac md hinh, trong do:
M A / > £ 4 £ ^
HIMH 1: GlA DONG CtfA THCTC TE VA GlA Dtf BAO VCB
0.0 0 2 0 4
HINH 2: SAI SO TddNG DOI COA DCT BAO
00 02
HiNH 3: GlA DOMG CCTA THCtC TE VA GlA Dtf BAO VIB
jj
gia diing dubao 1 - - dUbao2
00 02 04 06 08 10 ngay (scale)
2 0 1-5
1-0 05 0 0 - 0 5 - 1 0
HINH 4
y.
SAI s o TCfdtiG DO! CUA D
fi-
iBAo
Sfll56 1 saisd2
i f;
1 j [*i^t:
0 4 0 6 OB 3-0
Mguon' Tinh loan cua n h o m tac gis
Trong nghien ciiu nay, cdc tac gia da sii dung dff lieu download tff trang https://
www.vndirect.com.vn ciia hai ma cd phieu: VCB vd VIB tff ngay 01/01/2017- 01/01/2020 (Bdi viel sd dung cdch vie't sd thdp phdn theo chudn qud'c ti").
KET QUA NGHIEN COU Hinh 1 so sdnh ket qua dy bao vdi gia ddng cffa thyc te' cda cd phieu VCB trong^ba trffdng hdp: dffdng net lien Id gia CO phie'u thffc te' (sau khi da scaled), dffdng net ".." Id gid dy bao theo md hinh dy bdo 1, nghia la chiing ta sff dung mang nd-ron vdi 3 Idp an vdi ham kich hoat Id ham "Tanh", vd dffdng net " - "
the hien ke't qua dff bdo khi sff dung md hinh 2.
Tfl do thi cd the tha'y, ke't qua dy bdo thu dffdc khi su" dung mo hinh 2 td't hdn so vdi md hinh 1. PTinh 2 cho tha';^, sai sd hfdng ddi cua 2 md hinh dy bdo. Cf ca hai md hinh, sai sd' cd xu hffdng giam dan khi sd' diem dff lieu tang len.
Tffdng tff nhff vay, khi dp dung 2 rad hinh ANN tren cho bd dfl Ueu cua VIB, ta cd ket qua nhff Hinh 3, 4.
Hinh 3 md ta dd thi ciia gid thyc te (dffdng net lien) so vdi gid dy bdo khi su' dung md hinh 1 (dffdng net "..") vd md hinh 2 (dffdng net " - " ) . Hinh 4 la do thi so sanh sai sd tffdng dd'i gifla ke't qua thu dffdc tff md hinh 1 (dffdng net lien) vdi ket qua thu dffdc tff md hinh 2 (dffdng net"..").
Tffdng ty tru'dng hdp ciia md cd phieu VCB, md hinh 2 chd ket qua tdt hdn vdi sai so nho hdn so vdi md hinh l.Tuy nhien, sai sd ciia kdt qua dff bao trong vi du_ nay Idn hdn so vdi khi sff dung bd sd lieu cua ngan hang VCB Dieu ndy cd the' giai thich do gia cd phieu thyc te VIB cd bie'n ddng tffdng ddi Idn, trong dff lieu cd nhidu didm bat thffdng (gia tang vgt hoac gidm dot ngdt), anh hffdng den do chinh xac cua md hinh dff bao.
Bang 1 la ket qud binh phffdng sai so' tu}
dy bao khi ap dung 2 IT cd phie'u cac ngan hdng Ke't qua cho tha'y, cdc hieu qua vd md hinh 2 ci.
hdn md hinh 1.
Bang 2 vd Bang 3 dffa r bdo cu the cho 10 ngdy cut (khdng tinh Thff 7, Chu nhdi.
^!ng b i n h 'J a cdc
• ha'p
dir
^019
•ai so'
tydng dd'i tffdng ffng chd tffng gid tri dy bao. Ket qua thu dffdc tff bd so lieu cua VCB cd sai so' thd'p hdn nhieu (chi khoang 2%) so vdi sai sd' khi ap dung phydng phdp cho bd so lieu cua VIB (khoang tren 10%). Tuy nhien, ca hai mffc sai sd nay ddu cho tha'y, md hinh dff bao Id phu hdp.
KET LUAN
Nghien cffu da dffa ra dffdc mot phffdng phdp dy bdo bang mo hinh mang nd-ron nhan tao cho gia cd phie'u tai mgt thdi diem tffdng lai khi cd Iffdng dff lieu du Idn trong qua khff, dong thdi cd ddnh gia sai sd cho cac md hinh de xua't. Ket qua dat dffdc cho tha'y sff phii hdp cua mo hinh, ddc biet neu xet ve xu hffdng bien ddng ciia gia.
Han che ddng ke ciia md hinh la mdi chi sff dung duy nha't dfl kien gia ddng cffa trong qua khff de dy bdo gia trdng tffdng lai, md chffa de cap tdi cac ye'u td khdc cung cd anh hffdng rat Idn de'n thi trffdng cd phidu, nhff: truyen thdng xa hdi, gia ca ciia cac ddi thii canh tranh, bien dgng thi trffdng, chi so thi trffdng... bdi viec Iffdng hoa cdc yeu to' nay khdng he ddn gian.
Vi vay, hffdng nghien cffu tiep theo ciia nhdm tdc gia la se dffa ra cac md hinh dy bao bang ANNs, trong dd cd sff dung mgt hoac mgt vdi cac nhan to' khdc ngodi gia cd phieu d qua khfl.Q
BANG 1: TRONG BINH SAI SO CGA Dtf BAO
MSE MAPE MAPE -2
VCB-m6 Wnhl 0.00456 0.25988 0.06355
VCB- mfi hinh 2 0.00286 0.19804 0.05734
VIB -md hinhl 0 00589 0.26409 0.31683
VIB- md binb 2 0-00356 0.20213 0.27746
BAtSG 2: SO S A N H GlA THCfC TE VA GlA DCf BAO 10 NGAY CGA VCB Ngay
31/12/2019 30/12/2019 27/12/2019 26/12/2019 25/12/2019 24/12/2019 23/12/2019 20/12/2019 19/12/2019 18/12/2019 MAPE
Gia thi^c t^
90 20 91 00 90 50 t^9.40 89.90 90.00 90.90 88 30 88 00 86 70
Dif b^o 1 88,20 87.63 87.67 88 26 87.86 87.71 87.08 88 46 88 57 89 37
Saiso(%) 2 22 371 3 13 1 28 2 27 2.54 4.20 0 18 0.65 3.08 2.33
Di^b^o2 88.91 88.31 88.25 88.82 88.38 88.20 87 53 88 79 88.86 89.62
Saistf(%) 1.42 2.96 2.48 0.65 1.69 2.00 3,71 0,55 0.98 3 37 1.98 BAtiG 3:
Ngay 31/12/2019 30/12/2019 27/12/2019 26/12/2019 25/12/2019 24/12/2019 23/12/2019 20/12/2019 19/12/2019 18/12/2019 MAPE
SO S A N H GlA THdC TE VA GlA DCf BAO 10 NGAY CCIA VIB Gia ttnjfc t^
17.30 17-50 17.60 17 70 17.80 17 40 17.40 17.20 17.10 17-00
Dif bao 1 15.68 14-55 14.23 1381 13 42 15 57 15-65 17.26 18.14 19.13
sai sS'(%) 9.39 16.86 19.14 22.00 24.63 10.50 10.08 0 32 6.09 12 50 13.15
Dtf bao 2 17 14 15.87 15-40 14.90 14.44 16.73 16.76 18.36 19 25 20.25
sais6'(%) 0.92 9.33 12.50 15.80 18.86 3,88 3.66 6.72 12.60 19.13 10.34
TAI LmV THAM KHAO
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2. Kimoto, T., Asakawa, K., Yoda, M., and Takeoka, M. (1990). Stock market prediction system with modular neural network. Proceedings of the International Joint Conference on Neural Networks, 1-6
3. McCulloch & Pitts (1943). A Logical Calculus of Ideas Immanent in Nervous Activity, Bulletin of Mathematical Biophysics, 115-133
4. Lucas Nunno (2014). Stock Market Price Prediction Using Linear and Polynomial Regression Models, University of New Mexico Computer Science Department Albuquerque, New Mexico
5. Parsinejad Shahbaz, Bagheri Ahmad, Ebrahimi Atani Reza, Javadi Moghaddam Jalal (2014).
Stock market forecasting using artificial neural networks, European Online Journal of Natural and Social Sciences, 2(3s)
6. Phua, P.K.H. Ming, D., Lin, W. (2000). Neural Network with Genetic Algorithms Prediction, Fifth Conference of the Association of Asian-Pacific Operations Research Societies, 5-7th July, Singapore
7. Rumelhart, David £.; McClelland, James L. (1986). Parallel Distributed Processing : Explorations in the Mlcrostructure of Cognition, Volume 1 : Foundations, Cambridge: MIT Press
8. Tsai, Tsiao (2010). Combining multiple selections for stock prediction: union, intersection and multi-intersection approachs, Decis. Support Syst, 50(1), 258-269
