Tap chi KHKT Mo - Dia chdt, sd30, 4/2010, tr. 75-80
PHAN TICH BIEN DANG CONG TRINH VOI MO HINH HOI QUY
PtiAN VAN HIEN, Trudng Dgi hgc Md - Dia chdt
DINH XUAN VINH, Cdng ty CP Tu vdn ddu tu vd xdy dung HUDCIC
Tom tat: Md hinh hdi quy la cdng cu thdng ki todn hgc dugc ung dung rpng rdi trong cdc ngdnh kinh ti qude ddn vd trin todn the gidi. Ung dung md hinh hdi quy di phdn tich dir lieu quan trdc biin dgng cdng trinh da dugc thi gidi img dung vd dgt dugc thdnh qua rdt khich li. Thdng qua cdc phirong phdp hdi quy da dir bdo dugc nhiiu tai hog thiin nhiin tham khdc di tim cdch hgn chi chimg. Bdi bdo ndy trinh bdy phuong phdp hdi quy bgc thang vd img dung di phdn tich biin dgng cua cdc cdng trinh thdp tdng tgi khu do thi Vdn Qudn, Hd Ddng, Hd Npi. Sd lieu quan trdc dirge thu thdp trong 4 ndm lien tuc du di phdn dnh khd todn diin sir biin dgng cua cdng trinh.
1. Md hinh hdi quy
Phan tich bien dang cdng trinh lien quan den xu ly va phan tich sd lieu quan trac, giai thich vat ly bien dang va du bao bien dang.
Cdng tac nay cd the chia lam hai phan, mdt la phan tich hinh hgc bien dang va hai la giai thich vat ly bien dang. Phan tich hinh hgc bien dang la mieu ta su bien ddi ve kich thudc va hinh dang cLia the bien dang, tuc la xac djnh trang thai khdng gian va dac tinh thdi gian bien dang ciia the bien dang. Giai thich vat ly bien dang la xac djnh mdi quan he giua bien dang vdi nguyen nhan bien dang, tuc la giai thich nguyen nhan bien dang.
Trong thdng ke toan hgc, cd hai dang md hinh diing de phan tich the gidi thuc, la md hinh tham sd va md hinh phi tham sd. Md hinh phi tham sd dua tren cac thuat toan du bao thdng ke loan hgc. Mdt dang ciia md hinh phi tham sd la hdi quy tirng bude (stepwise regression-SR) hay cdn ggi la hdi quy bac thang.
Phan tich hdi quy la mdt phuong phap thdng ke ma gia trj ky vgng cua mdt hay nhieu bien ngau nhien dugc dir doan dua vao dieu kien ciia cac bien ngau nhien (da tinh toan) khac. Phan tich hdi qui khdng chi la triing khdp dudng cong (lua chgn mdt dudng cong ma viia khdp nhat vdi mdt tap diem du lieu); nd cdn phai trimg khdp vdi mdt md binh cd cac thanh phan ngau nhien va xac djnh {deterministic and stochastic components). Thanh phan xac djnh dugc ggi la bd du doan {predictor) va thanh phin ngau nhien dugc ggi la phan sai sd {error term). Phan tich hdi quy vira la mdt phuang
phap thdng ke toan hgc, viia la mdt phuong phap giai thich vat ly bien dang, cho nen cd the diing de du bao bien dang. Tinh toan hdi quy don bien hay da bien deu la giai he phuong trinh tuyen tinh theo nguyen ly binh phuong nhd nhat, md hinh ham sd dugc bieu dien dudi dang ma tran
Y = xp + 8. (1)
Trong md hinh nay, Y la nhan bien, tuc vector trj do bien dang, vector bieu dien thanh phan nhan bien la Y^ =(y|,y2,...,y„), n la sd lugng cac trj do; phuong trinh (1) cd nhieu bien X va mdi bien cd mdt tham sd /? can phai udc tinh; vector ciia sai sd ngau nhien s la do lech CLia trj do (SSTP trj do), s'=(S|.s,....,sJ.
Trudng hgp trj do la cac thanh phan ngau nhien va tuan theo luat phan phdi chuan N{0,a^]. ta cd the ap dung quy trinh Gauss - Markov. Md hinh ngau nhien la
X . . = E{8.s^}=a=Q,,
Md hinh ham sd va md hinh ngau nhien tren la chuan muc de xay dung md hinh hdi quy. X la ma tran cd dang
(3)
Ma tran (3) bieu thi m nhan tir anh hudng bien dang. Mdi nhan tu anh huong bien dang bieu thj mdt tri do ciia mdt tu bien hoac ham sd
(2)
1 X , , X , , •
^ 1 ^ 2
j \, \l •
" X,,,
' • X 2 m
\:n
cua nd. Chung tao thanh cac phin tu cua ma tran X. tuong img vdi nhan biln cd tit ca n nhdm:
(3 la vector he sd hdi quy, P^ =(Pn,(3,....,p,J.
trong dd:
Pfi la he sd tung do gdc;
PI la he sd do ddc cua Y theo biln x-^ giu cac bien X2,x^,...,x^ khdng ddi;
p, la he sd do ddc ciia Y theo bien X2 giir cac bien Xi,x3,...,x^ khdng ddi;
P,„ la he sd do ddc cua Y theo bien x^ gitr cac bien Xi,X2,-,'x^_i khdng ddi.
He sd do ddc p, the hien su thay ddi trong trj trung binh cua Y tren mdi don vj thay ddi cua Xj khdng ke den sir thay ddi ciia
X2,X3 x^. vi the. cac P, cdn dugc ggi la cac he sd hdi quy rieng phan.
Ddi vdi phuong trinh hdi quy tuyen tinh da bien. ta di tim udc lugng p theo phuong phap binh phuong nho nhat. sao cho:
Z [ y , - y , ) ' Y - Y mm.
Vector nghiem P = (X'X)"'XW.
Do chinh xac hau nghiem cua an Ip3=^oQ„p=<^f,(x^X)-'.
Cac phan tu nam tren dudng cheo chinh ciia ma Iran hiep phuong sai ^ . . chinh la cac phuong sai ciia cac udc lugng Pjtuc la
Gia trj hdi quy hau nghiem
Y = Y + V = Xp = X(X 'X)"' X W = H Y.
Ma tran H dugc ggi la ma tran "mii"" [2], Nguyen ly cua md hinh hdi quy tuyin tinh da bien va each giai gidng nhir md hinh binh sai gian tiep va each giai thdng thudng trong trie dia. Tuy nhien khac d ehd sd nhan tir anh hudng bien dang trong mo hinh hdi quy tuyin ti'nh da bien chua dugc xac djnh trudc, can phai sir d^ng mdt phuong phap nhat djnh de thdng qua hdi quy .xac djnh. lam cho md hinh hdi quy tdi uu.
Trong phan tich hoi quy tu}ln tinh. chirng ta dua \ao khai niem sau: Tdng binh phuong sd hieu chinh Q. tdng binh phuong do lech tdng thi S va tdng binh phuong hdi quy U. Ta cd:
Y - Y = ( Y - Y ) + ( Y - Y ) . (4)
Cac khai niem dugc djnh nghTa nhu sau:
n
s = ( Y - Y f ( Y - Y ) = X ( y i - y ) ' .
1=1
Q = (Y-Y7(Y-Y)=X(yi-y)' v'^v,
U = (Y-Yf(Y-Y)=J(y,- rf.
trong dd, v = —Vy va yj la tri hdi quy cua nhan bien.
Trong hdi quy, he sd xac djnh R la mdt chi sd thdng ke do ludng iniic do tuong quan giua nhan tir anh hudng bien dang vdi gia trj bien dang do dugc [4] . He sd xac djnh gan bang 0 tuc la giua nhan tu anh hudng bien dang va gia trj bien dang do dugc khdng cd lien he gi vdi nhau. Neu he sd gan bang -1 hoac +1 thi nhan til' anh hudng bien dang va gia trj bien dang do dugc cd mdi lien he rat Idn. He sd xac djnh dugc tinh bang ty sd cua hai he sd bien thien sau: R- = U/S.
2. Phu-ong phap hdi quy bac thang
Hdi quy bac thang la thuat toan dua vao kien thirc chuyen nganh va tu lieu quan trac, trudc tien xay dung md hinh hdi quy tuyen tinh don bien, thdng qua danh gia su phii hgp ciia md hinh hdi quy, tien hanh thu nap hoac loai bo dan cac nhan tir anh hudng bien dang, cuoi cimg cd dugc md hinh hdi quy tdt nhat.
Phuong phap danh gia su phii hgp ciia mo hinh hdi quy theo thdng ke toan hgc gdm tinh he sd xac djnh R, dimg kiem djnh thdng ke danh gia tdng the md hinh, tinh sai sd chuan ciia udc lugng va kiem djnh thdng ke tirng bien ddc lap rieng biet. Trong trac dja, ta quan tam tdi kiem djnh tdng the md hinh hdi quy va kiem dinh tinh trdi ciia tirng nhan tu' anh hudng biln dang (nhu nhiet do, thdi gian, ap suat...) tdi nhan biln (la gia trj biln dang do dugc).
Trong phuong trinh hdi quy. cac nhan tir anh hudng bien dang thudng cd tuong quan lan
nhau. nghTa la cd mdi lien he nao do vdi nhau.
Viec cd tuong quan chat che gitra cac biln trong md hinh hdi quy tao ra hien tugng da cdng tuyen. lam cho phuong sai ciia cac udc lugng he sd hdi quy cd gia trj rit Idn. Hien tugng da cdng tuyen con gay sai diu ciia he sd hoi quy, thay vi he sd duo'ng, tire mire nude cao khien cho bien dang cua dap Idn, lai cho ket qua am, tire mire nude cao khien cho dap bien dang ft di.
Thdng qua cac bude kiem dinh ti'nh trdi cua tiing nhan tu anh hudng bien dang va kiem dinh tinh trdi cua nhan tii' anh hudng bien dang tang them khi dua vao phuong trinh hdi quy, cd the quy nap thanh cac bude hdi quy bac thang sau:
a. So tuyen nhan tii' anh hudng bien dang;
b. Xac djnh phuo'ng trinh hdi quy tuyen tinh don bien dau tien. Gia thiet chgn m bien ddc lap lam nhan tir anh hudng bien dang, mdi nhan tir nay lap mdt phuong trinh hdi quy tuyen tinh don bien, tdng cdng cd m phuong trinh. Tinh tdng binh phuo'ng sd hieu chinh Q cua mdi phuong trinh. Chgn phuong trinh hdi quy coQk : mi n{Qi l,m thi thu nap phuong trinh hdi quy cd Qi^ dd sau khi kiem djnh tinh trdi cua nd;
c. Xac djnh phuong trinh hdi quy hai bien tdt nhat tren co' sd phuong trinh hdi quy tuyen tinh don bien, lan lugt tang them nhan tu anh hudng bien dang, lap dugc (m-l) phuong trinh hdi quy tuyen tinh hai bien. Tinh dugc (m-l) tdng binh phuong hdi quy lech AQ, xet hieu AQi = max{AQj}, i = iLm.
Nhan tu tang them thii'j dd la nhan tir "dgi tu\en"". tien hanh kiem dinh tinh trdi cua nd.
nlu thdng qua thi thu nap vao phuong trinh. Dd la phuong trinh hdi quy tuyen tinh hai bien tdt nhit. Neu khdng thdng qua thi dirng lai d phuong trinh hdi quy don bien;
d. Xac dinh phuo'ng trinh hdi quy tuyen tinh ba bien tdt nhat. Qua trinh nay gidng bude (c). Luc do thanh lap (m-2) phuong trinh hdi quy tuyen ti'nh ba bien, tinh (m-2) gia trj tdng binh phuong hdi quy lech AQ. Chu y la sau khi chgn nhan tir anh hudng bien dang thu' ba, van can kiem djnh tinh trdi cua hai nhan tii' anh hudng bien dang da chgn trudc dd, nhir da biet, dieu nay tranh hien tugng da cdng tuyen. Neu kiem dinh khdng dugc thdng qua, ta dii'ng d md hinh hdi quy tuyen tinh hai bien;
e. Neu ba nhan tu anh hudng bien dang deu cd tinh trdi ddi vdi bien phu thudc Y (lugng bien dang), thi theo phuong phap tren tiep tuc tuyen chgn nhan tu anh hudng bien dang thir tu, thu nam,... Cir nhu vay cho den khi khdng the tang them dugc nhan tu mdi va cung khdng the loai bd dugc nhan tii' da chgn nao nua thi dirng. Ket qua ta cd md hinh hdi quy tdt nhat.
3. Ung dung trong thyc tien
Trong bai nay, chiing tdi gidi thieu mdt thuc nghiem dugc tien hanh can than tai khu dd thi mdi Van Quan, Ha Ddng, Ha Ndi. Khu nha thap tang thudc khu dd thj mdi cd chieu cao 3,5 tang, xay thd. Tien hanh quan trac liin tii' 05/9/2005 din 31/3/2009. Su dung may thuy chuan quang hgc do chinh xac cao Ni007, mia Inva. Trong qua trinh quan trac, may va mia thudng xuyen dugc kiem nghiem do chinh xac.
mdi chu ky quan trac deu lay mau nude ngam va nhiet do khdng khi tai khu vuc.
Hinh 1. Gieng quan trac nude ngam Hinh 2 Vet nut tren cdng ti inh do lun
50
10
Hinh 3. Nhiet do khdng khi tai khu vuc quan trie tir 05/9/2005 din 31/3/2009
-10.60 - -10.65
" j ^ w ^ u ^ u ^ i j ^ u ^ i j ^ i / ^ y D
r s i i-st r s i ""^ i~^ <-g
c o o o c o c o c o c o 2 ; 2 ) r s | r s | f N l r N l » - . j r > j i - s l r v | CO rsJ m r n « i rvj
u*^ r^ r i T-i r H <¥\ w^ C\ r^ r^ m rsi r g rsi r\j r\| rsi 'N
Hinh 4. Muc nude ngim tai khu vuc quan trac tii' 05/9/2005 den 31/3/2009 Nhan tu anh hudng bien dang chgn la nhiet
do khdng khi tai khu vuc quan trac, muc nude ngam tai khu vuc quan trac va thdi gian quan trac. Thdi gian quan trac bieu trung cho vj tri diem kiem tra bien dang theo thdi gian, nen vi phan bac mdt cua nd la tdc do lun va vi phan bac hai cua nd la gia tdc lun. Xay dung phuong trinh hdi quy tuyen tinb ddi vdi bien nhiet do t, bien mire nude ngam y va bien thai gian 9. Ta cd phuong trinh hdi quy tuyen tinh ddi vdi bien nhiet do:
Y, =Po,+P|X,+£, .
Phuong trinh hdi quy tuyen tinh ddi vdi bien mire nude nsam;
¥„ P„,, -Fp.X., -h8.^.
Phuong trinh hdi quy tuyen tinh ddi vdi bien thdi gian (x,g = x^):
Y B = P i i o + P . - . > ; e + P 4 > ^ 2 e + S e -
Dua tren day sd lieu quan trac, ta cd cac phuong trinh hdi quy sau:
- Ddi vdi bien nhiet do:
Yt =6610,1642 + 0,4724 Xt+58,3859.
He sd xac dinh R^ = 0,0026 = 0,26%, tiic la 0,26% su bien thien cua do lun do dugc dugc giai thich bang su bien thien cua nhiet do. He sd do ddc PI cho biet khi do cao tang them 1 mm thi nhiet do tang trung binh 0,4724 do. Sai so hau nghiem cua hdi quy la 58,3859 mm. Sai so hau nghiem ciia udc lugng he sd p, la S,, =
1,2150. Gia trj kiem djnh tinh trdi ddi vdi p, la T = 0,38877, tuong ung muc y nghTa 69.89%.
gia trj nay vugt qua xa tieu chuan muc y nghTa a-trong kilm djnh thdng ke {5%). Do vay, ta loai bd nhan tu nhiet do khdi md hinh hdi quy, tire la nhiet do khdng cd tac ddng gi tdi do lun cdng trinh thip tang tai dd thj mdi Van Quan.
- Ddi vdi bien mire nude ngim:
Y^ =9876,1124 + 309,3856 x^ +56.5974.
R; = 0,0628 = 6.28%, tire la 6,28% su biln thien cua do lun do dugc dugc giai thich bang su bien thien ciia muc nude ngim. He sd do ddc
p,cho biet khi do cao giam them 1 mm thi mire nude ngam giam them 309,38m. Sai sd hau nghiem ciia hdi quy la 56,5974 mm. Sai sd hau nghiem ciia udc lugng he sd p, la Sp, = 158,29.
Gia trj kiem djnh tinh trdi ddi vdi p, la T = - 1.9545, tuong ung muc y nghTa a = 5.55%). He sd .xac djnh qua thap va sai sd udc lugng hau nghiem Sp, qua cao, nen ta loai bd nhan tir mirc nude ngam khdi md hinh hdi quy.
- Ddi vdi bien thai gian:
Y„ =6694,9641-1,4108 Xe+0,0024 x,o +6.7862
R^= 0,9809 = 98,09%. tire la 98% sir biln thien ciia do liin do dugc dugc giai thich bang sir gia tang ciia thdi gian. He sd do ddc p, cho biit khi do cao giam them 1 mm thi thdi gian tang them trung binh 1,3644 tuin va p^ cho biet gia toe liin la 0,0024. Sai sd hau nghiem ciia hdi quy la 6,7862 mm. Sai sd hau nghiem cua udc lugng he sd p, la Sp, = 0.0422. he sd P4 la Sp^
= 0,0002. Gia trj kiem djnh tinh trdi ddi vdi p, la T = - 33,4372, tuong irng muc y nghTa 6,6.10"^^ %, vdi P4 la T = 9,8588, tuong img mirc y nghTa a=2,3.10"'^ %>, gia trj nay rat nhd so vd'i tieu chuan {5%),
Cung can ndi them, cdng trinh lun theo thdi gian ddng nghTa vdi muc do cd ket cua dat nen theo thdi gian (liin cd ket). Cudi ciing ta ve dugc dudng hdi quy do kin theo thai gian.
mm(M155)
6750 6700 6650 6600 6550 6500 6450 6400
c
'•'i-S^f—^—^^
D i X ) < ^ 0 0 ^ O v O r s J C 0 ' 3 - O v i > r s i 0 0 ' ' 3 " O < i > r s l 0 0 T H T H f N i m m ^ ' ^ L n « x i y 3 r ^ r ^ c o o < T > 0 0
Time: week
^ 0 " ^ t-( r g rsi r-i rH r-H
r^ 00 -ct m r o • ^ r~* r-i r-t
0 "X) r g 00 - ^ LTi m KD ^£> r^
f-i r-i r-i t-l r-t
Hinh 5 Trj do (net cham) va trj hdi quy (net lien) bieu dien do liin diem 155 khu Van Quan 4. Ket luan
Vdi hai bien nhiet do va muc nude ngam dugc do tai khu vuc quan trac, cd the ket luan:
do kin cdng trinh thap tang khdng lien quan den nhiet do khdng khi va muc nude ngam. Cdng trinh lun do lire nen cd ket cua dat ngam, ket qua do qua trinh thoat nude Id rdng cua dat ngam theo thdi gian.
Mot van de thuc te dat ra la cd phai mdt sd cdng trinh xay dung d Ha Ndi bj kin nhieu la do khai thac nude ngam ?. Qua thuc nghiem nay cho thiy. muc nude ngam tai khu dd thi Van
Quan khdng bien ddi nhieu nen anh hudng khdng rd ret den do liin cdng trinh.
Hdi quy bac thang la thuat toan cd dien va truyen thdng vdi nhung bien tra ldi cd y nghTa thdng ke tinh toan den trong md hinh. Ket qua phan tich cd the tuo'ng ddi phii hgp vdi thuc te.
Phuong phap hdi quy bac thang dugc sir dung thudng xuyen tren the gidi va cd irng dung tdt ddi vdi phan tich bien dang cdng trinh. Tuy nhien, hdi quy bac thang se bat luc vdi dtr lieu cd thdi gian quan trac ngan, dac biet khi dTr lieu cd sd chu ky quan trac k hon sd bien du bao.
TAI LIEU THAM KHAO
[1] Hoang Thanh Huong. Doan Huy va Tudng Chinh. 2003, Xir ly sd lieu quan trac bien dang cdng trinh. (tieng Trung Qude). Nha xuat ban Dai hgc Vu Han.
[2] Peter J. Huber. 1981, Robust Statistics, Published by John Wiley & Sons, Inc.
[3] Rand R. W'ilco.x, 2005. Introduction to Robust Estimation and Hypothesis Testing, Second Edition. Elsevier Academic Press.
[4] Ricardo A. Maronna. R. Douglas Martin and Victor J. Yohai, 2006, Robust Statistics: Theory and Methods. John Wiley & Sons. Ltd ISBN:
0-470-01092-4 SUMMARY
The model regression in constructions deformation analysis Phan Van Hien, University of Mining and Geology
Dinh Xuan Vinh, HUD-CIC construction and investment consultants joint stock company Regression model is a mathematically statistic tool which has been applied widely in variety of areas around the world. Application model regression to analyze data monitoring deformation of the already world applications and achieve results very encouraging. Through the method of regression was predicted to be much the scourge of natural catastrophic to try to liink them. This paper represents the stepwise regression (RS) which has been applied into analysing the deformation of the low-floor construction in Van Quan, Ha Dong, Hanoi. Observation data was collected during four consecutive years enough to reflect the comprehensive deformation of the works.
Ngudi phan bien: Do Nggc Dudng
NGHIEN CUtJ THUAT TOAN FUZZY PID... (tiep theo trang 47)
TAI LIEU THAIVI KHAO
[1]. SIMATIC S7- Fuzzy Control. User Manual.
SIEMENS.
[2]. Lutz Wendt, Taschenbuch der Regelungstecbnik mit MATLAB und Simulink., Veriag Harri Deutsch 2007.
[3]. K.H Borelbach, G.Kreamer, W.Mock und Nows, Automatisierungstechnik mit der SIMATIC 85 und 87. 4 Auflage 2003.
[4]. Nguyen Doan Phudc, Phan Xuan Minh. Ly thuyet dieu khien md. Nha xuat ban khoa hoc va ky thuat 1999.
[5]. Nguyen Doan Phudc, Phan Xuan Minh, Vti Van Ha, 2000. Tu ddng hda vdi SIMATIC 87-300. Nha xuat ban khoa hoc va ky thuat.
[6]. Nguyen Phiing Quang, 2005. Matlab va Simulink - NXB Khoa hoc - Ky thuat.
[7]. http://wl.siemens.com ttp SUMIVIARY
Researching Fuzzy-PID algorithm
in the programmable controller to control heat objects Dang Van Chi, Nguyen Chi Dung, Dao Hieu
University of Mining and Geology
This paper presents an application of Fuzzy-PID algorithm installed for the programmable controller S7- 300 to control heat objects. The results of researching were examined in detail by computer simulations and experiments in the laboratory.
Ngudi phan bien: Cung Quang Khang
