A Study of Fuzzy Linear Regression

Transcription

1 A tuy of Fuzzy Lr Rgrsso Dr. J-Hu J Drtt of ttsts Gr Vy tt Uvrsty A, Mg, 4940 Dr. w- og Mr. J.. o Drtt of A Mtts Nto g- Uvrsty T, Tw, R.O... Itrouto W oft us rgrsso yss to o t rtos tw t rsos t xtory vrs. I trto rgrsso yss, rsus r ssu to u to ro rrors. Tus, sttst tqus r to rfor stto fr rgrsso yss. Howvr, t rsus r sots u to t ftss of t o strutur or rs osrvtos. T urtty ts ty of rgrsso o os fuzzss, ot ro. Z 965 roos fuzzy sts, fuzzss s rv or ttto fuzzy t yss s o rsgy ortt. I orr to osr t fuzzss rgrsso yss, T t. 98 frst roos stuy of fuzzy r rgrsso F o. Ty osr t rtr sttos of F os ur two ftors, y t gr of t fttg t vguss of t o. T stto ros wr t trsfor to r rogrg LP s o ts two ftors. t sur of st ft y rsus ur fuzzy osrto s ot rst T s ro, Do 988 roos t fuzzy st-squrs ro, w s fuzzy xtso of t orry st squrs s o w f st o t s of fuzzy urs. I gr, t fuzzy rgrsso tos rougy v to two tgors. T frst s s o T s LP ro. T so tgory s s o t fuzzy st-squrs ro. I sto, w trou t fuzzy ur ts orto, s st foru, two fuzzy r rgrsso os tr st squrs stts. I sto 3, w trou ty fuzzy ur osytr ouy r tv fuzzy rgrsso o, Yg Ko s st foru, st squrs stts w rt to rs futos. I sto 4, w trto tos of ttg oss outrs fu ots to rv t vrg vus, rsus oo st foru for t fuzzy r rgrsso os..

2 I sto 5, w us t tort rsuts t rvous trs to yz t T s 987 t. T rvto of so ortt forus s gv s.. Itrouto to Fuzzy Lr Rgrsso. Fuzzy ur Its Orto Fuzzy t s tur ty of t, o-rs t or t wt sour of urtty ot us y ross. Ts of t s sy to f tur gug, so s, syotrs, vrots ootrs, t. Fuzzy urs v us to rrst fuzzy t. Ts r so us to o fuzzss of t. Lt R o-so Eu s wt ts or ot y.. A fuzzy ur s ur sotuous ovx futo F: R [0,] wt { x R F x } o-ty. I otr wors, fuzzy ur A s f s ovx orz fuzzy st of t r R so tt tr xsts xty o xo R wt F x o, ts rs Fx s ws otuous. Dfto. Zr [.6-63] Lt L R rsg, s futos fro R to [0,] wt L0; Lx< for x>0; Lx>0 for x<; L0 for x L 0. T fuzzy ur A s of -ty f for, α >0, β > 0 R, x L x, Ax α wr s t tr or ovu of A x R x, β α β r ft rgt srs, rstvy. yoy, A s ot y, α, β. If α β, A,α, α s sytr fuzzy ur, ot y A, α. For st, t gr gotr rtrsts of t rs futo of t or utz fuzzy ur, t trgur fuzzy ur, r sow t foowg: x α x, Ax α x x α, β Aotr x, t xot fuzzy ur, ts rs futo

3 x x Ax s x x s Dfto. Duos980 x, x, wr s s t sr. Lt A, α, β B, α, β two -ty fuzzy urs. T y t xtso r, t foowg ortos r f:. AB, α α, β β α, β λ, λα, λβ w λ >0. λ A λ, 3. λ A λ, α, β λ λβ, λα RL 4. B, β, α RL 5. A B α, β, w λ <0,, β, α RL, α β, β α Dfto.3 Eu st foru Lt A, α B, α two sytr fuzzy urs, t t st tw A B s f s: D α α Lt A, α, β B st tw A B s f s:., α, β two -ty fuzzy urs, t t D w α α wα β β wβ. Wr w > 0, w > 0, w > 0 r rtrry wgts. α. Gr Fuzzy Lr Rgrsso Mo β osr t foowg gr fuzzy r rgrsso o t Mo I: y Ao A x A x... A x,,,.3 wr x r r urs, y [ s, s ] r fuzzy urs, s t tr or o vu, s s t sr, A [ r, r ] r t fuzzy rgrsso rtrs, w s t s rs futo s y. How sou w stt A f t st tw two fuzzy urs r uf? W y trt yl s y R s s t ft rgt ots of t s t, rstvy. For t ft t ots { yl, x, x,..., x,,..., } sry to t rgt t ots { yr, x, x,..., x,,..., } w y us t r rgrsso o y β o βx... β x to ot t foowg stts, rstvy, yl Lo L x L x... L x,,,...,

4 yr Ro R x R x... R x,,,..., L R R L T, A [ r, r] wr, r. Usg ts wy to stt t rgrsso rtrs, A, t osr t vtg of usg t rs futo to sr t t. T fuzzy ot wr ot us t stto of rtrs. I orr to ot or rort stts of fuzzy rgrsso rtrs, A, t st squrs to t st tw two fuzzy urs sou osr. Bs o t fto.3, w us orry st-squrs to to stt t fuzzy rtrs t gr fuzzy r rgrsso o.3, Mo I. Assug tt y, s A, r v s rs futo, ftr rort trsto, w of x > 0. T.3 xrss s, s o, ro, r x, r x..., r x Aorg to t Eu st foru of., t st-squrs stts of r r t vus of, r w z t vu of D wr D [ o x... x s ro r x... r x ] Lt v r ot t gt of vtor v r, t y usg vtor trx xrssos D rwrtt s D r wr s sg trx,,,...,, r r, r,..., r,,,...,, s, s,..., s. o o Lt 0 0 t t soutos of r w z r foows: r D r s.4 T ov to us rgrsso wt rst to tr sr. T stto rsuts r ot rt to t rs futos. But, t tr r t yss, ts to rov ttr rsuts t stto of fuzzy rtr vus..3 ytr Douy Lr Atv Fuzzy Rgrsso Mo Ur t strutur of o I, f w us t Eu st foru t stsqurs to to o r rgrsso wt rst to tr sr rstvy, t t stts sow tt t trs srs r ot rt. But, D Urso Gst 000 t tt t y of t srs s soow t o t gtu of t stt trs. Trfor, ty roos t ouy r tv fuzzy rgrsso o t Mo II to ot t rtr stts.

5 Ty osr sytr fuzzy urs wt trgur rs futo. Wr fuzzy ur, y, s, s oty tf y t two rtrs tr s ft rgt sr. Mo II s f s foows: * * ε.5 * * * ε s.6 wr s trx otg t ut vrs t trx,,,..., s ou vtor otg t rgrsso rtrs of t o * frst o rfrr to s or rgrsso o,,,..., r t vtor of t osrv trs t vtor of t trot trs, rstvy, * ot vg sos, s, s,..., s r t vtor of t ssg srs t vtor of t trot srs, rstvy, ot vg so, s -vtor of s, r t rgrsso rtrs for t so rgrsso o rfrr to s sr rgrsso o. Arty, t ov o s s o two r os. T frst o trots t trs of t fuzzy osrvtos, t so o ys t srs, y ug otr r o ovr t frst o. Osrv tt rtv vrs X r t to osrto Eq..6 troug t osrv trs. T o s to t to out oss r rtos tw t sz of t srs t gtu of t stt trs. Ts s oft t s r wor tos, wr og trs srs s y for st, t urtty or fuzzss wt surt ou o ts gtu. D Urso Gst us t Eu st foru of. t stsqurs to to ot t stts of, su tt t vu of D s z, wr D * * Lt 0, 0 0, ty ot t foowg qutos: Bs o t qutos.7, ty ot t foowg st-squrs trtv soutos of, :

6 .8 T rvto of t rursv soutos of, : fro t frst quto of.7, w sy ot, susttutg t to t so tr qutos of.7, w ot: wr,,,. Fro.0, w ot, susttutg t to.9, w ot sf qurt quto of : M M M 0 Wr M, M 3 By sovg t qurt quto of, w ot, M 3. M ± M 4M M 3, t orrsog soutos of M, T st-squrs stts wr ot y susttutg ts two sts of â,, to D su tt t vu of D s z. Bs o t qutos of â,,, w ou tt o ttr wt of rs futo of t rsos fuzzy ur, y, s, t stts of rtrs r t s. Trfor, ts st squrs stts o ot osr otr oss ss of fuzzy urs. 3. ty of Fuzzy Lr Rgrsso 3. osytr Douy Lr Atv Fuzzy Rgrsso Mo W w v ur rs xtory vrs X,,..., fuzzy t vr Y,, q wr s t tr, q, rstvy, t ft rgt srs, o to orort t oss fu of t gtu of t trs o t srs, t to out D Urso Gst, 000, 00, 00. If t fuzzy rsos urs y q, ] r [

7 osytr wt trgur rs futo. D Urso 003 roos fuzzy rgrsso o t Mo III w s xrss t trx for: * ε * 3. P P * λ * * P 3. * * * q q ρ q g 3.3 wr s trx otg t vtor ott to rs ut * vrs;, r vtors of t osrv trs trot trs, * rstvy; P, P r vtors of osrv ft srs trot ft srs, * rstvy; q, q r vtors of osrv rgt srs trot rgt srs, rstvy; s vtor of rgrsso rtrs for t rgrsso o for ;,, g, r rgrsso rtrs for t rgrsso os for P q ; s vtor of s; ε, λ, ρ r vtors of rsus. Ts o s s o tr su-os. T frst o trots t trs of t fuzzy t, t otr two su-os r ut ovr t frst o y t srs. Ts foruto ows t o to osr oss rtos tw t sz of t srs t gtu of t stt trs, s t s oft ssry r s stus. Mo III osytr ouy r tv fuzzy rgrsso o. D Urso us t Eu st foru of. t st-squrs to to ot t stts of,,, g, su tt t vu of D s z, wr D * * π P P π * q q π q π P P P π q q q g g g π 3.4 q π, π, π q r rtrry ostv wgts. Rursv soutos to t ov syst r fou y qutg to zros t rt rvts wt rst to t rtrs,,, g, : â [ π P π q gπ ] π π g π q P P ĝ q ĥ q g q q 3.5 Wr,,, g, r t trtv st-squrs stts ot t t of t trtv ross. T otzto rour os ot gurt t ttt of t go u, oy o o. For ts rso, t s suggst to tz t trtv q

8 gort y osrg svr oss strtg ots orr to t stty of t souto. Bs o t qutos of,,, g,, w ou tt t stts of rtrs r ot rt to t rs futo of t rsos fuzzy ur. 3. Yg Ko s Dst Foru Ur t strutur of Mo I, II, III t us of Eu st, t stsqurs stts r ot to osr t oss fft of t rs futo of fuzzy rsos urs. I ts sto, w w t t Yg Ko s 996 st foru to try to f t st-squrs stts w r rt to t rs futo of fuzzy rsos urs. Dfto 3. Yg Ko s st foru996 Lt F R ot t st of -ty fuzzy urs. Df w ty of st for y A, α, β, B, α, β F R s foows: A, B α α rβ rβ 3.6 wr L ω ω r 0 0 R ω ω Yg Ko 996 so rov tt F R, s ot tr s. If A B r sytr ty fuzzy urs t r A, B 3 α α. If A B r sytr trgur ty of fuzzy urs t L x x x 0 x fuzzy urs t L x x 0. If A B r xot ty of / x x Γ. or wt t 0 0 st forus of.., t st foru of 3.6 vo t sutv o of t wgts w > 0, w > 0, w > 0 r rtrry wgts. α 3.3 T Lst qurs Estts Bs o Yg Ko s Dst I ts sto, w w osr ty of rsos fuzzy urs us t st foru of 3.6 to f st squrs stts of rgrsso rtrs. Ur t strutur of Mo I, f w v sytr ty fuzzy rsos urs y, s, t r 3.6. T su of squr rror D xrss vtor for: D r r 3 6 r r 4 r Lt 0 0 t r β

9 r 4 r t soutos of r w z D r s foows: r 3.7 Trfor, ur t strutur of Mo I, o ttr wtr w us t st foru of. or 3.6, w ot t s st squrs stts ty r r ot rt wt tr ss rs futos. Nxt, t us osr Mo II D Urso Gst 000, ouy r tv fuzzy rgrsso o, t su of squr rror D xrss vtor for: D [ ] [ ] Lt 0, 0 0, ftr gty tous ot utos s Ax I w ot t foowg st squrs stts: â,, K ± K 4KK K, 3 â 3 X wr K, 3 K, K 3 3, 3,,,. T st-squrs stts wr ot y susttutg ts two sts of â,, to D su tt t vu of D s z. Bs o t qutos of â,,, w ou tt ts st squrs stts o rt to t rs futo of t rsos fuzzy ur, y, s. Ur t strutur of Mo III D Urso 00 osr osytr ty of rsos fuzzy urs, t su of squr rror D xrss vtor for: D [ ] P [ g ] rq

10 3 6 3 rg r g rg P rg q r r g P q r P rq r P P q q Lt 0, 0, 0, 0 0, ftr gty tous g ot utos w ot t foowg qutos: rg r g P P rg q rg q r r g 0 P 0 P 0 r rq r r g r 0 r rq r r g r 0 t qutos r too ot to f gr soutos of,,, g, w ust st t foowg rursv qutos try to us tts softwr to f oss soutos. [3 P P 3 rg r g rg r q r g q r g r] P g rq r r P rq rg r Fro t ov qutos, t s ovous tt t st squrs stts r rt to t rs futo of t rsos fuzzy ur, y, s. 4. Dgost of Outrs Ifus 4. Dgost of Outrs Ifus Lr Rgrsso Mo Atoug rsu yss s usfu ssssg o ft, rturs fro t rgrsso o r oft y t fttg ross. For x, tr y outrs tr t rsos or xtory vrs tt v osr fft o t yss. Osrvtos tt sgfty fft frs rw fro t t r s to fut. Mtos for ssssg fu r tyy s o t g t vtor of rtr stts w osrvtos r t.

11 T vrg x x s ssot wt t t t ot surs, t t s of t xtory vrs, ow fr t osrvto s fro t otr - osrvtos. For t ot wt g vrg, ros 0, ts t s oss outr. T rsus y y r us to tt oss outrs for t rsos vr y, wr ŷ s t t rt y vu. A rg vu of ts t t t ot ou outr. O y so us y y to tt oss outrs, wr y s t rt y vu w t t osrvto s ro fro t yss. A rg vu of so ts t t D t ot ou outr. I trto r rgrsso yss, o y us t oo st, ΥΥ s s to tt oss fut t ots wr Υ s t rt Y vtor vu w t t osrvto s ro fro t yss, s t ur of rtrs, s s t squr rror. A rg vu of D ts tt t t ot ou fut osrvto. O of t vtgs of usg oo st s tt o ttr wt surt uts r us t xtory rsos vrs, t vu of D w ot fft. 4. Dgost of Outrs Ifus Fuzzy Lr Rgrsso Mo I ts sto, w w osr t Mo I s.3 rv t orrsog forus of,, D to tt oss outrs fut t ots. For Mo II s.5.6 Mo III s 3., 3., 3.3, w wr ot to rv y forus of,, D to tt oss outrs fut t ots. Bs o t Eu st, w ot s t rvtos Ax A. s x s x r 4. x s xr 4. s wr x s t rsu fro t tr of fuzzy ur s xr s t rsu fro t srs of fuzzy ur. â r r f.4.

12 ry, s o t Yg Ko s st w ot s t rvtos Ax A. s y, y s, y y Fro , t rto tw r t s s gr r rgrsso o. Tt s, rg vu of, ts t t t ot ou outr. I orr to rv foru sr to t oo s st ur t fuzzy vrot, w to f w ty of st tw fuzzy vtors. Lt F R ot t st of -ty fuzzy urs, ~ F R { X, X,..., X X F R } s t st of fuzzy so vtors. ~ Bs o t st fto F R, w f w st F R. L 4. Lt : F R F R R tr, for y two fuzzy vtors ~ X, X,..., X, Υ Y, Y,..., Y F R, f ~, Υ X, Y 4.5 t ~ ~ s tr F R. If s ot tr t so os ~ s t roof Ax 3. W s s tr, f oo s st D s foows: ~ Υ, Υ r r D s s t w ot s t rvto Ax 4 D s s wr s. W s Yg Ko s tr, f oo s st ~ Υ, Υ D s D s foows: 4.6 { } r r r r s t w ot s Ax A.4

13 wr s D s s Atoug t forus oos t s, t vus of s r ffrt. I gr, s 4.7 s rgr t t vu of s 4.6, trfor t oo st ut 4.6 s rgr t t oo st ut 4.7. Fro w w tt D s fft y t vrg vu rsu. Ts s t s s t trto rgrsso yss. w wr ot to rv sr forus s for Mo II III, t st w o s to t t ot r t rut t vus of, D, t. 5. Dt Ayss I ts sto, w w us t T s t 987, s T to ustrt t tort rsuts w w ot t rvous stos. T t st ots tr t vrs, o fuzzy rsos vr t t ots. W oy osr xot fuzzy rsos vus. T vtg of usg xot rs futo s tt w oy to oos rort vu Not: s t vu of ty fuzzy urs to rft t struto of rsos vr. If t vus of rsos vr t to f outs t trv of xstg t t w oos sr vu. Otrws, w w oos rgr vu to sr t rs futo. w wr ot to rv t st squrs stts for o III w oy osr xot rs futo, w w us o I II to o t yss. Ts sow t rsuts of usg t Eu st, Yg Ko s st ffrt vus. I t, t ots t st squrs stts, t su of squr rsus, t vrg vu, t vus of, t OOK st, D. ur t Eu s st foru, t vu w ot fft t rsuts of usg o I II, trfor w oy gv t rsuts of s T 3.

14 T : T s Dt 987 s # Prtors Fuzzy Rsos Vr x Y, r x x , , , , , , , , , ,44 T : Mo I,, Lst-qurs Estts Ur Eu Dst s #, s, s D 96,4 93.0, ,47.48, , , , , , , * * 0.5* 6 94, , , , * * 0.50* 8 6,78.65, * ,5 0.89, , , ,3.5,7.9,5.03 r 8.0,.64,.0, T 3: Mo II,, Lst-qurs Estts Ur Eu Dst s #, s, s D 96, , ,47.04, ,33 50., * 0.66* 4 06, , , , * 74.67* 90.38* , , * 0.63* 7 07,4 08., * ,78.7, * 0.6* 9 08,5.48, * , , ,3.43,7.6,5.40, 0.9, 4.88, 36.98

15 T 4: Mo I,., Lst-qurs Estts Ur Yg Ko s Dst s #, s, s D 96,4 93.0, ,47.48, , , , , , , * * 0.5* 6 94, , , , * * 0.55* 8 6,78.65, * ,5 0.89, , , ,3.5,7.9,5.03, r 8.0,.64,.0,.85, T 5: Mo II,., Lst-qurs Estts Ur Yg Ko s Dst s #, s, s D 96,4 94.9, ,47.3, , , * * , , , , * , , * * , , * , , * 498.7* 0.3* 9 08,5.8, , , ,3.30,7.4,5.9, 0.43, , 67.3

16 T 6: Mo I,, Lst-qurs Estts Ur Yg Ko s Dst s #, s, s D 96,4 93.0, ,47.48, , , , , , , * * 0.5* 6 94, , , , * * 0.55* 8 6,78.65, * ,5 0.89, , , ,3.5,7.9,5.03, r 8.0,.64,.0,.85, 49.9 T 7: Mo II,, Lst-qurs Estts Ur Yg Ko s Dst s #, s, s D 96, , ,47.33, ,33 5.8, * * , , ,79 9., * , , * 493.7* , , * , , * * 0.3* 9 08,5.65, , , ,3.9,7.45,5.7, 0.43, , 87.75

17 T 8: Mo I, 3, Lst-qurs Estts Ur Yg Ko s Dst s #, s, s D 96,4 93.0, ,47.48, , , , , , , * * 0.5* 6 94, , , , * * 0.55* 8 6,78.65, * ,5 0.89, , , ,3.5,7.9,5.03, r 8.0,.64,.0,.85, T 9: Mo II, 3, Lst-qurs Estts Ur Yg Ko s Dst s #, s, s D 96, , ,47.33, ,33 5.3, * 339.6* , , ,79 9., * , , * 498.5* , , * , , * 479.* 0.3* 9 08,5.67, , , ,3.9,7.44,5.7, 0.43, , 97.3

18 T 0: Mo I, 0, Lst-qurs Estts Ur Yg Ko s Dst s #, s, s D 96,4 93.0, ,47.48, , , , , , , * * 0.5* 6 94, , , , * * 0.54* 8 6,78.65, * ,5 0.89, , , ,3.5,7.9,5.03, r 8.0,.64,.0,.85, 58.0 T : Mo II, 0, Lst-qurs Estts Ur Yg Ko s Dst s #, s, s D 96, , ,47.3, , , * * , , , , * , , * 545.3* , , * , , * 50.7* 0.3* 9 08,5.86, , , ,3.30,7.40,5.9, 0.43, -.96, 83.9

19 5. Dsusso Fro T 3, t stts of tr sr ur o I r ttr t tos stts o II. I tory, f w usyg Ko s st, t stts of tr sr ur o II sou fft y t vu of. But, s o Ts 5,7,9,, w fou tt ffrt vus o ot fft vry u o t stts. I tory, t st foru vus o ot fft t stts of o I rtrs. But, ty o fft t rtr stts o II. Bs o Ts 3 5, t usg of ffrt foru s or fft o t rtr stts o II. s #5 #7 v rgr vrg vus, ty r oss outrs fro t rtors. I o I, s o t vu of t ss o oss outrs fro t rsos vr. Howvr, s o t vus of Ts,4,6,8, s #5,7,8 r oss outrs fro t rsos vr. I o II ur t Eu st, T 3 sows tt s #3,5,6,8,9 r t fv oss outrs fro t rsos vr. But, ur t Yg Ko s st, Ts 5,7,9, sow tt oy s #3,6,8 r t tr oss outrs fro t rsos vr. Ur o I, s o Ts,4,6,8,0, s #5,7 v rgr D vus ty r fut osrvtos. Ur o II Eu st, t 3 sows tt s #3,6,8 v rgr D vus. But, o II us Yg Ko s st, oy s #8 s rg D vu s fut ot s Ts 5,7,9,. If w us xot rs futo for our fuzzy urs usyg Ko s st, ow to st oos t vu to o fuzzy r rgrsso ur o II? T sst ru s to oos t vu su tt t rsu su of squrs,, s sst. Bs o ts 5,7,9,, w s t st o s.

20 APPENDIX A.: T rvto of â,, 3.8 D ] [ ] [ Lt 0 D, 0 D 0 D, w ot 0 D 3 3 A.. 0 D A.. 0 D A..3 Fro A.., w ot â 3 3 X, susttutg â to A.. A..3, w ot 0 D A..4 0 D A..5 wr,,,. Fro A..5, w ot susttutg t to A..4 w ot qurt quto of, 0 3 K K K. T souto s 3 4 K K K K K ± A.: T rvto of 4., 4.3, 4.4 I. Bs o Eu st foru, w v r x s x

21 s x, r x trfor s r s x x x s xr x x s II. Bs o Yg Ko s st foru, w v x [ s x x r] [ s x x r] 3 x [ s x r] 3 s x [ s x xr ] [ s x xr ] 3 x s xr 3 A.3: Proof of L 4. s I orr to rov ~ s tr, w to rov t foowg tr rorts: ~ ~ ~., Υ F R,, Υ 0. If, Υ 0 t Υ. ~ ~ ~., Υ F R,, Υ Υ,. ~ ~ ~ ~ 3.,Υ,Ζ F R,, Υ, Ζ Ζ, Υ. s tr, t s sy to sow tt rorts r stsf. W to sow tt rorty 3 s stsf: ~, Υ X, Y X, Z Z, Y ~ ~, Ζ Ζ, Υ ~ ~, Ζ Ζ, Υ ~ ~ ~ Trfor,, Υ, Ζ Ζ, Υ. X, Z Z, Y

22 Assu tt F R, ~ squ F R,.., ε > 0 for, >, X, X < s ot tr. Lt { }, Ν X, X ~, uy, > < ε. T, ~, < ε. H,, { X } s uy squ F R. Trfor, X F R, X X. Lt X, X,..., X. X X for ε > 0, Ν ε for >, w v X, X <,,,...,. Lt x{,,..., }, ~ T for >, w v, X, X < ε. Tt s,. A.4: T rvto of qutos ~ Ur t Eu st:, D Υ Υ Y, Y s s x x xr xr s s s s ~ Ur Yg Ko s st:, D Υ Υ Y, Y s s { x x [ x xr x xr ] s s 3 [ x xr x xr ] } s s

23 REFERENE. D. Duos, H. Pr, Fuzzy ts ysts: Tory Atos, A Pusrs, Nw Yor, Zr, H. J., 996, Fuzzy t Tory Its Atos, Kuwr A Prss, Dorrt. 3. Drr, N. R. t, H. 980, A Rgrsso Ayss, Wy, Nw Yor. 4. D Urso, P. Gst, T. 000, A Lst-squrs Aro to Fuzzy Lr Rgrsso Ayss, outto ttsts Dt Ayss 34, D Urso, P., 003, Lr Rgrsso Ayss for Fuzzy/rs Iut Fuzzy/rs Outut Dt, outto ttsts Dt Ayss 4, T, H., 987, Fuzzy Dt Ayss y Possst Lrr Mos, Fuzzy ts ysts 4, T, H., U,., As, K., 98, Fuzzy Lr Rgrsso Mo, IEEE Trs. ysts M yrt, Xu, R. L,., 00, Mutso Lst-squrs Fttg Wt Fuzzy Mo, Fuzzy ts ysts 9, Yg, M.. Ko,. H., 996, O ss of -urs ustrg Prours for Fuzzy Dt, Fuzzy ts ysts 84, Yg, M.. Lu, H. H., 003, Fuzzy Lst-squrs Agors for Itrtv Fuzzy Lr Rgrsso Mos, Fuzzy ts ysts 35, PENA, D 005, A Nw ttsts for Ifu Lr Rgrsso, Totrs VOL. 47, NO., -.