Исследование непараметрического алгоритма распознавания образов: Методические указания

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khklZ\e_gguobaklbebah\Zgguowe_f_glh\b gZ[ex^Z_fuohi_jZlhjhfgZ“aZ[blhf”ihf_oZfbwdjZg_L<-^bkie_y W
Knhjfbjh\Zggh_Weywlh]h baf_jyxlkyaZ\bkbfhklb\_jhylghkl_cijZ\bevgh]hh[gZjm`_gbyρih beh`ghclj_\h]bρelhlaZrmfe_gghklbnhgZρad nbh[jZaZρad h[j, ijbaZ^Zgghckljmdlmj_rmfZ Bah[jZ`_gb_nhjfbjm_lkyijh]jZfghGZwdjZg_L<-^bkie_y Wey ]jZnbq_kdh]hj_`bfZjZ[hluWeykbf\hevgh]h^bkie_y[m^_f\dZq_kl\_ we_f_glZbah[jZ`_gbybkihevah\ZlvagZdhf_klhkbf\heZ Bah[jZ`_gb_[m^_ljbkh\Zlvkyijh[_eZfbbih^oh^ysbfb kbf\heZfbgZijbf_j beb[md\hc“O”<h[hbokemqZyolZdb_ fbgbfZevgu_^bkdj_lubah[jZ`_gby[m^_fgZau\Zlvwe_f_glZfb wdjZgZBah[jZ`_gbyh[jZah\^_ebfgZfhghkljmdlmgu_– gZjbkh\Zggu_dZd[udbklvxihklhygghclhesbgub ihebkljmdlmjgu_–gZjbkh\Zggu_dbklyfbjZaghclhesbgujbk >Ze__[m^_fbkihevah\Zlvlhevdhfhghkljmdlmjgu_bah[jZ`_gby >eyhibkZgbykljmdlmjubah[jZ`_gbch[jZaZbihf_obrmfZ  bkihevam_fke_^mxsb_iZjZf_lju: 1)iehsZ^vkljmdlmjgh]hwe_f_glZbah[jZ`_gbygZijbf_j: 



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H[eZklbbaf_g_gby Pad nbPad h[j . 





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1). hlghr_gb_fkb]gZeihjh]D9  hlghr_gb_fihjh]rmf: V/σx , eb[hkb]gZe/rmf: Z/σx  nmgdpb_cjZkij_^_e_gbyrmfZϕ(x) k^bki_jkb_cσx2 b gme_\uffZlh`b^Zgb_f >Ze__[m^_fjZkkfZljb\Zlvrmfbf_xsbcghjfZevgh_ jZkij_^_e_gb_ >eylh]hqlh[u^eykemqZcghc\_ebqbguZobf_xs_c ^bki_jkbxσx2b X Zihevah\ZlvkynhjfmeZfbbeblZ[ebpZfb klZg^Zjlgh]hghjfZevgh]h jZkij_^_e_gbykemqZcghc\_ebqbguN01 c σN b N = 0 1

ϕ(N01)=

exp[–N012/2]

2π

gZ^hkemqZcgmx\_ebqbgmZop_gljbjh\ZlvdZbghjfbjh\Zlvd σx Lh]^Zk\yavnmgdpbbjZkij_^_e_gbykmffuZok\_ebqbgZfb Zb9[m^_llZdZydZdgZjbkZ\_jhylghklbaZdjZrb\Zgby hij_^_eylkybgl_]jZeZfb ∞

z2 − 2

 V 1 PN01 >  = e dz ∫ ρad n= σx  2π V  σ

(4)



x

∞ z −  V − a 1 2 > = P N e dz ∫ ρad h[j=  01 σ  2π V−a x   σ 2

(5)



x

Banhjfme b \b^ghqlh: 0 < Pad n=< 0,5 

V →∞ σx

V →0 σx

ρad h[j >= ρad n

B\k_]^Z



>eyZ > V: ρad h[j V −a < −(3 ÷ 5) σx 



→ijbσo → ijZdlbq_kdbijb

V −a > (3 ÷ 5) σx <_ebqbgZ\_jhylghklbPh[j\u[bjZ_lkybkoh^ybaf_lh^bdb ijh\_^_gbywdki_jbf_glZ >eyZ < V: ρad h[j 

→ijbσo → ijZdlbq_kdbijb

F_lh^bdZihlhq_qgh]hbaf_j_gby GZqZevgu_mkeh\by3ad n, Pad h[j, ( 



V V −a ), ( ) bkljmdlmjZ σx σx

Sr H^gZbawlbo\_ebqbgy\ey_lky\ZjbZglhc S h[j Zj]mf_glhf bkdhfhcaZ\bkbfhklbhklZevgu_\_ebqbgu–aZ^Zggu_ dhgklZglu Nhjfbjm_lkyF~(30 ÷100) ij_^ty\e_gbcj_ZebaZpbc dZjlbgdbjZ[hq_]hihey^eyi_j\h]hagZq_gby\ZjbZglu Nbdkbjmxlkyj_amevlZluijbgylbyj_r_gbyhi_jZlhjhfbih nhjfmeZf \uqbkeyxlky\_ebqbguρih bρell_gZoh^blkyi_j\Zy lhqdZbkdhfhcaZ\bkbfhklb AZl_f[_jzlky\lhjh_agZq_gb_\ZjbZglubkgh\Z^ZzlkyF ij_^ty\e_gbcknhjfbjh\Zgguoih\_jhylghklbdZjlbghd IhemqZ_lkyke_^mxsZy–\lhjZylhqdZdjb\hcbl^>eywlhc f_lh^bdb3h[j=0,5. rmfZ

F_lh^bdZ]jmiih\h]hbaf_j_gby GZijbf_jieZgbjm_lkyihemqblvdjb\mxaZ\bkbfhklb: ρih=f(

S r a V  =const, S h[j V σx ↑

\ZjbZglZ

=const)

iZjZf_lju djb\hc

L_i_jvwdki_jbf_glkhklhblbaFh[sij_^ty\e_gbckj_^b dhlhjuo^he`ghkh^_j`ZlvkyDk_jbcihFdd «D ij_^ty\e_gbcj_ZebaZpbcbah[jZ`_gbyjZ[hq_]hiheykh^_j`Zs_]h h[jZakaZ^Zggufd–fagZq_gb_f

a K  . ∑ M k = M  V  k =1 

Weylh]hqlh[u\uqbke_gbyihnhjfme_ bf_eb h^bgZdh\mxlhqghklvgZ^hqlh[udZ`^Zyk_jbybf_eZh^bgZdh\h_

dhebq_kl\hij_^ty\e_gbcFd K FRQVWghqlh[uij_^ty\e_gby kemqZcghq_j_^h\ZebkvbajZaguok_jbcDjhf_lh]hgZ^h_sz h[_ki_qblvqlh[uqZklvij_^ty\e_gbcg_kh^_j`ZeZh[jZaZ L_i_jvfh`gh\aylvPh[jf_gvr_Bih\b^bfhfm ikboheh]bq_kdb\hkijbylb_kemqZcghklbihy\e_gbyg_ihy\e_gby  h[jZaZkhojZgblkymhi_jZlhjZijbmf_gvr_gbbPh[j\iehlv^h \_ebqbgu.qlhiha\heblagZqbl_evghkhdjZlblvh[s__\j_fy wdki_jbf_glZ >eyj_ZebaZpbbmdZaZgguolj_[h\Zgbcfh`ghij_^eh`blv gZijbf_jke_^mxsbcZe]hjblfnhjfbjh\Zgbyij_^ty\e_gbc “Hj]ZbaZpby fZkkb\Z ba D kqzlqbdh\ qbkeZ ^ey i hl 0 ^hD gp M[i] := C ij_^ty\e_gbc \ dZ`^hc k_jbb b aZg_k_gb_ \ dp dZ`^uc kqzlqbd bkoh^gh]h qbkeZ K ^ey i hl 0 ^hD “<\h^ fZkkb\Z ba D gZqZevguo agZq_gbc a/V. gp \\h^ P[i] dp _keb {kemqZcgh_qbkeh[0,1]}>1/K lhb^lbf_ldZ bgZq_b^lbf_ldZ \kz_keb Fih^ijh]jZffZ nhjfbjh\Zgby bah[jZ`_gby jZ[hq_]h ihey [_a h[jZaZ FL :={kemqZcgh_qbkeh[0,1]} “kemqZcguc \u[hj agZq_gby Jad h[j , gZijbf_j ijb D L := INT(L*10) “INT–p_eZy qZklv qbkeZ _keb M[L] = 0 lhb^lbf_ldZ \kz_keb M[l ]:=M[L]–1 Pad h[j:=P[L] ih^ijh]hjZffZnhjfbjh\Zgby bah[jZ`_gby jZ[hq_]h ihey kh^_j`Zs_]h h[jZa 







k Jad h[j=P[L].





Ih^]hlh\dZdjZ[hl_ – KhklZ\blvf_lh^bdm\uiheg_gby\u[jZggh]h\ZjbZglZ eZ[hjZlhjghcaZ^Zqb – Hij_^_eblvh[eZklvbaf_g_gbybqbkehlhq_dgZqZevguomkeh\bc – KhklZ\blv^eywlh]h\ZjbZglZijh]jZffmjZ[hluW


–fh^mevnhjfbjh\Zgbyj_ZebaZpbbbah[jZ`_gbyjZ[hq_]h ihey –fh^mev^bZeh]ZWeybkdexq_gbyjZkk_ygby\gbfZgbyhi_jZlhjZdZ`^Zyk_jby ij_^ty\e_gbchlebqZxsZykybkoh^gufbmkeh\byfb^he`gZ ijh^he`Zlvkyg_[he__–fbgkihke_^mxsbfi_j_ju\hf–5 fbg Ke_^m_l\\_klbij_^\Zjbl_evgmxk_jbx^eyh[mq_gby hi_jZlhjZbklZ[bebaZpbb_]hjZ[hluWlZk_jby\ijhlhdhe_ j_amevlZlh\bkke_^h\Zgbyg_nbdkbjm_lky Ihihemq_gguf^Zggufkljhylkydjb\u_aZ\bkbfhkl_cJihb JelhlgZqZevguomkeh\bcbijhba\h^blkybokjZ\g_gb_knhjfmehc (1). Bg^b\b^mZevgu_\ZjbZglueZ[hjZlhjguoaZ^Zq 6.1

H[jZa:

6.2 6.3 6.4 6.5

Sh[j=5, Sr=1, Sr/ S ÷ h[j=1, V/σx= var =0.1 ÷ 3 a/V= const =0.5 H[jZalhl`_a/V= const =1.1 H[jZalhl`_a/V= const =2 H[jZalhl`_a/σx= var =0.1 ÷ 3, V/σx= const =0.1 H[jZalhl`_ V/σx= const =1

6.6–Lh`_qlh–gh^eyh[jZaZ: Ijbeh`_gb_ Bgl_]jZe: ∞

Q(y)=1/

2π ∫ e

− z2 2

dz

y

Fh`ghgZclbihlZ[ebpZf[4],eb[h\uqbkeblvjZaeh`_gb_f\jy^ :e]hjblf>ey =< y >= 3 y

P(y)= 1/ 2π ∫ e

−t 2 2

−∞ ∞

Q(y)= 1/ 2π

∫e

−t 2 2

2

dt = 0.5–1/ 2π e

y

1:

y 2n+1 1⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ (2n +1)

dt = 0.5+1/ 2π e−y / 2 ⋅ ∑ − y2 / 2

y 2n+1 ⋅∑ 1⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ (2n +1)

\\h^< z := exp(–Y*Y/2)/sqr(2*π) r := 1 s := 0 n := 0 k := 2*n+1 r :=r*k w := s s := s+(Yk)/r n :=n+1 _kebs–w >= 10–10 lh goto 1 bgZq_ P := 0.5+z*s Q := 0.5–z*s

Lhqghklv\uqbke_gbcm\_ebqb\Z_lkyijbmf_gvr_gbbY LZ[ebpZ: Y PjZkkqblZggu_ JlZ[ebqgu_ 0,01 0,50398936 0,503989 0,1 0,53982784 0,539828 1 0,84134775 0,841345

3

0,9986501

0,99866501

IJH=J:FF: 10 20 30 40 50 60 70 80 90

INPUT “Y=”; Y LET Z = EXP(–Y*Y/2)/SQR(2*PI) LET R = 1 : LET S =0 : LET N = 0 : LET W = S LET K = 2*N+1 : LET R = R*K LET S = S+(Y ↑ K)/R : LET N = N+1 IF S–W >= 1E–10 THEN GOTO 40 LET P = 0.5+S*Z PRINT “Y=”; Y ; TAB2 ; “P=”; P ; TAB2 ; “Q=”; 1–P GOTO 10

>eyY > wlhlZe]hjblfg_jZ[hlZ_lldqbkehrlj_[m_lky [hevr_^himklbfh]h^eyjZajy^ghck_ldbWey Y > 0 y

F(y) = 1/ 2π ∫ e 0

−t 2 2

3 5 dt =1/ 2π  y − y + y 2



i=

0

 y7 + ...  2 ⋅1!⋅3 2 ⋅ 2!⋅5 23 ⋅ 3!⋅7 

1

2

−

3

A3 A0 A1 A2 H[hagZqbfghf_jZi = 0,1,2,… qe_gh\jy^ZA0,A1,A2,A3… y 2 i+1 y2 , B = –B , n = 1,2,3,… n n–1 2n 2 i ⋅ i! B An= n , A0 = y, B0 = y 2n + 1

Bi =

1:

\\h^Y a := Y b := Y s := Y n := 0 n := n+1 b := –b*(x2)/(2*n) a := b/(2*n+1) s := s+a _keb lhgoto 1 bgZq_

F := s/sqr(2*π) Q := 0.5–F Lhqghklv\uqbke_gbyJm\_ebqb\Z_lkyijbmf_gvr_gbbY. LZ[ebpZ Y PjZkqzlgu_ JlZ[ebqgu_ 0,5 0,69146246 0,691462 1 0,84134475 0,841345 3 0,9986501 0,9986501 4.5 0,99999676 0,9999966023 5 1,0000007 0,99999971335 IJH=J:FF: 10 20 30 40 50 60 70 80 90 100 110 120

INPUT “Y=”; Y LET A = Y : LET B = Y : LET S = Y LET N = 0 LET N = N+1 LET B = –B*(X ↑ 2)/(2*N) LETA = B/(2*N+1) LET S = S+A IF ABS(A) >= 1E–10 THEN GOTO 40 LET Q = S/SQR(2*PI) LET Q = 0.5+Q PRINT “Y=”; Y, “Q=”; Q GOTO 10

:e]hjblf: Lhqghklv\uqbke_gbyQihZe]hjblfZfbaZf_lgh mf_gvrZ_lkyijb[hevrboY. >eyY > 2lhqgu_\uqbke_gbyh[_ki_qb\Z_ljZaeh`_gb_ bgl_]jZeZ\_jhylghkl_c\p_igmx^jh[v ∞

Q(y)= 1/ 2π

∫e y

−t 2 2

e−y ⋅ 2π y + 2

dt =

\\h^F \\h^Y z := Y pbde^eyN = M ^hrZ]–1

1 1 y +

2 y +

3 y +

4 y + ...

z := Y+z /N dhg_ppbdeZ Q := EXP(–Y*Y/2)/SQR(2*π)/z P := 1–Q Lhqghklv\uqbke_gbyQihwlhfmZe]hjblfmaZf_lgh mf_gvrZ_lkyebrvijbY < 0,5. F \j_fykqzlZk Y PjZkqzlgh_ JlZ[ebqgh_ 0,1 0,53804137 0,2 0,57925653 0,3 0,61791142 F \j_fykqzlZk Y PjZkqzlgh_ JlZ[ebqgh_ 0,1 0,53969777 0,539828 0,2 0,57925969 0,579260 0,3 0,6179142 0,617911 IJH=J:FF: 10 20 30 40 50 60 70 80 90

INPUT “M=”; M INPUT “Y=”; Y LET Z = Y FOR N = M TO 1 STEP –1 LET Z = Y+N/Z NEXT LET Q = EXP(–Y*Y/2)/SQR(2*PI)/Z PRINT “Y=”; Y ; TAB2 ; “Q=”; Q ; TAB2 ; “P=”; 1–Q GOTO 20