🥇Ukuhlafuna uhlengahlengiso

Kule nqaku, siya kuhlalutya izibalo zethiyori zenguqu imisebenzi yokubuyisela umgca в umsebenzi woguqulo lwelogiti eguqukileyo (ebizwa ngokuba ngumsebenzi wempendulo yolungiselelo). Emva koko, sebenzisa i-arsenal eyona ndlela inokwenzeka, ngokuhambelana nemodeli yokuguqulwa kwezinto, sifumana umsebenzi wokulahlekelwa Ilahleko yoLungiselelo, okanye ngamanye amazwi, sizakuchaza umsebenzi apho iparameters zevektha yobunzima zikhethwa kumzekelo wobuyiselo. .

Ulwandlalo lwenqaku:

Masiphinde unxulumano lomgca phakathi kweenguqu ezimbini
Masichonge imfuneko yenguqu imisebenzi yokubuyisela umgca в umsebenzi impendulo yolungiselelo
Masiqhube iinguqu kunye nemveliso umsebenzi impendulo yolungiselelo
Masizame ukuqonda ukuba kutheni eyona ndlela incinci yesikwere imbi xa ukhetha iiparamitha msebenzi Ilahleko yoLungiselelo
Sisebenzisa eyona ndlela inokwenzeka ukumisela imisebenzi yokukhetha iparameter :
5.1. Ityala 1: umsebenzi Ilahleko yoLungiselelo kwizinto ezinobizo lweklasi 0 и 1:

5.2. Ityala 2: umsebenzi Ilahleko yoLungiselelo kwizinto ezinobizo lweklasi -1 и +1:

Inqaku lizaliswe yimizekelo elula apho zonke izibalo kulula ukuzenza ngomlomo okanye ephepheni; kwezinye iimeko, i-calculator inokufuneka. Ngoko zilungiselele :)

Eli nqaku lijoliswe ngokukodwa kwizazinzulu zedatha kunye nenqanaba lokuqala lolwazi kwiziseko zokufunda koomatshini.

Inqaku liya kunika kwakhona ikhowudi yokuzoba iigrafu kunye nokubala. Yonke ikhowudi ibhalwe ngolwimi i-python 2.7. Makhe ndichaze kwangaphambili "ngento entsha" yenguqulelo esetyenzisiweyo - le yenye yeemeko zokuthatha ikhosi eyaziwayo ukusuka. I-Yandex kwiqonga lemfundo le-intanethi elaziwa ngokulinganayo Coursera, yaye, njengoko ubani enokucinga, umbandela wawulungiselelwe ngokusekelwe kwesi sifundo.

01. Ukuxhomekeka kumgca othe ngqo

Kunengqiqo ukubuza umbuzo- ingaba ukuxhomekeka komgca kunye nokuhlehliswa komgaqo kunento yokwenza nayo?

Ilula! Uhlengahlengiso lolungiselelo yenye yeemodeli ezizezomdidi womgca. Ngamagama alula, umsebenzi womhleli womgca kukuqikelela amaxabiso ekujoliswe kuwo ukusuka kwizinto eziguquguqukayo (i-regressors) . Kukholelwa ukuba ukuxhomekeka phakathi kweempawu kunye namaxabiso ekujoliswe kuwo umgca. Yiyo ke loo nto igama lomdidiyeli - linear. Ukuyibeka ngokurhabaxa kakhulu, imodeli yohlengahlengiso lolungiselelo isekwe kwingqikelelo yokuba kukho unxulumano lomda phakathi kweempawu. kunye namaxabiso ekujoliswe kuwo . Olu lunxibelelwano.

Kukho umzekelo wokuqala kwi-studio, kwaye, ngokuchanekileyo, malunga nokuxhomekeka kwe-rectilinear yamanani afundwayo. Kwinkqubo yokulungiselela inqaku, ndifumene umzekelo osele ubeke abantu abaninzi kumda - ukuxhomekeka kwangoku kumbane. ("Uhlalutyo olusetyenzisiweyo lokuhlehla", N. Draper, G. Smith). Siza kuyijonga nalapha.

Ngokutsho Umthetho ka-Ohm:

phi - amandla angoku, - I-Voltage, - ukuchasa.

Ukuba besingazi Umthetho ka-Ohm, ngoko ke sinokufumana ukuxhomekeka empirically ngokutshintsha kunye nokulinganisa , ngelixa uxhasa ilungisiwe. Emva koko siya kubona ukuba igrafu yokuxhomekeka ukusuka inika umgca othe ngqo ngaphezulu okanye ngaphantsi kwimvelaphi. Sithi "ngaphezulu okanye ngaphantsi" kuba, nangona ubudlelwane buchanekile ngokwenene, imilinganiselo yethu inokuba neempazamo ezincinci, kwaye ngoko ke amanqaku kwigrafu angeke awele ngokuthe ngqo kumgca, kodwa aya kusasazeka ngeenxa zonke.

Igrafu 1 "Ukuxhomekeka" ukusuka »

Ikhowudi yokuzoba itshathi

import matplotlib.pyplot as plt
%matplotlib inline

import numpy as np

import random

R = 13.75

x_line = np.arange(0,220,1)
y_line = []
for i in x_line:
    y_line.append(i/R)
    
y_dot = []
for i in y_line:
    y_dot.append(i+random.uniform(-0.9,0.9))


fig, axes = plt.subplots(figsize = (14,6), dpi = 80)
plt.plot(x_line,y_line,color = 'purple',lw = 3, label = 'I = U/R')
plt.scatter(x_line,y_dot,color = 'red', label = 'Actual results')
plt.xlabel('I', size = 16)
plt.ylabel('U', size = 16)
plt.legend(prop = {'size': 14})
plt.show()

02. Imfuneko yokuguqula i-linear regression equation

Makhe sijonge omnye umzekelo. Makhe sicinge ukuba sisebenza ebhankini kwaye umsebenzi wethu kukuqinisekisa ukuba kunokwenzeka ukuba umboleki abuyisele imali mboleko ngokuxhomekeke kwizinto ezithile. Ukwenza lula umsebenzi, siya kuqwalasela izinto ezimbini kuphela: umvuzo wenyanga mboleki kunye nemali yokubuyisela imali yenyanga.

Umsebenzi unemiqathango kakhulu, kodwa ngalo mzekelo sinokuqonda ukuba kutheni kunganelanga ukusebenzisa imisebenzi yokubuyisela umgca, kwaye ufumanise ukuba zeziphi iinguqu ekufuneka zenziwe kunye nomsebenzi.

Makhe sibuyele kumzekelo. Kuyaqondwa ukuba xa uphezulu umvuzo, kokukhona umboleki uya kukwazi ukwaba inyanga nenyanga ukubuyisela imali mboleko. Kwangaxeshanye, kuluhlu oluthile lomvuzo olu dlelwane luya kuba lungqamana. Ngokomzekelo, makhe sithathe uluhlu lomvuzo ukusuka kwi-60.000 RUR ukuya kwi-200.000 RUR kwaye sicinge ukuba kuluhlu lomvuzo oluchaziweyo, ukuxhomekeka kobungakanani bentlawulo yenyanga kubungakanani bomvuzo kumgca. Masithi uluhlu oluchaziweyo lwemivuzo lubonakaliswe ukuba umlinganiselo womvuzo-kwintlawulo awukwazi ukuwela ngaphantsi kwe-3 kwaye umboleki kufuneka abe ne-5.000 RUR kwindawo yokugcina. Kwaye kuphela kule meko, siya kucinga ukuba umboleki uya kubuyisela imali mboleko ebhankini. Emva koko, i-equation yohlengahlengiso yomgca iya kuthatha ifom:

apho , , , - umvuzo -umboleki, - intlawulo-mboleko -th umboleki.

Ukutshintsha umvuzo kunye nentlawulo yemali-mboleko kunye neeparamitha ezisisigxina kwi-equation Ungagqiba ekubeni uyikhuphe okanye uyale imali-mboleko.

Ukujonga phambili, siyaqaphela ukuba, kunye neeparamitha ezinikiweyo umsebenzi wokubuyisela umgca, esetyenziswa kwi imisebenzi yempendulo yolungiselelo iya kuvelisa amaxabiso amakhulu aya kwenza ukubala kube nzima ukumisela amathuba okubuyisela imali mboleko. Ngoko ke, kucetywayo ukunciphisa i-coefficients yethu, masithi, ngamaxesha angama-25.000. Olu tshintsho kwii-coefficients aluyi kutshintsha isigqibo sokukhupha imali mboleko. Masikhumbule le ngongoma kwixesha elizayo, kodwa ngoku, ukwenza kucace ngakumbi oko sithetha ngako, makhe siqwalasele imeko kunye nabathathu abanokuba ngababoleki.

ITheyibhile 1 “Abanokuba ngababoleki”

Ikhowudi yokwenziwa kwetafile

import pandas as pd

r = 25000.0
w_0 = -5000.0/r
w_1 = 1.0/r
w_2 = -3.0/r

data = {'The borrower':np.array(['Vasya', 'Fedya', 'Lesha']), 
        'Salary':np.array([120000,180000,210000]),
       'Payment':np.array([3000,50000,70000])}

df = pd.DataFrame(data)

df['f(w,x)'] = w_0 + df['Salary']*w_1 + df['Payment']*w_2

decision = []
for i in df['f(w,x)']:
    if i > 0:
        dec = 'Approved'
        decision.append(dec)
    else:
        dec = 'Refusal'
        decision.append(dec)
        
df['Decision'] = decision

df[['The borrower', 'Salary', 'Payment', 'f(w,x)', 'Decision']]

Ngokuhambelana nedatha etafileni, uVasya, kunye nomvuzo we-120.000 RUR, ufuna ukufumana imali mboleko ukuze akwazi ukuyibuyisela ngenyanga kwi-3.000 RUR. Sinqume ukuba ukuze sivume ukubolekwa imali, umvuzo kaVasya kufuneka udlule ngokuphindwe kathathu inani lentlawulo, kwaye kufuneka kusekho i-5.000 RUR eseleyo. UVasya uyanelisa le mfuneko: . Nokuba i-106.000 RUR ihleli. Nangona into yokuba xa ubala siye sawanciphisa amathuba Amaxesha angama-25.000, umphumo wawufanayo-imali-mboleko inokuvunywa. UFedya naye uya kufumana imali-mboleko, kodwa uLesha, nangona efumana kakhulu, kuya kufuneka anqande ukutya kwakhe.

Masizobe igrafu yale meko.

Itshathi yesi-2 “Ukuhlelwa kwababoleki”

Ikhowudi yokuzoba igrafu

salary = np.arange(60000,240000,20000)
payment = (-w_0-w_1*salary)/w_2


fig, axes = plt.subplots(figsize = (14,6), dpi = 80)
plt.plot(salary, payment, color = 'grey', lw = 2, label = '$f(w,x_i)=w_0 + w_1x_{i1} + w_2x_{i2}$')
plt.plot(df[df['Decision'] == 'Approved']['Salary'], df[df['Decision'] == 'Approved']['Payment'], 
         'o', color ='green', markersize = 12, label = 'Decision - Loan approved')
plt.plot(df[df['Decision'] == 'Refusal']['Salary'], df[df['Decision'] == 'Refusal']['Payment'], 
         's', color = 'red', markersize = 12, label = 'Decision - Loan refusal')
plt.xlabel('Salary', size = 16)
plt.ylabel('Payment', size = 16)
plt.legend(prop = {'size': 14})
plt.show()

Ngoko, umgca wethu othe tye, owakhiwe ngokuhambelana nomsebenzi , yahlula ababoleki “ababi” kwabalungileyo. Abo baboleki abaneminqweno engahambelani namandla abo bangaphezu komgca (uLesha), ngelixa abo, ngokweeparamitha zemodeli yethu, banako ukubuyisela imali mboleko bangaphantsi komgca (Vasya noFedya). Ngamanye amazwi, sinokuthi oku: umgca wethu othe ngqo wahlula ababoleki kwiiklasi ezimbini. Masizichaze ngolu hlobo lulandelayo: eklasini Siya kuhlela abo baboleki ekunokwenzeka ukuba bayibuyise imali-mboleko njengabo okanye Siza kubandakanya abo baboleki abanokuthi bangakwazi ukuhlawula imali-mboleko.

Makhe sishwankathele izigqibo ngalo mzekelo ulula. Makhe sithathe inqaku kunye, nokutshintsha ulungelelwaniso lwenqaku kwi-equation ehambelanayo yomgca , qwalasela izinto ezintathu onokukhetha kuzo:

Ukuba inqaku liphantsi komgca kwaye siyabela iklasi , ngoko ixabiso lomsebenzi iyakuba positive ukusuka до . Oku kuthetha ukuba sinokucinga ukuba amathuba okubuyisela imali-mboleko angaphakathi . Ukuba likhulu kwexabiso lomsebenzi, kokukhona liphezulu amathuba.
Ukuba inqaku lingaphezulu komgca kwaye siyabela iklasi okanye , ngoko ixabiso lomsebenzi lizakuba libi ukusuka до . Emva koko siya kucinga ukuba amathuba okuhlawula amatyala angaphakathi kwaye, okukhona ixabiso elipheleleyo elipheleleyo lomsebenzi, kokukhona ukuzithemba kwethu kuphezulu.
Inqaku likumgca othe ngqo, kumda phakathi kweeklasi ezimbini. Kule meko, ixabiso lomsebenzi iyakulingana kunye namathuba okubuyisela imali-mboleko iyalingana .

Ngoku, makhe sicinge ukuba asinazo izinto ezimbini, kodwa zininzi, kwaye hayi ezintathu, kodwa amawaka ababoleki. Emva koko endaweni yomgca othe ngqo siya kuba nayo m-ubukhulu indiza kunye ne-coefficients asiyi kukhutshwa emoyeni obhityileyo, kodwa sithathwe ngokwemigaqo yonke, kunye nesiseko sedatha eqokelelweyo kubaboleki abaye okanye abangayibuyiselanga imali mboleko. Kwaye ngokwenene, qaphela ukuba ngoku sikhetha ababoleki sisebenzisa ii-coefficients esele zaziwa . Ngapha koko, umsebenzi wemodeli yohlengahlengiso lwenkqubo kukumisela ngokuchanekileyo iiparamitha , apho ixabiso lomsebenzi welahleko Ilahleko yoLungiselelo iya kuthatha ubuncinci. Kodwa malunga nokuba i-vector ibalwa njani , siya kufumana ngakumbi kwicandelo lesi-5 lenqaku. Okwangoku, sibuyela kwilizwe lesithembiso - kwibhanki yethu kunye nabathengi bakhe abathathu.

Enkosi kumsebenzi siyazi ukuba ngubani onokubolekwa kwaye ngubani ofuna ukwaliwa. Kodwa awukwazi ukuya kumlawuli ngolwazi olunjalo, kuba bafuna ukufumana kuthi ithuba lokubuyisela imali-mboleko ngumboleki ngamnye. Kwenziwe ntoni? Impendulo ilula - kufuneka siguqule umsebenzi ngandlela thile , amaxabiso abo alele kuluhlu Kumsebenzi amaxabiso awo aya kulala kuluhlu . Kwaye umsebenzi onjalo ukhona, ubizwa ngokuba umsebenzi wempendulo yolungiselelo okanye uguqulo lwelogit eguqukileyo. Ukudibana:

Makhe sibone inyathelo ngenyathelo ukuba isebenza njani umsebenzi impendulo yolungiselelo. Qaphela ukuba siya kuhamba kwicala elichaseneyo, okt. siyakuthatha ukuba siyalazi ixabiso elinokwenzeka, elikuluhlu ukusuka до kwaye emva koko siya "kukhulula" eli xabiso kulo lonke uluhlu lwamanani ukusuka до .

03. Sifumana umsebenzi wokuphendula we-logistic

Inyathelo 1. Guqula amaxabiso anokubakho kuluhlu

Ngexesha lokuguqulwa komsebenzi в umsebenzi impendulo yolungiselelo Siza kushiya umhlalutyi wethu wamatyala yedwa kwaye sikhenkethe kubabhuki endaweni yoko. Hayi, kunjalo, asiyi kubheja, yonke into enomdla kuthi kukho intsingiselo yeli binzana, umzekelo, ithuba ngu-4 ukuya ku-1. I-Odds, eyaziwayo kubo bonke ababheji, lumlinganiselo "wempumelelo" ukuya ku- " ukusilela”. Ngokwemigaqo enokwenzeka, izinto ezinokuthi zenzeke ngamathuba okuba isiganeko senzeke sahlulwe ngokuba nokwenzeka kokuba isiganeko singenzeki. Masibhale phantsi ifomula yamathuba okuba kwenzeke isiganeko :

phi - amathuba okuba kwenzeke isiganeko, — amathuba okuba kwenzeke isiganeko

Umzekelo, ukuba kunokwenzeka ukuba ihashe eliselula, elinamandla nelidlalayo elibizwa ngokuba yi "Veterok" lizakubetha ixhegokazi elidala neligqabileyo eligama lingu "Matilda" kugqatso lilingana , ke amathuba okuphumelela "Veterok" aya kuba к kwaye ngokuphambeneyo, ukwazi izinto ezingathandekiyo, akuyi kuba nzima kuthi ukubala okunokwenzeka :

Ke, siye safunda "ukuguqulela" amathuba okunokwenzeka, athatha amaxabiso kuwo до . Masithathe elinye inyathelo kwaye sifunde “ukuguqulela” amathuba okuba kumgca-manani uphela ukusuka до .

Inyathelo 2. Guqula amaxabiso anokubakho kuluhlu

Eli nyathelo lilula kakhulu - masithathe i-logarithm ye-odds kwisiseko senombolo ka-Euler kwaye sifumana:

Ngoku siyazi ukuba , uze ubale ixabiso iya kuba lula kakhulu kwaye, ngaphezu koko, iya kuba yinto entle: . Lena Yinyaniso.

Ngomdla wokwazi, makhe sijonge ukuba , emva koko silindele ukubona ixabiso elibi . Siyajonga: . Ilungile lo nto.

Ngoku siyayazi indlela yokuguqula ixabiso elinokwenzeka ukusuka до ecaleni komgca manani wonke ukusuka до . Kwinqanaba elilandelayo siya kwenza ngokuchaseneyo.

Okwangoku, siyaqaphela ukuba ngokuhambelana nemithetho yelogarithm, ukwazi ixabiso lomsebenzi , ungabala amathuba:

Le ndlela yokumisela izinto ezingathandekiyo iya kuba luncedo kuthi kwinqanaba elilandelayo.

Inyathelo lesi-3. Masivelise ifomula yokumisela

Ngoko safunda, sisazi , fumana amaxabiso omsebenzi . Nangona kunjalo, enyanisweni, sifuna ngokuchaseneyo - ukwazi ixabiso fumana . Ukwenza oku, makhe sijike kwingcamango efana ne-inverse odds function, ethi:

Kwinqaku asiyi kuyifumana le fomyula ingentla, kodwa siya kuyijonga ngokusebenzisa amanani avela kumzekelo ongentla. Siyazi ukuba nge-Odds ka-4 ukuya ku-1 (), amathuba okuba isiganeko senzeke ngu-0.8 (). Masenze enye indawo: . Oku kungqamana nokubala kwethu ebesikwenza ngaphambili. Masiqhubele phambili.

Kwinqanaba lokugqibela siye safumanisa ukuba NONE Sifumana:

Yahlula zombini inani kunye nedinomineyitha nge , Emva koko:

Ukuba kunokwenzeka, ukuqinisekisa ukuba asenzanga mpazamo naphi na, masenze enye itshekhi encinci. Kwinqanaba lesi-2, thina ngenxa uzimisele ukuba . Emva koko, ukutshintsha ixabiso kumsebenzi wempendulo yolungiselelo, silindele ukufumana . Sitshintsha kwaye sifumana:

Sivuyisana nawe, mfundi othandekayo, sisanda kukhangela kwaye sivavanya umsebenzi wokuphendula. Makhe sijonge igrafu yomsebenzi.

Igrafu 3 "Umsebenzi wempendulo yoLungiselelo"

Ikhowudi yokuzoba igrafu

import math

def logit (f):
    return 1/(1+math.exp(-f))

f = np.arange(-7,7,0.05)
p = []

for i in f:
    p.append(logit(i))

fig, axes = plt.subplots(figsize = (14,6), dpi = 80)
plt.plot(f, p, color = 'grey', label = '$ 1 / (1+e^{-w^Tx_i})$')
plt.xlabel('$f(w,x_i) = w^Tx_i$', size = 16)
plt.ylabel('$p_{i+}$', size = 16)
plt.legend(prop = {'size': 14})
plt.show()

Kuncwadi unokufumana kwakhona igama lalo msebenzi njenge umsebenzi we-sigmoid. Igrafu ibonisa ngokucacileyo ukuba olona tshintsho luphambili kukwenzeka kwento eyeyeklasi lwenzeka phakathi koluhlu oluncinci ngokwentelekiso. , kwindawo ethile до .

Ndicebisa ukuba sibuyele kumhlalutyi wethu wamatyala kwaye simncede abale amathuba okubuyiswa kwemali mboleko, kungenjalo usengozini yokushiywa ngaphandle kwebhonasi :)

ITheyibhile 2 “Abanokuba ngababoleki”

Ikhowudi yokwenziwa kwetafile

proba = []
for i in df['f(w,x)']:
    proba.append(round(logit(i),2))
    
df['Probability'] = proba

df[['The borrower', 'Salary', 'Payment', 'f(w,x)', 'Decision', 'Probability']]

Ngoko ke, sizimisele ukuba nokwenzeka kokubuyisela imali-mboleko. Ngokubanzi, oku kubonakala kuyinyaniso.

Enyanisweni, amathuba okuba uVasya, kunye nomvuzo we-120.000 RUR, uya kukwazi ukunika i-3.000 RUR kwibhanki rhoqo ngenyanga isondele kwi-100%. Ngendlela, kufuneka siqonde ukuba ibhanki inokukhupha imali mboleko kuLesha ukuba umgaqo-nkqubo webhanki ubonelela, umzekelo, ukuboleka abathengi abanakho ukubuyiswa kwemali mboleko engaphezulu kwe-0.3. Kuphela nje kule meko ibhanki iya kudala indawo yokugcina ilahleko enokwenzeka.

Kufuneka kwakhona kuqatshelwe ukuba umlinganiselo womvuzo wokuhlawula ubuncinane ubuncinane be-3 kunye nomda we-5.000 RUR uthathwe kwisilingi. Ke ngoko, asikwazanga ukusebenzisa i-vector yobunzima kwimo yayo yokuqala . Sidinga ukunciphisa kakhulu i-coefficients, kwaye kule meko sahlulahlula i-coefficient nganye ngama-25.000, oko kukuthi, ngokwenene, silungelelanise umphumo. Kodwa oku kwenziwa ngokukodwa ukwenza lula ukuqonda umbandela kwinqanaba lokuqala. Ebomini, asiyi kudinga ukusungula kunye nokulungelelanisa i-coefficients, kodwa sifumane. Kumacandelo alandelayo enqaku siza kufumana ii-equations apho iiparameters zikhethwa khona .

04. Ubuncinci indlela yokumisela i-vector yobunzima kumsebenzi wempendulo yolungiselelo

Sele siyayazi le ndlela yokukhetha i-vector yobunzima , njengoko indlela yezikwere ezincinci (LSM) kwaye eneneni, kutheni ke ngoko singayisebenzisi kwiingxaki zokuhlelwa kokubini? Ngokwenene, akukho nto ikuthintelayo ekusebenziseni MNC, kuphela le ndlela kwiingxaki zokuhlela inika iziphumo ezingachanekanga kakhulu kune Ilahleko yoLungiselelo. Kukho isiseko sethiyori kule nto. Masiqale sijonge umzekelo omnye olula.

Makhe sicinge ukuba iimodeli zethu (usebenzisa MSE и Ilahleko yoLungiselelo) sele iqalile ukukhetha i-vector yobunzima kwaye siye sayeka ukubala kwinqanaba elithile. Akukhathaliseki nokuba phakathi, ekupheleni okanye ekuqaleni, eyona nto iphambili kukuba sele sinamaxabiso athile e-vector yobunzima kwaye masicinge ukuba kweli nyathelo, i-vector yobunzima. kuzo zombini iimodeli akukho mahluko. Emva koko thatha iintsimbi ezibangelwayo kwaye uzifake endaweni yazo umsebenzi impendulo yolungiselelo () kwinto ethile eyeyeklasi . Sivavanya iimeko ezimbini xa, ngokuhambelana nevektha ekhethiweyo yobunzima, imodeli yethu iphosakele kakhulu kwaye ngokuchaseneyo - imodeli iqinisekile ukuba into leyo yeyodidi. . Makhe sibone ukuba zeziphi izohlwayo eziya kukhutshwa xa usebenzisa MNC и Ilahleko yoLungiselelo.

Ikhowudi yokubala izohlwayo ngokuxhomekeke kumsebenzi welahleko osetyenzisiweyo

# класс объекта
y = 1
# вероятность отнесения объекта к классу в соответствии с параметрами w
proba_1 = 0.01

MSE_1 = (y - proba_1)**2
print 'Штраф MSE при грубой ошибке =', MSE_1

# напишем функцию для вычисления f(w,x) при известной вероятности отнесения объекта к классу +1 (f(w,x)=ln(odds+))
def f_w_x(proba):
    return math.log(proba/(1-proba)) 

LogLoss_1 = math.log(1+math.exp(-y*f_w_x(proba_1)))
print 'Штраф Log Loss при грубой ошибке =', LogLoss_1

proba_2 = 0.99

MSE_2 = (y - proba_2)**2
LogLoss_2 = math.log(1+math.exp(-y*f_w_x(proba_2)))

print '**************************************************************'
print 'Штраф MSE при сильной уверенности =', MSE_2
print 'Штраф Log Loss при сильной уверенности =', LogLoss_2

Ityala lempazamo — imodeli yabela iklasi into ngokunokwenzeka kwe-0,01

Isohlwayo ekusebenziseni MNC iya kuba:

Isohlwayo ekusebenziseni Ilahleko yoLungiselelo iya kuba:

Ityala lokuzithemba ngamandla — imodeli yabela iklasi into ngokunokwenzeka kwe-0,99

Isohlwayo ekusebenziseni MNC iya kuba:

Isohlwayo ekusebenziseni Ilahleko yoLungiselelo iya kuba:

Lo mzekelo ubonisa kakuhle ukuba xa kukho impazamo enkulu umsebenzi welahleko Los Loss imohlwaya imodeli kakhulu ngaphezu MSE. Ngoku masiqonde ukuba yintoni imvelaphi yethiyori ekusebenziseni umsebenzi welahleko Los Loss kwiingxaki zokuhlela.

05. Ubuninzi bendlela enokwenzeka kunye nokuhlehla kwenkqubo

Njengoko kwakuthenjisiwe ekuqaleni, eli nqaku lizaliswe yimizekelo elula. Kwi-studio kukho omnye umzekelo kunye neendwendwe ezindala - ababoleki bebhanki: uVasya, uFedya noLesha.

Ukuba kunokwenzeka, ngaphambi kokwenza umzekelo, mandikukhumbuze ukuba ebomini sijongana nesampulu yoqeqesho lwamawaka okanye izigidi zezinto ezinamashumi okanye amakhulu eempawu. Nangona kunjalo, apha amanani athathiweyo ukuze akwazi ukungena lula kwintloko yesazi sedatha ye-novice.

Makhe sibuyele kumzekelo. Makhe sicinge ukuba umlawuli webhanki wagqiba ekubeni akhuphe imali mboleko kubo bonke abasweleyo, nangona i-algorithm yamxelela ukuba angayikhuphi kuLesha. Kwaye ngoku ixesha elaneleyo lidlulile kwaye siyazi ukuba yiyiphi kula magorha amathathu abuyisele imali mboleko kwaye ayihlawulanga. Yintoni eyayilindeleke: uVasya noFedya babuyisela imali mboleko, kodwa uLesha akazange. Ngoku makhe sicinge ukuba esi siphumo siya kuba yisampula entsha yoqeqesho kuthi kwaye, ngexesha elifanayo, ngathi yonke idatha kwizinto ezichaphazela amathuba okubuyisela imali mboleko (umvuzo womboleki, ubungakanani bentlawulo yenyanga) yanyamalala. Emva koko, ngokuqondayo, sinokucinga ukuba wonke umboleki wesithathu akayibuyiseli imali mboleko ebhankini, okanye ngamanye amazwi, ukuba nokwenzeka komboleki olandelayo ukuba abuyisele imali mboleko. . Le ngqikelelo enengqondo inokuqinisekiswa kwethiyori kwaye isekelwe eyona ndlela inokwenzeka, ngokufuthi kuncwadi ebizwa ngalo umgaqo wamathuba aphezulu.

Okokuqala, makhe siqhelane nesixhobo sokucinga.

Isampulu enokwenzeka lithuba lokufumana kanye isampuli enjalo, ukufumana kanye oko kuqwalaselwe/iziphumo, oko kukuthi. imveliso yamathuba okufumana umphumo ngamnye wesampula (umzekelo, ukuba imboleko kaVasya, iFedya kunye neLesha ibuyiswe okanye ayibuyiswanga ngexesha elifanayo).

Umsebenzi onokwenzeka inxulumanisa ukubakho kwesampulu kumaxabiso eparameters zonikezelo.

Kwimeko yethu, isampuli yoqeqesho sisikimu seBernoulli ngokubanzi, apho ukuguquguquka okungahleliwe kuthatha amaxabiso amabini kuphela: okanye . Ke ngoko, ukubakho kwesampulu kunokubhalwa njengento enokwenzeka yeparameter ngolu hlobo:

Elingeno lingasentla lingatolikwa ngolu hlobo lulandelayo. Ithuba elidibeneyo lokuba uVasya noFedya baya kubuyisela imali mboleko ilingana , ithuba lokuba uLesha AYI kubuyisela imali-mboleko iyalingana (kuba ibingeyiyo imbuyekezo yemali-mboleko eyenzeka), ngoko ke amathuba adibeneyo azo zontathu iziganeko ziyalingana. .

Eyona ndlela inokwenzeka yindlela yokuqikelela iparameter engaziwayo ngokwandisa imisebenzi enokwenzeka. Kwimeko yethu, kufuneka sifumane ixabiso elinjalo , apho ifikelela ubuninzi bayo.

Ivela phi eyona ngcamango-ukukhangela ixabiso leparameter engaziwayo apho umsebenzi wokucingela ufikelela kubuninzi? Imvelaphi yombono ivela kwingcamango yokuba isampuli kuphela komthombo wolwazi olukhoyo kuthi malunga nabemi. Yonke into esiyaziyo malunga nabemi imelwe kwisampulu. Ke ngoko, konke esinokukutsho kukuba isampulu yeyona mbonakalo ichanekileyo yoluntu olukhoyo kuthi. Ke ngoko, kufuneka sifumane iparameter apho isampulu ekhoyo iba yeyona inokwenzeka.

Ngokucacileyo, sijongene nengxaki yokuphucula apho kufuneka sifumane eyona ndawo iphezulu yomsebenzi. Ukufumana inqaku eligqithiseleyo, kuyimfuneko ukuqwalasela imeko yomyalelo wokuqala, oko kukuthi, ukulinganisa i-derivative yomsebenzi kwi-zero kwaye uxazulule i-equation ngokubhekiselele kwipharamitha efunwayo. Nangona kunjalo, ukukhangela i-derivative yemveliso yezinto ezininzi kunokuba ngumsebenzi omde; ukunqanda oku, kukho ubuchule obukhethekileyo-ukutshintshela kwilogarithm. imisebenzi enokwenzeka. Kutheni le nto olo tshintsho lunokwenzeka? Masinikele ingqalelo kwinto yokuba asijongi isiphelo salo msebenzi ngokwawo, kunye nenqanaba eliphezulu, oko kukuthi, ixabiso leparameter engaziwayo , apho ifikelela ubuninzi bayo. Xa ufudukela kwi-logarithm, i-extremum point ayitshintshi (nangona i-extremum ngokwayo iya kwahluka), ekubeni i-logarithm ingumsebenzi we-monotonic.

Makhe, ngokuhambelana noku ngasentla, siqhubeke siphuhlisa umzekelo wethu ngeemali-mboleko ezivela kuVasya, Fedya noLesha. Okokuqala masiqhubele phambili kwi ilogarithm yomsebenzi wokucingela:

Ngoku sinokuyahlula ngokulula ibinzana nge :

Kwaye ekugqibeleni, qwalasela imeko yomyalelo wokuqala- silinganisa i-derivative yomsebenzi kwi-zero:

Ngoko ke, uqikelelo lwethu olucacileyo lwamathuba okubuyisela imali-mboleko yathetheleleka ngokwethiyori.

Kulungile, kodwa kufuneka senze ntoni ngolu lwazi ngoku? Ukuba sicinga ukuba wonke umboleki wesithathu akayibuyiseli imali ebhankini, ngoko ke lo mva uya kubhanga. Kulungile, kodwa kuphela xa kuvavanywa ukuba kunokwenzeka ukubuyiswa kwemali mboleko elinganayo Asizange sithathele ingqalelo izinto ezichaphazela ukubuyiswa kwemali mboleko: umvuzo womboleki kunye nobukhulu bentlawulo yenyanga. Masikhumbule ukuba ngaphambili sasibala amathuba okubuyisela imali-mboleko ngumxhasi ngamnye, sithathela ingqalelo ezi zinto zinye. Kusengqiqweni ukuba sifumane amathuba anokwenzeka ngokwahlukileyo kukulingana rhoqo .

Makhe sichaze ukuba nokwenzeka kweesampulu:

Ikhowudi yokubala iisampulu ezinokwenzeka

from functools import reduce

def likelihood(y,p):
    line_true_proba = []
    for i in range(len(y)):
        ltp_i = p[i]**y[i]*(1-p[i])**(1-y[i])
        line_true_proba.append(ltp_i)
    likelihood = []
    return reduce(lambda a, b: a*b, line_true_proba)
        
    
y = [1.0,1.0,0.0]
p_log_response = df['Probability']
const = 2.0/3.0
p_const = [const, const, const]


print 'Правдоподобие выборки при константном значении p=2/3:', round(likelihood(y,p_const),3)

print '****************************************************************************************************'

print 'Правдоподобие выборки при расчетном значении p:', round(likelihood(y,p_log_response),3)

Isampulu enokubakhona ngexabiso elingaguqukiyo :

Isampulu enokwenzeka xa kubalwa ukuba nokwenzeka kwentlawulo yemali-mboleko kuthathelwa ingqalelo izinto :

Ukuba nokwenzeka kwesampulu enokubalwa ngokuxhomekeke kwizinto ezithe zavela ziphezulu kunokuba nokwenzeka ngexabiso elithe gqolo elinokwenzeka. Ithetha ntoni le nto? Oku kuphakamisa ukuba ulwazi malunga nemiba yenze ukuba kube lula ukukhetha ngokuchanekileyo amathuba okubuyiselwa kwemali mboleko kumxhasi ngamnye. Ngoko ke, xa ukhupha imali mboleko elandelayo, kuya kulunga ngakumbi ukusebenzisa imodeli ecetywayo ekupheleni kwecandelo lesi-3 lenqaku lokuvavanya amathuba okubuyiswa kwetyala.

Kodwa ke, ukuba sifuna ukwandisa Isampulu yamathuba omsebenzi, ngoko kutheni ungasebenzisi i-algorithm ethile eya kuvelisa amathuba okuba uVasya, uFedya noLesha, umzekelo, alingane no-0.99, 0.99 kunye no-0.01, ngokulandelanayo. Mhlawumbi i-algorithm enjalo iya kwenza kakuhle kwisampulu yoqeqesho, kuba iya kuzisa ixabiso lesampulu yokunokwenzeka kufutshane , kodwa, okokuqala, i-algorithm enjalo iya kuba nobunzima ngokukwazi ukwenza ngokubanzi, kwaye okwesibini, le algorithm ngokuqinisekileyo ayizukuba ngumgca. Kwaye ukuba iindlela zokulwa nokugqithisa (ngokulinganayo amandla obuthakathaka obuthakathaka) ngokucacileyo azibandakanyi kwisicwangciso seli nqaku, makhe sihambe ngenqaku lesibini ngokubanzi. Ukwenza oku, phendula nje umbuzo olula. Ngaba amathuba okuba uVasya noFedya babuyisele imali mboleko iyafana, ngokuqwalasela izinto ezaziwa kuthi? Ukusuka kwimbono yengqiqo yesandi, ngokuqinisekileyo akunjalo, ayikwazi. Ngoko uVasya uya kuhlawula i-2.5% yomvuzo wakhe ngenyanga ukubuyisela imali mboleko, kunye noFedya - phantse i-27,8%. Kwakhona kwigrafu ye-2 "Ukuhlelwa komthengi" sibona ukuba uVasya ude kakhulu kumgca ohlula iiklasi kuneFedya. Kwaye ekugqibeleni, siyazi ukuba umsebenzi kuba uVasya noFedya bathatha amaxabiso ahlukeneyo: 4.24 yeVasya kunye ne-1.0 yeFedya. Ngoku, ukuba uFedya, umzekelo, ufumene umyalelo wobukhulu okanye wacela imali mboleko encinci, ngoko ke amathuba okubuyisela imali mboleko yeVasya kunye neFedya iya kufana. Ngamanye amazwi, ukuxhomekeka komgca akunakuqhathwa. Kwaye ukuba ngenene sibale amathuba , kwaye akazange abakhuphe emoyeni omncinci, sinokuthi ngokukhuselekileyo ukuba amaxabiso ethu Okungcono kusivumele ukuba siqikelele ukuba nokwenzeka kwembuyekezo yemali-mboleko ngumboleki ngamnye, kodwa ekubeni siye savumelana ukuba sicinge ukuba ukumiselwa kwe-coefficients. lwenziwa ngokwemigaqo yonke, ngoko siya kucinga njalo - ii-coefficients zethu zisivumela ukuba sinike uqikelelo olungcono lwamathuba :)

Nangona kunjalo, siphumelele. Kweli candelo kufuneka siqonde ukuba i-vector yobunzima inqunywe njani , okuyimfuneko ukuvavanya amathuba okubuyisela imali-mboleko ngumboleki ngamnye.

Makhe sishwankathele ngokufutshane ukuba yeyiphi na i-arsenal esiya kuyikhangela iingxaki :

1. Sicinga ukuba ubudlelwane phakathi kokuguquguquka okujoliswe kuyo (ixabiso lokubikezela) kunye nesiphumo esichaphazela umphumo sinomgca. Ngenxa yoko, isetyenziswa umsebenzi wokubuyisela umgca uhlobo lwe , umgca ohlula izinto (abaxhasi) kwiiklasi и okanye (abaxumi abakwaziyo ukuhlawula imali-mboleko kunye nabo bangenako). Kwimeko yethu, i-equation inefomu .

2. Sisebenzisa umsebenzi welog eguqukileyo uhlobo lwe ukugqiba ukuba nokwenzeka kwento eyeyeklasi .

3. Siluthathela ingqalelo iseti yoqeqesho lwethu njengokuphunyezwa kwenkqubo eqhelekileyo Iinkqubo zeBernoulli, oko kukuthi, into nganye uguquko olungenamkhethe luyenziwa, olunokwenzeka (eyayo into nganye) ithatha ixabiso 1 kunye nokunokwenzeka - 0.

4. Siyazi ukuba yintoni na ekufuneka siyenzile Isampulu yamathuba omsebenzi kuthathelwa ingqalelo izinto ezamkelweyo ukuze isampulu ekhoyo ibe yeyona ibambekayo. Ngamanye amazwi, kufuneka sikhethe iiparameters apho isampuli iya kuvakala kakhulu. Kwimeko yethu, iparameter ekhethiweyo yithuba lokubuyisela imali mboleko , nto leyo ixhomekeke kwii-coefficients ezingaziwayo . Ngoko ke kufuneka sifumane i-vector yobunzima , apho ukwenzeka kwesampulu kuya kuba kuninzi.

5. Siyazi ukuba senze ntoni na isampula imisebenzi enokwenzeka unokusebenzisa eyona ndlela inokwenzeka. Kwaye siyawazi onke amaqhinga anamaqhinga okusebenza ngale ndlela.

Yile ndlela eyenzeka ngayo ukuba yintshukumo enamanyathelo amaninzi :)

Ngoku khumbula ukuba ekuqaleni kwenqaku sifuna ukufumana iindidi ezimbini zemisebenzi yelahleko Ilahleko yoLungiselelo kuxhomekeke kwindlela iiklasi zento ezonyulwa ngayo. Kwenzekile ukuba kwiingxaki zokuhlela kwiiklasi ezimbini, iiklasi zichazwe njenge и okanye . Ngokuxhomekeke kwi-notation, imveliso iya kuba nomsebenzi wokulahlekelwa ohambelanayo.

Ityala 1. Ukuhlelwa kwezinto zibe и

Ngaphambili, xa kujongwa ukuba nokwenzeka kwesampulu, apho amathuba okubuyiswa kwetyala ngumboleki kubalwa ngokusekwe kwimiba kwaye kunikwe i-coefficients. , sisebenzise ifomula:

Enyanisweni yintsingiselo imisebenzi yempendulo yolungiselelo kwivektha enikiweyo yobunzima

Ke akukho nto isithintelayo ekubhaleni isampulu yamathuba emisebenzi ngolu hlobo lulandelayo:

Kwenzeka ukuba ngamanye amaxesha kunzima kwabanye abahlalutyi be-novice ukuba baqonde ngokukhawuleza ukuba lo msebenzi usebenza njani. Makhe sijonge imizekelo emi-4 emifutshane eya kucoca izinto:

1. ukuba (oko kukuthi, ngokwesampulu yoqeqesho, into yeklasi +1), kunye ne-algorithm yethu misela ukuba nokwenzeka kokuhlela into ngokodidi ilingana no 0.9, ke eliqhekeza lesampulu enokubakho liza kubalwa ngolu hlobo lulandelayo:

2. ukuba , kwaye , ngoko ke ubalo luya kuba ngolu hlobo:

3. ukuba , kwaye , ngoko ke ubalo luya kuba ngolu hlobo:

4. ukuba , kwaye , ngoko ke ubalo luya kuba ngolu hlobo:

Kucacile ukuba umsebenzi onokwenzeka uya kwandiswa kwiimeko 1 kunye ne-3 okanye kwimeko jikelele - kunye namaxabiso aqikelelweyo ngokuchanekileyo anokubakho kokwabela into kwiklasi. .

Ngenxa yokuba xa kujongwa ithuba lokwabela iklasi into enokwenzeka Asiyazi kuphela i-coefficients , ngoko siya kubakhangela. Njengoko kukhankanyiwe ngasentla, le yingxaki yokwandisa apho kuqala kufuneka sifumane i-derivative yomsebenzi onokwenzeka ngokubhekiselele kwivector yobunzima. . Nangona kunjalo, okokuqala kunengqiqo ukwenza lula umsebenzi wethu: siya kukhangela i-derivative yelogarithm. imisebenzi enokwenzeka.

Kutheni emva kwe-logarithm, ngo imisebenzi yempazamo yolungiselelo, sitshintshe uphawu ukusuka phezu . Yonke into ilula, kuba kwiingxaki zokuvavanya umgangatho womzekelo kuyinto yesiko ukunciphisa ixabiso lomsebenzi, siphindaphinda icala lasekunene lentetho ngo. kwaye ngokufanelekileyo, endaweni yokwandisa, ngoku sinciphisa umsebenzi.

Ngokwenyani, ngoku, phambi kwamehlo akho, umsebenzi welahleko uthathwe ngobuhlungu - Ilahleko yoLungiselelo kwiseti yoqeqesho eneeklasi ezimbini: и .

Ngoku, ukufumana i-coefficients, sifuna nje ukufumana i-derivative imisebenzi yempazamo yolungiselelo kwaye emva koko, usebenzisa iindlela zokwandisa amanani, ezinje ngokwehla komgangatho okanye ukwehla komgangatho westochastic, khetha awona mlinganiso uphezulu. . Kodwa, ngenxa yomthamo omkhulu wenqaku, kucetywayo ukuba wenze ulwahlulo ngokwakho, okanye mhlawumbi oku kuya kuba sisihloko kwinqaku elilandelayo kunye ne-arithmetic eninzi ngaphandle kwemizekelo ecacileyo.

Ityala 2. Ukuhlelwa kwezinto zibe и

Indlela apha iya kufana neeklasi и , kodwa indlela ngokwayo kwimveliso yomsebenzi welahleko Ilahleko yoLungiselelo, ziya kuhonjiswa ngakumbi. Masiqalise. Kumsebenzi onokwenzeka siya kusebenzisa umsebenzisi "ukuba... ngoko...". Oko kukuthi, ukuba Into yeklasi , emva koko ukubala ukwenzeka kwesampulu sisebenzisa ukwenzeka , ukuba into iyeyeklasi , emva koko sitshintshela kwinto enokwenzeka . Nantsi indlela umsebenzi onokuthi ujongeke ngayo:

Makhe sichaze kwiminwe yethu ukuba isebenza njani. Makhe siqwalasele iimeko ezi-4:

1. ukuba и , emva koko ukwenzeka kwesampulu kuya "kuhamba"

2. ukuba и , emva koko ukwenzeka kwesampulu kuya "kuhamba"

3. ukuba и , emva koko ukwenzeka kwesampulu kuya "kuhamba"

4. ukuba и , emva koko ukwenzeka kwesampulu kuya "kuhamba"

Kucacile ukuba kwiimeko 1 kunye ne-3, xa izinto ezinokwenzeka zichazwe ngokuchanekileyo yi-algorithm, umsebenzi onokwenzeka izakwenziwa nkulu, oko kukuthi, yilento kanye ebesifuna ukuyifumana. Nangona kunjalo, le ndlela inzima kwaye ngokulandelayo siza kuthathela ingqalelo inqaku elihlangeneyo. Kodwa kuqala, makhe senze i-logarithm umsebenzi onokwenzeka ngotshintsho lophawu, kuba ngoku sizakuwenza mncinci.

Masitshintshe endaweni yoko intetho :

Masenze lula igama elichanekileyo phantsi kwelogarithm sisebenzisa ubuchule obulula be-arithmetic kwaye sifumane:

Ngoku lixesha lokususa umqhubi "ukuba... ngoko...". Qaphela ukuba xa into ungowaseklasini , ngoko kwintetho ephantsi kwelogarithm, kwidinomineyitha, ephakanyiselwe emandleni , ukuba into iyeyeklasi , emva koko i-$ e $ iphakanyiswe kumandla . Ke ngoko, inqaku lesidanga linokwenziwa lula ngokudibanisa iimeko zombini zibe nye: . Ke umsebenzi wemposiso yolungiselelo iya kuthatha ifom:

Ngokuhambelana nemithetho yelogarithm, sijika iqhezu kwaye sibeke uphawu ""(minus) kwilogarithm, sifumana:

Nanku umsebenzi welahleko ilahleko yezinto, esetyenziswa kwiseti yoqeqesho kunye nezinto ezabelwe iiklasi: и .

Ewe, ngeli xesha ndithatha ikhefu kwaye sigqibezela inqaku.

Umsebenzi wangaphambili wombhali "Ukuzisa i-equation yohlengahlengiso kwifom ye-matrix"

Izinto ezincedisayo

1. Uncwadi

1) Uhlalutyo olusetyenzisiweyo lokuhlehla / N. Draper, G. Smith - 2nd ed. – M.: Finance and Statistics, 1986 (inguqulelo esuka kwisiNgesi)

2) Ithiyori enokwenzeka kunye nezibalo zemathematika / V.E. Gmurman - 9th ed. - M.: Isikolo samabanga aphakamileyo, ngo-2003

3) Ithiyori enokwenzeka / N.I. Chernova-Novosibirsk: Novosibirsk State University, 2007

4) Uhlalutyo lweshishini: ukusuka kwidatha ukuya kulwazi / Paklin N. B., Oreshkov V. I. - 2nd ed. - St. Petersburg: Peter, 2013

5) Inzululwazi yeNzululwazi yeDatha yeNzululwazi ukusuka ekuqaleni / uJoel Gras - eSt. Petersburg: BHV Petersburg, 2017

6) Izibalo ezisebenzayo kwiingcali zeSayensi zeDatha / P. Bruce, E. Bruce - iSt. Petersburg: BHV Petersburg, 2018

2. Iintetho, iikhosi (ividiyo)

1) Undoqo weyona ndlela iphezulu yamathuba, uBoris Demeshev

2) Ubuninzi bendlela enokwenzeka kwimeko eqhubekayo, uBoris Demeshev

3) Uhlengahlengiso lolungiselelo. Vula ikhosi ye-ODS, uYury Kashnitsky

4) Isifundo 4, uEvgeny Sokolov (ukusuka kwimizuzu engama-47 yevidiyo)

5) Ukulungiswa kwezinto, Vyacheslav Vorontsov

3. Imithombo ye-Intanethi

1) Ukuhlelwa komgca kunye neemodeli zokuhlehla

2) Indlela yokuqonda ngokulula ukuLawulwa koLungiselelo

3) Umsebenzi wempazamo yoLungiselelo

4) Iimvavanyo ezizimeleyo kunye nefomula yeBernoulli

5) I-Ballad ye-MMP

6) Eyona ndlela inokwenzeka

7) Iifomyula kunye neepropathi zelogarithms

8) Kutheni inani ?

9) Umdidiyeli womgca

umthombo: www.habr.com

Ukuhlafuna kwi-logistic regression