Fomura yekudzoreredza zvinhu

Fomura Yekudzoreredza Zvinhu

Kudzoreredza logistic ndeimwe yenzira dzakakurumbira muhuwandu hwezviverengero nesainzi yedata yekuenzanisa hukama huripo pakati pezvinhu zvakasiyana-siyana zvakazvimiririra (zvinofanotaura) uye shanduko inoenderana nechikamu, kunyanya binary (semuenzaniso, hongu/kwete, kubudirira/kukundikana, kurwara/hutano). Kusiyana nekudzokorora kwakarongeka, uko kunogadzira mavalue anoenderera mberi, logistic regression yakagadzirirwa kufungidzira mukana wechiitiko, saka mhedzisiro yekupedzisira iri pakati pe0 kusvika 1. Muchinyorwa chino, tichakurukura fomura yekudzokorora logistic, zvinoreva chikamu chimwe nechimwe, uye maitirwo ekuchidudzira.

Sei Kudzoreredzwa Kwezvinhu Kuchidiwa?

Kana tikashandisa linear regression kufanotaura mikana, modhi inogona kuburitsa ma values ​​​​ari pasi pe 0 kana pamusoro pe 1, izvo zviri pachena kuti hazvina musoro pa probability. Logistic regression inogadzirisa dambudziko iri nekushandisa nonlinear function iyo inobatanidza mhedzisiro yakaverengerwa (inogona kuva chero kukosha) kune probability value iri pakati pe 0 ne 1. Basa rinonyanya kushandiswa i logistic kana sigmoid function.

Semuenzaniso, ngatitii tinoda kufanotaura kana mutengi achachinja zvichienderana nezera rake, nguva yaakanyoresa, uye nguva yaanoshandisa. Mhedzisiro yakafanotaurwa ine mikana miviri chete: kuchinjika (1) kana kusachinja (0). Kuchinjika kweLogistic kwakakodzera mamiriro ezvinhu akadai.

Fomura Yekutanga yeKudzoreredza Kwezvinhu

Chinangwa che logistic regression ndechekuratidza mukana \( p \) wekuti \( Y = 1 \) (chiitiko chinoitika), tichifunga nezve kukosha kwe predictor variable \( X \).

Mamodheru ekugadzirisa maitiro anowanzo nyorwa mumhando mbiri dzakakosha:

1) Fomu reMhinduro (Sigmoid)

\[
p = P(Y=1 \pakati X) = \frac{1}{1 + e^{-z}}
\]

dengan

\[
z = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_k X_k
\]

Information:
– \( p \) ndiyo mukana wechiitiko (semuenzaniso: churn = 1).
– \( e \) ndiyo nhamba yaEuler (inenge 2,71828).
– \( z \) musanganiswa wezvinoreva zvinhu zvinofanotaura zvinhu nenzira yakatsetseka.
– \( \beta_0 \) ndiyo intercept (inogara iripo).
– \( \beta_1, \beta_2, \ldots, \beta_k \) ndiwo ma coefficients ekudzoreredza.
– \( X_1, X_2, \ldots, X_k \) ma "variables" akazvimiririra.

Basa re sigmoid rinovimbisa kuti chero kukosha kwe \( z \), kukosha kwe \( p \) kunoramba kuri pakati pe0 ne1.

2) Fomu reLogit (Matambudziko eLog)

Imwe fomu yakakosha zvikuru iform ye logit, inova iyo logarithm yematambudziko:

\[
\text{logit}(p) = \ln\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_k X_k
\]

Information:
– \( \frac{p}{1-p} \) inonzi mikana (mukana wekuenzanisa).
– \( \ln \) ndiyo logarithm yechisikigo.

Chimiro che "logist" chinotsanangura kuti "logistic regression" inotevedzera "log odds" se "linear function" ye "predictors". Izvi zvinoita kuti kududzira ma "coefficients" kuve pachena, kunyanya kana tichitarisa "odds ratios".

Kunzwisisa Matambudziko neMatambudziko

Kuti tinyatsonzwisisa fomura ye logistic regression, tinofanira kusiyanisa pakati pe probability ne odds.

– Mikana \( p \): mukana wekuti chiitiko chiitike (0 kusvika 1).
- Mikana: kuenzanisa mukana wekuti chimwe chinhu chiitike kuti chisaitike:

\[
\text{odds} = \frac{p}{1-p}
\]

Muenzaniso: kana \( p = 0{,}8 \), zvino:

\[
\text{odds} = \frac{0{,}8}{0{,}2} = 4
\]

Izvi zvinoreva kuti chiitiko ichi chine mukana wakapetwa ka4 kupfuura kusaitika.

Mu logistic regression, coefficient \( \beta \) inowanzo tsanangurwa kuburikidza ne odds ratio:

\[
\text{OR} = e^{\beta}
\]

– Kana \( \beta > 0 \), ipapo \( e^{\beta} > 1 \): chinofanotaura chiitiko chinowedzera mikana yechiitiko.
– Kana \( \beta < 0 \), ipapo \( e^{\beta} < 1 \): chinofanotaura chinoderedza mikana yechiitiko. - Kana \( \beta = 0 \), ipapo \( e^{\beta} = 1 \): hapana chinokanganisa mikana. Semuenzaniso, kana \( \beta_1 = 0{,}7 \), ipapo: \[ e^{0{,}7} \approx 2{,}01 \] Izvi zvinoreva kuti kuwedzera kweyuniti imwe neimwe mu \( X_1 \) kuchawedzera mikana yechiitiko nenguva dzinenge 2,01 (tichifunga kuti zvimwe zvinoshanduka zvinoramba zviripo). Muenzaniso weModhi Yekudzoreredza Rondedzero Yakareruka Ngatitii tine imwe chete inofanotaura inochinja \( X \), semuenzaniso nhamba yemaawa ekudzidza pasvondo, yekufanotaura kupasa bvunzo (pasa = 1, kundikana = 0). Modhi:

\[ \text{logit}(p) = \beta_0 + \beta_1 X \] Kana mhedzisiro inofungidzirwa iri: - \( \beta_0 = -4 \) - \( \beta_1 = 0{,}8 \) Zvadaro: \[ z = -4 + 0{,}8X \] \[ p = \frac{1}{1 + e^{-(-4 + 0{,}8X)}} = \frac{1}{1 + e^{4 - 0{,}8X}} \] Kana \( X = 6 \) maawa ekudzidza: \[ z = -4 + 0{,}8(6) = 0{,}8 \] \[ p = \frac{1}{1 + e^{-0{,}8}} \approx 0{,}69 \] Dudziro: nemaawa matanhatu ekudzidza pasvondo, mukana wekudzidza wakapfuura nechikamu chinosvika 69%. Kufungidzira kweCoefficient: Sei Isiri Nzira yeLeast Squares? Mukudzokorora kwemutsara, macoefficient anowanzo kuverengerwa uchishandisa nzira ye least squares. Zvisinei, mukudzokorora kwelogistic, hukama huripo pakati pezvinofanotaura nezvinogoneka hauna kurongeka, saka nzira ye least squares haina kunaka. Kudzokorora kwelogistic kunowanzo shandisa Maximum Likelihood Estimation (MLE) kuti uwane kukosha kwecoefficient \( \beta \) iyo inowedzera mukana wedata rakaonekwa. Muchidimbu, mukana wekuti zvionekwe zviri zviviri \( y_i \in \{0,1\} \) uye fungidziro \( p_i \) ndeizvi: \[ L(\beta) = \prod_{i=1}^{n} p_i^{y_i}(1-p_i)^{(1-y_i)} \] Inowanzo shandurwa kuita mukana wekuti zvive nyore kuverenga: \[ \ell(\beta) = \sum_{i=1}^{n} \left[ y_i \ln(p_i) + (1-y_i)\ln(1-p_i) \right] \] Kukosha kwe \( \beta \) kunosarudzwa kuti kuwedzere \( \ell(\beta) \). Nzira dzenhamba dzakadai seNewton-Raphson kana gradient descent dzinowanzo shandiswa nesoftware yestatistical. Zvakanakira uye Zvipingamupinyi zveLogistic Regression Zvakanakira 1. Mhedzisiro yacho iri muchimiro chezvingangoitika saka zviri nyore kushandura kuita sarudzo. 2. Kududzirwa kwema coefficients kwakajeka kuburikidza nechiyero che odds. 3. Yakakodzera matambudziko ekuisa muzvikamu zve binary uye inogona kuwedzerwa ku multinomial/ordinal. Miganhu 1. Inofungidzira hukama hwakatsetseka pakati pezvinofanotaura uye log odds, kwete zvakananga kune zvingangoitika. 2. Inogona kuva dambudziko kana paine multicollinearity kana data risina kuenzana zvakanyanya. 3. Kune mapatani ehukama akaomarara, dzimwe nzira dzisiri dzemutsara (semuenzaniso, random forest kana neural network) dzinogona kunge dziri nani. Mhedziso Fomura yekudzokorora logistic inosanganisa musanganiswa wemitsara yezvinochinja-chinja zve predictor nebasa re sigmoid kuti ibudise mikana. Chimiro chinowanzo shandiswa ndechekuti: \[ p = \frac{1}{1 + e^{-(\beta_0 + \beta_1 X_1 + \cdots + \beta_k X_k)}} \] kana muchimiro che logit: \[ \ln\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 X_1 + \cdots + \beta_k X_k \] Nekunzwisisa mafomu maviri aya efomura, tinogona kuvaka mamodheru ekufungidzira ematambudziko akasiyana-siyana ekupatsanura binary uku tichidudzira simba rezvinochinja-chinja kuburikidza nechiyero che odds \( e^{\beta} \). Kudzokorora logistic kunoramba kuri hwaro hwakakosha mukuongorora data nekuti kuri nyore, kune simba, uye kunodudzira—uye kazhinji ndiyo nhanho yekutanga usati waedza mamodheru akaomarara. Kana uchida, ndinogona kuwedzera muenzaniso wekuverenga nedata diki (tafura), kana muenzaniso wekushandisa logistic regression muPython/R pamwe nekududzirwa kwezvakabuda.