Mokhoa oa Likwere tse Nyenyane: Selelekela le Ts'ebeliso Tlhahlobong ea Lintlha
Pendahuluan
Mokhoa oa Least Squares ke o mong oa mekhoa ea motheo le e sebelisoang haholo tlhahlobong ea data, haholo-holo lipalo-palo le lipalo tse sebelisitsoeng. Mokhoa ona o ikemiselitse ho hakanya liparamente tse fokotsang kakaretso ea lisekoere tsa liphapang tse bonoang ho tsoa mohlaleng o reriloeng. Sehloohong sena, re tla hlahloba likhopolo tsa motheo tsa mokhoa oa lisekoere tse nyane, lits'ebetso tsa ona masimong a fapaneng, le mehato e sebetsang ea ho o kenya tšebetsong.
Mehopolo ea Motheo ea Mokhoa oa Likwere tse Nyenyane
Mokhoa oa Least Squares o ka hlalosoa habonolo ka ho khutlela morao ka mola. A re re re na le data ka mokhoa oa lipara \((x_i, y_i)\) moo \( i = 1, 2, …, n \). Mohlala o otlolohileng oo re batlang ho o haha o ka hlalosoa tjena:
\[ y = \beta_0 + \beta_1 x + \epsilon \]
moo \( \beta_0 \) le \( \beta_1 \) e leng diparamithara tseo re batlang ho di hakanya, ha \( \epsilon \) e le phoso kapa masalla a lebelletsweng ho ba le karolelano ya lefela.
Sepheo sa mokhoa oa lisekoere tse nyane ke ho fokotsa ts'ebetso e latelang ea sepheo:
\[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i – \beta_0 – \beta_1 x_i)^2 \]
Ho Fumana Liparamithara tse Molemohali: Mokhoa oa Lipalo
Ho fumana boleng ba paramethara bo fokotsang mosebetsi o ikemiseditseng \( S \), re hloka ho bala di-derivative tse sa fellang tsa \( S \) mabapi le \( \beta_0 \) le \( \beta_1 \), ebe re rarolla equation e latelang:
\[ \frac{\partial S}{\partial \beta_0} = -2 \sum_{i=1}^n (y_i – \beta_0 – \beta_1 x_i) = 0 \]
\[ \frac{\partial S}{\partial \beta_1} = -2 \sum_{i=1}^n x_i (y_i – \beta_0 – \beta_1 x_i) = 0 \]
Ka ho rarolla tsamaiso ena ea li-equation tse otlolohileng, re ka fumana likhakanyo \(\hat{\beta_0}\) le \(\hat{\beta_1}\):
\[ \hat{\beta_1} = \frac{n \sum_{i=1}^n x_i y_i – \sum_{i=1}^n x_i \sum_{i=1}^n y_i}{n \sum_{i=1}^n x_i^2 – (\sum_{i=1}^n x_i)^2} \]
\[ \hat{\beta_0} = \bar{y} – \hat{\beta_1} \bar{x} \]
moo \(\bar{y}\) le \(\bar{x}\) e leng karolelano ea \(y\) le \(x\) ka ho latellana.
Tšebeliso ea Mokhoa oa Liteko tse Nyenyane
1. Moruo le Lichelete
Mokhoa oa lisekoere tse nyane o sebelisoa haholo thutong ea moruo ho etsa mohlala oa likamano lipakeng tsa li-variable tsa moruo. Mohlala, setsebi sa moruo se ka 'na sa batla ho etsa mohlala oa phello ea sekhahla sa ho hloka mosebetsi ho infleisheneng. A sebelisa mokhoa oa lisekoere tse nyane, setsebi se ka hlahisa mohlala oa regression o amanang le li-variable tse peli le ho etsa liqeto tsa lipalo-palo mabapi le matla le mofuta oa kamano.
2. Mahlale a Sechaba
Mahlaleng a kahisano, mokhoa oa lisekoere tse nyane o atisa ho sebelisoa liphuputsong le lipatlisisong tsa kelello ho ithuta kamano pakeng tsa boitšoaro ba batho le lintho tse ling tse fapaneng. Mohlala oa khale ke ho khutlela morao ho bonolo ho amanang le boemo ba thabo ea motho le chelete ea hae ea selemo le selemo.
3. Boenjiniere
Boenjiniere, mokhoa oa lisekoere tse nyane o ka sebelisoa bakeng sa ho lekanya lisebelisoa le ts'ebetso ea matšoao. Mohlala, ts'ebetsong ea litšoantšo tsa dijithale, mokhoa ona o sebelisoa ho fokotsa lerata litšoantšong ka ho kenya mohlala o thehiloeng ho data e bonoeng.
4. Thuto ea Boemo ba Leholimo le Thuto ea Boemo ba Leholimo
Litsebi tsa boemo ba leholimo li sebelisa mokhoa ona ho sekaseka lintlha tse amanang le mocheso, pula, kapa liphetoho tse ling tsa boemo ba leholimo. Ka mehlala ea ho khutlela morao, li ka bolela esale pele mekhoa ea boemo ba leholimo ho latela lintlha tsa nalane, e leng se thusang ho etsa likhakanyo tse nepahetseng haholoanyane.
Ho kenya tšebetsong ka mokhoa o sebetsang ka Python
Ho kenya tshebetsong mokhoa wa dikwere tse nyane ka ho fetisisa, haholoholo ho kgutlisa mola o bonolo, re ka sebedisa puo ya lenaneo la Python ka thuso ya dilaeborari tsa `numpy` le `matplotlib`. Mohlala wa khoutu ke ona o bontshang tshebetso ena:
"'python
kenya numpy joalo ka np
kenya matplotlib.pyplot joalo ka plt
Lintlha tsa mohlala
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 3, 5, 7, 11])
Karolelano ea x le y
karolelano_x = np. karolelano(x)
mean_y = np. mean(y)
Bala liparamente
nomoro = np.sum((x – mean_x) (y – mean_y))
denominator = np.sum((x – mean_x) 2)
b1 = nomoro / denominator
b0 = mean_y – b1 mean_x
Ponelopele y
y_pred = b0 + b1 x
Liphetho tsa pale
plt.scatter(x, y, mmala='putsoa', label='Lintlha tsa ho Sheba')
plt.plot(x, y_pred, mmala='khubelu', label='Mola wa Kgatelelo')
plt.xlabel('x')
plt.ylabel('y')
Plt.legend()
bonts'a ()
print(f”Li-coefficient tsa ho khutlela morao: b0 = {b0}, b1 = {b1}”)
``
Qetello
Mokhoa oa lisekoere tse nyane ke motheo o matla le oa bohlokoa oa lipalo-palo le tlhahlobo ea data. Bokhoni ba oona ba ho fokotsa liphoso le ho eketsa ho lekana ha mohlala bo etsa hore o be molemo haholo masimong a fapaneng, ho tloha moruong ho ea boenjiniere le mahlaleng a kahisano. Le hoja mohopolo oa motheo o le bonolo, mokhoa ona o ka atolosoa ho ea ho mehlala e rarahaneng haholoanyane joalo ka ho khutlela morao ho sa lateleng, mehlala ea liphello tse tsoakiloeng, le ho ithuta ka mochini. Ka kutloisiso e ntle ea mokhoa oa lisekoere tse nyane le mokhoa o lekaneng oa ho itloaetsa, re ka ntlafatsa ho nepahala ha tlhahlobo ea rona ea data le ho etsa liqeto tse nang le tsebo e ngata.
Ke tšepa hore sengoloa sena se fana ka kakaretso e hlakileng le e felletseng ea mokhoa oa lisekoere tse nyane le ts'ebeliso ea ona.
