Nzira yeLeast Squares: Nhanganyaya uye Mashandisirwo muKuongorora Data
Pendauluan
Nzira yeLeast Squares ndeimwe yenzira dzakakosha uye dzinoshandiswa zvakanyanya mukuongorora data, kunyanya muhuwandu hwemasvomhu nemasvomhu anoshandiswa. Nzira iyi ine chinangwa chekufungidzira maparamita anoderedza huwandu hwemasquare ekutsauka kwakaonekwa kubva kumuenzaniso wakapihwa. Muchinyorwa chino, tichaongorora pfungwa dzekutanga dzenzira yeleast squares, mashandisirwo ayo muminda yakasiyana-siyana, uye matanho anoshanda ekuishandisa.
Pfungwa Dzekutanga dzeNzira Yezvikwere Zvidiki
Nzira yeLeast Squares inogona kutsanangurwa zviri nyore kuburikidza ne linear regression. Ngatitii tine data muchimiro chemapeya \((x_i, y_i)\) apo \( i = 1, 2, …, n \). Muenzaniso we linear watinoda kuvaka unogona kuratidzwa se:
\[ y = \beta_0 + \beta_1 x + \epsilon \]
apo \( \beta_0 \) uye \( \beta_1 \) ndiwo maparamita atinoda kufungidzira, nepo \( \epsilon \) iri chikanganiso kana kuti chisara chinotarisirwa kuva neavhareji ye zero.
Chinangwa chenzira ye least squares ndechekuderedza basa rinotevera rechinangwa:
\[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i – \beta_0 – \beta_1 x_i)^2 \]
Kuwana Maparamita Akakodzera: Nzira Yemasvomhu
Kuti tiwane ma parameter values anodzikisa basa rechinangwa \( S \), tinofanira kuverenga chikamu che derivatives che \( S \) maererano ne \( \beta_0 \) uye \( \beta_1 \), tobva tagadzirisa equation inotevera:
\[ \frac{\partial S}{\partial \beta_0} = -2 \sum_{i=1}^n (y_i – \beta_0 – \beta_1 x_i) = 0 \]
\[ \frac{\partial S}{\partial \beta_1} = -2 \sum_{i=1}^n x_i (y_i – \beta_0 – \beta_1 x_i) = 0 \]
Nekugadzirisa iyi sisitimu yekuenzanisa kwakatwasuka, tinogona kuwana vanoongorora \(\hat{\beta_0}\) uye \(\hat{\beta_1}\):
\[ \hat{\beta_1} = \frac{n \sum_{i=1}^n x_i y_i – \sum_{i=1}^n x_i \sum_{i=1}^n y_i}{n \sum_{i=1}^n x_i^2 – (\sum_{i=1}^n x_i)^2} \]
\[ \hat{\beta_0} = \bar{y} – \hat{\beta_1} \bar{x} \]
apo \(\bar{y}\) uye \(\bar{x}\) ari avhareji ye \(y\) uye \(x\) zvichiteerana.
Kushandiswa kweMethod yeLeast Squares
1. Zvehupfumi neMari
Nzira ye least squares inoshandiswa zvakanyanya mu econometrics kuratidza hukama pakati pezvinhu zvehupfumi. Semuenzaniso, nyanzvi yezvehupfumi ingangoda kuratidza mhedzisiro yehuwandu hwekushaya mabasa pa inflation. Achishandisa nzira ye least squares, nyanzvi inogona kugadzira modhi ye regression inobatanidza zvinhu zviviri izvi uye kuita fungidziro dzehuwandu nezvesimba uye hunhu hwehukama.
2. Sainzi Yemagariro Evanhu
Musainzi yezvemagariro evanhu, nzira ye "least squares" inowanzoshandiswa muongororo nekutsvaga kwepfungwa kudzidza hukama huripo pakati pemaitiro evanhu nezvimwe zvinhu. Muenzaniso wekare ndewekudzokera shure kuri nyore kunobatanidza mwero wemufaro wemunhu nemari yaanowana pagore.
3. Uinjiniya
Muinjiniya, nzira ye "least squares" inogona kushandiswa pakugadzirisa zvishandiso uye kugadzirisa masaini. Semuenzaniso, mukugadzirisa mifananidzo yedhijitari, nzira iyi inoshandiswa kuderedza ruzha mumifananidzo nekuisa modhi zvichibva padata rakaonekwa.
4. Meteorology uye Mamiriro ekunze
Nyanzvi dzezvemamiriro ekunze dzinoshandisa nzira iyi kuongorora data rine chekuita netembiricha, mvura inonaya, kana zvimwe zvinhu zvinochinja mamiriro ekunze. Nekushandisa mamodheru ekudzoreredza mamiriro ekunze, vanogona kufanotaura mamiriro ekunze zvichibva pane zvakaitika kare, zvichibatsira kugadzira kufanotaura kwakarurama.
Kushanda Kunoita nePython
Kuti tishandise nzira ye "least squares" mukuita, kunyanya "linear regression" iri nyore, tinogona kushandisa mutauro wePython programming tichibatsirwa ne "numpy" uye "matplotlib" raibhurari. Heino muenzaniso wekodhi unoratidza maitiro aya:
"'python
import numpy se np
import matplotlib.pyplot as plt
Ruzivo rwemuenzaniso
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 3, 5, 7, 11])
Avhareji ye x na y
zvinoreva_x = np. zvinoreva(x)
zvinoreva_y = np. zvinoreva(y)
Verenga maparamita
nhamba = np.sum((x – mean_x) (y – mean_y))
dhinominata = np.sum((x – mean_x) 2)
b1 = nhamba / dhinomineta
b0 = mean_y – b1 mean_x
Kufanotaura y
y_pred = b0 + b1 x
Mhedzisiro yezvirongwa
plt.scatter(x, y, ruvara='bhuruu', label='Ruzivo rweKucherechedza')
plt.plot(x, y_pred, color='red', label='Regression Line')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.show ()
print(f”Regression coefficients: b0 = {b0}, b1 = {b1}”)
``
Mhedziso
Nzira ye least squares ihwaro hwakasimba uye hwakakosha mukuongorora manhamba nedata. Kugona kwayo kuderedza zvikanganiso uye kuwedzera kukodzera kwemuenzaniso kunoita kuti ibatsire zvakanyanya muminda yakasiyana-siyana, kubva kuhupfumi kusvika kuinjiniya nesainzi yemagariro evanhu. Kunyange zvazvo pfungwa huru iri nyore, nzira iyi inogona kuwedzerwa kune mamodheru akaomarara akadai se nonlinear regression, mamodheru ane mhedzisiro yakasanganiswa, uye kudzidza kwemuchina. Nekunzwisisa kwakanaka nzira ye least squares uye kudzidzira kwakakwana, tinogona kuvandudza kururama kwekuongorora kwedu data uye kuita sarudzo dzine ruzivo rwakawanda.
Ndinovimba chinyorwa chino chinopa ruzivo rwakajeka uye rwakakwana rwenzira ye least squares uye mashandisirwo ayo.
