Ụzọ Kachasị Obere: Okwu Mmalite na Ngwa na Nyocha Data
Pendahuluan
Usoro Least Squares bụ otu n'ime usoro kachasị mkpa ma a na-ejikarị eme ihe na nyocha data, ọkachasị na ọnụọgụgụ na mgbakọ na mwepụ etinyere. Ebumnuche usoro a bụ ịtụle paramita ndị na-ebelata mkpokọta nke square nke ọdịiche a hụrụ site na ihe nlereanya akwadoro. N'isiokwu a, anyị ga-enyocha echiche ndị bụ isi nke usoro least square, ojiji ya n'ọtụtụ ngalaba, na usoro bara uru maka itinye ya n'ọrụ.
Echiche Ndị Dị Mkpa nke Usoro Obere Ihe Ndị Kacha N'ime
Enwere ike ịkọwa usoro Least Squares site na regression linear. Ka e were ya na anyị nwere data n'ụdị abụọ \((x_i, y_i)\) ebe \(i = 1, 2, …, n \). Enwere ike ịkọwa ụdị ahịrị anyị chọrọ iwuli dị ka:
\[ y = \beta_0 + \beta_1 x + \epsilon \]
ebe \( \beta_0 \) na \( \beta_1 \) bụ paramita anyị chọrọ ịtụle, ebe \( \epsilon \) bụ njehie ma ọ bụ ihe fọdụrụ nke a na-atụ anya na ọ ga-enwe nkezi nke efu.
Ebumnuche nke usoro obere akụkụ bụ ibelata ọrụ ebumnuche ndị a:
\[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i – \beta_0 – \beta_1 x_i)^2 \]
Ịchọta Paramita Kachasị Mma: Ụzọ Mgbakọ na Mwepụ
Iji chọta ụkpụrụ paramita ndị na-ebelata ọrụ ebumnuche \(S\), anyị kwesịrị ịgbakọ akụkụ nke ihe ndị mejupụtara \(S\) ma e jiri ya tụnyere \( \beta_0 \) na \( \beta_1 \), wee dozie nha nha ndị a:
\[ \frac{\partial S}{\partial \beta_0} = -2 \sum_{i=1}^n (y_i – \beta_0 – \beta_1 x_i) = 0 \]
\[ \frac{\partial S}{\partial \beta_1} = -2 \sum_{i=1}^n x_i (y_i – \beta_0 – \beta_1 x_i) = 0 \]
Site na idozi usoro nhazi ahịrị a, anyị nwere ike ịchọta ndị na-eme atụmatụ \(\hat{\beta_0}\) na \(\hat{\beta_1}\):
\[ \hat{\beta_1} = \frac{n \sum_{i=1}^n x_i y_i – \sum_{i=1}^n x_i \sum_{i=1}^n y_i}{n \sum_{i=1}^n x_i^2 – (\sum_{i=1}^n x_i)^2} \]
\[ \okpu{\beta_0} = \okpu{y} – \okpu{\beta_1} \okpu{x} \]
ebe \(\bar{y}\) na \(\bar{x}\) bụ nkezi nke \(y\) na \(x\) n'otu n'otu.
Ojiji nke Usoro Least Squares
1. Akụnụba na Ego
A na-ejikarị usoro nke obere square eme ihe na akụnụba iji gosipụta mmekọrịta dị n'etiti mgbanwe akụ na ụba. Dịka ọmụmaatụ, onye nyocha akụ na ụba nwere ike ịchọ igosipụta mmetụta nke ọnụego enweghị ọrụ na mmụba ọnụ ahịa. Site na iji usoro nke obere square, onye nyocha nwere ike ịmepụta ụdị nlọghachi azụ nke metụtara mgbanwe abụọ ahụ ma mee nkwubi okwu ọnụ ọgụgụ gbasara ike na ọdịdị nke mmekọrịta ahụ.
2. Sayensị Mmekọrịta Mmadụ na Nwanyị
Na sayensị mmekọrịta mmadụ na ibe ya, a na-ejikarị usoro nke least squares eme ihe na nyocha na nnyocha uche iji mụọ mmekọrịta dị n'etiti omume mmadụ na ihe ndị ọzọ dị iche iche. Ihe atụ a ma ama bụ mgbanwe dị mfe nke na-ejikọta ọkwa obi ụtọ nke onye ọ bụla na ego ha na-enweta kwa afọ.
3. Injinia
N'ihe gbasara injinia, enwere ike iji usoro nke pere mpe square mee ihe maka nhazi ngwa na nhazi mgbaàmà. Dịka ọmụmaatụ, na nhazi onyonyo dijitalụ, a na-eji usoro a ebelata mkpọtụ na onyonyo site na itinye ihe nlereanya dabere na data a hụrụ.
4. Ọmụmụ Meteorology na Ihu Igwe
Ndị ọkachamara n'ihe gbasara ihu igwe na-eji usoro a enyocha data metụtara okpomọkụ, mmiri ozuzo, ma ọ bụ mgbanwe ihu igwe ndị ọzọ. Site na ụdị mgbanwe ihu igwe, ha nwere ike ịkọ amụma ihu igwe dabere na data akụkọ ihe mere eme, na-enyere aka ịmepụta amụma ziri ezi karị.
Mmejuputa Bara Uru na Python
Iji tinye usoro obere akụkụ n'ọrụ, ọkachasị mgbanwe dị mfe n'usoro, anyị nwere ike iji asụsụ mmemme Python site n'enyemaka nke ọbá akwụkwọ 'numpy' na 'matplotlib'. Nke a bụ ihe atụ koodu nke na-egosi usoro a:
"'Python
bubata numpy dika np
mbubata matplotlib.pyplot ka plt
Ihe atụ data
x = np.array([1, 2, 3, 4, 5])
y = np.array ([2, 3, 5, 7, 11])
Nkezi nke x na y
mean_x = np.mean(x)
mean_y = np.mean(y)
Gbakọọ paramita
onye na-enyocha ọnụọgụ = np.sum((x – mean_x) (y – mean_y))
ihe nkesa = np.sum((x – mean_x) 2)
b1 = ihe ngụta / ihe ngụta
b0 = mean_y – b1 mean_x
Amụma y
y_pred = b0 + b1 x
Nsonaazụ nke atụmatụ ahụ
plt.scatter(x, y, agba='acha anụnụ anụnụ', akara='Data Nlele')
plt.plot(x, y_pred, agba='uhie', label='Ahịrị Nlaghachi')
plt.xlabel('x')
plt.ylabel('y')
plt.akụkọ ()
plt.show ()
bipụta(f”Nhazi Nlaghachi: b0 = {b0}, b1 = {b1}”)
““
Mmechi
Usoro nke obere square bụ ntọala dị ike ma dị mkpa na ọnụọgụgụ na nyocha data. Ike ya ibelata mmejọ na ime ka ụdị dabara adaba dịkwuo mma na-eme ka ọ baa uru nke ukwuu n'ọtụtụ ubi, site na akụnụba ruo na injinia na sayensị mmekọrịta mmadụ na ibe ya. Ọ bụ ezie na echiche bụ isi dị mfe, enwere ike ịgbatị usoro ahụ gaa na ụdị ndị dị mgbagwoju anya dịka nlọghachi na-abụghị ahịrị, ụdị mmetụta dị iche iche, na mmụta igwe. Site na nghọta dị mma nke usoro nke obere square na omume zuru oke, anyị nwere ike imeziwanye izi ezi nke nyocha data anyị ma mee mkpebi ndị ka mma.
Enwere m olileanya na isiokwu a ga-enye nkọwa doro anya na zuru oke nke usoro kacha nta na ojiji ya.
