13.3-dars: Koeffitsiyentlarni talqin qilish

Koeffitsiyent birligi = y birligi / x birligi

narx (ming $) ~ maydon (m^2):  w = 1.52  →  "+1 m^2 = +1.52 ming $"
maydon sm^2 da berilsa:        w = 0.000152  →  bir xil model, boshqa son!

Koeffitsiyent kattaligi MUHIMLIK emas

x ni standartlashtiring: z = (x - o'rtacha) / SD
Keyin: w_std = w × SD_x            (y standartlashtirilmagan bo'lsa)

Talqin: "x bir SD ga o'zgarsa, y qancha o'zgaradi"
Bu — belgilarni O'ZARO solishtirish uchun yagona to'g'ri yo'l

Diqqat: 0/1 belgilar uchun SD ma'nosi cheklangan (ulushga bog'liq)

Multikollinearlik — belgilar o'zaro kuchli bog'liq

Oqibatlari:
  · koeffitsiyentlar BEQAROR (namuna biroz o'zgarsa keskin o'zgaradi)
  · standart xatolar katta, p-qiymatlar yuqori
  · ishoralar mantiqqa zid bo'lishi mumkin
  · BASHORAT sifati odatda ZARAR KO'RMAYDI

VIF_j = 1 / (1 - R_j^2),  R_j^2 — x_j ni qolgan belgilarga regressiya
  VIF < 5   — muammo yo'q
  5..10     — ehtiyot
  > 10      — jiddiy

1. Belgilarni BIRLASHTIRISH (ikki kanal byudjeti → umumiy byudjet)
2. Birini olib tashlash (mantiqan kamroq muhimini)
3. Ridge regularizatsiyasi 13.7-bob — koeffitsiyentlarni barqarorlashtiradi
4. PCA bilan yangi ortogonal belgilar 10.9-bob — talqin yo'qoladi
5. Ko'proq ma'lumot yig'ish
6. Hech narsa qilmaslik — agar faqat bashorat kerak bo'lsa

log(y) ~ x        →  w = "x bir birlikka oshsa, y ≈ 100·w % o'zgaradi"  (kichik w uchun)
                     aniq: (exp(w) - 1) × 100%
y ~ log(x)        →  "x 1% oshsa, y ≈ w/100 birlikka o'zgaradi"
log(y) ~ log(x)   →  w = ELASTIKLIK: "x 1% oshsa, y w% o'zgaradi"

Diqqat: log fazoda o'qitilgan modelni asl birlikka qaytarganda
        exp(bashorat) MEDIANANI beradi, o'rtachani emas (qiyshiqlik tuzatishi kerak)

One-hot (drop_first=True): bitta daraja TAYANCH bo'ladi
  hudud: markaz (tayanch), shimol, janub, chekka
  w_shimol = "shimol markazdan qancha farq qiladi"

Tayanch darajani o'zgartirsangiz — barcha koeffitsiyentlar o'zgaradi (model bir xil!)
Tayanch: eng ko'p uchraydigan yoki eng mantiqiy daraja

import numpy as np
import statsmodels.api as sm
from statsmodels.stats.outliers_influence import variance_inflation_factor

# standartlashtirilgan koeffitsiyentlar
w_std = model.coef_ * X.std(axis=0, ddof=1)

# VIF
Xc = sm.add_constant(X)
vif = [variance_inflation_factor(Xc, i) for i in range(1, Xc.shape[1])]

# log talqini
# log(y) ~ x:      (np.exp(w) - 1) * 100  → foiz o'zgarish
# log(y) ~ log(x): w → elastiklik

# kategoriya
pd.get_dummies(df["hudud"], drop_first=True)      # tayanch — birinchi daraja
QOIDA: birlikni yoz · VIF tekshir · CI ber · tayanch darajani ayt · sabab dema

w birligi = y birligi / x birligi · solishtirish uchun w × SD_x
VIF > 10 — koeffitsiyent beqaror (bashorat emas, talqin muammosi)
log(y)~x: ~100w% · y~log(x): w/100 birlik · log-log: elastiklik
Kategoriya: tayanch darajaga nisbatan farq

"""Koeffitsiyent kattaligi muhimlik emas (real numpy/sklearn)."""

import numpy as np
from sklearn.linear_model import LinearRegression


def main() -> None:
    rng = np.random.default_rng(6)
    n = 2000
    maydon = rng.lognormal(np.log(70), 0.33, n).clip(28, 260)
    markaz = (rng.random(n) < 0.3).astype(float)
    yosh = rng.integers(0, 45, n).astype(float)
    narx = 18 + 1.52 * maydon + 38 * markaz - 0.62 * yosh + rng.normal(0, 13, n)
    nomlar = ["maydon", "markaz", "yosh"]
    X = np.column_stack([maydon, markaz, yosh])

    print("=== 1. Xom koeffitsiyentlar ===")
    m = LinearRegression().fit(X, narx)
    for nom, w in zip(nomlar, m.coef_):
        print(f"  {nom:<8}: {w:+8.3f}")
    print("  ('markaz eng muhim' degan xulosa — xato)")

    print("\n=== 2. Birlikni o'zgartiramiz: maydon sm^2 da ===")
    X2 = X.copy()
    X2[:, 0] = X2[:, 0] * 10_000
    m2 = LinearRegression().fit(X2, narx)
    print(f"  maydon koeffitsiyenti: {m2.coef_[0]:.8f}")
    print(f"  model bir xil: bashoratlar teng = "
          f"{np.allclose(m.predict(X), m2.predict(X2))}")

    print("\n=== 3. Standartlashtirilgan koeffitsiyentlar ===")
    sd = X.std(axis=0, ddof=1)
    for nom, w, s in zip(nomlar, m.coef_, sd):
        print(f"  {nom:<8}: w = {w:+8.3f}, SD = {s:6.2f}, "
              f"w × SD = {w * s:+7.2f}")
    print("  talqin: belgi bir SD ga o'zgarsa narx qancha o'zgaradi")

    print("\n=== 4. Tartib o'zgardi ===")
    tartib = sorted(zip(nomlar, np.abs(m.coef_ * sd)), key=lambda t: -t[1])
    print(f"  standartlashgan bo'yicha: {[n for n, _ in tartib]}")
    print(f"  xom koeffitsiyent bo'yicha: "
          f"{[n for n, _ in sorted(zip(nomlar, np.abs(m.coef_)), key=lambda t: -t[1])]}")
    print("  ⭐ Solishtirish faqat standartlashgan koeffitsiyentlar bilan")


if __name__ == "__main__":
    main()

=== 1. Xom koeffitsiyentlar ===
  maydon  :   +1.527
  markaz  :  +37.905
  yosh    :   -0.635
  ('markaz eng muhim' degan xulosa — xato)

=== 2. Birlikni o'zgartiramiz: maydon sm^2 da ===
  maydon koeffitsiyenti: 0.00015270
  model bir xil: bashoratlar teng = True

=== 3. Standartlashtirilgan koeffitsiyentlar ===
  maydon  : w =   +1.527, SD =  25.15, w × SD =  +38.41
  markaz  : w =  +37.905, SD =   0.47, w × SD =  +17.63
  yosh    : w =   -0.635, SD =  13.11, w × SD =   -8.33
  talqin: belgi bir SD ga o'zgarsa narx qancha o'zgaradi

=== 4. Tartib o'zgardi ===
  standartlashgan bo'yicha: ['maydon', 'markaz', 'yosh']
  xom koeffitsiyent bo'yicha: ['markaz', 'maydon', 'yosh']
  ⭐ Solishtirish faqat standartlashgan koeffitsiyentlar bilan

"""Korrelyatsiyalangan belgilar va VIF (real numpy/sklearn/statsmodels)."""

import numpy as np
import statsmodels.api as sm
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import KFold, cross_val_score
from statsmodels.stats.outliers_influence import variance_inflation_factor


def yarat(seed: int, n: int = 500):
    rng = np.random.default_rng(seed)
    insta = rng.gamma(4, 250, n)
    face = 0.9 * insta + rng.normal(0, 120, n)      # juda kuchli bog'liq
    tv = rng.gamma(3, 400, n)
    sotuv = 5000 + 1.1 * (insta + face) + 0.8 * tv + rng.normal(0, 900, n)
    return np.column_stack([insta, face, tv]), sotuv


def main() -> None:
    X, y = yarat(0)
    nomlar = ["instagram", "facebook", "tv"]

    print("=== 1. Korrelyatsiya ===")
    print(f"  corr(instagram, facebook) = {np.corrcoef(X[:, 0], X[:, 1])[0, 1]:.3f}")
    print(f"  corr(instagram, tv)       = {np.corrcoef(X[:, 0], X[:, 2])[0, 1]:.3f}")

    print("\n=== 2. VIF ===")
    Xc = sm.add_constant(X)
    for i, nom in enumerate(nomlar, start=1):
        print(f"  {nom:<10}: VIF = {variance_inflation_factor(Xc, i):6.2f}")

    print("\n=== 3. Koeffitsiyentlar turli namunalarda ===")
    for seed in [0, 1, 2, 3]:
        Xs, ys = yarat(seed)
        m = LinearRegression().fit(Xs, ys)
        print(f"  namuna {seed}: instagram {m.coef_[0]:+7.3f}, "
              f"facebook {m.coef_[1]:+7.3f}, tv {m.coef_[2]:+6.3f}")
    print("  (haqiqiy ta'sir: instagram va facebook uchun 1.1, tv uchun 0.8)")

    print("\n=== 4. Bashorat zarar ko'rmaydi ===")
    cv = KFold(5, shuffle=True, random_state=0)
    r2_hammasi = cross_val_score(LinearRegression(), X, y, cv=cv, scoring="r2").mean()
    birlashgan = np.column_stack([X[:, 0] + X[:, 1], X[:, 2]])
    r2_birlashgan = cross_val_score(LinearRegression(), birlashgan, y, cv=cv,
                                    scoring="r2").mean()
    print(f"  uchta belgi bilan:  CV R^2 = {r2_hammasi:.4f}")
    print(f"  birlashtirilgan:    CV R^2 = {r2_birlashgan:.4f}")
    mb = LinearRegression().fit(birlashgan, y)
    print(f"  birlashgan koeffitsiyent: {mb.coef_[0]:.3f} (haqiqiy 1.1), "
          f"tv {mb.coef_[1]:.3f}")
    print("  ⭐ Multikollinearlik — talqin muammosi, bashorat emas")


if __name__ == "__main__":
    main()

=== 1. Korrelyatsiya ===
  corr(instagram, facebook) = 0.962
  corr(instagram, tv)       = 0.028

=== 2. VIF ===
  instagram : VIF =  13.50
  facebook  : VIF =  13.51
  tv        : VIF =   1.00

=== 3. Koeffitsiyentlar turli namunalarda ===
  namuna 0: instagram  +1.185, facebook  +1.033, tv +0.762
  namuna 1: instagram  +1.046, facebook  +1.071, tv +0.784
  namuna 2: instagram  +1.087, facebook  +1.209, tv +0.890
  namuna 3: instagram  +0.654, facebook  +1.508, tv +0.764
  (haqiqiy ta'sir: instagram va facebook uchun 1.1, tv uchun 0.8)

=== 4. Bashorat zarar ko'rmaydi ===
  uchta belgi bilan:  CV R^2 = 0.6102
  birlashtirilgan:    CV R^2 = 0.6124
  birlashgan koeffitsiyent: 1.111 (haqiqiy 1.1), tv 0.762
  ⭐ Multikollinearlik — talqin muammosi, bashorat emas

"""log(y), log(x) va log-log koeffitsiyentlari (real numpy/sklearn)."""

import numpy as np
from sklearn.linear_model import LinearRegression


def main() -> None:
    rng = np.random.default_rng(8)
    n = 4000

    print("=== 1. log(y) ~ x: foizli o'zgarish ===")
    tajriba = rng.uniform(0, 20, n)
    # har yil tajriba maoshni ~7% oshiradi (ko'paytiruvchi model)
    maosh = 400 * np.exp(0.07 * tajriba) * rng.lognormal(0, 0.18, n)
    m = LinearRegression().fit(tajriba.reshape(-1, 1), np.log(maosh))
    w = m.coef_[0]
    print(f"  koeffitsiyent = {w:.4f}")
    print(f"  taxminiy talqin: har yil +{w * 100:.2f}%")
    print(f"  aniq talqin:     har yil +{(np.exp(w) - 1) * 100:.2f}%  (haqiqiy +7.25%)")

    print("\n=== 2. y ~ log(x): kamayuvchi qaytim ===")
    reklama = rng.uniform(100, 10_000, n)
    sotuv = 2000 + 850 * np.log(reklama) + rng.normal(0, 400, n)
    m2 = LinearRegression().fit(np.log(reklama).reshape(-1, 1), sotuv)
    print(f"  koeffitsiyent = {m2.coef_[0]:.1f}")
    print(f"  talqin: reklama 1% oshsa, sotuv +{m2.coef_[0] / 100:.1f} birlik")
    print(f"  reklama 2 barobar oshsa: +{m2.coef_[0] * np.log(2):.0f} birlik")

    print("\n=== 3. log-log: elastiklik ===")
    narx = rng.uniform(5, 60, n)
    talab = 10_000 * narx ** (-1.3) * rng.lognormal(0, 0.15, n)
    m3 = LinearRegression().fit(np.log(narx).reshape(-1, 1), np.log(talab))
    print(f"  elastiklik = {m3.coef_[0]:.3f} (haqiqiy -1.3)")
    print("  talqin: narx 1% oshsa, talab "
          f"{abs(m3.coef_[0]):.2f}% kamayadi")

    print("\n=== 4. Asl birlikka qaytish tuzog'i ===")
    log_bashorat = m.predict(tajriba.reshape(-1, 1))
    oddiy = np.exp(log_bashorat)
    qoldiq_var = np.var(np.log(maosh) - log_bashorat, ddof=2)
    tuzatilgan = np.exp(log_bashorat + qoldiq_var / 2)
    print(f"  haqiqiy o'rtacha maosh:    {maosh.mean():8.2f}")
    print(f"  exp(bashorat) o'rtachasi:  {oddiy.mean():8.2f}  ← past (median beradi)")
    print(f"  qiyshiqlik tuzatilgan:     {tuzatilgan.mean():8.2f}")
    print("  ⭐ Log fazodan qaytishda tuzatish kerak")


if __name__ == "__main__":
    main()

=== 1. log(y) ~ x: foizli o'zgarish ===
  koeffitsiyent = 0.0700
  taxminiy talqin: har yil +7.00%
  aniq talqin:     har yil +7.25%  (haqiqiy +7.25%)

=== 2. y ~ log(x): kamayuvchi qaytim ===
  koeffitsiyent = 853.4
  talqin: reklama 1% oshsa, sotuv +8.5 birlik
  reklama 2 barobar oshsa: +592 birlik

=== 3. log-log: elastiklik ===
  elastiklik = -1.303 (haqiqiy -1.3)
  talqin: narx 1% oshsa, talab 1.30% kamayadi

=== 4. Asl birlikka qaytish tuzog'i ===
  haqiqiy o'rtacha maosh:      882.05
  exp(bashorat) o'rtachasi:    868.50  ← past (median beradi)
  qiyshiqlik tuzatilgan:       882.18
  ⭐ Log fazodan qaytishda tuzatish kerak

"""Tayanch daraja va to'liq koeffitsiyentlar hisoboti (real pandas/statsmodels)."""

import numpy as np
import pandas as pd
import statsmodels.api as sm


def yarat(seed: int = 15, n: int = 1500) -> pd.DataFrame:
    rng = np.random.default_rng(seed)
    hudud = rng.choice(["markaz", "shimol", "janub", "chekka"], n,
                       p=[0.3, 0.25, 0.25, 0.2])
    maydon = rng.lognormal(np.log(70), 0.33, n).clip(28, 260)
    yosh = rng.integers(0, 45, n).astype(float)
    qosh = pd.Series(hudud).map({"markaz": 40.0, "shimol": 8.0,
                                 "janub": 0.0, "chekka": -14.0}).to_numpy()
    narx = 18 + 1.52 * maydon - 0.62 * yosh + qosh + rng.normal(0, 13, n)
    return pd.DataFrame({"hudud": hudud, "maydon": maydon, "yosh": yosh, "narx": narx})


def moslash(df: pd.DataFrame, tayanch: str):
    darajalar = [d for d in ["markaz", "shimol", "janub", "chekka"] if d != tayanch]
    X = pd.concat([df[["maydon", "yosh"]],
                   pd.DataFrame({f"hudud_{d}": (df["hudud"] == d).astype(float)
                                 for d in darajalar})], axis=1)
    return sm.OLS(df["narx"], sm.add_constant(X)).fit(), X.columns.tolist()


def main() -> None:
    df = yarat()

    print("=== 1. Tayanch = markaz ===")
    nat, ustunlar = moslash(df, "markaz")
    for nom, w, se in zip(["kesma"] + ustunlar, nat.params, nat.bse):
        print(f"  {nom:<14}: {w:+8.2f} ± {se:5.2f}")
    print("  hudud koeffitsiyentlari — MARKAZGA nisbatan")

    print("\n=== 2. Tayanch = janub ===")
    nat2, ustunlar2 = moslash(df, "janub")
    for nom, w, se in zip(["kesma"] + ustunlar2, nat2.params, nat2.bse):
        print(f"  {nom:<14}: {w:+8.2f} ± {se:5.2f}")
    print(f"  model bir xil: R^2 {nat.rsquared:.4f} va {nat2.rsquared:.4f}")

    print("\n=== 3. Farqlar saqlanadi ===")
    print(f"  markaz - shimol (1-model): {-nat.params['hudud_shimol']:+.2f}")
    print(f"  markaz - shimol (2-model): "
          f"{nat2.params['hudud_markaz'] - nat2.params['hudud_shimol']:+.2f}")

    print("\n=== 4. Hisobot ko'rinishi ===")
    ci = nat.conf_int()
    print("  Belgilar (tayanch hudud: markaz), narx ming $ da:")
    talqin = {
        "maydon": "har m^2 uchun",
        "yosh": "har yil uchun",
        "hudud_shimol": "markazga nisbatan",
        "hudud_janub": "markazga nisbatan",
        "hudud_chekka": "markazga nisbatan",
    }
    for nom in ustunlar:
        w = nat.params[nom]
        lo, hi = ci.loc[nom]
        print(f"   {nom:<14} {w:+7.2f}  [95% CI {lo:+7.2f}, {hi:+7.2f}]  "
              f"{talqin[nom]}")
    print("  ⭐ Tayanch daraja hisobotda ko'rsatilishi SHART")


if __name__ == "__main__":
    main()

=== 1. Tayanch = markaz ===
  kesma         :   +59.00 ±  1.31
  maydon        :    +1.51 ±  0.01
  yosh          :    -0.63 ±  0.03
  hudud_shimol  :   -32.19 ±  0.96
  hudud_janub   :   -39.12 ±  0.91
  hudud_chekka  :   -54.53 ±  1.00
  hudud koeffitsiyentlari — MARKAZGA nisbatan

=== 2. Tayanch = janub ===
  kesma         :   +19.89 ±  1.35
  maydon        :    +1.51 ±  0.01
  yosh          :    -0.63 ±  0.03
  hudud_markaz  :   +39.12 ±  0.91
  hudud_shimol  :    +6.93 ±  0.97
  hudud_chekka  :   -15.41 ±  1.01
  model bir xil: R^2 0.9121 va 0.9121

=== 3. Farqlar saqlanadi ===
  markaz - shimol (1-model): +32.19
  markaz - shimol (2-model): +32.19

=== 4. Hisobot ko'rinishi ===
  Belgilar (tayanch hudud: markaz), narx ming $ da:
   maydon           +1.51  [95% CI   +1.48,   +1.54]  har m^2 uchun
   yosh             -0.63  [95% CI   -0.68,   -0.58]  har yil uchun
   hudud_shimol    -32.19  [95% CI  -34.07,  -30.30]  markazga nisbatan
   hudud_janub     -39.12  [95% CI  -40.91,  -37.33]  markazga nisbatan
   hudud_chekka    -54.53  [95% CI  -56.49,  -52.57]  markazga nisbatan
  ⭐ Tayanch daraja hisobotda ko'rsatilishi SHART

# "markaz (38) maydondan 1.5-bob muhimroq"                          # ⚠️
w_std = m.coef_ * X.std(axis=0, ddof=1)                           # ✅

# koeffitsiyent ishorasidan xulosa                                # ⚠️
variance_inflation_factor(sm.add_constant(X), i)                  # ✅

# "log(maosh) modeli: +0.07 so'm"                                 # ⚠️
# "+7.25% (exp(0.07) - 1)"                                        # ✅

bashorat = np.exp(model.predict(X))                               # ⚠️
bashorat = np.exp(pred + qoldiq_var / 2)                          # ✅

print("shimol: +8.0")                                             # ⚠️
print("shimol: +8.0 (markazga nisbatan)")                         # ✅

pd.get_dummies(df["hudud"])            # kesma bilan kollinear    # ⚠️
pd.get_dummies(df["hudud"], drop_first=True)                      # ✅

print(f"maydon: {w:.2f}")                                         # ⚠️
print(f"maydon: {w:.2f} [CI {lo:.2f}, {hi:.2f}]")                 # ✅

1.  # koeffitsiyent birligi?
2.  # standartlashtirilgan koeffitsiyent formulasi?
3.  # multikollinearlik nimaga zarar qiladi?
4.  # VIF formulasi?
5.  # VIF chegarasi?
6.  # log(y) ~ x talqini?
7.  # log-log talqini?
8.  # y ~ log(x) talqini?
9.  # exp(bashorat) nimani beradi?
10. # kategoriya koeffitsiyenti nimaga nisbatan?
11. # dummy tuzog'i nima?
12. # multikollinearlikni qanday hal qilish mumkin?

1.  # "markaz koeffitsiyenti 38 — eng muhim belgi"

2.  # VIF 22 bo'lgan belgi koeffitsiyentidan xulosa

3.  # "log(narx) modeli: +0.12 ming so'm"

4.  bashorat = np.exp(model.predict(X))

5.  print("shimol: +8.0")