モデル適合度の定量化
Rで学ぶ回帰入門
Richie Cotton
Data Evangelist at DataCamp
ヘラブナとパーチのモデル
ヘラブナ
パーチ
決定係数
「r 二乗」や「R 二乗」とも呼ばれる。
応答変数の分散のうち、説明変数から予測できる割合
1は完全適合0は最悪の適合
summary()
「Multiple R-Squared」の値を見る
mdl_bream <- lm(mass_g ~ length_cm, data = bream)
summary(mdl_bream)
# 一部の出力を省略
Residual standard error: 74.15 on 33 degrees of freedom
Multiple R-squared: 0.8781, Adjusted R-squared: 0.8744
F-statistic: 237.6 on 1 and 33 DF, p-value: < 2.2e-16
glance()
library(broom)
library(dplyr)
mdl_bream %>%
glance()
# A tibble: 1 × 12
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.878 0.874 74.2 238. 1.22e-16 1 -199. 405. 409.
# ... with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
mdl_bream %>%
glance() %>%
pull(r.squared)
0.8780627
相関の二乗にすぎない
bream %>%
summarize(
coeff_determination = cor(length_cm, mass_g) ^ 2
)
coeff_determination
1 0.8780627
残差標準誤差 (RSE)
予測と観測応答の「典型的」な差
応答変数と同じ単位を持つ。
summary() の再確認
「Residual standard error」の値を見る
summary(mdl_bream)
# 一部の出力を省略
Residual standard error: 74.15 on 33 degrees of freedom
Multiple R-squared: 0.8781, Adjusted R-squared: 0.8744
F-statistic: 237.6 on 1 and 33 DF, p-value: < 2.2e-16
glance() の再確認
library(broom)
library(dplyr)
mdl_bream %>%
glance()
# A tibble: 1 x 11
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC deviance df.residual
<dbl> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl> <dbl> <int>
1 0.878 0.874 74.2 238. 1.22e-16 2 -199. 405. 409. 181452. 33
mdl_bream %>%
glance() %>%
pull(sigma)
74.15224
RSEの計算: 残差の二乗
bream %>%
mutate(
residuals_sq = residuals(mdl_bream) ^ 2
)
species mass_g length_cm residuals_sq
1 Bream 242 23.2 138.9571
2 Bream 290 24.0 260.7586
3 Bream 340 23.9 5126.9926
4 Bream 363 26.3 1318.9197
5 Bream 430 26.5 390.9743
6 Bream 450 26.8 547.9380
...
RSEの計算: 二乗和の合計
bream %>%
mutate(
residuals_sq = residuals(mdl_bream) ^ 2
) %>%
summarize(
resid_sum_of_sq = sum(residuals_sq)
)
resid_sum_of_sq
1 181452.3
RSEの計算: 自由度
自由度は、観測数からモデル係数数を引いたもの。
bream %>%
mutate(
residuals_sq = residuals(mdl_bream) ^ 2
) %>%
summarize(
resid_sum_of_sq = sum(residuals_sq),
deg_freedom = n() - 2
)
resid_sum_of_sq deg_freedom
1 181452.3 33
RSEの計算: 比の平方根
bream %>%
mutate(
residuals_sq = residuals(mdl_bream) ^ 2
) %>%
summarize(
resid_sum_of_sq = sum(residuals_sq),
deg_freedom = n() - 2,
rse = sqrt(resid_sum_of_sq / deg_freedom)
)
resid_sum_of_sq deg_freedom rse
1 181452.3 33 74.15224
RSEの解釈
mdl_bream の RSE は 74。
予測されたヘラブナ質量と観測質量の差は、典型的に約 74g。
二乗平均平方根誤差 (RMSE)
残差標準誤差
bream %>%
mutate(
residuals_sq = residuals(mdl_bream) ^ 2
) %>%
summarize(
resid_sum_of_sq = sum(residuals_sq),
deg_freedom = n() - 2,
rse = sqrt(resid_sum_of_sq / deg_freedom)
)
二乗平均平方根誤差
bream %>%
mutate(
residuals_sq = residuals(mdl_bream) ^ 2
) %>%
summarize(
resid_sum_of_sq = sum(residuals_sq),
n_obs = n(),
rmse = sqrt(resid_sum_of_sq / n_obs)
)
実践してみましょう！
Rで学ぶ回帰入門
