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決定係数を非線形モデルで非推奨の理由

Posted at

決定係数について、非線形モデルに対する評価指標として良くないと先日知りました。
あまり詳しく書きませんが、リンク集とコメントを少し。

非推奨の理由

もともとはここに書かれていた内容を見ました。

上記で大事だと思った部分を太字強調しました。

For linear models, the sums of the squared errors always add up in a specific manner: SS Regression + SS Error = SS Total.
This seems quite logical. The variance that the regression model accounts for plus the error variance adds up to equal the total variance. Further, R-squared equals SS Regression / SS Total, which mathematically must produce a value between 0 and 100%.
In nonlinear regression, SS Regression + SS Error do not equal SS Total! This completely invalidates R-squared for nonlinear models, and it no longer has to be between 0 and 100%.

そもそも決定係数とは

で、決定係数を復習しました。以下のサイトが非常にわかりやすい。

決定係数に関して数式を見ると非線形モデルで成立しないことがわかります。全変動の式展開時に推定値の代入ができない

元論文

元論文「An evaluation of R2 as an inadequate measure for nonlinear models in pharmacological and biochemical research: a Monte Carlo approach」で詳しく書かれている模様。あまり見られていない。

代替指標を探す際に以下の文言が気になりました。

The bias-corrected AICc and also BIC performed significantly better in this respect.

# TensorFlow と指標

そういえば、tfのような非線形のNNにはMetricsにいないですね(v2.9.1時点)。

tfa にはあった。

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