Jackknife variance estimates for random forest

In statistics, jackknife variance estimates for random forest are a way to estimate the variance in random forest models, in order to eliminate the bootstrap effects.

Jackknife variance estimates

The sampling variance of bagged learners is:

${\displaystyle V(x)=Var[{\hat {\theta }}^{\infty }(x)]}$

Jackknife estimates can be considered to eliminate the bootstrap effects. The jackknife variance estimator is defined as:^[1]

${\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\hat {\theta }}_{(-i)}-{\overline {\theta }})^{2}}$

In some classification problems, when random forest is used to fit models, jackknife estimated variance is defined as:

${\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\overline {t}}_{(-i)}^{\star }(x)-{\overline {t}}^{\star }(x))^{2}}$

Here, ${\displaystyle t^{\star }}$denotes a decision tree after training, ${\displaystyle t_{(-i)}^{\star }}$ denotes the result based on samples without ${\displaystyle ith}$ observation.

Examples

E-mail spam problem is a common classification problem, in this problem, 57 features are used to classify spam e-mail and non-spam e-mail. Applying IJ-U variance formula to evaluate the accuracy of models with m=15,19 and 57. The results shows in paper( Confidence Intervals for Random Forests: The jackknife and the Infinitesimal Jackknife ) that m = 57 random forest appears to be quite unstable, while predictions made by m=5 random forest appear to be quite stable, this results is corresponding to the evaluation made by error percentage, in which the accuracy of model with m=5 is high and m=57 is low.

Here, accuracy is measured by error rate, which is defined as:

${\displaystyle ErrorRate={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij},}$

Here N is also the number of samples, M is the number of classes, ${\displaystyle y_{ij}}$ is the indicator function which equals 1 when ${\displaystyle ith}$ observation is in class j, equals 0 when in other classes. No probability is considered here. There is another method which is similar to error rate to measure accuracy:

${\displaystyle logloss={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij}log(p_{ij})}$

Here N is the number of samples, M is the number of classes, ${\displaystyle y_{ij}}$ is the indicator function which equals 1 when ${\displaystyle ith}$ observation is in class j, equals 0 when in other classes. ${\displaystyle p_{ij}}$ is the predicted probability of ${\displaystyle ith}$ observation in class ${\displaystyle j}$.This method is used in Kaggle^[2] These two methods are very similar.

Modification for bias

When using Monte Carlo MSEs for estimating ${\displaystyle V_{IJ}^{\infty }}$ and ${\displaystyle V_{J}^{\infty }}$, a problem about the Monte Carlo bias should be considered, especially when n is large, the bias is getting large:

${\displaystyle E[{\hat {V}}_{IJ}^{B}]-{\hat {V}}_{IJ}^{\infty }\approx {\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}}$

To eliminate this influence, bias-corrected modifications are suggested:

${\displaystyle {\hat {V}}_{IJ-U}^{B}={\hat {V}}_{IJ}^{B}-{\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}}$

${\displaystyle {\hat {V}}_{J-U}^{B}={\hat {V}}_{J}^{B}-(e-1){\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}}$

References