# oobQuantilePredict

Quantile predictions for out-of-bag observations from bag of regression trees

## Syntax

## Description

returns
a vector of medians of the predicted responses at all out-of-bag observations
in `YFit`

= oobQuantilePredict(`Mdl`

)`Mdl.X`

, the predictor data, and using `Mdl`

,
which is a bag of regression trees. `Mdl`

must be
a `TreeBagger`

model
object and `Mdl.OOBIndices`

must be nonempty.

uses
additional options specified by one or more `YFit`

= oobQuantilePredict(`Mdl`

,`Name,Value`

)`Name,Value`

pair
arguments. For example, specify quantile probabilities or trees to
include for quantile estimation.

`[`

also returns a sparse
matrix of response
weights using any of the previous syntaxes.`YFit`

,`YW`

]
= oobQuantilePredict(___)

## Input Arguments

`Mdl`

— Bag of regression trees

`TreeBagger`

model object (default)

Bag of regression trees, specified as a `TreeBagger`

model object created by the `TreeBagger`

function.

The value of

`Mdl.Method`

must be`regression`

.When you train

`Mdl`

using the`TreeBagger`

function, you must specify the name-value pair`'OOBPrediction','on'`

. Consequently,`TreeBagger`

saves required out-of-bag observation index matrix in`Mdl.OOBIndices`

.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

`Quantile`

— Quantile probability

`0.5`

(default) | numeric vector containing values in [0,1]

Quantile probability, specified as the comma-separated pair
consisting of `'Quantile'`

and a numeric vector containing
values in the interval [0,1]. For each observation (row) in `Mdl.X`

, `oobQuantilePredict`

estimates
corresponding quantiles for all probabilities in `Quantile`

.

**Example: **`'Quantile',[0 0.25 0.5 0.75 1]`

**Data Types: **`single`

| `double`

`Trees`

— Indices of trees to use in response estimation

`'all'`

(default) | numeric vector of positive integers

Indices of trees to use in response estimation, specified as
the comma-separated pair consisting of `'Trees'`

and `'all'`

or
a numeric vector of positive integers. Indices correspond to the cells
of `Mdl.Trees`

; each cell therein contains a tree
in the ensemble. The maximum value of `Trees`

must
be less than or equal to the number of trees in the ensemble (`Mdl.NumTrees`

).

For `'all'`

, `oobQuantilePredict`

uses
the indices `1:Mdl.NumTrees`

.

**Example: **`'Trees',[1 10 Mdl.NumTrees]`

**Data Types: **`char`

| `string`

| `single`

| `double`

`TreeWeights`

— Weights to attribute to responses from individual trees

numeric vector of nonnegative values

Weights to attribute to responses from individual trees, specified
as the comma-separated pair consisting of `'TreeWeights'`

and
a numeric vector of `numel(`

nonnegative
values. * trees*)

*is the value of the*

`trees`

`Trees`

name-value
pair argument.The default is `ones(size(`

.* trees*))

**Data Types: **`single`

| `double`

## Output Arguments

`YFit`

— Estimated quantiles

numeric matrix

Estimated quantiles for out-of-bag observations, returned as
an * n*-by-

`numel(``tau`

)

numeric
matrix. *is the number of observations in the training data (*

`n`

`numel(Mdl.Y)`

) and *is the value of the*

`tau`

`Quantile`

name-value pair argument.
That is, `YFit(``j`

,`k`

)

is
the estimated `100*``tau`

(`k`

)

percentile
of the response distribution given `X(``j`

,:)

and
using `Mdl`

.`YW`

— Response weights

sparse matrix

Response weights,
returned as an *n*-by-*n* sparse
matrix. *n* is the number of responses in the training
data (`numel(Mdl.Y)`

). `YW(:,`

specifies
the response weights for the observation in * j*)

`Mdl.X(``j`

,:)

.`oobQuantilePredict`

predicts quantiles using linear
interpolation of the empirical cumulative distribution function (cdf).
For a particular observation, you can use its response weights to
estimate quantiles using alternative methods, such as approximating
the cdf using kernel smoothing.

## Examples

### Predict Out-of-Bag Medians Using Quantile Regression

Load the `carsmall`

data set. Consider a model that predicts the fuel economy (in MPG) of a car given its engine displacement.

`load carsmall`

Train an ensemble of bagged regression trees using the entire data set. Specify 100 weak learners and save out-of-bag indices.

rng(1); % For reproducibility Mdl = TreeBagger(100,Displacement,MPG,'Method','regression',... 'OOBPrediction','on');

`Mdl`

is a `TreeBagger`

ensemble.

Perform quantile regression to predict the out-of-bag median fuel economy for all training observations.

oobMedianMPG = oobQuantilePredict(Mdl);

`oobMedianMPG`

is an `n`

-by-1 numeric vector of medians corresponding to the conditional distribution of the response given the sorted observations in `Mdl.X`

. `n`

is the number of observations, `size(Mdl.X,1)`

.

Sort the observations in ascending order. Plot the observations and the estimated medians on the same figure. Compare the out-of-bag median and mean responses.

[sX,idx] = sort(Mdl.X); oobMeanMPG = oobPredict(Mdl); figure; plot(Displacement,MPG,'k.'); hold on plot(sX,oobMedianMPG(idx)); plot(sX,oobMeanMPG(idx),'r--'); ylabel('Fuel economy'); xlabel('Engine displacement'); legend('Data','Out-of-bag median','Out-of-bag mean'); hold off;

### Estimate Out-of-Bag Prediction Intervals Using Percentiles

Load the `carsmall`

data set. Consider a model that predicts the fuel economy of a car (in MPG) given its engine displacement.

`load carsmall`

Train an ensemble of bagged regression trees using the entire data set. Specify 100 weak learners and save out-of-bag indices.

rng(1); % For reproducibility Mdl = TreeBagger(100,Displacement,MPG,'Method','regression',... 'OOBPrediction','on');

Perform quantile regression to predict the out-of-bag 2.5% and 97.5% percentiles.

`oobQuantPredInts = oobQuantilePredict(Mdl,'Quantile',[0.025,0.975]);`

`oobQuantPredInts`

is an `n`

-by-2 numeric matrix of prediction intervals corresponding to the out-of-bag observations in `Mdl.X`

. `n`

is number of observations, `size(Mdl.X,1)`

. The first column contains the 2.5% percentiles and the second column contains the 97.5% percentiles.

Plot the observations and the estimated medians on the same figure. Compare the percentile prediction intervals and the 95% prediction intervals, assuming the conditional distribution of `MPG`

is Gaussian.

[oobMeanMPG,oobSTEMeanMPG] = oobPredict(Mdl); STDNPredInts = oobMeanMPG + [-1 1]*norminv(0.975).*oobSTEMeanMPG; [sX,idx] = sort(Mdl.X); figure; h1 = plot(Displacement,MPG,'k.'); hold on h2 = plot(sX,oobQuantPredInts(idx,:),'b'); h3 = plot(sX,STDNPredInts(idx,:),'r--'); ylabel('Fuel economy'); xlabel('Engine displacement'); legend([h1,h2(1),h3(1)],{'Data','95% percentile prediction intervals',... '95% Gaussian prediction intervals'}); hold off;

### Estimate Out-of-Bag Conditional Cumulative Distribution Using Quantile Regression

Load the `carsmall`

data set. Consider a model that predicts the fuel economy of a car (in MPG) given its engine displacement.

`load carsmall`

Train an ensemble of bagged regression trees using the entire data set. Specify 100 weak learners and save the out-of-bag indices.

rng(1); % For reproducibility Mdl = TreeBagger(100,Displacement,MPG,'Method','regression',... 'OOBPrediction','on');

Estimate the out-of-bag response weights.

[~,YW] = oobQuantilePredict(Mdl);

`YW`

is an n-by-n sparse matrix containing the response weights. `n`

is the number of training observations, `numel(Y)`

. The response weights for the observation in `Mdl.X(j,:)`

are in `YW(:,j)`

. Response weights are independent of any specified quantile probabilities.

Estimate the out-of-bag, conditional cumulative distribution function (ccdf) of the responses by:

Sorting the responses is ascending order, and then sorting the response weights using the indices induced by sorting the responses.

Computing the cumulative sums over each column of the sorted response weights.

[sortY,sortIdx] = sort(Mdl.Y); cpdf = full(YW(sortIdx,:)); ccdf = cumsum(cpdf);

`ccdf(:,j)`

is the empirical out-of-bag ccdf of the response, given observation `j`

.

Choose a random sample of four training observations. Plot the training sample and identify the chosen observations.

[randX,idx] = datasample(Mdl.X,4); figure; plot(Mdl.X,Mdl.Y,'o'); hold on plot(randX,Mdl.Y(idx),'*','MarkerSize',10); text(randX-10,Mdl.Y(idx)+1.5,{'obs. 1' 'obs. 2' 'obs. 3' 'obs. 4'}); legend('Training Data','Chosen Observations'); xlabel('Engine displacement') ylabel('Fuel economy') hold off

Plot the out-of-bag ccdf for the four chosen responses in the same figure.

figure; plot(sortY,ccdf(:,idx)); legend('ccdf given obs. 1','ccdf given obs. 2',... 'ccdf given obs. 3','ccdf given obs. 4',... 'Location','SouthEast') title('Out-of-Bag Conditional Cumulative Distribution Functions') xlabel('Fuel economy') ylabel('Empirical CDF')

## More About

### Out-of-Bag

In a bagged ensemble, observations are *out-of-bag* when
they are left out of the training sample for a particular learner.
Observations are *in-bag* when they are used
to train a particular learner.

When bagging learners, a practitioner takes a bootstrap sample
(that is, a random sample with replacement) of size *n* for
each learner, and then trains the learners using their respective
bootstrap samples. Drawing *n* out of *n* observations
with replacement omits on average about 37% of observations for each
learner.

The out-of-bag ensemble error, the ensemble error estimated using out-of-bag observations only, is an unbiased estimator of the true ensemble error.

### Quantile Random Forest

*Quantile random forest* [2] is
a quantile-regression method that uses a random forest [1] of regression trees to model the conditional
distribution of a response variable, given the value of predictor
variables. You can use a fitted model to estimate quantiles in the
conditional distribution of the response.

Besides quantile estimation, you can use quantile regression to estimate prediction intervals or detect outliers. For example:

To estimate 95% quantile prediction intervals, estimate the 0.025 and 0.975 quantiles.

To detect outliers, estimate the 0.01 and 0.99 quantiles. All observations smaller than the 0.01 quantile and larger than the 0.99 quantile are outliers. All observations that are outside the interval [

*L*,*U*] can be considered outliers:$$L={Q}_{1}-1.5*IQR$$

and

$$U={Q}_{3}+1.5*IQR,$$

where:

*Q*_{1}is the 0.25 quantile.*Q*_{3}is the 0.75 quantile.*IQR*=*Q*_{3}–*Q*_{1}(the*interquartile range*).

### Response Weights

*Response weights* are
scalars that represent the conditional distribution of the response
given a value in the predictor space. The observations in the bootstrap
samples and the leaves that the training and test observations share
induce response weights.

Given the observation *x*, the response weight
for observation *j* in the training sample using
tree *t* in the ensemble is

$${w}_{tj}(x)=\frac{I\{{X}_{j}\in {S}_{t}(x)\}}{{\displaystyle \sum _{k=1}^{{n}_{\text{train}}}I\{{X}_{k}\in {S}_{t}(x)\}}},$$

where:

*I*{*h*} is the indicator function.*S*(_{t}*x*) is the leaf of tree*t*containing*x*.*n*_{train}is the number of training observations.

In other words, the response weights of a particular tree form the conditional relative frequency distribution of the response.

The response weights for the entire ensemble are averaged over the trees:

$${w}_{j}^{\ast}(x)=\frac{1}{T}{\displaystyle \sum _{t=1}^{T}{w}_{tj}(x)}.$$

## Algorithms

`oobQuantilePredict`

estimates out-of-bag quantiles
by applying `quantilePredict`

to all observations in the
training data (`Mdl.X`

). For each observation, the
method uses only the trees for which the observation is out-of-bag.

For observations that are in-bag for all trees in the ensemble, `oobQuantilePredict`

assigns
the sample quantile of the response data. In other words, `oobQuantilePredict`

does
not use quantile regression for out-of-bag observations. Instead,
it assigns `quantile(Mdl.Y,`

,
where * tau*)

*is the value of the*

`tau`

`Quantile`

name-value
pair argument.## References

[1] Meinshausen, N. “Quantile Regression
Forests.” *Journal of Machine Learning Research*,
Vol. 7, 2006, pp. 983–999.

[2] Breiman, L. “Random Forests.” *Machine
Learning*. Vol. 45, 2001, pp. 5–32.

## Version History

**Introduced in R2016b**

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