# Supervised harvesting of expression trees

- Trevor Hastie
^{1, 2}, - Robert Tibshirani
^{2, 1}Email author, - David Botstein
^{3}and - Patrick Brown
^{4}

**2**:research0003.1

**DOI: **10.1186/gb-2001-2-1-research0003

© BioMed Central Ltd. 2001

**Received: **24 August 2000

**Accepted: **21 November 2000

**Published: **10 January 2001

## Abstract

### Background

We propose a new method for supervised learning from gene expression data. We call it 'tree harvesting'. This technique starts with a hierarchical clustering of genes, then models the outcome variable as a sum of the average expression profiles of chosen clusters and their products. It can be applied to many different kinds of outcome measures such as censored survival times, or a response falling in two or more classes (for example, cancer classes). The method can discover genes that have strong effects on their own, and genes that interact with other genes.

### Results

We illustrate the method on data from a lymphoma study, and on a dataset containing samples from eight different cancers. It identified some potentially interesting gene clusters. In simulation studies we found that the procedure may require a large number of experimental samples to successfully discover interactions.

### Conclusions

Tree harvesting is a potentially useful tool for exploration of gene expression data and identification of interesting clusters of genes worthy of further investigation.

## Background

In this paper we introduce 'tree harvesting' - a general method for supervised learning from gene expression data. The scenario is as follows. We have real-valued expression measurements for thousands of genes, measured over a set of samples. The number of samples is typically 50 or 100, but will be larger in the future. An outcome measurement is available for each sample, such as a survival time or cancer class. Our objective is to understand how the genes relate to the outcome.

The generic problem of predicting an outcome measure from a set of features is called 'supervised learning'. If the outcome is quantitative, the term 'regression' is used; for a categorical outcome, 'classification'. There are many techniques available for supervised learning: for example, linear regression, discriminant analysis, neural networks, support vector machines, and boosting. However, these are not likely to work 'off the shelf', as expression data present special challenges. The difficulty is that the number of inputs (genes) is large compared with the number of samples, and they tend to be highly correlated. Hastie *et al*. [1] describe one simple approach to this problem. Here we build a more ambitious model that includes gene interactions.

Our strategy is first to cluster the genes via hierarchical clustering, and then to consider the average expression profiles from all of the clusters in the resulting dendrogram as potential inputs into our prediction model. This has two advantages. First, hierarchical clustering has become a standard descriptive tool for expression data (see, for example, [2]), so by 'harvesting' its clusters, the components of our prediction model will be convenient for interpretation. Second, by using clusters as inputs, we bias the inputs towards correlated sets of genes. This reduces the rate of overfitting of the model. In fact we go further, and give preference to larger clusters, as detailed below.

The basic method is described in the next section for a quantitative output and squared error. We then generalize it to cover other settings such as survival data and qualitative responses. Tree harvesting is illustrated in two real examples and a simulation study is described to investigate the performance of the method. Finally, we generalize tree harvesting further, allowing nonlinear expression effects.

## Results

### Tree harvesting

As our starting point, we have gene expression data *x*_{
ij
} for genes *i* = 1,2, ...*p* and samples *j* = 1,2 ...*n*, and a response measure *y* = (*y*_{1}, *y*_{2}, ... *y*_{
n
}) for each sample (each *y*_{
j
} may be vector-valued). The response measure can take many forms: for example, a quantitative measure such as percentage response to a treatment, a censored survival time, or one of *K* cancer classes. The expression data *x*_{
ij
} may be from a cDNA microarray, in which case it represents the log red to green ratio of a target sample relative to a reference sample. Or *x*_{
ij
}might be the expression level from an oligonucleotide array.

The basic method has two components: a hierarchical clustering of the gene expression profiles, and a response model. The average expression profile for each cluster provides the potential features (inputs) for the response model.

We denote a cluster of genes by *X*_{
c
}, and the corresponding average expression profile by
_{
c
} = (
_{
c
}_{,1},
_{
c
}_{,2}, ...
_{
c
}_{,n}). Starting with *p* genes, a hierarchical clustering agglomerates genes in *p* - 1 subsequent steps, until all genes fall into one big cluster. Hence it generates a total of *p* + (*p* - 1) = 2*p* - 1 clusters, which we denote by *c*_{1}, *c*_{2}, ... *c*_{2p - 1}.

The response model approximates the response measurement by some of the average gene expression profiles and their products, with the potential to capture additive and interaction effects. To facilitate construction of the interaction model, we translate each *x*_{
ij
} to have minimum value 0 over the samples:

*c*, using these translated values. The translation is done solely to make interactions in the model more interpretable. Note that the untranslated values are used in the clustering.

For a quantitative response *y*_{
j
}, *j* = 1,2,...*n*, the model takes the form:

where β_{
k
} and *β*_{
kk'
} are parameters that are estimated by minimizing the sum of squared errors ∑_{
j
} (*y*_{
j
} -
_{
j
})^{2}. As each
has minimum value 0, the product terms represent positive or negative synergy between the genes involved.

*p*- 1 average expression profiles . At each stage we consider all products consisting of a term in and a term in , and add in the term that most improves the fit of the model in terms of a score statistic

*S*. We continue until some maximum number of terms have been added to the model.

_{ j }=

_{0}+

_{1},

*j*, where

_{0},

_{1}are found by least squares. At the second stage, the possible additions to the model are

_{2},

*j*or

_{12}

*,j*,

*j*for some cluster

*c*

_{2}.

In general, this algorithm can produce terms involving the products of three or more average expression profiles. However, the user can put an explicit limit on the order of interaction, *I*, allowed in the model. For simplicity of interpretation, in the examples here we set *I* = 2, meaning that products are limited to pairwise products. This is achieved by only considering single terms (non-products) in
as candidates in the second step.

Models with pairwise interactions as in Equation 2 are often used in statistical applications. The interactions are usually included in an ad hoc basis, after the important additive terms have been included. An exception is the MARS (multivariate additive regression spline) procedure of Friedman [3]. This is a general adaptive learning method, which builds up an interaction model as products of piecewise linear functions of the inputs. The model is built up in the same way as in the tree-harvest procedure. MARS is a very popular methodology and inspired some of the ideas in this paper.

There are crucial computational details that make this algorithm run fast enough for practical applications. First, before the forward stepwise process is started, we need the average expression profiles for all of the 2*p* - 1 clusters. This is achieved in a natural recursive fashion using the tree structure available after a hierarchical clustering: the average expression profile in a node is the weighted average of the two average profiles of the daughter nodes, where the weights are the sizes of the daughter nodes. Other node specific statistics, such as variances and within-variances can be computed in a similar way.

Second, in the second step of the algorithm we must search over all 2*p* - 1 clusters to find the term that most improves the fit of the model. This is achieved by orthogonalizing the candidate average expression profiles with respect to the terms already in the model, and then computing a score test for each candidate term. With a quantitative response and least squares, this process gives exactly the contribution of each candidate term to the model. For survival, classification, and other likelihood-based models, it is a widely used approximation.

### Additional features and issues

#### Data normalization

As with most sets of microarray experiments, the data for each experiment come from different chips and hence must first be normalized to account for chip variation. We assume that the values for each experiment *j* have been centered, that is *x*_{
ij
} →*x*_{
ij
} - (1/*p*) ∑_{
i
}*x*_{
ij
}.

#### Choice of clustering method and criterion

The tree-harvest procedure just starts with a set of clusters, and these can be provided by any clustering method. We have chosen to base the procedure on hierarchical clustering, because of its popularity and effectiveness for microarray data (see, for example, [2]). The sets of clusters are conveniently arranged in a hierachical manner, and are nested within one another. Specifically if the clustering tree is cut off at two different levels, producing say four and five clusters, respectively, then the four clusters are nested within the five. Hence one can look at clusterings of all different resolutions at the same time. This feature is convenient for interpretation of the tree-harvest results, and is not a property of most other clustering methods. Despite this, other clustering methods might prove to have advantages for use in the tree-harvest procedure, including K-means clustering, self-organizing maps [4], and procedures that allow overlapping clusters (for example, gene shaving [1]). The choice of clustering criterion will also effect the results. Again, we have followed Eisen *et al*. [2] and used average linkage clustering, applied to the correlation matrix of the genes. The use of correlation makes the clustering invariant to scaling of the individual genes. Expanding the final clusters (see below) alleviates some of the sensitivity of the results to the choice of clustering method and criterion.

#### Biasing towards larger clusters

*S*

_{ c }, we chose the largest cluster

*c*' whose score

*S*

_{ c }

_{'}is within a factor (1 - α) of the best, that is satisfying

*S*

_{ c }

_{'}≥ (1 - α)

*S*

_{ c }. The parameter α may be chosen by the user: we chose α = 0.10 in our examples. The cluster

*c*' often contains some or all of the genes in

*c*, but this is not a requirement. Although this biases the selection towards larger clusters, a single gene can still be chosen if its contribution is spectacular and unique.

#### Model size selection and cross-validation

Having built a harvest model with some large number of terms, *M*, we carry out a backward deletion, at each stage discarding the term that causes the smallest increase in sum of squares. We continue until the model contains only the constant term. This gives a sequence of models with numbers of terms 1,2, ... *M*, and we wish to select a model size, and hence one of these models. The model size is chosen by *K*-fold cross-validation. The data is split into *K* parts. For each *k* = 1,2, ... *K* the harvest procedure is trained on all of the data except the *k*th part, and then data in the *k*th part is predicted from the trained model. The results are averaged over *k* = 1,2, ... *K*. This is illustrated in the examples in the next two sections.

#### Expanding the clusters

Hierarchical clustering uses a sequence of discrete partitions of genes. Hence, for a given cluster, there may be genes not in that cluster that are more highly correlated with the cluster's average expression profile than some of the genes in the cluster. To account for this, we simply look for such genes in the final set of clusters and report them as 'extra genes' belonging to each cluster.

### Tree harvesting for general response variables

The tree-harvest method can be applied to most commonly occurring types of response data. Given responses *y* =(*y*_{1}, *y*_{2}, ... *y*_{
n
}), we form a model-based approximation η =(η_{1}, η_{2}, ... η_{
n
}) to minimize a loss function:

Each quantity η_{
j
} is a function of the average gene expression profiles, having the form given in Equation 2:

Some common response types and loss functions

Response type | Loss function |
---|---|

Quantitative | Sum of squares ∑ |

Censored survival time | Partial log-likelihood |

Categorical | Multinomial log-likelihood |

_{ k },

*β*

_{ k,k' }}, addition of each new term to the model is based on the size of the score statistic:

and similarly for *β*_{
k,k'
}. The censored survival time and categorical response models are illustrated in the next two sections.

### Survival of lymphoma patients

*et al*. [5]. The column labels refer to different patients, and the row labels identify the genes. We have applied hierarchical clustering to the genes and a separate clustering to the samples. Each clustering produces a (non-unique) ordering, one that ensures that the branches of the corresponding dendrogram do not cross. Figure 1 displays the original data, with rows and columns ordered accordingly.

For each of the 36 patients, a (possibly censored) survival time is available; these range from 1.3 to 102.4 months, and 19 of the 36 patients died in the study period. An appropriate response model is Cox's proportional hazards model [6]. This has the form:

Here *z*_{
j
} = (*z*_{1j}, *z*_{2j}, ... *z*_{
mj
}) are *m* risk factors (features) for sample *j*, and *h*(*t*|*z*_{
j
}) denotes the hazard function for an individual with feature values *z*; *h*_{0}(*t*) is the baseline hazard function for an individual with risk factors *z* = 0. The unknown function *r*(*z*_{
j
}) represents the log-relative risk of dying at any time *t* for an individual with *z* = *z*_{
j
} versus an individual with *z* = 0. In the tree harvest model, the features (*z*_{1j}, *z*_{2j}, ... *z*_{
mj
}) are average expression profiles and we take *r*(*z*_{
j
}) to be of the form:

as in Equation 2. The tree-harvest algorithm computes an approximate score test from the partial likelihood, to decide which term is entered at each stage.

Results of tree harvesting applied to lymphoma data

Node | Parent | Score | -2log-likelihood | Size | |
---|---|---|---|---|---|

1 | 3005 | 0 | 2.980 | 104.34 | 8 |

2 | 2236 | 3005 | 2.784 | 94.91 | 3 |

3 | 443 | 0 | 2.579 | 84.12 | 2 |

4 | s2461 | 3005 | 2.948 | 70.06 | 1 |

5 | s2188 | 3005 | 2.658 | 60.16 | 1 |

Cox survival model fit to all five terms: | |||||

Coef | exp(coef) | se(coef) | z |
| |

z1 | 4.118 | 61.442 | 0.921 | 4.47 | 7.7e-06 |

z2 | 1.072 | 2.922 | 0.293 | 3.66 | 2.5e-04 |

z3 | 2.195 | 8.976 | 0.528 | 4.15 | 3.3e-05 |

z4 | 1.079 | 2.941 | 0.281 | 3.83 | 1.3e-04 |

z5 | -0.667 | 0.513 | 0.221 | -3.02 | 2.5e-03 |

Some explanation is needed. At each stage the 'Node' refers to the cluster whose average expression profile is chosen for addition to the model. 'Parent' is the number of the cluster, already in the model, that is to be multiplied by the Node average expression profile; Parent = 0 refers to the constant function 1. Nodes starting with 's' for Node or Parent indicate single genes. 'Score' is the score value achieved by addition of the term; it is roughly a Gaussian variate, so that values ≥ 2 are reasonably large.

Overall the resulting model has the form:

*p*= 2.4 x 10

^{-5}).

If each of the 3,624 genes is ranked from lowest (1) to highest (3,624) value of the Cox score statistic, the average rank of the eight genes in the cluster 3005 is 3,574.5. Hence these genes are among the strongest individually for predicting survival, but are not the eight strongest genes. Rather they are a set of genes with very similar expression profiles, highly correlated with survival.

### Human tumor data

*et al*. [7] and Scherf

*et al*. [8], omitting the two prostate tumors and the one unknown class. There are expression values for 6,830 genes for each of the tumors, with the distribution across cancer classes shown in Table 3.

Distribution of gene expression across cancer class

Breast | CNS | Colon | Leukemia | Melanoma | NSCLC | Ovarian | Renal |
---|---|---|---|---|---|---|---|

9 | 5 | 7 | 8 | 8 | 9 | 6 | 9 |

Here, the tree-harvest method builds a multiple logistic regression (MLR) model in a stepwise fashion, using similar steps to those used for the Cox model for survival data. The goal here is to model the probability of the tumor class, given the expression values. In general terms, if the class variable is denoted by *y* taking values in {1,2, ..., *J*} and the predictor variables by *x*_{1},*x*_{2}, ..., *x*_{
p
} a linear MLR model has the form:

As before, the *x*_{1} will be cluster averages, possibly individual genes, or pairwise products of these. The logistic transform is a natural scale on which to model the *K* probabilities; the inverse transformation:

guarantees that the probabilities sum to 1 and are positive. The model is usually fit by multinomial maximum likelihood. Because the response is really multidimensional, we do not expect a single *x* to be able to distinguish all the cancer classes; this would imply that a single gene average creates an ordering that separates the cancer classes. Typically several are required.

At each stage, the tree-harvest algorithm considers augmenting the current fitted MLR model with a new term, candidates being any of the node averages, individual genes, or products of these with terms already in the model. As before, a score statistic is used, appropriate for the multinomial model.

Results of tree harvesting applied to human tumor data

Node | Parent | Score | -2log-likelihood | Size | |
---|---|---|---|---|---|

1 | 1177 | 0 | 6.48 | 197.53 | 6 |

2 | 3843 | 0 | 1.97 | 132.34 | 4 |

3 | 2008 | 0 | 1.78 | 79.34 | 3 |

4 | 1665 | 3843 | 0.85 | 71.01 | 3 |

5 | 5009 | 0 | 0.69 | 51.91 | 68 |

6 | 5087 | 2008 | 0.59 | 9.32 | 9 |

7 | 820 | 3843 | 0.55 | 0.00 | 2 |

### Simulations

We carried out a simulation experiment to assess how well tree harvesting discovers 'true' structure. To ensure that the gene expression measurements were realistic in magnitude and correlation, we used the matrix of 3624 × 36 lymphoma expression measurements for our study. Artificial survival and censoring times were then generated, to produce a simulated dataset for harvesting.

Simulation results

Scenario | Average | Average | Proportion | Proportion | Average |
---|---|---|---|---|---|

number | number | of harvest | of true | correlation | |

in | in | genes in | genes in | ||

true | estimate | true | harvest | ||

3,624 total genes, 36 samples. Relative risk = 2.0 in additive scenarios | |||||

| 1.0 | 2.4 | 0.80 | 0.80 | 0.86 |

2 < | 3.4 | 4.8 | 0.60 | 0.60 | 0.91 |

10 < | 26.2 | 6.4 | 0.60 | 0.19 | 0.77 |

Interaction | 3.4 | 2.6 | 0.28 | 0.21 | 0.65 |

3,624 total genes, 36 samples. Relative risk = 1.0 in additive scenarios | |||||

| 1.0 | 1.6 | 0.24 | 0.60 | 0.61 |

2 < | 3.4 | 4.6 | 0.13 | 0.20 | 0.58 |

10 < | 26.2 | 3.8 | 0.40 | 0.21 | 0.61 |

1,622 total genes, 129 samples. Relative risk = 2.0 in additive scenarios | |||||

| 1.00 | 1.60 | 0.77 | 1.00 | 0.97 |

2 < | 2.80 | 3.00 | 0.93 | 1.00 | 0.99 |

10 < | 64.6 | 16.6 | 0.94 | 0.66 | 0.91 |

Interaction | 9.4 | 7.6 | 0.85 | 0.87 | 0.86 |

For the interaction scenario, we randomly chose one cluster *c*_{1} with between two and ten genes, and then chose the second cluster *c*_{2} to be the cluster containing between two and ten genes whose average expression profile had the smallest correlation with that for *c*_{1}. This made the two clusters as independent as possible, giving the harvest procedure the most chance of discovering their interaction. The survival data were then generated with relative risk function 4
+ 4
+ 3[
- *r*] where *r* is the projection of
on
and
. Tree harvesting was allowed to enter three terms.

The results are shown in the top panel of Table 5. The numbers are averages over five simulations. The columns show the average number of genes in the true cluster, average number of genes in the cluster found by tree harvesting, the proportion of the genes found by tree harvesting that are in the true cluster, and vice versa. The final column shows the average absolute correlation of the average expression profile of the true cluster with the estimated cluster. For the interaction scenario, these quantities refer to the pooled set of genes that make up the interaction. If more than one interaction was found, the one having greatest overlap with the true interacting clusters is reported. We see that tree harvesting returns clusters that are a little too large when the true cluster is a single gene, and too small when the true cluster is large. In the additive scenario, it does a fairly good job at discovering the true cluster or one similar to it. However, it correctly discovers interactions only about a quarter of the time. A greater number of samples are needed to accurately find interactions among such a large set of genes. On the other hand, the correlations in the rightmost column are all quite high, indicating that tree harvesting is able to find clusters that are nearly as good as the true ones.

The middle panel of Table 5 shows the results for the additive scenarios when the relative risk is lowered to 1.0. As expected, they are somewhat worse, although the average correlations are still around 0.60.

To investigate whether a greater number of samples would improve the detection of interactions, we applied the same methodology to a set of 129 samples and 1,622 genes, from an unpublished study of breast cancer (T. Sorlie, C. Perou, and collaborators, personal communication). As before, we used the expression values and simulated sets of synthetic survival times. The results are shown in the bottom panel of Table 5. Now the tree-harvest procedure does a good job of recovering the interactions. The greater number of samples, together with the smaller number of genes, resulted in a significant improvement in performance.

### Nonlinear tree-harvest models

In the harvest procedure described above, the effect of gene expression is modeled linearly. Thus, in modeling each term we assume that increasing or decreasing gene expression has a consistent effect on the outcome. However, it is biologically plausible for a gene to have a nonlinear effect: for example, increasing expression may correlate with longer survival, but only up to some level. Beyond that level, the same or worse survival might result.

To allow for nonlinear effects, flexible bases of functions could be used for each gene. However, with a large number of genes this would tend to overfit quickly. Hence we allow a simple quadratic function for each gene:

We first orthogonalize *b*(*x*) with respect to the linear term for the same gene, and then allow the transformed expression *b*(*x*) in place of the expression *x* in the tree-harvest model. In detail, the model has the same form as Equation 2:

If a quadratic term is multiplied by a positive coefficient, then the effect of a gene has a 'U' shape, decreasing and then increasing. For a negative coefficient, the effect is an inverted 'U'. A product interaction between two quadratic terms would indicate a strong synergistic effect between the two genes, with direction of expression (below or above average) ignored. When the nonlinear option is used in harvesting, the procedure tries both linear and nonlinear terms at each stage, and chooses the one with maximum score.

### Lymphoma data continued

Results of nonlinear tree harvest procedure applied to lymphoma data

Node | Parent | Score | -2Log-likelihood | Size | Nonlinear? | |
---|---|---|---|---|---|---|

1 | 3005 | 2.980 | 104.34 | 8 | No | |

2 | s2597 | 0 | 3.891 | 91.18 | 1 | Yes |

3 | s1021 | 3005 | 3.919 | 81.59 | 1 | Yes |

4 | s583 | 3005 | 3.314 | 72.39 | 1 | Yes |

Cox model fit to all 4 terms | ||||||

coef | exp(coef) | se(coef) | z |
| ||

z1 | 3.107 | 22.36 | 0.6551 | 4.74 | 2.1 × 10 | |

z2 | 0.794 | 2.21 | 0.1990 | 3.99 | 6.6 × 10 | |

z3 | 0.380 | 1.46 | 0.0954 | 3.98 | 6.8 × 10 | |

z4 | 0.238 | 1.27 | 0.0729 | 3.27 | 1.1 × 10 |

In the second cluster, for example (marked '2' in Figure 9), we see that survival time is greatest for moderate expression levels, and is worse for very low or very high levels.

Overall, the lack of significant improvement of the nonlinear model over the linear model gives greater confidence that the linear shape for each term is appropriate in this example. However, quadratic models may well be useful for other gene expression experiments.

## Conclusions

The tree harvest procedure is a promising, general method for supervised learning from gene expression data. It aims to find additive and interaction structure among clusters of genes, in their relation to an outcome measure. This procedure, and probably any procedure with similar aims, requires a large number of samples to uncover successfully such structure. In the real data examples, the method was somewhat hampered by the paucity of available samples. We plan to try tree harvesting on larger gene expression datasets, as they become available. We used a forward stepwise strategy involving sum and products of the average gene expression of chosen clusters. We chose this strategy because it produces interpretable, biologically plausible models. Other models could be built from the average gene expression of clusters, including tree-based models or boosting methods (see, for example, Friedman *et al*. [9]).

## Additional data

Additional data available with the online version of this article include clusters from the harvest model applied to lymphoma data.

## Declarations

## Authors’ Affiliations

## References

- Hastie T, Tibshirani R, Eisen M, Alizadeh A, Levy R, Staudt L, Botstein D, Brown P: 'Gene shaving' as a method of identifying distinct sets of genes with similar expression patterns. Genome Biology. 2000, 1: research0003.1-0003.21.View ArticleGoogle Scholar
- Eisen M, Spellman P, Brown P, Botstein D: Cluster analysis and display of genome-wide expression patterns. Proc Natl Acad Sci USA. 1998, 95: 14863-14868.View ArticleGoogle Scholar
- Friedman J: Multivariate adaptive regression splines. Annl Stat. 1991, 19: 1-141.View ArticleGoogle Scholar
- Tamayo P, Slonim D, Mesirov J, Zhu Q, Kitareewan S, Dmitrovsky E, Lander E, Golub T: Interpreting patterns of gene expression with self-organizing maps: methods and application to hematopoietic differentiation. Proc Natl Acad Sci USA. 1999, 96: 2907-2912.View ArticleGoogle Scholar
- Alizadeh A, Eisen M, Davis RE, Ma C, Lossos I, Rosenwal A, Boldrick J, Sabet H, Tran T, Yu X, et al: Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling. Nature. 2000, 403: 503-511. 10.1038/35000501.View ArticleGoogle Scholar
- Cox D: Regression models and life tables (with discussion). J Roy Stat Soc B. 1972, 74: 187-220.Google Scholar
- Ross D, Scherf U, Eisen M, Perou C, Spellman P, Iyer V, Rees C, Jeffery S, Van de Rijn M, Waltham M, et al: Systematic variation in gene expression patterns in human cancer cell lines. Nat Genet. 2000, 24: 227-235. 10.1038/73432.View ArticleGoogle Scholar
- Scherf U, Ross DT, Waltham M, Smith LH, Lee JK, Kohn KW, Eisen MB, Reinhold WC, Myers TG, Andrews DT, et al: A gene expression database for the molecular pharmacology of cancer. Nat Genet. 2000, 24: 236-244. 10.1038/73439.View ArticleGoogle Scholar
- Friedman J, Hastie T, Tibshirani R: Additive logistic regression: a statistical view of boosting. Annl Stat. 2000, 28: 337-374.View ArticleGoogle Scholar