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Liu CVPR 2013 - Semi-supervised Node Splitting for Random Forest Construction

Liu CVPR 2013 - Semi-supervised Node Splitting for Random Forest Construction
Liu CVPR 2013 - Semi-supervised Node Splitting for Random Forest Construction

Semi-supervised Node Splitting for Random Forest Construction

Xiao Liu?,Mingli Song?,Dacheng Tao?,Zicheng Liu?,Luming Zhang?,Chun Chen?and Jiajun Bu?College of Computer Science,Zhejiang University?

Centre for Quantum Computation and Intelligent Systems,University of Technology Sydney?

Microsoft Research,Redmond?

{ender liux,brooksong,zglumg,chenc,bjj}@https://www.sodocs.net/doc/1c18405958.html,?

dacheng.tao@https://www.sodocs.net/doc/1c18405958.html,?,zliu@https://www.sodocs.net/doc/1c18405958.html,?

Abstract

Node splitting is an important issue in Random Forest but robust splitting requires a large number of training sam-ples.Existing solutions fail to properly partition the fea-ture space if there are insuf?cient training data.In this pa-per,we present semi-supervised splitting to overcome this limitation by splitting nodes with the guidance of both la-beled and unlabeled data.In particular,we derive a non-parametric algorithm to obtain an accurate quality mea-sure of splitting by incorporating abundant unlabeled da-ta.To avoid the curse of dimensionality,we project the data points from the original high-dimensional feature s-pace onto a low-dimensional subspace before estimation.

A uni?ed optimization framework is proposed to select a coupled pair of subspace and separating hyperplane such that the smoothness of the subspace and the quality of the splitting are guaranteed simultaneously.The proposed al-gorithm is compared with state-of-the-art supervised and semi-supervised algorithms for typical computer vision ap-plications such as object categorization and image segmen-tation.Experimental results on publicly available datasets demonstrate the superiority of our method.

1.Introduction

Random Forest(RF)has been applied to various comput-er vision tasks including target tracking,object categoriza-tion and image segmentation.Though it is one of the state-of-the-art classi?ers,its promising performance depends heavily on the size of the labeled data.Because labeling training samples is very time consuming,only a small size of labeled training set is given in some tasks,which usually leads to an obvious performance drop.Thus,sometimes the insuf?ciency of labeled data is a severe challenging issue in the construction of RF.A popular solution to overcome this problem is to introduce abundant unlabeled data to guide the learning,which is known as semi-supervised learning (SSL).However,though many approaches have been given on SSL,few of them are applicable to RF.The only existing representative attempt is the Deterministic Annealing based Semi-Supervised Random Forests(DAS-RF)[14],which treated the unlabeled data as additional variables for margin maximization between different classes.Similar to many other margin maximization methods,?nding the exact so-lution of DAS-RF is NP-hard.Although an ef?cient deter-ministic annealing optimization is used to search an approx-imate solution,it cannot provide robust results ef?ciently. In spite of this,it has been pointed out[26]that the ef-fectiveness of margin maximization based methods for SSL depends heavily on speci?c data distribution which is usu-ally dif?cult to be satis?ed in many applications.Hence,it is desirable to?nd a method that allows RF to utilize the unlabeled data without losing its?exibility.

In this paper,by analyzing the construction of an RF using a small size of labeled training dataset,we?nd that the performance bottleneck is located in the node splitting. From this insight,we tackle the aforementioned problem by introducing abundant unlabeled data to guide the split-ting.Based on kernel density estimation and the law of total probability,we derive a nonparametric algorithm to utilize abundant unlabeled data to obtain an accurate quality mea-sure for node splitting.In particular,to avoid the curse of dimensionality,the data points are projected from the origi-nal high-dimensional feature space onto a low-dimensional subspace before estimating the categorical distributions.Fi-nally,a uni?ed optimization framework is proposed to se-lect a coupled pair of subspace and separating hyperplane for each node such that the smoothness of the subspace and the quality of the splitting are guaranteed simultaneously.

Our contribution is three-fold:

?We experimentally show that node splitting quality is the performance bottleneck for constructing RF with a small size labeled training set.

?We show that partitioning an arbitrary feature space

2013 IEEE Conference on Computer Vision and Pattern Recognition

with a hyperplane can be treated as projecting the data points from the original high-dimensional space onto the one-dimensional subspace that is perpendicular to the separating hyperplane.Thus a uni?ed optimiza-tion framework is presented to choose a coupled pair of subspace and hyperplane such that the subspace is smooth and the hyperplane can effectively partition the feature space.

?We present an ef?cient nonparametric estimation-based semi-supervised splitting method to construct R-F.

2.Related Work

Node splitting is the key issue of tree-based classi?ers. Payne et al.[16]tried to build optimal binary trees such that the least number of tests are required to approach the leaf nodes.However,their tree construction is based on recursive dynamic programming and the algorithm is fea-sible only for a small number of features.Hya?l et al.[12] showed that the optimal sense of minimizing the expect-ed number of tests required to classify an unknown sample is an NP-complete problem.Wu et al.[22]suggested a histogram-based splitting criterion for decision design.The histogram of training data is plotted on each feature axis.

A threshold is selected to partition the classes.A limita-tion of this method is that only few features(usually one) are considered at each stage such that the interaction be-tween features cannot be observed.Rounds[19]proposed Komogorov-Smirnov(K-S)distance and test as the splitting criterion.He suggested that the K-S distance between parts of the partition should be as large as possible.Breiman et al.[5]proposed to use the Gini index as the impurity mea-sure for internal nodes.The goodness of a split is de?ned by the decrease in impurity.Suen et al.[21]proposed an entropy-based splitting criterion for decision tree construc-tion.The entropy measure was later used to construct the well-known ID3decision tree[17].An improved version [18],namely C4.5tree,was proposed in which the normal-ized information gain is used as the criterion of splitting.

All the semi-supervised learning(SSL)methods rely on the smoothness assumption:if two points are close,the corresponding outputs should be similar[7,24].Howev-er,since the feature points are usually in high-dimensional space,one may have to face the curse of dimensionality to directly use the unlabeled data for estimation and thus there is insuf?cient number of observations to obtain a good estimation.Determined by how to address this problem, previous feasible discriminative SSL methods can be cat-egorized into two families.The?rst family relies on the low density separation assumption:the decision boundary should lie in a low-density region.A classi?cation margin for both unlabeled and labeled data is de?ned and maxi-mized through global optimization.Typical methods in this family are Transductive Support Vector Machine(TSVM) [13]and DAS-RF[14].The second family assumes that the high-dimensional data roughly lie on a low-dimensional manifold such that the unlabeled data can be ef?ciently used to infer the structure of the manifold without being troubled by the curse of dimensionality.A typical method in this family is LapSVM[2].Both of the above two families pre-dict the labels of unlabeled data as additional optimization variables,while the proposed method follows another line, i.e.,we project the data onto a low-dimensional subspace such that a small amount of data can give an accurate esti-mation.

3.Pre-analysis

The lack of training data in?uences RF construction in two ways:1)the depth of the forest is limited and2)the best splitting may not be chosen.Since the?rst in?uence is inevitable,we focus our effort on the second one in this paper.

Before doing this,it is necessary to understand the in-?uence of splitting.Two RFs were constructed for com-parison:the?rst RF was constructed conventionally only using a very small size labeled training set which may lead to bad splitting.The second RF was constructed using the same training set but an additional labeled set was used ex-clusively for better splitting.We constructed both RFs with 100trees and used the popular entropy gain maximization criterion for splitting.

We used the Satimage and the Pendigits datasets from the LibSVM repository[6].The comparison results of the two RFs are shown in Figure1,where it can be seen that there are obvious performance improvements as a result of better splitting.From the comparisons above,it is obvious that the splitting quality is the performance bottleneck of RF construction when the size of the training set is small.For this reason,it is necessary to focus our effort on the node splitting strategies for RF construction.

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Figure1.Recognition accuracy on(a)the Satimage dataset,and (b)the Pendigits dataset.The?rst RF was constructed conven-tionally only using a very small size of training data which may lead to bad splitting.The second RF was constructed using the same training set but an additional larger labeled set was used for better splitting.

4.Semi-supervised Node Splitting

RF consists of multiple decision trees:F= {t1,t2,???,t N}and each is independently trained and test-ed.

In the RF construction stage,the algorithm learns a clas-si?cation function?:X→Y using the training sam-ples{x i∈X}i=1???l and the corresponding labels{y i∈Y}i=1???l,where X??M is the feature space and Y= {1???K}is the label set.The trees are usually grown to the greatest possible extent without pruning.Each internal n-ode of RF is binary split with a partition criterion.Each leaf node of RF is a voter which votes for the class into which the most samples fall.

During testing,given a test case x,RF gives the proba-bility estimation for each class as follows

p(k∣x)=1

N

N

i=1

p i(k∣x),(1)

where p i(k∣x)is the probability estimation of class k given by the i t?tree.It is estimated by calculating the ratio that class k gets votes from the leaves in the i t?tree

p i(k∣x)=

l i,k

∑K

j=1

l i,j

,(2)

where l i,k is the number of leaves in the i t?tree that vote for class k.The overall decision function of RF is de?ned as

?=argmax

k∈Y

p(k∣x).(3) 4.1.Semi-supervised Splitting

Considering both accuracy and time cost,oblique linear split is the most popular split strategy.An oblique linear split is expressed as a function of the hyperplane

W?x=θ,(4) where W∈?M andθ∈?are the parameters.When there is only one non-zero element in W,the split strategy focus-es on a single attribute in one node[4,10].Otherwise,the combination of multiple attributes is considered[14,11].

Given the data falling into a node and a candidate hy-perplane,a quality measure needs to be de?ned such that one can search for the best hyperplane to maximize the s-plitting quality.There are four common criteria to evaluate the splitting quality,i.e.,information gain[17],normalized information gain[18],Gini index[5],and Bayesian classi?-cation error[9].As the discussion given in Appendix A,the key procedure of all the four criteria is the estimation of p k, which re?ects the categorical distribution of the k t?class.

Traditionally,we can use the law of total probability to calculate p k

p k=

R

p(k∣x)dp(x).(5) Since both p(x)and p(k∣x)are unknown,?nite labeled samples are used to make the estimation

p k=

1

∑∣R∣

i=1

w R

i

∣R∣

i=1

p(k∣x R

i

)w R

i

,(6)

where w R

i

is the i t?sample falling into a node R and w R

i is its weight.Since all the samples are labeled,p(k∣x R

i

)is either1or0

p(k∣x R

i

)=[y R

i

=k].(7) The problem with the fully supervised splitting is that,

although the distribution p(k∣x R

i

)is given by the labeled sample,the sparse labeled data cannot give a good approx-imation of the marginal distribution which may lead to a worse choice of the separating hyperplane.If only given few labeled samples,for example,one would choose to par-tition the two-dimensional space with the hyperplane shown in Figure2(a).However,its estimation of the probability distribution is not good.In contrast,given more data,a bet-ter splitting can be found,as shown in Figure2(b).

(a)(b)

(c)(d)

Figure2.The red triangles and the blue circles are labeled samples of two classes while the black squares are unlabeled data.If only given a small number of labeled data,one may partition the fea-ture space with the hyperplane shown in(a).When given more la-beled data,one could?nd a better splitting strategy as in(b).Even the abundant data are unlabeled one can still choose the appropri-ate separating hyperplane as in(c)by combining the law of total probability and the kernel-based density estimation.As shown in (d),our method goes a bit further.We carry out the kernel-based density estimation in a one-dimensional subspace of the original feature space.Through this means,the curse of dimensionality can be avoided.

Unfortunately,the insuf?ciency of labeled training da-ta usually leads to a sparse distribution and a bad approxi-mation like Figure2(a).Our solution to overcome the this

limitation is to introduce abundant unlabeled samples to es-timate p k .The law of total probability is still used to calcu-late the probability distribution p k over different classes

?p k =1

∑∣R ∣

i =1w R i

∣R ∣

∑i =1

?p (k ∣x R i )w R i .(8)

Since we now have many more data points,a much bet-ter approximation for the marginal distribution of p (x )in

R can be obtained.A new problem to arise is that the pos-teriori distribution ?p (k ∣x R i )of unlabeled data is unknown.Since there are no priors of the categories,we can estimate ?p (k ∣x R i )as the probability density ratio

?p (k ∣x R i )=p (x R i ∣k )

∑K

j =1p (x R i ∣j ).(9)For p (x R i ∣k ),we apply a kernel-based density estima-tion with Gaussian kernel [20]

K ?(u )=??d (2π)?d/2exp {

?1

2?

2u T u }

,

(10)

where ?is the bandwidth to be determined and d is the data dimension.We then have the following estimation for an unlabeled sample

p (x R i ∣k )=

1n k

y j =k

K ?(x R i ?x j ),(11)where n k is the number of samples that are labeled k .

4.2.Kernel Based Density Estimation on Low-dimensional Subspace

In this part,we derive a convenient formula to select the bandwidth ?.The asymptotic mean squared error (AMISE)criterion [20]is applied to select the optimal bandwidth AMISE (H )=1

4μ22

(K )∫

[tr {H T H p (u )H }]2du (12)

+

∣∣K ∣∣22

n det(H )

,where H is the bandwidth matrix,n is the sample size,H p (u )is the Hessian of the density p (u )to be estimated and μ22(K )is the squared variance of the kernel.In our case,K is the Gaussian kernel and the bandwidth matrix is H =?I d .Since the true density p (x ∣k )is unknown,we adopt the commonly used rule-of-thumb [20]to replace the unknown true density by a reference density q (x ),e.g.,a Gaussian distribution with its covariance matrix equal to the sample covariance.To simplify the calculation,we conduct whitening preprocessing before applying the estimation so that the sample covariance is the identity matrix,and thus q (x )=N (0,I ).Based on the above discussions,AMISE for the conditional density p (x ∣y =k )can be simpli?ed as

AMISE (?)=2d +d 2

2d

+4πd/2+12d πd/2n k ?d

.(13)

Taking the derivative with respect to ?and setting it as 0,we obtain the optimal bandwidth

?opt =(

4

(d +2)n k )1/(4+d ).(14)

Figure 2(c)shows a case where,by combining the law of total probability and the kernel-based density estimation,the appropriate separating hyperplane with abundant unla-beled data can be chosen.

Since the feature points are usually in very high-dimensional space,directly estimating the densities is still not a good idea,not only because of the intractable time complexity but also because there are insuf?cient labeled samples to make a good estimation in the high-dimensional space.

Our solution to this problem is to estimate the densities in a low-dimensional subspace rather than in the original high-dimensional feature space.Note that another way of looking at the hyperplane partition is that the original da-ta are projected onto the one-dimensional subspace that is perpendicular to the separating hyperplane.In particular,by using a hyperplane W ?x =θto partition the feature space,the algorithm makes a projection with the projection function

z =W ?x.

(15)

Furthermore,if the data can be well separated by the hyperplane,it is reasonable to believe that the metric in the subspace will re?ect the intrinsic distance between data points.Thus,the labels of both labeled and unlabeled data distribute smoothly along the subspace.We can therefore estimate the density p (k ∣z )instead of p (k ∣x ).In this case,the Gaussian kernel becomes

K ?(u )=1

?√2πexp {?12?2u 2}

,

(16)and the optimal choice of the bandwidth is ?opt =(4

3n k

)1/5.

(17)

We want to search for a coupled pair of subspace and separating hyperplane such that the smoothness of the sub-space and the quality of the splitting are guaranteed simul-taneously.An alternative optimization strategy is adopted to couple the two procedures by iterating the following two updating steps:

?Project the data onto the given subspace and estimate the categorical distribution.

?Search for a separating hyperplane according to the quality measure.Set the projection subspace as the perpendicular direction to the selected hyperplane.

Notice that we do not have to search for the optimal sep-arating hyperplane,but only need to choose the best one from a candidate sets.We list the details of the proposed method in Algorithm1.

Algorithm1Semi-supervised Splitting

Input:Labeled training data X l that fall into R and the cor-responding labels Y l.

Input:Unlabeled training data X u that fall into R. Output:The parameters of the chosen hyperplane W R and θR.

1:Randomly generate the set of parametersΩfor candi-date hyperplanes.

2:Search for a hyperplane with parameters W0andθ0 that maximize the quality measure considering only la-beled data.

3:Set the iteration number t=0.

4:repeat

5:Set t=t+1.

6:Project all the samples onto the subspace that is per-pendicular to the separating hyperplane:z=W t?x. 7:for each labeled samples x i∈X l do

8:Use the given label as posterior distribution p(k∣z i)=[y i=k].

9:end for

10:for each unlabeled samples x i∈X u do

11:Calculate p(k∣z i)with the kernel-based density es-timation.

12:end for

13:Measure the split quality of each hyperplane.

14:Choose the hyperplane parameters W t andθt that maximize the quality measure.

15:until the chosen separating hyperplane is stable or the algorithm reaches enough iterations.

16:Set W R=W t andθR=θt.

17:Return W R andθR.

5.Random Forest Construction Based on

Semi-supervised Splitting

In the RF construction stage,an individual training set for each tree is generated from the original training set us-ing bootstrap aggregation.The samples which are not cho-sen for training are called Out-Of-Bag(OOB)samples of the tree and can be used for calculating the Out-Of-Bag-Error(OOBE),which is an unbiased estimation of the gen-eralization error.It has been shown that the OOBE is an unbiased estimation of the generalization error[3].Leistner et al.[14]?rst proposed the‘airbag’algorithm to use the OOBE to measure the performance of RF to detect whether the variation to the original algorithm would harm the sys-tem rather than assist.They decided whether to discard or retain a new forest by comparing its OOBE with the previ-ous forest.A shortcoming of the‘airbag’approach is that it regards the RF as a whole,so some trees with high perfor-mance may be dragged down by bad trees.

To overcome the limitation of‘airbag’algorithm,we propose to independently compare the OOBE of single de-cision trees in the supervised and semi-supervised models. In each model,we use a set of labeled bootstrap data to con-struct the supervised decision tree and use the same set of la-beled data and a set of unlabeled bootstrap data to construct the semi-supervised tree.The tree with smaller OOBE will be retained.The overall learning and control procedures of the proposed algorithm are shown in Algorithm2. Algorithm2Semi-supervised Splitting

Input:Labeled training data X l and the corresponding la-bels Y l.

Input:Unlabeled training data X u.

Input:The size of the forest N.

Output:The learned RF F.

1:Initialize an empty forest F.

2:for the i t?decision tree in F do

3:Generate a new labeled set X i l and a new unlabeled set X i u using the bootstrap aggregation.

4:Train the tree with only labeled samples:t i l= trainT ree(X i l).

5:Compute the OOBE:e i l=oobe(t i l,X l?X i l).

6:Train the tree with both labeled and unlabeled sam-ples:t i u=semiT ree(X i l,X i u).

7:Compute the OOBE:e i u=oobe(t i u,X l?X i l). 8:if e i l>e i u then

9:F=F∪t i u.

10:else

11:F=F∪t i l.

12:end if

13:end for

14:Return F.

6.Experiment and Analysis

We compared the proposed semi-supervised splitting with different splitting criteria on typical machine learning tasks.We show that by introducing abundant unlabeled da-ta,obvious accuracy improvement can be achieved.We also applied the proposed semi-supervised splitting RF for ob-ject categorization and image segmentation.Our method achieves state-of-the-art categorization performance on the Caltech-101dataset[15]and segmentation performance on the MSRC dataset[8].

6.1.Data Classi?cation

To quantitatively evaluate the improvement over the tra-ditional splitting criteria,we test our method on the Satim-age and Pendigits datasets.The Satimage dataset has 4435training samples and 2000testing samples while the Pendigits dataset has 7494training samples and 3498test-ing samples.We implement Breiman’s Random Forests [4]with the four different splitting criteria,i.e.,information gain,normalized information gain,Gini index and Bayesian classi?cation error.The traditional versions of these criteria are used as baselines.The proposed semi-supervised split-ting was applied to the four criteria.For each dataset,we randomly chose a part of the training data as the labeled da-ta and left the remainder as unlabeled data.The ratios of the chosen data range from 0.01to 0.1.The weights of the labeled samples were set at 1.0while the weights of the un-labeled samples were set through cross-validation.We built the RF with 100trees and 10hyperplanes were random-ly generated as candidates in the internal node of RF.To avoid computation cost in testing,only two attributes were considered in each hyperplane,and the RF was constructed without pruning.The training-testing procedures were re-peated 10times.We show the comparison results in Figure 3and it is clear that substantial performance improvement is achieved as a result of using our method.

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Semiísupervised normalized information gain Semiísupervised Gini index

Semiísupervised Bayesian classification error (b)

Figure 3.The classi?cation accuracy of Random Forests with tra-ditional splitting criteria (dashed lines)and the proposed semi-supervised splitting (solid lines).The results on the Satimage dataset and the Pendigits dataset are shown in (a)and (b)respec-tively.

We also compared the proposed semi-supervised split-ting RF with the state-of-the-art semi-supervised and super-vised classi?ers:RF [4],TSVM [13],SVM [6]and DAS-RF [14].The comparisons are shown in Figure 4.It can

be seen that the proposed method performs the best among them.

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Figure 4.The classi?cation accuracy of the proposed method with state-of-the-art classi?ers.The results on the Satimage dataset and the Pendigits dataset are shown in (a)and (b)respectively.

6.2.Object Categorization

We used the Caltech-101dataset for the object cate-gorization experiment.The Caltech-101dataset is com-posed of 101categories.Each class contains 31to 800im-ages;most of the images are medium resolution,i.e .about 300×300pixels.Following the common experiment setup,we randomly chose 15images per category as the labeled training data and 15images as the unlabeled training data.The remaining images were used for testing.We followed the ScSPM approach [23]in which each image was repre-sented as a 21504-dimension feature vector.PCA was then applied to the resulting vector.Maintaining 99.5%ener-gy,we obtained a 4000-dimensional feature vector for each image.We constructed a RF with 100trees and used 100candidate hyperplanes in the internal node.The weights of the unlabeled data were set at 0.5while the weights of the labeled data were set at 1.0.Normalized information gain is used as the splitting criterion to construct the RF.The proposed method and the traditional RF achieved 69.3%and 64.4%categorization accuracy respectively while the accuracy of SVM was 65.8%.Using the same features,our method brought an improvement of 4.9percent over the tra-ditional RF and 3.5percent over the SVM,by introducing abundant unlabeled data.

6.3.Image Segmentation

We applied the proposed semi-supervised splitting RF for the task of image segmentation in the 9-class MSRC

dataset.The9classes are:building,grass,tree,cow,sky, aeroplane,face,car and bicycle.We randomly chose150 images as the labeled training data and150images as the unlabeled training data from a total of480images,leaving the remainder as the testing data.We used the SLIC[1]to over-segment each image into200superpixels.The label of a superpixel was assigned as the majority of the pixel-level ground truth.We used the9-dimensional color momen-t,48-bins histogram of RGB,9-bins histogram of gradient [25]and59-bins histogram of LBP as the features.We con-structed a RF with100trees,each having a maximum depth of15.During splitting,10hyperplanes were randomly gen-erated as candidates and the hyperplane that maximized the information gain was chosen.We set the weight of a la-beled superpixel at1.0and set the weight of an unlabeled superpixel at0.3through cross-validation.Similar to most popular segmentation approaches,we applied a CRF stage after having learnt the class posteriors of the superpixels. The class posteriors from RF were used as unary potentials and pairwise potentials were de?ned over adjacent super-pixels.The parameters of the CRF model were trained us-ing the same150labeled training data used to construct the RF.When testing,the alpha-expansion graph-cut algorithm was used to infer the CRF model.We show the segmenta-tion results of the proposed semi-supervised splitting RF in Figure5.The original images and the ground truths are also shown.The segmentation accuracy of the proposed method and traditional RF is87.5%and83.6%respectively.As can be seen,our method is more accurate than the alternative method.

7.Conclusion and Future Work

In this paper,we introduced a semi-supervised splitting method that uses abundant unlabeled data to guide the node splitting of random forests.We derived a nonparametric al-gorithm to estimate the categorical distributions of the inter-nal nodes such that an accurate quality measure of splitting can be obtained.Our method can be combined with many popular splitting criteria,and the experimental results show that it brings obvious performance improvements to all of them.

In the future,we would like to investigate the problem of constructing RFs without labeled data.A uni?ed splitting framework that can handle both labeled and unlabeled data would be the extension. ACKNOWLEDGEMENTS

This work was supported in part by the Nation-al Natural Science Foundation of China(61170142), by the National Key Technology R&D Program under Grant(2011BAG05B04),by the Program of Internation-al S&T Cooperation(2013DFG12841),and by the Fun-damental Research Funds for the Central Universities (2013FZA5012).M.Song is the corresponding author. Appendix A

We consider four usual choices:information gain,nor-malized information gain,Gini index and Bayesian classi?-cation error.

Information Gain:

The information gain is de?ned as the subtraction of en-tropies before and after splitting

△H(R,W,θ)=H(R)?

∣R l∣

∣R∣

H(R l)?

∣R r∣

∣R∣

H(R r),(18)

where R is an internal node,R l and R r are its left and right child respectively.H(R)is the Shannon entropy of R

H(R)=?

K

k=1

p k log p k,(19)

where p k is the probability distribution of the k t?classes in R.

Normalized Information Gain:

The normalized entropy gain is de?ned as the quotient of the information gain and a normalized factor

△N(R,W,θ)=

△H(R,W,θ)

?

(

∣R l∣

∣R∣

log∣R l∣∣R∣+∣R r∣∣R∣log∣R r∣∣R∣

).(20)

Gini Index:

The Gini index is a measure of the impurity of a node and its de?nition is

G(R)=

K

k=1

p k(1?p k).(21)

When used for qualifying splitting,the following func-tion should be maximized

△G(R,W,θ)=G(R)?

∣R l∣

∣R∣

G(R l)?

∣R r∣

∣R∣

G(R r).(22)

Bayesian Classi?cation Error:

The Bayesian classi?cation error of a node is de?ned as

C(R)=1.0?argmax

k∈1???K

p k.(23)

When used for qualifying splitting,the following func-tion needs to be maximized

△C(R,W,θ)=C(R)?

∣R l∣

∣R∣

C(R l)?

∣R r∣

∣R∣

C(R r).(24)

From the above discussion,it is noticeable that all the four criteria heavily depends on the estimation of the prob-ability distribution p k.

Figure5.The original images,ground truths,the proposed segmentation results and traditional RF-based segmentation results on the MSRF dataset are shown from the top to the bottom row.The black area in the ground truth images is not labeled.

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混凝土规范相关问题专家答疑整理版

混凝土规范相关问题专家答疑整理版 问题①:混凝土规范4.2.3条:横向钢筋用作受剪、受扭、受冲切承载力计算时,其数值大于360N/mm2时应取360N/mm2。 可是我在看朱总2013年真题解析时,箍筋HRB500,可是计算时取fyv=435N/mm2 是我规范理解错了吗?往曾工指点迷津 采纳回答:(曾晓峰) 1.[规范]就是[规定],如其它无更专业的柤关规定与之矛盾,和规范自身无明显错误外,都应以它(规范)为准. 2.例:[假如]混凝土规范4.2.3条:"横向钢筋用作受剪、受扭、受冲切承载力计算时,其数值大于[300N/mm2时应取300N/mm2。]在规范未修改时,谁能说它错?且都应无条件执行,且该条还是[强条]. ②结构的层间刚度比,抗剪承载力需要满足抗震规范的不?(未解决,等待高手在线解答) ③抗规5.2.5条文说明: “扭转效应明显与否一般可由考虑耦联的振型分解反应谱法分析结果判断,例如前三个振型中,二个水平方向的振型参与系数为同一个量级,即存在明显的扭转效应。” 这句话应该怎样理解呢?(未解决,等待高手在线解答) ④高规“大底盘多塔结构”的疑问 2.1.15 未通过结构缝分开的裙房上部具有两个或者两个以上塔楼的结构。

条文说明:一般情况下,在地下室连为整体的多塔楼结构可不作为本规程10.6节规定的复杂结构。 10.6.3.4 大底盘多塔结构,可按本规程第5.1.14条规定的整体和分塔计算模型分别验算整体结构和各塔楼结构扭转为主的第一周期与平动为主的第一周期的比值,并应符合本规程第3.4.5条的有关要求。 请问 1:对于仅三面有土的地下室,上部两个塔楼用结构缝分开,地下室不分缝,这个是复杂结构,需要满足10.6.3.4对吧? 2:对于1的多塔楼结构需要整体建模和分塔建模分别计算,周期比和位移比都满足要求,那其他参数呢?是哪个模型满足还是都满足?配筋取包络吗? 3:对于1的的多塔结构,整体建模就是把整个模型按实际建吧,那分塔建模呢?是指分几个模型建还是在整体模型中定义多塔? 采纳解答:(曾晓峰) 1.对于仅三面有土的地下室,上部两个塔楼用结构缝分开,地下室不分缝,应属于大底盘多塔结构。因当地震水平力从无土的对方作用而来时,在该方向等于无地下室,该地下室相当于通常情况下的裙房。 2. A.多塔楼结构需要整体建模和分塔建模分别计算和相关要求可参《全国民用建筑措施》(混凝土) 12.5.3较为详细。

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新用户名及密码输入)——病毒库更新——确定。 2防护状态如果是黄色的,就关闭更新。 设置——(切换至高级——是——)显示所有高级设置——系统更新——点重大更新——选择无更新——确定——点防护状态——此处——运行系统更新——请通过Windows u pdate服务器获得补丁更新——关闭下载的网页。 或点防护状态——此处——安装——立即安装-等待安装完成关闭、变绿了吧。

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三、关闭系统恢复功能 这是一个非常重要的步骤,最近很多反馈表明系统还原会影响到SSD或者TRIM的正常操作,进而影响SSD的读写能力。 ▲关闭系统还原 1、右键单击我的电脑选择属性 2、选择系统保护 3、设置项 4、关闭系统还原 四、核实TRIM是否开启 确认TRIM是否开启一直是一个比较困难的事情,因为我们必须非常密切的监控电脑的状态才能判断,而往往磁盘TRIM都是瞬间的事,所以很难察觉。我们可以通过如下方式来看TRIM是否开启成功:

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问题5:2009定额装饰工程中的B.F.8楼地面、墙面块料面层铲除是否包含块料面层水泥砂浆基层打底(找平)的拆除? 答:2009定额装饰装修工程B.F.8拆除、铲除中的楼地面、墙面块料面层铲除未包含其水泥砂浆找平层、打底层的拆除。水泥砂浆的拆除套用2009定额装饰装修工程B.F.8拆除、铲除中水泥砂浆铲除相应定额项目。

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C:\ProgramFiles\Rising\RFW\RegGuide.exe C:\ProgramFiles\Rising\RFW\rfwProxy.exe 3.上面的步骤设置完成后,点击“确定”即可。 二、将浏览器的选项,不要选择或者是打上叉尝试。 1.您可以双击打开ESETNOD32-点击F5进入高级设置-点击HTTP-点击WEB浏览器,查看里面是否有浏览器的选项,请您不要选择或者是打上叉尝试。

2.您可以双击打开ESETNOD32-点击F5进入高级设置-点击HTTP-点击WEB浏览器-点击主动模式-查看里面是否有浏览器的选项,请您不要选择尝试。 三、如果以上两种方法还是无法解决,在“开始”-“程序”-“附件”,打开“命令提示符”程序。然后输入netshwinsockreset回车后会重置Winsock协议,按照提示重启电脑后即可解决问题。

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