11institutetext: Orange Innovation, Lannion, France

# Constructing Variables Using Classifiers as an Aid to Regression: An
Empirical Assessment

Colin Troisemaine  Vincent Lemaire

###### Abstract

This paper proposes a method for the automatic creation of variables (in the
case of regression) that complement the information contained in the initial
input vector. The method works as a pre-processing step in which the
continuous values of the variable to be regressed are discretized into a set
of intervals which are then used to define value thresholds. Then classifiers
are trained to predict whether the value to be regressed is less than or equal
to each of these thresholds. The different outputs of the classifiers are then
concatenated in the form of an additional vector of variables that enriches
the initial vector of the regression problem. The implemented system can thus
be considered as a generic pre-processing tool. We tested the proposed
enrichment method with 5 types of regressors and evaluated it in 33 regression
datasets. Our experimental results confirm the interest of the approach.

##  1 Introduction

Learning techniques can be divided into two main categories according to their
main purpose: those used to describe data (descriptive methods) and those used
to predict a (more or less) observable phenomenon (predictive methods).
Predictive methods use a set of labeled data to predict and explain one or
more (more or less) observable phenomena.

In the case of regression, this involves predicting the value of a numerical
variable (noted $y$), for example the amount of an invoice, using a set of
explanatory variables (a vector noted $X$). In the case of machine learning,
we seek to learn a function $f$ such that $y=f(X)$ using a machine learning
algorithm and a training set, a set of $N$ input-output pairs
$(X_{i},y_{i}),i={1,...,N}$. During this modeling phase, it is often necessary
to create new variables that better describe the problem and enable the model
to achieve better performance. This is what we call the "engineering process
of creating new explanatory variables" [19]. The idea here is here that the
new variables (a vector, in this case $X^{\prime}$) will provide additional
information. By automating the generation of these "new variables", more
useful and meaningful information can be extracted from the data, in a
framework that can be applied to any problem. This allows the machine learning
engineer to devote more time on more useful tasks.

The aim of the article presented here is to propose a method for automatically
creating variables (in the case of regression) that complement the information
contained in the vector $X$ in order to predict the values of the dependent
variable $y$. The proposed method first transforms the regression problem into
several classification sub-problems, then integrates the results in the form
of additive variables ($X^{\prime}$). The ’augmented’ vector,
$X^{\prime\prime}=X\cup X^{\prime}$, is then fed into standard regressors to
measure the contribution of the created variables. The advantages of this
approach are presented in the form of a detailed experimental study.

##  2 Proposal

The proposed method relies on transforming the regression problem into a
classification problem to construct the additional variables. This idea has
been proposed before, but our approach differs from the state of the art, as
explained below in section 2.1. Having presented this difference, section 2.2
describes the proposed method in detail and section 2.3 describes the choices
made for its implementation.

###  2.1 Related work

Solving a regression problem using classification models is an approach that
has already been explored. This process has been described in numerous papers
[2, 3, 14, 15, 17, 20] and generally consists of two main steps: (i)
discretization (also called partitioning or binning) of the target variable to
enable the use of classifiers on the dataset; (ii) prediction of the
regression is then usually performed by calculating the mean ([2, 3, 14, 17])
or median ([3, 15, 20]) of the instances within the fragment of discretized
output predicted by the classifier.

The method we propose and present below differs from these works in that the
classifiers used by the method are solely aimed at adding complementary
explanatory variables to the initial explanatory variables (native variables).
The augmented vector is then positioned at the input of a standard regressor.
This regressor directly predicts the target variable without any further
transformation or estimation operations. As the next section will show, the
proposed method is linked to a conditional estimate of the density function of
$y$. It would then be interesting to test other, less computationally-
intensive methods of variable creation in the same framework [18, 13, 21, 16]
in future work. But the general principle remains the same.

###  2.2 General principle

The idea is to take opportunistic advantage of the progress made in recent
years by classifiers in the literature. The principle of the first step of the
proposed method is to transform the regression problem into one (or more)
classification problem(s) by discretizing the space of variation of the
variable to be regressed. This step consists in defining thresholds ($S$) on
the space of the variable to be regressed (see figure 1).

Figure 1: Example of thresholds

These thresholds will be defined using the values of the variable to be
regressed in the training set. They will be used to define membership classes
($C=\\{C_{1},...,C_{i},...,C_{S}\\})$. Examples include classes defined on the
basis of value inferiority thresholds ($C_{i}:=\mathbb{1}_{y\leq y_{i}}$) or
classes defined on the basis of membership of value intervals
($C_{i}:=\mathbb{1}_{y\in]y_{i},y_{i+1}]}$), etc.

Once the classes have been encoded and one (or more) classifiers have been
trained (using the training set), it is then possible to predict the
membership of individuals in the previously defined classes. The predictions
of the classifier(s) on each individual will then be used to create a new
"extended" data set, either for the training set or the test set.

The figure 2 illustrates the extension of the vector $X$ (with $d$ components)
to the vector $X^{\prime\prime}$ (with $d+S$ components). The section 2.3 will
describe in more detail what the vector
$X^{\prime}=\\{X^{\prime}_{1},...,X^{\prime}_{i},...,X^{\prime}_{S}\\}$ from
the prediction of the classifier(s) is composed of.

Figure 2: Data set extension.

Once the $X$ vector has been "extended", the regression model can be trained
and predicted on the new data. We can then compare the regressor’s performance
when trained on the original data set (using only $X$) with its performance
when using the extended set ($X^{\prime\prime}=X\cup X^{\prime}$). The
assumption being that the regression model will have a better intuition of the
general position of individuals in the space of the variable to be regressed
and will produce better results.

Figure 3 summarizes the process followed by the proposed method: (i) the first
step is to transform the regression problem into a classification problem. To
do this, the target variable $y$ is first discretized (see section 2.3), then
classes are defined using the thresholds thus defined. (ii) In a second step,
the classifiers are trained using the original descriptive variables $X$ and
the new classes derived from $y$. Then the prediction, of the classifiers
using the initial vector $X$, is used to extract new features, i.e.
$X^{\prime}$. (iii) Finally, the regression model can be trained using the
vector $X^{\prime\prime}=X\cup X^{\prime}$. The proposed method is thus based
on a discretization and a class encoding mechanism. These two processes can be
implemented in different ways. An example of implementation is given below in
the section 2.3.

Figure 3: Diagram of the general principle of the method.

###  2.3 Implementation

One of the most important aspects of the proposed method is the association of
classes with instances. These classes will be used to train the classifiers.
This process is based on two steps: the definition of thresholds and the
encoding of classes. For the definition of threshold placement, which amounts
to an unsupervised discretization of the variable to be regressed, there are
numerous possibilities in the literature, such as “EqualWidth”, “EqualFreq”,
etc. For class encoding, it is possible to formulate the problem either as an
$S$ class classification problem (where $S$ denotes the number of thresholds),
or as $S$ binary classification problems, or as a multi-label classification
problem.

The choice of discretization method and the number of thresholds are linked,
and this choice represents a trade-off between (i) classification efficiency
and (ii) information input for the initial regression problem. In the case of
a single classifier predicting the membership interval of individuals, it
seems obvious that the larger S is, the more precise the information provided
to the regressor will be, but the more difficult the problem will become to
learn. On the other hand, if we formulate the problem as $S$ binary
classification problems, each classifier will have a problem of the same
difficulty to solve, regardless of the value of $S$. Preliminary tests carried
out on the set of datasets described in Section 3.1 have confirmed this
behavior. The final choices for the various elements associated with the
method proposed in this article are described below.

For the discretization method \- The “Equal Frequency" method was chosen
because, unlike the “Equal Width” method, it does not run the risk of creating
intervals that contain no individuals. It also ensures that there are no
classification problems where the minority class (extreme threshold on the
left, see Figure 1) represents less than $\frac{1}{S+1}$ percent of the
individuals.

For the nature of the classification problem \- Preliminary tests, posing the
problem as $S$ binary classification problems gave better performance, both in
learning and in deployment. The results are also more robust (better
generalization), which is important if we want the added variables to benefit
the regressor. The classifiers used in the rest of this article are Scikit-
Learn’s Random Forests [8] with 100 trees (but any other powerful and robust
classifier could be used) with their default parameters.

For the number of thresholds \- The first intuition (confirmed by the
experiments) is that the greater the number of thresholds defined, the greater
the potential performance gain. Indeed, the more thresholds defined, the finer
the discretization and the closer the classifiers’ prediction will be to the
true regression value. In the experimental phase of this article, we will
illustrate the performance of our method as a function of the number of
thresholds defined. The number of thresholds defined on the space of the
variable to be regressed is therefore a hyper-parameter of our method.

For the definition of classes associated with thresholds \- We chose classes
associated with value inferiority thresholds ($C_{i}:=\mathbb{1}_{y\leq
y_{i}}$) (see Section 2.2 and Figure 1).

For the variables extracted from the classifiers - Each classifier will be
trained to predict whether the data provided is below or above the threshold
at which it is assigned a class. As the method is intended to be generic, it
was decided to extract the conditional probabilities predicted by each
classifier. Indeed, this information is available for the vast majority of
classifiers in the literature. Conditional probabilities111Note: as the
selected classifiers are random forests, the conditional probability
corresponds to the proportion of trees that voted for the class in question.
of class 1 (i.e. if $y\leq y_{i}$) predicted by each classifier will therefore
make up the vector
$X^{\prime}=\\{X^{\prime}_{1},....,X^{\prime}_{i},...,X^{\prime}_{S}\\}=\\{P(C_{1}=1|X),...,P(C_{i}=1|X),...,P(C_{S}=1|X)\\}$;
$X^{\prime}$ will then be added to the initial data vector ($X$).

##  3 Experimental protocol

### Code availability

The code for experiments is available at the following url:
https://github.com/ColinTr/ClassificationForRegression

###  3.1 Data sets and pre-processing

To carry out the analysis of the proposed method, we selected a large
collection of regression datasets from the UCI Repository [10] or from Kaggle
[1]. A total of 33 datasets were used, of which 23 consisted of more than
10,000 individuals. The remaining 10 datasets range from 1,030 to 9,568
individuals (see Table 1). This selection was largely influenced by the
article by M. Fernandez-Delgado et al. [11]. The selected datasets were,
according to the categorization of Fernandez-Delgado et al. "large and
difficult". Some of these datasets included several subsets of data with
different target variables to be regressed, and therefore different regression
problems. It was decided not to include multiple regression problems from the
same datasets in order to avoid giving more weight to certain datasets and
creating a bias in the comparison of methods that might favour one method over
another. The reader can refer to the "Dataset name" column of the table 1,
which identifies the regression problem used for each dataset.

Pre-processing: As in [11], two pre-processings were carried out before
entering the process described in Figure 3 (see Section 2.2): (i) recoding of
categorical variables using full disjunctive coding; (ii) deletion of date
variables, constant variables, individual identifiers, collinear variables and
other variables potentially to be ’regressed’. Three further pre-processings
were performed on each learning fold. These are, in order of completion: (i)
normalization of numerical variables (centering - reduction); (ii)
normalization of the variable to be regressed (centering - reduction) then
transformation using Box-Cox [4]; (iii) creation of thresholds for class
definition as described above. For each of these three pre-processings, the
associated statistics were calculated on the training set and then applied to
the training and test sets. In the following results presentation, the RMSE
results (see metric used below) will be given without performing the inverse
Box-Cox transformation. Finally, examples with missing values have been
removed from the initial dataset. Dataset statistics after pre-processing are
presented in Table 1.

Train-Test split and model optimization: Each dataset has been split into a
"10-fold cross validation", resulting in 10 training and test sets. For the
three models requiring parameter optimization (AR, FA and XGB see Section
3.3): 30% of the training set was reserved for optimizing the model parameters
in a "grid-search" process, often avoiding over-training. Then, provided with
the "right" training parameters, the model was trained using the remaining 70%
of the training set. Finally, once the model had been trained for this fold,
its performance was measured. This rigorous process was carried out for all
datasets, all folds and all $S$ values, resulting in the training of thousands
of models, but allowing a rigorous test of the proposed method.

Data set | #individuals | # variables | target | Data set | #individuals | # variables | target
---|---|---|---|---|---|---|---
3Droad | 434,874 | 3 | altitude | geo-lat | 1,059 | 116 | latitude
air-quality-CO | 1,230 | 8 | PT08.S1(CO) | greenhouse-net | 955,167 | 15 | synthetic var
airfoil | 1,503 | 5 | scaled sound | KEGG-reaction | 65,554 | 27 | edge count
appliances-energy | 19,735 | 26 | appliances | KEGG-relation | 54,413 | 22 | clustering coef
beijing-pm25 | 41,758 | 14 | PM2.5 | online-news | 39,644 | 59 | shares
temp-forecast-bias | 7,752 | 22 | Next_Tmax | video-transcode | 68,784 | 26 | utime
bike-hour | 17,379 | 17 | count | pm25-chengdu-us | 27,368 | 20 | PM_US Post
blog-feedback | 60,021 | 18 | target | park-total-UPDRS | 5,875 | 16 | total_UPDRS
buzz-twitter | 583,250 | 70 | discussions | physico-protein | 45,730 | 9 | RMSD
combined-cycle | 9,568 | 4 | PE | production-quality | 29,184 | 17 | quality
com-crime | 1,994 | 122 | Violent crimes | CT-slices | 53,500 | 384 | reference
com-crime-unnorm | 2,215 | 134 | Violent crimes | gpu-kernel-perf | 241,600 | 14 | Run1 (ms)
compress-stren | 1,030 | 8 | compressive strength | SML2010 | 4,137 | 26 | indoor temp
cond-turbine | 11,934 | 15 | gt turbine | seoul-bike-sharing | 8,760 | 9 | Rented Bike Count
cuff-less | 61,000 | 2 | APB | uber-location-price | 205,670 | 5 | fare amount
electrical-grid-stab | 10,000 | 12 | stab | year-prediction | 515,345 | 90 | Year
facebook-comment | 40,949 | 53 | target |  |  |  |
Table 1: Description of the 33 data sets. The first column indicates the name
of the dataset (or sub-dataset) from UCI or Kaggle. The second and third
columns give the initial number of individuals and (explanatory) variables in
the dataset (after pre-processing). Finally, the fourth column indicates the
name of the target variable to be regressed.

###  3.2 Metric used for results

When comparing regression models, the aim is to find a model without over- or
under-fitting that achieves a low generalization error, which characterizes
its predictive performance. Various metrics are proposed in the literature,
such as the root mean square error (RMSE), the mean absolute error (MAE), the
R-Square, etc… There seems to be no consensus on the "best" metric to use,
although some papers do offer comparisons [5]. In the rest of the results,
RMSE has been chosen and is defined as:
$RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}$, where $n$ is
the number of individuals, $y_{i}$ the desired value and $\hat{y}_{i}$ the
output of a regressor.

###  3.3 Tested Regressors

Five regressors were used in the experiments, using different frameworks. They
are briefly described below, although they are well known in the machine
learning community.

$\bullet$ Linear Regression (LR): Regression refers to the process of
estimating a continuous numerical variable using other variables that are
correlated with it. This means that regression models take the form of
$y=f(X)$ where $y$ can take on an infinite number of values. In this article,
we have chosen a multivariate linear regression whose model, $f$, is
determined by the method of least squares. We have used Scikit-Learn version
0.24.2. This model does not require parameter optimization.

$\bullet$ Regression Tree (DT): Decision trees are used to predict a real
quantity which, in the case of regression, is a numerical value. Algorithms
for building decision trees are usually constructed by dividing the tree from
the root to the leaves, choosing at each step an input variable that achieves
the best partitioning of the set of individuals. In the case of regression
trees, the aim is to maximize the inter-class variance (i.e. to have subsets
whose values of the target variable are as widely spread as possible). In this
article, we used Scikit-Learn version 0.24.2. This model requires parameter
optimization.

$\bullet$ Random Forest (RF): Decision-tree forests (or Random Forest (RF)
regressors) were first proposed by Ho in 1995 [12] and then extended by Leo
Breiman and Adele Cutler [8]. This algorithm combines the concepts of random
subspaces and bagging. The decision tree forest algorithm performs training on
multiple decision trees trained on slightly different subsets of data. In this
article, we have used Scikit-Learn version 0.24.2. This model requires
parameter optimization.

$\bullet$ XGBoost (XGB): XGBoost [9] is a boosting method. It sequentially
combines weak learners that would individually perform poorly to improve the
prediction of the full algorithm. A weak learner is a regressor with poor
regression performance. In this boosting algorithm, high weights will be
associated with weak learners having good accuracy, and conversely, lower
weights with weak learners having poor accuracy. In the training phase, high
weights are associated with data that has been poorly "learned", so that the
next weak learner in the sequence will focus more on that data. In this
article, we have used XGBoost version 1.4.1. This model requires parameter
optimization.

$\bullet$ Selective Naïve Bayes (SNB): In the context of regression, a naive
Bayes regressor (NB) whose variables are weighted by weights (NBP) can be
obtained in two steps. First, for each explanatory variable, a 2D grid is
created to estimate $P(X,y)$ (see for example [6]). Then, in a second step,
all variables are grouped together in a Forward Backward algorithm [7] to
estimate their informativeness in the context of a Naive Bayes regressor. At
the end of the second step, the final model (to be deployed) is a Naïve Bayes
(using the 2D discretization found in the first step) where the variables are
weighted (the weights are found in the second step). In this article we have
used the Weighted Naïve Bayes produced by Khiops library (www.khiops.org).
This model does not require parameter optimization.

##  4 Results

$\bullet$ Illustrative results -  Figure 4 illustrates the behavior of the
proposed method: on the left for linear regression and the "KEGG Metabolic
Reaction" dataset, in the middle and on the right for the "SML 2010" dataset,
respectively for the Selective Naïve Bayes regressor and the Random Forest.
These 3 figures are fairly representative of the results obtained, with a more
or less marked decrease in the RMSE versus the value of $S$ and the regressor
considered. This decrease is most pronounced at the beginning of the curve and
then levels off for a less pronounced increase beyond $S=16$. The following
section presents the results obtained for each data set.

Figure 4: $S$ (horizontal axis) versus RMSE (vertical axis). The dotted lines
represent the initial performance of the regressor in blue for the test set
and in orange for the training set. Solid lines represent the regressor’s
performance with the proposed method, using the same color code.

$\bullet$ Results tables on all datasets -  The results obtained are presented
in detail in Table 2 for a number of thresholds equal to 32 ($S=32$). For each
dataset, this table gives the test results (average results over the 10 test
folds (see Section 3.1)) for each of the five regressors for which the
addition of the vector $X^{\prime}$ was tested. In this table, a value in bold
indicates a significant difference between results with the $X$ vector
"Native" and the $X^{\prime}$ vector "Aug" (Increased) according to paired
Student’s t test (p-value at 5%). The last row of the table shows the number
of losses, ties and wins for "Aug" versus "Native". We can see that the
addition of the $X^{\prime}$ vector essentially benefits 3 of the regressors:
the linear regression, the regression tree and the naive Bayes regressor. For
the random forest and XGboost, the gains and/or losses are fairly limited.

| Linear Regr. | Decision Tree | Random Forest | XGBoost | SNB
---|---|---|---|---|---
Dataset name | Native | Aug | Native | Aug | Native | Aug | Native | Aug | Native | Aug
3Droad | 0,9867 | 0,0930 | 0,1213 | 0,0831 | 0,0908 | 0,0792 | 0,0935 | 0,0781 | 0,5580 | 0,0827
air-quality-CO | 0,3269 | 0,2672 | 0,3255 | 0,2727 | 0,2667 | 0,2642 | 0,2682 | 0,2633 | 0,3428 | 0,2768
airfoil | 0,7176 | 0,2351 | 0,4373 | 0,3194 | 0,2987 | 0,2865 | 0,2814 | 0,2752 | 0,7709 | 0,2963
appliances-energy | 0,8260 | 0,4865 | 0,7431 | 0,5267 | 0,5456 | 0,5164 | 0,5790 | 0,5203 | 0,8354 | 0,5206
beijing-pm25 | 0,7833 | 0,4180 | 0,6013 | 0,3948 | 0,4168 | 0,3874 | 0,4149 | 0,3885 | 0,8082 | 0,3931
temp-forecast-bias | 0,4749 | 0,2847 | 0,4668 | 0,2848 | 0,3237 | 0,2771 | 0,3085 | 0,2773 | 0,4782 | 0,2907
bike-hour | 0,7163 | 0,2547 | 0,3294 | 0,2564 | 0,2415 | 0,2499 | 0,2208 | 0,2500 | 0,4858 | 0,2534
blog-feedback | 0,8307 | 0,6434 | 0,6741 | 0,6694 | 0,6481 | 0,6618 | 0,6444 | 0,6652 | 0,7013 | 0,6601
buzz-twitter | 0,7248 | 0,2183 | 0,2299 | 0,2265 | 0,2170 | 0,2200 | 0,2162 | 0,2205 | 0,2238 | 0,2191
combined-cycle | 0,2702 | 0,1967 | 0,2647 | 0,2014 | 0,2079 | 0,1952 | 0,2041 | 0,1950 | 0,2570 | 0,1995
com-crime | 0,6312 | 0,5485 | 0,6418 | 0,5552 | 0,5525 | 0,5505 | 0,5665 | 0,5503 | 0,5612 | 0,5523
com-crime-unnorm | 0,6751 | 0,6031 | 0,7036 | 0,6251 | 0,6163 | 0,6157 | 0,6186 | 0,6124 | 0,6453 | 0,6340
compress-stren | 0,6331 | 0,2732 | 0,4519 | 0,3173 | 0,3137 | 0,2896 | 0,2790 | 0,2955 | 0,5962 | 0,2959
cond-turbine | 0,3002 | 0,1096 | 0,1764 | 0,1094 | 0,1117 | 0,1064 | 0,1062 | 0,1094 | 0,4481 | 0,1096
cuff-less | 0,7996 | 0,5791 | 0,5165 | 0,6255 | 0,5025 | 0,6105 | 0,5012 | 0,6225 | 0,5450 | 0,5922
electrical-grid-stab | 0,5961 | 0,3156 | 0,5545 | 0,3245 | 0,3352 | 0,3068 | 0,2634 | 0,3092 | 0,5951 | 0,3157
facebook-comment | 0,6828 | 0,4242 | 0,4655 | 0,4429 | 0,4162 | 0,4384 | 0,4095 | 0,4397 | 0,5014 | 0,4347
geo-lat | 0,8752 | 0,8066 | 0,9594 | 0,8538 | 0,8321 | 0,8287 | 0,8540 | 0,8353 | 0,8668 | 0,8408
greenhouse-net | 0,6492 | 0,5418 | 0,5397 | 0,5686 | 0,5158 | 0,5638 | 0,5137 | 0,5618 | 0,6016 | 0,5599
KEGG-reaction | 0,1803 | 0,0845 | 0,0919 | 0,0861 | 0,0837 | 0,0835 | 0,0835 | 0,0840 | 0,1172 | 0,0855
KEGG-relation | 0,6342 | 0,1750 | 0,2454 | 0,1748 | 0,1782 | 0,1700 | 0,1809 | 0,1715 | 0,4314 | 0,1738
online-news | 1,1822 | 1,0677 | 0,9516 | 0,9622 | 0,9177 | 0,9521 | 0,9124 | 0,9521 | 0,9228 | 0,9942
video-transcode | 0,3975 | 0,0561 | 0,0846 | 0,0652 | 0,0604 | 0,0597 | 0,0590 | 0,0568 | 0,3704 | 0,0627
pm25-chengdu-us-post | 0,8022 | 0,4283 | 0,5578 | 0,4164 | 0,4111 | 0,4084 | 0,4125 | 0,4079 | 0,7716 | 0,4131
park-total-UPDRS | 0,9472 | 0,7816 | 0,9416 | 0,8057 | 0,8020 | 0,7893 | 0,8196 | 0,7860 | 0,9428 | 0,8117
physico-protein | 0,8453 | 0,5179 | 0,7676 | 0,5338 | 0,5528 | 0,5218 | 0,5744 | 0,5249 | 0,8494 | 0,5330
production-quality | 0,4872 | 0,2807 | 0,3886 | 0,2830 | 0,2836 | 0,2779 | 0,2794 | 0,2781 | 0,3790 | 0,2832
CT-slices | 1,9533 | 0,0504 | 0,1332 | 0,0586 | 0,0544 | 0,0418 | 0,0693 | 0,0355 | 0,1104 | 0,0376
gpu-kernel-perf | 0,6347 | 0,0696 | 0,0300 | 0,0557 | 0,0246 | 0,0501 | 0,0228 | 0,0500 | 0,5927 | 0,0620
SML2010 | 0,2536 | 0,1157 | 0,2179 | 0,1036 | 0,1241 | 0,0857 | 0,1058 | 0,0920 | 0,3473 | 0,1221
seoul-bike-sharing | 0,6990 | 0,4839 | 0,5801 | 0,5217 | 0,5101 | 0,5149 | 0,5171 | 0,5139 | 0,5891 | 0,5169
uber-location-price | 0,9998 | 0,4589 | 0,6153 | 0,4789 | 0,4635 | 0,4699 | 0,4705 | 0,4688 | 0,8419 | 0,4879
year-prediction | 0,8569 | 0,8079 | 0,8783 | 0,8137 | 0,8059 | 0,8062 | 0,7929 | 0,8131 | 0,8822 | 0,8138
Moyenne | 0,7083 | 0,3842 | 0,4754 | 0,3945 | 0,3856 | 0,3842 | 0,3831 | 0,3850 | 0,5749 | 0,3917
Déf / Egal / Vict | 0 / 4 / 29 | 3 / 2 / 28 | 8 / 16 / 9 | 10 / 14 / 9 | 2 / 2 / 29
Table 2: Results for each dataset in Test (average results over the 10 test
folds (see Section 3.1)) for $S=32$.

The penultimate row of the table shows the average RMSE obtained on all the
data sets (purely indicative), confirming the interest of the proposed method
for three of the five regressors. This average should be treated with caution.
As each dataset is a problem of unique difficulty, the error scale differs
between each dataset. So, if the new RMSE mean in test is higher, this does
not necessarily mean that the regressors perform worse on average. For this
reason, in the next section we present another view of the results for further
analysis.

$\bullet$ Critical diagram -  We present in Figure 5 critical diagrams
comparing the results obtained by each regressor through the Nemenyi post-hoc
test, performed after a Friedman signed rank test of RMSE values, taking into
account all the data sets.

Figure 5: Critical diagram: on the left, the regressors without the proposed
method, in the center, the regressors with and without the proposed method,
and on the right, the regressors only with the proposed method (indicated by a
’+’).

From the graph on the left of Figure 5 it can be seen that the XGBoost
regressor performs best, achieving the highest mean RMSE rank, closely
followed by Random Forest (RF). These two regressors outperform the other
three regressors by a wide margin, with linear regression being the worst
performing model. It is closely followed by the naive Bayes regressor and the
regression tree.The figure 5 in the middle compares the performance of all the
regressors, with and without the proposed method: the first point we can
observe is that, statistically, all the regressors with $X^{\prime}$ have a
better average rank than their counterparts with only $X$. This is an
encouraging result: no regressor has been negatively affected by the proposed
method, since all regressors that have used this method are better ranked than
their basic versions. Again, there are different groups, and the reader can
compare the change in the ranking of different regressors by comparing the
figure on the left with the one in the middle. Finally, the graph on the right
of figure 5 shows that: when the proposed method is used, the regressors are
no longer differentiable according to the Nemenyi test (only one group
compared to the left figure), even though they have different mean ranks.

##  5 Conclusion

This paper has proposed a method for automatically generating variables (in
the case of regression) that complement the information contained in the
initial input vector, the explanatory variables. The method works as a pre-
processing step in which the continuous values of the variable to be regressed
are discretized into a set of intervals. These intervals are used to define
threshold values. Classifiers are then trained to predict whether the value to
be regressed is less than or equal to each of these thresholds. The different
outputs of the classifiers are then concatenated into an additional vector of
variables that enriches the initial vector of explanatory variables native to
the regression problem. The results are encouraging, although they mainly
benefit three of the five regressors for which the method has been tested. A
first improvement would be to extract a more informative vector from the
classifiers, such as tree leaf identifiers. A second improvement could be to
design a neural architecture that combines all the steps of the proposed
method.

The proposed method also opens up certain perspectives. The attentive reader
will have noticed that the vector $X^{\prime}$, as defined in the proposed
implementation, is in fact an estimate of the conditional cumulative density
of $y$, knowing $X$. It would therefore be possible to dispense with
regressors altogether, via an expectation calculation, in the case where $S$
is sufficiently high to have a fairly good estimate of this cumulative
density. This last point is likely to be the subject of future work.

## References

  * [1] Kaggle. https://www.kaggle.com
  * [2] Ahmad, A., Halawani, S.M., Albidewi, I.: Novel ensemble methods for regression via classification problems. Expert Syst. Appl. 39, 6396–6401 (2012)
  * [3] Ahmad, A., Khan, S.S., Kumar, A.: Learning regression problems by using classifiers. Journal of Intelligent & Fuzzy Systems 35(1), 945–955 (2018)
  * [4] Atkinson, A.: The box-cox transformation: review and extensions. LSE Research Online Documents on Economics (2020)
  * [5] Botchkarev, A.: A new typology design of performance metrics to measure errors in machine learning regression algorithms. Interdisciplinary Journal of Information, Knowledge, and Management 14, 45–79 (01 2019). https://doi.org/10.28945/4184
  * [6] Boullé, M.: MODL: a Bayes optimal discretization method for continuous attributes. Machine Learning 65(1), 131–165 (2006)
  * [7] Boullé, M.: Compression-based averaging of selective naive Bayes classifiers. Journal of Machine Learning Research 8, 1659–1685 (2007)
  * [8] Breiman, L.: Random forests. Machine Learning 45, 5–32 (10 2001). https://doi.org/10.1023/A:1010950718922
  * [9] Chen, T., Guestrin, C.: Xgboost: A scalable tree boosting system. In: the 22nd ACM SIGKDD International Conference. pp. 785–794 (08 2016). https://doi.org/10.1145/2939672.2939785
  * [10] Dua, D., Graff, C.: UCI machine learning repository (2017), http://archive.ics.uci.edu/ml
  * [11] Fernández-Delgado, M., Sirsat, M., Cernadas, E., Alawadi, S., Barro, S., Febrero-Bande, M.: An extensive experimental survey of regression methods. Neural Networks 111 (2019)
  * [12] Ho, T.K.: Random decision forests. In: Proceedings of 3rd International Conference on Document Analysis and Recognition. vol. 1, pp. 278–282 vol.1 (1995). https://doi.org/10.1109/ICDAR.1995.598994
  * [13] Holmes, M.P., Gray, A.G., Isbell, C.L.: Fast nonparametric conditional density estimation. In: Uncertainty in Artificial Intelligence (2007)
  * [14] Janssen, F., Fürnkranz, J.: Separate-and-conquer regression. Tech. Rep. TUD-KE-2010-01, TU Darmstadt, Knowledge Engineering Group (2010), http://www.ke.tu-darmstadt.de/publications/reports/tud-ke-2010-01.pdf
  * [15] Janssen, F., Fürnkranz, J.: Heuristic rule-based regression via dynamic reduction to classification. In: Walsh, T. (ed.) Proceedings of IJCAI. pp. 1330–1335 (2011), http://ijcai.org/papers11/Papers/IJCAI11-225.pdf
  * [16] Langford, J., Oliveira, R., Zadrozny, B.: Predicting conditional quantiles via reduction to classification. arXiv preprint arXiv:1206.6860 (2012)
  * [17] Memon, S.A., Zhao, W., Raj, B., Singh, R.: Neural regression trees. In: 2019 International Joint Conference on Neural Networks (IJCNN). pp. 1–8 (2019). https://doi.org/10.1109/IJCNN.2019.8852133
  * [18] Rothfuss, J., Ferreira, F., Walther, S., Ulrich, M.: Conditional density estimation with neural networks: Best practices and benchmarks (2019)
  * [19] Sondhi, P.: Feature construction methods: a survey. sifaka. cs. uiuc. edu 69, 70–71 (2009)
  * [20] Torgo, L., Gama, J.: Regression using classification algorithms. Intelligent Data Analysis 1(1), 275–292 (1997)
  * [21] Tutz, G.: Ordinal trees and random forests: Score-free recursive partitioning and improved ensembles. arXiv:2102.00415 [stat.ME] (2021)


# The stellar metallicity distribution in intermediate latitude fields with
BATC and SDSS data

Xiyan Peng1, Cuihua Du1 , Zhenyu Wu2
1College of Physical Sciences, Graduate university of the Chinese Academy of
Sciences, Beijing 100049, P. R. China
2National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, P. R. China<EMAIL_ADDRESS>
(Received)

###### Abstract

Based on the Beijing-Arizona-Taiwan-Connecticut (BATC) and Sloan Digital Sky
Survey (SDSS) photometric data, we adopt SEDs fitting method to evaluate the
metallicity distribution for $\sim$40, 000 main-sequence stars in the Galaxy.
According to the derived photometric metallicities of these sample stars, we
find that the metallicity distribution shift from metal-rich to metal-poor
with the increase of distance from the Galactic center. The mean metallicity
is about of $-1.5\pm 0.2$ dex in the outer halo and $-1.3\pm 0.1$ dex in the
inner halo. The mean metallicity smoothly decreases from $-0.4$ to $-0.8$ in
interval $0<r\leqslant 5$ kpc. The fluctuation in the mean metallicity with
Galactic longitude can be found in interval $4<r\leqslant 8$ kpc. There is a
vertical abundance gradients d[Fe/H]/dz $\sim-0.21\pm 0.05$ dex kpc-1 for the
thin disk ($z\leqslant 2$ kpc). At distance $2<z\leqslant 5$ kpc, where the
thick disk stars are dominated, the gradients are about of $-0.16\pm 0.06$ dex
kpc-1, it can be interpreted as a mixture of stellar population with different
mean metallicities at all $z$ levels. The vertical metallicity gradient is
$-0.05\pm 0.04$ dex kpc-1 for the halo ($z$ $>$ 5 kpc). So there is little or
no metallicity gradient in the halo.

###### keywords:

Galaxy: structure-Galaxy: metallicity-Galaxy: formation.

††pagerange: The stellar metallicity distribution in intermediate latitude
fields with BATC and SDSS data–References††pubyear: 2002

##  1 Introduction

The structure, formation and evolution of the Galaxy are very important issues
in contemporary astrophysics. The basic components of Galaxy are the thin
disk, the thick disk, the halo and central bulge, albeit that the inter-
relationships and distinction among different components remain subject to
some debate (Gilmore 1983; 1984; Lemon et al. 2004). Recent studies, based on
accurate large-area surveys, have revealed that the Galaxy is marked by
numerous irregular substructure such as the Sagittarius dwarf tidal stream in
the halo (Ivezic et al. 2000; Yanny et al. 2000; Vivas et al. 2001; Majewski
et al. 2003) and the Monoceros stream closer to the Galactic plane (Newberg et
al. 2002; Rocha-Pinto et al. 2003). Carollo et al. (2007) shown that the the
halo is clearly divisible into two broadly overlapping structural components -
an inner and outer halo from a local kinematic analysis. It is now apparent
that our Galaxy is much more complex system than we thought before. The
formation of galaxies was long thought to be a steady process resulting in a
smooth distribution of stars (Bahcall & Soneira 1981; Gilmore et al. 1989;
Majewski 1993). But the view of the formation of the Galaxy has changed
dramatically since the discoveries of complex substructures (Newberg et al.
2002; Belokurov et al. 2007). The presence of these lumpy and complex
substructure are in qualitatively agreement with models for the formation of
the stellar halo through the accretion and merging of nearby dwarf galaxies.
Numerical simulations also suggest that this merger process plays a crucial
role in setting the structure and motions of stars within galaxies (Bullock &
Johnston 2005).

The abundance distribution is particular importance to understanding the
formation and chemical evolution of the Galaxy (Freeman & Bland-Hawthorn
2002). Researchers have long sought to constrain models for the Galactic
formation and evolution on the basis of observation of the stellar and
clusters populations that it contains. Specific models of galaxy formation
make specific predictions about the stellar abundance distribution. For
example, stars on more radial orbits are more metal-poor than stars on planar
orbits. This may indicate that the Milky Way formation began with a relatively
rapid collapse of the initial proto-galactic cloud, which means the halo stars
formed during the initial collapse, the disk stars formed after the gas had
settled into the galactic plane. But the global collapse theory was unable to
account for the lack of an abundance gradient in the Galactic halo (Searle &
Zinn 1978). The current view is that the Galactic halo formed at least partly
through the accretion of small satellite galaxies or merger of larger systems
(Freeman et al., 2002), which is well-supported by studies of stellar
kinematics and spatial distribution (Yannny et al. 2003; Juric et al. 2008).
For the thick disk, it may be one of the most significant components for
studying signatures of galaxy formation because it presents a snap frozen
relic of the state of the early disk (Freeman 2002). An intrinsic abundance
gradient in the thick disk would favor a scenarios which the thick disk was
formed either in the slow late stages of the early Galactic collapse or the
gradual kinematical diffusion of disk stars. On the contrary, a irregular
metallicity distribution or absence of gradient would favor the thick disk
having formed via the kinematical heating of thin disk or from merger debris
(Siegel et al. 2009).

The metallicity distribution of the Galaxy is best probed directly through
spectroscopic surveys (Yoss et al. 1987; Allende-Prieto et al. 2006). However,
it has the advantage of using the photometric metallicity of many more stars
out to limiting magnitude of photometric survey. Accurate determination of the
properties of the Galactic components requires surveys with sufficient sky
coverage to assess the overall geometry, sufficient depth for mapping stars to
larger distance and sufficient information to obtain reasonable distance
estimates for these stars (De Jong et al. 2010). Over the past few years,
numerous surveys have been used to investigate the existence and size of the
Galactic abundance gradient in the disk and halo. The existence of a radial
gradient in the Galaxy is now well established. An average gradient of about
$-0.06$ dex kpc-1 is observed in the Galactic disk for most of the elements
(Chen et al, 2003). De Jong et al. (2010) provided evidence for a radical
metallicity gradients in the Galactic stellar halo. However, there is
considerable disagreement about whether there is a vertical metallicity
gradient among field and/or open cluster stars of the Galaxy.

The BATC multicolor photometric survey accumulated a large data base which is
very useful for studying the Galactic structure and formation. Du et al.
(2003) provided some information on the density distribution of the main
components of the Galaxy, which can present constraints on the parameters of
models of the Galactic structure. Later, they use F and G dwarfs from the BATC
data to study the metal-abundance information (Du et al., 2004). With the new
improved observation and improved knowledge regarding galaxy formation, it
becomes possible to further discuss the metallicity gradient from different
observation direction of the Galaxy.

In this paper, we attempt to study the metallicity gradient of the Milky Way
galaxy using the 21 BATC photometric survey fields combined with the SDSS
photometric data. The outlines of this paper is as follows: The BATC
photometric system and data reduction are introduced briefly in Section 2. In
Sect. 3 we describe the theoretical model atmospheric spectra and synthetic
photometry. The metallicity distribution is discussed in Sect. 4. Finally in
Sect. 5 we summarize our main conclusions in this study.

##  2 Observations and data

###  2.1 BATC photometric system and SDSS photometric system

The BATC survey performs photometric observations with a large field multi-
colour system. There are 15 intermediate-band filters in the BATC filter
system, which covers an optical wavelength range from 3000 to 10000 Å (Fan et
al. 1996; Zhou et al. 2001). The 60/90 cm f/3 Schmidt Telescope of National
Astronomical Observatories (NAOC) was used in the BATC program, with a Ford
Aerospace 2048 $\times$ 2048 CCD camera at its main focus. The field of view
of the CCD is $58^{\prime}$ $\times$ $58^{\prime}$ with a pixel scale of
$1\arcsec{\mbox{}.}7$. The BATC magnitudes adopt the monochromatic AB
magnitudes as defined by Oke & Gunn (1983). The PIPELINE II reduction
procedure was performed on each single CCD frame to get the point spread
function (PSF) magnitude of each point source (Zhou et al. 2003). The detailed
description of the BATC photometric system and flux calibration of the
standard stars can be found in Fan et al. (1996) and Zhou et al. (2001, 2003).
In order to apply more color information to accurately estimate the
photometric stellar metallicity, we combine the BATC colors with the SDSS
colors message for the sample stars.

The SDSS used a dedicated 2.5 m telescope which has an imaging camera and a
pair of spectrographs. The imaging camera (Gunn et al. 1998) contained 30 2048
$\times$ 2048 CCDs in the focal plane of the telescope. The flux densities of
observed objects were measured almost simultaneously in five broad bands [
$u$, $g$, $r$, $i$, $z$] (Fukugita et al. 1996; Gunn et al. 1998; Hongg et al.
2001). For distinguishing explicitly between BATC and SDSS filter names, we
refer to the SDSS filters and magnitudes as $u^{\prime}$, $g^{\prime}$,
$r^{\prime}$, $i^{\prime}$, $z^{\prime}$. The photometric pipeline (Luption et
al. 2001) detected the objects, matched the data from the five filters and
measured instrumental fluxes, positions and shape parameters. The shape
parameters allowed the classification of objects as point source or extended.
The magnitudes derived from fitting a PSF are currently accurate to about 2
percents in $g$, $r$ and $i$, and 3-5 percents in $u$ and $z$ for bright
($\leq$ 20 mag) point source. In Table 1, we list the parameters of the BATC
and SDSS filters. Col. (1) and Col. (2) represent the ID of the BATC and SDSS
filters, Col. (3) and Col. (4) the central wavelengths and FWHM of the 20
filters, respectively. The reddening extinction for each star are determined
from the SDSS catalog.

Table 1: Parameters of the BATC and SDSS filters  No. | Filter | Wavelength | FWHM
---|---|---|---
|  | (Å) | (Å)
1 | $a$ | 3371.5 | 359
2 | $b$ | 3906.9 | 291
3 | $c$ | 4193.5 | 309
4 | $d$ | 4540.0 | 332
5 | $e$ | 4925.0 | 374
6 | $f$ | 5266.8 | 344
7 | $g$ | 5789.9 | 289
8 | $h$ | 6073.9 | 308
9 | $i$ | 6655.9 | 491
10 | $j$ | 7057.4 | 238
11 | $k$ | 7546.3 | 192
12 | $m$ | 8023.2 | 255
13 | $n$ | 8484.3 | 167
14 | $o$ | 9182.2 | 247
15 | $p$ | 9738.5 | 275
1 | $u^{\prime}$ | 3543 | 569
2 | $g^{\prime}$ | 4770 | 1387
3 | $r^{\prime}$ | 6231 | 1373
4 | $i^{\prime}$ | 7625 | 1526
5 | $z^{\prime}$ | 9134 | 9500

###  2.2 Direction and data reduction

Accurate determination of the properties of components of the Milky Way
requires surveys with sufficient sky coverage to assess the overall geometry
(De Jong et al. 2010). Most previous investigations about the Galactic
metallicity distribution use only one or a few selected lines-of-sight
directions (Du et al. 2004; Siegel et al. 2009; Karatas et al. 2009). In this
paper, the BATC photometry survey presented 21 intermediate-latitude fields in
the multiple directions. These fields used in this paper are towards the
Galactic center, the anticentre, the antirotation direction at median and high
latitudes, $|b|>35^{\circ}$. Since metallicity distribution at high Galactic
latitudes are not strongly related to the radial distribution, they are well
suited to study the vertical distribution of the Galaxy. Table 2 lists the
locations of the observed fields and their general characteristics. In Table
2, column 1 represents the BATC field name, columns 2 and 3 represent the
Galactic longitude and latitude, columns 4 and 5 represent the limit magnitude
and the number of sample star used in this study. As shown in the Table 2, the
most photometric depth of our data is 21.0 mag in $i$ band. In total, there
are about 40, 000 sample stars in our study.

Table 2: Relative information for the BATC observation fields Observed field |  $l$ (deg) |  $b$ (deg) |  $i$ (Comp) | star number
---|---|---|---|---
T485 | 175.7 | 37.8 | 21.0 | 1550
T518 | 238.9 | 39.8 | 19.5 | 1584
T288 | 189.0 | 37.5 | 20.0 | 2115
T477 | 175.7 | 39.2 | 20.0 | 2001
T328 | 160.3 | 41.9 | 19.5 | 1666
T349 | 224.1 | 35.3 | 20.5 | 2436
TA26 | 191.1 | 44.4 | 20.0 | 1285
T291 | 167.8 | 46.4 | 20.0 | 1670
T362 | 245.7 | 53.4 | 20.0 | 1237
T330 | 147.2 | 68.3 | 20.5 | 1020
U085 | 121.6 | 60.2 | 21.0 | 1679
T521 | 56.1 | -36.8 | 20.5 | 4430
T491 | 62.9 | -44.0 | 20.0 | 2824
T359 | 79.7 | -37.8 | 20.5 | 2932
T350 | 251.3 | 67.3 | 19.5 | 1353
T534 | 91.6 | 51.1 | 21.0 | 2044
T193 | 59.8 | -39.7 | 20.0 | 2830
T516 | 125.0 | -62.0 | 20.0 | 1113
T329 | 169.9 | 50.4 | 21.0 | 1704
TA01 | 135.7 | -62.1 | 20.5 | 1342
T517 | 188.6 | -38.2 | 20.0 | 1296
Figure 1: Two-color diagrams for the BATC T329 field stars. The panel (a) is
the distribution of sample stars in the ($j-m$)/($g-j$) two-colour diagram,
and panel (b) is the ($j-m$)/($g-j$) two-colour diagram after removing those
objects that lay significantly off the stellar locus. The line denotes the
stellar locus from equation (1).

Here, because our fields in this work have also been observed by the Sloan
Digital Space Survey (SDSS-DR7) and each object type (stars-galaxies-QSO) has
been given. Thus, we can obtain a relative reliable star catalogue. In these
star sample, it is complete to 20.5 mag with an error of less than 0.1 mag in
the BATC $i$ band. Owing to the BATC observing strategy, data of stars
brighter than $m$ = 14 mag, are saturated, and star counts are not completed
for visual magnitudes fainter than $m$ = 21.0 mag. So our work is restricted
to the magnitudes range 15 $\leqslant$ $i$ $<$ 21\. From two-color diagrams
for all objects in any field as an example, we can find that most stars are
plotted on a diagonal band. In Fig. 1 show the BATC ($g-j$) versus ($j-m$)
two-color diagram for T329 field stars. The line which can be described by
equation (1) donates the stellar locus of the sample in the T329 field .

$\displaystyle(j-m)=-0.10245+0.81172(g-j)$ (1)

The sample shows a main-sequence (MS) stellar locus but has significant
contamination from giants, manifested in the broader distribution overlying
the narrow stellar locus. In order to determine the MS star sample, we use
multi-color selection criteria outlined in Karaali et al. (2003) and Juric et
al. (2008), which remove objects based on their location relative to the
dominant stellar locus. For example, Juric et al. (2008) applied an extra
procedure which consists of rejecting objects at distance larger than 0.3 mag
from the stellar locus in order to remove hot dwarfs, low-red shift quasars,
and white /red dwarf unresolved binaries from their sample. The procedure does
well for high latitude field data from SDSS (Karaali et al. 2003; Yaz et al.
2010). Fig. 1b gives the cleaning sample stars after rejecting those objects
which is lay significantly off the stellar locus.

##  3 THEORETICAL MODEL AND CALIBRATION FOR METALLICITY

###  3.1  theoretical stellar library and synthetic photometric

A homogeneous and complete stellar library can match any ambitious goals
imposed on a standard library. Lejeune et al. (1997) presented a hybrid
library of synthetic stellar spectra. The library covers a wide range of
stellar parameters $T_{\rm eff}$= 50, 000 K to 2, 000 K in intervals of 250 K,
log $g$= $-1.02$ to 5.50 in main increments of 0.5, and [M/H]= $-5.0$ to +1.0.
For each model in the library, a flux spectrum is given for the same set of
1221 wavelength points covering the range 9.1 to 160, 000 nm, with a mean
resolution of 20Å in the visible. The spectra are thus in a format which has
proved to be adequate for synthetic photometry of wide- and intermediate-band
systems (Du et al. 2004).

On the basis of the theoretical library, we calculate synthetic colors of the
BATC and SDSS photometric system. Here, we synthesize colors for simulated
stellar spectra with $T_{\rm eff}$ and log $g$ characteristic of main
sequences (log $g$ = 3.5, 4.0, 4.5 for dwarfs) and 19 values of metallicity
([M/H]= $-5.0,-4.5,-4.0,-3.5$,
$-3.0,-2.5,-2.0,-1.5,-1.0,-0.5$,$-0.3,-0.2,-0.1$, 0.0, +0.1, +0.2, +0.3, +0.5
and +1.0), where [M/H] denotes metallicity relative to hydrogen. The synthetic
$i$th filter magnitude can be calculated with equation (2).

$m=-2.5~{}{\rm log}\frac{\int{F_{\lambda}\phi_{i}({\lambda}){\rm
d}\lambda}}{\int{\phi_{i}({\lambda}){\rm d}\lambda}}-48.60,$ (2)

Where ${F_{\lambda}}$ is the flux per unit wavelength, $\phi_{i}$ is the
transmission curve of the $i$th filter of the BATC or SDSS filter system (Du
et al. 2004) .

The bluer colors are sensitive to metallicity down to the lowest observed
metallicities because most of the line-blanketing from heavy elements occurs
in the shorter wavelength regions. In contrast, the redder colors are
primarily sensitive to temperature index. The BATC $a$, $b$ bands contain the
Balmier jump, a stellar spectral feature which is sensitive to surface
gravity. Since our sample includes only main sequences, it conveys little
gravity information. It should be mentioned that, although the metallicity or
temperature derived from synthetic photometry is not very accurate for a
single star, perhaps which can be distorted by a poor point, it is meaningful
for the statistic analysis of sample stars.

###  3.2  METALLICITY AND PHOTOMETRIC PARALLAX

The most accurate measurements of stellar metallicity are based on
spectroscopic observation. Despite the recent progress in the availability of
stellar spectra (e.g. SDSS-III and RAVE), the stellar number detected in
photometric surveys is much more than spectroscopic observation. So
photometric methods have also often been used to give the stellar metallicity.
For example, Sandage (1969) detailed a technique using UBV photometry indices
to measure approximate abundance. Karaali (2003) evaluated the metal abundance
by ultraviolet-excess photometric parameter using CCD UBVI data. Karaali
(2005) extended this method to the SDSS photometry. Ivezic et al. (2008)
obtained the mean metallicity of stars as a function of $u-g$ and $g-r$ colors
of SDSS data. For the BATC multicolor photometric system, there are 15
intermediate-band filters covering an optical wavelength range from 3000 to
10000 Å. There are 5 filters for the SDSS photometric system. So the SEDs of
20 filters for every object are equivalent to a low resolution spectrum.

The sample SEDs simulation with template SEDs can be used to derive the
parameter of sample stars (Du et al. 2004). The standard minimization,
computing and minimizing the deviations between the photometric SEDs of the
star and the templates SEDs obtained with the same photometric system, is used
in the fitting process. The minimum indicated the best fit to the observed SED
by the set of spectra (Du et al. 2004).

${\chi^{2}=\sum\limits_{l=1}^{N_{filt}=20}\left[\frac{m_{obs,l}-m_{temp,l}}{\sigma_{l}}\right]^{2}},$
(3)

where ${m_{obs,l}}$, $m_{temp,l}$ and $\sigma_{l}$ are the observed magnitude,
template magnitude and their uncertainty in filter $l$, respectively, and
$N_{filt}$ is the total number of filters in the photometry.

According to the results of SEDs fitting, the metallicity and temperature of
about 40, 000 sample stars are obtained in 21 fields. In addition, we extract
the spectroscopic metallicities from sdss DR7 database for our studied 21
fields, and there are about 870 stars for which they also have photometric
metallicities from our method. Using these stars, we present a calibration of
our SED fitting method. After applying calibration, it is reliable for the
derived photometric metallicities from SEDs fitting method. In Figure 2., we
present the difference of photometric and spectroscopic metallicity as a
function of $(g-r)$ color.  The uncertainties of metallicity obtained from
comparing SEDs between photometry and theoretical models are due to the
observational error and the finite grid of the models. For the metal-poor
stars ([Fe/H]$<-1.0$), the metallicity uncertainty is about 0.5 dex, and 0.2
dex for the stars [Fe/H]$>-0.5$ (Du et al. 2004). The metallicity distribution
diagram for all sample stars was given in Fig. 3. One local maximum appear at
[Fe/H] from -0.5 to 0 dex, and a tail down to -3.0 dex.

Figure 2: The difference of photometric and spectroscopic metallicity for 870
stars as a function of $(g-r)$ color is shown. Figure 3: The metallicity
distribution for all the sample stars selected in our study is shown.

The stellar type can be derived from effective temperature of dwarfs, then the
stellar distance relative to the sun can be obtained by equation (4).

$\displaystyle m_{v}-M_{v}=5lgr-5+A_{v}$ (4)

where $m_{v}$ is the visual magnitude, absolute magnitude $M_{v}$ can be
obtained according to the stellar type. The reddening extinction $A_{v}$ is
small for most fields. We adopted the absolute magnitude versus stellar type
relation for main-sequence stars from Lang (1992). $r$ is the stellar
distances. The vertical distance of the star to the galactic plane can be
evaluated by equation (5):

$\displaystyle z=rsinb$ (5)

A variety of errors affect the determination of stellar distances. The first
source of errors is from photometric uncertainty less than 0.1 mag in the BATC
$i$ band; the second from the misclassification, which should be small due to
the multicolor photometry. For luminosity class V, types F/G, the absolute
magnitude uncertainty is about 0.3 mag. In addition, there may exist an error
from the contamination of binary stars in our sample. We neglect the effect of
binary contamination on distance derivation due to the unknown but small
influence from mass dsitribution in binary components (Kroupa et al. 1993;
Ojha et al. 1996).

Figure 4:  The metallicity distribution for the T291 field in different
distance range is shown. In the short distance, r $<$ 1kpc, most stars are in
the range [0, $-0.5$], in the larger distance, 15 $<$ r $<$ 20 kpc, most stars
are poorer than $-1$.

##  4 METALLICITY DISTRIBUTION

It is well known that the chemical abundance of stellar population contains
much information about the population,s early evolution. The stellar
metallicity distribution in the Galaxy has been the subject of photometric and
spectroscopic surveys (Gilmore et al. 1985; Ratnatunga et al. 1989; Friel
1988). In this study, we want to explore possible stellar metallicity
distribution variation with the observation direction. A method of SED
combination for the SDSS and BATC photometries has been adopted to give the
steller metallicity distribution. At first, the metallicity for the sample
stars can be derived by comparing SEDs between photometry and the theoretical
models. The SED fitting method is described in Section 3\. 2. The mean
metallicity distribution for each field is determined in the following
distance intervals (in kpc): $15<r\leqslant 20$, $10<r\leqslant 15$,
$8<r\leqslant 10$, $6<r\leqslant 8$, $5<r\leqslant 6$, $4<r\leqslant 5$,
$3<r\leqslant 4$, $2.5<r\leqslant 3$, $2<r\leqslant 2.5$, $1.5<r\leqslant 2$,
$1<r\leqslant 1.5$, $0.25<r\leqslant 1$. As an example the metallicity
distributions as a function of vertical distance for the field T291 is
presented in Fig. 4. From the figures (Fig. 4), it is clear that there is a
number-shift from metal-rich stars to metal-poor ones with the increasing of
distance. In star counts the younger metal rich stars are confined to regions
close to the Galactic mid-plane, while the older, metal-poorer stars with a
larger scale height dominated at larger vertical distances from the Galactic
plane.

###  4.1 Metallicity variation with Galactic longitude

Mean metallicity distribution as a function of Galactic longitude for
different distance intervals are presented in Fig. 5. The mean metallicity
shift from metal-rich to metal-poor with the increase of distance from the
Galactic center can be found in Fig. 5. The solid points represent the south
galactic latitude fields, The open square points represent the north galactic
latitude fields. As shown in Fig. 5, the mean metallicity in interval
$10<r\leqslant 20$ kpc is around of $-1.5$ dex. In intervals $8<r\leqslant 10$
kpc our result indicates that the mean metallicity is $\sim$ $-1.3$. The mean
metallicity in intervals $5<r\leqslant 8$ kpc is about $-1.0$ and the mean
metallicity smoothly decreases from $-0.6$ to $-0.8$ in intervals
$2.5<r\leqslant 5$ kpc. Our results are consistent with the results of Siegel
et al. (2009) and Karatas et al. (2009). Siegel et al. find a monometallic
thick disk and halo with abundances of [Fe/H] = $-0.8$ and $-1.4$
respectively. Karatas et al. derive mean abundance values of [Fe/H]= $-0.77\pm
0.36$ dex for the thick disk, and [Fe/H] = $-1.42\pm 0.98$. The mean
metallicity decreases from $-0.4$ to $-0.6$ in intervals $0<r\leqslant 2.5$
kpc. The mean metallicity in interval $0.25<r\leqslant 1$ kpc is [Fe/H] $\sim$
$-0.3$, which is consistent with the result of Yaz et al. (2010).

As shown in Fig. 5, at larger distance, r $>$ 10 kpc, compared to the typical
error bars, the mean metallicity distributions variation with Galactic
longitude is almost flat. For $4<r\leqslant 8$ kpc, there is a fluctuation in
the mean metallicity with Galactic longitude. The overall distribution of mean
metallicity has a maxminum at $l$ $\sim$ 200∘. For $2<r\leqslant 8$ kpc, the
T517 filed (Galactic coordinates : $l=188.6^{\circ},b=-38.2^{\circ}$;
Equatorial coordinates : $\alpha=58.59^{\circ}$, $\delta=-0.35^{\circ}$) and
the TA26 field (Galactic coordinates : $l=191.1^{\circ},b=44.4^{\circ}$;
Equatorial coordinates : $\alpha=139.956^{\circ}$, $\delta=33.745^{\circ}$)
show metal rich character related to other fields. This feature may reflect a
fluctuation from streams (such as Monocers stream) which are accreted from
nearby galaxies. Juric et al.(2008) detect two overdensities in the thick disk
region. Klement et al.(2009) also find individual stream from the SSPP in the
direction with central coordinates (Equatorial coordinates): $\alpha$ =
$58.58^{\circ}$ and $\delta$ = $-4.99^{\circ}$. Maybe the deviant behaviors of
the two fields result from systematical error in the observation. In the work
of AK et al. (2007), they find that the metallicity distributions for both
(relatively) short and large vertical distances show systematic fluctuations.
The scaleheight of thick disk varies with the observed direction were found in
the works of Du et al. (2006) and Bilir et al. (2008).

Figure 5: The mean metallicity distribution as a function of galactic
longitude in different distance intervals are shown.

###  4.2 THE VERTICAL METALLICITY GRADIENT

Detailed information about the vertical metallicity gradient can provide
important clue about the formation scenario of stellar population. Here, we
used the mean metallicity to described the metallicity distribution function.
As an example, The distribution trend of mean metallicity [Fe/H] with height
above the galactic plane [$z$] for the T291 field is shown in Fig. 6. The
metallicity gradients for all the fields in different $z$ intervals $z<2$ kpc,
$2<z\leqslant 5$ kpc and $5<z\leqslant 15$ kpc are given in Fig. 7 and
detailed in Table 3. In Table 3, Column 1 represents the BATC field name,
Columns 2 - 7 represent gradient and error of gradient in different $z$
distance: $z\leqslant 2$ kpc, $2<z\leqslant 5$ kpc and $5<z\leqslant 15$ kpc,
respectively.

From the Fig. 7 we can find that the variation of the gradient for the halo
with galactic longitude is flat and the mean gradient of halo is about
$-0.05\pm 0.04$ dex kpc-1 ($5<z\leqslant 15$ kpc), which is essentially in
agreement with the conclusion of Yaz et al. (2010) and Du et al. (2004). Du et
al.(2004) find the small or zero gradient d[Fe/H]/dz = $-0.06\pm 0.09$ in the
halo. Yaz et al. (2010) find $d[M/H]/dz=-0.01$ dex kpc-1 for the inner
spheroid. The result of Karaali et al. (2003) is slightly steeper than the
value of our result. Karaali et al. (2003) find that there is a metallicity
gradient d[Fe/H]/dz $\sim-0.1$ dex kpc-1 in the inner halo ($5<z\leqslant 8$
kpc ) and zero in the outer part ($8<z\leqslant 10$ kpc). From Fig. 6 we can
find that the incompleteness of the star sample causes significant statistical
uncertainties at large distance. Probably, there is little or no metallicity
gradient in the halo. It is consistent with the merger or accretion origin of
the outer halo.

As shown in Fig. 7, at distance $0<z<2$ kpc, the mean vertical abundance
gradient is about d[Fe/H]/dz $\sim-0.21\pm 0.05$ dex kpc-1. The value for the
vertical metallicity at distance $0<z<2$ kpc is in agreement with the
canonical metallicity gradients with the same $z$ distances. For example, Yaz
et al. (2009) find the metallicity gradient is d[Fe/H]/dz $\sim-0.3$ dex kpc-1
for short distance. The metallicity gradient is found to be d[Fe/H]/dz
$\sim-0.37$ dex kpc-1 for $z$ $<$ 4 kpc in the work of Du et al. (2004). The
result of Karaali et al. (2003) can be described as d[Fe/H]/dz $\sim-0.2$ dex
kpc-1 for the thin and thick disk.

At distance $2<z\leqslant 5$ kpc, where the thick disk stars dominated, the
gradient is about $-0.16\pm 0.06$ dex kpc-1 in our work which is consistent
with the work of karaali et al. (2003) and less than the value of Du et al.
(2004). Du et al. (2004) point out that the metallicity gradient is d[Fe/H]/dz
$\sim$ $-0.37$ dex kpc-1. In our study, the thick disk gradient is interpreted
as different contribution from three components of the Galaxy at different $z$
distance. The existence of a clear vertical metallicity of the thick disk
would be an important clue about the origin of the thick disk. However, it is
an open question for the formation of the thick disk component. A number of
models have been put forward since the confirmation if its existence. Chen et
al. (2001) support that the thick disk formed through the heating of a
preexisting thin disk, with the heating mechanism being the merging of a
satellite galaxy. Here, we also favor the thick disk having formed via the
kinematical heating of thin disk and from merger debris. Thus, there is a
irregular metallicity distribution or absence of intrinsic gradient.

Figure 6: The mean metallicity as a function of vertical distance $z$ for the T291 field. The metallicity gradients of the thin disk, thick disk and halo are $-0.23\pm 0.03$, $-0.18\pm 0.07$, $-0.05\pm 0.01$ dex kpc-1, respectively.  Figure 7: The metallicity gradients distribution for all fields in this study are shown for the intervals $z$ $<$ 2 kpc , 2 kpc $<$ $z$ $<$ 5kpc and 5 kpc $<$ $z$ $<$ 15kpc. Table 3:  The gradient distribution in different distance interval for the selected fields | 0-2 (kpc) |  | 2-5 (kpc) |  | 5-15 (kpc) |
---|---|---|---|---|---|---
Observed field | gradient | error | gradient | error | gradient | error
T193 | -0.136 | 0.017 | -0.080 | 0.020 | -0.020 | 0.022
T288 | -0.258 | 0.027 | -0.267 | 0.032 | -0.007 | 0.037
T291 | -0.228 | 0.028 | -0.185 | 0.071 | -0.045 | 0.014
T328 | -0.110 | 0.027 | -0.126 | 0.032 | -0.137 | 0.043
T329 | -0.170 | 0.028 | -0.100 | 0.042 | -0.037 | 0.015
T330 | -0.171 | 0.042 | -0.217 | 0.055 | -0.080 | 0.010
T349 | -0.180 | 0.017 | -0.225 | 0.050 | -0.029 | 0.031
T350 | -0.267 | 0.033 | -0.178 | 0.074 | -0.069 | 0.035
T359 | -0.203 | 0.017 | -0.126 | 0.023 | -0.054 | 0.023
T362 | -0.237 | 0.037 | -0.126 | 0.060 | -0.037 | 0.022
T477 | -0.181 | 0.025 | -0.275 | 0.030 | -0.053 | 0.022
T485 | -0.247 | 0.033 | -0.119 | 0.044 | 0.028 | 0.040
T491 | -0.194 | 0.017 | -0.179 | 0.020 | -0.070 | 0.027
T516 | -0.291 | 0.039 | -0.204 | 0.061 | -0.005 | 0.026
T517 | 0.054 | 0.041 | -0.173 | 0.059 | -0.101 | 0.067
T518 | -0.163 | 0.035 | -0.026 | 0.045 | -0.051 | 0.052
T521 | -0.197 | 0.012 | -0.175 | 0.013 | -0.045 | 0.011
T534 | -0.254 | 0.028 | -0.110 | 0.035 | -0.035 | 0.012
TA01 | -0.199 | 0.037 | -0.192 | 0.046 | -0.047 | 0.015
TA26 | 0.307 | 0.030 | -0.187 | 0.038 | -0.085 | 0.047
U085 | -0.146 | 0.024 | -0.148 | 0.037 | -0.035 | 0.013

##  5 CONCLUSIONS AND SUMMARY

In this work, based on the BATC and SDSS photometric data, we evaluated the
stellar metallicity distribution for 40, 000 main-sequence stars in the Galaxy
by adopting SEDs fitting method. These selected fields are towards the
Galactic center, the anticentre, the antirotation direction at median and high
latitudes . The metallicity distribution could be obtained up to distances
$r=20$ kpc, which covers the thin disk, thick disk and halo. We determined the
mean stellar metallicity as a function of vertical distance in different
direction. It can be clearly seen that the metallicity distribution shift from
metal-rich to metal-poor with the increase of distance from the Galactic
center. The mean metallicity is about $-1.5\pm 0.2$ dex in intervals
$10<r\leqslant 20$ kpc and $-1.3\pm 0.1$ dex in interval $8<r\leqslant 10$
kpc. The mean metallicity smoothly decreases from $-0.6$ to $-0.8$ in interval
$2.5<r\leqslant 5$ kpc, while the mean metallicity decreases from $-0.4$ to
$-0.6$ in interval $0<r\leqslant 2.5$ kpc. In addition, a fluctuation in the
mean metallicity with Galactic longitude can be found and the overall
distribution has a maximum at about $l$ $\sim$ 200∘ in interval $4<r\leqslant
8$ kpc. Maybe this feature can be related with the substructure or streams
(such as Monoceros stream) which are accreted from nearby galaxies. At the
same time, we find the vertical abundance gradients for the thin disk ($0<z<2$
kpc) is d[Fe/H]/dz $\sim-0.21\pm 0.05$ dex kpc-1, and a vertical gradient
$-0.16\pm 0.06$ dex kpc-1 at distances $2<z\leqslant 5$ kpc where the thick
disk stars are dominated. Here, we consider the thick disk gradient may be the
result from the different contributions from three components of the Galaxy at
different $z$ distance. The vertical gradient d[Fe/H]/dz $\sim-0.05\pm 0.04$
dex kpc-1 is found in distance $5<z\leqslant 15$ kpc. So, there is little or
no gradient in the halo. These results are in agreement with the values in the
literature (Yoss et al. 1987; Trefzger et al. 1995; Karaali et al. 2003; AK et
al. 2007; Yaz et al. 2010). It is possible that additional observational
investigations (some projects aimed at spectroscopic sky surveys such as
SEGUE, LAMOST, GAIA) will give more evidence for the metallicity gradient of
the Galaxy and therefore provide a powerful clue to the disk and halo
formation.

## Acknowledgments

We especially thank the anonymous referee for numerous helpful comments and
suggestions which have significantly improved this manuscript. This work was
supported by the joint fund of Astronomy of the National Nature Science
Foundation of China and the Chinese Academy of Science, under Grants 10778720
and 10803007. This work was also supported by the GUCAS president fund.

## References

  * Ak et al. (2007) AK S., Bilir S., Karaali S., Buser R., 2007, AN, 328, 169
  * Ak et al. (2007) AK S., Bilir S., Karaali S., Buser R., Cabrera-Lavers A., 2007, NewA, 12, 605
  * Allende et al. (2006) Allende Prieto C., Beers T. C., Wilhelm R., Newberg H. J., Rockosi C. M., Yanny B., Lee Y. S., 2006, ApJ, 636, 804
  * Bahcall et al. (1981) Bahcall J. N., Soneira R. M., 1981, ApJS, 47, 357
  * Belokurov et al. (2003) Belokurov V. A., Evans N. W. 2003, MNRAS, 341, 569
  * Bilir et al. (2008) Bilir S., Cabrera - Lavers A., Karaali S., Ak S., Yaz E., López-Corredoira M., 2008, PASP, 25, 69
  * Bullock et al. (2005) Bullock J. S., Johnston K. V., 2005, ApJ, 635, 931
  * Du et al. (2003) C.-H. Du, et al. 2003, A&A, 407, 541
  * Du et al. (2004) C.-H. Du, et al. 2004, AJ, 128, 2265
  * Du et al. (2006) C.-H. Du, et al. 2006, MNRAS, 372, 1304
  * Carollo et al. (2007) Carollo D., et al. 2007, Nature., 450, 1020
  * Chen et al. (2001) Chen B. et al., 2001, ApJ, 553, 184
  * Chen et al. (2003) Chen L., Hou J. L., Wang J. J., 2003, AJ, 125, 1397
  * De et al. (2010) De Jong Jelte T. A.; Yanny B., Rix Hans-Walter, Dolphin A. E., Martin N. F., Beers T. C., 2010, ApJ, 714, 663
  * Fan et al. (1996) Fan X. H. et al., 1996, AJ, 112, 628
  * Freeman et al. (2002) Freeman K., Bland-Hawthorn J., 2002, ARA&A, 40, 487
  * Friel (1988) Friel E. D., 1988, AJ, 95, 1727
  * Fukugita et al. (1996) Fukugita M., Ichikawa T., Gunn J. E., Doi M., Shimasaku K., Schneider D. P., 1996, AJ, 111, 1748
  * Gilmore et al. (1963) Gilmore G., Reid N., 1983, MNRAS, 202, 1025
  * Gilmore (1984) Gilmore G., 1984, MNRAS, 207, 223
  * Gilmore G & Wyse (1985) Gilmore G., Wyse R. F. G., 1985, AJ, 90, 2015
  * Gilmore et al. (1989) Gilmore G., Wyse R. F. G., Kuijken K., 1989, ARA&A, 27, 555
  * Gunn et al. (1998) Gunn J. E. et al., 1998, AJ, 116, 3040
  * Hogg et al. (2001) Hogg D. W., Finkbeiner D. P., Schlegel D. J., Gunn J. E., 2001, AJ, 122, 2129
  * Ivezic Z. et al. (2000) Ivezic Z. et al., 2000, AJ, 120, 963
  * Ivezic Z. et al. (2008) Ivezic Z. et al., 2008, ApJ, 684, 287
  * Juric et al. (2008) Juric M. et al., 2008, ApJ, 673, 864
  * Karaali et al. (2003) Karaali S., AK S. G., Bilir S., Karatas Y., Gilmore G., 2003, MNRAS, 343, 1013
  * Karaali et al. (2003) Karaali S., Bilir S., Karatas Y., Ak S. G., 2003, PASA, 20, 165
  * Karaali et al. (2005) Karaali S., Bilir S., Tuncel S., 2005, PASA, 22, 24
  * Karatas et al. (2009) Karatas Y., Kilic M., Günes O., Limboz F., 2009, PASP, 26, 1
  * Klement et al. (2009) Klement R. et al., 2009, ApJ, 698, 865
  * Kroupa et al. (1993) Kroupa P., Tout C. A., Gilmore G., 1993, MNRAS, 262, 545
  * Lang et al. (1992) Lang K. R., 1992, Astrophysical Data I, Planets and Stars. Spinger-Verlag, Berlin
  * Belokurov et al. (2003) Belokurov V. A., Evans N. W. 2003, MNRAS, 341, 569
  * Lang et al. (1992) Lang K. R., 1992, Astrophysical Data I, Planets and Stars. Spinger-Verlag, Berlin
  * Lejeune et al. (1997) Lejeune T., Cuisinier F., Buser R., 1997, A&AS, 125, 229
  * Lemon et al. (2004) Lemon D. J., Wyse Rosemary F. G., Liske J., Driver S. P., Horne K., 2004, MNRAS, 347, 1043
  * López -Corredoira et al. (2002) López -Corredoira M., Cabrera - Lavers A., Garzón F., Hammersley P. L., 2002, A&A, 394, 883
  * Lupton et al. (2001) Lupton R. H., Gunn J. E., Ivezic Z., Knapp G. R., Kent S., 2001, ASPC, 238, 269
  * Majewski et al. (2003) Majewski S. R., Skrutskie M. F., Weinberg M. D., Ostheimer J. C., 2003, ApJ, 599, 1082
  * Majewski et al. (1993) Majewski S. R., 1993, ARA&A, 31, 575
  * Newberg et al. (2002) Newberg H. J. et al., 2002, ApJ, 569, 245
  * Ojha et al. (1996) Ojha D. K., Bienayme O., Robin A. C., Creze M., Mohan V., 1996, A&A, 311, 456
  * Oke et al. (1983) Oke J. B., Gunn J. E., 1983, ApJ, 266, 713
  * Ratnatunga et al. (1989) Ratnatunga K. U., Freeman K. C., 1989, ApJ, 339, 126
  * Rocha-Pintoet al. (2003) Rocha-Pinto H. J., Majewski S. R., Skrutskie M. F., Crane J. D., 2003, ApJ, 594, 115
  * Sandage et al. (1969) Sandage A., 1969, ApJ, 158, 1115
  * Siegel et al. (2009) Siegel M. H., Karatas Y., Reid I. N., Yüksel R., I. Neill, 2009, MNRAS, 395, 1569
  * Stetson (1987) Stetson P. B., 1987, PASP, 99, 191
  * Searle et al. (1978) Searle L., Zinn R., 1978, ApJ, 225, 357
  * Trefzger et al. (1995) Trefzger C. F., Pel J. W., Gabi S., 1995, A&A, 304, 381
  * Vivas et al. (2001) Vivas A. K. et al., 2001, ApJ, 554, 33
  * Xia et al. (2002) Xia L. F. et al., 2002, PASP, 114, 1349
  * Yanny et al. (2000) Yanny B. et al., 2000, ApJ, 540, 825
  * Yoss et al. (1987) Yoss K. M., Neese C. L., Hartkopf W. I., 1987, AJ, 94, 1600
  * Yaz et al. (2010) Yaz E., Karaali S., 2010, NewA, 15, 234
  * York et al. (2000) York D. et al., 2000, AJ, 120, 1579
  * Zhou et al. (2003) Zhou X. et al., 2003, A&A, 397, 361