In the network technology era, the collected data are growing more and more complex, and become larger than before. In this article, we focus on estimates of the linear regression parameters for symbolic interval data. We propose two approaches to estimate regression parameters for symbolic interval data under two different data models and compare our proposed approaches with the existing methods via simulations. Finally, we analyze two real datasets with the proposed methods for illustrations.
In classical statistical analysis, the collected data usually have exact value. But, in network technology era, the collected data are usually symbolic type. Diday [
Billard and Diday [
This paper is organized as follows. Section 2 gives a introduction for symbolic interval data, Model 1 and Model 2. In Section 3, we propose two methods to estimate regression coefficient for symbolic interval data. In Section 4, the comparisons of the proposed methods and some existing methods are performed via simulations. In Section 5, we analyze two real datasets with the proposed approaches. Finally, we make some concluding remarks in Section 6.
In this article, we study on symbolic regression of interval-valued data. First of all, we introduce the symbolic interval data. In classical data, the exact value of the interested variables usually can be observed. In the network technology era, the collected data are growing more and more complex, and no longer a single point. Diday [
Y = ( Y 1 , ⋯ , Y n ) ′ , X i = ( X i 1 , ⋯ , X i p ) ′ and X = ( X ′ 1 , ⋯ , X ′ n ) ′ . Thus, the observed data are { ( Y i , X i 1 , ⋯ , X i p ) : i = 1 , ⋯ , n } .
This model considered the linear regression model for symbolic interval data as
Y i = β 0 + β 1 X i 1 + ⋯ + β p X i p + ε i , (1)
⇔ [ Y i U Y i L ] = [ β 0 + β 1 X i 1 U + ⋯ + β p X i p U + ε i U β 0 + β 1 X i 1 L + ⋯ + β p X i p L + ε i L ] , (2)
where i = 1 , ⋯ , n , ( β 1 , ⋯ , β p ) are the parameters of interest, and ε i is the error term. Here, we assume X i j L < X i j U , which is also considered by Billard and Diday [
In typical statistical analysis, the linear regression model is
Y i * = β 0 + β 1 X i 1 * + ⋯ + β p X i p * + ε i , i = 1 , ⋯ , n , (3)
where Y i * and X i j * are single points, ( β 1 , ⋯ , β p ) are the parameters of interest, and ε i is the error term. In practice, ( Y i * , X i j * ) may not be observed due to privacy issues or some reasons. Usually, the proxies Y i = [ c i , d i ] , X i j = [ a i j , b i j ] of ( Y i * , X i j * ) can be collected. Note that Y i * ∈ Y i and X i j * ∈ X i j , i = 1 , ⋯ , n , j = 1 , ⋯ , p . Thus, the collected data is { ( Y i , X i 1 , ⋯ , X i p ) : i = 1 , ⋯ , n } . In this model, the length of Y i does not depend on the lengths of X i .
Based on Model 1, we propose the endpoints least squares estimation approach to estimate ( β 0 , ⋯ , β p ) . We assume that X i j L < X i j U , i = 1 , ⋯ , n , j = 1 , ⋯ , p , which is also considered by Billard and Diday [
Y i U = β 0 + β 1 X i 1 U + β 2 X i 2 U + ⋯ + β p X i p U + ε i U , Y i L = β 0 + β 1 X i 1 L + β 2 X i 2 L + ⋯ + β p X i p L + ε i L , (4)
where i = 1 , ⋯ , n . To identify the order of Y i L and Y i U , we apply the centre method [
Then, compute Y ^ i U c and Y ^ i L c as
Y ^ i U c = β ^ 0 c + β ^ 1 c X i 1 U + β ^ 2 c X i 2 U + ⋯ + β ^ p c X i p U , Y ^ i L c = β ^ 0 c + β ^ 1 c X i 1 L + β ^ 2 c X i 2 L + ⋯ + β ^ p c X i p L . (5)
When Y ^ i L c < Y ^ i U c , Y i L = c i and Y i U = d i . When Y ^ i U c < Y ^ i L c , Y i U = c i and Y i L = d i . Then we would obtain the estimates of β by the endpoints least squares estimate as
β ^ U = ( X ′ U X U ) − 1 X ′ U Y U , β ^ L = ( X ′ L X L ) − 1 X ′ L Y L , (6)
where Y U = ( Y 1 U , ⋯ , Y n U ) ′ , Y L = ( Y 1 L , ⋯ , Y n L ) ′ , X U = ( X 1 U , ⋯ , X n U ) ′ , X L = ( X 1 L , ⋯ , X n L ) ′ , X i U = ( X i 1 U , ⋯ , X i p U ) and X i L = ( X i 1 L , ⋯ , X i p L ) . Then, set
β ^ = β ^ U + β ^ L 2 . (7)
β ^ is the estimator of β in model 1.
The method 2 is provided for the model 2, which allows the length of Y i does not depend on the lengths of X i . The centre method [
min β ∑ i = 1 n W i * ( Y i c − β 0 − β 1 X i 1 c − β 2 X i 2 c − ⋯ − β p X i p c ) 2 , (8)
where W i * = W i k / ∑ i = 1 n W i k , i = 1 , ⋯ , n , k = 1 , 2 , 3 . As the results of (8), the minimizer β ^ is the estimator of β in model 2. Through some examinations in simulations, we suggest three weighted functions of the length of the interval data in the following. Denote the length of interval: Y i r = | d i − c i | , X i j r = | b i j − a i j | , M Y r = max i ( Y i r ) and M X j r = max i ( X i j r ) , i = 1 , ⋯ , n , j = 1 , ⋯ , p . The first weighted function is designed as
W i 1 = ( a 1 * + b 1 * × exp ( − Y i r M Y r ) ) + ∑ j = 1 p ( c 1 * + d 1 * × exp ( − X i j r M X j r ) ) , (9)
where i = 1 , ⋯ , n and a 1 * , b 1 * , c 1 * , d 1 * are positive constants. The weighted function is exponential decline as the lengths of interval data increasing. The second weighted function is given as
W i 2 = ( a 2 * + b 2 * × ( − Y i r M Y r ) ) + ∑ j = 1 p ( c 2 * + d 2 * × ( − X i j r M X j r ) ) , (10)
where i = 1 , ⋯ , n , a 2 * , b 2 * , c 2 * , d 2 * are positive constants, b 2 * < a 2 * and d 2 * < c 2 * . The weighted function is linear decline as the lengths of interval data increasing. Define the standardized lengths of interval data Y i r M Y r and X i j r M X j r as S Y i r and S X i j r . Let 1 n ∑ i = 1 n S Y i r and 1 n ∑ i = 1 n S X i j r be S ¯ Y r and S ¯ X j r . The third weighted function is designed as
W i 3 = ( 1 2 a 3 * exp ( − | S Y i r − S ¯ Y r a 3 * | ) ) − 1 − 2 a 3 * + ∑ j = 1 p ( ( 1 2 a 3 * exp ( − | S X i j r − S ¯ X j r a 3 * | ) ) − 1 − 2 a 3 * ) , (11)
where i = 1 , ⋯ , n and a 3 * is a positive constant. The weighted function is decreasing when the standardized length less than the average of the standardized length and increasing when the standardized length is more than the average of the standardized length. We will compare all methods via simulations in Section 4.
In this section, we compare our proposed methods, endspoints least squares estimator (M1) and interval weighted least squares estimator (M2), with the existing methods, CM [
For model 1: we first generate 2 independent values from N ( 0, σ X j 2 ) , and let
X i j U be the larger one and X i j L be the smaller one, where i = 1 , ⋯ , n , j = 1 , 2 . The error term ε i L ~ N ( 0, σ ε L 2 ) and ε i U ~ N ( 0, σ ε U 2 ) , i = 1 , ⋯ , n . Then, we generate Y i L and Y i U as
Y i L = β 0 + β 1 X i 1 L + β 2 X i 2 L + ε i L , Y i U = β 0 + β 1 X i 1 U + β 2 X i 2 U + ε i U . (12)
The β = ( β 0 , β 1 , β 2 ) are set as ( 10,10,5 ) and ( 10, − 10,5 ) . ( σ X 1 2 , σ X 2 2 , σ ε L 2 , σ ε U 2 ) are set as ( 1,1,0.1,0.1 ) . In
For model 2: we first generate the single points X i j * ~ N ( 0, σ X j 2 ) , j = 1 , 2 , ε i ~ N ( 0, σ ε 2 ) and set
Y i * = β 0 + β 1 X i 1 * + β 2 X i 2 * + ε i , (13)
where i = 1 , ⋯ , n . To construct the interval data, the range is generated from a uniform distribution, and denote the upper range of ( Y i * , X i j * ) by ( Y i U h , X i j U h ) and the lower range of ( Y i * , X i j * ) by ( Y i L h , X i j L h ) , i = 1 , ⋯ , n , j = 1 , 2 . Therefore, we could built the interval-valued data as [ c i , d i ] = [ Y i * − Y i L h , Y i * + Y i U h ] and [ a i j , b i j ] = [ X i j * − X i j L h , X i j * + X i j U h ] , i = 1 , ⋯ , n , j = 1 , 2 . Thus, we obtain the interval data ( Y i , X i 1 , X i 2 ) , i = 1 , ⋯ , n . For the settings, β = ( β 0 , β 1 , β 2 ) are set as ( 10,10,5 ) and ( 10, − 10,5 ) , and ( σ X j 2 , σ ε 2 ) are set as ( 1,0.5 ) , j = 1 , 2 . Y i L h , Y i U h , X i j L h and X i j U h are generated from uniform distribution such as Y i L h and Y i U h from U ( 0, b 1 h ) , X i 1 L h and X i 1 U h from U ( 0, b 2 h ) , and X i 2 L h and X i 2 U h from U ( 0, b 3 h ) , i = 1 , ⋯ , n . Note that M2 (1) is the interval weighted LSE with the first weighted function, W1, and ( a 1 * , b 1 * , c 1 * , d 1 * ) = ( 1,0.2,1,0.15 ) ; M2 (2) is the method with the second weighted function, W2, and ( a 2 * , b 2 * , c 2 * , d 2 * ) = ( 1,0.2,1,0.15 ) ; M2 (3) is the method with the third weighted function, W3, and a 3 * = 0.5 ; M2 (4) is the method with the third weighted function, W3, and a 3 * = 1 ; M2 (5) is the
β ^ | Method | n = 50 | n = 100 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Bias | SD | JackSD | MSE | CP | Bias | SD | JackSD | MSE | CP | ||
β ^ 0 | CM | −0.0014 | 0.0139 | 0.0146 | 0.0002 | 0.966 | 0.0008 | 0.0100 | 0.0102 | 0.0001 | 0.952 |
CRM | −0.0014 | 0.0139 | 0.0146 | 0.0002 | 0.966 | 0.0008 | 0.0100 | 0.0102 | 0.0001 | 0.952 | |
SCM | −0.0014 | 0.0138 | 0.0144 | 0.0002 | 0.964 | 0.0008 | 0.0099 | 0.0101 | 0.0001 | 0.954 | |
M1 | −0.0011 | 0.0160 | 0.0163 | 0.0003 | 0.944 | 0.0008 | 0.0113 | 0.0114 | 0.0001 | 0.944 | |
M2 (1) | −0.0014 | 0.0139 | 0.0146 | 0.0002 | 0.966 | 0.0008 | 0.0100 | 0.0102 | 0.0001 | 0.954 | |
M2 (2) | −0.0014 | 0.0139 | 0.0146 | 0.0002 | 0.966 | 0.0008 | 0.0100 | 0.0102 | 0.0001 | 0.954 | |
M2 (3) | −0.0007 | 0.0173 | 0.0182 | 0.0003 | 0.946 | 0.0006 | 0.0119 | 0.0126 | 0.0001 | 0.968 | |
M2 (4) | −0.0007 | 0.0165 | 0.0173 | 0.0003 | 0.942 | 0.0007 | 0.0114 | 0.0120 | 0.0001 | 0.964 | |
M2 (5) | −0.0007 | 0.0162 | 0.0170 | 0.0003 | 0.944 | 0.0007 | 0.0112 | 0.0118 | 0.0001 | 0.964 | |
β ^ 1 | CM | −0.0011 | 0.0215 | 0.0214 | 0.0005 | 0.934 | 0.0005 | 0.0149 | 0.0146 | 0.0002 | 0.950 |
CRM | −0.0011 | 0.0215 | 0.0214 | 0.0005 | 0.934 | 0.0005 | 0.0149 | 0.0146 | 0.0002 | 0.950 | |
SCM | −0.0009 | 0.0162 | 0.0162 | 0.0003 | 0.938 | 0.0003 | 0.0115 | 0.0112 | 0.0001 | 0.950 | |
M1 | −0.0007 | 0.0156 | 0.0156 | 0.0002 | 0.934 | 0.0003 | 0.0109 | 0.0107 | 0.0001 | 0.952 | |
M2 (1) | −0.0011 | 0.0215 | 0.0214 | 0.0005 | 0.934 | 0.0005 | 0.0149 | 0.0146 | 0.0002 | 0.948 | |
M2 (2) | −0.0011 | 0.0215 | 0.0214 | 0.0005 | 0.934 | 0.0005 | 0.0150 | 0.0146 | 0.0002 | 0.948 | |
M2 (3) | −0.0018 | 0.0255 | 0.0269 | 0.0007 | 0.944 | 0.0001 | 0.0183 | 0.0179 | 0.0003 | 0.930 | |
M2 (4) | −0.0017 | 0.0244 | 0.0256 | 0.0006 | 0.944 | 0.0002 | 0.0174 | 0.0171 | 0.0003 | 0.930 | |
M2 (5) | −0.0017 | 0.0241 | 0.0252 | 0.0006 | 0.940 | 0.0002 | 0.0171 | 0.0168 | 0.0003 | 0.938 | |
β ^ 2 | CM | 0.0011 | 0.0206 | 0.0212 | 0.0004 | 0.946 | 0.0006 | 0.0151 | 0.0145 | 0.0002 | 0.936 |
CRM | 0.0011 | 0.0206 | 0.0212 | 0.0004 | 0.946 | 0.0006 | 0.0151 | 0.0145 | 0.0002 | 0.936 | |
SCM | 0.0010 | 0.0157 | 0.0161 | 0.0002 | 0.952 | 0.0004 | 0.0116 | 0.0112 | 0.0001 | 0.932 | |
M1 | 0.0008 | 0.0151 | 0.0154 | 0.0002 | 0.950 | 0.0004 | 0.0110 | 0.0106 | 0.0001 | 0.932 | |
M2 (1) | 0.0011 | 0.0206 | 0.0212 | 0.0004 | 0.948 | 0.0006 | 0.0152 | 0.0145 | 0.0002 | 0.938 | |
M2 (2) | 0.0011 | 0.0207 | 0.0212 | 0.0004 | 0.950 | 0.0007 | 0.0152 | 0.0145 | 0.0002 | 0.942 | |
M2 (3) | 0.0003 | 0.0254 | 0.0262 | 0.0006 | 0.946 | 0.0004 | 0.0189 | 0.0178 | 0.0004 | 0.922 | |
M2 (4) | 0.0005 | 0.0243 | 0.0251 | 0.0006 | 0.946 | 0.0004 | 0.0181 | 0.0170 | 0.0003 | 0.920 | |
M2 (5) | 0.0005 | 0.0238 | 0.0246 | 0.0006 | 0.946 | 0.0004 | 0.0178 | 0.0167 | 0.0003 | 0.928 |
β ^ | Method | n = 50 | n = 100 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Bias | SD | JackSD | MSE | CP | Bias | SD | JackSD | MSE | CP | ||
β ^ 0 | CM | 0.0008 | 0.0148 | 0.0147 | 0.0002 | 0.944 | 0.0002 | 0.0103 | 0.0102 | 0.0001 | 0.954 |
CRM | 0.0008 | 0.0148 | 0.0147 | 0.0002 | 0.944 | 0.0002 | 0.0103 | 0.0102 | 0.0001 | 0.954 | |
SCM | 0.0359 | 0.3585 | 0.3813 | 0.1298 | 0.976 | 0.0058 | 0.2479 | 0.2592 | 0.0615 | 0.970 | |
M1 | 0.0012 | 0.0161 | 0.0164 | 0.0003 | 0.942 | 0.0002 | 0.0117 | 0.0114 | 0.0001 | 0.942 | |
M2 (1) | 0.0008 | 0.0148 | 0.0147 | 0.0002 | 0.942 | 0.0002 | 0.0103 | 0.0102 | 0.0001 | 0.950 | |
M2 (2) | 0.0008 | 0.0148 | 0.0147 | 0.0002 | 0.942 | 0.0002 | 0.0103 | 0.0102 | 0.0001 | 0.950 | |
M2 (3) | 0.0002 | 0.0180 | 0.0184 | 0.0003 | 0.932 | 0.0004 | 0.0126 | 0.0127 | 0.0002 | 0.940 | |
M2 (4) | 0.0003 | 0.0170 | 0.0174 | 0.0003 | 0.938 | 0.0003 | 0.0120 | 0.0120 | 0.0001 | 0.938 | |
M2 (5) | 0.0004 | 0.0167 | 0.0170 | 0.0003 | 0.940 | 0.0003 | 0.0118 | 0.0117 | 0.0001 | 0.940 | |
β ^ 1 | CM | −0.0007 | 0.0217 | 0.0212 | 0.0005 | 0.938 | 0.0004 | 0.0142 | 0.0145 | 0.0002 | 0.960 |
CRM | −0.0007 | 0.0217 | 0.0212 | 0.0005 | 0.938 | 0.0004 | 0.0142 | 0.0145 | 0.0002 | 0.960 | |
SCM | 3.5157 | 0.9454 | 0.8777 | 13.2539 | 0.000 | 3.4392 | 0.6212 | 0.6107 | 12.2137 | 0.000 | |
M1 | −0.0005 | 0.0159 | 0.0154 | 0.0003 | 0.940 | 0.0004 | 0.0104 | 0.0106 | 0.0001 | 0.960 | |
M2 (1) | −0.0007 | 0.0217 | 0.0212 | 0.0005 | 0.936 | 0.0004 | 0.0142 | 0.0145 | 0.0002 | 0.962 | |
M2 (2) | −0.0007 | 0.0217 | 0.0212 | 0.0005 | 0.936 | 0.0004 | 0.0142 | 0.0145 | 0.0002 | 0.962 | |
M2 (3) | −0.0008 | 0.0271 | 0.0262 | 0.0007 | 0.910 | −0.0002 | 0.0180 | 0.0178 | 0.0003 | 0.938 | |
M2 (4) | −0.0008 | 0.0257 | 0.0249 | 0.0007 | 0.914 | −0.0001 | 0.0170 | 0.0169 | 0.0003 | 0.938 | |
M2 (5) | −0.0008 | 0.0252 | 0.0245 | 0.0006 | 0.918 | −0.0001 | 0.0167 | 0.0166 | 0.0003 | 0.938 | |
β ^ 2 | CM | 0.0012 | 0.0217 | 0.0213 | 0.0005 | 0.932 | −0.0005 | 0.0150 | 0.0145 | 0.0002 | 0.918 |
CRM | 0.0012 | 0.0217 | 0.0213 | 0.0005 | 0.932 | −0.0005 | 0.0150 | 0.0145 | 0.0002 | 0.918 | |
SCM | 0.9000 | 0.4932 | 0.4847 | 1.0532 | 0.516 | 0.8954 | 0.3160 | 0.3233 | 0.9017 | 0.190 | |
M1 | 0.0008 | 0.0158 | 0.0156 | 0.0002 | 0.932 | −0.0003 | 0.0110 | 0.0106 | 0.0001 | 0.928 | |
M2 (1) | 0.0012 | 0.0217 | 0.0213 | 0.0005 | 0.932 | −0.0005 | 0.0150 | 0.0145 | 0.0002 | 0.920 | |
M2 (2) | 0.0012 | 0.0217 | 0.0213 | 0.0005 | 0.930 | −0.0005 | 0.0150 | 0.0145 | 0.0002 | 0.922 | |
M2 (3) | 0.0009 | 0.0265 | 0.0265 | 0.0007 | 0.942 | −0.0007 | 0.0172 | 0.0180 | 0.0003 | 0.948 | |
M2 (4) | 0.0010 | 0.0250 | 0.0251 | 0.0006 | 0.948 | −0.0007 | 0.0165 | 0.0170 | 0.0003 | 0.946 | |
M2 (5) | 0.0010 | 0.0245 | 0.0247 | 0.0006 | 0.948 | −0.0006 | 0.0162 | 0.0166 | 0.0003 | 0.948 |
β ^ | Method | n = 50 | n = 100 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Bias | SD | JackSD | MSE | CP | Bias | SD | JackSD | MSE | CP | ||
β ^ 0 | CM | 0.0004 | 0.0097 | 0.0105 | 0.0001 | 0.966 | −0.0001 | 0.0071 | 0.0072 | 0.0001 | 0.952 |
CRM | 0.0004 | 0.0097 | 0.0105 | 0.0001 | 0.966 | −0.0001 | 0.0071 | 0.0072 | 0.0001 | 0.952 | |
SCM | 0.0004 | 0.0097 | 0.0103 | 0.0001 | 0.962 | 0.0000 | 0.0071 | 0.0071 | 0.0001 | 0.944 | |
M1 | 0.0006 | 0.0137 | 0.0148 | 0.0002 | 0.956 | 0.0002 | 0.0103 | 0.0101 | 0.0001 | 0.950 | |
M2 (1) | 0.0004 | 0.0097 | 0.0105 | 0.0001 | 0.962 | −0.0001 | 0.0071 | 0.0072 | 0.0001 | 0.952 | |
M2 (2) | 0.0004 | 0.0097 | 0.0105 | 0.0001 | 0.962 | −0.0001 | 0.0071 | 0.0072 | 0.0001 | 0.956 | |
M2 (3) | 0.0005 | 0.0118 | 0.0128 | 0.0001 | 0.950 | −0.0001 | 0.0088 | 0.0088 | 0.0001 | 0.942 | |
M2 (4) | 0.0005 | 0.0113 | 0.0122 | 0.0001 | 0.968 | −0.0001 | 0.0084 | 0.0084 | 0.0001 | 0.940 | |
M2 (5) | 0.0004 | 0.0110 | 0.0120 | 0.0001 | 0.972 | −0.0001 | 0.0083 | 0.0083 | 0.0001 | 0.944 | |
β ^ 1 | CM | 0.0005 | 0.0149 | 0.0151 | 0.0002 | 0.950 | −0.0004 | 0.0103 | 0.0103 | 0.0001 | 0.940 |
CRM | 0.0005 | 0.0149 | 0.0151 | 0.0002 | 0.950 | −0.0004 | 0.0103 | 0.0103 | 0.0001 | 0.940 | |
SCM | 0.0004 | 0.0123 | 0.0119 | 0.0002 | 0.940 | −0.0003 | 0.0084 | 0.0083 | 0.0001 | 0.936 | |
M1 | 0.0002 | 0.0131 | 0.0130 | 0.0002 | 0.938 | −0.0003 | 0.0087 | 0.0089 | 0.0001 | 0.948 | |
M2 (1) | 0.0005 | 0.0149 | 0.0151 | 0.0002 | 0.948 | −0.0004 | 0.0103 | 0.0103 | 0.0001 | 0.938 | |
M2 (2) | 0.0005 | 0.0149 | 0.0151 | 0.0002 | 0.948 | −0.0004 | 0.0104 | 0.0103 | 0.0001 | 0.938 | |
M2 (3) | 0.0001 | 0.0171 | 0.0183 | 0.0003 | 0.938 | −0.0007 | 0.0118 | 0.0124 | 0.0001 | 0.964 | |
M2 (4) | 0.0001 | 0.0165 | 0.0175 | 0.0003 | 0.938 | −0.0007 | 0.0114 | 0.0119 | 0.0001 | 0.956 | |
M2 (5) | 0.0001 | 0.0162 | 0.0172 | 0.0003 | 0.932 | −0.0006 | 0.0112 | 0.0117 | 0.0001 | 0.956 | |
β ^ 2 | CM | −0.0005 | 0.0155 | 0.0153 | 0.0002 | 0.934 | −0.0005 | 0.0102 | 0.0103 | 0.0001 | 0.942 |
CRM | −0.0005 | 0.0155 | 0.0153 | 0.0002 | 0.934 | −0.0005 | 0.0102 | 0.0103 | 0.0001 | 0.942 | |
SCM | −0.0005 | 0.0126 | 0.0120 | 0.0002 | 0.938 | −0.0002 | 0.0084 | 0.0082 | 0.0001 | 0.934 | |
M1 | −0.0004 | 0.0130 | 0.0131 | 0.0002 | 0.954 | −0.0001 | 0.0090 | 0.0088 | 0.0001 | 0.938 | |
M2 (1) | −0.0005 | 0.0155 | 0.0153 | 0.0002 | 0.934 | −0.0005 | 0.0102 | 0.0103 | 0.0001 | 0.940 | |
M2 (2) | −0.0005 | 0.0155 | 0.0153 | 0.0002 | 0.936 | −0.0005 | 0.0102 | 0.0103 | 0.0001 | 0.938 | |
M2 (3) | −0.0001 | 0.0180 | 0.0185 | 0.0003 | 0.936 | −0.0001 | 0.0120 | 0.0125 | 0.0001 | 0.952 | |
M2 (4) | −0.0002 | 0.0174 | 0.0178 | 0.0003 | 0.936 | −0.0001 | 0.0115 | 0.0119 | 0.0001 | 0.948 | |
M2 (5) | −0.0002 | 0.0172 | 0.0175 | 0.0003 | 0.934 | −0.0001 | 0.0114 | 0.0117 | 0.0001 | 0.948 |
β ^ | Method | n = 50 | n = 100 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Bias | SD | JackSD | MSE | CP | Bias | SD | JackSD | MSE | CP | ||
β ^ 0 | CM | −0.0002 | 0.0106 | 0.0104 | 0.0001 | 0.942 | −0.0004 | 0.0069 | 0.0072 | 0.0000 | 0.952 |
CRM | −0.0002 | 0.0106 | 0.0104 | 0.0001 | 0.942 | −0.0004 | 0.0069 | 0.0072 | 0.0000 | 0.952 | |
SCM | 0.0191 | 0.3548 | 0.3789 | 0.1262 | 0.982 | −0.0132 | 0.2404 | 0.2588 | 0.0579 | 0.972 | |
M1 | −0.0006 | 0.0151 | 0.0147 | 0.0002 | 0.936 | −0.0003 | 0.0093 | 0.0101 | 0.0001 | 0.960 | |
M2 (1) | −0.0002 | 0.0106 | 0.0104 | 0.0001 | 0.940 | −0.0004 | 0.0069 | 0.0072 | 0.0000 | 0.952 | |
M2 (2) | −0.0002 | 0.0106 | 0.0104 | 0.0001 | 0.940 | −0.0004 | 0.0069 | 0.0072 | 0.0000 | 0.952 | |
M2 (3) | 0.0000 | 0.0121 | 0.0130 | 0.0001 | 0.950 | −0.0004 | 0.0089 | 0.0090 | 0.0001 | 0.964 | |
M2 (4) | 0.0000 | 0.0116 | 0.0123 | 0.0001 | 0.946 | −0.0003 | 0.0084 | 0.0084 | 0.0001 | 0.962 | |
M2 (5) | 0.0000 | 0.0114 | 0.0120 | 0.0001 | 0.946 | −0.0003 | 0.0082 | 0.0083 | 0.0001 | 0.960 | |
β ^ 1 | CM | 0.0002 | 0.0152 | 0.0152 | 0.0002 | 0.932 | −0.0007 | 0.0093 | 0.0103 | 0.0001 | 0.962 |
CRM | 0.0002 | 0.0152 | 0.0152 | 0.0002 | 0.932 | −0.0007 | 0.0093 | 0.0103 | 0.0001 | 0.962 | |
SCM | 3.4912 | 0.8982 | 0.8697 | 12.9956 | 0.000 | 3.4429 | 0.6252 | 0.6125 | 12.2446 | 0.000 | |
M1 | 0.0000 | 0.0129 | 0.0132 | 0.0002 | 0.940 | −0.0009 | 0.0080 | 0.0088 | 0.0001 | 0.972 | |
M2 (1) | 0.0002 | 0.0153 | 0.0152 | 0.0002 | 0.932 | −0.0007 | 0.0093 | 0.0103 | 0.0001 | 0.966 | |
M2 (2) | 0.0002 | 0.0153 | 0.0152 | 0.0002 | 0.930 | −0.0007 | 0.0092 | 0.0103 | 0.0001 | 0.966 | |
M2 (3) | 0.0000 | 0.0183 | 0.0188 | 0.0003 | 0.948 | −0.0008 | 0.0120 | 0.0128 | 0.0001 | 0.956 | |
M2 (4) | 0.0001 | 0.0174 | 0.0179 | 0.0003 | 0.946 | −0.0008 | 0.0112 | 0.0121 | 0.0001 | 0.956 | |
M2 (5) | 0.0001 | 0.0171 | 0.0175 | 0.0003 | 0.944 | −0.0008 | 0.0110 | 0.0118 | 0.0001 | 0.958 | |
β ^ 2 | CM | 0.0010 | 0.0151 | 0.0154 | 0.0002 | 0.956 | −0.0003 | 0.0098 | 0.0103 | 0.0001 | 0.954 |
CRM | 0.0010 | 0.0151 | 0.0154 | 0.0002 | 0.956 | −0.0003 | 0.0098 | 0.0103 | 0.0001 | 0.954 | |
SCM | 0.9018 | 0.4700 | 0.4722 | 1.0342 | 0.502 | 0.8680 | 0.3307 | 0.3245 | 0.8627 | 0.228 | |
M1 | 0.0006 | 0.0130 | 0.0133 | 0.0002 | 0.942 | −0.0003 | 0.0085 | 0.0089 | 0.0001 | 0.954 | |
M2 (1) | 0.0010 | 0.0151 | 0.0154 | 0.0002 | 0.958 | −0.0003 | 0.0098 | 0.0103 | 0.0001 | 0.956 | |
M2 (2) | 0.0010 | 0.0151 | 0.0154 | 0.0002 | 0.958 | −0.0003 | 0.0098 | 0.0103 | 0.0001 | 0.956 | |
M2 (3) | 0.0004 | 0.0179 | 0.0184 | 0.0003 | 0.944 | −0.0005 | 0.0120 | 0.0127 | 0.0001 | 0.968 | |
M2 (4) | 0.0005 | 0.0171 | 0.0176 | 0.0003 | 0.948 | −0.0005 | 0.0114 | 0.0120 | 0.0001 | 0.960 | |
M2 (5) | 0.0006 | 0.0168 | 0.0173 | 0.0003 | 0.948 | −0.0005 | 0.0112 | 0.0118 | 0.0001 | 0.956 |
method with the third weighted function, W3, and a 3 * = 2 . The simulation results are shown in Tables 5-8. From the results, the interval weighted least squares estimation with W3 has better performance than others.
β ^ | Method | n = 50 | n = 100 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Bias | SD | JackSD | MSE | CP | Bias | SD | JackSD | MSE | CP | ||
β ^ 0 | CM | −0.0064 | 0.1896 | 0.1816 | 0.0360 | 0.934 | −0.0072 | 0.1271 | 0.1261 | 0.0162 | 0.934 |
CRM | −0.0064 | 0.1896 | 0.1816 | 0.0360 | 0.934 | −0.0072 | 0.1271 | 0.1261 | 0.0162 | 0.934 | |
SCM | −0.0110 | 0.2009 | 0.1940 | 0.0405 | 0.934 | −0.0058 | 0.1332 | 0.1340 | 0.0178 | 0.950 | |
M1 | −0.0102 | 0.1996 | 0.1919 | 0.0399 | 0.938 | −0.0073 | 0.1313 | 0.1330 | 0.0173 | 0.948 | |
M2 (1) | −0.0064 | 0.1894 | 0.1816 | 0.0359 | 0.934 | −0.0071 | 0.1271 | 0.1261 | 0.0162 | 0.934 | |
M2 (2) | −0.0064 | 0.1894 | 0.1817 | 0.0359 | 0.934 | −0.0070 | 0.1272 | 0.1261 | 0.0162 | 0.932 | |
M2 (3) | −0.0056 | 0.1874 | 0.1758 | 0.0352 | 0.928 | −0.0044 | 0.1250 | 0.1222 | 0.0156 | 0.936 | |
M2 (4) | −0.0055 | 0.1864 | 0.1783 | 0.0348 | 0.936 | −0.0050 | 0.1242 | 0.1239 | 0.0154 | 0.950 | |
M2 (5) | −0.0054 | 0.1860 | 0.1778 | 0.0346 | 0.934 | −0.0052 | 0.1239 | 0.1235 | 0.0154 | 0.952 | |
β ^ 1 | CM | −0.1036 | 0.1823 | 0.1822 | 0.0440 | 0.882 | −0.1033 | 0.1257 | 0.1271 | 0.0265 | 0.846 |
CRM | −0.1036 | 0.1823 | 0.1822 | 0.0440 | 0.882 | −0.1033 | 0.1257 | 0.1271 | 0.0265 | 0.846 | |
SCM | −0.4285 | 0.1946 | 0.1983 | 0.2214 | 0.412 | −0.4211 | 0.1336 | 0.1362 | 0.1951 | 0.110 | |
M1 | −0.2033 | 0.1814 | 0.1825 | 0.0742 | 0.770 | −0.2045 | 0.1258 | 0.1268 | 0.0577 | 0.646 | |
M2 (1) | −0.1035 | 0.1821 | 0.1822 | 0.0439 | 0.882 | −0.1033 | 0.1257 | 0.1271 | 0.0265 | 0.846 | |
M2 (2) | −0.1035 | 0.1820 | 0.1823 | 0.0438 | 0.884 | −0.1033 | 0.1258 | 0.1271 | 0.0265 | 0.846 | |
M2 (3) | −0.0908 | 0.1820 | 0.1774 | 0.0414 | 0.900 | −0.0888 | 0.1258 | 0.1232 | 0.0237 | 0.860 | |
M2 (4) | −0.0913 | 0.1804 | 0.1803 | 0.0409 | 0.904 | −0.0898 | 0.1248 | 0.1249 | 0.0236 | 0.858 | |
M2 (5) | −0.0915 | 0.1798 | 0.1797 | 0.0407 | 0.902 | −0.0902 | 0.1245 | 0.1245 | 0.0236 | 0.860 | |
β ^ 2 | CM | −0.0481 | 0.1774 | 0.1836 | 0.0338 | 0.924 | −0.0459 | 0.1236 | 0.1273 | 0.0174 | 0.944 |
CRM | −0.0481 | 0.1774 | 0.1836 | 0.0338 | 0.924 | −0.0459 | 0.1236 | 0.1273 | 0.0174 | 0.944 | |
SCM | −0.3586 | 0.1912 | 0.1991 | 0.1651 | 0.562 | −0.3495 | 0.1337 | 0.1364 | 0.1400 | 0.270 | |
M1 | −0.0974 | 0.1768 | 0.1837 | 0.0408 | 0.908 | −0.0976 | 0.1246 | 0.1271 | 0.0250 | 0.902 | |
M2 (1) | −0.0481 | 0.1774 | 0.1836 | 0.0338 | 0.924 | −0.0458 | 0.1237 | 0.1273 | 0.0174 | 0.944 | |
M2 (2) | −0.0481 | 0.1774 | 0.1837 | 0.0338 | 0.926 | −0.0457 | 0.1238 | 0.1274 | 0.0174 | 0.942 | |
M2 (3) | −0.0421 | 0.1752 | 0.1786 | 0.0325 | 0.924 | −0.0390 | 0.1246 | 0.1233 | 0.0170 | 0.942 | |
M2 (4) | −0.0424 | 0.1737 | 0.1816 | 0.0320 | 0.928 | −0.0394 | 0.1236 | 0.1251 | 0.0168 | 0.952 | |
M2 (5) | −0.0425 | 0.1731 | 0.1810 | 0.0318 | 0.932 | −0.0396 | 0.1232 | 0.1247 | 0.0167 | 0.950 |
β ^ | Method | n = 50 | n = 100 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Bias | SD | JackSD | MSE | CP | Bias | SD | JackSD | MSE | CP | ||
β ^ 0 | CM | 0.0083 | 0.1782 | 0.1809 | 0.0318 | 0.942 | 0.0075 | 0.1232 | 0.1269 | 0.0152 | 0.946 |
CRM | 0.0083 | 0.1782 | 0.1809 | 0.0318 | 0.942 | 0.0075 | 0.1232 | 0.1269 | 0.0152 | 0.946 | |
SCM | 0.0093 | 0.1795 | 0.1831 | 0.0323 | 0.942 | 0.0079 | 0.1237 | 0.1282 | 0.0154 | 0.946 | |
M1 | 0.0120 | 0.1871 | 0.1902 | 0.0351 | 0.950 | 0.0094 | 0.1260 | 0.1328 | 0.0160 | 0.960 | |
M2 (1) | 0.0082 | 0.1780 | 0.1809 | 0.0317 | 0.942 | 0.0075 | 0.1233 | 0.1268 | 0.0152 | 0.946 | |
M2 (2) | 0.0080 | 0.1779 | 0.1810 | 0.0317 | 0.942 | 0.0074 | 0.1234 | 0.1269 | 0.0153 | 0.944 | |
M2 (3) | 0.0109 | 0.1750 | 0.1755 | 0.0307 | 0.936 | 0.0093 | 0.1221 | 0.1227 | 0.0150 | 0.952 | |
M2 (4) | 0.0106 | 0.1737 | 0.1782 | 0.0303 | 0.944 | 0.0093 | 0.1211 | 0.1244 | 0.0148 | 0.954 | |
M2 (5) | 0.0105 | 0.1732 | 0.1777 | 0.0301 | 0.948 | 0.0093 | 0.1208 | 0.1241 | 0.0147 | 0.954 | |
β ^ 1 | CM | 0.1013 | 0.1874 | 0.1861 | 0.0454 | 0.896 | 0.1068 | 0.1236 | 0.1273 | 0.0267 | 0.852 |
CRM | 0.1013 | 0.1874 | 0.1861 | 0.0454 | 0.896 | 0.1068 | 0.1236 | 0.1273 | 0.0267 | 0.852 | |
SCM | 0.2656 | 0.1867 | 0.1868 | 0.1054 | 0.706 | 0.2655 | 0.1256 | 0.1273 | 0.0863 | 0.476 | |
M1 | 0.1924 | 0.1847 | 0.1859 | 0.0711 | 0.824 | 0.1983 | 0.1245 | 0.1266 | 0.0548 | 0.668 | |
M2 (1) | 0.1013 | 0.1873 | 0.1861 | 0.0453 | 0.894 | 0.1067 | 0.1236 | 0.1273 | 0.0267 | 0.852 | |
M2 (2) | 0.1014 | 0.1873 | 0.1862 | 0.0454 | 0.892 | 0.1067 | 0.1237 | 0.1274 | 0.0267 | 0.852 | |
M2 (3) | 0.0854 | 0.1847 | 0.1814 | 0.0414 | 0.898 | 0.0905 | 0.1234 | 0.1238 | 0.0234 | 0.870 | |
M2 (4) | 0.0862 | 0.1836 | 0.1843 | 0.0411 | 0.910 | 0.0918 | 0.1223 | 0.1257 | 0.0234 | 0.874 | |
M2 (5) | 0.0865 | 0.1832 | 0.1837 | 0.0410 | 0.908 | 0.0924 | 0.1219 | 0.1253 | 0.0234 | 0.870 | |
β ^ 2 | CM | −0.0424 | 0.1838 | 0.1861 | 0.0356 | 0.938 | −0.0605 | 0.1273 | 0.1284 | 0.0199 | 0.926 |
CRM | −0.0424 | 0.1838 | 0.1861 | 0.0356 | 0.938 | −0.0605 | 0.1273 | 0.1284 | 0.0199 | 0.926 | |
SCM | 0.0683 | 0.1804 | 0.1849 | 0.0372 | 0.938 | 0.0445 | 0.1270 | 0.1276 | 0.0181 | 0.938 | |
M1 | −0.0880 | 0.1819 | 0.1858 | 0.0408 | 0.926 | −0.1052 | 0.1262 | 0.1281 | 0.0270 | 0.866 | |
M2 (1) | −0.0425 | 0.1838 | 0.1861 | 0.0356 | 0.936 | −0.0605 | 0.1271 | 0.1284 | 0.0198 | 0.928 | |
M2 (2) | −0.0427 | 0.1839 | 0.1862 | 0.0356 | 0.938 | −0.0605 | 0.1271 | 0.1285 | 0.0198 | 0.928 | |
M2 (3) | −0.0340 | 0.1775 | 0.1807 | 0.0327 | 0.942 | −0.0539 | 0.1213 | 0.1244 | 0.0176 | 0.930 | |
M2 (4) | −0.0342 | 0.1765 | 0.1835 | 0.0323 | 0.942 | −0.0541 | 0.1209 | 0.1262 | 0.0175 | 0.934 | |
M2 (5) | −0.0342 | 0.1762 | 0.1829 | 0.0322 | 0.944 | −0.0542 | 0.1207 | 0.1258 | 0.0175 | 0.934 |
β ^ | Method | n = 50 | n = 100 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Bias | SD | JackSD | MSE | CP | Bias | SD | JackSD | MSE | CP | ||
β ^ 0 | CM | 0.0022 | 0.3220 | 0.3403 | 0.1037 | 0.958 | −0.0049 | 0.2336 | 0.2342 | 0.0546 | 0.958 |
CRM | 0.0022 | 0.3220 | 0.3403 | 0.1037 | 0.958 | −0.0049 | 0.2336 | 0.2342 | 0.0546 | 0.958 | |
SCM | 0.0014 | 0.3907 | 0.4086 | 0.1526 | 0.958 | −0.0024 | 0.2686 | 0.2790 | 0.0722 | 0.958 | |
M1 | 0.0110 | 0.3735 | 0.4028 | 0.1396 | 0.960 | −0.0088 | 0.2768 | 0.2763 | 0.0767 | 0.944 | |
M2 (1) | 0.0024 | 0.3219 | 0.3402 | 0.1036 | 0.958 | −0.0048 | 0.2335 | 0.2341 | 0.0546 | 0.958 | |
M2 (2) | 0.0026 | 0.3221 | 0.3404 | 0.1038 | 0.958 | −0.0048 | 0.2335 | 0.2342 | 0.0546 | 0.958 | |
M2 (3) | 0.0025 | 0.3123 | 0.3278 | 0.0975 | 0.964 | −0.0034 | 0.2348 | 0.2243 | 0.0551 | 0.950 | |
M2 (4) | 0.0022 | 0.3103 | 0.3317 | 0.0963 | 0.968 | −0.0032 | 0.2329 | 0.2266 | 0.0543 | 0.952 | |
M2 (5) | 0.0020 | 0.3097 | 0.3306 | 0.0959 | 0.970 | −0.0032 | 0.2322 | 0.2261 | 0.0539 | 0.950 | |
β ^ 1 | CM | −0.3965 | 0.3453 | 0.3375 | 0.2765 | 0.762 | −0.4048 | 0.2310 | 0.2318 | 0.2172 | 0.560 |
CRM | −0.3965 | 0.3453 | 0.3375 | 0.2765 | 0.762 | −0.4048 | 0.2310 | 0.2318 | 0.2172 | 0.560 | |
SCM | −1.4872 | 0.3941 | 0.4075 | 2.3672 | 0.038 | −1.4679 | 0.2776 | 0.2777 | 2.2317 | 0.000 | |
M1 | −0.7594 | 0.3384 | 0.3391 | 0.6912 | 0.376 | −0.7755 | 0.2281 | 0.2317 | 0.6534 | 0.094 | |
M2 (1) | −0.3962 | 0.3450 | 0.3375 | 0.2760 | 0.760 | −0.4046 | 0.2311 | 0.2318 | 0.2171 | 0.560 | |
M2 (2) | −0.3962 | 0.3449 | 0.3377 | 0.2759 | 0.758 | −0.4048 | 0.2313 | 0.2319 | 0.2173 | 0.558 | |
M2 (3) | −0.3405 | 0.3463 | 0.3258 | 0.2359 | 0.782 | −0.3448 | 0.2180 | 0.2235 | 0.1664 | 0.638 | |
M2 (4) | −0.3450 | 0.3435 | 0.3299 | 0.2371 | 0.784 | −0.3483 | 0.2174 | 0.2261 | 0.1686 | 0.646 | |
M2 (5) | −0.3472 | 0.3424 | 0.3288 | 0.2378 | 0.788 | −0.3500 | 0.2173 | 0.2255 | 0.1697 | 0.634 | |
β ^ 2 | CM | −0.1956 | 0.3263 | 0.3392 | 0.1447 | 0.918 | −0.2005 | 0.2171 | 0.2334 | 0.0873 | 0.866 |
CRM | −0.1956 | 0.3263 | 0.3392 | 0.1447 | 0.918 | −0.2005 | 0.2171 | 0.2334 | 0.0873 | 0.866 | |
SCM | −1.2065 | 0.4099 | 0.4052 | 1.6236 | 0.150 | −1.2011 | 0.2662 | 0.2746 | 1.5135 | 0.002 | |
M1 | −0.3728 | 0.3251 | 0.3386 | 0.2447 | 0.800 | −0.3865 | 0.2173 | 0.2321 | 0.1966 | 0.6180 | |
M2 (1) | −0.1952 | 0.3264 | 0.3393 | 0.1446 | 0.918 | −0.2002 | 0.2172 | 0.2333 | 0.0873 | 0.866 | |
M2 (2) | −0.1950 | 0.3268 | 0.3395 | 0.1448 | 0.916 | −0.2001 | 0.2174 | 0.2334 | 0.0873 | 0.868 | |
M2 (3) | −0.1684 | 0.3224 | 0.3284 | 0.1323 | 0.916 | −0.1666 | 0.2172 | 0.2246 | 0.0749 | 0.876 | |
M2 (4) | −0.1701 | 0.3203 | 0.3332 | 0.1315 | 0.920 | −0.1695 | 0.2152 | 0.2274 | 0.0750 | 0.882 | |
M2 (5) | −0.1709 | 0.3195 | 0.3320 | 0.1313 | 0.918 | −0.1709 | 0.2144 | 0.2267 | 0.0752 | 0.882 |
β ^ | Method | n = 50 | n = 100 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Bias | SD | JackSD | MSE | CP | Bias | SD | JackSD | MSE | CP | ||
β ^ 0 | CM | −0.0178 | 0.3619 | 0.3383 | 0.1313 | 0.922 | 0.0132 | 0.2236 | 0.2340 | 0.0502 | 0.950 |
CRM | −0.0178 | 0.3619 | 0.3383 | 0.1313 | 0.922 | 0.0132 | 0.2236 | 0.2340 | 0.0502 | 0.950 | |
SCM | −0.0209 | 0.3720 | 0.3508 | 0.1388 | 0.924 | 0.0111 | 0.2353 | 0.2425 | 0.0555 | 0.954 | |
M1 | −0.0126 | 0.4002 | 0.3984 | 0.1604 | 0.932 | 0.0195 | 0.2711 | 0.2729 | 0.0739 | 0.950 | |
M2 (1) | −0.0178 | 0.3619 | 0.3382 | 0.1313 | 0.920 | 0.0132 | 0.2237 | 0.2339 | 0.0502 | 0.948 | |
M2 (2) | −0.0178 | 0.3623 | 0.3385 | 0.1316 | 0.918 | 0.0133 | 0.2241 | 0.2340 | 0.0504 | 0.946 | |
M2 (3) | −0.0252 | 0.3384 | 0.3247 | 0.1152 | 0.944 | 0.0160 | 0.2204 | 0.2237 | 0.0488 | 0.958 | |
M2 (4) | −0.0254 | 0.3377 | 0.3282 | 0.1147 | 0.948 | 0.0161 | 0.2186 | 0.2258 | 0.0481 | 0.956 | |
M2 (5) | −0.0254 | 0.3376 | 0.3275 | 0.1146 | 0.948 | 0.0161 | 0.2179 | 0.2253 | 0.0477 | 0.954 | |
β ^ 1 | CM | 0.4027 | 0.3299 | 0.3323 | 0.2710 | 0.746 | 0.4105 | 0.2236 | 0.2293 | 0.2185 | 0.564 |
CRM | 0.4027 | 0.3299 | 0.3323 | 0.2710 | 0.746 | 0.4105 | 0.2236 | 0.2293 | 0.2185 | 0.564 | |
SCM | 0.9541 | 0.3214 | 0.3312 | 1.0136 | 0.198 | 0.9371 | 0.2257 | 0.2271 | 0.9291 | 0.008 | |
M1 | 0.7353 | 0.3168 | 0.3298 | 0.6410 | 0.384 | 0.7375 | 0.2230 | 0.2262 | 0.5936 | 0.100 | |
M2 (1) | 0.4026 | 0.3298 | 0.3323 | 0.2709 | 0.746 | 0.4103 | 0.2235 | 0.2292 | 0.2183 | 0.562 | |
M2 (2) | 0.4028 | 0.3300 | 0.3324 | 0.2711 | 0.746 | 0.4104 | 0.2236 | 0.2294 | 0.2184 | 0.566 | |
M2 (3) | 0.3523 | 0.3233 | 0.3215 | 0.2286 | 0.774 | 0.3552 | 0.2189 | 0.2196 | 0.1741 | 0.620 | |
M2 (4) | 0.3564 | 0.3209 | 0.3255 | 0.2300 | 0.770 | 0.3586 | 0.2172 | 0.2218 | 0.1758 | 0.636 | |
M2 (5) | 0.3583 | 0.3200 | 0.3247 | 0.2308 | 0.770 | 0.3603 | 0.2165 | 0.2214 | 0.1767 | 0.636 | |
β ^ 2 | CM | −0.1949 | 0.3207 | 0.3344 | 0.1408 | 0.896 | −0.1944 | 0.2349 | 0.2325 | 0.0929 | 0.846 |
CRM | −0.1949 | 0.3207 | 0.3344 | 0.1408 | 0.896 | −0.1944 | 0.2349 | 0.2325 | 0.0929 | 0.846 | |
SCM | 0.1531 | 0.3144 | 0.3230 | 0.1223 | 0.924 | 0.1431 | 0.2248 | 0.2245 | 0.0710 | 0.906 | |
M1 | −0.3567 | 0.3215 | 0.3350 | 0.2306 | 0.804 | −0.3593 | 0.2345 | 0.2322 | 0.1841 | 0.690 | |
M2 (1) | −0.1948 | 0.3202 | 0.3344 | 0.1405 | 0.898 | −0.1943 | 0.2346 | 0.2325 | 0.0928 | 0.846 | |
M2 (2) | −0.1949 | 0.3200 | 0.3346 | 0.1404 | 0.902 | −0.1944 | 0.2345 | 0.2326 | 0.0928 | 0.846 | |
M2 (3) | −0.1596 | 0.3161 | 0.3215 | 0.1254 | 0.898 | −0.1702 | 0.2314 | 0.2223 | 0.0825 | 0.860 | |
M2 (4) | −0.1622 | 0.3131 | 0.3253 | 0.1243 | 0.902 | −0.1715 | 0.2298 | 0.2244 | 0.0822 | 0.864 | |
M2 (5) | −0.1634 | 0.3119 | 0.3245 | 0.1239 | 0.902 | −0.1721 | 0.2292 | 0.2240 | 0.0821 | 0.866 |
In this section, we apply our proposed methods to analyze two datasets, mushroom data and medical data, which are interval data corresponding to Model 1 or Model 2. The first data which we used to analyze is a mushroom data, which is from the Fungi of California Species Index. The complete data can be downloaded from the internet site, http://www.mykoweb.com/CAF/species_index.html. Three features are represented by three variables Y = the width of the pileus cap, X1 = the length of the stipe, and X2 = the thickness of the stipe. These measurements in the dataset are interval value (in cm). There were 311 observations from the Fungi of California Species Index. Because the lengths of the variables should depend on each other, the dataset belongs to Model 1. By the method 1 and method 2 with the same settings in simulations, we analyze the dataset and present the results in
The next data which we used to analyze is a medical data, which is from Billard and Diday [
In the network technology era, the collected data are growing more and more complex, and become larger than before. It brings the difficulty to analyze by using the standard statistical tools. Diday [
β ^ | Method | Beta ^ | JackSD | 95% CI |
---|---|---|---|---|
β ^ 0 | CM | 0.3436 | 0.2510 | [−0.1484, 0.8356] |
CRM | 0.3436 | 0.2510 | [−0.1484, 0.8356] | |
SCM | 0.1703 | 0.2160 | [−0.2531, 0.5937] | |
M1 | 0.5468 | 0.2434 | [0.0697, 1.0239] | |
M2 (1) | 0.3449 | 0.2509 | [−0.1469, 0.8367] | |
M2 (2) | 0.3458 | 0.2513 | [−0.1467, 0.8383] | |
M2 (3) | 0.0392 | 0.2302 | [−0.4120, 0.4904] | |
M2 (4) | 0.0612 | 0.2226 | [−0.3751, 0.4975] | |
M2 (5) | 0.0705 | 0.2260 | [−0.3725, 0.5135] | |
β ^ 1 | CM | 0.4786 | 0.0750 | [0.3316, 0.6256] |
CRM | 0.4786 | 0.0750 | [0.3316, 0.6256] | |
SCM | 0.5276 | 0.0733 | [0.3839, 0.6713] | |
M1 | 0.5043 | 0.0681 | [0.3708, 0.6378] | |
M2 (1) | 0.4772 | 0.0748 | [0.3306, 0.6238] | |
M2 (2) | 0.4759 | 0.0746 | [0.3297, 0.6221] | |
M2 (3) | 0.7009 | 0.1530 | [0.4010, 1.0008] | |
M2 (4) | 0.6718 | 0.1413 | [0.3949, 0.9487] | |
M2 (5) | 0.6577 | 0.1349 | [0.3933, 0.9221] | |
β ^ 2 | CM | 1.9618 | 0.2868 | [1.3997, 2.5239] |
CRM | 1.9618 | 0.2868 | [1.3997, 2.5239] | |
SCM | 1.8746 | 0.3122 | [1.2627, 2.4865] | |
M1 | 1.5679 | 0.2645 | [1.0495, 2.0863] | |
M2 (1) | 1.9672 | 0.2850 | [1.4086, 2.5258] | |
M2 (2) | 1.9727 | 0.2828 | [1.4184, 2.5270] | |
M2 (3) | 1.0400 | 0.7273 | [−0.3855, 2.4655] | |
M2 (4) | 1.1740 | 0.6514 | [−0.1027, 2.4507] | |
M2 (5) | 1.2390 | 0.6126 | [0.0383, 2.4397] |
β ^ | Method | Beta ^ | JackSD | 95% CI |
---|---|---|---|---|
β ^ 0 | CM | 142.4305 | 6.9138 | [128.8797, 155.9813] |
CRM | 142.4305 | 6.9138 | [128.8797, 155.9813] | |
SCM | 78.336 | 9.4546 | [59.8053, 96.8667] | |
M1 | 139.4896 | 6.5044 | [126.7412, 152.2380] | |
M2 (1) | 142.4262 | 6.9291 | [128.8454, 156.0070] | |
M2 (2) | 142.3836 | 6.9511 | [128.7597, 156.0075] | |
M2 (3) | 148.2383 | 6.4528 | [135.5910, 160.8856] | |
M2 (4) | 147.8034 | 6.4676 | [135.1271, 160.4797] | |
M2 (5) | 147.5888 | 6.4773 | [134.8935, 160.2841] | |
β ^ 1 | CM | 0.6834 | 0.0834 | [0.5199, 0.8469] |
CRM | 0.6834 | 0.0834 | [0.5199, 0.8469] | |
SCM | 0.6580 | 0.1388 | [0.3860, 0.9300] | |
M1 | 0.6691 | 0.0832 | [0.5060, 0.8322] | |
M2 (1) | 0.6846 | 0.0836 | [0.5208, 0.8485] | |
M2 (2) | 0.6867 | 0.0838 | [0.5225, 0.8509] | |
M2 (3) | 0.5848 | 0.0736 | [0.4406, 0.7291] | |
M2 (4) | 0.5929 | 0.0735 | [0.4488, 0.7370] | |
M2 (5) | 0.5968 | 0.0735 | [0.4527, 0.7409] | |
β ^ 2 | CM | 0.0003 | 0.0003 | [−0.0003, 0.0009] |
CRM | 0.0003 | 0.0003 | [−0.0003, 0.0009] | |
SCM | 0.0036 | 0.0004 | [0.0029, 0.0044] | |
M1 | −0.0003 | 0.0004 | [−0.0011, 0.0005] | |
M2 (1) | 0.0003 | 0.0003 | [−0.0003, 0.0009] | |
M2 (2) | 0.0003 | 0.0003 | [−0.0003, 0.0009] | |
M2 (3) | 0.0003 | 0.0003 | [−0.0003, 0.0009] | |
M2 (4) | 0.0003 | 0.0003 | [−0.0003, 0.0009] | |
M2 (5) | 0.0003 | 0.0003 | [−0.0003, 0.0009] |
with W3 has better performance for model 2 data. Finally, we analyze two real datasets for illustration. Furthermore, the results coincide with the results in simulation studies.
The authors declare no conflicts of interest regarding the publication of this paper.
Hsieh, J.-J. and Pan, C.-C. (2018) Linear Regression Analysis for Symbolic Interval Data. Open Journal of Statistics, 8, 885-901. https://doi.org/10.4236/ojs.2018.86059