A Rough ν-Twin Support Vector Regression Machine 5
(ν-TWSVR-1)
min g
1
P
(w
1
, b
1
, ε
1
, ξ) =
c
1
2
(kw
1
k
2
+ b
2
1
) +
1
2
kY − (Aw
1
+ eb
1
)k
2
+ c
2
(ν
1
ε
1
+
1
l
e
T
ξ)
s.t. Y − (Aw
1
+ eb
1
) ≥ −ε
1
e − ξ,
ξ ≥ 0, ε
1
≥ 0.
(4)
(ν-TWSVR-2)
min g
2
P
(w
2
, b
2
, ε
2
, η) =
c
3
2
(kw
2
k
2
+ b
2
2
) +
1
2
kY − (Aw
2
+ eb
2
)k
2
+ c
4
(ν
2
ε
2
+
1
l
e
T
η)
s.t. (Aw
2
+ eb
2
) − Y ≥ −ε
2
e − η,
η ≥ 0, ε
2
≥ 0.
(5)
where parameters ν
1
, ν
2
, c
1
, c
2
, c
3
and c
4
are user specified positive parameters. Defining G = [A, e], υ
1
=
[w
T
1
, b
1
]
T
and υ
2
= [w
T
2
, b
2
]
T
, using the K.K.T optimality conditions, we can obtain the Wolfe dual of (ν-TWSVR-
1) as follows.
(Dual ν-TWSVR-1)
min g
1
D
(α) =
1
2
α
T
G(c
1
I + G
T
G)
−1
G
T
α − Y
T
G(c
1
I + G
T
G)
−1
G
T
α + Y
T
α
s.t. 0 ≤ α ≤
c
2
l
e,
e
T
α ≤ c
2
ν
1
.
(6)
where α is the vector of Lagrangian multipliers.
In a similar way, the dual of the problem (ν-TWSVR-2) can be obtained as
(Dual ν-TWSVR-2)
min g
2
D
(λ) =
1
2
λ
T
G(c
3
I + G
T
G)
−1
G
T
λ + Y
T
G(c
3
I + G
T
G)
−1
G
T
λ − Y
T
λ
s.t. 0 ≤ λ ≤
c
4
l
e,
e
T
λ ≤ c
4
ν
2
.
(7)
where λ is vector of Lagrangian multipliers.
After solving the optimization problem (6) and (7), we can obtain the augmented vector υ
1
= [w
T
1
, b
1
]
T
, and
υ
2
= [w
T
2
, b
2
]
T
as follows,
υ
1
= (c
1
I + G
T
G)
−1
G
T
(Y − α),
υ
2
= (c
3
I + G
T
G)
−1
G
T
(Y + λ).
The final regressor is then obtained by
f(x) =
1
2
(f
1
(x) + f
2
(x)) =
1
2
(w
1
+ w
2
)
T
x +
1
2
(b
1
+ b
2
).
For the non-linear case, the ν-TWSVR considers the following kernel generated hyperplane f
1
(x) = K(x
T
, A
T
)u
1
+
b
1
and f
2
(x) = K(x
T
, A
T
)u
2
+ b
2
, which are obtained by solving the following pairs of QP problems,
(Kernel ν-TWSVR-1)
min g
1
P
(u
1
, b
1
, ε
1
, ξ) =
c
1
2
(ku
1
k
2
+ b
2
1
) +
1
2
k(Y − (K(A, A
T
)u
1
+ eb
1
)k
2
+ c
2
(ν
1
ε
1
+
1
l
e
T
ξ)
s.t. Y − (K(A, A
T
)u
1
+ eb
1
) ≥ −ε
1
e − ξ,
ξ ≥ 0, ε
1
≥ 0.
(8)
(Kernel ν-TWSVR-2)
min g
2
P
(u
2
, b
2
, ε
2
, η) =
c
3
2
(ku
2
k
2
+ b
2
2
) +
1
2
k(Y − (K(A, A
T
)u
2
+ eb
2
)k
2
+ c
4
(ν
2
ε
2
+
1
l
e
T
η)
s.t. (K(A, A
T
)u
2
+ eb
2
) − Y ≥ −ε
2
e − η,
η ≥ 0, ε
2
≥ 0.
(9)
Similar to the linear case, the (Kernel ν-TWSVR-1) and (Kernel ν-TWSVR-2) are solved in their dual forms using
the K.K.T optimality conditions.
剩余23页未读，继续阅读
weixin_38500948
- 粉丝: 3
- 资源: 915
我的内容管理
展开
最新资源
- Hadoop生态系统与MapReduce详解
- MDS系列三相整流桥模块技术规格与特性
- MFC编程：指针与句柄获取全面解析
- LM06：多模4G高速数据模块，支持GSM至TD-LTE
- 使用Gradle与Nexus构建私有仓库
- JAVA编程规范指南：命名规则与文件样式
- EMC VNX5500 存储系统日常维护指南
- 大数据驱动的互联网用户体验深度管理策略
- 改进型Booth算法：32位浮点阵列乘法器的高速设计与算法比较
- H3CNE网络认证重点知识整理
- Linux环境下MongoDB的详细安装教程
- 压缩文法的等价变换与多余规则删除
- BRMS入门指南：JBOSS安装与基础操作详解
- Win7环境下Android开发环境配置全攻略
- SHT10 C语言程序与LCD1602显示实例及精度校准
- 反垃圾邮件技术：现状与前景
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
