Assignment on Data Science PDF

Data Science

Table of Contents
1. Question....................................................................................................................................3
2. Question....................................................................................................................................5
3. Question....................................................................................................................................6
4. Question....................................................................................................................................7
References......................................................................................................................................10

1. Question
Ridge regression of same weighted vectors wεRm+1
Lagrange multipliers
argmin 1
2 ∑i=1
n
¿¿-wi x(i))
||W||22≤t
a)
State the constraints
Let us consider the optimization problems of constraints which are used for the Lagrange
multiplier that considers λ¿=0. The constraint yk(x*) does not constraint the minimum of yj(x)
=0 andλ =0, where a small change of jth constraint.
From ( y(t)-w.x(t))2, the w value is used to apply for (|w|22-t).
λ=o
△w=¿)2 + 0
2 [|W|22-t])=0
b)
Optimization problem using language of multiplier
λ=0 λ>0
argmin 1
2 ∑i=1
n
¿¿-wi x(i)) +
λ
2 ||W||22
λ=0
¿¿)2 + 0
2 [|W|22])
=∑i=1
n
¿¿)2
A constraint optimization problem is used for the objective functions of constraints. In other case
of particular constraints, the optimization problem considers the convex function of w1 x. The
optimization of the Lagrange can be considered as w.x(i) is the other one which can be used for
w.x(t), as the passive definite of which follows the variable (Adebayo Olorunsola, 2014).

Let us consider x,
F(x) = w.x(T)
From matrix x, Quadratic can use,
∂
∂x (w.x(T))=wx+x(T)x = (w+wT) x
In other case, it can be used for the Lagrange multiplier,
∂
∂w (wT Iw)= (I+I) w = 2w, λ=1
1
2 ∑i=1
n
¿¿) 2 + 1
2 [|W|22] = 0)
1
2 ¿) 2+ [|W|22] = 0)]
∂2
∂x wT Iw=2>0 can be strictly used in w.
C)
Analytical matrix algebra
The analytical matrix can be used for the formulate matrix. The regression ridge of RSS is
expressed on the following condition,
RSS (wi λ) = (y-xw) T (y-xw) +λwTw
Let us consider the minimizing straight forward application of matrix, from the calculus part that
includes,
∂
∂xRSS (wi λ¿=2(x T x) w-2xTy + 2λw=0
That are the simplified expressions,
2(x T x) w + 2λw= 2 x T y
(x T x+λI) w = x T y
Right estimators,
∇w=¿ ((x T x+λI)-1 x T y)
∇w=¿ (x T x)-1)-1 w OLS
D)
Relationship between t and λ