A Note on “ Geraghty contraction type mappings”
tutorial4
1. EENGM0014 Mathematics for Signal Processing and
Communications
Tutorial 4
Soon Yau Cheong
University of Bristol
28 Oct 2016
Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 1 / 12
2. Data Fitting
Estimate a, b, c, d, e, ...
Can be solved with 2 methods:
1 Pseudo inverse of
over-determined equations
2 Gradient descent
Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 2 / 12
3. Overdetermined Equations
Assume y = a + bx + cx2
+ dx3
Say (x0, y0) corresponds to coordinates of first point and (xm, ym) for m-th
point. We form matrices by substituting (x,y) into polynomial equation:
1 x0 x2
0 x3
0
1 x1 x2
1 x3
1
...
...
...
...
1 xM−1 x2
M−1 x3
M−1
a
b
c
d
=
y0
y1
...
yM−1
MX = Y
Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 3 / 12
5. Apply pseudo inverse to our problem:
X = (MT
M)−1
MT
Y
or use Matlab function pinv
X = pinv(M) ∗ Y =
a
b
c
d
Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 5 / 12
6. Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 6 / 12
7. Gradient
First Order Partial Derivative
If y is function of X = {x1, x2, ..., xn} then
X (y) =
∂y
∂X
=
∂y
∂x1
∂y
∂x2
...
∂y
∂xn
where ∂y
∂x1
is partial derivative of y with respect to x1, with other variables
in X being held constant.
∂y
∂xi
= lim
h→∞
f (x1, ..., xi + h, ..., xn) − f (x1, ..., xi , ..., xn)
h
Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 7 / 12
8. Hessian Matrix
Second Order Partial Derivative
H =
∂2y
∂x2
1
∂2y
∂x1∂x2
· · · ∂2y
∂x1∂xn
∂2y
∂x2∂x1
∂2y
∂x2
2
· · · ∂2y
∂x2∂xn
...
...
...
...
∂2y
∂xn∂x1
∂2y
∂xn∂x2
· · · ∂2y
∂x2
n
Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 8 / 12
9. Minima
Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 9 / 12
11. Saddle Point
Minimum in one direction
Maximum in orthogonal
direction
Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 11 / 12
12. Weak Minimum
Function does not necessary
decrease in all directions
Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications28 Oct 2016 12 / 12
