Least absolute deviations¶

Nikolai Shokhirev, http://www.numericalexpert.com/

In [1]:

import numpy as np

In [2]:

def d1(c, x):
    return sum(abs(x-c))

Data points: \(x_{1},\, x_{2},\, x_{3},\,\ldots\,,\, x_{N}\)

In [3]:

def generate_nods(N):
    x = np.linspace(-1.0, 1.0, num=N)
    e = np.random.uniform(-0.1, 0.1, size=N) 
    #x.sort()
    return x+e

In [4]:

def generate_points(x, z, f):
    ''' ordinates for z '''
    d = np.zeros_like(z)
    for i in range(size(z)):
        d[i] = f(z[i], x)
    return d

In [5]:

N = 5
x = generate_nods(N)
print x    
d = generate_points(x, x, d1)
print d

[-0.96905703 -0.40652249  0.07371904  0.41039564  0.93449811]
[ 4.88831843  3.2007148   2.72047328  3.05714988  4.62945728]

Points for plotting

In [6]:

n = 481
z = np.linspace(-1.2, 1.2, num=n)
y = generate_points(x, z, d1)

In [7]:

plot(z, y)
plot(x, d, 'bo')
title('$d_{1}(x)$')
xlabel('$x$')

Out[7]:

<matplotlib.text.Text at 0x5231cf0>

In [8]:

def dd1(c, x):
    return sum(copysign(-1, x-c))

In [9]:

xx = np.zeros(N+2)
xx[0] = -1.2
xx[N+1] = 1.2
for i in range(N):
    xx[i+1] = x[i]
yy = -5
for i in range(1,N+1):
    plot([xx[i-1],xx[i]], [yy,yy],'b')
    yy += 2
plot([-1.2,1.2], [0,0],'r--') 
ylim(-6,6)
title('Derivative')
xlabel('$x$')

Out[9]:

<matplotlib.text.Text at 0x552ab70>

The \(abs\) function can be approximated by a smooth function:

And we have \[ \bar{d}_{1}(c)=\sum_{i=1}^{N}\varphi\left(x_{i}-c\right) \]

In [10]:

def de(c, x, eps=0.0):
    s = 0.0
    for a in x:
        s += sqrt((a-c)**2 + eps**2)
    return s

In [11]:

def generate_points2(x, z, f, e):
    ''' ordinates for z '''
    d = np.zeros_like(z)
    for i in range(size(z)):
        d[i] = f(z[i], x, e)
    return d

In [12]:

y2 = generate_points2(x, z, de, 0.05)

In [13]:

plot(z, y2)
title('Smoothed $d_{1}S')
xlabel('$x$')

Out[13]:

<matplotlib.text.Text at 0x53cc2b0>

In [14]:

def dde(c, x, eps=0.0):
    s = 0.0
    for a in x:
        s += (c-a)/sqrt((a-c)**2 + eps**2)
    return s

In [15]:

dy2 = generate_points2(x, z, dde, 0.05)
plot(z, dy2)
plot([-1.2,1.2], [0,0],'r--')
title('Smoothed derivative')
xlabel('$x$')

Out[15]:

<matplotlib.text.Text at 0x55398b0>

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