In [1]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

Weighted median¶

In the video we have discussed that for MAPE metric the best constant prediction is weighted median with weights

$w_i = \frac{\sum_{j=1}^N \frac{1}{x_j}}{x_i}$

for each object $x_i$ .

This notebook exlpains how to compute weighted median. Let's generate some data first, and then find it's weighted median.

In [2]:

N = 5
x = np.random.randint(low=1, high=100, size=N)
x

Out[2]:

array([63, 52, 70, 17, 76])

Compute normalized weights:

In [3]:

inv_x = 1.0/x
inv_x

Out[3]:

array([ 0.01587302,  0.01923077,  0.01428571,  0.05882353,  0.01315789])

In [4]:

w = inv_x/sum(inv_x)
w

Out[4]:

array([ 0.13078104,  0.15844626,  0.11770294,  0.48465916,  0.1084106 ])

Now sort the normalized weights. We will use argsort (and not just sort) since we will need indices later.

In [5]:

idxs = np.argsort(w)
sorted_w = w[idxs]
sorted_w

Out[5]:

array([ 0.1084106 ,  0.11770294,  0.13078104,  0.15844626,  0.48465916])

Compute cumulitive sum of sorted weights

In [6]:

sorted_w_cumsum = np.cumsum(sorted_w)
plt.plot(sorted_w_cumsum); plt.show()
print ('sorted_w_cumsum: ', sorted_w_cumsum)

sorted_w_cumsum:  [ 0.1084106   0.22611354  0.35689458  0.51534084  1.        ]

Now find the index when cumsum hits 0.5:

In [7]:

idx = np.where(sorted_w_cumsum>0.5)[0][0]
idx

Out[7]:

Finally, your answer is sample at that position:

In [8]:

pos = idxs[idx]
x[pos]

Out[8]:

In [9]:

print('Data: ', x)
print('Sorted data: ', np.sort(x))
print('Weighted median: %d, Median: %d' %(x[pos], np.median(x)))

Data:  [63 52 70 17 76]
Sorted data:  [17 52 63 70 76]
Weighted median: 52, Median: 63

Thats it!

If the procedure looks surprising for you, try to do steps 2--5 assuming the weights are $w_i=\frac{1}{N}$ . That way you will find a simple median (not weighted) of the data.