Fading Function

For a damped window model, consider the fading function f(t) = 2^−λt, where t is the time-point and λ is a user-defined parameter. What is the weight of an instance x observed at time-point T(T > t)? Calculate the weight of the instance x at t0, t1, t2, t3, t4 since time t0. Plot a graph of hte weight v/s the time-point.

import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
sns.set()

lam = 1
def f(t):
    return 2**(-lam*t)

timepoints = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
weights = []
for timepoint in timepoints:
    weight = f(timepoint)
    weights.append(weight)

df = pd.DataFrame(weights, columns=[['Weight']])
df['Timepoint'] = df.index
df

	Weight	Timepoint
0	1.000000	0
1	0.500000	1
2	0.250000	2
3	0.125000	3
4	0.062500	4
5	0.031250	5
6	0.015625	6
7	0.007812	7
8	0.003906	8
9	0.001953	9
10	0.000977	10

fig, ax = plt.subplots()
fig.set_size_inches(15, 8.27)
plt.title('Weight of an instance X over different timepoints according to the fading function f(t) = 2^−λt')
plt.xlabel('Time')
plt.ylabel('Weight')
plt.plot(df['Timepoint'].values, df['Weight'].values, marker='o')

[<matplotlib.lines.Line2D at 0x138043f0>]

png

Fading Function

As Time increases, the weight of the instance X gets smaller and smaller