56  does order matter?

Let’s say we want to take the derivative of a signal, but it is too rough, so some smoothing is required. Does the order of operations matter? Which should be done first? Smoothing or differentiation?

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns
sns.set(style="ticks", font_scale=1.5)  # white graphs, with large and legible letters
from matplotlib.dates import DateFormatter
import matplotlib.dates as mdates
# %matplotlib widget

filename = "dead_sea_1d.csv"
df = pd.read_csv(filename)
df['date'] = pd.to_datetime(df['date'])
df = df.set_index('date')

df['grad_of_level'] = np.gradient(df['level'], 1.0)
df['smooth_of_level'] = df['level'].rolling('365D', center=True).mean()
df['grad_of_smooth_of_level'] = np.gradient(df['smooth_of_level'], 1.0)
df['smooth_of_grad_of_level'] = df['grad_of_level'].rolling('365D', center=True).mean()

fig, ax = plt.subplots(1, 1, figsize=(8,6))
ax.plot(df['level'], label="original")
ax.plot(df['smooth_of_level'], label="smoothed")
ax.legend()
ax.set(ylabel="sea level (m)");

Now let’s see the derivatives.

fig, ax = plt.subplots(1, 1, figsize=(8,6))
ax.plot(df['grad_of_smooth_of_level'], label="first smooth data, then take gradient")
ax.plot(df['smooth_of_grad_of_level'], label="first take gradient of data, then smooth it")
ax.legend()
ax.set(ylabel="rate of change (m/day)");