Normalizing a monotonically increasing function and calculate std

Question

I have an increasing function like so:

I plan on breaking it up into intervals (between the red lines). I want to rotate the segment horizontal and calculate the standard deviation.

I know this might seem silly, but I essentially want to calculate the variation after normalizing the increasing ramp per segment. What method can I use to achieve this?

My initial thoughts are to take calculate the slope and draw a line from the beginning to the end of the segment with that slope. Then, calculate the delta of each data point with respect to the line.

Answer 1

Just had quick play with writing some code. I grabbed some numbers from your sketched function using https://apps.automeris.io/wpd/, giving me:

csvdata = """\
x,y
22937.2,1822.1
22942.9,1822.2
22950.0,1822.4
22959.6,1822.5
22967.8,1822.5
22976.8,1822.6
22987.1,1822.6
22995.5,1822.7
23004.7,1822.7
23014.1,1822.7
23025.1,1822.7
23034.2,1822.8
23043.1,1822.9
23049.8,1823.0
23057.9,1823.2
23064.0,1823.3
23070.0,1823.5
23078.7,1823.6
23086.6,1823.7
23096.3,1823.9
23104.0,1824.0
23112.9,1824.1
23122.6,1824.1
23131.5,1824.1
23141.3,1824.1
23153.3,1824.0
23164.6,1824.1
"""

Then I got that data into Python using Pandas, performed the regression I suggested and plot the output using:

from io import StringIO

import numpy as np
import scipy.stats
import pandas as pd
import matplotlib.pyplot as plt

# get data
df = pd.read_csv(StringIO(csvdata))

# basic univariate linear regression
res = scipy.stats.linregress(df.x, df.y)
print(res)

# estimate y at each x
yp = res.intercept + df.x * res.slope
# calculate standard error
sd = np.std(yp - df.y)

plt.plot(df.x, df.y, label="function")
plt.plot(df.x, yp, label="slope that minimises residual")
plt.title(f"SD of residual = {sd:.2f}")
plt.legend()

giving me: