How to Interpret Lomb Scargle periodogram

Question

lomb scargle periodogram:

import numpy as np
import pandas as pd
from matplotlib import pyplot as plt
from scipy import signal
from astropy.time import Time
from astropy.timeseries import LombScargle

m5_data = pd.read_csv('tmp')
m5_data.head()

     date    sales
0  2011-01-29  32631.0
1  2011-01-30  31749.0
2  2011-01-31  23783.0
3  2011-02-01  25412.0
4  2011-02-02  19146.0

"""Converting dates to MJD """
tmp_str = m5_data.iloc[:,0].astype(pd.StringDtype())
t_date = np.array(tmp_str.values,dtype = 'str')
t_date = Time(t_date, format='isot', scale='utc')
t_date.format = 'mjd'

y = m5_data.iloc[:,1]
m5_ls = LombScargle(t_date, y)
m5_frequency, m5_power = m5_ls.autopower()
plt.plot(m5_frequency,m5_power)
plt.show()

1. What does the x axis represent? Is it frequencies in days or is it frequencies in 1/day?
2. So the spike at 2 does that mean there is a period of every two days or does it mean there is a period 1/2 days, which is a fractional day and I do not understand how I could get fractional days when I only have 1 observation per day.

The overall goal is to find if the data is periodic. Given this data is sales of a product from a store I would expect that if the data is periodic that the periods would be 7days(weekly), 30days(monthly), 182days(semi annually)

So the spike at 2 does not make sense to me.

Answer 1

The x-axis of a periodogram is usually a frequency, measured in inverse time units. It is possible to plot the inverse of frequency, in which case the axis would be non-linear.

In your case, the x-axis looks linear, and the the code you present appears to be using something labelled as a frequency on the x-axis. Therefore I would assume (though I would read the documentation for the functions you are using) that the x-axis is frequency. The units will be based on whatever units are in your input. i.e. It is just the inverse of whatever you are giving it for the times - in this case, $d^{-1}$.

If your sampling rate is once per day, then you are getting no reliable information at all about frequencies $>1$ d$^{-1}$. The stuff you should be concentrating on lies between $0\leq f < 1$ d$^{-1}$. Even in this range you won't have good fidelity on signals with $0.5<f<1$ d$^{-1}$ because they are undersampled. You can also expect to get aliasing signals in the range $0\leq f <1$ that are caused by true signals (including those with higher frequencies) aliasing with the 1 d$^{-1}$ frequency of your sampling.

It looks to me, by eye, like you might have a genuine signal at $1/7$ d$^{-1}$, corresponding to a weekly variation. There may be a weaker signal at much lower frequencies that might correspond to a monthly signal. The 2 signals either side of 1 day look like aliases of the weekly signal with the sampling frequency.

I would highly recommend reading VanderPlas 2018 for a thorough discussion of the Lomb-Scargle periodogram and its interpretation.