Roulette Statistics

This demo will show the distribution of roulette amplitudes. Higher-order amplitudes are always small, and other analyses may have to take this into account.

We will only need pandas and submodule to parse roulette amplitudes from CosmoSim.

import pandas as pd
from CosmoSim.RouletteAmplitudes import RouletteAmplitudes

We will look at the roulette amplitudes already produced for the machine learning demo, so we simply load the dataset as a pandas DataFrame.

filename = "ML/roulette.csv"
dataset = pd.read_csv( filename )
ramp = RouletteAmplitudes( dataset.columns )

The RouletteAmplitudes class gives us a method to check the number of roulette amplitudes, that is the maximum order $m$ .

print( ramp.getNterms() )

This is a bit rich, so we set our own limit, say 5. Feel free to change it.

maxm = 5

Now we can run through $m$ and $s$ to compute the mean and standard deviation for each amplitude, as follows.

ls = []
for m in range(maxm+1):
    for s in range((m+1)%2,m+2,2):
        avg = dataset[f"alpha[{m}][{s}]"].mean()
        sd = dataset[f"alpha[{m}][{s}]"].std()
        s = pd.Series( { "$m$" : m
                       , "$s$" : s
                       , "$\\bar\\alpha$" : avg
                       , "$s_\\alpha$" : sd
                       , "$\\bar\\beta$" : dataset[f"beta[{m}][{s}]"].mean()
                       , "$s_\\beta$" : dataset[f"beta[{m}][{s}]"].std()
                       } )
        ls.append( s )
df = pd.DataFrame( ls )
df["$m$"] = df["$m$"].astype(int)
df["$s$"] = df["$s$"].astype(int)
display( df )

We will leave the analysis for others; just observe that $\beta_0^m=0$ for all $m$ , which can be shown theoretically too.