Statistics decribing ranges

stats.md

SEE UPDATE BELOW

There are a few statistics to look at. The most obvious are the ones that compare the different parameters. These take a single parameter and compare it to the true value. I calculate 3 different statistics.

• The Fractional Delta. This is essentially the percent difference of the model and the input values, but no absolute value is taken. This should be read as a percentage over/under estimation of the true value.
• The Factor above or below. Read as the name implies.
• Absolute fractional difference. Basically the standard percent error

Looking at the spreads, in dex, for the diagnostic plots, we see a that the spread decreases. In fact the 16%, 50%, and 84% values for the 2-d rms spread in decades (factor) for MENV/MASSC vs MDOT/MASSC and MENV/MDOT v MASSC are

Plot	16%	50%	84%
MENV/MASSC vs MDOT/MASSC	0.51 (3)	1.04 (11)	1.84 (68)
MENV/MDOT v MASSC	0.22 (1.7)	0.57 (3.7)	1.41 (25.5)

Offner et al (2012) Table 3

These tables are analagous to Table 3 from Offner 2012.

Value	Median	RMS
Median MENV is	0.77	445
Median MDOT* is	0.548	0.00115
Median MSTAR is	0.957	8.26
Best MENV is	0.83	528
Best MDOT* is	0.838	0.00141
Best MSTAR is	0.971	8.34

Value	Median	RMS
Median MENV/MSTAR is	0.878	23.3
Median MDOT/MSTAR is	0.673	6.49e-05
Median MDOT/MENV is	0.995	8.67e-05
Best MENV/MSTAR is	0.789	27.5
Best MDOT/MSTAR is	0.985	8.18e-05
Best MDOT/MENV is	1.09	9.83e-05

Offner's table does not include MDOT (the infall rate) since it is dependent on the density. In Robitaille's models, MDOT is used to parametrize the envelope density.

Value	Median	RMS
MSTAR	1.1	0.4
MENV	3.2	3.2

Fractional delta

After removing values with input values equal to 0

function: 1 - (median or best)/input

Method	Median	Mean	Stdev
Median MENV/MSTAR - True is	0.422	-6.63	50.3
Median MDOT/MSTAR - True is	0.458	-17.1	115
Median MDOT/MENV - True is	0.121	-9.54	39.7
Best MENV/MSTAR - True is	0.366	-25.7	198
Best MDOT/MSTAR - True is	0.322	-41.4	226
Best MDOT/MENV - True is	0.0672	-27.4	205
Factor (above or below)

---

After remove values with input values equal to 0

function:

• take f = (median or best)/input
• if f_i < 1 then f_i = 1/f_i

Method	Median	Mean	Stdev
Median MENV/MSTAR - True is	3.76	6.21e+05	5.68e+06
Median MDOT/MSTAR - True is	4.8	80.4	388
Median MDOT/MENV - True is	2.91	21.7	65.7
Best MENV/MSTAR - True is	5.52	8.26e+06	9.68e+07
Best MDOT/MSTAR - True is	6.15	508	4.88e+03
Best MDOT/MENV - True is	3.97	112	1.02e+03

Absolute fractional difference

function = |(median or best) - input|/input

Method	Median	Mean	Stdev
Median MENV/MSTAR - True is	0.824	7.52	50.2
Median MDOT/MSTAR - True is	0.913	18	115
Median MDOT/MENV - True is	0.858	10.2	39.5
Best MENV/MSTAR - True is	0.959	26.6	198
Best MDOT/MSTAR - True is	0.951	42.3	226
Best MDOT/MENV - True is	0.937	28.1	205

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Update with MENV, MDOT, MSTAR

This shows the errors for the parameters themselves, not the ratios.

Fractional delta

After removing values with input values equal to 0

function: 1 - (median or best)/input

Method	Median	Mean	Stdev
Median MENV - True is	0.485	-6.76	54.3
Median MDOT - True is	0.546	-11	51.6
Median MSTAR - True is	0.0888	-0.134	1.3
Best MENV - True is	0.395	-21.4	159
Best MDOT - True is	0.499	-35.6	219
Best MSTAR - True is	0.0521	-0.17	1.26

Factor (above or below)

After remove values with input values equal to 0

function:

• take f = (median or best)/input
• if f_i < 1 then f_i = 1/f_i

Method	Median	Mean	Stdev
Median MENV - True is	3.89	2.37e+05	2.32e+06
Median MDOT - True is	7.32	74.2	327
Median MSTAR - True is	1.39	1.97	1.5
Best MENV - True is	6.41	8.6e+06	1.07e+08
Best MDOT - True is	8.07	588	5.11e+03
Best MSTAR - True is	1.39	2.1	1.71

Absolute fractional difference

function = |(median or best) - input|/input or Percent difference / 100

Method	Median	Mean	Stdev
Median MENV - True is	0.861	7.7	54.2
Median MDOT - True is	0.95	12	51.3
Median MSTAR - True is	0.299	0.57	1.17
Best MENV - True is	0.975	22.3	159
Best MDOT - True is	0.97	36.6	219
Best MSTAR - True is	0.303	0.606	1.12