Re: Best linear Fit to slope data with Fixed starting point/value.
- To: mathgroup at smc.vnet.net
- Subject: [mg63624] Re: Best linear Fit to slope data with Fixed starting point/value.
- From: "Ray Koopman" <koopman at sfu.ca>
- Date: Sat, 7 Jan 2006 05:35:41 -0500 (EST)
- References: <dpg13b$pmj$1@smc.vnet.net><dplhkk$ejr$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Ray Koopman wrote:
> [...]
> Is 2.842170943040401*^-14 some sort of quantum number?
After some prompting -- thank you, Daniel -- I realized that it's
the value of the low-order bit in the machine representation of the
first 1785 data values; i.e., the ones in [128, 256).
For quantized data such as this, an observed value y does not mean
exactly y, but an interval (y-h,y+h), where h is the quantization
error (half the value of the low-order bit). When h is not small
relative to the random error, it may be better to fit the regression
by maximizing the sum of the logs of the likelihoods, where the
likelihood of an observed value y is Pr[y-h < a + b*x + s*e < y+h]
= Pr[((y-h)-(a+b*x))/s < e < ((y+h)-(a+b*x))/s],
e is a zero-mean, fixed-scale random variable, and a, b, & s
are to-be-estimated parameters of the regression.
For instance, if e is taken to be normal then we could maximize
Sum[ Log@Erf[(y[[i]]-h[[i]]-a-b*x[[i]])/s,
(y[[i]]+h[[i]]-a-b*x[[i]])/s], {i,n}],
or something equivalent to it. (It might help to write s = E^t
and maximize with respect to t instead of s.)