Goodness-of-Fit

We created models to predict the transit time of magnetic clouds from the Sun to 1 AU and performed goodness-of-fit tests on them to see which ones most accurately matched our observations. This has helped us to determine the best way to track a magnetic cloud back to its eruption time on the Sun.

Our goodness-of-fit program uses a weighted sum of squares (chi-square) approach to test whether our models match our observations any better than random guessing. The weighted sum of squares is the summation of the square of the difference between predicted values and observed values divided by the error in each observed measurement. Since it is difficult to determine the error in each value, we allow the user to specify the error as a percentage of the observed measurements. If no error is specified, a standard of ten percent is assumed in all measurements. Since we place no constraints on the values that our observations can take, we use no free parameters and thus our degree of freedom is equal to the sample size. If the outputted probability is greater than 80%, it is considered to be a very good match. A probability of less than 65% is considered suspect and less than 25% is considered a poor match.

The first model tested determines how long it will take a cloud to travel from the Sun to Earth, given its speed and acceleration as determined by LASCO, as well as the time at which LASCO first saw the cloud. The model assumes that the cloud begins with the given initial velocity and accelerates with the given acceleration out to 12 solar radii. After 12 solar radii, the cloud travels at a constant speed. Unfortunately, this model ignores the fact that not all clouds travel with a constant acceleration. In fact, we ran across a few instances in which the cloud was traveling at a sufficiently slow speed with a negative acceleration that the model predicted it would actually stop and then head back to the Sun. Thus, we also tested this model using only LASCO speeds.

The other model we tested used the sheath speed of a cloud and its arrival time at 1 AU to determine when it lifted off the surface of the Sun. We assumed the cloud traveled at a constant speed for the entirety of the trip.

The following table presents our results:

	Error Assumed in Measurements	Probability of Accurately Predicting Transit Times
LASCO speeds and accelerations	10%	0%
	20%	0%
	30%	0%
	95%	4.40%
	100%	12.06%
LASCO speeds only	10%	0%
	25%	0%
	50%	0%
	75%	46.37%
	85%	85.29%
Sheath Speeds	10%	0%
	20%	19.83%
	25%	87.20%
Cloud Speeds	10%	0%
	20%	1.1%
	27%	71.68%

Essentially what this means, is that even if we look at a time period of two days before the time LASCO speeds and accelerations predict a cloud would have launched off the Sun through two days after, we only have a 12.06% of having that progenitor actually erupt and release the corresponding cloud. The sheath speeds, however seem to be much more accurate. Within about a day, we can accurately predict the launch time of an event 8 out of 10 times. Although not ideal, it appears that the sheath speed is the best guage of transit times that we have at this time.