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Interpreting the IDL Code's Results on the Relevance of SGT

Iver H. Cairns and P. A. Robinson,
School of Physics, University of Sydney, NSW 2006, Australia
Tel: +61-2-9351-3961, Fax: +61-2-9351-7726, e-mail: cairnsphysics.usyd.edu.au

(16 January 2001 -revised 24 June 2002)

The IDL code sgt.pro, written by Dr. Pritha Das, analyzes a data file for the consistency or not of a dataset with SGT. The pages titled ``Necessary Parameters'' and ``How to Run the IDL code'', as well as the pages containing the IDL code in both text and html format, should be consulted on how to run the code. These pages address the output of the code and its interpretation.

The preliminary outputs of the code are plots of the wave data as functions of time and/or the position coordinate (called ``diff'' hereafter) and a plot of the raw P(log A) where A is the input field quantity. The main output of the code, however, is the plot of the probability distribution P(X) and associated statistical tests. Here X is the renormalized wave coordinate derived by extracting sliding time-averages of the SGT quantities $\mu$ and $\sigma$ , which are the average and standard deviation of the logarithim of the wave field, respectively, as a function of time or diff [1].

Pure SGT predicts that the P(X) distribution should be Gaussian with zero mean and unit standard deviation (solid curve in Fig. 6) [1]. The observed P(X) distribution is plotted with plus signs. Good visual agreement between predictions and theory is the first step in checking whether SGT is applicable or not. (The scatter plot of values of X versus $\log E$ can be used to assess biases in the observed distribution.) The second is to assess the statistical significance of this agreement, by interpreting the results of the standard $\chi^{2}$ and Kolmogorov-Smirnov tests. Press et al.'s book titled ``Numerical Recipes'' [8] describes these tests.

The IDL code performs the standard $\chi^{2}$ test on the numbers of counts in each bin (in X) that are observed and predicted by pure SGT, restricted to bins with 10 or more counts in our example. The output gives the value of $\chi^{2}$ , the number of degrees of freedom, and the associated significance probability $P(\chi^{2})$ . Here $P(\chi_{obs}^{2})$ is the probability that a value of $\chi^{2}$ greater than the observed value $\chi_{obs}^{2}$ would be found even if the theory is correct i.e. , that purely random fluctuations could produce the observed value. Very good agreement between the observed and predicted distributions then corresponds to $P(\chi^{2})$ in the range 0.1 - 1. Values $\stackrel{\scriptstyle>}{\scriptstyle\sim}10^{-3}$ are in reasonable agreement.

The IDL code also performs the standard Kolmogorov-Smirnov test on the unbinned distribution P(X), by comparing the maximum distance between the observed and predicted cumulative distributions of P(X). Outputs are the Kolmogorov-Smirnov distance d_K-S and the significance probability P(K-S). Here P(K-S) is the probability that a value of d_K-S greater than the observed value would be found even if the theory is correct. Very good agreement between the observed and predicted distributions then corresponds to P(K-S) in the range 0.1 - 1. Values $\stackrel{\scriptstyle>}{\scriptstyle\sim}10^{-3}$ are in reasonable agreement.

Our experience on several projects is that often the predicted and observed P(X) distributions are very similar by eye for a wide range of analysis parameters (such as the time/diff interval, number of points to construct the sliding estimates of $\mu$ and $\sigma$ over, the size of the bins in X) but that the significance probabilities range from very small values $\stackrel{\scriptstyle<}{\scriptstyle\sim}10^{-6}$ to values $\stackrel{\scriptstyle>}{\scriptstyle\sim}0.1$ . At least, this argues for very robust qualitative agreement of the observations with SGT. The difference between high and low significance probabilities can be due to constructing sliding quantities over too large or too small a range in time/diff, the typically relatively small number of samples in many datasets, the bin size, or to observational difficulties (e.g., the spacecraft moving into a different physical region or intrinsic time variations in the plasma).

One other matter is whether another theory can do as well or better than SGT. The usual ``secular growth'' theory for wave growth [2], in which the plasma is uniform and waves grow exponentially with constant growth rate until saturated by a nonlinear process, predicts a flat $P(\log E)$ distribution and a flat P(X) distribution. This theory can therefore be easily compared with the observations. Thus far, it is strongly inconsistent with data in several applications [1,3-7].

Those desiring to discuss these matters in more detail should read the references below or contact either Iver Cairns (cairns@physics.usyd.edu.au) or Peter Robinson (p.robinson@physics.usyd.edu.au). Their fax number is +61-2-9351-7726 and their telephone numbers are +61-2-9351-3961 and +61-2-9351-3779, respectively.