Handbook of Frequency Stability Analysis

5.3.2 Chi-Squared Confidence IntervalsChi-squared statistics can be applied to calculate single and double-sided confidence intervals at any desiredconfidence factor. These calculations are based on a determination of the number of degrees of freedom for theestimated noise type. Most stability plots show ±1σ error bars for its overlapping Allan deviation plot.The error bars for the modified Allan and time variances are also determined by Chi-squared statistics, using thenumber of MVAR degrees of freedom for the particular noise type, averaging factor, and number of data points.During the Run function, noise type estimates are made at each averaging factor (except the last, where the noise typeof the previous averaging factor is used).Sample variances are distributed according to the expression22 edf ⋅ sχ = , (44)2σwhere χ² is the Chi-square, s² is the sample variance, σ² is the true variance, and edf is the equivalent number ofdegrees of freedom (not necessarily an integer). The edf is determined by the number of analysis points and the noisetype. Procedures exist for establishing single- or double-sided confidence intervals with a selectable confidencefactor, based on χ² statistics, for many of its variance functions. The general procedure is to choose a single- ordouble-limited confidence factor, p, calculate the corresponding χ² value, determine the edf from the variance type,noise type and number of analysis points, and thereby set the statistical limit(s) on the variance. For double-sidedlimits,σedfedf= s ⋅ and σ = s ⋅. (45)χ ( p, edf ) χ (1 − p, edf )2 2 2 2min2 max25.4. Degrees of FreedomThe equivalent number of χ² degrees of freedom (edf) associated with a statistical variance (or deviation) estimatedepends on the variance type, the number of data points, and the type of noise involved. In general, the progressionfrom the original two-sample (Allan) variance to the overlapping, total, and Thêo1 variances has provided larger edfsand better confidence. The noise type matters because it determines the extent that the points are correlated. Highlycorrelated data have a smaller edf than does the same number of points of uncorrelated (white) noise. An edfdetermination therefore involves (1) choosing the appropriate algorithm for the particular variance type, (2)determining the dominant power law noise type of the data, and (3) using the number of data points to calculate thecorresponding edf.5.4.1. AVAR, MVAR, TVAR, and HVAR EDFThe equivalent number of χ² degrees of freedom (edf) for the Allan variance (AVAR), the modified Allan variance(MVAR) and the related time variance (TVAR), and the Hadamard variance (HVAR) is found by a combinedalgorithm developed by C.A. Greenhall, based on its generalized autocovariance function [2].38