MIX software fits mixture distributions to grouped data by the
method of maximum likelihood. Mixtures of up to 15 components can be
fitted, and the data can be grouped over as many as 80 intervals.
Mathematically, the mixed probability density function g is a
weighted sum of k component densities, where k is
assumed to be known.
The component densities can be normal, lognormal, gamma, exponential or Weibull distributions. The parameters are the mixing proportions and the means and standard deviations of the component distributions. The parameters can be constrained in many different ways: specified parameters can be held fixed, the coefficients of variation can be held constant, the means can be constrained to be equal, to be equally spaced or to lie on a von Bertalanffy growth curve, to give just a few examples. Constraints can help to resolve identifiability problems when there is extensive overlapping of component distributions. MIX computes standard errors for all the estimates. The data can be truncated on the left, the right, or both sides.
Special features of MIX make it particularly convenient for analysing length-frequency distributions as mixtures of age groups. The prototype of MIX was introduced by Macdonald and Pitcher (1979) for the analysis of fisheries length-frequency data, and this remains an important application (Macdonald 1987). Figures 1 - 3 show examples of length-frequency analyses and more can be found in the Demonstration Examples. See Titterington et al. (1985) for a more general discussion of mixture distributions and their applications.
Figure 1. An example of fisheries length-frequency analysis. The five components correspond to the five age-groups in the population, the thick line is their sum, the mixture distribution. The abscissa unit is length in cm. The triangles mark the mean lengths of the age-groups. Data from Macdonald (1987).
An efficient quasi-Newton algorithm gives fast results. MIX 3.1 features improved numerical methods and is more reliable and less sensitive to the choice of initial values than were earlier releases.
MIX is available for Apple Macintosh and DOS microcomputers. A version for Windows may be released in at some future date.
The DOS version requires at least 640K RAM and one disk drive. A floating-point coprocessor is highly recommended but not required. Anyone purchasing the DOS version of MIX 3.1 will receive a free upgrade to MIX for Windows as soon as it is available.
The Windows 3.1/95/NT version requires a Pentium processor, or an Intel 386 or 486 processor with floating-point coprocessor installed.
The Macintosh version is supplied as a "fat" binary, accelerated for Power Macintosh. It will also run on a 68020, 68030 or 68040 processor with FPU, 2 Mb RAM and System 6.x.x or higher. MIX will not run on a Macintosh with 68LC040 processor.
All versions of MIX 3.1 feature high-resolution screen graphics, giving the fitting process a strong visual orientation. The DOS version of MIX includes drivers for most popular dot-matrix and laser printers. Graphics from the Macintosh version can be saved as SimpleText PICT files which can be edited and printed from any graphics software. The Windows version can print graphics directly or save them in .BMP or .WMF format.
MIX 3.1 allows subsampling data to be incorporated in the analysis along with the mixed data. In fisheries length-frequency applications, for example, some fish from specified length classes can be aged, and the length-specific age distributions analysed simultaneously with the mixed length-frequency distribution. This means you can get more information where it is most needed, at the lengths where the age-groups overlap the most.
MIX 3.1 will compute the optimal subsampling design for a given mixed sample and subsample size. In fisheries length-frequency applications, this will give the recommended number to subsample from each length class to get the most information from the least amount of age determination. Figure 2 shows the optimal allocation of subsampling for the data of Figure 1, if 10% of the fish are to be aged.
Figure 2. Optimal allocation for a 10% subsample of the data from Figure 1.
Figure 3 shows a mixture of 6 gamma components fitted to Atlantic cod length-frequency data that would have been impossible to fit without the help of subsampling data.
Figure 3. A highly overlapped mixture, fitted with the help of subsampling data. The mixed sample included 287 fish from age groups 3 to 8; up to 3 fish were randomly selected from each length class for age determination, 55 in total. The means were constrained to lie on a von Bertalanffy growth curve. Data from Stephen Smith, Dept. of Fisheries and Oceans, Canada.
Click here to see how the Atlantic cod data were fitted.
The MIX 3.1 User's Guide is a valuable reference on the analysis of mixture distributions and length-frequency distributions in particular. The demonstration examples have been chosen to illustrate the range of application of MIX 3.1 and serve as a tutorial for MIX 3.1. Some examples, such as the scale mixture of normal distributions in Figure 4, are very easy to fit; others, like a mixture of exponential distributions, can be difficult.
MIX is special-purpose software intended to solve problems that are inherently difficult. If you have any problems running the demonstration examples, or analysing your own data, you may contact Peter Macdonald (firstname.lastname@example.org) at Ichthus Data Systems for free technical support. If you send us your data file and a copy of the input/output log for a session, either on disk or by e-mail, we will do our best to find a solution.
Figure 4. A scale mixture of three normal distributions (equal means, different standard deviations) fitted by MIX 3.1.