The GDMS offers three different types of Time Domain Analysis; VCV Least Squares Regression modelling, VCV Least Squares Regression modelling with automated re-estimation and Random Walk Least Squares. VCV regression modelling allows a linear model to be fitted to the data in a fast and precise manner, enabling any general data trends to be seen. It also allows the user to detect and remove any erroneous data, thus allowing the most optimal model to be calculated. In addition, an automated re-estimation model offers the capability of automatically detecting and removing erroneous data points, thereby computing the line of best fit that most accurately models the data. The third method of Time Domain Analysis is the Random Walk Least Squares model, which allows a weighting matrix to be specified that detects this underlying trend that may be occurring and affecting the accuracy of the data. Time Domain Analysis is implemented as a separate class inside the GDMS called cepLs and all calculations are encapsulated therein. A given solution is essentially reached by solving the Least Squares equation equation 2-7 and a set of residuals are calculated by equation 2-13. Each calculation produces three matrices, X which is the solution to the Least Squares matrix equation and contains the unknown co-efficients m and c of the line equation (see equation 2-1). The second matrix indicates whether or not the given data point was weighted out of the calculations and finally, the matrix of residuals is also produced. The first method of Time Domain Analysis offered by the GDMS is known as VCV Least Squares Regression modelling. This method allows the user to specify a VCV weighting matrix to be used in the calculations, thus giving the user the maximum amount of flexibility in determining to what degree a given point in the data set will influence the line of best fit. Another method of VCV analysis offered is that of automatic re-weighting. In this type of analysis, the most accurate regression model is calculated by the process of automated re-weighing using the residual space which is discussed in depth in chapter 2, Time Domain analysis. The implementation of this process can also be seen in the following diagram

eqimgTD-ReweightFlow.gif

One important implementation problem that was discovered when applying this method was that in some cases the solution did not converge, and thus the program was remained in an infinite loop because the exit condition was never met. Another problem occurred where it appeared that the re-weighting process continued until every point in the data set had been weighted out. Both of these problems stemmed from the fact that in our implementation of this re-weighting algorithm, for over all design reasons, no point from data was ever entirely removed from the system but set it to 0 inside the weighting matrix, which achieved the same effect in the data domain. In the residual domain, however, the weighted out points were still being included due to the fact that the calculation of residuals does not require the use of a weighting matrix (see equation 2-13). The solution to this problem was to pre-multiply the residuals with the weighting matrix and removing any residual equal to zero before calculating the thresholds for determining whether a data point was to be weighted out or not. The third method of Time Domain Analysis offered by the GDMS is the calculation of Random Walk Least Squares solutions. This method differers from the first two, in that a Random Walk weighting matrix is not a strictly diagonal matrix. At this time there is limited support for Random Walk Time Domain Analysis and although the GDMS is capable of producing a solution, the user must specify the Random Walk weighting matrix to be used. This is largely due to the fact that at this time there is still no general consensus as to how a Random Walk matrix is to be produced. As one of the goals of GDMS speed of operations, great care has been taken to optimise the calculation speed of solutions in the Time Domain. This includes employing the use of passing values by reference to and from functions where ever possible, thus saving the time that it takes to replicate the parameters being passed in memory. The main optimisation, however, takes place in the area of matrix multiplication. It can be seen from equations 2-7 and 2-13 that this is the most often used operation in calculating a Least Squares solution and is by far the slowest of all the matrix operations. This is due to the fact that in a normal matrix multiplication if, for example, we were multiplying matrix A by matrix B where A is m x n and B is n x o it would take m x n x o additions plus m x n x o multiplications to reach as solution. Conversely, if certain properties are known about the matrices that are being used, the number of computations can be greatly reduced. For instance, in VCV Time Domain analysis we know that the weighting matrix P is always strictly diagonal, that is, every value of Pij = 0 where i ≠ j. This means that when we calculate ATP, for instance, where A is n x 2 and P is n x n we need only calculate AijPii for each element in the resulting matrix. Thus the problem has been reduced from being 2 x n2 additions and 2 x n2 multiplications to just 2 x n multiplications. Similar calculation reduction can also be achieved by using the properties of the design, or A matrix. By using the property that the second row the the design matrix is always one, and the fact that anything multiplied by one is itself, we can further reduce the the number of calculations made in our previous example to just n multiplications. To put this into perspective, if we were carry out this matrix multiplication on an Intel Pentium 4 desktop processor at 2.0 GHz where each double precision multiplication operation takes 8 instruction cycles and each double precision addition takes 6 instruction cycles(http://developer.intel.com/design/pentium4/manuals/248966.htm, 2002). If we assume that matrix A is a 1000 x 2 matrix and P is 1000 x 1000, it would take 2 x 10002 multiplications 2 x 10002 additions and 14 ms to complete this operation. Conversely, by using the optimised method the number of operations required to compute this matrix equation reduces to 1000 multiplications or 50 &mgr;s. The GDMS offerers much functionality in the way of Time Domain Analysis, especially when VCV weighting matrices are employed. It does currently, however, only have limited support of Random Walk Time Domain Analysis. It is hoped that once an agreement has been reached as to how to specify a Random Walk Matrix it will be fully integrated into this product.