Fig 1.
Graphical illustration of sampling from a simplex-truncated bivariate normal distribution using the LIN-ESS algorithm.
Coordinates are transformed from (x1, x2) (thin black dashed lines) to (y1, y2) (thin black solid lines) so that the origin is located at the mean μ of the distribution. With the previous sample’s location indicated by yt, an ellipse y*(θ), 0 ≤ θ < 2π (shown in red) is obtained from yt and sampling a vector . Values of θ on arcs of the ellipse that fall within the distribution’s domain (thick red arcs) are identified, and one value θ* is randomly sampled from a uniform distribution on the range of θ values that fall within the domain. The resulting location y*(θ*) becomes the next sample yt+1 from the distribution. Notice that this sampling procedure has a 100% acceptance rate. Figure adapted from [13].
Fig 2.
Application of the inclusion-exclusion principle to express the target region enclosed by the non-negative space under a simplex in two dimensions (x1 ≥ 0, x2 ≥ 0, x1 + x2 ≤ 1) in terms of domains that are truncated by no more than two constraints.
Figure adapted from [12].
Table 1.
Transformation matrices Tv, and constraint vectors cv and dv, that define the truncation cv < Tvx < dv for six relevant regions shown in Fig 2.
Dummy inequalities −∞ < xi < ∞ were introduced into the transformation matrices T1, T2 and T3 to ensure that these matrices have rank equal to their dimension. The order of rows in T23, c23 and d23 has been switched so that this matrix and these vectors possess the form that would be obtained from a generalised two-step algorithm for ST-MNDs of any dimension described later in the “Materials and methods” section.
Fig 3.
Comparison of estimates of Z, and estimates of the elements of and
, obtained from implementation of the three methods presented in this paper, for 100 different simplex-truncated bivariate normal distributions
.
Fig 4.
Comparison of estimates of Z, and estimates of the elements of and
, obtained from implementation of the three methods presented in this paper, for 100 different simplex-truncated trivariate normal distributions
.
Fig 5.
Comparison of estimates of Z, and estimates of the elements of and
, obtained from implementation of the three methods presented in this paper, for 100 different simplex-truncated multivariate normal distributions
of dimension n = 4.
Notice that the semi-analytical method (plots in the right column) is starting to become inaccurate.
Fig 6.
Comparison of estimates of Z, and estimates of the elements of and
, obtained from implementation of the three methods presented in this paper, for 100 different simplex-truncated multivariate normal distributions
of dimension n = 5.
Notice that the semi-analytical method (plots in the central column) becomes rather inaccurate, but increasing the accuracy of the integrals of hyperrectangularly-truncated MNDs calculated within this method vastly improves the estimates overall (plots in the right-most column). However, for one of the distributions tested, the covariance matrix was very poorly estimated by the semi-analytical method with increased accuracy (bottom-right plot, results not shown), with the values of some covariance matrix elements predicted to be more than ten orders of magnitude higher than their values predicted by the other methods.
Fig 7.
Comparison of estimates of Z, and estimates of the elements of and
, obtained from implementation of the two sampling methods presented in this paper, for 100 different simplex-truncated multivariate normal distributions
of dimension n = 6 (plots in the left column), n = 7 (plots in the central column), and n = 7 for a higher thinning ratio used within the Gessner et al. method (plots in the right column).
Notice that increasing the thinning ratio in the Gessner et al. method improves the match of the two sampling methods (compare plots in the central and right columns).
Fig 8.
Comparison of computation times (dots and error bars indicating the median and 68% central credible interval of 100 different values, respectively), for the present implementation of the three methods for calculating the integral, mean and covariance of simplex-truncated multivariate normal distributions.
Notice that the semi-analytical method was the fastest for distributions of low dimension n ≤ 3. As the dimension of the distribution increases, the Gessner et al. method becomes increasingly recommended because of its high efficiency, the latter of which is due primarily to this method’s ability to sample without rejection. For the naive rejection sampling and semi-analytical methods, results are only shown for distributions of dimension n ≤ 5 and n ≤ 7 respectively, as the computational times of both of these methods are excessive at higher dimensions.
Table 2.
Summary of the advantages and disadvantages of the three methods presented in this manuscript, for calculations on the simplex-truncated multivariate normal distribution.