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| 2. |
- Jonsson, Jakob
(författare)
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On the number of Euler trails in directed graphs
- 2002
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Ingår i: Mathematica Scandinavica. - 0025-5521 .- 1903-1807. ; 90:2, s. 191-214
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Tidskriftsartikel (refereegranskat)abstract
- Let G be an Eulerian digraph with all in- and out-degrees equal to 2, and let pi be an Euler trail in G. We consider an intersection matrix L(pi) with the property that the determinant of L(pi) + I is equal to the number of Euler trails in G; I denotes the identity matrix. We show that if the inverse of L(pi) exists, then L-1 (pi) = L(sigma) for a certain Euler trail sigma in G. Furthermore, we use properties of the intersection matrix to prove some results about how to divide the set of Euler trails in a digraph into smaller sets of the same size.
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| 3. |
- Jonsson, Jakob
(författare)
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On the topology of simplicial complexes related to 3-connected and Hamiltonian graphs
- 2003
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 104:1, s. 169-199
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Tidskriftsartikel (refereegranskat)abstract
- Using techniques from Robin Forman's discrete Morse theory, we obtain information about the homology and homotopy type of some graph complexes. Specifically, we prove that the simplicial complex Delta(n)(3) of not 3-connected graphs on it vertices is homotopy equivalent to a wedge of (n - 3) (.) (n - 2)!/2 spheres of dimension 2n - 4, thereby verifying a conjecture by Babson, Bjorner, Linusson, Shareshian, and Welker. We also determine a basis for the corresponding nonzero homology group in the CW complex of 3-connected graphs. In addition, we show that the complex Gamma(n) of non-Hamiltonian graphs on it vertices is homotopy equivalent to a wedge of two complexes, one of the complexes being the complex Delta(n)(2) of not 2-connected graphs on it vertices. The homotopy type of Delta(n)(2) has been determined, independently, by the five authors listed above and by Turchin. While Gamma(n) and Delta(n)(2) are homotopy equivalent for small values on it, they are nonequivalent for n = 10.
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| 4. |
- Lindbom, Lars, et al.
(författare)
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Perl-speaks-NONMEM (PsN) – a Perl module for NONMEM related programming
- 2004
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Ingår i: Computer Methods and Programs in Biomedicine. - : Elsevier BV. - 0169-2607 .- 1872-7565. ; 75:2, s. 85-94
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Tidskriftsartikel (refereegranskat)abstract
- The NONMEM program is the most widely used nonlinear regression software in population pharmacokinetic/pharmacodynamic (PK/PD) analyses. In this article we describe a programming library, Perl-speaks-NONMEM (PsN), intended for programmers that aim at using the computational capability of NONMEM in external applications. The library is object oriented and written in the programming language Perl. The classes of the library are built around NONMEM's data, model and output files. The specification of the NONMEM model is easily set or changed through the model and data file classes while the output from a model fit is accessed through the output file class. The classes have methods that help the programmer perform common repetitive tasks, e.g. summarising the output from a NONMEM run, setting the initial estimates of a model based on a previous run or truncating values over a certain threshold in the data file. PsN creates a basis for the development of high-level software using NONMEM as the regression tool.
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| 5. |
- Ribbing, Jakob, et al.
(författare)
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Power, Selection Bias and Predictive Performance of the Population Pharmacokinetic Covariate Model
- 2004
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Ingår i: Journal of Pharmacokinetics and Pharmacodynamics. - 1567-567X .- 1573-8744. ; 31:2, s. 109-134
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Tidskriftsartikel (refereegranskat)abstract
- Identification and quantification of covariate relations is often an important part of population pharmacokinetic/pharmacodynamic (PK/PD) modelling. The covariate model is regularly built in a stepwise manner. With such methods, selection bias may be a problem if only statistically significant covariates are accepted into the model. Competition between multiple covariates may further increase selection bias, especially when there is a moderate to high correlation between the covariates. This can also result in a loss of power to find the true covariates. The aim of this simulation study was to investigate the effect on power, selection bias and predictive performance of the covariate model, when altering study design and system-related quantities. Data sets with 20-1000 subjects were investigated. Five covariates were created by sampling from a multivariate standard normal distribution. The true covariate was set up to have no, low, moderate and high correlation to the other four covariates, respectively. Data sets, in which each individual had two or three PK observations, were simulated using a one-compartment i.v. bolus model. The true covariate influenced clearance according to one of several magnitudes. Different magnitudes of residual error and inter-individual variability in the structural model parameters were also introduced to the simulation model. A total of 7400 replicate data sets were simulated independently for each combination of the above conditions. Models with one of the five simulated covariates influencing clearance and the model without any covariate were fitted to the data. The probability of selecting (according to a pre-specified P-value) the different covariates, along with the estimated covariate coefficient, was recorded. The results show that selection bias is very high for small data sets (< or = 50 subjects) simulated with a weak covariate effect. If selected under these circumstances, the covariate coefficient is on average estimated to be more than twice its true value, making the covariate model useless for predictive purposes. Surprisingly, even though competition from false covariates caused substantial loss in the power of selecting the true covariate, the already high selection bias increased only marginally. This means that the bias due to competition is negligible if statistical significance is also required for covariate selection. Bias and predictive performance are direct functions of power, only indirectly affected by study design and system-related quantities. Mainly because of selection bias, low-powered covariates can be expected to harm the predictive performance when selected. For the same reason these low-powered covariates may falsely appear to be clinically relevant when selected. If the aim of an analysis is predictive modelling, we do not recommend stepwise selection or significance testing of covariates to be performed on small or moderately sized data sets (<50-100 subjects).
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