UPPSALA DISSERTATIONS IN MATHEMATICS 113
Optimal adaptive designs and adaptive randomization techniques for clinical trials
Yevgen Ryeznik
Department of Mathematics Uppsala University UPPSALA 2019 Dissertation presented at Uppsala University to be publicly examined in Ång 4101, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 26 April 2019 at 09:00 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Associate Professor Franz König (Center for Medical Statistics, Informatics and Intelligent Systems, Medical University of Vienna).
Abstract Ryeznik, Y. 2019. Optimal adaptive designs and adaptive randomization techniques for clinical trials. Uppsala Dissertations in Mathematics 113. 94 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2748-0.
In this Ph.D. thesis, we investigate how to optimize the design of clinical trials by constructing optimal adaptive designs, and how to implement the design by adaptive randomization. The results of the thesis are summarized by four research papers preceded by three chapters: an introduction, a short summary of the results obtained, and possible topics for future work. In Paper I, we investigate the structure of a D-optimal design for dose-finding studies with censored time-to-event outcomes. We show that the D-optimal design can be much more efficient than uniform allocation design for the parameter estimation. The D-optimal design obtained depends on true parameters of the dose-response model, so it is a locally D-optimal design. We construct two-stage and multi-stage adaptive designs as approximations of the D-optimal design when prior information about model parameters is not available. Adaptive designs provide very good approximations to the locally D-optimal design, and can potentially reduce total sample size in a study with a pre-specified stopping criterion. In Paper II, we investigate statistical properties of several restricted randomization procedures which target unequal allocation proportions in a multi-arm trial. We compare procedures in terms of their operational characteristics such as balance, randomness, type I error/ power, and allocation ratio preserving (ARP) property. We conclude that there is no single “best” randomization procedure for all the target allocation proportions, but the choice of randomization can be done through computer-intensive simulations for a particular target allocation. In Paper III, we combine the results from the papers I and II to implement optimal designs in practice when the sample size is small. The simulation study done in the paper shows that the choice of randomization procedure has an impact on the quality of dose-response estimation. An adaptive design with a small cohort size should be implemented with a procedure that ensures a “well-balanced” allocation according to the D-optimal design at each stage. In Paper IV, we obtain an optimal design for a comparative study with unequal treatment costs and investigate its properties. We demonstrate that unequal allocation may decrease the total study cost while having the same power as traditional equal allocation. However, a larger sample size may be required. We suggest a strategy on how to choose a suitable randomization procedure which provides a good trade-off between balance and randomness to implement optimal allocation. If there is a strong linear trend in observations, then the ARP property is important to maintain the type I error and power at a certain level. Otherwise, a randomization- based inference can be a good alternative for non-ARP procedures.
Keywords: optimal designs, optimal adaptive designs, randomization in clinical trials, restricted randomization, adaptive randomization, unequal cost, allocation ratio preserving randomization procedures, heterogeneous costs, multi-arm clinical trials, randomization-based inference, unequal allocation
Yevgen Ryeznik, Department of Mathematics, Box 480, Uppsala University, SE-75106 Uppsala, Sweden.
© Yevgen Ryeznik 2019
ISSN 1401-2049 ISBN 978-91-506-2748-0 urn:nbn:se:uu:diva-377552 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-377552) 'HGLFDWHG WR P\ IDPLO\ DQG P\ WHDFKHUV
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Decreasing hazard (b=1.5) (β 0 ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● =1.9, 1.0 ●●● ●●●●●●●●●●● ●●●● ● ●●●●●●●● ●●●●●●●●●● ●●●●●● ●●● ●●●●●●●● ●●●●●●●● ●●●●●●●● ●●●● ●●●●●● ●●●●●● ●●●● ●●● ●●●● ●●● ●●● ● ●● ●● ●● Shape I ●● ● ● β ●●● ●●● ●● 1 ●● =0.6, 0.5 ● ●● ●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●● ●●●● ●●●● ●●●●● ● ● ●●●● ●●● ●● β ●●● 2 =2.8) 0.0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● (β 0 =3.4, 1.0 ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●●●●●●● ●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●● ● ●●●●● ●●●●●● ●●●●● ●●●●●● ●●●●● ● ● Shape II ● ● ● β ●● ●●●● ●●●● ●● ●●● ● 1 ● = − 0.5 ●●●●●●●●●●●●●●●●● 7.6, ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ● β ●● ● 2 ● ● ● =9.4) ●● ●●●● ●●●● ●●● 0.0 ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● (β 0 =3.5, 1.0 ●● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● Design Points Shape III β 1 =4.7, 0.5 ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ● ●●● ●●●● ●●●● β ●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●● ●●●●●●●●● ●● 2 ●●●●●●●●●●●●● = − 0.0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 3.1) (β 0 =3.1, 1.0 ●● ● ● ● ● ● ● ●●● ● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● Shape IV β 1 =4.2, 0.5 ●●● ● ● ●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●● ● ●● ●● ●● ●● β ●●●●● ●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●● ●●●●●● ●●●●●●● ●●● 2 ●●●●●●●●● = − 0.0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 2.1) 110 30 50 70 90 110 130 150 1103050709011013015011030507090110130150110 30 50 70 90 110 130 150 1103050709011013015011030507090110130150 Censoring Time )LJXUH 'RSWLPDO GHVLJQV IRU WKH FRQVLGHUHG GRVHUHVSRQVH UHODWLRQVKLSV ZLWK GLIIHUHQW YDOXHV RI WKH FHQVRULQJ WLPH GHFUHDVHV WKH 'RSWLPDO GHVLJQ VWUXFWXUH DSSURDFKHV WR WKH XQLIRUP DOORFDWLRQ GHVLJQ )RU HYHU\ JLYHQ VKDSH RI WKH GRVHUHVSRQVH 6KDSHV , ,, ,,, DQG ,9 WKH 'RSWLPDO GHVLJQ LV PRVW VKLIWHG IURP WKH XQLIRUP GHVLJQ ZKHQ WKH KD]DUG LV LQFUHDVLQJ 0 2EYLRXVO\ 0 &RPSDULVRQ RI )L[HG DQG $GDSWLYH 'HVLJQV $ VLPXODWLRQ VWXG\ WR FRPSDUH WZR QRQDGDSWLYH GHVLJQV XQLIRUP DQG ORFDOO\ 'RSWLPDO DQG D WZRVWDJH DGDSWLYH 'RSWLPDO GHVLJQ XVLQJ 0/( XSGDWLQJ KDV EHHQ SHUIRUPHG 7ZR VFHQDULRV IRU WKH VDPSOH VL]H KDYH EHHQ FRQVLGHUHG GHVLJQV ZLWK D IL[HG VDPSOH VL]H DQG GHVLJQV ZLWK DQ HDUO\ VWRSSLQJ FULWHULRQ 'HVLJQV ZLWK D )L[HG 6DPSOH 6L]H 7KUHH GHVLJQV VLQJOHVWDJH XQLIRUP VLQJOHVWDJH ORFDOO\ 'RSWLPDO DQG WZR VWDJH DGDSWLYH 'RSWLPDO ZHUH LQYHVWLJDWHG XQGHU VHYHUDO H[SHULPHQWDO VFH QDULRV XVLQJ VLPXODWLRQ UXQV IRU HDFK GHVLJQVFHQDULR FRPELQDWLRQ 7KH Increasing hazard (b=0.4) Increasing hazard (b=0.57721) Increasing hazard (b=0.65) Increasing hazard (b=0.8) Constant hazard (b=1) Decreasing hazard (b=1.5) (β 0 1.0 =1.9, 0.8 Shape I β 1 0.6 =0.6, 0.4 β 2 =2.8) 0.2 0.0 ( β 0 1.0 =3.4, 0.8 Shape II β 1 0.6 =− 7.6, 0.4 β 2 0.2 =9.4) 0.0 ( β 0 1.0 =3.5, 0.8 Shape III β 1 0.6 =4.7, 0.4 β 2 =− 0.2 3.1) 0.0 (β 0 1.0 =3.1, 0.8 Shape IV β 1 0.6 =4.2, 0.4 β 2 =− 0.2 2.1) 0.0 110 30 50 70 90 110 130 150 110 30 50 70 90 110 130 150 110 30 50 70 90 110 130 150 110 30 50 70 90 110 130 150 1103050709011013015011030507090110130150 Censoring Time Rel. Efficiency (Uniform vs. D−optimal) Prob. of Event (D−optimal) Prob. of Event (Uniform) ∗ )LJXUH 'HIILFLHQF\ RI D WKUHHSRLQW XQLIRUP GHVLJQ ξU UHODWLYH WR WKH 'RSWLPDO GHVLJQ ξD Shape I (β0=1.9, β1=0.6, β2=2.8) 2.0 1.5 1.0 0.5 0.0 Shape II (β0=3.4, β1=− 7.6, β2=9.4) 2.0 1.5 [0,0.1) 1.0 [0.1,0.2) 0.5 [0.2,0.3) [0.3,0.4) 0.0 [0.4,0.5) Shape III (β0=3.5, β1=4.7, β2=− 3.1) [0.5,0.6) 2.0 [0.6,0.7) Parameter b Parameter [0.7,0.8) 1.5 [0.8,0.9) 1.0 [0.9,1) NA 0.5 0.0 Shape IV (β0=3.1, β1=4.2, β2=− 2.1) 2.0 1.5 1.0 0.5 0.0 11030507090110130150 Censoring Time ∗ )LJXUH 'HIILFLHQF\ RI D WKUHHSRLQW XQLIRUP GHVLJQ ξU UHODWLYH WR WKH 'RSWLPDO GHVLJQ ξD VFHQDULRV LQFOXGHG FKRLFHV RI GRVH UHVSRQVH UHODWLRQVKLSV )LJ GLI IHUHQW FKRLFHV RI WKH WRWDO VDPSOH VL]H n = 150; 300; 450 GLIIHUHQW DPRXQWV RI FHQVRULQJ LQ WKH VWXG\ DQG IRU WKH DGDSWLYH GHVLJQ GLIIHUHQW LQLWLDO FRKRUW VL]HV )RU FHQVRULQJ ZH FRQVLGHUHG W\SH , FHQVRULQJ VFKHPHV ZLWK SDUDPH WHU τ VHOHFWHG LQ VXFK D ZD\ WKDW WKH WRWDO DYHUDJH 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∼ ZKHQ n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●● ●● ●● ● ●● ●● ●● ●● ●● ● ●● ●● ● ●● ●● ● ● ●● ●● 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● BUD 3.0 BUD (2) MaxEnt (0.05) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● 2.5 ●●●●●●●●●●● ●●●● BUD (3) MaxEnt (0.1) 2.0 ●● ●● ● 1.5 ● BUD (4) MaxEnt (0.25) 1.0 ● BUD (5) MaxEnt (0.5) MWUD CRD MaxEnt (1) 1.6 DBCD (1) MWUD (2) ● ●●●●● ●●●●●● ●●●●● ●●●●● ●●●●● ●●●●● ●●●●●● ●●●●●● ●●●●● ●●●●●● ●●●●● ●●●●● ●●●●●● ●●●●●● ●●●●● ●●●●●● ●●●●●● ●●●●●● ●●●●●● ● ● 1.2 ●●● ●●● ● ●●● ●●● ● ●●● ● ●●● ● ●●● ● ●●● ●●● ●●● ● ●●● ●●● ● ●●● ● ●●● ●●● ●●● ● ●●● ●●● ●●● ●●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● DBCD (10) MWUD (4) 0.8 ● DBCD (2) MWUD (6) DBCD (4) MWUD (8) DL DBCD (5) PBD (1) 5 DL (2) ● PBD (2) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 4 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●● DL (4) PBD (3) ●●●●●●●●●●● 3 ●●●●●● ●●●● ●●●● DL (6) PBD (4) 2 ●●●● 1 ●● DL (8) PBD (5) Momentum of probability mass Momentum of probability DBCD 6 4 ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● 2 ●● ●●●●●●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● MaxEnt 8 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 6 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● 4 ●●●●●●●●●●●● ●●●●●●●●● ●●●●●● 2 ●●●● 0 0 50 100 150 200 Sample size )LJXUH 0RPHQWXP RI SUREDELOLW\ PDVV YV DOORFDWLRQ VWHS IRU WKH DOORFDWLRQ CRD 0.50 0.25 0.00 −0.25 −0.50 PBD 0.15 ●●● ●●●●● ●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0.10 ●●●● ● ●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●● ● ●●●●●●●●●●●● ●● 0.05 ●●● ●●●● 0.00 ●●●●●●●●● BUD BUD (2) MaxEnt (0.05) 0.03 ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● BUD (3) MaxEnt (0.1) 0.02 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●● 0.01 ●●●● BUD (4) MaxEnt (0.25) ●●●●● 0.00 ●●●● BUD (5) MaxEnt (0.5) MWUD CRD MaxEnt (1) 0.15 DBCD (1) MWUD (2) 0.10 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● DBCD (10) MWUD (4) ●●●●●●● ●● 0.05 ●● ● DBCD (2) MWUD (6) 0.00 ● DBCD (4) MWUD (8) DL DBCD (5) PBD (1) ● ● ● ●● DL (2) PBD (2) 0.03 ● ●● ● ●●● ● ● ●●●●● DL (4) PBD (3) 0.02 ●●●●●●●●●●●● ●●●●●●●●● Average forcing index forcing Average ● ●●●●●●●●● ●●●●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0.01 ● DL (6) PBD (4) ● 0.00 ● DL (8) PBD (5) DBCD ●●●●●●● ●●●●●●● ●●●●●●●●●●●● ●● ●●●●●●●●●● ● ●●●●●●●●●●●● ● ●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●● 0.10 ●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0.05 ● ●●●●●●● ●● 0.00 ●●●●● MaxEnt 0.6 0.4 0.2 0.0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0 50 100 150 200 Sample size )LJXUH $YHUDJH IRUFLQJ LQGH[ YV DOORFDWLRQ VWHS IRU WKH DOORFDWLRQ DV EHVW VRPH GHVLJQV H J %8' 0:8' DQG '%&' H[KLELW JRRG EDO DQFHUDQGRPQHVV WUDGHRII DFURVV FRQVLGHUHG VFHQDULRV )LJ LV D KHDW PDS SORW RI WKH ³RYHUDOO´ EDODQFHUDQGRPQHVV SHUIRU PDQFH LQGH[ FRPSXWHG XVLQJ IRUPXOD IRU D WULDO ZLWK D UDQJH RI VDPSOH VL]HV XS WR m = 200 ,Q WKLV ILJXUH WKH xD[LV VKRZV WKH DOORFDWLRQ VWHS DQG WKH yD[LV GLVSOD\V WKH GHVLJQV 7KH HQWULHV DUH FRORUHG DFFRUGLQJ WR WKH PDJQLWXGH RI WKH YDOXH RI ³RYHUDOO´ SHUIRUPDQFH LQGH[ IRU D JLYHQ GHVLJQ DW D JLYHQ DOORFDWLRQ VWHS 7KH UHG FRORU UHSUHVHQWV WKH ORZHVW YDOXH RI WKH LQGH[ LH PRVW GHVLUDEOH SHUIRUPDQFH 2QH FDQ VHH KRZ WKH GHVLJQV SHUIRUP RYHU WLPH ² DV VDPSOH VL]H LQFUHDVHV WKH SHUIRUPDQFH RI PRVW GHVLJQV LPSURYHV 7KH 0:8' ZLWK α =4, 6, RU 8 DQG WKH %8' ZLWK λ =2KDYH RYHUDOO EHVW SHUIRUPDQFH 6LPLODU KHDW PDS SORWV IRU RWKHU DOORFDWLRQ WDUJHWV DQG KDYH EHHQ REWDLQHG DV ZHOO $ FORVHU ORRN DW WKH KHDW PDS SORWV KDV VKRZQ UDQNLQJ RI WKH GHVLJQV LQ WHUPV RI WKHLU ³RYHUDOO´ EDODQFHUDQGRPQHVV SHUIRUPDQFH LQGH[ IRU D WULDO ZLWK VDPSOH VL]H m = 200 7KH YDOXHV RI WKH LQGH[ DUH DOZD\V EHWZHHQ DQG %RWK WKH &5' DQG 0D[(QW η =1 ZHUH DOZD\V DW WKH ERWWRP RI WKH UDQNLQJ 0:8' α =6 DQG 0:8' α =8 KDG RYHUDOO WKH EHVW SHUIRUPDQFH WKH\ ZHUH DOZD\V DPRQJ WRS GHVLJQV 7KH %8' ZLWK λ =2, 3, 4, RU 5 ZHUH DPRQJ WRS GHVLJQV IRU DQG DOORFDWLRQV EXW QRW LQ WKH FDVH :LWKLQ WKH FODVV RI '%&' D GHVLJQ ZLWK γ =10KDG WKH EHVW SHUIRUPDQFH DFURVV DOO DOORFDWLRQ VFHQDULRV LW ZDV DPRQJ WRS GHVLJQV LQ WKH FDVH :LWKLQ WKH FODVV RI '/8' D GHVLJQ ZLWK a =2ZDV EHVW DFURVV DOO DOORFDWLRQ VFHQDULRV KRZHYHU LW ZDV QHYHU LQ WRS 9DULDELOLW\ RI WKH $OORFDWLRQ 3URSRUWLRQV )LJ VKRZV ER[SORWV RI VLPXODWHG GLVWULEXWLRQV RI WUHDWPHQW DOORFDWLRQ SURSRUWLRQV IRU WKH GHVLJQV IRU WKH VFHQDULR ZKHQ WKH WDUJHW DOORFDWLRQLV DQG WKH VDPSOH VL]H m = 200 $OO ER[SORWV DUH V\PPHWULF DQG FHQWHUHG DW WKH FRUUHVSRQGLQJ WDUJHW DOOR FDWLRQ 7KH &5' LV PRVW YDULDEOH DQG 0D[(QW η =1 DV ZHOO DV WKH 3%'V ZKLFK DFKLHYH SHUIHFW EDODQFH DIWHU VXEMHFWV KDYH ]HUR YDULDELOLW\ 2WKHU GHVLJQV KDYH YDULDELOLW\ EHWZHHQ WKH &5' DQG 0D[(QW η =1 7KH PRUH UDQGRP D GHVLJQ LV WKH JUHDWHU LV LWV YDULDELOLW\ DQG PRUH OLNHO\ DUH GHYLDWLRQV IURP WKH WDUJHW DOORFDWLRQ 6LPLODU SDWWHUQV DUH REVHUYHG IRU WKH DOORFDWLRQ WDU JHWV DQG sample size = 25 sample size = 50 sample size = 100 sample size = 200 w =1:1:1:1 0.2 0.1 BUD ● CRD ● ● ● ● 0.0 DBCD DL 0.4 w =2:1:1:2 MaxEnt MWUD 0.3 PBD 0.2 0.1 ● BUD (2) ● MaxEnt (0.05) ● BUD (3) ● MaxEnt (0.1) 0.0 ● ● ● ● ● BUD (4) ● MaxEnt (0.25) ● BUD (5) ● MaxEnt (0.5) w =4:3:2:1 0.6 ● CRD ● MaxEnt (1) ● DBCD (1) ● MWUD (2) 0.4 ● DBCD (10) ● MWUD (4) ● DBCD (2) ● MWUD (6) Average forcing index forcing Average 0.2 ● DBCD (4) ● MWUD (8) ● DBCD (5) ● PBD (1) 0.0 ● ● ● ● ● DL (2) ● PBD (2) 0.5 w =37:21:21:21 ● DL (4) ● PBD (3) ● ● 0.4 DL (6) PBD (4) ● DL (8) ● PBD (5) 0.3 0.2 0.1 0.0 ● ● ● ● 0.02 0.04 0.06 0.08 0.10 0.02 0.04 0.06 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 Cumulative momentum of probability mass )LJXUH $YHUDJH FXPXODWLYH PRPHQWXP RI SUREDELOLW\ PDVV xD[LV YV DYHUDJH IRUFLQJ LQGH[ yD[LV PBD (5) PBD (4) PBD (3) PBD (2) PBD (1) MWUD (8) MWUD (6) MWUD (4) MWUD (2) MaxEnt (1) MaxEnt (0.5) MaxEnt (0.25) MaxEnt (0.1) 0.6 MaxEnt (0.05) DL (8) 0.4 Design DL (6) 0.2 DL (4) DL (2) DBCD (5) DBCD (4) DBCD (2) DBCD (10) DBCD (1) CRD BUD (5) BUD (4) BUD (3) BUD (2) 1 25 50 75 100 125 150 175 200 Sample size )LJXUH +HDW PDS SORW RI WKH ³RYHUDOO´ EDODQFHUDQGRPQHVV SHUIRUPDQFH LQGH[ * IRU WKH DOORFDWLRQ ● ● ● ●●● ●●● ● ● ●●● ● ● ● ● ● ●●●●●● ●●● ●●●●●● ● ● ● ● ●●● ●●●●●● ● ● ●●●●●●●●●● ●●●●●● ●●● ● ●●● ●●●●●●●●●● ●●●●●●●● ●●● ● ●●●●●●●●●●●● ●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●● ●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●● ● 0.4 ● ●●●●●● ● ●●●●●●●●●● ●●●●●● ●●●●●● ● ●●● ●●●●●●●●●●●●●●● ●●●●●● ●●●●●● ●●●●●● ●●●●●● ●●● ● ●●● ●●● ● ●●● ● ● ● ●●● ● ● ●●● ● ●●● ● ● ●●● ● ρ 0.3 ● ● 1 0.2 0.1 ● ● ● ●●● ● ●●●●●● ●●● ● 0.4 ● ● ● ●●●●●●●●●● ●●● ● ●●●●●●●●●●●●●●● ●●● ●●● ●●● ●●●●●● ● ● ●●● ●●● ● ● ● ●●●●●● ●●●●●● ●●● ●●● ● ●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●● ●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ρ 0.3 ● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●● ● ● ●●●● ● ● ● ●●● ●●●●●● ●●●●●● ●●● ●●● ● ● ● ● ●●● ●●●●●● 2 ● ● ●●● ● ●●●●●●●●●● ● ● ● 0.2 ● ● 0.1 0.4 ● ●●● ● ●●●●●● ● ● ρ 0.3 ● ●●● ● ● ● ● ●●● ●●● ● ●●●●●●●●●● ●●● ● ● ●●●● ●●● ●●●●●● ● 3 ● ●●● ● ●●● ●●●●●● ● ●●●●●● ●●●●●● ●●●●●●●●●● ● ●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●● ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● 0.2 ●●● ● ●●● ●●● ●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●● ●●● ● ●●● ●●●●●●●●●●● ●●● ● Allocation proportion ● ● ● ● ● ●●● ● ● ● ● 0.1 ● 0.4 0.3 ρ4 ● ● ● ● 0.2 ●●●●●●● ● ● ● ● ●●●●●●●●●●● ●●●●●● ● ●●●●●● ● ● ● ●●●●●●●●●● ●●●●●●●●●● ●●● ● ●●●●●●●●●● ●●● ●●●●●●●●●● ●●● ●●●●●● ●●●●●●●●●● ●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ● 0.1 ● ●●● ● ●●●●●● ●●● ● ● ●●● ●●● ●●● ● CRD DL (2) DL (4) DL (6) DL (8) BUD (2) BUD (3) BUD (4) BUD (5) PBD (1) PBD (2) PBD (3) PBD (4) PBD (5) DBCD (1) DBCD (2) DBCD (4) DBCD (5) MWUD (2) MWUD (4) MWUD (6) MWUD (8) DBCD (10) MaxEnt (1) MaxEnt (0.1) MaxEnt (0.5) MaxEnt (0.05) MaxEnt (0.25) Design )LJXUH 'LVWULEXWLRQV RI WUHDWPHQW DOORFDWLRQ SURSRUWLRQV IRU WKH DOORFDWLRQ $53 3URSHUW\ 7KH $53 SURFHGXUHV VWXGLHG LQ WKLV SDSHU DUH WKH &5' WKH 3%' WKH %8' DQG WKH '/8' 7KH $53 SURSHUW\ RI WKH &5' DQG 3%' IROORZV GLUHFWO\ IURP WKH GHILQLWLRQV RI WKHVH SURFHGXUHV 7KH $53 SURSHUW\ RI WKH %8' DQG '/8' SURFHGXUHV FDQ EH HDVLO\ HVWDEOLVKHG E\ XVLQJ D WHFKQLTXH FDOOHG ³$53 H[WHQ VLRQ EDVHG RQ PDSSLQJ´ GHVFULEHG LQ GHWDLO E\ >@ 7KH QRQ$53 SURFHGXUHV LQ WKH FXUUHQW SDSHU DUH '%&' 0D[(QW DQG 0:8' )LJ VKRZV VLPXODWHG XQFRQGLWLRQDO WUHDWPHQW DOORFDWLRQ SUREDELOLWLHV IRU UDQGRPL]DWLRQ GHVLJQV '%&' γ 0D[(QW η DQG 0:8' α IRU WKH WDUJHW DOORFDWLRQ DQG IRU VDPSOH VL]HV XS WR m = 200 )RU DQ $53 SURFHGXUH WKH XQFRQGLWLRQDO UDQGRPL]DWLRQ SUREDELOLWLHV VKRXOG EH FRQVWDQW DQG DW HDFK DOORFDWLRQ VWHS +RZHYHU IURP )LJ DOO FRQVLGHUHG GHVLJQV H[KLELW GLIIHUHQW GHJUHHV RI IOXFWXDWLRQ LQ WKH SUREDELOLWLHV 7KH '%&' γ GRHV QRW SUHVHUYH DOORFDWLRQ UDWLR DW LQLWLDO DOOR FDWLRQV EXW LW VWDELOL]HV DV WKH VDPSOH VL]H LQFUHDVHV %\ FRQWUDVW 0:8' α DQG 0D[(QW η KDYH V\VWHPDWLF YDULDWLRQV UHJDUGOHVV RI WKH YDOXH RI WKH VDP SOH VL]H )RU 0:8' α VPDOOHU YDOXHV RI α FRUUHVSRQG WR JUHDWHU YDULD WLRQV ZKHUHDV IRU 0D[(QW η ODUJHU YDOXHV RI η UHVXOW LQ JUHDWHU YDULDWLRQV ,Q HVVHQFH PRUH EDODQFHG GHVLJQV KDYH KLJKHU YDULDWLRQV LQ WKH XQFRQGLWLRQDO DOORFDWLRQ SUREDELOLWLHV 7KH H[WUHPH FDVH LV 0D[(QW η =1 ,WV SUREDELOL WLHV IOXFWXDWH RYHU WKH HQWLUH LQWHUYDO 7KH GHVLJQV 0:8' α =2 DQG 0D[(QW η =0.5 VHHP WR H[KLELW VLPLODU SDWWHUQV )RU 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Ɍɜɨɹ ɥɸɛɨɜ ɿ ɩɿɞɬɪɢɦɤɚ ɞɨɩɨɦɨɝɥɢ ɦɟɧɿ ɞɿɫɬɚɬɢɫɹ ɰɿɽʀ ɬɨɱɤɢ ɦɨɝɨ ɠɢɬɬɹ ± ɛɟɡ ɰɶɨɝɨ ɦɿɣ ɲɥɹɯ ɛɭɜ ɛɢ ɧɚɛɚɝɚɬɨ ɜɚɠɱɟ ɿ ɰɿɥɤɨɦ ɦɨɠɥɢɜɨ ɡɨɜɫɿɦ ɿɧɲɢɦ Ʌɸɛɥɸ ɬɟɛɟ Ɍɚɧɸɲɤɚ ɿ ȱɥɸɲɤɚ ɹ ɞɭɠɟ ɜɞɹɱɧɢɣ ɜɚɦ ɡɚ ɜɚɲɿ ɥɸɛɨɜ ɪɨɡɭɦɿɧɧɹ ɩɿɞ ɬɪɢɦɤɭ ɿ ɜɢɬɪɢɦɤɭ ɭɜɟɫɶ ɰɟɣ ɱɚɫ ə ɦɨɠɭ ɬɿɥɶɤɢ ɭɹɜɥɹɬɢ ɹɤɚ ɰɟ ɜɚɠɤɚ ɩɪɚɰɹ ± ɛɭɬɢ ɞɪɭɠɢɧɨɸ ɿ ɫɢɧɨɦ ɜɱɟɧɨɝɨ ɨɫɨɛɥɢɜɨ ɬɚɤɨɝɨ ɳɨ ɩɪɚɰɸɽ ɧɚɞ ɞɢɫɟɪɬɚɰɿɽɸ ɚɥɟ ɭ ɜɚɫ ɜɢɫɬɚɱɢɥɨ ɫɢɥ ɿ ɬɟɪɩɿɧɧɹ ɩɪɨɣɬɢ ɩɨɜɧɢɣ ɲɥɹɯ ɡɿ ɦɧɨɸ ɪɚɡɨɦ ɿ ɹ ɞɭɠɟ ɰɿɧɭɸ ɰɟ ɐɹ ɞɢɫɟɪɬɚɰɿɹ ± ɰɟ ɪɟɡɭɥɶɬɚɬ ɧɚɲɢɯ ɫɩɿɥɶɧɢɯ ɡɭɫɢɥɶ ɿ ɜɚɲ ɜɧɟɫɨɤ ɧɟ ɦɟɧɲ ɜɚɠɥɢɜɢɣ ɚɧɿɠ ɦɿɣ ə ɥɸɛɥɸ ɜɚɫ ɿ ɞɭɠɟ ɳɚɫɥɢɜɢɣ ɛɭɬɢ ɪɚɡɨɦ ɿɡ ɜɚɦɢ 5HIHUHQFHV >@ $OHVVDQGUR %DOGL $QWRJQLQL :LOOLDP ) 5RVHQEHUJHU >@ 9DOHULL 9 )HGRURY DQG 3HWHU +DFNO 0RGHO2ULHQWHG 'HVLJQ RI ([SHULPHQWV 6SULQJHU9HUODJ 1HZ 6HULHV LQ 3UREDELOLW\ DQG 6WDWLVWLFV -RKQ :LOH\ 6RQV ,QF HGLWLRQ >@ 5REHUW - 0D\HU (ULF 9DQ &XWVHP $OIUHGR )DOFRQH DQG HW DO 5DQGRPL]HG WULDO RI WDV IRU UHIUDFWRU\ PHWDVWDWLF FRORUHFWDO FDQFHU 7KH 1HZ (QJODQG -RXUQDO RI 0HGLFLQH ± >@ 3KLOLS 3DOOPDQQ $OXQ : %HGGLQJ %DEDN &KRRGDUL2VNRRHL 0XQ\DUDG]L 'LPDLUR /DXUD )OLJKW /LVD 9 +DPSVRQ -DQH +ROPHV $GULDQ 3 0DQGHU /DQJ¶R 2GRQGL 0DWWKHZ 5 6\GHV 6RID6 9LOODU -DPHV 0 6 :DVRQ &KULVWRSKHU - :HLU *UDKDP 0 :KHHOHU &KULVWLQD WULDOV 7KHUDSHXWLF ,QQRYDWLRQ DQG 5HJXODWRU\ 6FLHQFH ± >@ 7HUU\ 0 7KHUQHDX $ 3DFNDJH IRU 6XUYLYDO $QDO\VLV LQ 6 YHUVLRQ >@ $WXO 9HUPD &KHQ \DQJ -LDQJ 7LPRWK\ 5 %HWWV DQG HW DO $SSURDFKHV WR FDWKHWHU DEODWLRQ IRU SHUVLVWHQW DWULDO ILEULOODWLRQ 7KH 1HZ (QJODQG -RXUQDO RI 0HGLFLQH ± >@ -L[LDQ :DQJ ([SRVXUH5HVSRQVH 0RGHOOLQJ 0HWKRGV DQG 3UDFWLFDO ,PSOHPHQWDWLRQ &KDSPDQ +DOO&5& %RFD 5DWRQ )/ >@ :HQOH =KDR &ROLQ &(YHUHWW UPPSALA DISSERTATIONS IN MATHEMATICS Dissertations at the Department of Mathematics Uppsala University 1. Torbjörn Lundh: Kleinian groups and thin sets at the boundary. 1995. 2. Jan Rudander: On the first occurrence of a given pattern in a semi-Markov process. 1996. 3. Alexander Shumakovitch: Strangeness and invariants of finite degree. 1996 4. Stefan Halvarsson: Duality in convexity theory applied to growth problems in complex analysis. 1996. 5. Stefan Svanberg: Random walk in random environment and mixing. 1997. 6. Jerk Matero: Nonlinear elliptic problems with boundary blow-up. 1997. 7. Jens Blanck: Computability on topological spaces by effective domain representations. 1997. 8. Jonas Avelin: Differential calculus for multifunctions and nonsmooth functions. 1997. 9. Hans Garmo: Random railways and cycles in random regular graphs. 1998. 10. Vladimir Tchernov: Arnold-type invariants of curves and wave fronts on surfaces. 1998. 11. Warwick Tucker: The Lorenz attractor exists. 1998. 12. Tobias Ekholm: Immersions and their self intersections. 1998. 13. Håkan Ljung: Semi Markov chain Monte Carlo. 1999. 14. Pontus Andersson: Random tournaments and random circuits. 1999. 15. Anders Andersson: Lindström quantifiers and higher-order notions on finite structures. 1999. 16. Marko Djordjević: Stability theory in finite variable logic. 2000. 17. Andreas Strömbergsson: Studies in the analytic and spectral theory of automorphic forms. 2001. 18. Olof-Petter Östlund: Invariants of knot diagrams and diagrammatic knot invariants. 2001. 19. Stefan Israelsson: Asymptotics of random matrices and matrix valued processes. 2001. 20. Yacin Ameur: Interpolation of Hilbert spaces. 2001. 21. Björn Ivarsson: Regularity and boundary behavior of solutions to complex Monge-Ampère equations. 2002. 22. Lars Larsson-Cohn: Gaussian structures and orthogonal polynomials. 2002. 23. Sara Maad: Critical point theory with applications to semilinear problems without compactness. 2002. 24. Staffan Rodhe: Matematikens utveckling i Sverige fram till 1731. 2002. 25. Thomas Ernst: A new method for q-calculus. 2002. 26. Leo Larsson: Carlson type inequalities and their applications. 2003. 27. Tsehaye K. Araaya: The symmetric Meixner-Pollaczek polynomials. 2003. 28. Jörgen Olsén: Stochastic modeling and simulation of the TCP protocol. 2003. 29. Gustaf Strandell: Linear and non-linear deformations of stochastic processes. 2003. 30. Jonas Eliasson: Ultrasheaves. 2003. 31. Magnus Jacobsson: Khovanov homology and link cobordisms. 2003. 32. Peter Sunehag: Interpolation of subcouples, new results and applications. 2003. 33. Raimundas Gaigalas: A non-Gaussian limit process with long-range dependence. 2004. 34. Robert Parviainen: Connectivity Properties of Archimedean and Laves Lattices. 2004. 35. Qi Guo: Minkowski Measure of Asymmetry and Minkowski Distance for Convex Bodies. 2004. 36. Kibret Negussie Sigstam: Optimization and Estimation of Solutions of Riccati Equations. 2004. 37. Maciej Mroczkowski: Projective Links and Their Invariants. 2004. 38. Erik Ekström: Selected Problems in Financial Mathematics. 2004. 39. Fredrik Strömberg: Computational Aspects of Maass Waveforms. 2005. 40. Ingrid Lönnstedt: Empirical Bayes Methods for DNA Microarray Data. 2005. 41. Tomas Edlund: Pluripolar sets and pluripolar hulls. 2005. 42. Göran Hamrin: Effective Domains and Admissible Domain Representations. 2005. 43. Ola Weistrand: Global Shape Description of Digital Objects. 2005. 44. Kidane Asrat Ghebreamlak: Analysis of Algorithms for Combinatorial Auctions and Related Problems. 2005. 45. Jonatan Eriksson: On the pricing equations of some path-dependent options. 2006. 46. Björn Selander: Arithmetic of three-point covers. 2007. 47. Anders Pelander: A Study of Smooth Functions and Differential Equations on Fractals. 2007. 48. Anders Frisk: On Stratified Algebras and Lie Superalgebras. 2007. 49. Johan Prytz: Speaking of Geometry. 2007. 50. Fredrik Dahlgren: Effective Distribution Theory. 2007. 51. Helen Avelin: Computations of automorphic functions on Fuchsian groups. 2007. 52. Alice Lesser: Optimal and Hereditarily Optimal Realizations of Metric Spaces. 2007. 53. Johanna Pejlare: On Axioms and Images in the History of Mathematics. 2007. 54. Erik Melin: Digital Geometry and Khalimsky Spaces. 2008. 55. Bodil Svennblad: On Estimating Topology and Divergence Times in Phylogenetics. 2008. 56. Martin Herschend: On the Clebsch-Gordan problem for quiver representations. 2008. 57. Pierre Bäcklund: Studies on boundary values of eigenfunctions on spaces of constant negative curvature. 2008. 58. Kristi Kuljus: Rank Estimation in Elliptical Models. 2008. 59. Johan Kåhrström: Tensor products on Category O and Kostant’s problem. 2008. 60. Johan G. Granström: Reference and Computation in Intuitionistic Type Theory. 2008. 61. Henrik Wanntorp: Optimal Stopping and Model Robustness in Mathematical Finance. 2008. 62. Erik Darpö: Problems in the classification theory of non-associative simple algebras. 2009. 63. Niclas Petersson: The Maximum Displacement for Linear Probing Hashing. 2009. 64. Kajsa Bråting: Studies in the Conceptual Development of Mathematical Analysis. 2009. 65. Hania Uscka-Wehlou: Digital lines, Sturmian words, and continued fractions. 2009. 66. Tomas Johnson: Computer-aided computation of Abelian integrals and robust normal forms. 2009. 67. Cecilia Holmgren: Split Trees, Cuttings and Explosions. 2010. 68. Anders Södergren: Asymptotic Problems on Homogeneous Spaces. 2010. 69. Henrik Renlund: Recursive Methods in Urn Models and First-Passage Percolation. 2011. 70. Mattias Enstedt: Selected Topics in Partial Differential Equations. 2011. 71. Anton Hedin: Contributions to Pointfree Topology and Apartness Spaces. 2011. 72. Johan Björklund: Knots and Surfaces in Real Algebraic and Contact Geometry. 2011. 73. Saeid Amiri: On the Application of the Bootstrap. 2011. 74. Olov Wilander: On Constructive Sets and Partial Structures. 2011. 75. Fredrik Jonsson: Self-Normalized Sums and Directional Conclusions. 2011. 76. Oswald Fogelklou: Computer-Assisted Proofs and Other Methods for Problems Regarding Nonlinear Differential Equations. 2012. 77. Georgios Dimitroglou Rizell: Surgeries on Legendrian Submanifolds. 2012. 78. Granovskiy, Boris: Modeling Collective DecisionMaking in Animal Groups. 2012 79. Bazarganzadeh, Mahmoudreza: Free Boundary Problems of Obstacle Type, a Numerical and Theoretical Study. 2012 80. Qi Ma: Reinforcement in Biology. 2012. 81. Isac Hedén: Ga-actions on Complex Affine Threefolds. 2013. 82. Daniel Strömbom: Attraction Based Models of Collective Motion. 2013. 83. Bing Lu: Calibration: Optimality and Financial Mathematics. 2013. 84. Jimmy Kungsman: Resonance of Dirac Operators. 2014. 85. Måns Thulin: On Confidence Intervals and Two-Sided Hypothesis Testing. 2014. 86. Maik Görgens: Gaussian Bridges – Modeling and Inference. 2014. 87. Marcus Olofsson: Optimal Switching Problems and Related Equations. 2015. 88. Seidon Alsaody: A Categorical Study of Composition Algebras via Group Actions and Triality. 2015. 89. Håkan Persson: Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables. 2015. 90. Djalal Mirmohades: N-complexes and Categorification. 2015. 91. Shyam Ranganathan: Non-linear dynamic modelling for panel data in the social sciences. 2015. 92. Cecilia Karlsson: Orienting Moduli Spaces of Flow Trees for Symplectic Field Theory. 2016. 93. Olow Sande: Boundary Estimates for Solutions to Parabolic Equations. 2016. 94. Jonathan Nilsson: Simple Modules over Lie Algebras. 2016. 95. Marta Leniec: Information and Default Risk in Financial Valuation. 2016. 96. Arianna Bottinelli: Modelling collective movement and transport network formation in living systems. 2016. 97. Katja Gabrysch: On Directed Random Graphs and Greedy Walks on Point Processes. 2016. 98. Martin Vannestål: Optimal timing decisions in financial markets. 2017. 99. Natalia Zabzina: Mathematical modelling approach to collective decision- making. 2017. 100. Hannah Dyrssen: Valuation and Optimal Strategies in Markets Experiencing Shocks. 2017. 101. Juozas Vaicenavicius: Optimal Sequential Decisions in Hidden-State Models. 2017. 102. Love Forsberg: Semigroups, multisemigroups and representations. 2017. 103. Anna Belova: Computational dynamics – real and complex. 2017. 104. Ove Ahlman: Limit Laws, Homogenizable Structures and Their Connections. 2018. 105. Erik Thörnblad: Degrees in Random Graphs and Tournament Limits. 2018. 106. Yu Liu: Modelling Evolution. From non-life, to life, to a variety of life. 2018. 107. Samuel Charles Edwards: Some applications of representation theory in homogeneous dynamics and automorphic functions. 2018. 108. Jakob Zimmermann: Classification of simple transitive 2-representations.2018. 109. Azza Alghamdi: Approximation of pluricomplex Green functions – A probabilistic approach. 2018. 110. Brendan Frisk Dubsky: Structure and representations of certain classes of infinite-dimensional algebras. 2018. 111. Björn R. H. Blomqvist: Gaussian process models of social change. 2018. 112. Konstantinos Tsougkas: Combinatorial and analytical problems for fractals and their graph approximations. 2019. 113. Yevgen Ryeznik: Optimal adaptive designs and adaptive randomization techniques for clinical trials. 2019.