### Serial composition of two Gotoh transducers

The composite transducer, , where is the GotohTransducer. (See TransducerLegend for key.)

The diagrams on this page are admittedly pretty horrendous. My motivation for looking at these is to see if there is some way to approximate the following transducer using the simpler GotohTransducer. This would essentially correspond to a "renormalization" of the transducer's parameter space, which in turn might illuminate a classic problem in statistical alignment: the derivation of convex gap penalties from an evolutionary indel model with affine gaps. See RenormalizedTransducerComposition for more details.

digraph G {

SS [shape=doublecircle, color=red]; EE [shape=doublecircle, color=red];

WW [shape=doubleoctagon, color=red]; WX [shape=doubleoctagon, color=red]; XW [shape=doubleoctagon, color=red]; XX [shape=doubleoctagon, color=red];

SI [shape=house, color=red]; IM [shape=house, color=red]; II [shape=house, color=red]; MI [shape=house, color=red];

MM [shape=rect, color=red];

MD [shape=invhouse, color=red]; DW [shape=invhouse, color=red]; DX [shape=invhouse, color=red];

SS->SI;SS->SW;SS->SX; SI->SI->SW;SI->SX; SW->IM;SW->WW;SW->XW; SX->ID;SX->WX;SX->XX;

IM->II;IM->IW;IM->IX; II->II;II->IW;II->IX; IW->IM;IW->WW;IW->XW; IX->ID;IX->WX;IX->XX; ID->IW;ID->IX;

WW->EE;WW->MM; WX->EE;WX->MD; XW->EE;XW->DW; XX->EE;XX->DX;

MM->MI;MM->MW;MM->MX; MI->MI;MI->MW;MI->MX; MD->MW;MD->MX; MW->IM;MW->WW;MW->XW; MX->ID;MX->WX;MX->XX;

DW->WW;DW->XW; DX->WX;DX->XX;

DW->IM; DX->ID; ID->II; MD->MI;

label="Serial composition of two Gotoh transducers"; }

We can also consider the jointly normalized HMM that is the composition of the SingletTransducer with the above composed transducer (c.f. GotohPairHMM). Notationally this is , where is the SingletTransducer.

This diagram assumes S/M symmetry in the GotohTransducer and S/I symmetry in the SingletTransducer (so e.g. "SSI" and "IMI" are the same state).

digraph G {

SSS [shape=doublecircle, color=red]; EEE [shape=doublecircle, color=red];

SSI [shape=house, color=red, label="SSI/IMI"]; SIM [shape=house, color=red, label="SIM/IIM"]; SII [shape=house, color=red, label="SII/III"];

SID [label="SID/IID"];

IMM [shape=rect, color=red];

IMD [shape=invhouse, color=red]; IDW [shape=invhouse, color=red]; IDX [shape=invhouse, color=red];

IMM->SSS;

SSS->SSI [label=d]; SSS->SIM [label=ed]; SSS->SID [label=fd]; SSS->IMM [label=eeg]; SSS->IMD [label=feg]; SSS->IDW [label=efg]; SSS->IDX [label=ffg]; SSS->EEE [label="h(e+f)(e+f)"];

SSI->SSI [label=v]; SSI->SIM [label=wd]; SSI->SID [label=xd]; SSI->IMM [label=weg]; SSI->IMD [label=xeg]; SSI->IDW [label=wfg]; SSI->IDX [label=xfg]; SSI->EEE [label="h(w+x)(w+x)"];

SIM->SII [label=d]; SIM->SIM [label=ev]; SIM->SID [label=fv]; SIM->IMM [label=ewg]; SIM->IMD [label=fwg]; SIM->IDW [label=exg]; SIM->IDX [label=fxg]; SIM->EEE [label="h(e+f)(w+x)"];

SII->SII [label=v]; SII->SIM [label=wv]; SII->SID [label=xv]; SII->IMM [label=wwg]; SII->IMD [label=xwg]; SII->IDW [label=wxg]; SII->IDX [label=xxg]; SII->EEE [label="h(w+x)(w+x)"];

SID->SII [label=p]; SID->SIM [label=qv]; SID->SID [label=rv]; SID->IMM [label=qwg]; SID->IMD [label=rwg]; SID->IDW [label=qxg]; SID->IDX [label=rxg]; SID->EEE [label="h(q+r)(w+x)"];

IMD->SSI [label=p]; IMD->SIM [label=qd]; IMD->SID [label=rd]; IMD->IMM [label=qeg]; IMD->IMD [label=reg]; IMD->IDW [label=qfg]; IMD->IDX [label=rfg]; IMD->EEE [label="h(q+r)(e+f)"];

IDW->SIM [label=p]; IDW->IMM [label=qg]; IDW->IDW [label=rg]; IDW->EEE [label="h(q+r)"];

IDX->SID [label=p]; IDX->IMD [label=qg]; IDX->IDX [label=rg]; IDX->EEE [label="h(q+r)"];

label="Serial composition of singlet & two Gotoh transducers"; }

-- IanHolmes - 28 Feb 2007

Topic revision: r12 - 2007-03-01 - IanHolmes

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