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update the wiki for linked exchangeabilities
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doc/Command-Reference.md

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| Option | Usage and meaning |
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|----------|------------------------------------------------------------------------------|
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| `--link-exchange-rates` | Turn on linked exchangeability estimation for a profile mixture model. Note that the model must have specified `GTR20` exchangeabilities for eg.`GTR20+C20+G`. This option also produces a nexus file `GTRPMIX.nex` with the exchangeability matrix obtained from the optimization. This file can be later used for phylogenetic inference with the use of the `-mdef` flag|
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| `--gtr20-model` | Specify the initial exchangeabilities for linked exchangeability estimation. Note that this must be used with `--link-exchange-rates.` |
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| `--link-exchange` | Turn on linked exchangeability estimation for a profile mixture model. Note that the model must have specified `GTR20` exchangeabilities for eg.`GTR20+C20+G`. This option also produces a nexus file `GTRPMIX.nex` with the exchangeability matrix obtained from the optimization. This file can be later used for phylogenetic inference with the use of the `-mdef` flag|
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| `--init-exchange` | Specify the initial exchangeabilities for linked exchangeability estimation. Note that this must be used with `--link-exchange`. |
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### Example usages:
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* Estimate linked exchangeabilities for a protein alignment `prot.phy` under C60+G model and a guide tree `guide.treefile`, where optimization is initialized from LG exchangeabilities
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iqtree -s prot.phy -m GTR20+C60+G --link-exchange-rates --gtr20-model LG -te guide.treefile
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iqtree -s prot.phy -m GTR20+C60+G --link-exchange --init-exchange LG -te guide.treefile
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>**NOTE**: For better and faster performance, read the [recommendations](Complex-Models#linked-gtr-exchangeabilities-models) provided in the Complex Models section.
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>**NOTE**: For better and faster performance, read the [recommendations](Estimating-amino-acid-substitution-models#estimating-linked-exchangeabilities) provided in the Estimating amino acid substitution models section.
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Rate heterogeneity

doc/Estimating-amino-acid-substitution-models.md

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To start with, we show an example:
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iqtree2 -s <alignment> -m GTR20+C60+G4 --link-exchange-rates -te <guide_tree> -me 0.99
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iqtree2 -s <alignment> -m GTR20+C60+G4 --link-exchange -te <guide_tree> -me 0.99
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Here, IQ-TREE applies a (freely-estimated) 20x20 rate matrix `GTR20` with the
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[profile mixture model](Substitution-Models#protein-mixture-models) `C60` (other model such as C10 can also be used) and Gamma rate heterogeneity across sites. The option `--link-exchange-rates` tells
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[profile mixture model](Substitution-Models#protein-mixture-models) `C60` (other model such as C10 can also be used) and Gamma rate heterogeneity across sites. The option `--link-exchange` tells
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IQ-TREE to link GTR20 rates across all 60 mixture classes: without this option
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IQ-TREE will estimate 60 GTR20 matrices!
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iqtree2 -s <alignment> -mdef <.GTRPMIX.nex file> -m GTRPMIX+C60+G4
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The optimizer in IQ-TREE by default initializes exchangeability rates to be all equal, which are the least biased but may make the subsequent optimization quite slow. If users have a good guess of the rate values, the option `--gtr20-model` can be used. For example, `--gtr20-model LG` will intialize the exchangeability to that
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The optimizer in IQ-TREE by default initializes exchangeability rates to be all equal, which are the least biased but may make the subsequent optimization quite slow. If users have a good guess of the rate values, the option `--init-exchange` can be used. For example, `--init-exchange LG` will intialize the exchangeability to that
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of the LG model before optimization. Choosing good starting values can make estimation considerably faster. Apart from LG, users can specify any matrix, including those defined by the `-mdef` option with a [NEXUS model file](Complex-Models#nexus-model-file). Another use of this option is to _test the robustness_ of the optimizer with different starting points.
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Note that the user can estimate exchangeabilities jointly with weights of the profiles, branch lengths, and rates. This can be very time-consuming. If the goal is to optimize exchangeabilities, one can fix the other parameters to reasonable estimates (for eg. fixing branch lengths and rates has been shown to perform adequately for the estimation of exchangeabilities).

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