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cobraa (coalescent based reconstruction of ancient admixture): a hidden Markov model to infer ghost admixture using just a diploid sequence.

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# *cobraa*

A method for coalescence-based reconstruction of archaic admixture. *cobraa* is a hidden Markov model that uses a diploid sequence to infer population size changes and archaic admixture with an unsampled population. It is an extension of the PSMC framework, which infers population size changes and assumes no admixture (i.e. panmixia).

The model of admixture that *cobraa* seeks to infer is as follows. Going forwards in time, an ancestral population splits cleanly into two populations $A$ and $B$ at time $T_2$; $A$ and $B$ remain in isolation until time $T_1$, at which point there is an admixture event where the admixed population derives $\gamma$ percent of its ancestry from $B$ and $1-\gamma$ from $A$. All populations are allowed to vary in size over time. Thus, the model parameters are N_A(t), N_B(t), $\gamma$, $T_1$, and $T_2$. For convenience, the size changes of the admixed population (more recent than $T_1$), and the ancestral population (more ancient than $T_2$) are modelled as changes in $N_A(t)$.

After the parameters have been inferred, *cobraa-path* (an extension of *cobraa*) can be used to infer regions of the genome that derive from $A$ or $B$ (or both).

Care should be used when interpretting the fit of *cobraa*. We recommend comparing the fit of the model to a panmictic model as in PSMC, to see if admixture is well supported.

## Installation

*cobraa* is written in Python. You will need numpy, numba, pandas, joblib, scipy, psutil and matplotlib. You can install these in a `conda` environment with:

```
conda create --name cobraa
conda activate cobraa
conda install numpy numba pandas joblib scipy psutil matplotlib
git clone https://github.com/trevorcousins/cobraa.git
```

To check whether the installation worked, you can run with some test data:

```python /path/to/installation/cobraa/cobraa.py -in /path/to/installation/cobraa/simulations/constpopsize.mhs -D 10 -its 1 -o testing```

This should save an output file to `testingfinal_parameters.txt`, and print log information to stdout. The `/path/to/installation/` should be changed to the path from which you performed the `git clone` operation.

If you are having problems, see the Troubleshooting section.

## Quick start

### Input files

*cobraa* takes multi-hetsep (mhs) files as introduced by Stephan Schiffels. To generate these files, you can use [his tutorial](https://github.com/stschiff/msmc-tools/blob/master/msmc-tutorial/guide.md). If you have a CRAM/BAM file, you can use my [Snakefile](https://github.com/trevorcousins/cobraa/blob/main/reproducibility/mhsfiles/Snakefile) as guide for how to process this into mhs files.

### Inference of population size changes and admixture

You can run *cobraa* with the following command line:

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b -ts {ts} -te {te} -its <its> -o <outprefix> | tee <logfile>```

`D` is the number of discrete time interval windows, `b` is the genomic bin size, `ts` is $T_1$, `te` is $T_2$, and `its` is the number of iterations (see the Advanced section for a more detailed explanation). `<infiles>` is a string (separated by a space if more than one) that points to the mhs files. The inferred parameters will be saved to `<outprefix>final_parameters.txt` and a log file will be saved to `<logfile>`. The output file contains `D` rows and 4 columns, which are the left time boundary, the right time boundary, and the coalescence rate per discrete time window in population $A$, and the coalescence rate per discrete time window in population $B$. To scale the times into generations, you must divide by `mu`. We get the effective population size by taking the inverse of the coalescence rate and dividing this by `mu`. Note that population $B$ only exists between `ts` and `te`, and we must have `ts`<`te`<`D`.

You can plot inference in Python using the following code:
```
final_parameters_file = "PATH_TO_COBRAA_OUTPUT"
final_params = np.loadtxt(final_parameters_file)
time_array = list(final_params[:,1])
time_array.insert(0,0)
time_array = np.array(time_array)
plt.stairs(edges=(time_array/mu)*gen,values=(1/final_params[:,2])/mu,label=population,linewidth=4,linestyle="solid",baseline=None)
plt.xscale('log')
plt.ylabel('$N_A(t)$')
plt.xlabel('Years')
```

Alternatively, you can plot this in R using the following code:
```
mu = 1.25e-8
gen = 30
final_parameters_file<-"PATH_TO_COBRAA_OUTPUT"
inference<-read.table(final_parameters_file, header=FALSE)
time = inference[,1]/mu*gen
N_A = (1/inference[,3])/mu
plot(time,N_A,log="x",type="n",xlab="Years",ylab="Time")
lines(time, N_A, type="s", col="red")

```

See the [Inference Tutorial](https://github.com/trevorcousins/cobraa/blob/main/tutorial/Inference_tutorial.ipynb) notebook for a specific example, and the Advanced section for a more detailed explanation of the hyperparameters.

### Decoding

You can decode the HMM (that is, get the inferred coalescence times and admixture regions across the genome) with *cobraa-path* by doing:

```python /path/to/installation/cobraa/cobraa.py -in <infile> -D <D> -o <outprefix> -decode -decode_downsample <decode_downsize> -path | tee <logfile.txt>```

The argument `-decode` tells cobraa to decode the HMM, as opposed to inferring the $N_e$ parameters. The argument `-path` tells the algorithm to use the *cobraa-path* model as opposed to the *cobraa* model (you could decode with *cobraa* by omitting `-path` in the above command line, but this would only give you coalescence times). The decoding file is large - you can reduce disc space by saving the posterior probabilities only at every X base pairs with the `-decode_downsample` argument, where `X = b*decode_downsize` (`b` is the genomic binsize, see above or Advanced). For example, if `b=100` and you want to save every the posterior every 10kb, you should do `-decode_downsample 100` (as `b*decode_downsample = 100*100 = 10000`). When decoding you must provide the command line with the inferred model parameters, as assuming an incorrect demography can induce bias. See `-lambda_A_fg`, `lambda_B_fg`, `-gamma_fg` in the Advanced section for more information.  See the [Snakefile](https://github.com/trevorcousins/cobraa/blob/main/reproducibility/inference_realdata/Snakefile) I used in real data for an example.

Interpretting the output is not obvious, see the [Inference Tutorial](https://github.com/trevorcousins/cobraa/blob/main/tutorial/Inference_tutorial.ipynb) for how to parse this file.

# Usage

The commandlines we used for inference in real data in our paper are provided in this [Snakefile](https://github.com/trevorcousins/cobraa/blob/main/reproducibility/inference_realdata/Snakefile), and can be used as a general usage guide.

# Advanced

Here is a more in depth explanation of the hyperparameters for *cobraa*. Quick preliminaries: *cobraa* works in coalescent units, so uses `theta` and `rho`. `theta` is the scaled mutation rate and is equal to `4*N*mu` where `mu` is the rate per generation per base; `rho` is the scaled recombination rate and is equal to `4*N*r` where `r` is the rate per generation per neighbouring base pairs. In the previous sentence, `N` is the "long-term effective population size". The inferred parameters are scaled to time in years by a user-provided mutation rate per base pair per generation (`mu`) and generation time (`gen`), and to real units of `N` also by `mu`. Time is discretised into `D` windows, where the `D+1` boundaries are ~logarithmically spaced (see below).

-D 
The number of discrete time intervals.

Default behaviour is to use `D=32`.

-ts, -te 
The time indices of $T_1$ and $T_2$, respectively.

We must have `ts`<`te`<`D`. If you want to fit an unstructured model (i.e. panmictic, exactly as in PSMC), then you can do `-ts None -te None`, or just omit these arguments as default behaviour is to assume `ts=None`, `te=None`. If you are fitting a structured model with `D=32`, and want to fit populations size and admixture parameters with a divergence at the 18th index and an admixture time at the 10th index, then you would do:

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D 32 -b -ts 10 -te 18 -its <its> -o <outprefix> | tee <logfile>```

-o 
The output prefix (if inferring $N(t)$), or the output file (if decoding).

If you are inferring parameters, and you set `-o /path/to/installation/cobraa_inference/structure_` then the file describing the inferred parameters will be saved to `/path/to/installation/cobraa_inference/structure_final_parameters`. Default behaviour is to save to the current working directory. 
If you are decoding and you set `-o /path/to/installation/cobraa_inference/TMRCA_paths.txt.gz`, then the matrix of posterior distributions will be saved to `/path/to/installation/cobraa_inference/TMRCA_paths.txt.gz`. You must give this argument if decoding.

-theta 
The scaled mutation rate, `theta = 4*N*mu`.

Default behaviour is to calculate this from the data with theta=number_hets/(sequence_length-number_masked_bases). It is not subsequently updated as part of the expectation-maximisation (EM) algorithm. Usage: if you instead want to give a particular value of theta, e.g. 0.001, you can do:

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b -its <its> -o <outprefix> -theta 0.001 | tee <logfile>```

-mu_over_rho_ratio 
The ratio of the mutation rate to the recombination rate (this parameter should really be called "theta_over_rho_ratio").

If you know the ratio of the mutation to recombination rate, then the starting value of rho is `rho=theta/mu_over_rho_ratio`. 
Default behaviour is to set `mu_over_rho_ratio=1.5`.

In humans, the genome wide average rate of `mu` and `r` are currently taken to be 1.25e-8 and 1e-8, respectively. This ratio will not be true in all species and it requires a bit of thought. Note that if `r>mu` then it is not recommended to use *cobraa* or PSMC, as the algorithm is not able to accurately infer the coalescence times because a typical stretch of the genome will experience a recombination before a mutation. If there are no mutations on a genomic segment then there is no way for its age to be estimated.

Usage: if you want to give a particular value of `mu_over_rho_ratio`, e.g. 2, you can do:

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b -its <its> -o <outprefix> -mu_over_rho_ratio 2 | tee <logfile>```

-rho 
The scaled recombination rate, `rho = 4*N*r`.

Default behaviour is to set `rho=theta/mu_over_rho_ratio` (see above), then update rho every iteration as part of the EM algorithm. Useage: if you have a particularly good idea about what rho is, e.g. 0.0005, you can give it with

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b -its <its> -o <outprefix> -rho 0.0005 | tee <logfile>```

-rho_fixed 
Do not infer rho as part of the EM algorithm.

By default rho is updated as part of the EM algorithm. This has been shown not to be particularly accurate, but historically has been standard practise. If you do not want to update rho, you can add the argument `-rho_fixed` which will force rho to remain at its starting value. Usage:

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b -its <its> -o <outprefix> -rho_fixed | tee <logfile>```

-b 
Genomic bin size.

You can bin the genome into windows of size `b`, which enforces the assumption that recombinations can occur only between windows. The primary advantage of doing this is speeding up the code by a factor of `b`, and also reducing RAM by a factor of `b`. Default behaviour is `b=100`. In humans, `b=1` and `b=100` seem to be indistinguishable, and one could probably get away with using even `b=200` or possibly even greater. Usage: if you want to use a bin size of 100 you can do:

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b 100 -its <its> -o <outprefix> -rho_fixed | tee <logfile>```

-its 
Number of iterations of the EM algorithm.

Default behaviour is `its=20`.

-thresh 
Stop iterating the EM algorithm after the change in log-likelihood is less than thresh.

Typically, PSMC or MSMC/MSMC2 are run for 20 or 30 iterations. Heng Li demonstrates that this is sufficient (atleast in humans) for a satisfactory "goodness of fit" (see Figure S12 in the original [paper](https://pubmed.ncbi.nlm.nih.gov/21753753/)), and for recovering $N_e$ parameters. However you might wish for a more specific convergence criteria, such as the change in log-likelihood of the EM algorithm to be less than a particular value: you can achieve this with `thresh`. 
Default behaviour is to run for 30 iterations - if both `its` and `thresh` are given, then the algorithm will terminate when the first of either is satisfied. Usage: if you want a `thresh` of 1 then you can do

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b -its <its> -o <outprefix> -thresh 1 | tee <logfile>```

-spread_1, spread_2 
The spread of the discrete time window boundaries.

The time window boundaries are given by the following equation: 
$\tau_i = \omega exp\left( \frac{i}{D}log\left(1+\frac{\psi}{\omega}\right)-1\right)$ 
Where $\omega$ = spread_1 controls the dispersion of intervals in recent time, and $\psi$ = spread_2 controls the dispersion in ancient time. In humans, I recommend `spread_1=0.05` and `spread_2=50`, which spans ~10kya to ~5Mya. This may not be optimal in other species and it is probably a good idea to experiment with different combinations. 
Default behaviour is to set `spread1=0.05` and `spread2=50`. Usage: if you want `spread_1=0.05` and `spread_2=50`

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b -its <its> -o <outprefix> -spread_1 0.05 -spread_2 50 | tee <logfile>```

-lambda_{A,B}_segments 
Fix some adjacent time windows to have the same coalescence rate.

If fitting an unstructured model, cobraa will fit changes in population $A$. If fitting a structured model, cobraa will fit changes in population $A$ and $B$. In either population, you may want to force adjacent intervals to have the same coalescent rates. If you want to use `D=32` with `ts=10` and `te=18` and fit population $A$ without constraints (i.e. "free") but want to enforce the size of $B$ to be constant, you can do

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D 32 -b -its <its> -o <outprefix> -lambda_A_segments 32*1 -labda_B_segments 1*32 -ts <ts> -te <te> | tee <logfile>```

so the argument is parsed as "32 free intervals of length 1" for $A$ and "1 free interval of length 32" for $B$. Note that population $B$ is only valid between `ts` and `te`. Default behaviour is to infer $A$ freely and lock $B$ constant. To infer $B$ freely you can do `-lambda_B_segments {ts}*0,{te-ts}*1,{D-te}*0`, so for `D=32` with `ts=10` and `te=18` this would be `-lambda_B_segments 10*0,8*1,14*0`

-lambda_{A,B}_fg 
The first guess for the inverse coalescence rates, in each discrete time interval.

A comma separated list of floats that are taken are used as the starting guess for the inverse coalescence rates. You should also use this to decode with the inferred population size parameters. The length of this list must be equal to the number of segments as specified by the `-lambda_segments` argument. If you have 10 discrete time intervals and know your inverse coalescence rates are [1,1,5,5,5,5,5,1,1,1] (a population that experiences a five-fold bottleneck) you can do:

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b -its <its> -o inference/final_parameters.txt -lambda_A_fg 1,1,5,5,5,5,5,1,1,1 | tee <logfile>```

-gamma_fg 
The first guess for the admixture fraction.

When running inference, we found that as part of the EM algorithm the optimisation sometimes get stuck in local maxima. For this reason it is good to experiment with different starting values of `gamma`. Default behaviour to use `gamma_fg=0.2`. You should also use this for decoding with the inferred parameters. If you want to try inference with a first guess of `gamma=0.3` then you can do:

```python /path/to/installation/cobraa/cobraa.py -in <infiles> -D <D> -b -its <its> -o <outprefix> -gamma_fg 0.3 | tee <logfile>```

If searching for a structured model, we recommend setting the first guess for `gamma` away from 0, otherwise the algorithm will likely will get stuck in the local optima corresponding to panmixia (`gamma=0`).

## Bounding parameter searches

To control runaway behaviour, you can set bounds on the parameters used. To control the lower and upper inverse coalescence rates, you can use `-lambda_lwr_bnd` and `-lambda_upr_bnd`, respectively. We can further control lambda behaviour in the structured period, with `-lambda_lwr_struct` and `-lambda_upr_struct`. We can also control the bounds on `gamma` with `-gamma_lwr` and `-gamma_upr`.

# Simulation

If you want to simulate a demography, you probably want to use msprime or SLiM as these are extremely powerful and flexible. For the simulations in our paper, I used this [Snakefile](https://github.com/trevorcousins/cobraa/blob/main/reproducibility/simulations/Snakefile); this could be used as a guide. I also provide functionality to simulate directly from the *cobraa* HMM (this is necessarily simulating from the SMC' model, which is marginally a very good approximation to the full coalescent with recombination - see [Wilton et al 2015](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4423375/)). An example command line to simulate a constant population size is:

```python /path/to/installation/cobraa/simulate_HMM.py -D 10 -theta 0.001 -rho 0.0005 -o_mhs simulations/sim1_variants.mhs -o_coal simulations/sim1_coal.txt.gz -spread_1 0.1 -spread_2 50 -L 1000000 -gamma 0.3 -ts 5 -te 8 ```

`D`, `theta`, `rho`, `b`, `spread_1` and `spread_2`, are as described above. `L` is an integer for the sequence , `gamma` is the admixture fraction, `ts` is the index of $T_1$, and `te` is the index of $T_2$. This will save an mhs file detailing the variant positions to `simulations/sim1_variants.mhs`, and a file detailing the coalescent data (pairwise coalescence times across the genome) to `simulations/sim1_coal.txt.gz`. The third column of this file is the index of the coalescent time between the genomic positions described in the first and second column.

To simulate with a changing population size, in $A$ or $B$, you must give an argument `-lambda_A` or `-lambda_B`, which is a comma separated string where each value is the inverse coalescence rate in each time interval. For example if you want a simulation with 32 discrete time windows with $A$ and $B$ of relative size 1, with 30% admixture, then you can do:

```python /path/to/installation/cobraa/simulate_HMM.py -D 32 -lambda_A 1,1,1,1,1,1,1,1,1,1,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,1,1,1,1,1,1,1,1,1,1,1,1,1,1 -theta 0.001 -rho 0.0005 -o_mhs simulations/sim2_variants.mhs -o_coal simulations/sim2_coal.txt.gz -L 1000000```

```python /path/to/installation/simulate_HMM.py -L 100000 -gamma 0.3 -theta 0.001 -rho 0.0005 -D 32 -ts 10 -te 18 -lambda_A 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 -lambda_A 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1```

Note that population $B$ only exists between `ts` and `te`, so input parameters before and after then are meaningless.

## Troubleshooting

### Installation

We use *cobraa* with the following versions of each package: 
```
python==3.10.5
numba==0.55.2
joblib==1.1.0
matplotlib==3.5.2
pandas==1.4.3
psutil==5.9.1
scipy==1.8.1
numpy==1.26.4

```
If your installation attempt fails, please try with versions listed above.
In the above commands you will of course need to change the `/path/to/installation/` path. For a particular example, if I am in my home directory (`/home/trevor`) and do `git clone https://github.com/trevorcousins/cobraa.git`, then the code is ready to run at `/home/trevor/cobraa/cobraa.py` .

### Division by Zero error

If your heterozygosity is very high (e.g. `theta>0.01`), the default binsize of 100 will not be appropriate and you'll get an error. Instead, reduce the binsize such that there being more than ~3 heterozygotes per bin is rare. E.g. if theta=0.05 then we expect a heterozygous position every 20 base pairs on average, so a suitable bin size is 5 or 10 (add `-b 5` or `-b 10` to your command line).

### Other

If you are still having problems with the code, please submit a new issue. If you have questions about the research, please email me at trevorsimoncousins(/at/)gmail.com and [Richard Durbin](https://www.gen.cam.ac.uk/directory/richard-durbin).

## Citation

If you use cobraa, please cite the following [paper](https://www.nature.com/articles/s41588-025-02117-1):

*Cousins, Trevor, Aylwyn Scally, and Richard Durbin. "A structured coalescent model reveals deep ancestral structure shared by all modern humans." Nature Genetics (2025)

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