IQ-tree's method for least-squares AU implementation

[Lucas Marques]

Hi,

I wonder which method IQ-tree uses to minimise the least-squares function for 'c' and 'd' estimation (during AU test p-value calculation).

Does it implement some kind of numeric method or is the function minimised in a more analytical manner? And which is it?

Being an amateur programmer with relatively poor backgrounds in mathematics, I've been struggling to figure out the most efficient way to calculate 'c' and 'd' and would very much appreciate any hint.

Thanks in advance, 

Lucas

[Bui Quang Minh]

Hi Lucas, There is a closed formula for this, which I implemented in IQ-TREE. Have a look at phylotesting.cpp. For your convenience I paste here the method (copied from the comments in the code):

/*

    Problem: solve the following linear system equation:

    a_1*x + b_1*y = c_1

    a_2*x + b_2*y = c_2

    ....

    a_n*x + b_n*y = c_n

    

becomes minimizing weighted least square:

    sum_k { w_k*[ c_k - (a_k*x + b_k*y) ]^2 }

the solution is:

    x = [(sum_k w_k*b_k*c_k)*(sum_k w_k*a_k*b_k) - (sum_k w_k*a_k*c_k)(sum_k w_k*b_k^2)] / 

        [ (sum_k w_k*a_k*b_k)^2 - (sum_k w_k*a_k^2)*(sum_k w_k*b_k^2) ]

    

    y = [(sum_k w_k*a_k*c_k)*(sum_k w_k*a_k*b_k) - (sum_k w_k*b_k*c_k)(sum_k w_k*a_k^2)] / 

        [ (sum_k w_k*a_k*b_k)^2 - (sum_k w_k*a_k^2)*(sum_k w*k*b_k^2) ]

    

    @param n number of data points

    @param w weight vector of length n

    @param a a value vector of length n

    @param b b value vector of length n

    @param c c value vector of length n

    @param[out] x x-value

    @param[out] y y-value

    @return least square value

*/

Cheers,

Minh

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Bui Quang Minh

Center for Integrative Bioinformatics Vienna (CIBIV)

Campus Vienna Biocenter 5, VBC5, Ebene 1

A-1030 Vienna, Austria

Phone: ++43 1 4277 74326

Email: minh.bui (AT) univie.ac.at

[Lucas Marques]

Thanks for your quick reply, Minh. This will be very helpful. I'll take a close look at the code now. 

Best regards,

Lucas