Author: bugman
Date: Thu Apr 29 15:38:02 2010
New Revision: 11169
URL: http://svn.gna.org/viewcvs/relax?rev=11169&view=rev
Log:
Updated and corrected all the Hessian method docstrings.
Modified:
1.3/maths_fns/n_state_model.py
Modified: 1.3/maths_fns/n_state_model.py
URL:
http://svn.gna.org/viewcvs/relax/1.3/maths_fns/n_state_model.py?rev=11169&r1=11168&r2=11169&view=diff
==============================================================================
--- 1.3/maths_fns/n_state_model.py (original)
+++ 1.3/maths_fns/n_state_model.py Thu Apr 29 15:38:02 2010
@@ -1164,24 +1164,11 @@
def d2func_population(self, params):
"""The Hessian function for optimisation of the flexible population
N-state model.
- This function should be passed to the optimisation algorithm. It
accepts, as an array, a vector of parameter values and, using these, returns
the chi-squared Hessian corresponding to that coordinate in the parameter
space. If no RDC errors are supplied, then the SSE (the sum of squares
error) Hessian is returned instead. The chi-squared Hessian is simply the
SSE Hessian normalised to unit variance (the SSE divided by the error
squared).
-
- @param params: The vector of parameter values.
- @type params: numpy rank-1 array
- @return: The chi-squared or SSE Hessian.
- @rtype: numpy rank-2 array
- """
-
- raise RelaxImplementError
-
-
- def d2func_tensor_opt(self, params):
- """The Hessian function for optimisation of the alignment tensor
from RDC and/or PCS data.
-
Description
===========
- This function should be passed to the optimisation algorithm. It
accepts, as an array, a vector of parameter values and, using these, returns
the chi-squared gradient corresponding to that coordinate in the parameter
space. If no RDC errors are supplied, then the SSE (the sum of squares
error) gradient is returned instead. The chi-squared gradient is simply the
SSE gradient normalised to unit variance (the SSE divided by the error
squared).
+ This function should be passed to the optimisation algorithm. It
accepts, as an array, a vector of parameter values and, using these, returns
the chi-squared Hessian corresponding to that coordinate in the parameter
space. If no RDC/PCS errors are supplied, then the SSE (the sum of squares
error) Hessian is returned instead. The chi-squared Hessian is simply the
SSE Hessian normalised to unit variance (the SSE divided by the error
squared).
+
Indices
=======
@@ -1201,8 +1188,8 @@
To calculate the chi-squared gradient, a chain of equations are
used. This includes the chi-squared gradient, the RDC gradient and the
alignment tensor gradient.
- The chi-squared gradient
- ------------------------
+ The chi-squared Hessian
+ -----------------------
The equation is::
___
@@ -1220,64 +1207,173 @@
- d2Dij(theta)/dthetaj.dthetak is the RDC or PCS Hessian for
parameters j and k.
- The RDC gradient
- ----------------
-
- The only parameters are the tensor components.
-
- Amn second partial derivatives
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- The alignment tensor element partial derivative is::
+ The RDC Hessian
+ ---------------
+
+ pc-pd second partial derivatives
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The probability parameter second partial derivative is::
d2Dij(theta)
- ------------ = 0
+ ------------ = 0.
+ dpc.dpd
+
+
+ pc-Anm second partial derivatives
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The probability parameter-tensor element second partial derivative
is::
+
+ d2Dij(theta) T dAi
+ ------------ = dj . mu_jc . ---- . mu_jc.
+ dpc.dAmn dAmn
+
+
+ Amn-Aop second partial derivatives
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The alignment tensor element second partial derivative is::
+
+ d2Dij(theta)
+ ------------ = 0.
dAmn.dAop
- The PCS gradient
- ----------------
-
- Amn partial derivative
- ~~~~~~~~~~~~~~~~~~~~~~
-
- The alignment tensor element partial derivative is::
+ The PCS Hessian
+ ---------------
+
+ pc-pd second partial derivatives
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The probability parameter second partial derivative is::
+
+ d2delta_ij(theta)
+ ----------------- = 0.
+ dpc.dpd
+
+
+ pc-Anm second partial derivatives
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The probability parameter-tensor element second partial derivative
is::
+
+ d2delta_ij(theta) T dAi
+ ----------------- = djc . mu_jc . ---- . mu_jc.
+ dpc.dAmn dAmn
+
+
+ Amn-Aop second partial derivatives
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The alignment tensor element second partial derivative is::
d2delta_ij(theta)
----------------- = 0
dAmn.dAop
- The alignment tensor gradient
- -----------------------------
-
- The five unique elements of the tensor {Axx, Ayy, Axy, Axz, Ayz} all
have the same partial derivative of::
-
- dAi | 0 0 0 |
- ---- = | 0 0 0 |.
- dAmn | 0 0 0 |
-
-
-
- Stored data structures
- ======================
-
- There are a number of data structures calculated by this function
and stored for subsequent use in the Hessian function. This include the back
calculated RDC and PCS gradients and the alignment tensor gradients.
-
- dDij(theta)/dthetak
- -------------------
-
- The back calculated RDC gradient. This is a rank-3 tensor with
indices {k, i, j}.
-
- ddeltaij(theta)/dthetak
+ The alignment tensor Hessian
+ ----------------------------
+
+ The five unique elements of the tensor {Axx, Ayy, Axy, Axz, Ayz} all
have the same second partial derivative of::
+
+ d2Ai | 0 0 0 |
+ --------- = | 0 0 0 |.
+ dAmn.dAop | 0 0 0 |
+
+
+ @param params: The vector of parameter values. This is unused as
it is assumed that func_population() was called first.
+ @type params: numpy rank-1 array
+ @return: The chi-squared or SSE Hessian.
+ @rtype: numpy rank-2 array
+ """
+
+ raise RelaxImplementError
+
+
+ def d2func_tensor_opt(self, params):
+ """The Hessian function for optimisation of the alignment tensor
from RDC and/or PCS data.
+
+ Description
+ ===========
+
+ This function should be passed to the optimisation algorithm. It
accepts, as an array, a vector of parameter values and, using these, returns
the chi-squared Hessian corresponding to that coordinate in the parameter
space. If no RDC/PCS errors are supplied, then the SSE (the sum of squares
error) Hessian is returned instead. The chi-squared Hessian is simply the
SSE Hessian normalised to unit variance (the SSE divided by the error
squared).
+
+ Indices
+ =======
+
+ For this calculation, six indices are looped over and used in the
various data structures. These include:
+ - k, the index over all parameters,
+ - i, the index over alignments,
+ - j, the index over spin systems,
+ - c, the index over the N-states (or over the structures),
+ - m, the index over the first dimension of the alignment tensor
m = {x, y, z}.
+ - n, the index over the second dimension of the alignment tensor
n = {x, y, z},
+
+
+ Equations
+ =========
+
+ To calculate the chi-squared gradient, a chain of equations are
used. This includes the chi-squared gradient, the RDC gradient and the
alignment tensor gradient.
+
+
+ The chi-squared Hessian
-----------------------
- The back calculated PCS gradient. This is a rank-3 tensor with
indices {k, i, j}.
-
- dAi/dAmn
- --------
-
- The alignment tensor gradients. This is a rank-3 tensor with
indices {5, n, m}.
+ The equation is::
+ ___
+ d2chi^2(theta) \ 1 / dDij(theta) dDij(theta)
d2Dij(theta) \
+ --------------- = 2 > ---------- | ----------- . ----------- -
(Dij-Dij(theta)) . --------------- |.
+ dthetaj.dthetak /__ sigma_i**2 \ dthetaj dthetak
dthetaj.dthetak /
+ ij
+
+ where:
+ - theta is the parameter vector,
+ - Dij are the measured RDCs or PCSs,
+ - Dij(theta) are the back calculated RDCs or PCSs,
+ - sigma_ij are the RDC or PCS errors,
+ - dDij(theta)/dthetak is the RDC or PCS gradient for parameter k.
+ - d2Dij(theta)/dthetaj.dthetak is the RDC or PCS Hessian for
parameters j and k.
+
+
+ The RDC Hessian
+ ---------------
+
+ The only parameters are the tensor components.
+
+
+ Amn-Aop second partial derivatives
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The alignment tensor element second partial derivative is::
+
+ d2Dij(theta)
+ ------------ = 0.
+ dAmn.dAop
+
+
+ The PCS Hessian
+ ---------------
+
+ Amn-Aop second partial derivatives
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The alignment tensor element second partial derivative is::
+
+ d2delta_ij(theta)
+ ----------------- = 0
+ dAmn.dAop
+
+
+ The alignment tensor Hessian
+ ----------------------------
+
+ The five unique elements of the tensor {Axx, Ayy, Axy, Axz, Ayz} all
have the same second partial derivative of::
+
+ d2Ai | 0 0 0 |
+ --------- = | 0 0 0 |.
+ dAmn.dAop | 0 0 0 |
@param params: The vector of parameter values. This is unused as
it is assumed that func_population() was called first.
r11169 - /1.3/maths_fns/n_state_model.py