Module model_free

Functions

calc_jw(data, params)
Spectral density function.

calc_S2_jw(data, params)
Spectral density function.

calc_S2_te_jw(data, params)
Spectral density function.

calc_S2f_S2_ts_jw(data, params)
Spectral density function.

calc_S2f_tf_S2_ts_jw(data, params)
Spectral density function.

calc_S2f_S2s_ts_jw(data, params)
Spectral density function.

calc_S2f_tf_S2s_ts_jw(data, params)
Spectral density function.

calc_diff_djw_dGj(data, params, j)
Spectral density gradient.

calc_ellipsoid_djw_dGj(data, params, j)
Spectral density gradient.

calc_diff_S2_djw_dGj(data, params, j)
Spectral density gradient.

calc_ellipsoid_S2_djw_dGj(data, params, j)
Spectral density gradient.

calc_diff_S2_te_djw_dGj(data, params, j)
Spectral density gradient.

calc_ellipsoid_S2_te_djw_dGj(data, params, j)
Spectral density gradient.

calc_diff_djw_dOj(data, params, j)
Spectral density gradient.

calc_diff_S2_djw_dOj(data, params, j)
Spectral density gradient.

calc_diff_S2_te_djw_dOj(data, params, j)
Spectral density gradient.

calc_S2_djw_dS2(data, params, j)
Spectral density gradient.

calc_S2_te_djw_dS2(data, params, j)
Spectral density gradient.

calc_S2_te_djw_dte(data, params, j)
Spectral density gradient.

calc_diff_S2f_S2_ts_djw_dGj(data, params, j)
Spectral density gradient.

calc_ellipsoid_S2f_S2_ts_djw_dGj(data, params, j)
Spectral density gradient.

calc_diff_S2f_tf_S2_ts_djw_dGj(data, params, j)
Spectral density gradient.

calc_ellipsoid_S2f_tf_S2_ts_djw_dGj(data, params, j)
Spectral density gradient.

calc_diff_S2f_S2_ts_djw_dOj(data, params, j)
Spectral density gradient.

calc_diff_S2f_tf_S2_ts_djw_dOj(data, params, j)
Spectral density gradient.

calc_S2f_S2_ts_djw_dS2(data, params, j)
Spectral density gradient.

calc_S2f_S2_ts_djw_dS2f(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2_ts_djw_dS2f(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2_ts_djw_dtf(data, params, j)
Spectral density gradient.

calc_S2f_S2_ts_djw_dts(data, params, j)
Spectral density gradient.

calc_diff_S2f_S2s_ts_djw_dGj(data, params, j)
Spectral density gradient.

calc_ellipsoid_S2f_S2s_ts_djw_dGj(data, params, j)
Spectral density gradient.

calc_diff_S2f_tf_S2s_ts_djw_dGj(data, params, j)
Spectral density gradient.

calc_ellipsoid_S2f_tf_S2s_ts_djw_dGj(data, params, j)
Spectral density gradient.

calc_diff_S2f_S2s_ts_djw_dOj(data, params, j)
Spectral density gradient.

calc_diff_S2f_tf_S2s_ts_djw_dOj(data, params, j)
Spectral density gradient.

calc_S2f_S2s_ts_djw_dS2f(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2s_ts_djw_dS2f(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2s_ts_djw_dS2s(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2s_ts_djw_dtf(data, params, j)
Spectral density gradient.

calc_S2f_S2s_ts_djw_dts(data, params, j)
Spectral density gradient.

calc_diff_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2_te_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_diff_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2_te_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dGjdS2(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2_d2jw_dGjdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dGjdS2(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2_te_d2jw_dGjdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dGjdte(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2_te_d2jw_dGjdte(data, params, j, k)
Spectral density Hessian.

calc_diff_d2jw_dOjdOk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dOjdOk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dOjdOk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dOjdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dOjdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dOjdte(data, params, j, k)
Spectral density Hessian.

calc_S2_te_d2jw_dS2dte(data, params, j, k)
Spectral density Hessian.

calc_S2_te_d2jw_dte2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_tf_S2_ts_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_tf_S2_ts_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dGjdS2(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dGjdS2f(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dGjdS2f(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_tf_S2_ts_d2jw_dGjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dGjdtf(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_tf_S2_ts_d2jw_dGjdtf(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dGjdts(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdts(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dOjdOk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dOjdOk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dOjdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dOjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dOjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dOjdtf(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dOjdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2_ts_d2jw_dS2dts(data, params, j, k)
Spectral density Hessian.

calc_S2f_tf_S2_ts_d2jw_dS2fdtf(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2_ts_d2jw_dS2fdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_tf_S2_ts_d2jw_dtf2(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2_ts_d2jw_dts2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_tf_S2s_ts_d2jw_dGjdGk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_tf_S2s_ts_d2jw_dGjdOj(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dGjdS2f(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dGjdS2f(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_tf_S2s_ts_d2jw_dGjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dGjdS2s(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdS2s(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dGjdtf(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_tf_S2s_ts_d2jw_dGjdtf(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dGjdts(data, params, j, k)
Spectral density Hessian.

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdts(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dOjdOk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dOjdOk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dOjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dOjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dOjdS2s(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dOjdtf(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dOjdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2s_ts_d2jw_dS2fdS2s(data, params, j, k)
Spectral density Hessian.

calc_S2f_tf_S2s_ts_d2jw_dS2fdtf(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2s_ts_d2jw_dS2fdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2s_ts_d2jw_dS2sdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_tf_S2s_ts_d2jw_dtf2(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2s_ts_d2jw_dts2(data, params, j, k)
Spectral density Hessian.

Variables

__package__ = 'lib.spectral_densities'

Imports: sum

Function Details

calc_jw(data, params)

Spectral density function.

Calculate the spectral density values for the original model-free formula with no parameters {} with or without diffusion tensor parameters.

The formula is:

               _k_
            2  \                1
   J(w)  =  -   >  ci . ti ------------.
            5  /__         1 + (w.ti)^2
               i=-k

calc_S2_jw(data, params)

Spectral density function.

Calculate the spectral density values for the original model-free formula with the single parameter {S2} with or without diffusion tensor parameters.

The formula is:

                 _k_
            2    \                1
   J(w)  =  - S2  >  ci . ti ------------.
            5    /__         1 + (w.ti)^2
                 i=-k

calc_S2_te_jw(data, params)

Spectral density function.

Calculate the spectral density values for the original model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.

The model-free formula is:

               _k_
            2  \           /      S2             (1 - S2)(te + ti)te    \ 
   J(w)  =  -   >  ci . ti | ------------  +  ------------------------- |.
            5  /__         \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
               i=-k

calc_S2f_S2_ts_jw(data, params)

Spectral density function.

Calculate the spectral density values for the extended model-free formula with the parameters {S2f, S2, ts} with or without diffusion tensor parameters.

The model-free formula is:

               _k_
            2  \           /      S2            (S2f - S2)(ts + ti)ts   \ 
   J(w)  =  -   >  ci . ti | ------------  +  ------------------------- |.
            5  /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
               i=-k

calc_S2f_tf_S2_ts_jw(data, params)

Spectral density function.

Calculate the spectral density values for the extended model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The model-free formula is:

               _k_
            2  \           /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ 
   J(w)  =  -   >  ci . ti | ------------  +  -------------------------  +  ------------------------- |.
            5  /__         \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
               i=-k

calc_S2f_S2s_ts_jw(data, params)

Spectral density function.

Calculate the spectral density values for the extended model-free formula with the parameters {S2f, S2s, ts} with or without diffusion tensor parameters.

The model-free formula is:

                  _k_
            2     \           /      S2s           (1 - S2s)(ts + ti)ts    \ 
   J(w)  =  - S2f  >  ci . ti | ------------  +  ------------------------- |.
            5     /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                  i=-k

calc_S2f_tf_S2s_ts_jw(data, params)

Spectral density function.

Calculate the spectral density values for the extended model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The model-free formula is:

               _k_
            2  \           /   S2f . S2s        (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ 
   J(w)  =  -   >  ci . ti | ------------  +  -------------------------  +  ------------------------- |.
            5  /__         \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
               i=-k

calc_diff_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the original model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free gradient is:

                _k_
   dJ(w)     2  \        dti    1 - (w.ti)^2
   -----  =  -   >  ci . ---  ----------------.
    dGj      5  /__      dGj  (1 + (w.ti)^2)^2
                i=-k

calc_ellipsoid_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the original model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free gradient is:

                _k_
   dJ(w)     2  \   /      dti    1 - (w.ti)^2       dci           1       \ 
   -----  =  -   >  | ci . ---  ----------------  +  --- . ti ------------ |.
    dGj      5  /__ \      dGj  (1 + (w.ti)^2)^2     dGj      1 + (w.ti)^2 /
                i=-k

calc_diff_S2_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free gradient is:

                  _k_
   dJ(w)     2    \        dti    1 - (w.ti)^2
   -----  =  - S2  >  ci . ---  ----------------.
    dGj      5    /__      dGj  (1 + (w.ti)^2)^2
                  i=-k

calc_ellipsoid_S2_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free gradient is:

                  _k_
   dJ(w)     2    \   /      dti    1 - (w.ti)^2       dci           1       \ 
   -----  =  - S2  >  | ci . ---  ----------------  +  --- . ti ------------ |.
    dGj      5    /__ \      dGj  (1 + (w.ti)^2)^2     dGj      1 + (w.ti)^2 /
                  i=-k

calc_diff_S2_te_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free gradient is:

                _k_
   dJ(w)     2  \        dti  /      1 - (w.ti)^2                      (te + ti)^2 - (w.te.ti)^2   \ 
   -----  =  -   >  ci . ---  | S2 ----------------  +  (1 - S2)te^2 ----------------------------- |.
    dGj      5  /__      dGj  \    (1 + (w.ti)^2)^2                  ((te + ti)^2 + (w.te.ti)^2)^2 /
                i=-k

calc_ellipsoid_S2_te_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free gradient is:

                _k_
   dJ(w)     2  \   /      dti  /      1 - (w.ti)^2                      (te + ti)^2 - (w.te.ti)^2   \ 
   -----  =  -   >  | ci . ---  | S2 ----------------  +  (1 - S2)te^2 ----------------------------- |
    dGj      5  /__ \      dGj  \    (1 + (w.ti)^2)^2                  ((te + ti)^2 + (w.te.ti)^2)^2 /
                i=-k

                         dci      /      S2             (1 - S2)(te + ti)te    \ \ 
                      +  --- . ti | ------------  +  ------------------------- | |.
                         dGj      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 / /

calc_diff_djw_dOj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the O partial derivative of the original model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free gradient is:

               _k_
   dJ(w)     2 \   dci           1
   -----  =  -  >  --- . ti ------------.
    dOj      5 /__ dOj      1 + (w.ti)^2
               i=-k

calc_diff_S2_djw_dOj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the O partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free gradient is:

                  _k_
   dJ(w)     2    \   dci           1
   -----  =  - S2  >  --- . ti ------------.
    dOj      5    /__ dOj      1 + (w.ti)^2
                  i=-k

calc_diff_S2_te_djw_dOj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the O partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free gradient is:

                _k_
   dJ(w)     2  \   dci      /      S2             (1 - S2)(te + ti)te    \ 
   -----  =  -   >  --- . ti | ------------  +  ------------------------- |.
    dOj      5  /__ dOj      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
                i=-k

calc_S2_djw_dS2(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2 partial derivative of the original model-free formula with the single parameter {S2} with or without diffusion tensor parameters.

The model-free gradient is:

                _k_
   dJ(w)     2  \                1
   -----  =  -   >  ci . ti ------------.
    dS2      5  /__         1 + (w.ti)^2
                i=-k

calc_S2_te_djw_dS2(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2 partial derivative of the original model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.

The model-free gradient is:

                _k_
   dJ(w)     2  \           /      1                 (te + ti)te         \ 
   -----  =  -   >  ci . ti | ------------  -  ------------------------- |.
    dS2      5  /__         \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
                i=-k

calc_S2_te_djw_dte(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the te partial derivative of the original model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.

The model-free gradient is:

                        _k_
   dJ(w)     2          \               (te + ti)^2 - (w.te.ti)^2
   -----  =  - (1 - S2)  >  ci . ti^2 -----------------------------.
    dte      5          /__           ((te + ti)^2 + (w.te.ti)^2)^2
                        i=-k

calc_diff_S2f_S2_ts_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \        dti  /      1 - (w.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ 
   -----  =  -   >  ci . ---  | S2 ----------------  +  (S2f - S2)ts^2 ----------------------------- |.
    dGj      5  /__      dGj  \    (1 + (w.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                i=-k

calc_ellipsoid_S2f_S2_ts_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \   /      dti  /      1 - (w.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ 
   -----  =  -   >  | ci . ---  | S2 ----------------  +  (S2f - S2)ts^2 ----------------------------- |
    dGj      5  /__ \      dGj  \    (1 + (w.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                i=-k

                         dci      /      S2            (S2f - S2)(ts + ti)ts   \ \ 
                      +  --- . ti | ------------  +  ------------------------- | |.
                         dGj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2_ts_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \        dti  /      1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ 
   -----  =  -   >  ci . ---  | S2 ----------------  +  (1 - S2f)tf^2 -----------------------------  +  (S2f - S2)ts^2 ----------------------------- |.
    dGj      5  /__      dGj  \    (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                i=-k

calc_ellipsoid_S2f_tf_S2_ts_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \   /      dti  /      1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ 
   -----  =  -   >  | ci . ---  | S2 ----------------  +  (1 - S2f)tf^2 -----------------------------  +  (S2f - S2)ts^2 ----------------------------- |
    dGj      5  /__ \      dGj  \    (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                i=-k

                         dci      /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ \ 
                      +  --- . ti | ------------  +  -------------------------  +  ------------------------- | |.
                         dGj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2_ts_djw_dOj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the O partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \   dci      /      S2            (S2f - S2)(ts + ti)ts   \ 
   -----  =  -   >  --- . ti | ------------  +  ------------------------- |.
    dOj      5  /__ dOj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                i=-k

calc_diff_S2f_tf_S2_ts_djw_dOj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the O partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \   dci      /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ 
   -----  =  -   >  --- . ti | ------------  +  -------------------------  +  ------------------------- |.
    dOj      5  /__ dOj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                i=-k

calc_S2f_S2_ts_djw_dS2(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2 partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \           /      1                 (ts + ti).ts        \ 
   -----  =  -   >  ci . ti | ------------  -  ------------------------- |.
    dS2      5  /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                i=-k

calc_S2f_S2_ts_djw_dS2f(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2f partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} with or without diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \                 (ts + ti).ts
   -----  =  -   >  ci . ti -------------------------.
   dS2f      5  /__         (ts + ti)^2 + (w.ts.ti)^2
                i=-k

calc_S2f_tf_S2_ts_djw_dS2f(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2f partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The formula is:

                  _k_
   dJ(w)       2  \           /       (tf + ti).tf                  (ts + ti).ts        \ 
   -----  =  - -   >  ci . ti | -------------------------  -  ------------------------- |.
   dS2f        5  /__         \ (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                  i=-k

calc_S2f_tf_S2_ts_djw_dtf(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the tf partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The formula is:

                         _k_
   dJ(w)     2           \               (tf + ti)^2 - (w.tf.ti)^2
   -----  =  - (1 - S2f)  >  ci . ti^2 -----------------------------.
    dtf      5           /__           ((tf + ti)^2 + (w.tf.ti)^2)^2
                         i=-k

calc_S2f_S2_ts_djw_dts(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the ts partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The formula is:

                          _k_
   dJ(w)     2            \               (ts + ti)^2 - (w.ts.ti)^2
   -----  =  - (S2f - S2)  >  ci . ti^2 -----------------------------.
    dts      5            /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                          i=-k

calc_diff_S2f_S2s_ts_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The formula is:

                   _k_
   dJ(w)     2     \        dti  /       1 - (w.ti)^2                       (ts + ti)^2 - (w.ts.ti)^2   \ 
   -----  =  - S2f  >  ci . ---  | S2s ----------------  +  (1 - S2s)ts^2 ----------------------------- |.
    dGj      5     /__      dGj  \     (1 + (w.ti)^2)^2                   ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                   i=-k

calc_ellipsoid_S2f_S2s_ts_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The formula is:

                   _k_
   dJ(w)     2     \   /      dti  /       1 - (w.ti)^2                       (ts + ti)^2 - (w.ts.ti)^2   \ 
   -----  =  - S2f  >  | ci . ---  | S2s ----------------  +  (1 - S2s)ts^2 ----------------------------- |
    dGj      5     /__ \      dGj  \     (1 + (w.ti)^2)^2                   ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                   i=-k

                            dci      /     S2s             (1 - S2s)(ts + ti)ts   \ \ 
                         +  --- . ti | ------------  +  ------------------------- | |.
                            dGj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \        dti  /           1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
   -----  =  -   >  ci . ---  | S2f.S2s ----------------  +  (1 - S2f)tf^2 -----------------------------  +  S2f(1 - S2s)ts^2 ----------------------------- |
    dGj      5  /__      dGj  \         (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                i=-k

calc_ellipsoid_S2f_tf_S2s_ts_djw_dGj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Gj partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \   /      dti  /           1 - (w.ti)^2                        (tf + ti)^2 - (w.tf.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
   -----  =  -   >  | ci . ---  | S2f.S2s ----------------  +  (1 - S2f).tf^2 -----------------------------  +  S2f(1 - S2s).ts^2 ----------------------------- |
    dGj      5  /__ \      dGj  \         (1 + (w.ti)^2)^2                    ((tf + ti)^2 + (w.tf.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                i=-k

                         dci      /  S2f . S2s         (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ \ 
                      +  --- . ti | ------------  +  -------------------------  +  ------------------------- | |.
                         dGj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2s_ts_djw_dOj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the O partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The formula is:

                   _k_
   dJ(w)     2     \   dci      /     S2s             (1 - S2s)(ts + ti)ts   \ 
   -----  =  - S2f  >  --- . ti | ------------  +  ------------------------- |.
    dOj      5     /__ dOj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                   i=-k

calc_diff_S2f_tf_S2s_ts_djw_dOj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the O partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \   dci      /  S2f . S2s         (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ 
   -----  =  -   >  --- . ti | ------------  +  -------------------------  +  ------------------------- |.
    dOj      5  /__ dOj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                i=-k

calc_S2f_S2s_ts_djw_dS2f(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2f partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} with or without diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \           /     S2s            (1 - S2s)(ts + ti).ts   \ 
   -----  =  -   >  ci . ti | ------------  +  ------------------------- |.
   dS2f      5  /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                i=-k

calc_S2f_tf_S2s_ts_djw_dS2f(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2f partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The formula is:

                _k_
   dJ(w)     2  \           /     S2s                (tf + ti).tf              (1 - S2s)(ts + ti).ts   \ 
   -----  =  -   >  ci . ti | ------------  -  -------------------------  +  ------------------------- |.
   dS2f      5  /__         \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                i=-k

calc_S2f_tf_S2s_ts_djw_dS2s(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2s partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The formula is:

                   _k_
   dJ(w)     2     \           /      1                 (ts + ti).ts        \ 
   -----  =  - S2f  >  ci . ti | ------------  -  ------------------------- |.
   dS2s      5     /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                   i=-k

calc_S2f_tf_S2s_ts_djw_dtf(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the tf partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The formula is:

                         _k_
   dJ(w)     2           \               (tf + ti)^2 - (w.tf.ti)^2
   -----  =  - (1 - S2f)  >  ci . ti^2 -----------------------------.
    dtf      5           /__           ((tf + ti)^2 + (w.tf.ti)^2)^2
                         i=-k

calc_S2f_S2s_ts_djw_dts(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the ts partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The formula is:

                            _k_
   dJ(w)     2              \               (ts + ti)^2 - (w.ts.ti)^2
   -----  =  - S2f(1 - S2s)  >  ci . ti^2 -----------------------------.
    dts      5              /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                            i=-k

calc_diff_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the original model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \      /    dti   dti           3 - (w.ti)^2        d2ti     1 - (w.ti)^2   \ 
   -------  =  -   >  ci | -2 --- . ---  w^2.ti ----------------  +  ------- ---------------- |.
   dGj.dGk     5  /__    \    dGj   dGk         (1 + (w.ti)^2)^3     dGj.dGk (1 + (w.ti)^2)^2 /
                  i=-k

calc_ellipsoid_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the original model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   /      dti   dti           3 - (w.ti)^2       / dti   dci     dti   dci         d2ti   \   1 - (w.ti)^2        d2ci           1       \ 
   -------  =  -   >  | -2ci --- . ---  w^2.ti ----------------  +  | --- . ---  +  --- . ---  +  ci ------- | ----------------  +  ------- ti ------------ |.
   dGj.dGk     5  /__ \      dGj   dGk         (1 + (w.ti)^2)^3     \ dGj   dGk     dGk   dGj        dGj.dGk / (1 + (w.ti)^2)^2     dGj.dGk    1 + (w.ti)^2 /
                  i=-k

calc_diff_S2_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                    _k_
    d2J(w)     2    \      /    dti   dti           3 - (w.ti)^2        d2ti     1 - (w.ti)^2   \ 
   -------  =  - S2  >  ci | -2 --- . ---  w^2.ti ----------------  +  ------- ---------------- |.
   dGj.dGk     5    /__    \    dGj   dGk         (1 + (w.ti)^2)^3     dGj.dGk (1 + (w.ti)^2)^2 /
                    i=-k

calc_ellipsoid_S2_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                    _k_
    d2J(w)     2    \   /      dti   dti           3 - (w.ti)^2       / dti   dci     dti   dci         d2ti   \   1 - (w.ti)^2        d2ci           1       \ 
   -------  =  - S2  >  | -2ci --- . ---  w^2.ti ----------------  +  | --- . ---  +  --- . ---  +  ci ------- | ----------------  +  ------- ti ------------ |.
   dGj.dGk     5    /__ \      dGj   dGk         (1 + (w.ti)^2)^3     \ dGj   dGk     dGk   dGj        dGj.dGk / (1 + (w.ti)^2)^2     dGj.dGk    1 + (w.ti)^2 /
                    i=-k

calc_diff_S2_te_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \      /    dti   dti  /             3 - (w.ti)^2                    (te + ti)^3 + 3.w^2.te^3.ti(te + ti) - (w.te)^4.ti^3 \ 
   -------  =  -   >  ci | -2 --- . ---  | S2.w^2.ti ----------------  +  (1 - S2)te^2 ---------------------------------------------------- |
   dGj.dGk     5  /__    \    dGj   dGk  \           (1 + (w.ti)^2)^3                            ((te + ti)^2 + (w.te.ti)^2)^3              /
                  i=-k

                            d2ti   /      1 - (w.ti)^2                      (te + ti)^2 - (w.te.ti)^2   \ \ 
                        +  ------- | S2 ----------------  +  (1 - S2)te^2 ----------------------------- | |.
                           dGj.dGk \    (1 + (w.ti)^2)^2                  ((te + ti)^2 + (w.te.ti)^2)^2 / /

calc_ellipsoid_S2_te_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   /      dti   dti  /             3 - (w.ti)^2                    (te + ti)^3 + 3.w^2.te^3.ti(te + ti) - (w.te)^4.ti^3 \ 
   -------  =  -   >  | -2ci --- . ---  | S2.w^2.ti ----------------  +  (1 - S2)te^2 ---------------------------------------------------- |
   dGj.dGk     5  /__ \      dGj   dGk  \           (1 + (w.ti)^2)^3                            ((te + ti)^2 + (w.te.ti)^2)^3              /
                  i=-k

                           / dti   dci     dti   dci         d2ti   \ /      1 - (w.ti)^2                      (te + ti)^2 - (w.te.ti)^2   \ 
                        +  | --- . ---  +  --- . ---  +  ci ------- | | S2 ----------------  +  (1 - S2)te^2 ----------------------------- |
                           \ dGj   dGk     dGk   dGj        dGj.dGk / \    (1 + (w.ti)^2)^2                  ((te + ti)^2 + (w.te.ti)^2)^2 /


                            d2ci      /      S2             (1 - S2)(te + ti)te    \ \ 
                        +  ------- ti | ------------  +  ------------------------- | |.
                           dGj.dGk    \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 / /

calc_diff_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the original model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   dci   dti     1 - (w.ti)^2
   -------  =  -   >  --- . --- . ----------------.
   dGj.dOj     5  /__ dOj   dGj   (1 + (w.ti)^2)^2
                  i=-k

calc_ellipsoid_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the original model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   / dci   dti     1 - (w.ti)^2        d2ci           1       \ 
   -------  =  -   >  | --- . --- . ----------------  +  ------- ti ------------ |.
   dGj.dOj     5  /__ \ dOj   dGj   (1 + (w.ti)^2)^2     dGj.dOj    1 + (w.ti)^2 /
                  i=-k

calc_diff_S2_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                    _k_
    d2J(w)     2    \   dci   dti     1 - (w.ti)^2
   -------  =  - S2  >  --- . --- . ----------------.
   dGj.dOj     5    /__ dOj   dGj   (1 + (w.ti)^2)^2
                    i=-k

calc_ellipsoid_S2_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                    _k_
    d2J(w)     2    \   / dci   dti     1 - (w.ti)^2        d2ci           1       \ 
   -------  =  - S2  >  | --- . --- . ----------------  +  ------- ti ------------ |.
   dGj.dOj     5    /__ \ dOj   dGj   (1 + (w.ti)^2)^2     dGj.dOj    1 + (w.ti)^2 /
                    i=-k

calc_diff_S2_te_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   dci   dti  /      1 - (w.ti)^2                      (te + ti)^2 - (w.te.ti)^2   \ 
   -------  =  -   >  --- . ---  | S2 ----------------  +  (1 - S2)te^2 ----------------------------- |.
   dGj.dOj     5  /__ dOj   dGj  \    (1 + (w.ti)^2)^2                  ((te + ti)^2 + (w.te.ti)^2)^2 /
                  i=-k

calc_ellipsoid_S2_te_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   / dci   dti  /      1 - (w.ti)^2                      (te + ti)^2 - (w.te.ti)^2   \ 
   -------  =  -   >  | --- . ---  | S2 ----------------  +  (1 - S2)te^2 ----------------------------- |
   dGj.dOj     5  /__ \ dOj   dGj  \    (1 + (w.ti)^2)^2                  ((te + ti)^2 + (w.te.ti)^2)^2 /
                  i=-k

                              d2ci      /      S2             (1 - S2)(te + ti)te    \ \ 
                          +  ------- ti | ------------  +  ------------------------- | |.
                             dGj.dOj    \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 / /

calc_diff_S2_d2jw_dGjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2 double partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \        dti    1 - (w.ti)^2
   -------  =  -   >  ci . ---  ----------------.
   dGj.dS2     5  /__      dGj  (1 + (w.ti)^2)^2
                  i=-k

calc_ellipsoid_S2_d2jw_dGjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2 double partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   /      dti    1 - (w.ti)^2       dci           1       \ 
   -------  =  -   >  | ci . ---  ----------------  +  --- . ti ------------ |.
   dGj.dS2     5  /__ \      dGj  (1 + (w.ti)^2)^2     dGj      1 + (w.ti)^2 /
                  i=-k

calc_diff_S2_te_d2jw_dGjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2 double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \        dti  /   1 - (w.ti)^2              (te + ti)^2 - (w.te.ti)^2   \ 
   -------  =  -   >  ci . ---  | ----------------  -  te^2 ----------------------------- |.
   dGj.dS2     5  /__      dGj  \ (1 + (w.ti)^2)^2          ((te + ti)^2 + (w.te.ti)^2)^2 /
                  i=-k

calc_ellipsoid_S2_te_d2jw_dGjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2 double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   /      dti  /   1 - (w.ti)^2              (te + ti)^2 - (w.te.ti)^2   \ 
   -------  =  -   >  | ci . ---  | ----------------  -  te^2 ----------------------------- |
   dGj.dS2     5  /__ \      dGj  \ (1 + (w.ti)^2)^2          ((te + ti)^2 + (w.te.ti)^2)^2 /
                  i=-k

                           dci      /      1                 (te + ti)te         \ \ 
                        +  --- . ti | ------------  -  ------------------------- | |.
                           dGj      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 / /

calc_diff_S2_te_d2jw_dGjdte(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - te double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                               _k_
    d2J(w)     4               \        dti                   (te + ti)^2 - 3(w.te.ti)^2
   -------  =  - (1 - S2) . te  >  ci . --- . ti . (te + ti) -----------------------------.
   dGj.dte     5               /__      dGj                  ((te + ti)^2 + (w.te.ti)^2)^3
                               i=-k

calc_ellipsoid_S2_te_d2jw_dGjdte(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - te double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                         _k_
    d2J(w)     2         \   /       dti                        (te + ti)^2 - 3(w.te.ti)^2       dci          (te + ti)^2 - (w.te.ti)^2   \ 
   -------  =  - (1 - S2) >  | 2ci . --- . te . ti . (te + ti) -----------------------------  +  --- . ti^2 ----------------------------- |.
   dGj.dte     5         /__ \       dGj                       ((te + ti)^2 + (w.te.ti)^2)^3     dGj        ((te + ti)^2 + (w.te.ti)^2)^2 /
                         i=-k

calc_diff_d2jw_dOjdOk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - Ok double partial derivative of the original model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \    d2ci          ti
   -------  =  -   >  ------- . ------------.
   dOj.dOk     5  /__ dOj.dOk   1 + (w.ti)^2
                  i=-k

calc_diff_S2_d2jw_dOjdOk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - Ok double partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                    _k_
    d2J(w)     2    \    d2ci          ti
   -------  =  - S2  >  ------- . ------------.
   dOj.dOk     5    /__ dOj.dOk   1 + (w.ti)^2
                    i=-k

calc_diff_S2_te_d2jw_dOjdOk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - Ok double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \    d2ci        /      S2             (1 - S2)(te + ti)te    \ 
   -------  =  -   >  ------- . ti | ------------  +  ------------------------- |.
   dOj.dOk     5  /__ dOj.dOk      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
                  i=-k

calc_diff_S2_d2jw_dOjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - S2 double partial derivative of the original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   dci            1
   -------  =  -   >  --- . ti ------------.
   dOj.dS2     5  /__ dOj      1 + (w.ti)^2
                  i=-k

calc_diff_S2_te_d2jw_dOjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - S2 double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   dci      /      1                 (te + ti)te         \ 
   -------  =  -   >  --- . ti | ------------  -  ------------------------- |.
   dOj.dS2     5  /__ dOj      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
                  i=-k

calc_diff_S2_te_d2jw_dOjdte(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - te double partial derivative of the original model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                          _k_
    d2J(w)     2          \   dci          (te + ti)^2 - (w.te.ti)^2
   -------  =  - (1 - S2)  >  --- . ti^2 -----------------------------.
   dOj.dte     5          /__ dOj        ((te + ti)^2 + (w.te.ti)^2)^2
                          i=-k

calc_S2_te_d2jw_dS2dte(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2 - te double partial derivative of the original model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)       2 \               (te + ti)^2 - (w.te.ti)^2
   -------  =  - -  >  ci . ti^2 -----------------------------.
   dS2.dte       5 /__           ((te + ti)^2 + (w.te.ti)^2)^2
                   i=-k

calc_S2_te_d2jw_dte2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the te - te double partial derivative of the original model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.

The model-free Hessian is:

                           _k_
   d2J(w)       4          \             (te + ti)^3 + 3.w^2.ti^3.te.(te + ti) - (w.ti)^4.te^3
   ------  =  - - (1 - S2)  >  ci . ti^2 -----------------------------------------------------.
   dte**2       5          /__                        ((te + ti)^2 + (w.te.ti)^2)^3
                           i=-k

calc_diff_S2f_S2_ts_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \      /    dti   dti  /             3 - (w.ti)^2                      (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
   -------  =  -   >  ci | -2 --- . ---  | S2.w^2.ti ----------------  +  (S2f - S2)ts^2 ---------------------------------------------------- |
   dGj.dGk     5  /__    \    dGj   dGk  \           (1 + (w.ti)^2)^3                              ((ts + ti)^2 + (w.ts.ti)^2)^3              /
                  i=-k

                            d2ti   /      1 - (w.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ \ 
                        +  ------- | S2 ----------------  +  (S2f - S2)ts^2 ----------------------------- | |.
                           dGj.dGk \    (1 + (w.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 / /

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   /      dti   dti  /             3 - (w.ti)^2                      (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
   -------  =  -   >  | -2ci --- . ---  | S2.w^2.ti ----------------  +  (S2f - S2)ts^2 ---------------------------------------------------- |
   dGj.dGk     5  /__ \      dGj   dGk  \           (1 + (w.ti)^2)^3                              ((ts + ti)^2 + (w.ts.ti)^2)^3              /
                  i=-k

                           / dti   dci     dti   dci         d2ti   \ /      1 - (w.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ 
                        +  | --- . ---  +  --- . ---  +  ci ------- | | S2 ----------------  +  (S2f - S2)ts^2 ----------------------------- |
                           \ dGj   dGk     dGk   dGj        dGj.dGk / \    (1 + (w.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 /


                            d2ci        /      S2            (S2f - S2)(ts + ti)ts   \ \ 
                        +  ------- . ti | ------------  +  ------------------------- | |.
                           dGj.dGk      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2_ts_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \      /    dti   dti  /             3 - (w.ti)^2                     (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
   -------  =  -   >  ci | -2 --- . ---  | S2.w^2.ti ----------------  +  (1 - S2f)tf^2 ----------------------------------------------------
   dGj.dGk     5  /__    \    dGj   dGk  \           (1 + (w.ti)^2)^3                             ((tf + ti)^2 + (w.tf.ti)^2)^3
                  i=-k

                                                            (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
                                          +  (S2f - S2)ts^2 ---------------------------------------------------- |
                                                                      ((ts + ti)^2 + (w.ts.ti)^2)^3              /


                            d2ti   /      1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ \ 
                        +  ------- | S2 ----------------  +  (1 - S2f)tf^2 -----------------------------  +  (S2f - S2)ts^2 ----------------------------- | |.
                           dGj.dGk \    (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 / /

calc_ellipsoid_S2f_tf_S2_ts_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   /      dti   dti  /             3 - (w.ti)^2                     (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
   -------  =  -   >  | -2ci --- . ---  | S2.w^2.ti ----------------  +  (1 - S2f)tf^2 ----------------------------------------------------
   dGj.dGk     5  /__ \      dGj   dGk  \           (1 + (w.ti)^2)^3                             ((tf + ti)^2 + (w.tf.ti)^2)^3
                  i=-k

                                                            (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
                                          +  (S2f - S2)ts^2 ---------------------------------------------------- |
                                                                      ((ts + ti)^2 + (w.ts.ti)^2)^3              /


                           / dti   dci     dti   dci         d2ti   \ /      1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2
                        +  | --- . ---  +  --- . ---  +  ci ------- | | S2 ----------------  +  (1 - S2f)tf^2 -----------------------------
                           \ dGj   dGk     dGk   dGj        dGj.dGk / \    (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2


                                                                                            (ts + ti)^2 - (w.ts.ti)^2   \ 
                                                                        +  (S2f - S2)ts^2 ----------------------------- |
                                                                                          ((ts + ti)^2 + (w.ts.ti)^2)^2 /


                            d2ci        /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ \ 
                        +  ------- . ti | ------------  +  -------------------------  +  ------------------------- | |.
                           dGj.dGk      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2_ts_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   dci   dti  /      1 - (w.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  --- . ---  | S2 ----------------  +  (S2f - S2)ts^2 ----------------------------- |.
   dGj.dOj     5  /__ dOj   dGj  \    (1 + (w.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   / dci   dti  /      1 - (w.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  | --- . ---  | S2 ----------------  +  (S2f - S2)ts^2 ----------------------------- |
   dGj.dOj     5  /__ \ dOj   dGj  \    (1 + (w.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

                              d2ci        /      S2            (S2f - S2)(ts + ti)ts   \ \ 
                          +  ------- . ti | ------------  +  ------------------------- | |.
                             dGj.dOj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2_ts_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   dci   dti  /      1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  --- . ---  | S2 ----------------  +  (1 - S2f)tf^2 -----------------------------  +  (S2f - S2)ts^2 ----------------------------- |.
   dGj.dOj     5  /__ dOj   dGj  \    (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

calc_ellipsoid_S2f_tf_S2_ts_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   / dci   dti  /      1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2                        (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  | --- . ---  | S2 ----------------  +  (1 - S2f)tf^2 -----------------------------  +  (S2f - S2)ts^2 ----------------------------- |
   dGj.dOj     5  /__ \ dOj   dGj  \    (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2                    ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

                              d2ci        /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ \ 
                          +  ------- . ti | ------------  +  -------------------------  +  ------------------------- | |.
                             dGj.dOj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2_ts_d2jw_dGjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2 double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \        dti  /   1 - (w.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  ci . ---  | ----------------  -  ts^2 ----------------------------- |.
   dGj.dS2     5  /__      dGj  \ (1 + (w.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2 double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   /      dti  /   1 - (w.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  | ci . ---  | ----------------  -  ts^2 ----------------------------- |
   dGj.dS2     5  /__ \      dGj  \ (1 + (w.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

                           dci      /      1                 (ts + ti)ts         \ \ 
                        +  --- . ti | ------------  -  ------------------------- | |.
                           dGj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2_ts_d2jw_dGjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \        dti         (ts + ti)^2 - (w.ts.ti)^2
   --------  =  -   >  ci . ---  ts^2 -----------------------------.
   dGj.dS2f     5  /__      dGj       ((ts + ti)^2 + (w.ts.ti)^2)^2
                   i=-k

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \   /      dti         (ts + ti)^2 - (w.ts.ti)^2       dci            (ts + ti)ts         \ 
   --------  =  -   >  | ci . ---  ts^2 -----------------------------  +  --- . ti ------------------------- |.
   dGj.dS2f     5  /__ \      dGj       ((ts + ti)^2 + (w.ts.ti)^2)^2     dGj      (ts + ti)^2 + (w.ts.ti)^2 /
                   i=-k

calc_diff_S2f_tf_S2_ts_d2jw_dGjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                     _k_
    d2J(w)        2  \        dti  /        (tf + ti)^2 - (w.tf.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
   --------  =  - -   >  ci . ---  | tf^2 -----------------------------  -  ts^2 ----------------------------- |.
   dGj.dS2f       5  /__      dGj  \      ((tf + ti)^2 + (w.tf.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=-k

calc_ellipsoid_S2f_tf_S2_ts_d2jw_dGjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                     _k_
    d2J(w)        2  \   /      dti  /        (tf + ti)^2 - (w.tf.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
   --------  =  - -   >  | ci . ---  | tf^2 -----------------------------  -  ts^2 ----------------------------- |
   dGj.dS2f       5  /__ \      dGj  \      ((tf + ti)^2 + (w.tf.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=-k

                              dci      /       (tf + ti)tf                   (ts + ti)ts         \ \ 
                           +  --- . ti | -------------------------  -  ------------------------- | |.
                              dGj      \ (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2_ts_d2jw_dGjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                                _k_
    d2J(w)     4                \        dti                   (tf + ti)^2 - 3(w.tf.ti)^2
   -------  =  - (1 - S2f) . tf  >  ci . --- . ti . (tf + ti) -----------------------------.
   dGj.dtf     5                /__      dGj                  ((tf + ti)^2 + (w.tf.ti)^2)^3
                                i=-k

calc_ellipsoid_S2f_tf_S2_ts_d2jw_dGjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                           _k_
    d2J(w)     2           \   /       dti                        (tf + ti)^2 - 3(w.tf.ti)^2       dci          (tf + ti)^2 - (w.tf.ti)^2   \ 
   -------  =  - (1 - S2f)  >  | 2ci . --- . tf . ti . (tf + ti) -----------------------------  +  --- . ti^2 ----------------------------- |.
   dGj.dtf     5           /__ \       dGj                       ((tf + ti)^2 + (w.tf.ti)^2)^3     dGj        ((tf + ti)^2 + (w.tf.ti)^2)^2 /
                           i=-k

calc_diff_S2f_S2_ts_d2jw_dGjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                                 _k_
    d2J(w)     4                 \        dti                   (ts + ti)^2 - 3(w.ts.ti)^2
   -------  =  - (S2f - S2) . ts  >  ci . --- . ti . (ts + ti) -----------------------------.
   dGj.dts     5                 /__      dGj                  ((ts + ti)^2 + (w.ts.ti)^2)^3
                                 i=-k

calc_ellipsoid_S2f_S2_ts_d2jw_dGjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                            _k_
    d2J(w)     2            \   /       dti                        (ts + ti)^2 - 3(w.ts.ti)^2       dci          (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  - (S2f - S2)  >  | 2ci . --- . ts . ti . (ts + ti) -----------------------------  +  --- . ti^2 ----------------------------- |.
   dGj.dts     5            /__ \       dGj                       ((ts + ti)^2 + (w.ts.ti)^2)^3     dGj        ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                            i=-k

calc_diff_S2f_S2_ts_d2jw_dOjdOk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - Ok double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \    d2ci        /      S2            (S2f - S2)(ts + ti)ts   \ 
   -------  =  -   >  ------- . ti | ------------  +  ------------------------- |.
   dOj.dOk     5  /__ dOj.dOk      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                  i=-k

calc_diff_S2f_tf_S2_ts_d2jw_dOjdOk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - Ok double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \    d2ci        /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ 
   -------  =  -   >  ------- . ti | ------------  +  -------------------------  +  ------------------------- |.
   dOj.dOk     5  /__ dOj.dOk      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                  i=-k

calc_diff_S2f_S2_ts_d2jw_dOjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - S2 double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} and {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   dci      /      1                 (ts + ti)ts         \ 
   -------  =  -   >  --- . ti | ------------  -  ------------------------- |.
   dOj.dS2     5  /__ dOj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                  i=-k

calc_diff_S2f_S2_ts_d2jw_dOjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \   dci             (ts + ti)ts
   --------  =  -   >  --- . ti -------------------------.
   dOj.dS2f     5  /__ dOj      (ts + ti)^2 + (w.ts.ti)^2
                   i=-k

calc_diff_S2f_tf_S2_ts_d2jw_dOjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                     _k_
    d2J(w)        2  \   dci      /       (tf + ti)tf                   (ts + ti)ts         \ 
   --------  =  - -   >  --- . ti | -------------------------  -  ------------------------- |.
   dOj.dS2f       5  /__ dOj      \ (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                     i=-k

calc_diff_S2f_tf_S2_ts_d2jw_dOjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                           _k_
    d2J(w)     2           \   dci          (tf + ti)^2 - (w.tf.ti)^2
   -------  =  - (1 - S2f)  >  --- . ti^2 -----------------------------.
   dOj.dtf     5           /__ dOj        ((tf + ti)^2 + (w.tf.ti)^2)^2
                           i=-k

calc_diff_S2f_S2_ts_d2jw_dOjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                            _k_
    d2J(w)     2            \   dci          (ts + ti)^2 - (w.ts.ti)^2
   -------  =  - (S2f - S2)  >  --- . ti^2 -----------------------------.
   dOj.dts     5            /__ dOj        ((ts + ti)^2 + (w.ts.ti)^2)^2
                            i=-k

calc_S2f_S2_ts_d2jw_dS2dts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2 - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                    _k_
    d2J(w)       2  \               (ts + ti)^2 - (w.ts.ti)^2
   -------  =  - -   >  ci . ti^2 -----------------------------.
   dS2.dts       5  /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                    i=-k

calc_S2f_tf_S2_ts_d2jw_dS2fdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                     _k_
    d2J(w)        2  \               (tf + ti)^2 - (w.tf.ti)^2
   --------  =  - -   >  ci . ti^2 -----------------------------.
   dS2f.dtf       5  /__           ((tf + ti)^2 + (w.tf.ti)^2)^2
                     i=-k

calc_S2f_S2_ts_d2jw_dS2fdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \               (ts + ti)^2 - (w.ts.ti)^2
   --------  =  -   >  ci . ti^2 -----------------------------.
   dS2f.dts     5  /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                   i=-k

calc_S2f_tf_S2_ts_d2jw_dtf2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the tf - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                            _k_
   d2J(w)       4           \             (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
   ------  =  - - (1 - S2f)  >  ci . ti^2 -----------------------------------------------------.
   dtf**2       5           /__                        ((tf + ti)^2 + (w.tf.ti)^2)^3
                            i=-k

calc_S2f_S2_ts_d2jw_dts2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the ts - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                             _k_
   d2J(w)       4            \             (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
   ------  =  - - (S2f - S2)  >  ci . ti^2 -----------------------------------------------------.
   dts**2       5            /__                        ((ts + ti)^2 + (w.ts.ti)^2)^3
                             i=-k

calc_diff_S2f_S2s_ts_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \      /    dti   dti  /                  3 - (w.ti)^2                        (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
   -------  =  -   >  ci | -2 --- . ---  | S2f.S2s.w^2.ti ----------------  +  S2f(1 - S2s)ts^2 ---------------------------------------------------- |
   dGj.dGk     5  /__    \    dGj   dGk  \                (1 + (w.ti)^2)^3                                ((ts + ti)^2 + (w.ts.ti)^2)^3              /
                  i=-k

                            d2ti   /           1 - (w.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ \ 
                        +  ------- | S2f.S2s ----------------  +  S2f(1 - S2s)ts^2 ----------------------------- | |.
                           dGj.dGk \         (1 + (w.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 / /

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   /      dti   dti  /                  3 - (w.ti)^2                        (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
   -------  =  -   >  | -2ci --- . ---  | S2f.S2s.w^2.ti ----------------  +  S2f(1 - S2s)ts^2 ---------------------------------------------------- |
   dGj.dGk     5  /__ \      dGj   dGk  \                (1 + (w.ti)^2)^3                                ((ts + ti)^2 + (w.ts.ti)^2)^3              /
                  i=-k

                           / dti   dci     dti   dci         d2ti   \ /           1 - (w.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
                        +  | --- . ---  +  --- . ---  +  ci ------- | | S2f.S2s ----------------  +  S2f(1 - S2s)ts^2 ----------------------------- |
                           \ dGj   dGk     dGk   dGj        dGj.dGk / \         (1 + (w.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /


                            d2ci        /   S2f.S2s         S2f(1 - S2s)(ts + ti)ts  \ \ 
                        +  ------- . ti | ------------  +  ------------------------- | |.
                           dGj.dGk      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \      /    dti   dti  /                  3 - (w.ti)^2                     (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
   -------  =  -   >  ci | -2 --- . ---  | S2f.S2s.w^2.ti ----------------  +  (1 - S2f)tf^2 ----------------------------------------------------
   dGj.dGk     5  /__    \    dGj   dGk  \                (1 + (w.ti)^2)^3                             ((tf + ti)^2 + (w.tf.ti)^2)^3
                  i=-k

                                                              (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
                                          +  S2f(1 - S2s)ts^2 ---------------------------------------------------- |
                                                                        ((ts + ti)^2 + (w.ts.ti)^2)^3              /


                   d2ti   /           1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
               +  ------- | S2f.S2s ----------------  +  (1 - S2f)tf^2 -----------------------------  +  S2f(1 - S2s)ts^2 ----------------------------- |.
                  dGj.dGk \         (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /

calc_ellipsoid_S2f_tf_S2s_ts_d2jw_dGjdGk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   /      dti   dti  /                  3 - (w.ti)^2                     (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
   -------  =  -   >  | -2ci --- . ---  | S2f.S2s.w^2.ti ----------------  +  (1 - S2f)tf^2 ----------------------------------------------------
   dGj.dGk     5  /__ \      dGj   dGk  \                (1 + (w.ti)^2)^3                             ((tf + ti)^2 + (w.tf.ti)^2)^3
                  i=-k

                                                              (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
                                          +  S2f(1 - S2s)ts^2 ---------------------------------------------------- |
                                                                        ((ts + ti)^2 + (w.ts.ti)^2)^3              /


                           / dti   dci     dti   dci         d2ti   \ /           1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2
                        +  | --- . ---  +  --- . ---  +  ci ------- | | S2f.S2s ----------------  +  (1 - S2f)tf^2 -----------------------------
                           \ dGj   dGk     dGk   dGj        dGj.dGk / \         (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2


                                                                                              (ts + ti)^2 - (w.ts.ti)^2   \ 
                                                                        +  S2f(1 - S2s)ts^2 ----------------------------- |
                                                                                            ((ts + ti)^2 + (w.ts.ti)^2)^2 /


                            d2ci        /   S2f.S2s          (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ \ 
                        +  ------- . ti | ------------  +  -------------------------  +  ------------------------- | |.
                           dGj.dGk      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2s_ts_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   dci   dti  /           1 - (w.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  --- . ---  | S2f.S2s ----------------  +  S2f(1 - S2s)ts^2 ----------------------------- |.
   dGj.dOj     5  /__ dOj   dGj  \         (1 + (w.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   / dci   dti  /           1 - (w.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  | --- . ---  | S2f.S2s ----------------  +  S2f(1 - S2s)ts^2 ----------------------------- |
   dGj.dOj     5  /__ \ dOj   dGj  \         (1 + (w.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

                              d2ci        /   S2f.S2s         S2f(1 - S2s)(ts + ti)ts  \ \ 
                          +  ------- . ti | ------------  +  ------------------------- | |.
                             dGj.dOj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   dci   dti  /           1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  --- . ---  | S2f.S2s ----------------  +  (1 - S2f)tf^2 -----------------------------  +  S2f(1 - S2s)ts^2 ----------------------------- |.
   dGj.dOj     5  /__ dOj   dGj  \         (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

calc_ellipsoid_S2f_tf_S2s_ts_d2jw_dGjdOj(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \   / dci   dti  /           1 - (w.ti)^2                       (tf + ti)^2 - (w.tf.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  -   >  | --- . ---  | S2f.S2s ----------------  +  (1 - S2f)tf^2 -----------------------------  +  S2f(1 - S2s)ts^2 ----------------------------- |
   dGj.dOj     5  /__ \ dOj   dGj  \         (1 + (w.ti)^2)^2                   ((tf + ti)^2 + (w.tf.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                  i=-k

                              d2ci        /   S2f.S2s          (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ \ 
                          +  ------- . ti | ------------  +  -------------------------  +  ------------------------- | |.
                             dGj.dOj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2s_ts_d2jw_dGjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \        dti  /       1 - (w.ti)^2                       (ts + ti)^2 - (w.ts.ti)^2   \ 
   --------  =  -   >  ci . ---  | S2s ----------------  +  (1 - S2s)ts^2 ----------------------------- |.
   dGj.dS2f     5  /__      dGj  \     (1 + (w.ti)^2)^2                   ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                   i=-k

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \   /      dti  /       1 - (w.ti)^2                       (ts + ti)^2 - (w.ts.ti)^2   \ 
   --------  =  -   >  | ci . ---  | S2s ----------------  +  (1 - S2s)ts^2 ----------------------------- |
   dGj.dS2f     5  /__ \      dGj  \     (1 + (w.ti)^2)^2                   ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                   i=-k

                           dci      /     S2s            (1 - S2s)(ts + ti)ts    \ \ 
                        +  --- . ti | ------------  +  ------------------------- | |.
                           dGj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_d2jw_dGjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \        dti  /       1 - (w.ti)^2              (tf + ti)^2 - (w.tf.ti)^2                       (ts + ti)^2 - (w.ts.ti)^2   \ 
   --------  =  -   >  ci . ---  | S2s ----------------  -  tf^2 -----------------------------  +  (1 - S2s)ts^2 ----------------------------- |.
   dGj.dS2f     5  /__      dGj  \     (1 + (w.ti)^2)^2          ((tf + ti)^2 + (w.tf.ti)^2)^2                   ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                   i=-k

calc_ellipsoid_S2f_tf_S2s_ts_d2jw_dGjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \   /      dti  /       1 - (w.ti)^2              (tf + ti)^2 - (w.tf.ti)^2                       (ts + ti)^2 - (w.ts.ti)^2   \ 
   --------  =  -   >  | ci . ---  | S2s ----------------  -  tf^2 -----------------------------  +  (1 - S2s)ts^2 ----------------------------- |
   dGj.dS2f     5  /__ \      dGj  \     (1 + (w.ti)^2)^2          ((tf + ti)^2 + (w.tf.ti)^2)^2                   ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                   i=-k

                            dci      /     S2s                (tf + ti)tf               (1 - S2s)(ts + ti)ts    \ \ 
                         +  --- . ti | ------------  -  -------------------------  +  ------------------------- | |.
                            dGj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2s_ts_d2jw_dGjdS2s(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2s double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                      _k_
    d2J(w)      2     \        dti  /   1 - (w.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
   --------  =  - S2f  >  ci . ---  | ----------------  -  ts^2 ----------------------------- |.
   dGj.dS2s     5     /__      dGj  \ (1 + (w.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                      i=-k

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdS2s(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - S2s double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                      _k_
    d2J(w)      2     \   /      dti  /   1 - (w.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
   --------  =  - S2f  >  | ci . ---  | ----------------  -  ts^2 ----------------------------- |
   dGj.dS2s     5     /__ \      dGj  \ (1 + (w.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                      i=-k

                           dci      /      1                 (ts + ti)ts         \ \ 
                        +  --- . ti | ------------  -  ------------------------- | |.
                           dGj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_d2jw_dGjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                                _k_
    d2J(w)     4                \        dti                   (tf + ti)^2 - 3(w.tf.ti)^2
   -------  =  - (1 - S2f) . tf  >  ci . --- . ti . (tf + ti) -----------------------------.
   dGj.dtf     5                /__      dGj                  ((tf + ti)^2 + (w.tf.ti)^2)^3
                                i=-k

calc_ellipsoid_S2f_tf_S2s_ts_d2jw_dGjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                           _k_
    d2J(w)     2           \   /       dti                        (tf + ti)^2 - 3(w.tf.ti)^2       dci          (tf + ti)^2 - (w.tf.ti)^2   \ 
   -------  =  - (1 - S2f)  >  | 2ci . --- . tf . ti . (tf + ti) -----------------------------  +  --- . ti^2 ----------------------------- |.
   dGj.dtf     5           /__ \       dGj                       ((tf + ti)^2 + (w.tf.ti)^2)^3     dGj        ((tf + ti)^2 + (w.tf.ti)^2)^2 /
                           i=-k

calc_diff_S2f_S2s_ts_d2jw_dGjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                                   _k_
    d2J(w)     4                   \        dti                   (ts + ti)^2 - 3(w.ts.ti)^2
   -------  =  - S2f(1 - S2s) . ts  >  ci . --- . ti . (ts + ti) -----------------------------.
   dGj.dts     5                   /__      dGj                  ((ts + ti)^2 + (w.ts.ti)^2)^3
                                   i=-k

calc_ellipsoid_S2f_S2s_ts_d2jw_dGjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Gj - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                              _k_
    d2J(w)     2              \   /       dti                        (ts + ti)^2 - 3(w.ts.ti)^2       dci          (ts + ti)^2 - (w.ts.ti)^2   \ 
   -------  =  - S2f(1 - S2s)  >  | 2ci . --- . ts . ti . (ts + ti) -----------------------------  +  --- . ti^2 ----------------------------- |.
   dGj.dts     5              /__ \       dGj                       ((ts + ti)^2 + (w.ts.ti)^2)^3     dGj        ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                              i=-k

calc_diff_S2f_S2s_ts_d2jw_dOjdOk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - Ok double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \    d2ci        /  S2f . S2s        S2f(1 - S2s)(ts + ti)ts  \ 
   -------  =  -   >  ------- . ti | ------------  +  ------------------------- |.
   dOj.dOk     5  /__ dOj.dOk      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                  i=-k

calc_diff_S2f_tf_S2s_ts_d2jw_dOjdOk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - Ok double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                  _k_
    d2J(w)     2  \    d2ci        /  S2f . S2s         (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ 
   -------  =  -   >  ------- . ti | ------------  +  -------------------------  +  ------------------------- |.
   dOj.dOk     5  /__ dOj.dOk      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                  i=-k

calc_diff_S2f_S2s_ts_d2jw_dOjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \   dci      /      S2s           (1 - S2s)(ts + ti)ts    \ 
   --------  =  -   >  --- . ti | ------------  +  ------------------------- |.
   dOj.dS2f     5  /__ dOj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                   i=-k

calc_diff_S2f_tf_S2s_ts_d2jw_dOjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - S2f double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _k_
    d2J(w)      2  \   dci      /      S2s               (tf + ti)tf               (1 - S2s)(ts + ti)ts    \ 
   --------  =  -   >  --- . ti | ------------  -  -------------------------  +  ------------------------- |.
   dOj.dS2f     5  /__ dOj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                   i=-k

calc_diff_S2f_tf_S2s_ts_d2jw_dOjdS2s(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - S2 double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} and {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                      _k_
    d2J(w)      2     \   dci      /      1                 (ts + ti)ts         \ 
   --------  =  - S2f  >  --- . ti | ------------  -  ------------------------- |.
   dOj.dS2s     5     /__ dOj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                      i=-k

calc_diff_S2f_tf_S2s_ts_d2jw_dOjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                           _k_
    d2J(w)     2           \   dci          (tf + ti)^2 - (w.tf.ti)^2
   -------  =  - (1 - S2f)  >  --- . ti^2 -----------------------------.
   dOj.dtf     5           /__ dOj        ((tf + ti)^2 + (w.tf.ti)^2)^2
                           i=-k

calc_diff_S2f_S2s_ts_d2jw_dOjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Oj - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                              _k_
    d2J(w)     2              \   dci          (ts + ti)^2 - (w.ts.ti)^2
   -------  =  - S2f(1 - S2s)  >  --- . ti^2 -----------------------------.
   dOj.dts     5              /__ dOj        ((ts + ti)^2 + (w.ts.ti)^2)^2
                              i=-k

calc_S2f_S2s_ts_d2jw_dS2fdS2s(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - S2s double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                    _k_
     d2J(w)      2  \           /      1                 (ts + ti).ts        \ 
   ---------  =  -   >  ci . ti | ------------  -  ------------------------- |.
   dS2f.dS2s     5  /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                    i=-k

calc_S2f_tf_S2s_ts_d2jw_dS2fdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                     _k_
    d2J(w)        2  \               (tf + ti)^2 - (w.tf.ti)^2
   --------  =  - -   >  ci . ti^2 -----------------------------.
   dS2f.dtf       5  /__           ((tf + ti)^2 + (w.tf.ti)^2)^2
                     i=-k

calc_S2f_S2s_ts_d2jw_dS2fdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                            _k_
    d2J(w)      2           \               (ts + ti)^2 - (w.ts.ti)^2
   --------  =  - (1 - S2s)  >  ci . ti^2 -----------------------------.
   dS2f.dts     5           /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                            i=-k

calc_S2f_S2s_ts_d2jw_dS2sdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2s - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                        _k_
    d2J(w)        2     \               (ts + ti)^2 - (w.ts.ti)^2
   --------  =  - - S2f  >  ci . ti^2 -----------------------------.
   dS2s.dts       5     /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                        i=-k

calc_S2f_tf_S2s_ts_d2jw_dtf2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the tf - tf double partial derivative of the extended model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                            _k_
   d2J(w)       4           \             (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
   ------  =  - - (1 - S2f)  >  ci . ti^2 -----------------------------------------------------.
   dtf**2       5           /__                        ((tf + ti)^2 + (w.tf.ti)^2)^3
                            i=-k

calc_S2f_S2s_ts_d2jw_dts2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the ts - ts double partial derivative of the extended model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The model-free Hessian is:

                               _k_
   d2J(w)       4              \             (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
   ------  =  - - S2f(1 - S2s)  >  ci . ti^2 -----------------------------------------------------.
   dts**2       5              /__                        ((ts + ti)^2 + (w.ts.ti)^2)^3
                               i=-k