Module jw_mf_comps

Functions

calc_S2_te_jw_comps(data, params)
Spectral density component function.

calc_S2f_S2_ts_jw_comps(data, params)
Spectral density component function.

calc_S2f_S2s_ts_jw_comps(data, params)
Spectral density component function.

calc_S2f_tf_S2_ts_jw_comps(data, params)
Spectral density component function.

calc_S2f_tf_S2s_ts_jw_comps(data, params)
Spectral density component function.

calc_diff_djw_comps(data, params)
Spectral density gradient component function.

calc_S2_te_djw_comps(data, params)
Spectral density gradient component function.

calc_diff_S2_te_djw_comps(data, params)
Spectral density gradient component function.

calc_S2f_S2_ts_djw_comps(data, params)
Spectral density gradient component function.

calc_diff_S2f_S2_ts_djw_comps(data, params)
Spectral density gradient component function.

calc_S2f_tf_S2_ts_djw_comps(data, params)
Spectral density gradient component function.

calc_diff_S2f_tf_S2_ts_djw_comps(data, params)
Spectral density gradient component function.

calc_S2f_S2s_ts_djw_comps(data, params)
Spectral density gradient component function.

calc_diff_S2f_S2s_ts_djw_comps(data, params)
Spectral density gradient component function.

calc_S2f_tf_S2s_ts_djw_comps(data, params)
Spectral density gradient component function.

calc_diff_S2f_tf_S2s_ts_djw_comps(data, params)
Spectral density gradient component function.

Imports: Float64, zeros

Function Details

calc_S2_te_jw_comps(data, params)

Spectral density component function.

Calculate the components of the spectral density value for the original model-free formula with
the parameters {S2, te}.

The model-free formula is

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

Replicated calculations are

    w_ti_sqrd = (w.ti)^2        (pre-calculated during initialisation),

    te_ti = te + ti,
    te_ti_te = (te + ti).te.


Calculations which are replicated in the gradient equations are

    fact_ti = 1 / (1 + (w.ti)^2)    (pre-calculated during initialisation),

    one_s2 = 1 - S2,

    te_ti_sqrd = (te + ti)^2,
    w_te_ti_sqrd = (w.te.ti)^2,

                       (te + ti)te
    fact_te  =  -------------------------.
                (te + ti)^2 + (w.te.ti)^2

calc_S2f_S2_ts_jw_comps(data, params)

Spectral density component function.

Calculate the components of the spectral density value for the extended model-free formula with
the parameters {S2f, S2, ts}.

The model-free formula is

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

Replicated calculations are

    w_ti_sqrd = (w.ti)^2        (pre-calculated during initialisation),

    ts_ti = ts + ti,
    ts_ti_ts = (ts + ti).ts.


Calculations which are replicated in the gradient equations are

    fact_ti = 1 / (1 + (w.ti)^2)    (pre-calculated during initialisation),

    s2f_s2 = S2f - S2,

    ts_ti_sqrd = (ts + ti)^2,
    w_ts_ti_sqrd = (w.ts.ti)^2,
    inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2),

                       (ts + ti)ts
    fact_ts  =  -------------------------.
                (ts + ti)^2 + (w.ts.ti)^2

calc_S2f_S2s_ts_jw_comps(data, params)

Spectral density component function.

Calculate the components of the spectral density value for the extended model-free formula with
the parameters {S2f, S2s, ts}.

The model-free formula is

                   _n_
             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=m

Replicated calculations are

    w_ti_sqrd = (w.ti)^2        (pre-calculated during initialisation),

    ts_ti = ts + ti,
    ts_ti_ts = (ts + ti).ts.


Calculations which are replicated in the gradient equations are

    fact_ti = 1 / (1 + (w.ti)^2)    (pre-calculated during initialisation),

    one_s2s = 1 - S2s,

    ts_ti_sqrd = (ts + ti)^2,
    w_ts_ti_sqrd = (w.ts.ti)^2,
    inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2),

                       (ts + ti)ts
    fact_ts  =  -------------------------.
                (ts + ti)^2 + (w.ts.ti)^2

calc_S2f_tf_S2_ts_jw_comps(data, params)

Spectral density component function.

Calculate the components of the spectral density value for the extended model-free formula with
the parameters {S2f, tf, S2, ts}.

The model-free formula is

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

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

Replicated calculations are

    w_ti_sqrd = (w.ti)^2        (pre-calculated during initialisation),

    tf_ti = tf + ti,
    ts_ti = ts + ti,
    tf_ti_tf = (tf + ti).tf,
    ts_ti_ts = (ts + ti).ts.


Calculations which are replicated in the gradient equations are

    fact_ti = 1 / (1 + (w.ti)^2)    (pre-calculated during initialisation),

    one_s2f = 1 - S2f,
    s2f_s2 = S2f - S2,

    tf_ti_sqrd = (tf + ti)^2,
    ts_ti_sqrd = (ts + ti)^2,
    w_tf_ti_sqrd = (w.tf.ti)^2,
    w_ts_ti_sqrd = (w.ts.ti)^2,
    inv_tf_denom = 1 / ((tf + ti)^2 + (w.tf.ti)^2),
    inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2),

                       (tf + ti)tf
    fact_tf  =  -------------------------,
                (tf + ti)^2 + (w.tf.ti)^2


                       (ts + ti)ts
    fact_ts  =  -------------------------.
                (ts + ti)^2 + (w.ts.ti)^2

calc_S2f_tf_S2s_ts_jw_comps(data, params)

Spectral density component function.

Calculate the components of the spectral density value for the extended model-free formula with
the parameters {S2f, tf, S2s, ts}.

The model-free formula is

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

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

Replicated calculations are

    w_ti_sqrd = (w.ti)^2        (pre-calculated during initialisation),

    tf_ti = tf + ti,
    ts_ti = ts + ti,
    tf_ti_tf = (tf + ti).tf,
    ts_ti_ts = (ts + ti).ts.


Calculations which are replicated in the gradient equations are

    fact_ti = 1 / (1 + (w.ti)^2)    (pre-calculated during initialisation),

    one_s2s = 1 - S2s,
    one_s2f = 1 - S2f,
    s2f_s2 = S2f(1 - S2s) = S2f - S2,

    tf_ti_sqrd = (tf + ti)^2,
    ts_ti_sqrd = (ts + ti)^2,
    w_tf_ti_sqrd = (w.tf.ti)^2,
    w_ts_ti_sqrd = (w.ts.ti)^2,
    inv_tf_denom = 1 / ((tf + ti)^2 + (w.tf.ti)^2),
    inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2),

                       (tf + ti)tf
    fact_tf  =  -------------------------,
                (tf + ti)^2 + (w.tf.ti)^2


                       (ts + ti)ts
    fact_ts  =  -------------------------.
                (ts + ti)^2 + (w.ts.ti)^2

calc_diff_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the original model-free formula
with no parameters {} or the parameter {S2} together with diffusion tensor parameters.

Replicated calculations are

                          1 - (w.ti)^2
    fact_ti_djw_dti  =  ----------------.
                        (1 + (w.ti)^2)^2

calc_S2_te_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the original model-free formula
with the parameters {S2, te}.

Replicated calculations are

                            (te + ti)^2 - (w.te.ti)^2
    fact_djw_dte  =  ti^2 -----------------------------.
                          ((te + ti)^2 + (w.te.ti)^2)^2

calc_diff_S2_te_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the original model-free formula
with the parameters {S2, te} together with diffusion tensor parameters.

Replicated calculations are

                          1 - (w.ti)^2
    fact_ti_djw_dti  =  ----------------,
                        (1 + (w.ti)^2)^2


                   (te + ti)^2 - (w.te.ti)^2
    fact_djw  =  -----------------------------,
                 ((te + ti)^2 + (w.te.ti)^2)^2


                               (te + ti)^2 - (w.te.ti)^2
    fact_te_djw_dti  =  te^2 -----------------------------,
                             ((te + ti)^2 + (w.te.ti)^2)^2


                            (te + ti)^2 - (w.te.ti)^2
    fact_djw_dte  =  ti^2 -----------------------------.
                          ((te + ti)^2 + (w.te.ti)^2)^2

calc_S2f_S2_ts_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the extended model-free formula
with the parameters {S2f, S2, ts}.

Replicated calculations are


                            (ts + ti)^2 - (w.ts.ti)^2
    fact_djw_dts  =  ti^2 -----------------------------.
                          ((ts + ti)^2 + (w.ts.ti)^2)^2

calc_diff_S2f_S2_ts_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the extended model-free formula
with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

Replicated calculations are

                          1 - (w.ti)^2
    fact_ti_djw_dti  =  ----------------,
                        (1 + (w.ti)^2)^2


                   (ts + ti)^2 - (w.ts.ti)^2
    fact_djw  =  -----------------------------,
                 ((ts + ti)^2 + (w.ts.ti)^2)^2


                               (ts + ti)^2 - (w.ts.ti)^2
    fact_ts_djw_dti  =  ts^2 -----------------------------,
                             ((ts + ti)^2 + (w.ts.ti)^2)^2


                            (ts + ti)^2 - (w.ts.ti)^2
    fact_djw_dts  =  ti^2 -----------------------------.
                          ((ts + ti)^2 + (w.ts.ti)^2)^2

calc_S2f_tf_S2_ts_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the extended model-free formula
with the parameters {S2f, tf, S2, ts}.

Replicated calculations are

                            (tf + ti)^2 - (w.tf.ti)^2
    fact_djw_dtf  =  ti^2 -----------------------------,
                          ((tf + ti)^2 + (w.tf.ti)^2)^2


                            (ts + ti)^2 - (w.ts.ti)^2
    fact_djw_dts  =  ti^2 -----------------------------.
                          ((ts + ti)^2 + (w.ts.ti)^2)^2

calc_diff_S2f_tf_S2_ts_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the extended model-free formula
with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

Replicated calculations are

                          1 - (w.ti)^2
    fact_ti_djw_dti  =  ----------------,
                        (1 + (w.ti)^2)^2


                      (tf + ti)^2 - (w.tf.ti)^2
    fact_tf_djw  =  -----------------------------,
                    ((tf + ti)^2 + (w.tf.ti)^2)^2


                      (ts + ti)^2 - (w.ts.ti)^2
    fact_ts_djw  =  -----------------------------,
                    ((ts + ti)^2 + (w.ts.ti)^2)^2


                               (tf + ti)^2 - (w.tf.ti)^2
    fact_tf_djw_dti  =  tf^2 -----------------------------,
                             ((tf + ti)^2 + (w.tf.ti)^2)^2


                               (ts + ti)^2 - (w.ts.ti)^2
    fact_ts_djw_dti  =  ts^2 -----------------------------,
                             ((ts + ti)^2 + (w.ts.ti)^2)^2


                            (tf + ti)^2 - (w.tf.ti)^2
    fact_djw_dtf  =  ti^2 -----------------------------,
                          ((tf + ti)^2 + (w.tf.ti)^2)^2


                            (ts + ti)^2 - (w.ts.ti)^2
    fact_djw_dts  =  ti^2 -----------------------------.
                          ((ts + ti)^2 + (w.ts.ti)^2)^2

calc_S2f_S2s_ts_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the extended model-free formula
with the parameters {S2f, S2s, ts}.

Replicated calculations are

                            (ts + ti)^2 - (w.ts.ti)^2
    fact_djw_dts  =  ti^2 -----------------------------.
                          ((ts + ti)^2 + (w.ts.ti)^2)^2

calc_diff_S2f_S2s_ts_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the extended model-free formula
with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

Replicated calculations are

                          1 - (w.ti)^2
    fact_ti_djw_dti  =  ----------------,
                        (1 + (w.ti)^2)^2


                   (ts + ti)^2 - (w.ts.ti)^2
    fact_djw  =  -----------------------------,
                 ((ts + ti)^2 + (w.ts.ti)^2)^2


                               (ts + ti)^2 - (w.ts.ti)^2
    fact_ts_djw_dti  =  ts^2 -----------------------------,
                             ((ts + ti)^2 + (w.ts.ti)^2)^2


                            (ts + ti)^2 - (w.ts.ti)^2
    fact_djw_dts  =  ti^2 -----------------------------.
                          ((ts + ti)^2 + (w.ts.ti)^2)^2

calc_S2f_tf_S2s_ts_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the extended model-free formula
with the parameters {S2f, tf, S2s, ts}.

Replicated calculations are

                            (tf + ti)^2 - (w.tf.ti)^2
    fact_djw_dtf  =  ti^2 -----------------------------,
                          ((tf + ti)^2 + (w.tf.ti)^2)^2


                            (ts + ti)^2 - (w.ts.ti)^2
    fact_djw_dts  =  ti^2 -----------------------------.
                          ((ts + ti)^2 + (w.ts.ti)^2)^2

calc_diff_S2f_tf_S2s_ts_djw_comps(data, params)

Spectral density gradient component function.

Calculate the components of the spectral density gradient for the extended model-free formula
with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

Replicated calculations are

                          1 - (w.ti)^2
    fact_ti_djw_dti  =  ----------------,
                        (1 + (w.ti)^2)^2


                      (tf + ti)^2 - (w.tf.ti)^2
    fact_tf_djw  =  -----------------------------,
                    ((tf + ti)^2 + (w.tf.ti)^2)^2


                      (ts + ti)^2 - (w.ts.ti)^2
    fact_ts_djw  =  -----------------------------,
                    ((ts + ti)^2 + (w.ts.ti)^2)^2


                               (tf + ti)^2 - (w.tf.ti)^2
    fact_tf_djw_dti  =  tf^2 -----------------------------,
                             ((tf + ti)^2 + (w.tf.ti)^2)^2


                               (ts + ti)^2 - (w.ts.ti)^2
    fact_ts_djw_dti  =  ts^2 -----------------------------,
                             ((ts + ti)^2 + (w.ts.ti)^2)^2


                            (tf + ti)^2 - (w.tf.ti)^2
    fact_djw_dtf  =  ti^2 -----------------------------,
                          ((tf + ti)^2 + (w.tf.ti)^2)^2


                            (ts + ti)^2 - (w.ts.ti)^2
    fact_djw_dts  =  ti^2 -----------------------------.
                          ((ts + ti)^2 + (w.ts.ti)^2)^2