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Package maths_fns :: Module jw_mf_comps
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Module jw_mf_comps

source code

Functions [hide private]
 
calc_S2_te_jw_comps(data, params)
Spectral density component function.		 source code
 
calc_S2f_S2_ts_jw_comps(data, params)
Spectral density component function.		 source code
 
calc_S2f_S2s_ts_jw_comps(data, params)
Spectral density component function.		 source code
 
calc_S2f_tf_S2_ts_jw_comps(data, params)
Spectral density component function.		 source code
 
calc_S2f_tf_S2s_ts_jw_comps(data, params)
Spectral density component function.		 source code
 
calc_diff_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_S2_te_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_diff_S2_te_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_S2f_S2_ts_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_diff_S2f_S2_ts_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_S2f_tf_S2_ts_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_diff_S2f_tf_S2_ts_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_S2f_S2s_ts_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_diff_S2f_S2s_ts_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_S2f_tf_S2s_ts_djw_comps(data, params)
Spectral density gradient component function.		 source code
 
calc_diff_S2f_tf_S2s_ts_djw_comps(data, params)
Spectral density gradient component function.		 source code

Imports: Float64, zeros


Function Details [hide private]

calc_S2_te_jw_comps(data, params)

source code  
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)

source code  
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)

source code  
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)

source code  
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          (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=m

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)

source code  
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         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=m

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)

source code  
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)

source code  
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)

source code  
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:

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

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

source code  
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)

source code  
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:

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


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

source code  
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)

source code  
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:

                      (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


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

source code  
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)

source code  
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:

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


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


                               (ts + ti)^2 - (w.ts.ti)^2
    fact_te_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)

source code  
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)

source code  
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:

                      (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


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

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