Integration.Steps {fptdApprox}

Subintervals and Integration Steps To Approximate First-Passage-Time Densities

Description

According to the First-Passage-Time Location (FPTL) function and the arguments in the function call, this function calculates suitable subintervals and integration steps in order to approximate the first-passage-time (f.p.t.) density.

Usage

Integration.Steps(sfptl, variableStep = TRUE, from.t0 = FALSE,
                  to.T = FALSE, n = 250, p = 0.2, alpha = 1)

Arguments

sfptl

an object of class “summary.fptl”.

variableStep

a logical value indicating whether a variable integration step is used.

from.t0

a logical value indicating whether the approximation should be calculated from the lower end of the interval considered, t_0, specified in the object used as sfptl argument.

to.T

a logical value indicating whether the approximation should be calculated to the upper end of the interval considered, T, specified in the object used as sfptl argument.

n

Number of points used to determine the integration step in subintervals [t_i^*, t_{max,i}^+], i=1, \ldots, m, from interesting instants provided by the FPTL function.

p

Ratio of n used to determine the integration step in subintervals [t_{max,i}^+, t_{i+1}^*], i=1, \ldots, m, [t_0, t_1^*] and [t_{max,m}^{+}, T].

alpha

Parameter used to determine the integration step in subintervals [t_{max,i}^+, t_{i+1}^*], i=1, \ldots, m, [t_0, t_1^*] and [t_{max,m}^{+}, T], in order to reduce the computational cost of approximating the f.p.t. density function in those cases where t_{i+1}^* - t_{max,i}^+ >> t_{max,i}^{-} - \, t_i^*, for some i, t_1^* - t_0 >> t_{max,1}^{-} - \thinspace t_1^* or T - t_{max,m}^+ >> t_{max,m}^{-} - \, t_m^*, respectively.

Details

Based on the information provided by the FPTL function contained in the sfptl object, this function computes and returns suitable subintervals and integration steps in order to approximate the density function of the f.p.t. variable according to the other arguments in the function call.

When the sfptl object is of length greater than 1, it comes from an unconditioned f.p.t. problem. Each component is associated with the same f.p.t. problem conditioned on different values of the initial distribution (equally spaced in the range of the distribution). Let x_{0,j}, j=1, \ldots, N, these values. For each initial value x_{0,j} let t_{i,j}^*, t_{max,i,j}^- and t_{max,i,j}^+, i=1,\ldots \ m_j, the interesting time instants provided by the FPTL function and stored in the instants component of the j-th list in the sfptl object. Then, the time instants \{t_{i,j}, i=1, 2, \ldots, 2m_j \}, where

t_{i,j} = \left\{ \begin{array}{ll} t_{(i+1)/2, \, j}^* & for \ i \ odd \\[7pt] t_{max, \, i/2, \thinspace j}^+ & for \ i \ even \end{array} \right. ,

provide a suitable partition of interval [t_0, T] to approximate the f.p.t density for the fixed value x_{0,j} of the initial distribution.

If the sfptl object is of length 1, it comes from a conditioned f.p.t. problem. In this case we denote the interesting time instants provided by the FPTL function and stored in the sfptl object by t_{i,1}^*, t_{max,i,1}^- and t_{max,i,1}^+.

In what follows, \lceil x \rceil is the integer part of x.

For each list in the sfptl object the function computes

h_{i,j} = \displaystyle{\frac{t_{max,i,j}^{+} - \, t_{i,j}^*}{n_{i,j}}} , i=1, \ldots, m_j,

where

n_{i,j} = \lceil n \, k_{i,j} \rceil

and

k_{i,j} = \displaystyle{\frac{t_{max,i,j}^+ - t_{i,j}^{*}}{t_{max,i,j}^{-} - \, t_{i,j}^*}} \ .

If variableStep = TRUE, for 0 < p and \alpha \leq 1, also computes

• h_{i,j}^* = \displaystyle{\frac{t_{i+1,j}^* - t_{max,i,j}^+}{n_{i,j}^*}} , i=1, \ldots, m_j-1, where

  n_{i,j}^* = \left\{ \begin{array}{cc} \lceil n \, p \, k_{i,j}^* \rceil & if \ k_{i,j}^* \leq 1 \\[7pt] \lceil n \, p \, {k_{i,j}^{*}}^{\alpha} \rceil & if \ k_{i,j}^* > 1 \end{array} \right.

  with

  k_{i,j}^* = \displaystyle{\frac{t_{i+1,j}^* - \, t_{max,i,j}^+}{t_{max,i,j}^{-} - \, t_{i,j}^*}} \ .

  If h_{i,j}^* < h_{i,j}, we then set t_{max,i,j}^+ equal to t_{i+1,j}^* and h_{i,j} is recalculated.

• h_{0,j}^* = \displaystyle{\frac{t_{1,j}^* - t_0}{n_{0,j}^*}} , where

  n_{0,j}^* = \left\{ \begin{array}{cc} \lceil n \, p \, k_{0,j}^* \rceil & if \ k_{0,j}^* \leq 1 \\[7pt] \lceil n \, p \, {k_{0,j}^{*}}^{\alpha} \rceil & if \ k_{0,j}^* > 1 \end{array} \right.

  with

  k_{0,j}^* = \displaystyle{\frac{t_{1,j}^* - \, t_0}{t_{max,1,j}^{-} - \, t_{1,j}^*}} \ ,

  when the sfptl object is of length 1 and from.t0 = TRUE, or the sfptl object is of length greater than 1.

  If h_{0,j}^* < h_{1,j}, we then set t_{1,j}^* equal to t_0 and h_{1,j} is recalculated.

• h_{m_j,j}^* = \displaystyle{\frac{T - \, t_{max,m_j,j}^+}{n_{m_j,j}^*}} , where

  n_{m_j,j}^* = \left\{ \begin{array}{cc} \lceil n \, p \, k_{m_j,j}^* \rceil & if \ k_{m_j,j}^* \leq 1 \\[7pt] \lceil n \, p \, {k_{m_j,j}^{*}}^{\alpha} \rceil & if \ k_{m_j,j}^* > 1 \end{array} \right.

  with

  k_{m_j,j}^* = \displaystyle{\frac{T - \, t_{max,m_j,j}^+}{t_{max,m_j,j}^{-} - \, t_{m_j,j}^*}} \ ,

  when the sfptl object is of length 1 and to.T = TRUE, or the sfptl object is of length greater than 1.

  If h_{m_j,j}^* < h_{m_j,j}, we then set t_{max,m_j,j}^+ equal to T and h_{m_j,j} is recalculated.
  

p \geq 0.1 and 0.75 \leq \alpha \leq 1 are recommended; otherwise, some integration steps can be excessively large.

If the sfptl object is of length 1 (conditioned f.p.t. problem), the suitable subintervals and integration steps that the function provides are:

• If variableStep = TRUE,

  • h_{i,1} in subintervals [t_{i,1}^*, t_{max,i,1}^+], i=1, \ldots, m_1.

  • h_{i,1}^* in subintervals [t_{max,i,1}^+, t_{i+1,1}^*], i=1, \ldots, m_1-1. In these subintervals is possible to avoid applying the numerical algorithm to approximate the f.p.t. density provided that the value of the approximate density at the time instant t_{max,i,1}^+ is almost 0.

  • h_{0,1}^* in subinterval [t_0, t_{1,1}^*], if from.t0 = TRUE.

  • h_{m_1,1}^* in subinterval [t_{max,m_1,1}^{+}, T], if to.T = TRUE.
    

• If variableStep = FALSE the function computes

  h = min \left\{ h_{i,1} \ , \ i=1, \ldots, m_1 \right\} \thinspace .

  Then

  • If from.t0 = FALSE and to.T = FALSE, h is readjusted to exactly split the interval [t_{1,1}^*, t_{max,m_1,1}^+].

  • If from.t0 = TRUE and to.T = FALSE, h is readjusted to exactly split the interval [t_0, t_{max,m_1,1}^+].

  • If from.t0 = FALSE and to.T = TRUE, h is readjusted to exactly split the interval [t_{1,1}^*, T].

  • If from.t0 = TRUE and to.T = TRUE, h is readjusted to exactly split the interval [t_0, T].

  h is a suitable fixed integration step in subintervals [t_{i,1}^*, t_{max,i,1}^+], i=1, \ldots, m_1, and [t_{max,i,1}^+, t_{i+1,1}^*], i=1, \ldots, m_1-1; in subintervals [t_0, t_{1,1}^*] if from.t0 = TRUE, and in [t_{max,m_1,1}^{+}, T] if to.T = TRUE. The endpoints of such subintervals are readjusted according to this integration step.
  

If the sfptl object is a list of length greater than 1 (unconditioned f.p.t problem), a common partition of the interval [t_0, T] is calculated from the suitable partitions of this interval for each fixed value of the initial distribution.

Let, in unified form, H_{r,j}, r=1, \ldots, 2m_j+1, the suitable integration steps (calculated for each j in similar manner to the case of the sfptl object is of length 1) in subintervals I_{r,j} = [t_{r-1,j}, t_{r,j}], with t_{0,j}=t_0 and t_{2m_j+1,j} = T, j=1, \ldots, N. Then, the ordered values of all time instants in the suitable partitions, t_{(1)}, \ldots, t_{(M)}, provide a common suitable partition of the interval [t_0, T] in subintervals [t_{(i-1)}, t_{(i)}], i=1, \ldots, M \negthinspace + \negthinspace 1, where t_{(0)} = t_0 and t_{(M \negthinspace + \negthinspace 1)} = T.

For this partition, the function computes

• H_{i} = min \, \{ H_{r,j} : j=1, \ldots, N, \ and \ [t_{(i-1)}, t_{(i)}] \subseteq I_{r,j} \} , \ i=2, \ldots, M.

• H_{1} = min \, \{ H_{r,j} : j=1, \ldots, N, \ and \ [t_0, t_{(1)}] \subseteq I_{r,j} \}, if from.t0 = TRUE.

• H_{M \negthinspace + \negthinspace 1} = min \, \{ H_{r,j} : j=1, \ldots, N, \ and \ [t_{(M)}, T] \subseteq I_{r,j} \}, if to.T = TRUE.
  

Thus,

• If variableStep = TRUE, the suitable subintervals and integrations steps that the function provides are

  • H_i in subintervals [t_{(i-1)}, t_{(i)}], i=2, \ldots, M.

  • H_1 in subinterval [t_0, t_{(1)}], if from.t0 = TRUE.

  • H_{M \negthinspace + \negthinspace 1} in subinterval [t_{(M)}, T], if to.T = TRUE.

  Each integration step is readjusted to exactly split the corresponding subinterval.
  

• If variableStep = FALSE, a suitable fixed integration step for any subinterval [t_{(i-1)}, t_{(i)}] is

  h = min \left\{ H_{i} : i=1, \ldots, M \negthinspace + \negthinspace 1 \right\}.

  In this case it is not possible to avoid applying the approximation algorithm in [t_{(i-1)}, t_{(i)}] \ \forall \ i=1, \ldots, M \negthinspace + \negthinspace 1 .
  

  Then

  • If from.t0 = FALSE and to.T = FALSE, h is readjusted to exactly split the interval [t_{(1)}, t_{(M)}].

  • If from.t0 = TRUE and to.T = FALSE, h is readjusted to exactly split the interval [t_0, t_{(M)}].

  • If from.t0 = FALSE and to.T = TRUE, h is readjusted to exactly split the interval [t_{(1)}, T].

  • If from.t0 = TRUE and to.T = TRUE, h is readjusted to exactly split the interval [t_0, T].

  h is a suitable fixed integration step in subintervals [t_{(i-1)}, t_{(i)}], i=2, \ldots, M, in subintervals [t_0, t_{(1)}] if from.t0 = TRUE, and in [t_{(M)}, T] if to.T = TRUE. The endpoints of such subintervals are readjusted according to this integration step.
  

Value

A two-component list:

H

A matrix of subintervals and integrations steps that we must consider in order to approximate the f.p.t. density according to the information contained in the sfptl object and the arguments in the function call.

skip

A list of logical vectors indicating, for each subinterval, the values of the initial distribution for which we must check whether it is possible to avoid applying the numerical algorithm.

Author(s)

Patricia Román-Román, Juan J. Serrano-Pérez and Francisco Torres-Ruiz.

References

Román, P., Serrano, J. J., Torres, F. (2008) First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Comput. Stat. Data Anal., 52, 4132–4146.

P. Román-Román, J.J. Serrano-Pérez, F. Torres-Ruiz. (2012) An R package for an efficient approximation of first-passage-time densities for diffusion processes based on the FPTL function. Applied Mathematics and Computation, 218, 8408–8428.

P. Román-Román, J.J. Serrano-Pérez, F. Torres-Ruiz. (2014) More general problems on first-passage times for diffusion processes: A new version of the fptdApprox R package. Applied Mathematics and Computation, 244, 432–446.

See Also

Approx.fpt.density to approximate f.p.t. densities from objects of class “summary.fptl” and create objects of class “fpt.density”.

summary.fptl to locate the f.p.t. variable and create objects of class “summary.fptl” from objects of class “fptl”.

FPTL to evaluate the FPTL function and create objects of class “fptl”.

Examples

## Continuing the summary.fptl(.) example:

Integration.Steps(yy)
Integration.Steps(yy, from.t0 = TRUE)
Integration.Steps(yy, to.T = TRUE, n = 100, p = 0.25)

Integration.Steps(zz)