Box Cumulative Sums

Description

A box cumulative sum is defined as the cumulative sum over a lower left rectangle. This function is primarily for use when the components are point probabilities for the number of crossings C and the longest run L, then component (c,l) in the result is the probability P(C \ge c, L \le l).

Usage

boxprobt(mtrx)

Arguments

Value

mpfr array

Examples

nill <- Rmpfr::mpfr(0, 120)
one <- Rmpfr::mpfr(1, 120)
two <- Rmpfr::mpfr(2, 120)
contents <- c(one,nill,nill, one,one,one, two,two,two)
mtrx3 <- Rmpfr::mpfr2array(contents, dim = c(3, 3))
print(mtrx3)
print(boxprobt(mtrx3))

Number of Crossings and Longest Run

Description

Auxiliary function for simclbin, computing the number of crossings (type=0) or longest run (type=2) in a sequence of independent normal observations. Crossings and runs are related to whether the observations are above a shift.

Usage

clshift(seri, shift = 0, type = 0)

Arguments

Value

number of crossings or longest run, numeric

Joint Distribution for Crossings and Runs, autocorrelated Sequence

Description

Joint probability distribution for the number of crossings C and the longest run L in a sequence of n autocorrelated Bernoulli observations with success probability p. To enhance precision, results are stored in mpfr arrays and the probabilities are multiplied by m^{n-1} for a multiplier m.

Usage

crossrunauto(
  nmax = 100,
  prob = 0.5,
  changeprob = 0.5,
  mult = 2,
  prec = 120,
  printn = FALSE
)

Arguments

Value

list of joint probabilities.

Examples

# p=0.6, independence
cr10.6 <- crossrunbin(nmax=10, prob=0.6, printn=TRUE)
cra10.6 <- crossrunauto(nmax=10, prob=0.6, changeprob=.6, printn=TRUE)
Rmpfr::asNumeric(cr10.6$pt[[10]])
Rmpfr::asNumeric(cr10.6$pt[[10]])
Rmpfr::asNumeric(cr10.6$pt[[10]]) - Rmpfr::asNumeric(cra10.6$pt[[10]]) # equal


# p=0.6, some dependence
cr10.6 <- crossrunbin(nmax=10, prob=0.6, printn=TRUE)
cra10.6.u.5 <- crossrunauto(nmax=10, prob=0.6, changeprob=.5, printn=TRUE)
round(Rmpfr::asNumeric(cr10.6$pt[[10]]),1)
round(Rmpfr::asNumeric(cra10.6.u.5$pt[[10]]),1) # not the same

Joint Distribution for Crossings and Runs

Description

Joint probability distribution for the number of crossings C and the longest run L in a sequence of n independent Bernoulli observations with success probability p. To enhance precision, results are stored in mpfr arrays and the probabilities are multiplied by m^{n-1} for a multiplier m.

Usage

crossrunbin(nmax = 100, prob = 0.5, mult = 2, prec = 120, printn = FALSE)

Arguments

Value

list of joint probabilities.

Examples

crb10.6 <- crossrunbin(nmax=10, prob=.6, printn=TRUE)
print(crb10.6$pt[[10]])

Joint Distribution for Crossings and Runs, Varying Success Probability.

Description

Joint probability distribution for the number of crossings C and the longest run L in a sequence of n independent Bernoulli observations with p ossibly varying success probability. To enhance precision, results are stored in mpfr arrays and the probabilities are multiplied by m^{n-1} for a multiplier m.

Usage

crossrunchange(
  nmax = 100,
  prob = rep(0.5, 100),
  mult = 2,
  prec = 120,
  printn = FALSE
)

Arguments

Value

list pt of joint probabilities. Cumulative probabilities qt within each row are also included. Further, mostly for code checking, lists pat and qat conditional on starting with a success, and pbt and qbt conditional of starting with a failure, are included.

Examples

prob10 <- c(rep(.5,5),rep(.7,5))
crchange10 <- crossrunchange(nmax=10, prob=prob10,printn=TRUE)
print(crchange10$pt[[10]])

Joint Distribution for Crossings and Runs Using the Empirical Median.

Description

Joint probability distribution for the number of crossings C and the longest run L in a sequence of n Bernoulli observations where the number of successes is fixed at m, m between 0 and n. For fixed n, the joint distribution is computed for all m, this makes the computation demanding in terms of time and storage requirements. The joint distribution is computed separately for sequences where the first observation is, or is not, a success. The results are mainly intended for use when n is even and m=n/2, but computation in this case requires that all distributions are computed previously for all m, for all shorter sequences (lower n). In the case of even n and m=n/2, the distributions for sequences starting or not with a success are identical, and only the distribution among sequences starting with a success is used. In that case, this may be interpreted as the joint distribution for sequences around the empirical median.

Usage

crossrunem(nmax = 100, prec = 120, printn = FALSE)

Arguments

Value

nfi, number of sequences with m successes, starting with a success, and nfn, number of sequences with m successes, not starting with a success. Three-dimensional Rmpfr arrays for each n up to nmax, with dimensions n (C=0 to n-1), n (L=1 to n) and n+1 (m=0 to n). For n even and m=n/2, only nfi, and the part corresponding to C=1 to n-1 and L=1 and m=n/2 is non-zero and should be used.

Examples

crem14 <- crossrunem(nmax=14, printn=TRUE)
Rmpfr::asNumeric(crem14$nfi[[14]][,,"m=7"]) # subsets of size 7=14/2
# restricted to possible values of C and L
Rmpfr::asNumeric(crem14$nfi[[14]][-1,1:7,"m=7"]) # same as stored data joint14em
Rmpfr::asNumeric(crem14$nfn[[14]][-1,1:7,"m=7"]) # the same

# subsets of sizes different from 14/2
# size 4, first observation included
Rmpfr::asNumeric(crem14$nfi[[14]][,,"m=4"])
# size 14-4=10, first observation not included
Rmpfr::asNumeric(crem14$nfn[[14]][,,"m=10"]) # the same

Continuation of an existing sequence of joint probabilities for crossings and longest run, based on the empirical median.

Description

Continuation of an existing sequence of the number of crossings C and the longest run L in a sequence of n independent continuous observations classified as above or below the empirical median. To enhance precision, results are stored in mpfr arrays and the probabilities are multiplied by choose(n,m)/2 where m=n/2, even n assumed. The probabilities are integers in this representation.

Usage

crossrunemcont(emstart, n1 = 61, nmax = 100, prec = 120, printn = FALSE)

Arguments

Value

nfi, number of sequences with m successes, starting with a success, and nfn, number of sequences with m successes, not starting with a success.

wrapper for crossrunbin, success probability=pnorm(shift).

Description

wrapper for crossrunbin, success probability=pnorm(shift).

Usage

crossrunshift(nmax = 100, shift = 0, mult = 2, prec = 120, printn = FALSE)

Arguments

Value

list pt of joint probabilities. Cumulative probabilities qt within each row are also included. Further, mostly for code checking, lists pat and qat conditional on starting with a success, and pbt and qbt conditional of starting with a failure, are included.

Examples

crs15 <- crossrunshift(nmax=15,printn=TRUE)
print(crs15$pt[[15]])

Joint Probabilities for Crossings and Runs, Symmetric Case

Description

Joint probability distribution for the number of crossings C and the longest run L in a sequence of n independent Bernoulli observations with success probability p. To enhance precision, results are stored in mpfr arrays and the probabilities are multiplied by m^{n-1} for a multiplier m. This is for the symmetric case with success probability 0.5, in which the multiplied probabilities are integers for the default value 2 of the multiplier.

Usage

crossrunsymm(nmax = 100, mult = 2, prec = 120, printn = FALSE)

Arguments

Value

pt, list of joint probabilities, multiplied with m^{n-1}. In addition cumulative probabilities qt within each row are also included.

Examples

crs10 <- crossrunsymm(nmax=10,printn=TRUE)

Row-wise Cumulative Sums

Description

Row-wise Cumulative Sums in mpfr Array.

Usage

cumsumm(mtrx)

Arguments

Value

mpfr array with row-wise cumulative sums, same dimension as the original array.

Examples

nill <- Rmpfr::mpfr(0, 120)
one <- Rmpfr::mpfr(1, 120)
two <- Rmpfr::mpfr(2, 120)
contents <- c(one,nill,nill, one,one,one, two,two,two)
mtrx3 <- Rmpfr::mpfr2array(contents, dim = c(3, 3))
print(mtrx3)
print(cumsumm(mtrx3))

Column-Wise Cumulative Sums

Description

Column-wise cumulative sums in mpfr array.

Usage

cumsummcol(mtrx)

Arguments

Value

mpfr array with column-wise cumulative sums, same dimension as the original array.

Examples

nill <- Rmpfr::mpfr(0, 120)
one <- Rmpfr::mpfr(1, 120)
two <- Rmpfr::mpfr(2, 120)
contents <- c(one,nill,nill, one,one,one, two,two,two)
mtrx3 <- Rmpfr::mpfr2array(contents, dim = c(3, 3))
print(mtrx3)
print(cumsummcol(mtrx3))

Exact Joint Probabilities for Low n

Description

Exact joint probabilities, for low n, of the number of crossings C and the longest run L in n independent Bernoulli observations with success probability p. Probabilites are multiplied by 2^{n-1}.

Usage

exactbin(n, p = 0.5, prec = 120)

Arguments

Value

mpfr array

Examples

exactbin(n=6)
exactbin(n=5, p=0.6)

Joint probabilities, n=100, success probability 0.6

Description

The joint probabilities of the number C og crossings (0, ... 99) and the longest run L (1, ..., 100) in a series of n=100 independent Bernoulli observations for success probability 0.6. The probabilities are stored in the "times" representations, multiplied by 2^{100-1}. Only the joint distributions for n=15, 60, 100 and success probabilities 0.5 and 0.6 are included in the package to avoid excessive storage, but many more cases may be generated by the function crossrunbin.

Usage

joint100.6

Format

matrix, 100 rows and 100 columns

Source

generated by the function crossrunbin and transformed from an Rmpfr array to a matrix

Joint probabilities, n=100, symmetric case

Description

The joint probabilities of the number C og crossings (0, ... 99) and the longest run L (1, ..., 100) in a series of n=100 independent Bernoulli observations for the symmetric case (success probability 0.5). The probabilities are stored in the "times" representations, multiplied by 2^{100-1} and are integers in the symmetric case. Only the joint distributions for n=15, 60, 100 and success probabilities 0.5 and 0.6 are included in the package to avoid excessive storage, but many more cases may be generated by the function crossrunsymm.

Usage

joint100symm

Format

matrix, 100 rows and 100 columns

Source

generated by the function crossrunsymm and transformed from an Rmpfr array to a matrix

Joint probabilities, n=14, success probability 0.6

Description

The joint probabilities of the number C og crossings (0, ... 13) and the longest run L (1, ..., 14) in a series of n=14 independent Bernoulli observations for success probability 0.6. The probabilities are stored in the "times" representations, multiplied by 2^{14-1}=8192. Only the joint distributions for n=14, 60, 100 and success probabilities 0.5 and 0.6 are included in the package to avoid excessive storage, but many more cases may be generated by the function crossrunbin.

Usage

joint14.6

Format

matrix, 14 rows and 14 columns

Source

generated by the function crossrunbin and transformed from an Rmpfr array to a matrix

Joint probabilities, n=14, around the empirical median

Description

Joint probabilities of the number C of crossings (1, ... 13) and the longest run L (1, ..., 17) in a series of n=60 Bernoulli observations around its empirical median. The probabilities are stored in the "times" representations, multiplied by (60 by 30)/2, the number of constellations starting above the median, and are integers. About the empirical median there is at least one crossing, and the longest run cannot exceed 14/2=7. Only the joint distributions for n=14, 60 are included in the package to avoid excessive storage, but many more cases may be generated by the function 'crossrunem. Since these computations are demanding in terms of storage and computation time, they are at present not performed for n much above 60.

Usage

joint14em

Format

matrix, 13 rows and 7 columns

Source

generated by the function crossrunsymm and transformed from an Rmpfr array to a matrix

Joint probabilities, n=14, symmetric case

Description

Joint probabilities of the number C of crossings (0, ... 13) and the longest run L (1, ..., 14) in a series of n=14 independent Bernoulli observations for the symmetric case (success probability 0.5). The probabilities are stored in the "times" representations, multiplied by 2^{14-1}=8192 and are integers in the symmetric case. Only the joint distributions for n=14, 60, 100 and success probabilities 0.5 and 0.6 are included in the package to avoid excessive storage, but many more cases may be generated by the function crossrunsymm.

Usage

joint14symm

Format

matrix, 14 rows and 14 columns

Source

generated by the function crossrunsymm and transformed from an Rmpfr array to a matrix

Joint probabilities, 60, success probability 0.6

Description

The joint probabilities of the number C og crossings (0, ... 59) and the longest run L (1, ..., 60) in a series of n=60 independent Bernoulli observations for success probability 0.6. The probabilities are stored in the "times" representations, multiplied by 2^{60-1}. Only the joint distributions for n=15, 60, 100 and success probabilities 0.5 and 0.6 are included in the package to avoid excessive storage, but many more cases are generated in the script crossrun1.R.

Usage

joint60.6

Format

matrix, 60 rows and 60 columns

Source

generated by the function crossrunbin and transformed from an Rmpfr array to a matrix

Joint probabilities, n=60, around the empirical median

Description

Joint probabilities of the number C of crossings (1, ... 59) and the longest run L (1, ..., 30) in a series of n=14 Bernoulli observations around its empirical median. The probabilities are stored in the "times" representations, multiplied by (14 by 7)/2=1716, the number of constellations starting above the median, and are integers. About the empirical median there is at least one crossing, and the longest runcannot exceed 60/2=30. Only the joint distributions for n=14, 60 are included in the package to avoid excessive storage, but many more cases may be generated by the function 'crossrunem. Since these computations are demanding in terms of storage and computation time, they are at present not performed for n much above 60. '#'

Usage

joint60em

Format

matrix, 59 rows and 30 columns

Source

generated by the function crossrunem and transformed from an Rmpfr array to a matrix

Joint probabilities, n=60, symmetric case

Description

The joint probabilities of the number C og crossings (0, ... 59) and the longest run L (1, ..., 60) in a series of n=60 independent Bernoulli observations for the symmetric case (success probability 0.5). The probabilities are stored in the "times" representations, multiplied by 2^{60-1} and are integers in the symmetric case. Only the joint distributions for n=15, 60, 100 and success probabilities 0.5 and 0.6 are included in the package to avoid excessive storage, but many more cases may be generated by the function crossrunsymm.

Usage

joint60symm

Format

matrix, 60 rows and 60 columns

Source

generated by the function crossrunsymm and transformed from an Rmpfr array to a matrix

Simulation of Independent Bernoulli Observations

Description

Simulation of a sequence of independent Bernoulli Observations. To reduce the amount of random draws, each simulation is based on a sequence of standard normal variables, and whether each observation is above a shift defined by the binomial probabilities assumed.

Usage

simclbin(nser = 100, nsim = 1e+05, probs = c(0.5, 0.6, 0.7, 0.8, 0.9))

Arguments

Value

a data frame with the number of crossings and longest run for each probability. For instance the variables nc0.5 and lr0.5 are the number of crossings and the longest run for success probability 0.5. One row for each simulation.

Examples

cl30simbin <- simclbin(nser=30, nsim=100)
mean(cl30simbin$nc0.5) # mean number of crossings, p=0.5
mean(cl30simbin$lr0.9) # mean longest run, p=0.9

Check of joint probabilities by simulations

Description

Simulation of a sequence of n=2m observations around the median in the sequence. To be used for checking the results of crossrunem.

Usage

simclem(m1 = 7, nsim = 1e+05)

Arguments

Value

data frame with cs, number of crossings and ls, longest run in the simulations.

Examples

simclem14 <- simclem(nsim=sum(joint14em))
print(table(simclem14)) # joint distributions in the simulations
print(joint14em) # for comparison