Package 'BayesTreePrior'

Simulation of the tree prior.

Description

This is the main function to use for simulating from the prior. There are 4 cases :

• Case #1: Unrealistic case where we assume that the number of variables and possible splits are infinite (therefore $P(T)$ is not dependent on the design matrix X) and $\beta=0$

• Case #2: Unrealistic case where we assume that the number of variables and possible splits are infinite (therefore $P(T)$ is not dependent on the design matrix X)

• Case #3: One variable with a finite number of observations (Seems to be equivalent to the multiple variables case when all variables are continuous)

• Case #4: General case

Case #1 will be used if no design matrix X or number of observations is given and $\beta = 0$. Case #2 will be used if no design matrix X or number of observations is given and $\beta \neq 0$. Case #3 will be used if no design matrix X is given but the number of observations is given. Case #4 will be used if the design matrix X is given. Note that case #4 is always slower, so if all your variables are continuous, it would be advisable to enter the number of uniques observations rather than the design matrix X, to be able to use case #3.

Usage

BayesTreePrior(alpha, beta, X = NULL, n_obs = NULL, n_iter = 500,
  minpart = 1, package = NULL, pvars = NULL, MIA = FALSE,
  missingdummy = FALSE)

Arguments

alpha	base parameter of the tree prior, $\alpha \in [0,1)$.
beta	power parameter of the tree prior, $\beta \geq 0$.
X	data.frame of the design matrix (Required for case #4).
n_obs	number of unique observations, $n_{obs}>1$ (Required for case #3).
n_iter	number of trees to generate, $n_{iter}>0$ (Used for case #2, #3 or #4).
minpart	the minimum number of observations required in one of the child to be able to split, $minpart>0$.
package	a optional string that can take the following values : "BayesTree", "tgp" or "bartMachine". It forces the algorithm to use the default paramameters used by the package specified ($minpart=5$ for BayesTree, $minpart = max(c(10,dim(X)[2]+1))$ for tgp and $minpart=1$ for bartMachine).
pvars	vector of probabilities for the choices of variables to split (Will automatically be normalized so that the sum equal to 1). It must be twice as large as the number of variables when $missingdummy$ is TRUE.
MIA	set to TRUE if you want Missing Incorporated in Attributes (MIA) imputation to be used.
missingdummy	set to TRUE if you want the NAs to be dummy coded.

Value

In case #1, it returns a list containing, in the following order: the expectation and the variance of the number of bottom nodes. In cases #2, #3 or #4, it returns a list containing, in the following order: the mean number of bottom nodes, the standard deviation of the number of bottom nodes, the mean of the depth, the standard deviation of the depth and a data.frame of vectors $(b_i,d_i)$, where $b_i$ is the number of bottom nodes and $d_i$ is the depth of the $i$th generated tree ($i=1, \ldots ,n_{iter}$).

References

Chipman, H. A., George, E. I., & McCulloch, R. E. (1998). Bayesian CART model search. Journal of the American Statistical Association, 93(443), 935-948.

Gramacy, R. B. (2007). tgp: an R package for Bayesian nonstationary, semiparametric nonlinear regression and design by treed Gaussian process models. Journal of Statistical Software, 19(9), 6.

Chipman, H. A., George, E. I., & McCulloch, R. E. (2010). BART: Bayesian additive regression trees. The Annals of Applied Statistics, 266-298.

Kapelner, A., & Bleich, J. (2013). bartMachine: A powerful tool for machine learning. stat, 1050, 8.

Twala, B. E. T. H., Jones, M. C., & Hand, D. J. (2008). Good methods for coping with missing data in decision trees. Pattern Recognition Letters, 29(7), 950-956.

Jolicoeur-Martineau, A. (expected 2016) Etude d'une loi a priori pour les arbres binaires de regression (Study on the prior distribution of binary regression trees) (Master thesis). UQAM university.

Examples

#Case 1 : Unrealistic case where we assume that the number of var/obs is infinite and beta=0
results1 = BayesTreePrior(.45,0)
#Case 2 : Unrealistic case where we assume that the number of var/obs is infinite
results2 = BayesTreePrior(.95,.5)
#Case 3 : One variable with a finite number of observations
results3 = BayesTreePrior(.95,.5,n_obs=150)
if (requireNamespace("MASS", quietly = TRUE)) {
    #Case 4 : General case, without missing data
    x1 = MASS::mcycle$times
    x2= MASS::mcycle$accel
    X = cbind(x1, x2)
    results4_nomiss = BayesTreePrior(.95,.5, data.frame(X), minpart=5, package="tgp")
    #Case 4 : General case, with missing data
    x1[sample(1:length(x1), 20)] <- NA
    x2[sample(1:length(x2), 20)] <- NA
    X = cbind(x1, x2)
    results4_miss = BayesTreePrior(.95,.5, data.frame(X), minpart=5, package="tgp", 
    MIA=TRUE, missingdummy=TRUE)
}

---

Simulation of the tree prior in the general case (Case #4).

Description

Generate $n_{iter}$ trees from the prior distribution in the general case (Case #4).

Usage

BayesTreePriorNotOrthogonal(alpha, beta, X, n_iter = 500, minpart = 1,
  pvars = NULL, MIA = FALSE, missingdummy = FALSE)

Arguments

alpha	base parameter of the tree prior, $\alpha \in [0,1)$.
beta	power parameter of the tree prior, $\beta \geq 0$.
X	data.frame of the design matrix.
n_iter	number of trees to generate, $n_{iter}>0$.
minpart	the minimum number of observations required in one of the child to be able to split, $minpart>0$.
pvars	vector of probabilities for the choices of variables to split (Will automatically be normalized so that the sum equal to 1). It must be twice as large as the number of variables when $missingdummy$ is TRUE.
MIA	set to TRUE if you want Missing Incorporated in Attributes (MIA) imputation to be used.
missingdummy	set to TRUE if you have dummy coded the NAs.

Value

Returns a list containing, in the following order: the mean number of bottom nodes, the standard deviation of the number of bottom nodes, the mean of the depth, the standard deviation of the depth and a data.frame of vectors $(b_i,d_i)$, where $b_i$ is the number of bottom nodes and $d_i$ is the depth of the $i$th generated tree ($i=1, \ldots ,n_{iter}$).

See Also

BayesTreePriorOrthogonalInf, BayesTreePriorOrthogonal

Examples

if (requireNamespace("MASS", quietly = TRUE)) {
    x1 = MASS::mcycle$times
    x1[sample(1:length(x1), 20)] <- NA
    x2= MASS::mcycle$accel
    x2[sample(1:length(x2), 20)] <- NA
    X = cbind(x1, x2)
    results1 = BayesTreePriorNotOrthogonal(.95,.5, data.frame(X), minpart=5)
    X_dummies = is.na(X) + 0
    results2 = BayesTreePriorNotOrthogonal(.95,.5, data.frame(cbind(X,X_dummies)), minpart=5, 
    MIA=TRUE, missingdummy=TRUE)
}

---

Simulation of the tree prior in the case where we have one single variable (Case #3).

Description

Generate $n_{iter}$ trees from the prior distribution in the case where we have one variable with a finite number of observations (Case #3).

Usage

BayesTreePriorOrthogonal(alpha, beta, n_obs, n_iter = 500)

Arguments

alpha	base parameter of the tree prior, $\alpha \in [0,1)$.
beta	power parameter of the tree prior, $beta \geq 0$.
n_obs	number of unique observations, $n_{obs}>1$.
n_iter	number of trees to generate, $n_{iter}>0$.

Value

Returns a list containing, in the following order: the mean number of bottom nodes, the standard deviation of the number of bottom nodes, the mean of the depth, the standard deviation of the depth and a data.frame of vectors $(b_i,d_i)$, where $b_i$ is the number of bottom nodes and $d_i$ is the depth of the $i$th generated tree ($i=1, \ldots ,n_{iter}$).

See Also

BayesTreePriorOrthogonalInf, BayesTreePriorNotOrthogonal

Examples

results1 = BayesTreePriorOrthogonal(.95,.5, 100)
results2 = BayesTreePriorOrthogonal(.95,.5, 250)

---

Simulation of the tree prior in the unrealistic case where we assume that the number of variables and possible splits are infinite (therefore P(T) is not dependent on the design matrix X) (Case #2).

Description

Generate $n_{iter}$ trees from the prior distribution in the unrealistic case where we assume that the number of variables and possible splits are infinite (therefore P(T) is not dependent on the design matrix X) (Case #2).

Usage

BayesTreePriorOrthogonalInf(alpha, beta, n_iter = 500)

Arguments

alpha	base parameter of the tree prior, $\alpha \in [0,1)$.
beta	power parameter of the tree prior, $beta \geq 0$.
n_iter	number of trees to generate, $n_{iter}>0$.

Value

Returns a list containing, in the following order: the mean number of bottom nodes, the standard deviation of the number of bottom nodes, the mean of the depth, the standard deviation of the depth and a data.frame of vectors $(b_i,d_i)$, where $b_i$ is the number of bottom nodes and $d_i$ is the depth of the $i$th generated tree ($i=1, \ldots ,n_{iter}$).

See Also

BayesTreePriorOrthogonal, BayesTreePriorNotOrthogonal

Examples

results = BayesTreePriorOrthogonalInf(.95,.5)

---

Expected value of the number of bottom nodes in the unrealistic case where we assume that the number of variables and possible splits are infinite (therefore P(T) is not dependent on the design matrix X) and $\beta=0$ (Case #1).

Description

Expected value of the number of bottom nodes in the unrealistic case where we assume that the number of variables and possible splits are infinite (therefore P(T) is not dependent on the design matrix X) and $\beta=0$ (Case #1).

Usage

E_alpha(alpha)

Arguments

alpha

base parameter of the tree prior, $alpha \in [0,1)$.

Value

Returns the expected value of the number of bottom nodes.

References

Jolicoeur-Martineau, A. (Currently in revision, expected 2016) Etude d'une loi a priori pour les arbres binaires de regression (Study on the prior distribution of binary regression trees) (Master thesis). UQAM university.

See Also

Var_alpha

Examples

E_alpha(.30)
E_alpha(.45)
E_alpha(.499)
E_alpha(.75)

---

Unique splits that leads to children with more than $minpart$ nodes.

Description

Unique splits that leads to children with more than $minpart$ nodes.

Usage

GetListUniqueSplits(x, minpart = 1, MIA = FALSE)

Arguments

x	vector containing the observations of a variable.
minpart	minimum number of observations in the children nodes.
MIA	set to TRUE if you want Missing Incorporated in Attributes (MIA) imputation to be used.

Value

If $MIA$ is TRUE and $minpart>1$, the possible splits could be different depending on whether we transfer the NAs to the left child or the right child; if this is the case then the function returns a list $(v1,v2)$, where $v1$ is the vector containing the unique splits that leads to $minpart$ nodes when transferring the NAs to the left child and $v2$ is the vector containing the unique splits that leads to children with more than $minpart$ nodes when transferring the NAs to the left child. Otherwise, it returns the vector containing the unique splits that leads to children with more than $minpart$ nodes.

Examples

GetListUniqueSplits(c(1,4,7,3,0,2,2,3,4,7,7,7),minpart=1)
GetListUniqueSplits(c(1,4,7,3,0,2,2,3,4,7,7,7),minpart=3)
GetListUniqueSplits(c(1,4,7,3,0,2,2,3,4,7,7,7,NA,NA,NA),minpart=1, MIA=TRUE)
GetListUniqueSplits(c(1,4,7,3,0,2,2,3,4,7,7,7,NA,NA,NA),minpart=3, MIA=TRUE)

---

Number of bottom nodes and depth in the case where we have one single variable (Case #3).

Description

Generate a tree and returns the number of bottom nodes and depth in the case where we have one variable with a finite number of observations (Case #3).

Usage

NumBotMaxDepth(alpha, beta, x_size, depth = 0)

Arguments

alpha	base parameter of the tree prior, $\alpha \in [0,1)$.
beta	power parameter of the tree prior, $beta \geq 0$.
x_size	number of possible splits, $x_{size}>0$.
depth	depth of the current node, $depth \geq 0$.

Value

Returns a vector containing the number of bottom nodes and depth

See Also

NumBotMaxDepth_inf, NumBotMaxDepthX

Examples

NumBotMaxDepth(.95,.5,500)

---

Number of bottom nodes and depth in the unrealistic case where we assume that the number of variables and possible splits are infinite (therefore P(T) is not dependent on the design matrix X) (Case #2).

Description

Generate a tree and returns the number of bottom nodes and depth in the unrealistic case where we assume that the number of variables and possible splits are infinite (therefore P(T) is not dependent on the design matrix X) (Case #2).

Usage

NumBotMaxDepth_inf(alpha, beta, depth = 0)

Arguments

alpha	base parameter of the tree prior, $\alpha \in [0,1)$.
beta	power parameter of the tree prior, $beta \geq 0$.
depth	depth of the current node, $depth \geq 0$.

Value

Returns a vector containing the number of bottom nodes and depth.

See Also

NumBotMaxDepth, NumBotMaxDepthX

Examples

NumBotMaxDepth_inf(.95,.5)

---

Number of bottom nodes and depth in the general case (Case #4).

Description

Generate a tree and returns the number of bottom nodes and depth in the general case (Case #4).

Usage

NumBotMaxDepthX(alpha, beta, X, depth = 0, minpart = 1, pvars = NULL,
  MIA = FALSE, missingdummy = FALSE)

Arguments

alpha	base parameter of the tree prior, $\alpha \in [0,1)$.
beta	power parameter of the tree prior, $beta \geq 0$.
X	data.frame of the design matrix.
depth	depth of the current node, $depth \geq 0$.
minpart	the minimum number of observations required in one of the child to be able to split, $minpart>0$.
pvars	vector of probabilities for the choices of variables to split (Will automatically be normalized so that the sum equal to 1). It must be twice as large as the number of variables when $missingdummy$ is TRUE.
MIA	set to TRUE if you want Missing Incorporated in Attributes (MIA) imputation to be used.
missingdummy	set to TRUE if you have dummy coded the NAs.

Value

Returns a vector containing the number of bottom nodes and depth

References

Twala, B. E. T. H., Jones, M. C., & Hand, D. J. (2008). Good methods for coping with missing data in decision trees. Pattern Recognition Letters, 29(7), 950-956.

See Also

NumBotMaxDepth_inf, NumBotMaxDepth

Examples

if (requireNamespace("MASS", quietly = TRUE)) {
    x1 = MASS::mcycle$times
    x1[sample(1:length(x1), 20)] <- NA
    x2= MASS::mcycle$accel
    x2[sample(1:length(x2), 20)] <- NA
    X = cbind(x1, x2)
    results1 = NumBotMaxDepthX(.95,.5, data.frame(X), minpart=5)
    X_dummies = is.na(X) + 0
    results2 = NumBotMaxDepthX(.95,.5, data.frame(cbind(X,X_dummies)), minpart=5, MIA=TRUE, 
    missingdummy=TRUE)
}

---

Probability of split of the tree prior.

Description

Probability of split of the tree prior.

Usage

p_split(alpha, beta, depth = 0)

Arguments

alpha	base parameter of the tree prior, $\alpha \in [0,1)$.
beta	power parameter of the tree prior, $\beta \geq 0$.
depth	depth of the current node, $depth \geq 0$.

Value

Returns the probability of split.

References

Chipman, H. A., George, E. I. et McCulloch, R. E. (1998). Bayesian CART model search. Journal of the American Statistical Association, 93(443), 935-948.

Examples

p_split(.95,.5)
p_split(.95,.5,1)
p_split(.95,.5,2)

---

Variance of the number of bottom nodes in the unrealistic case where we assume that the number of variables and possible splits are infinite (therefore P(T) is not dependent on the design matrix X) and $\beta=0$ (Case #1).

Description

Variance of the number of bottom nodes in the unrealistic case where we assume that the number of variables and possible splits are infinite (therefore P(T) is not dependent on the design matrix X) and $\beta=0$ (Case #1).

Usage

Var_alpha(alpha)

Arguments

alpha

base parameter of the tree prior, $\alpha \in [0,1)$.

Value

Returns the variance of the number of bottom nodes.

References

Jolicoeur-Martineau, A. (Currently in revision, expected 2016) Etude d'une loi a priori pour les arbres binaires de regression (Study on the prior distribution of binary regression trees) (Master thesis). UQAM university.

See Also

E_alpha

Examples

Var_alpha(.30)
Var_alpha(.45)
Var_alpha(.499)
Var_alpha(.75)

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