Maximizing the likelihood of the generalised additive index model with shape constraints

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Description

This function searches for a maximum likelihood estimator (mle) of the generalised additive index regression with shape constraints. A stochastic search strategy is used here.

Each index is a linear combination of some (or all) the covariates. Each additive component function of these index predictors is assumed to belong to one of the nine possible shape restrictions.

The output is an object of class scair which contains all the information needed to plot the estimator using the plot method, or to evaluate it using the predict method.

Usage

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scair(x,y,shape=rep("l",1), family=gaussian(), weights=rep(1,length(y)), 
  epsilon=1e-8, delta=0.1, indexgen=c("unif", "norm"), iter = 200, 
  allnonneg = FALSE)

Arguments

x

Observed covariates in R^d, in the form of an n x d numeric matrix.

y

Observed responses, in the form of a numeric vector of length n.

shape

A vector that specifies the shape restrictions for additive component function of each index (also called the ridge function), in the form of a string vector of length m. Here for the sake of identifiability, we require the number of indices m <= d. The shape constraints we considered (with their coresponding abbreviations used in shape) are listed below:

l: linear

in: monotonically increasing

de: monotonically decreasing

cvx: convex

cvxin: convex and increasing

cvxde: convex and decreasing

ccv: concave

ccvin: concave and increasing

ccvde: concave and decreasing

family

A description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. Currently only the following five common exponential families are allowed: Gaussian, Binomial, Poisson, and Gamma. By default the canonical link function is used.

weights

An optional vector of prior weights to be used when maximising the likelihood. It is a numeric vector of length n. By default equal weights are used.

epsilon

Positive convergence tolerance epsilon when performing the iteratively reweighted least squares (IRLS) method at each iteration of the active set algorithm in scar. See scar for more details.

delta

A tuning parameter used to avoid the perfect fit phenomenon, and to ensure identifiability. It represents the lower bound of the minimum eigenvalue of all possible A^T A subject to identiability conditions, where A is an index matrix. It should be smaller than 1. This parameter is NOT needed when d=1, or the prediction function is convex or concave, or all the entries of the index matrix are non-negative if all ridge functions are increasing or decreasing.

indexgen

It determines how the index matrices are generated in the stochastic search. If its value is "unif", then entries of the index matrices are drawn from uniform distribution; otherwise, if its value is "norm", entries are drawn from normal.

iter

Number of iterations of the stochastic search.

allnonneg

A boolean variable that specifies whether all the entries of the index matrices are non-negative. If it is true, then delta is no longer needed in case the ridge functions are either all increasing, or all decreasing.

Details

For i=1,...,n, let X_i be the d-dimensional covariates, Y_i be the corresponding one-dimensional response and w_i be its weight. The generalised additive index model can be written as

g(mu) = f(A^T x),

where x=(x_1,...,x_d)^T, g is a known link function, A is an d x m index matrix, and f is an additive function. Our task is to estimate both the index matrix and the additive function.

Assume the canonical link function is used here, then the maximum likelihood estimator of the generalised additive index model based on observations (X_1,Y_1), ..., (X_n,Y_n) is the function that maximises

[w_1 (Y_1 f(A^T X_1) -B(f(A^T X_1))) + ... + w_n (Y_n f(A^T X_n) -B(f(A^T X_n)))]/n

subject to the restrictions that for every j=1,...,m, the j-th additive component of f satisfies the constraint indicated by the j-th element of shape. Here B(.) is the log-partition function of the specified exponential family distribution, and w_i are the weights. For i.i.d. data, w_i should be 1 for each i.

For any given A, the optimization problem can solved using the active set algorithm implemented in scar. Therefore, this problem can be reduced to a finite-dimensional optimisation problem. Here we apply a simple stochastic search strategy is proposed, though other methods, such as downhill simplex, is also possible (and sometimes offers competitive performance). All the implementaton details can be found in Chen and Samworth (2014), where theoretical justification of our estimator (i.e. uniform consistency) is also given.

For the identifiability of additive index models, we refer to Yuan (2011).

Value

An object of class scair, with the following components:

x

Covariates copied from input.

y

Response copied from input.

shape

Shape vector copied from input.

weights

Vector of weights copied from input.

family

The exponential family copied from input.

componentfit

Value of the fitted component function at each observed index (computed using the estimated index matrix), in the form of an n x m numeric matrix, where the element at the i-th row and the j-th column is the value of f_j at the j-th coordinate of A^T X_i, with the identifiability condition satisfied (see details of scar).

constant

The estimated value of the constant c in the additive function f (see details of scar)).

deviance

Up to a constant, minus twice the maximised log-likelihood. Where applicable, the constant is chosen to make the saturated model to have zero deviance. See also glm.

nulldeviance

The deviance for the null model.

delta

A parameter copied from input.

iter

Total number of iterations of the stochastic search algorithm

allnonneg

specifies whether all entris of the index matrix is non-negative, copied from input.

Author(s)

Yining Chen and Richard Samworth

References

Chen, Y. and Samworth, R. J. (2014). Generalised additive and index models with shape constraints. arXiv:1404.2957.

Yuan, M. (2011). On the identifiability of additive index models. Statistica Sinica, 21, 1901-1911.

See Also

plot.scair, predict.scair, scar, decathlon

Examples

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## An example in the Gaussian additive index regression setting:
## Define the additive function f on the scale of the predictors
f<-function(x){
  return((0.5*x[,1]+0.25*x[,2]-0.25*x[,3])^2) 
}

## Simulate the covariates and the responses
## covariates are drawn uniformly from [-1,1]^3
set.seed(10)
d = 3
n = 500
x = matrix(runif(n*d)*2-1,nrow=n,ncol=d) 
y = f(x) + rnorm(n,sd=0.5)

## Single index model so no delta is required here
shape=c("cvx")
object = scair(x,y,shape=shape, family=gaussian(),iter = 100)

## Estimated index matrix
object$index

## Root squared error for the estimated index
sqrt(sum((object$index - c(0.5,0.25,-0.25))^2))

## Plot the estimatied additive function for the single index
plot(object)

## Evaluate the estimated prediction function at 10^4 random points 
## drawing from the interior of the support
testx = matrix((runif(10000*d)*1.96-0.98),ncol=d)
testf = predict(object,testx)

## and calculate the (estimated) absolute prediction error
mean(abs(testf-f(testx))) 

## Here we can treat the obtained index matrix as a warm start and perform 
## further optimization (on the second and third entry of the index)
## using e.g. the default R optimisation routine.
fn<-function(w){
    dev = Inf
    if (abs(w[1])+abs(w[2])>1) return(dev)
    else {
      wnew = matrix(c(1-abs(w[1])-abs(w[2]),w[1],w[2]),ncol=1)
      dev = scar(x %*% wnew, y, shape = "cvx")$deviance
      return (dev)
    } 
}
index23 = optim(object$index[2:3],fn)$par
newindex = matrix(c(1-sum(abs(index23)),index23),ncol=1); newindex

## Root squared error for the new estimated index
sqrt(sum((newindex - c(0.5,0.25,-0.25))^2))

## A further example is provided in decathlon dataset