WOCR: Fitting the weighted orthoginal components regression (WOCR)...
In xgsu/WOCR: Weighted Orthogonal Components Regression (WOCR) Analysis

Description Usage Arguments Details Value References See Also Examples

Fitting the weighted orthoginal components regression (WOCR) models

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WOCR(formula, data = NULL, scale = TRUE, model = "RR.d.lambda", a = 20,
  criterion = NULL, use.GenSA = TRUE, maxit.global = 100, eps = 1e-06,
  LB = NULL, UB = NULL, details = FALSE)

`formula`	An object of class `formula`, with the response on the left of a `~` operator, and the terms on the right. The function assumes the intercept to be zero with scaled data. If the interacept `~ +1` is included explicitly or by default, it will be automatically removed.
`data`	A data.frame in which to interpret the variables named in the `formula` argument.
`scale`	Logical indicator of whether you want to scale the data. Here, `scale=TRUE` should always be done unless the data have already scaled or standardized. This action involves standaradizing the columns in matrix X and center the response vector y. Note that this option also has an effect on the resultant prediction.
`model`	Specifies the WOCR model to be fitted. Six choices are possible: `RR.d.lambda`, `RR.gamma.lambda`, `PCR.d.c`, `PCR.gamma.c`, `PCR.d.a.c` and `PCR.gamma.a.c`.
`a`	The fixed shape parameter a in models `PCR.d.c` and `PCR.gamma.c` only. By default, a is set to 20.
`criterion`	Specifies the information criterion to be minimized. The appropriate choice is one from `"AIC"`, `"BIC"`, or `"GCV"`. By default, `criterion` is left as `NULL` and the recommended one is used.
`use.GenSA`	Logical value indicating if the generalized simulated annealing `GenSA` is used for optimizing the 2-D optimization problems in models `PCR.d.a.c` and `PCR.gamma.a.c`. The default is `TRUE`.
`maxit.global`	Maximum number of iterations allowed for the global optimization algorithm, which is either `GenSA` or `SANN` in `optim`. Default value is 100.
`LB`	The lower bounds for the search space in `GenSA` The default is `NULL` and recommended one is used.
`UB`	The upper bounds for the search space in `GenSA` The default is `NULL` and recommended one is used.
`details`	Logical value: if `TRUE`, detailed results will be printed out when running `coxphMIC`.
`epsilon`	Tolerance level for convergence. Default is 1e-6.
`maxit.local`	Maximum number of iterations allowed for the local optimizaiton algorithm `BFGS`. Default value is 100.

The main idea of WOCR is to parameterize the weights for orthogonal components with a simple function whose specification is up to one or two tuning parameters. The weighting strategy takes advantage of the inherent monotonicity among these orthogonal components.

To solve the 2-D smooth yet nonconvex optimization, two options are made available. The first is a simulated annealing (method="SANN" option in optim) global optimization algorithm is first applied. The resultant estimator is then used as the starting point for another local optimization algorithm, where the quasi-Newton BFGS method (method="BFGS" in optim) by default. Optionally, the generalized simulated annealing, implemented in GenSA, can be used instead. This latter approach tends to be slower. However, it does not need to be combined with another local optimization; besides, it often yields the same final solution with different runs. Thus, when use.GenSA=TRUE, the output includes opt.global only, without opt.local.

An object of class WOCR is returned, which may contain the following components depending on the options.

call: the matched call.
opt.global: Results from the preliminary run of a global optimization procedure (SANN as default or GenSA).
opt: Results from the final optimization procedure. This could be the results from the 1-D optimization optimize or from the second run of the local optimizaiton BFGS.
m: The rank of the design matrix X.
d: The vector of singular values.
gamma: The vector of coefficients, consisting of the inner product of each orthogonal component and the repsonse y.
formula: The formula used in fitting WOCR. A copy of formula will be useful for prediction later on.
mean.y: Mean value of the response y.
mean.x: Mean vector of columns of matrix X.
sd.x: Vector of standard deviations for each X variables or columns.
min.Q: Value of the minimized objective function.
lambda.hat: The 1-D or 2-D estimated tuning parameter(s).
beta.hat: The coefficients for the standardized X variables, instead of the orthogonal components.
w.hat: The estimated weights.
n.comp: The selected number of components for WOCR models PCR.d.c and PCR.gamma.c only.
y.hat: The fitted value for the response. The mean response is added back. So y.hat can be compared directly to the response y.

Su, X., Wonkye, Y., Wang, P., and Yin, X. (2016+). Weighted orthogonal component regression (WOCR). Submitted.

pcr

   data(BostonHousing1)
   fit.wocr <- WOCR(formula=cmedv~., data=BostonHousing1, model="RR.d.lambda")
   print(fit.wocr)
   plot(fit.wocr, horizontal=TRUE)