# Functional least angle regression.

### Description

This is the main function for the functional least angle regression algorithm. Under certain conditions, the function only needs the input of two arguments: `x`

and `y`

. This function can do both variable selection and parameter estimation.

### Usage

1 2 3 4 |

### Arguments

`x` |
The mixed scalar and functional variables. Note that each of the functional variables is expected to be stored in a matrix. Each row of the matrix should represent a sample or a curve. If there is only one functional variable, |

`y` |
The scalar variable. It can be a matrix or a vector. |

`method` |
The representative methods for the functional coefficients. The method could be one of the 'basis', 'gq' and 'raw' for basis function expression, Gaussian quadrature and representative data points, respectively. |

`max_selection` |
Number of maximum selections when stopping the algorithm. Set a reasonable number for this argument to increase the calculation speed. |

`cv` |
Choise of cross validation. At the moment, the only choice is the generalized cross validation, i.e., |

`lasso` |
Use lasso modification or not. In other words, can variables selected in the former iterations be removed in the later iterations. |

`check` |
Type of check methods for lasso modification. 1 means variance check, 2 means sign check. |

`select` |
If |

`VarThreshold` |
Threshold for removing variables based on variation explained. More specifically, one condition to remove a variable is that the variation explained by a variable is less than |

`SignThreshold` |
This is a similar argument to |

`normalize` |
Choice of normalization methods. This is to remove any effects caused by the different dimensions of functional variables and scalar variables. Currently we have |

`control` |
list of control elements for the functional coefficients. See |

### Value

`Mu` |
Estimated intercept from each of the iterations |

`Beta` |
Estimated functional coefficients from each of the iterations |

`alpha` |
Distance along the directions from each of the iterations |

`p2_norm` |
Normalization constant applied to each of the iterations |

`AllIndex` |
All the index. If one variable is removed, it will become a negative index. |

`index` |
All the index at the end of the selection. |

`CD` |
Stopping rule designed for this algorithm. The algorithm should stop when this value is very small. Normally we can observe an obvious and severe drop of the value. |

`resid` |
Residual from each of the iteration. |

`RowMeans` |
Point-wise mean of the functional variables and mean of the scalar variables. |

`RowSds` |
Point-wise sd of the functional variables and sd of the scalar variables. |

`yMean` |
Mean of the response variable. |

`ySD` |
SD of the response variable. |

`p0` |
The projections obtained from each iteration without normalization. |

`cor1` |
The maximum correlation obtained from the first iteration. |

`lasso` |
Weather have lasso step or not. |

`df` |
The degrees of freedom calculated at the end of each iteration. |

`Sigma2Bar` |
Estimated $sigma^2$. |

`StopStat` |
Conventional stopping criteria. |

`varSplit` |
The variation explained by each of the candidate variables at each iteration. |

`SignCheckF` |
The proportion of sign changing for each of the candidate variables at each iteration. |

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | ```
library(flars)
library(fda)
#### Ex1 ####
## Generate some data.
dataL=data_generation(seed = 1,uncorr = TRUE,nVar = 8,nsamples = 120,
var_type = 'm',cor_type = 3)
## Do the variable selection
out=flars(dataL$x,dataL$y,method='basis',max_selection=9,
normalize='norm',lasso=FALSE)
## Check the stopping point with CD
plot(2:length(out$alpha),out$CD) # plot the CD with the iteration number
## In simple problems we can try
(iter=which.max(diff(out$CD))+2)
#### Ex2 ####
## Generate some data.
# dataL=data_generation(seed = 1,uncorr = FALSE,nVar = 8,nsamples = 120,
# var_type = 'm',cor_type = 3)
## add more variables to the candidate
# for(i in 2:4){
# dataL0=data_generation(seed = i,uncorr = FALSE,nVar = 8,nsamples = 120,
# var_type = 'm',cor_type = 3)
# dataL$x=c(dataL$x,dataL0$x)
# }
# names(dataL$x)=paste0('v_',seq(length(dataL$x)))
## Do the variable selection
# out=flars(dataL$x,dataL$y,method='basis',max_selection=9,
# normalize='norm',lasso=FALSE)
#### Ex3 (small subset of a real data set) ####
data(RealDa, package = 'flars')
out=flars(RealDa$x,RealDa$y,method='basis',max_selection=9,
normalize='norm',lasso=FALSE)
# out=flars(RealDa$x,RealDa$y,method='basis',max_selection=9,
# normalize='norm',lasso=TRUE)
## Check the stopping point with CD
plot(2:length(out$alpha),out$CD) # plot the CD with the iteration number
## The value drops to very small compare to others at iteration six and
### stays low after that, so the algorithm may stop there.
``` |

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