This function can be used to generate a dataset for the function-on-function regression model $$ Y(u) = \int X(v) \beta(u,v) dv + \epsilon(u),$$ where \(Y(u)\) denotes the functional response, \(X(v)\) denotes the functional predictor, \(\beta(u,v)\) denotes the bivariate regression coefficient function, and \(\epsilon(u)\) is the error function.
fpqr_dgp(n, nphi, gpy, gpx, err.dist, out.p)A matrix containing the observations of simulated functional predictor variable.
A matrix containing the observations of simulated functional response variable.
A matrix containing the observations of simulated functional predictor variable without measurement error.
A matrix containing the observations of simulated functional response variable without measurement error.
A vector containing the intercept function.
A matrix containing the bivariate regression coefficient vector.
An integer, specifying the number of observations for each variable to be generated.
An integer, specifying the number of sub-random variables used in the generation of functional predictor. Default value is 10.
A vector, denoting the grid points of the functional response variable \(Y(u)\).
A vector, denoting the grid points of the functional predictor variable \(X(v)\).
Distribution of the error term. Possibilities are "normal","t","chisq", "cn".
An integer between 0 and 1, denoting the outlier percentage in the generated data when the error distribution is contaminated normel, that is, "cn".
Muge Mutis, Ufuk Beyaztas, Filiz Karaman, and Han Lin Shang
When generating the data for the function-on-function regression model, first, the functional predictor is generated using the following process: $$X_i(v) = \sum_{k=1}^{10} \frac{1}{k^{2}} \{ \zeta _{i1,k}\sqrt{2} \sin ( k \pi v ) \zeta _{i2,k} \sqrt{2} \cos (k \pi v) \}$$ where \(\zeta _{i1,k}\) and \(\zeta _{i2,k}\) for \(k=1,\ldots,10\) are independent random variables generated from the standard normal distribution. The true intercept and bivariate regression parameter functions are generated as \(\alpha ( u )=2e^\{- ( u-1 )^{2}\}\) and \(\beta ( v,u )= 4 \cos ( 2 \pi u ) \sin ( \pi v )\), respectively, Then, the elements of the functional response are generated as follows: $$ Y_{i} ( u ) = \alpha ( u ) + \int X_{i} ( v ) \beta( v, u) dv + \varepsilon _{i} ( u ),$$ where \(\varepsilon _{i} ( u )\) are generated from one of the following distributions:
When err.dist is "normal", then the error terms are generated from the standard normal distribution.
When err.dist is "t", then the error terms are generated from the t distribution with five degrees of freedom.
When err.dist is "chisq", then the error terms are generated from the chi-square distribution with one degree of freedom.
When err.dist is "cn", then the error terms are generated from the contaminated normal distribution.
C. Xiong, X. Liugen, and C. Jiguo (2021), "Robust penalized M-estimation for function-on-function linear regression"", Stat, 10(1), e390.
gpx <- (1:50)/50
gpy <- (1:60)/60
data <- fpqr_dgp(n = 250, gpy = gpy, gpx = gpx, err.dist = "normal")
y <- data$y
x <- data$x
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