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a research paper about FDSLRM modeling with supplementary materials - software, notebooks

Project: fdslrm
Views: 1041
Kernel: R (R-Project)

Authors: Andrej Gajdoš, Jozef Hanč, Martina Hančová
Faculty of Science, P. J. Šafárik University in Košice, Slovakia
email: [email protected]

EBLUP-NE for tourism

R-based computational tools - CVXR

Table of Contents

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R packages

CVXR: R-Embedded Object-Oriented Modeling Language for Convex Optimization

  • Purpose: R package for solving convex optimization tasks

  • Version: 0.99-5, 2019

  • URL: https://cvxr.rbind.io/

  • Computational parameters of CVXR:

  • solver - the convex optimization solver ECOS, OSQP, and SCS chosen according to the given problem * OSQP for convex quadratic problems * max_iter - maximum number of iterations (default: 4000). * eps_abs - absolute accuracy (default: 1e-3). * eps_rel - relative accuracy (default: 1e-4). * ECOS for convex second-order cone problems * maxit - maximum number of iterations (default: 100). * abstol - absolute accuracy (default: 1e-8). * reltol - relative accuracy (default: 1e-8). * feastol - tolerance for feasibility conditions (default: 1e-8). * abstol_inacc - absolute accuracy for inaccurate solution (default: 5e-5). * reltol_inacc - relative accuracy for inaccurate solution (default: 5e-5). * feastol_inacc - tolerance for feasibility condition for inaccurate solution (default: 1e-4). * SCS for large-scale convex cone problems * max_iters - maximum number of iterations (default: 5000). * eps - convergence tolerance (default: 1e-5). * alpha - relaxation parameter (default: 1.5). * scale - factor by which the data is rescaled, only used if normalize is TRUE (default: 1.0). * normalize - whether the heuristic data rescaling should be used (default: TRUE).

library(CVXR)
Attaching package: ‘CVXR’ The following object is masked from ‘package:stats’: power

Data and model

In this econometric FDSLRM application, we consider the time series data set, called visnights, representing total quarterly visitor nights (in millions) from 1998-2016 in one of the regions of Australia -- inner zone of Victoria state. The number of time series observations is n=76n=76. The data was adapted from Hyndman, 2018.

The Gaussian orthogonal FDSLRM fitting the tourism data has the following form:

X(t) ⁣= ⁣β1+β2cos⁡(2πt76)+β3sin⁡(2πt⋅276)++Y1cos⁡(2πt⋅1976)+Y2sin⁡(2πt⋅1976)+Y3cos⁡(2πt⋅3876)+w(t), t∈N, \begin{array}{rl} & X(t) & \! = \! & \beta_1+\beta_2\cos\left(\tfrac{2\pi t}{76}\right)+\beta_3\sin\left(\tfrac{2\pi t\cdot 2}{76}\right) + \\ & & & +Y_1\cos\left(\tfrac{2\pi t\cdot 19 }{76}\right)+Y_2\sin\left(\tfrac{2\pi t\cdot 19}{76}\right) +Y_3\cos\left(\tfrac{2\pi t\cdot 38}{76}\right) +w(t), \, t\in \mathbb{N}, \end{array}

where β=(β1, β2, β3)′∈R3,Y=(Y1,Y2,Y3)′∼N3(0,D),w(t)∼iid N(0,σ02),ν=(σ02,σ12,σ22,σ32)∈R+4\boldsymbol{\beta}=(\beta_1,\,\beta_2,\,\beta_3)' \in \mathbb{R}^3, \mathbf{Y} = (Y_1, Y_2, Y_3)' \sim \mathcal{N}_3(\boldsymbol{0}, \mathrm{D}), w(t) \sim \mathcal{iid}\, \mathcal{N} (0, \sigma_0^2), \boldsymbol{\nu}=(\sigma_0^2, \sigma_1^2, \sigma_2^2, \sigma_3^2) \in \mathbb{R}_{+}^4.

We identified the given and most parsimonious structure of the FDSLRM using an iterative process of the model building and selection based on exploratory tools of spectral analysis and residual diagnostics (for details see our Jupyter notebook tourism.ipynb).

# data - time series observation
x <- read.csv2('tourism.csv', header = FALSE)
x <- x[,2]
x <- as.numeric(as.vector(x[-1]))
t <- 1:length(x)

# auxilliary functions to create design matrices F, V and the projection matrix M_F
makeF <- function(times, freqs) {

        n <- length(times)
        f <- length(freqs)
        c1 <- matrix(1, n)
        F <- matrix(c1, n, 1)

        for (i in 1:f) {
                F <- cbind(F, cos(2 * pi * freqs[i] * times))
                F <- cbind(F, sin(2 * pi * freqs[i] * times))
        }

        return(F)
}

makeV <- function(times, freqs) {

        V <- makeF(times, freqs)
        V <- V[, -1]

        return(V)
}

makeM_F <- function(F) {

        n <- nrow(F)
        c1 <- rep(1, n)
        I <- diag(c1)
        M_F <- I - F %*% solve((t(F) %*% F)) %*% t(F)

        return(M_F)
}

# model parameters
n <- 76 
k <- 3 
l <- 3

# model - design matrices F, V
F <- makeF(t, c(1/76, 2/76))
F <- F[,-c(3,4)]

V <- makeV(t, c(19/76, 38/76))
V <- V[,-4]


# columns vj of V and their squared norm ||vj||^2
nv2 = colSums(V ^ 2)

# auxiliary matrices and vectors

# Gram matrices GF, GV
GF <- t(F) %*% F
GV <- t(V) %*% V
InvGF <- solve(GF)
InvGV <- solve(GV)

# projectors PF, MF, PV, MV
In <- diag(n)
PF <- F %*% InvGF %*% t(F)
PV <- V %*% InvGV %*% t(V)
MF <- In - PF
MV <- In - PV

# residuals e
e <- MF %*% x

Natural estimators

ANALYTICALLY

using formula (4.1) from Hancova et al 2019

ParseError: KaTeX parse error: \renewcommand{\arraystretch} when command \arraystretch does not yet exist; use \newcommand

1st\boldsymbol{1^{st}} stage of EBLUP-NE 

# auxilliary function to calculate the norm of a vector
norm_vec <- function(x) sqrt(sum(x^2))

# NE according to formula (4.1)
NE0 <- 1/(n-k-l) * t(e) %*% MV %*% (e)
NEj <- (t(e) %*% V) ^ 2 / nv2 ^ 2
NE <- c(NE0, NEj) 
print(NE)
print(norm_vec(NE))
[1] 0.10766780 0.00390562 0.23030625 0.02227313 [1] 0.2552345

CVXR

NE as a convex optimization problem

minimizef0(ν)=∣∣ee′−VDV′∣∣2+∣∣MVee′MV−ν0MFMV∣∣2$6pt]subject toν=(ν0,…,νl)′∈[0,∞)l+1 \begin{array}{ll} \textit{minimize} & \quad f_0(\boldsymbol{\nu})=||\mathbf{e}\mathbf{e}' - \mathrm{VDV'}||^2+||\mathrm{M_V}\mathbf{e}\mathbf{e}'\mathrm{M_V}-\nu_0\mathrm{M_F}\mathrm{M_V}||^2 $6pt] \textit{subject to} & \quad \boldsymbol{\nu} = \left(\nu_0, \ldots, \nu_l\right)'\in [0, \infty)^{l+1} \end{array} 

# the optimization variable, objective function
v <- Variable(l+1) 
fv <- sum_squares(e%*%t(e)-V%*%diag(v[2:(l+1)])%*%t(V)) + sum_squares(MV%*%e%*%t(e)%*%MV-v[1]%*%MF%*%MV)

# the optimization problem for NE
objective <- Minimize(fv)
constraints <- list(v >= 0)
prob <- Problem(objective, constraints)

# solve the NE problem
sol <- solve(prob)    
cat("NEcvxr =", as.vector(sol$getValue(v)))
cat("\n")
cat("norm =", norm_vec(sol$getValue(v)))
NEcvxr = 0.1076678 0.003905632 0.2303062 0.02227312 norm = 0.2552345

2nd\boldsymbol{2^{nd}} stage of EBLUP-NE

using formula (3.9) from Hancova et al 2019.

ν˚j=ρj2ν~j;j=0,1…,lρ0=1,ρj=νj∣∣vj∣∣2ν0+νj∣∣vj∣∣2 \mathring{\nu}_j = \rho_j^2 \tilde{\nu}_j; j = 0,1 \ldots, l\\ \rho_0 = 1, \rho_j = \dfrac{\nu_j||\mathbf{v}_j||^2}{\nu_0+\nu_j||\mathbf{v}_j||^2} 

# EBLUP-NE based on formula (3.9)
rho2 <- function(est) {
    result <- c(1)
    for(j in 2:length(est)) {
        result <- c(result, (est[j]*nv2[j-1]/(est[1]+est[j]*nv2[j-1])) ^ 2)
    }
    return(result)
}

EBLUPNE <- function(est) {
    result <- NE * rho2(est)
    return(result)
    }

# numerical results
print(rho2(NE))
[1] 1.0000000 0.3358855 0.9758415 0.8839736 
print(EBLUPNE(NE))
print(norm_vec(EBLUPNE(NE)))
[1] 0.107667801 0.001311841 0.224742406 0.019688860 [1] 0.2499818

NN-DOOLSE or MLE

1st\boldsymbol{1^{st}} stage of EBLUP-NE

KKT algorithm

using the the KKT algorithm (tab.3, Hancova et al 2019)

 ~

q=(e′e(e′v1)2⋮(e′vl)2) \qquad \mathbf{q} = \left(\begin{array}{c} \mathbf{e}' \mathbf{e}\\ (\mathbf{e}' \mathbf{v}_{1})^2 \\ \vdots \\ (\mathbf{e}' \mathbf{v}_{l})^2 \end{array}\right)

G=(n∗∣∣v1∣∣2∣∣v2∣∣2…∣∣vl∣∣2∣∣v1∣∣2∣∣v1∣∣40…0∣∣v2∣∣20∣∣v2∣∣4…0⋮⋮⋮…⋮∣∣vl∣∣200…∣∣vl∣∣4)\qquad\mathrm{G} = \left(\begin{array}{ccccc} \small n^* & ||\mathbf{v}_{1}||^2 & ||\mathbf{v}_{2}||^2 & \ldots & ||\mathbf{v}_{l}||^2 \\ ||\mathbf{v}_{1}||^2 & ||\mathbf{v}_{1}||^4 & 0 & \ldots & 0 \\ ||\mathbf{v}_{2}||^2 & 0 & ||\mathbf{v}_{2}||^4 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ldots & \vdots \\ ||\mathbf{v}_{l}||^2 & 0 & 0 & \ldots & ||\mathbf{v}_{l}||^4 \end{array}\right) 

NNMDOOLSE_kkt <- function(X, F, V, method = "NNDOOLSE") {
    n_star <- length(X)
    k <- ncol(F)
    l <- ncol(V)
    if(method == "NNMDOOLSE") {
        n_star <- n_star - k
    }
    
    MF <- makeM_F(F)
    u <- diag(t(V) %*% V)
    G <- rbind(c(n_star,u),cbind(u, diag(u^2)))
    b_comb <- expand.grid(rep(list(0:1), l))
    
    eps <- c(MF %*% X)
        q <- c(eps %*% eps, (eps %*% V)^2)
        K <- G
        s <- vector()
       
        for(i in 1:nrow(b_comb)) {
                K_inv <- matrix()
                b <- as.vector(unlist(b_comb[i,]))
                for(j in 1:length(b)) {
                        if(b[j] == 0) {
                                K[1,j+1] <- 0
                                K[j+1,j+1] <- -1
                        }
                        K_inv <- solve(K)
                }
                beta <- c(K_inv %*% q)
                if(all(beta >= 0)) {
                        s <- beta[1]
                        for(m in 1:l) {
                                if(b[m] == 0) {
                                        s <- c(s, 0)
                                } else {
                                        s <- c(s, beta[m+1])
                                }
                        }
                        break
                } else {
                        K <- G
                }
        }
    return(list("estimates" = s, "b" = b))
}

NN_DOOLSE <- NNMDOOLSE_kkt(x, F, V)
print(NN_DOOLSE$estimates)
print(norm_vec(NN_DOOLSE$estimates))
print(NN_DOOLSE$b)
[1] 0.103243097 0.001188697 0.227589325 0.020914669 [1] 0.2507885 [1] 1 1 1

CVXR

nonnegative DOOLSE as a convex optimization problem

ParseError: KaTeX parse error: Got function '\boldsymbol' with no arguments as subscript at position 97: …hbf{e}'-\Sigma_\̲b̲o̲l̲d̲s̲y̲m̲b̲o̲l̲{\nu}||^2 $6pt]…

NNDOOLSE_CVXR <- function(X, F, V) {
        n <- length(X)
        k <- ncol(F)
        l <- ncol(V)

        # GF <- t(F) %*% F
        # InvGF <- solve(GF)
        I <- diag(n)
        # PF <- F %*% InvGF %*% t(F)
        #MF <- I - PF
        MF <- makeM_F(F)
        MFV <- MF %*% V

        SX <- MF %*% X %*% t(X) %*% MF
        s <- Variable(l+1)
        p_obj <- Minimize(sum_squares(SX - (s[1] %*% I) - (V %*% diag(s[2:(l+1)]) %*% t(V))))
        constr <- list(s >= 0)
        prob <- Problem(p_obj, constr)

        sol <- solve(prob)

        return(as.vector(sol$getValue(s)))
}

NN_DOOLSEcvxr <- NNDOOLSE_CVXR(x, F, V)
print(NN_DOOLSEcvxr)
print(norm_vec(NN_DOOLSEcvxr))
[1] 0.103242984 0.001188655 0.227589319 0.020914661 [1] 0.2507885

CVXR

using equivalent (RE)MLE convex problem (proposition 5, Hancova et al 2019)

minimizef0(d)=−(n∗ ⁣−l)ln⁡d0−∑j=1lln⁡(d0−dj∣∣vj∣∣2+d0e′e−e′V diag{dj}V′e$6pt]subject tod0>max⁡{dj∣∣vj∣∣2,j=1,…,l}dj≥0,j=1,…lfor MLE: n∗=n, for REMLE: n∗=n−kback transformation:ν0=1d0,νj=djd0(d0−dj∣∣vj∣∣2) \begin{array}{ll} \textit{minimize} & \quad f_0(\mathbf{d})=-(n^*\!-l)\ln d_0 - \displaystyle\sum\limits_{j=1}^{l} \ln(d_0-d_j||\mathbf{v}_j||^2+d_0\mathbf{e}'\mathbf{e}-\mathbf{e}'\mathrm{V}\,\mathrm{diag}\{d_j\}\mathrm{V}'\mathbf{e} $6pt] \textit{subject to} & \quad d_0 > \max\{d_j||\mathbf{v}_j||^2, j = 1, \ldots, l\} \\ & \quad d_j \geq 0, j=1,\ldots l \\ & \\ & \quad\text{for MLE: } n^* = n, \text{ for REMLE: } n^* = n-k \\ \textit{back transformation:} & \quad \nu_0 = \dfrac{1}{d_0}, \nu_j = \dfrac{d_j}{d_0\left(d_0 -d_j||\mathbf{v}_j||^2\right)} \end{array} 

MLE_CVXR <- function(X, F, V){

        n <- length(X)
        l <- ncol(V)

        MF <- makeM_F(F)
        GV <- t(V) %*% V
        p <- n - l

        e <- as.vector(MF %*% X)
        ee <- as.numeric(t(e) %*% e)
        eV <- t(e) %*% V
        Ve <- t(V) %*% e
        d <- Variable(l+1)
        logdetS <- p * log(d[1]) + sum(log(d[1] - GV %*% d[2:(l+1)]))
        obj <- Maximize(logdetS - ((d[1] * ee) - (eV %*% diag(d[2:(l+1)]) %*% Ve)))
        constr <- list(d[2:(l+1)] >= 0, d[1] >= max_entries(GV %*% d[2:(l+1)]))
        p_MLE <- Problem(obj, constr)

        sol <- solve(p_MLE)

        s <- 1 / sol$getValue(d)[1]
        sj <- vector()
        for(j in 2:(l+1)) {
                sj <- c(sj, sol$getValue(d)[j]/(sol$getValue(d)[1] * (sol$getValue(d)[1] - sol$getValue(d)[j] * GV[j-1,j-1])))
        }
        result <- c(s, sj)

        return(as.vector(result))

}

MLEcvxr <- MLE_CVXR(x, F, V)
print(MLEcvxr)
print(norm_vec(MLEcvxr))
[1] 0.103243101 0.001188697 0.227589371 0.020914668 [1] 0.2507885

2nd\boldsymbol{2^{nd}} stage of EBLUP-NE

# numerical results
print(rho2(NN_DOOLSE$estimates))
[1] 1.00000000 0.09263222 0.97654516 0.88173780 
print(EBLUPNE(NN_DOOLSE$estimates)) 
print(norm_vec(EBLUPNE(NN_DOOLSE$estimates)))
[1] 0.1076678014 0.0003617863 0.2249044531 0.0196390615 [1] 0.2501204

NN-MDOOLSE or REMLE

using the result of the KKT algorithm (tab.3, Hancova et al 2019) from PY-estimation-tourism-SciPy.ipynb

1st\boldsymbol{1^{st}} stage of EBLUP-NE

KKT algorithm

NN_MDOOLSE <- NNMDOOLSE_kkt(x, F, V, method = "NNMDOOLSE")

print(NN_MDOOLSE$estimates)
print(norm_vec(NN_MDOOLSE$estimates))
print(NN_MDOOLSE$b)
[1] 0.107667801 0.001072257 0.227472886 0.020856449 [1] 0.252532 [1] 1 1 1

CVXR

nonnegative DOOLSE as a convex optimization problem

ParseError: KaTeX parse error: Got function '\boldsymbol' with no arguments as subscript at position 109: …hrm{M_F}\Sigma_\̲b̲o̲l̲d̲s̲y̲m̲b̲o̲l̲{\nu}\mathrm{M_…

NNMDOOLSE_CVXR <- function(X, F, V) {
    n <- length(X)
        k <- ncol(F)
        l <- ncol(V)

        # GF <- t(F) %*% F
        # InvGF <- solve(GF)
        I <- diag(n)
        # PF <- F %*% InvGF %*% t(F)
        #MF <- I - PF
        MF <- makeM_F(F)
        MFV <- MF %*% V

        SX <- MF %*% X %*% t(X) %*% MF
        s <- Variable(l+1)
        p_obj <- Minimize(sum_squares(SX - (s[1] %*% MF) - (MFV %*% diag(s[2:(l+1)]) %*% t(MFV))))
        constr <- list(s >= 0)
        prob <- Problem(p_obj, constr)

        sol <- solve(prob)

        return(as.vector(sol$getValue(s)))
}

NN_MDOOLSEcvxr <- NNMDOOLSE_CVXR(x, F, V)
print(NN_MDOOLSEcvxr)
print(norm_vec(NN_MDOOLSEcvxr))
[1] 0.107667592 0.001072172 0.227472872 0.020856431 [1] 0.2525319

CVXR

using equivalent (RE)MLE convex problem (proposition 5, Hancova et al 2019)

minimizef0(d)=−(n∗ ⁣−l)ln⁡d0−∑j=1lln⁡(d0−dj∣∣vj∣∣2+d0e′e−e′V diag{dj}V′e$6pt]subject tod0>max⁡{dj∣∣vj∣∣2,j=1,…,l}dj≥0,j=1,…lfor MLE: n∗=n, for REMLE: n∗=n−kback transformation:ν0=1d0,νj=djd0(d0−dj∣∣vj∣∣2) \begin{array}{ll} \textit{minimize} & \quad f_0(\mathbf{d})=-(n^*\!-l)\ln d_0 - \displaystyle\sum\limits_{j=1}^{l} \ln(d_0-d_j||\mathbf{v}_j||^2+d_0\mathbf{e}'\mathbf{e}-\mathbf{e}'\mathrm{V}\,\mathrm{diag}\{d_j\}\mathrm{V}'\mathbf{e} $6pt] \textit{subject to} & \quad d_0 > \max\{d_j||\mathbf{v}_j||^2, j = 1, \ldots, l\} \\ & \quad d_j \geq 0, j=1,\ldots l \\ & \\ & \quad\text{for MLE: } n^* = n, \text{ for REMLE: } n^* = n-k \\ \textit{back transformation:} & \quad \nu_0 = \dfrac{1}{d_0}, \nu_j = \dfrac{d_j}{d_0\left(d_0 -d_j||\mathbf{v}_j||^2\right)} \end{array} 

REMLE_CVXR <- function(X, F, V){

        n <- length(X)
        k <- ncol(F)
        l <- ncol(V)

        MF <- makeM_F(F)
        GV <- t(V) %*% V
        p <- n - l - k

        e <- as.vector(MF %*% X)
        ee <- as.numeric(t(e) %*% e)
        eV <- t(e) %*% V
        Ve <- t(V) %*% e
        d <- Variable(l+1)
        logdetS <- p * log(d[1]) + sum(log(d[1] - GV %*% d[2:(l+1)]))
        obj <- Maximize(logdetS - ((d[1] * ee) - (eV %*% diag(d[2:(l+1)]) %*% Ve)))
        constr <- list(d[2:(l+1)] >= 0, d[1] >= max_entries(GV %*% d[2:(l+1)]))
        p_remle <- Problem(obj, constr)

        sol <- solve(p_remle)

        s <- 1 / sol$getValue(d)[1]
        sj <- vector()
        for(j in 2:(l+1)) {
                sj <- c(sj, sol$getValue(d)[j]/(sol$getValue(d)[1] * (sol$getValue(d)[1] - sol$getValue(d)[j] * GV[j-1,j-1])))
        }

        result <- c(s, sj)

        return(as.vector(result))
}

REMLEcvxr <- REMLE_CVXR(x, F, V)
print(REMLEcvxr)
print(norm_vec(REMLEcvxr))
[1] 0.107667802 0.001072256 0.227473099 0.020856451 [1] 0.2525322

2nd\boldsymbol{2^{nd}} stage of EBLUP-NE

# numerical results
print(rho2(NN_MDOOLSE$estimates))
[1] 1.00000000 0.07537335 0.97554618 0.87683567 
print(EBLUPNE(NN_MDOOLSE$estimates)) 
print(norm_vec(EBLUPNE(NN_MDOOLSE$estimates)))
[1] 0.1076678014 0.0002943797 0.2246743801 0.0195298758 [1] 0.2499049

References

This notebook belongs to suplementary materials of the paper submitted to Statistical Papers and available at https://arxiv.org/abs/1905.07771.

Abstract of the paper

We propose a two-stage estimation method of variance components in time series models known as FDSLRMs, whose observations can be described by a linear mixed model (LMM). We based estimating variances, fundamental quantities in a time series forecasting approach called kriging, on the empirical (plug-in) best linear unbiased predictions of unobservable random components in FDSLRM.

The method, providing invariant non-negative quadratic estimators, can be used for any absolutely continuous probability distribution of time series data. As a result of applying the convex optimization and the LMM methodology, we resolved two problems −- theoretical existence and equivalence between least squares estimators, non-negative (M)DOOLSE, and maximum likelihood estimators, (RE)MLE, as possible starting points of our method and a practical lack of computational implementation for FDSLRM. As for computing (RE)MLE in the case of n n observed time series values, we also discovered a new algorithm of order O(n)\mathcal{O}(n), which at the default precision is 10710^7 times more accurate and n2n^2 times faster than the best current Python(or R)-based computational packages, namely CVXPY, CVXR, nlme, sommer and mixed.

We illustrate our results on three real data sets −- electricity consumption, tourism and cyber security −- which are easily available, reproducible, sharable and modifiable in the form of interactive Jupyter notebooks.