R/midas_nlpr.R

In mpiktas/midasr: Mixed Data Sampling Regression

##' Non-linear parametric MIDAS regression
##'
##' Estimate restricted MIDAS regression using non-linear least squares.
##'
##' @param formula formula for restricted MIDAS regression or \code{midas_r} object. Formula must include \code{\link{fmls}} function
##' @param data a named list containing data with mixed frequencies
##' @param start the starting values for optimisation. Must be a list with named elements.
##' @param Ofunction the list with information which R function to use for optimisation. The list must have element named \code{Ofunction} which contains character string of chosen
##' R function. Other elements of the list are the arguments passed to this function.  The default optimisation function is \code{\link{optim}} with arguments
##'  \code{method="Nelder-Mead"} and \code{control=list(maxit=5000)}. Other supported functions are \code{\link{nls}}, \code{\link{optimx}}.
##' @param ... additional arguments supplied to optimisation function
##' @return a \code{midas_r} object which is the list with the following elements:
##'
##' \item{coefficients}{the estimates of parameters of restrictions}
##' \item{midas_coefficients}{the estimates of MIDAS coefficients of MIDAS regression}
##' \item{model}{model data}
##' \item{unrestricted}{unrestricted regression estimated using \code{\link{midas_u}}}
##' \item{term_info}{the named list. Each element is a list with the information about the term, such as its frequency, function for weights, gradient function of weights, etc.}
##' \item{fn0}{optimisation function for non-linear least squares problem solved in restricted MIDAS regression}
##' \item{rhs}{the function which evaluates the right-hand side of the MIDAS regression}
##' \item{gen_midas_coef}{the function which generates the MIDAS coefficients of MIDAS regression}
##' \item{opt}{the output of optimisation procedure}
##' \item{argmap_opt}{the list containing the name of optimisation function together with arguments for optimisation function}
##' \item{start_opt}{the starting values used in optimisation}
##' \item{start_list}{the starting values as a list}
##' \item{call}{the call to the function}
##' \item{terms}{terms object}
##' \item{gradient}{gradient of NLS objective function}
##' \item{hessian}{hessian of NLS objective function}
##' \item{gradD}{gradient function of MIDAS weight functions}
##' \item{Zenv}{the environment in which data is placed}
##' \item{nobs}{the number of effective observations}
##' \item{convergence}{the convergence message}
##' \item{fitted.values}{the fitted values of MIDAS regression}
##' \item{residuals}{the residuals of MIDAS regression}
##'
##' @author Virmantas Kvedaras, Vaidotas Zemlys
##' @rdname midas_nlpr
##' @details Given MIDAS regression:
##'
##' \deqn{y_t = \sum_{j=1}^p\alpha_jy_{t-j} +\sum_{i=0}^{k}\sum_{j=0}^{l_i}\beta_{j}^{(i)}x_{tm_i-j}^{(i)} + u_t,}
##'
##' estimate the parameters of the restriction
##'
##' \deqn{\beta_j^{(i)}=g^{(i)}(j,\lambda).}
##'
##' Such model is a generalisation of so called ADL-MIDAS regression. It is not required that all the coefficients should be restricted, i.e the function \eqn{g^{(i)}}
##' might be an identity function. Model with no restrictions is called U-MIDAS model. The regressors \eqn{x_\tau^{(i)}} must be of higher
##' (or of the same) frequency as the dependent variable \eqn{y_t}.
##'
##' @importFrom stats as.formula formula model.matrix model.response terms lsfit time
##' @importFrom zoo index index2char
##' @export
midas_nlpr <- function(formula, data, start, Ofunction = "optim", ...) {
  Zenv <- new.env(parent = environment(formula))

  if (missing(data)) {
    ee <- NULL
  }
  else {
    ee <- data_to_env(data)
    parent.env(ee) <- parent.frame()
  }

  if (missing(start)) {
    stop("Please supply starting values.")
  }

  assign("ee", ee, Zenv)
  formula <- as.formula(formula)
  cl <- match.call()
  mf <- match.call(expand.dots = FALSE)
  mf$formula <- formula
  m <- match(c("formula", "data"), names(mf), 0L)
  mf <- mf[c(1L, m)]
  mf[[1L]] <- as.name("model.frame")
  mf[[3L]] <- as.name("ee")
  mf[[4L]] <- as.name("na.omit")
  names(mf)[c(2, 3, 4)] <- c("formula", "data", "na.action")

  mf <- eval(mf, Zenv)
  mt <- attr(mf, "terms")
  args <- list(...)
  y <- as.numeric(model.response(mf, "numeric"))
  X <- model.matrix(mt, mf)

  # Save ts/zoo information
  if (is.null(ee)) {
    yy <- eval(formula[[2]], Zenv)
  } else {
    yy <- eval(formula[[2]], ee)
  }

  y_index <- 1:length(yy)
  if (!is.null(attr(mf, "na.action"))) {
    y_index <- y_index[-attr(mf, "na.action")]
  }
  if (length(y_index) > 1) {
    if (sum(abs(diff(y_index) - 1)) > 0) warning("There are NAs in the middle of the time series")
  }

  ysave <- yy[y_index]

  if (inherits(yy, "ts")) {
    class(ysave) <- class(yy)
    attr(ysave, "tsp") <- c(time(yy)[range(y_index)], frequency(yy))
  }

  if (inherits(yy, c("zoo", "ts"))) {
    y_start <- index2char(index(ysave)[1], frequency(ysave))
    y_end <- index2char(index(ysave)[length(ysave)], frequency(ysave))
  } else {
    y_start <- y_index[1]
    y_end <- y_index[length(y_index)]
  }

  prepmd <- prep_midas_nlpr(y, X, mt, Zenv, cl, args, start, Ofunction)

  prepmd <- c(prepmd, list(lhs = ysave, lhs_start = y_start, lhs_end = y_end))

  class(prepmd) <- "midas_nlpr"

  midas_nlpr.fit(prepmd)
}

##' @method update midas_nlpr
##' @importFrom stats getCall update.formula setNames
##' @export
update.midas_nlpr <- function(object, formula., ..., evaluate = TRUE) {
  if (is.null(call <- getCall(object))) {
    stop("need an object with call component")
  }
  extras <- match.call(expand.dots = FALSE)$...
  if (!missing(formula.)) {
    call$formula <- update.formula(formula(object), formula.)
  }

  if (length(extras)) {
    existing <- !is.na(match(names(extras), names(call)))
    for (a in names(extras)[existing]) call[[a]] <- extras[[a]]
    if (any(!existing)) {
      call <- c(as.list(call), extras[!existing])
      call <- as.call(call)
    }
  }

  ## 1. If no start is supplied update the start from the call
  ## 2. If start is supplied intersect it with already fitted values.

  cf <- coef(object)
  ustart <- lapply(object$term_info, function(x) cf[x[["coef_index"]]])

  redo <- FALSE
  if (!("start" %in% names(extras))) {
    if (!("start" %in% names(call) && is.null(call$start))) {
      call$start <- ustart
      object$start_opt <- cf
    }
  } else {
    cstart <- eval(call$start, object$Zenv)
    ustart[names(cstart)] <- cstart
    call$start <- ustart
    object$start_opt <- unlist(ustart)
  }
  if (evaluate) {
    if (!missing(formula.) || "data" %in% names(extras) || "weight_gradients" %in% names(extras) || redo) {
      eval(call, parent.frame())
    } else {
      ## If we got here, we assume that we do not need to reevaluate terms.
      if (!is.null(extras$Ofunction)) {
        Ofunction <- eval(extras$Ofunction)
        extras$Ofunction <- NULL
      } else {
        Ofunction <- object$argmap_opt$Ofunction
      }
      dotargnm <- names(extras)
      if (length(dotargnm) > 0) {
        offending <- dotargnm[!dotargnm %in% names(formals(Ofunction))]
        if (length(offending) > 0) {
          stop(paste("The function ", Ofunction, " does not have the following arguments: ",
            paste(offending, collapse = ", "),
            sep = ""
          ))
        }
      }
      else {
        extras <- NULL
      }
      if (Ofunction != object$argmap_opt$Ofunction) {
        argmap <- c(list(Ofunction = Ofunction), extras)
      }
      else {
        argmap <- object$argmap_opt
        argmap$Ofunction <- NULL
        argnm <- union(names(argmap), names(extras))
        marg <- vector("list", length(argnm))
        names(marg) <- argnm
        marg[names(extras)] <- extras
        oldarg <- setdiff(names(argmap), names(extras))
        marg[oldarg] <- argmap[oldarg]
        argmap <- c(list(Ofunction = Ofunction), marg)
      }
      object$call <- call
      object$argmap_opt <- argmap
      midas_nlpr.fit(object)
    }
  }
  else {
    call
  }
}


##' Fit restricted MIDAS regression
##'
##' Workhorse function for fitting restricted MIDAS regression
##'
##' @param x \code{midas_r} object
##' @return \code{\link{midas_r}} object
##' @importFrom stats fitted.values
##' @author Vaidotas Zemlys
midas_nlpr.fit <- function(x) {
  args <- x$argmap_opt
  function.opt <- args$Ofunction
  args$Ofunction <- NULL

  if (!(function.opt %in% c("optim", "spg", "optimx", "nls", "dry_run"))) {
    stop("The optimisation function is not in the supported functions list. Please see the midasr:::midas_nlpr.fit code for the supported function list")
  }
  if (function.opt == "optim" | function.opt == "spg") {
    args$par <- x$start_opt
    args$fn <- x$fn0
    opt <- try(do.call(function.opt, args), silent = TRUE)
    if (inherits(opt, "try-error")) {
      stop("The optimisation algorithm of MIDAS regression failed with the following message:\n", opt, "\nPlease try other starting values or a different optimisation function")
    }
    par <- opt$par
    names(par) <- names(coef(x))
    x$convergence <- opt$convergence
  }
  if (function.opt == "optimx") {
    args$par <- x$start_opt
    args$fn <- x$fn0
    opt <- try(do.call(function.opt, args), silent = TRUE)
    if (inherits(opt, "try-error")) {
      stop("The optimisation algorithm of MIDAS regression failed with the following message:\n", opt, "\nPlease try other starting values or a different optimisation function")
    }
    bmet <- which.min(opt$value)
    par <- as.numeric(opt[bmet, 1:length(args$par)])
    names(par) <- names(coef(x))
    x$convergence <- opt$convcode[bmet]
  }
  if (function.opt == "nls") {
    rhs <- x$rhs
    y <- x$model[, 1]
    args$formula <- formula(y ~ rhs(p))
    args$start <- list(p = x$start_opt)
    opt <- try(do.call("nls", args), silent = TRUE)
    if (inherits(opt, "try-error")) {
      stop("The optimisation algorithm of MIDAS regression failed with the following message:\n", opt, "\nPlease try other starting values or a different optimisation function")
    }
    par <- coef(opt)
    names(par) <- names(coef(x))
    x$convergence <- opt$convInfo$stopCode
  }
  if (function.opt == "dry_run") {
    opt <- NULL
    par <- x$start_opt
  }

  x$opt <- opt
  x$coefficients <- par
  if (inherits(x, "midas_sp")) {
    bws <- par[1:length(x$bws)]
    x$bws <- bws
  }
  names(par) <- NULL
  x$fitted.values <- fitted.values(x)
  x$residuals <- as.vector(x$model[, 1] - x$fitted.values)
  x
}

## Prepare necessary objects for fitting of the non-linear parametric MIDAS regression
##
## y the response
## X the model matrix
## mt the terms of the formula
## Zenv the environment to evaluate the formula
## cl call of the function
## args additional argument
## start starting values
## Ofunction the optimisation function
## weight_gradients a list of gradient functions for weights
## lagsTable the lagstable from checkARstar
## unrestricted the unrestricted model
## guess_start if TRUE, get the initial values for non-MIDAS terms via OLS, if FALSE, initialize them with zero.
prep_midas_nlpr <- function(y, X, mt, Zenv, cl, args, start, Ofunction, guess_start = TRUE) {
  start <- start[!sapply(start, is.null)]

  if (!is.null(args$guess_start)) {
    guess_start <- args$guess_start
    args$guess_start <- NULL
  }
  terms.lhs <- as.list(attr(mt, "variables"))[-2:-1]
  rfd <- lapply(terms.lhs, dterm_nlpr, Zenv = Zenv)

  if (attr(mt, "intercept") == 1) {
    intc <- dterm_nlpr(expression(1), Zenv)
    intc$term_name <- "(Intercept)"
    rfd <- c(list(intc), rfd)
  }

  term_names <- sapply(rfd, "[[", "term_name")
  if (length(setdiff(names(start), intersect(names(start), term_names))) > 0) {
    stop("The names for the starting values should coincide with terms in formula")
  }
  names(rfd) <- term_names

  rfd[names(start)] <- mapply(function(tmi, st) {
    tmi[["full_start"]] <- unlist(st)
    if (is.list(st)) {
      # This is for pretty names, remove all the previous names
      for (i in 1:length(st)) names(st[[i]]) <- NULL
      if (!("r" %in% names(st))) stop("The starting values for the restriction should be in an element named r")
      if (setdiff(names(st), c("lstr", "mmm")) != "r") stop("The starting values for nlpr term should be in an element named either lstr or mmm")
      tmi[["start"]] <- st[["r"]]
      nlpr_name <- setdiff(names(st), "r")
      np <- st[[nlpr_name]]
      if (nlpr_name == "lstr" & length(np) != 4) stop("There should be 4 starting values for LSTR term")
      if (nlpr_name == "mmm" & length(np) != 2) stop("There should be 2 starting values for MMM term")
      tmi[["nlpr"]] <- eval(as.name(nlpr_name))
      bi <- build_indices(cumsum(sapply(st, length)), names(st))
      if (nlpr_name == "lstr") {
        lower <- rep(-Inf, length(unlist(st)))
        lower[bi$lstr[3]] <- 0
        tmi[["lower"]] <- lower
      }
      names(bi)[names(bi) == nlpr_name] <- "nlpr"
      tmi[["param_map"]] <- bi

      tmi
    } else {
      names(st) <- NULL
    }
    tmi[["full_start"]] <- unlist(st)
    tmi
  }, rfd[names(start)], start, SIMPLIFY = FALSE)

  nlpr_terms <- names(which(sapply(rfd, function(l) !is.null(l[["nlpr"]]))))

  rf <- lapply(rfd, "[[", "weight")

  fake_start_default <- lapply(rfd, "[[", "start")

  fake_pinds <- build_indices_list(fake_start_default)

  fake_coef2 <- function(p) {
    pp <- lapply(fake_pinds, function(x) p[x])
    res <- mapply(function(fun, param) fun(param), rf, pp, SIMPLIFY = FALSE)
    return(res)
  }

  fake_midas_coef <- fake_coef2(unlist(fake_start_default))

  if (sum(is.na(unlist(fake_midas_coef))) > 0) stop("Check your starting values, NA in midas coefficients")

  npx <- cumsum(sapply(fake_midas_coef, length))
  xinds <- build_indices(npx, names(fake_start_default))

  start_default <- lapply(rfd, "[[", "full_start")
  pinds <- build_indices_list(start_default)

  lower <- unlist(lapply(rfd, function(l) {
    if (is.null(l$lower)) {
      return(rep(-Inf, length(l$full_start)))
    }
    l$lower
  }))

  upper <- unlist(lapply(rfd, function(l) {
    if (is.null(l$upper)) {
      return(rep(Inf, length(l$full_start)))
    }
    l$upper
  }))

  pinds1 <- pinds[setdiff(names(pinds), nlpr_terms)]
  rf1 <- rf[setdiff(names(pinds), nlpr_terms)]

  rfd <- mapply(function(tmi, xind, pind) {
    tmi[["xind"]] <- xind
    tmi[["pind"]] <- pind
    tmi[["sd_x"]] <- sd(X[, xind], na.rm = TRUE)
    tmi
  }, rfd, xinds, pinds, SIMPLIFY = FALSE)


  coef_list <- function(p, pi, rfl) {
    pp <- lapply(pi, function(x) p[x])
    res <- mapply(function(fun, param) fun(param), rfl, pp, SIMPLIFY = FALSE)
    return(res)
  }
  coef1 <- function(p) unlist(coef_list(p, pi = pinds1, rfl = rf1))

  do_nlpr_term <- function(p, tfun, xind, wfun, pind, param_map, sd_x) {
    pr <- p[pind][param_map[["r"]]]
    pn <- p[pind][param_map[["nlpr"]]]
    tfun(X[, xind], wfun(pr), pn, sd_x)
  }

  xind1 <- unlist(xinds[setdiff(names(xinds), nlpr_terms)])
  X1 <- X[, xind1, drop = FALSE]

  check_p <- function(p) {
    return(TRUE)
  }
  use_bounds <- FALSE
  if (any(lower != -Inf) | any(upper != Inf)) {
    check_p <- function(p) {
      all(p >= lower) & all(p <= upper)
    }
    use_bounds <- TRUE
  }

  rhs <- function(p) {
    T2 <- lapply(rfd[nlpr_terms], function(l) {
      do_nlpr_term(p, l[["nlpr"]], l[["xind"]], l[["weight"]], l[["pind"]], l[["param_map"]], l[["sd_x"]])
    })
    cf1 <- coef1(p)
    X1 %*% cf1 + Reduce("+", T2)
  }

  fn0 <- function(p, ...) {
    if (check_p(p)) {
      r <- y - rhs(p)
      sum(r^2)
    } else {
      return(NA)
    }
  }

  hess <- function(x) numDeriv::hessian(fn0, x)


  control <- c(list(Ofunction = Ofunction), args)

  ## The default method is "Nelder-Mead" and number of maximum iterations is 5000
  if (!("method" %in% names(control)) & Ofunction == "optim") {
    if (use_bounds) {
      control$method <- "L-BFGS-B"
      control$lower <- lower
      control$upper <- upper
      control$control <- list(maxit = 1000)
    } else {
      control$method <- "Nelder-Mead"
      if (is.null(control$control$maxit)) control$control <- list(maxit = 5000)
    }
  }
  # Do a rename to conform to midas_r
  term_info <- lapply(rfd, function(l) {
    nm <- names(l)
    nm[nm == "xind"] <- "midas_coef_index"
    nm[nm == "pind"] <- "coef_index"
    names(l) <- nm
    l
  })


  list(
    coefficients = unlist(start_default),
    model = cbind(y, X),
    fn0 = fn0,
    rhs = rhs,
    opt = NULL,
    argmap_opt = control,
    start_opt = unlist(start_default),
    start_list = start,
    call = cl,
    terms = mt,
    hessian = hess,
    Zenv = Zenv,
    term_info = term_info,
    lower = lower,
    upper = upper,
    nobs = nrow(X)
  )
}

##' LSTR (Logistic Smooth TRansition)  MIDAS regression
##'
##' Function for fitting LSTR MIDAS regression without the formula interface
##' @param y model response
##' @param X prepared matrix of high frequency variable lags for LSTR term
##' @param z additional low frequency variables
##' @param weight the weight function
##' @param start_lstr the starting values for lstr term
##' @param start_x the starting values for weight function
##' @param start_z the starting values for additional low frequency variables
##' @param method a method passed to \link{optimx}
##' @param ... additional parameters to \link{optimx}
##' @return an object similar to \code{midas_r} object
##' @author Virmantas Kvedaras, Vaidotas Zemlys
##' @import numDeriv
##' @import optimx
##' @importFrom stats na.omit sd
##'
##' @export
##'
midas_lstr_plain <- function(y, X, z = NULL, weight, start_lstr, start_x, start_z = NULL, method = c("Nelder-Mead"), ...) {
  d <- ncol(X)

  if (!is.null(z) && !is.matrix(z)) z <- matrix(z, ncol = 1)

  model <- na.omit(cbind(y, X, z))

  y <- model[, 1]
  X <- model[, 2:(ncol(X) + 1)]
  if (is.null(z)) {
    z <- 0
  } else {
    z <- model[, (ncol(X) + 2):ncol(model)]
  }
  n <- nrow(model)

  sx <- length(start_x)
  sd_x <- sd(c(X))

  rhs <- function(p) {
    plstr <- p[1:4]
    pr <- p[5:(4 + sx)]
    if (is.null(z)) {
      pz <- 0
    } else {
      pz <- p[(5 + sx):length(p)]
    }
    lstr(X, weight(pr, ncol(X)), plstr, sd_x) + z %*% pz
  }

  fn0 <- function(p) {
    sum((y - rhs(p))^2)
  }

  start <- c(start_lstr, start_x, start_z)
  opt <- optim(start, fn0, method = method, ...)
  par <- opt$par
  call <- match.call()
  fitted.values <- as.vector(y - rhs(par))
  names(par) <- c(paste0("lstr", 1:length(start_lstr)), paste0("x", 1:length(start_x)), paste0("z", 1:length(start_z)))
  list(
    coefficients = par,
    midas_coefficients = weight(par[5:(4 + sx)], ncol(X)),
    lstr_coefficients = par[1:4],
    model = model,
    weights = weight,
    fn0 = fn0,
    rhs = rhs,
    opt = opt,
    call = call,
    hessian = function(x) numDeriv::hessian(fn0, x),
    fitted.values = fitted.values,
    residuals = as.vector(y - fitted.values),
    start = start
  )
}

##' MMM (Mean-Min-Max)  MIDAS regression
##'
##' Function for fitting MMM MIDAS regression without the formula interface
##' @param y model response
##' @param X prepared matrix of high frequency variable lags for MMM term
##' @param z additional low frequency variables
##' @param weight the weight function
##' @param start_mmm the starting values for MMM term
##' @param start_x the starting values for weight function
##' @param start_z the starting values for additional low frequency variables
##' @param method a method passed to \link{optimx}
##' @param ... additional parameters to \link{optimx}
##' @return an object similar to \code{midas_r} object
##' @author Virmantas Kvedaras, Vaidotas Zemlys
##' @import numDeriv
##' @import optimx
##' @importFrom stats na.omit sd
##'
##' @export
##'
midas_mmm_plain <- function(y, X, z = NULL, weight, start_mmm, start_x, start_z = NULL, method = c("Nelder-Mead"), ...) {
  d <- ncol(X)

  if (!is.null(z) && !is.matrix(z)) z <- matrix(z, ncol = 1)

  model <- na.omit(cbind(y, X, z))

  y <- model[, 1]
  X <- model[, 2:(ncol(X) + 1)]
  if (is.null(z)) {
    z <- 0
  } else {
    z <- model[, (ncol(X) + 2):ncol(model)]
  }
  n <- nrow(model)

  sx <- length(start_x)
  sd_x <- sd(c(X))

  rhs <- function(p) {
    pmmm <- p[1:2]
    pr <- p[3:(2 + sx)]
    if (is.null(z)) {
      pz <- 0
    } else {
      pz <- p[(3 + sx):length(p)]
    }
    mmm(X, weight(pr, ncol(X)), pmmm) + z %*% pz
  }

  fn0 <- function(p) {
    sum((y - rhs(p))^2)
  }

  start <- c(start_mmm, start_x, start_z)
  opt <- optim(start, fn0, method = method, ...)
  par <- opt$par
  call <- match.call()
  fitted.values <- as.vector(y - rhs(par))
  names(par) <- c(paste0("mm", 1:length(start_mmm)), paste0("x", 1:length(start_x)), paste0("z", 1:length(start_z)))
  list(
    coefficients = par,
    midas_coefficients = weight(par[3:(2 + sx)], ncol(X)),
    mmm_coefficients = par[1:2],
    model = model,
    weights = weight,
    fn0 = fn0,
    rhs = rhs,
    opt = opt,
    call = call,
    hessian = function(x) numDeriv::hessian(fn0, x),
    fitted.values = fitted.values,
    residuals = as.vector(y - fitted.values),
    start = start
  )
}

#' Compute LSTR term for high frequency variable
#'
#' @param X matrix, high frequency variable embedded in low frequency, output of mls
#' @param theta vector, restriction coefficients for high frequency variable
#' @param beta vector of length 4, parameters for LSTR term, slope and 3 LSTR parameters
#' @param sd_x vector of length 1, defaults to standard deviation of X.
#'
#' @return a vector
#' @export
#'
lstr <- function(X, theta, beta, sd_x = sd(c(X), na.rm = TRUE)) {
  xx <- X %*% theta
  b <- -beta[3] * (xx - beta[4]) / sd_x
  G <- 1 / (1 + exp(b))
  beta[1] * xx * (1 + beta[2] * G)
}

lstr_G <- function(X, theta, beta, sd_x = sd(c(X), na.rm = TRUE)) {
  xx <- X %*% theta
  b <- -beta[1] * (xx - beta[2]) / sd_x
  1 / (1 + exp(b))
}
#' Compute MMM term for high frequency variable
#'
#' @param X matrix, high frequency variable embedded in low frequency, output of mls
#' @param theta vector, restriction coefficients for high frequency variable
#' @param beta vector of length 2, parameters for MMM term, slope and MMM parameter.
#' @param ..., currently not used
#'
#' @return a vector
#' @export
#'
mmm <- function(X, theta, beta, ...) {
  mtr <- exp(beta[2] * X)
  mtr_denom <- apply(mtr, 1, sum)
  mmm_term <- ncol(X) * X * mtr / mtr_denom

  beta[1] * mmm_term %*% theta
}

dterm_nlpr <- function(fr, Zenv) {
  term_name <- as.character(fr)[1]
  weight_name <- ""
  rf <- function(p) p
  start <- 0
  freq <- 1
  lagstruct <- 0
  if (term_name %in% c("mls", "fmls", "dmls", "mlsd")) {
    type <- term_name
    term_name <- as.character(fr[[2]])

    wpos <- 5
    if (type == "mlsd") {
      freq <- NA
    } else {
      freq <- eval(fr[[4]], Zenv)
    }

    lags <- eval(fr[[3]], Zenv)
    nol <- switch(type,
      fmls = lags + 1,
      dmls = lags + 1,
      mls = length(lags),
      mlsd = length(lags)
    )
    lagstruct <- switch(type,
      fmls = 0:lags,
      dmls = 0:lags,
      mls = lags,
      mlsd = lags
    )
    start <- rep(0, nol)
    if (length(fr) > wpos - 1) {
      mf <- fr[-wpos]
      mf[[1]] <- fr[[wpos]]
      weight_name <- as.character(fr[[wpos]])

      noarg <- length(formals(eval(fr[[wpos]], Zenv)))
      if (noarg < 2) stop("The weight function must have at least two arguments")
      mf <- mf[1:min(length(mf), noarg + 1)]
      if (length(mf) > 3) {
        start_eval <- 4
        if (type == "mlsd") start_eval <- 5
        if (length(mf) >= start_eval) {
          for (j in start_eval:length(mf)) {
            mf[[j]] <- eval(mf[[j]], Zenv)
          }
        }
      }
      mf[[3]] <- nol
      rf <- function(p) {
        mf[[2]] <- p
        eval(mf, Zenv)
      }
    }
  }
  list(
    weight = rf,
    term_name = term_name,
    start = start,
    full_start = start,
    weight_name = weight_name,
    frequency = freq,
    lag_structure = lagstruct
  )
}