measure-of-accuracy {forecastSNSTS}

Mean squared or absolute h-step ahead prediction errors

Description

The function MSPE computes the empirical mean squared prediction errors for a collection of h-step ahead, linear predictors (h=1,\ldots,H) of observations X_{t+h}, where m_1 \leq t+h \leq m_2, for two indices m_1 and m_2. The resulting array provides

\frac{1}{m_{\rm lo} - m_{\rm up} + 1} \sum_{t=m_{\rm lo}}^{m_{\rm up}} R_{(t)}^2,

with R_{(t)} being the prediction errors

R_t := | X_{t+h} - (X_t, \ldots, X_{t-p+1}) \hat v_{N,T}^{(p,h)}(t) |,

ordered by magnitude; i.e., they are such that R_{(t)} \leq R_{(t+1)}. The lower and upper limits of the indices are m_{\rm lo} := m_1-h + \lfloor (m_2-m_1+1) \alpha_1 \rfloor and m_{\rm up} := m_2-h - \lfloor (m_2-m_1+1) \alpha_2 \rfloor. The function MAPE computes the empirical mean absolute prediction errors

\frac{1}{m_{\rm lo} - m_{\rm up} + 1} \sum_{t=m_{\rm lo}}^{m_{\rm up}} R_{(t)},

with m_{\rm lo}, m_{\rm up} and R_{(t)} defined as before.

Usage

MSPE(X, predcoef, m1 = length(X)/10, m2 = length(X), P = 1, H = 1,
  N = c(0, seq(P + 1, m1 - H + 1)), trimLo = 0, trimUp = 0)

MAPE(X, predcoef, m1 = length(X)/10, m2 = length(X), P = 1, H = 1,
  N = c(0, seq(P + 1, m1 - H + 1)), trimLo = 0, trimUp = 0)

Arguments

X

the data X_1, \ldots, X_T

predcoef

the prediction coefficients in form of a list of an array coef, and two integer vectors t and N. The two integer vectors provide the information for which indices t and segment lengths N the coefficients are to be interpreted; (m1-H):(m2-1) has to be a subset of predcoef$t. if not provided the necessary coefficients will be computed using predCoef.

m1

first index from the set in which the indices t+h shall lie

m2

last index from the set in which the indices t+h shall lie

P

maximum order of prediction coefficients to be used; must not be larger than dim(predcoef$coef)[1].

H

maximum lead time to be used; must not be larger than dim(predcoef$coef)[3].

N

vector with the segment sizes to be used, 0 corresponds to using 1, ..., t; has to be a subset of predcoef$N.

trimLo

percentage \alpha_1 of lower observations to be trimmed away

trimUp

percentage \alpha_2 of upper observations to be trimmed away

Value

MSPE returns an object of type MSPE that has mspe, an array of size H\timesP\timeslength(N), as an attribute, as well as the parameters N, m1, m2, P, and H. MAPE analogously returns an object of type MAPE that has mape and the same parameters as attributes.

Examples

T <- 1000
X <- rnorm(T)
P <- 5
H <- 1
m <- 20
Nmin <- 20
pcoef <- predCoef(X, P, H, (T - m - H + 1):T, c(0, seq(Nmin, T - m - H, 1)))

mspe <- MSPE(X, pcoef, 991, 1000, 3, 1, c(0, Nmin:(T-m-H)))

plot(mspe, vr = 1, Nmin = Nmin)