DepthProc (version 2.1.6)

deepReg2d: Simple deepest regression method.

Description

This function calculates deepest regression estimator for simple regression.

Usage

deepReg2d(x, y)

Arguments

x

Independent variable.

y

Dependent variable.

Author

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

Details

Function originates from an original algorithm proposed by Rousseeuw and Hubert. Let \( {Z} ^ {n} = {(x_1, y_1), ..., (x_n, y_n)} \subset {{R} ^ {d}} \) denotes a sample considered from a following semiparametric model: \( {{y}_{l}} = {{a}_{0}} + {{a}_{1}}{{x}_{1l}} + ... + {{a}_{(d - 1)l}}{{x}_{(d - 1)l}} + {{\varepsilon }_{l}}, {l = 1, ..., n} \), we calculate a depth of a fit \( \alpha = (a_{0}, ..., a_{d - 1}) \) as \( RD(\alpha, {{Z} ^ {n}}) = {u \ne 0}\min\sharp {l: \frac{ {{r}_{l}}(\alpha) }{ {{u} ^ {T}}{{x}_{l}} } < 0, l = 1, ..., n} \), where \( r(\cdot) \) denotes the regression residual, \( \alpha = (a_{0}, ..., a_{d - 1}) \), \( {u} ^ {T}{x}_{l} \ne 0 \). The deepest regression estimator \( DR(\alpha, {{Z} ^ {n}}) \) is defined as \( DR(\alpha, {{Z} ^ {n}}) = {\alpha \ne 0}\,\arg\max\,RD(\alpha, {{Z} ^ {n}}) \)

References

Rousseeuw J.P., Hubert M. (1998), Regression Depth, Journal of The American Statistical Association, vol.94.

Examples

# EXAMPLE 1
data(pension)
plot(pension)
abline(
  lm(Reserves ~ Income, data = pension),
  lty = 3,
  lwd = 2) # lm
abline(
  deepReg2d(pension[, 1], pension[, 2]),
  lwd = 2) # deepreg2d

# EXAMPLE 2
data(under5.mort)
data(inf.mort)
data(maesles.imm)
data2011 <- na.omit(
    cbind(under5.mort[, 22], inf.mort[, 22],
    maesles.imm[, 22]))

x <- data2011[, 3]
y <- data2011[, 2]
plot(
  x, y,
  cex = 1.2,
  ylab = "infant mortality rate per 1000 live birth",
  xlab = "against masles immunized percentage",
  main = "Projection Depth Trimmed vs. LS regressions"
)
abline(lm(x ~ y), lwd = 2, col = "black") # lm
abline(
  deepReg2d (x, y),
  lwd = 2, col = "red"
) # trimmed reg

legend(
  "bottomleft",
  c("LS", "DeepReg"),
  fill = c("black", "red"),
  cex = 1.4,
  bty = "n"
)