leverages {india} | R Documentation |
Leverages
Description
Computes leverage measures from a fitted model object.
Usage
leverages(model, ...)
## S3 method for class 'lm'
leverages(model, infl = lm.influence(model, do.coef = FALSE), ...)
## S3 method for class 'nls'
leverages(model, ...)
## S3 method for class 'ols'
leverages(model, ...)
## S3 method for class 'ridge'
leverages(model, ...)
## S3 method for class 'nls'
hatvalues(model, ...)
## S3 method for class 'ols'
hatvalues(model, ...)
## S3 method for class 'ridge'
hatvalues(model, ...)
Arguments
model |
|
infl |
influence structure as returned by |
... |
further arguments passed to or from other methods. |
Value
A vector containing the diagonal of the prediction (or ‘hat’) matrix.
For linear regression (i.e., for "lm"
or "ols"
objects) the prediction matrix assumes
the form
\bold{H} = \bold{X}(\bold{X}^T\bold{X})^{-1}\bold{X}^T,
in which case, h_{ii} = \bold{x}_i^T(\bold{X}^T\bold{X})^{-1}\bold{x}_i
for i=1,\dots,n
. Whereas
for ridge regression, the prediction matrix is given by
\bold{H}(\lambda) = \bold{X}(\bold{X}^T\bold{X} + \lambda\bold{I})^{-1}\bold{X}^T,
where \lambda
represents the ridge parameter. Thus, the diagonal elements of \bold{H}(\lambda)
,
are h_{ii}(\lambda) = \bold{x}_i^T(\bold{X}^T\bold{X} + \lambda\bm{I})^{-1}\bold{x}_i
, i=1,\dots,n
.
In nonlinear regression, the tangent plane leverage matrix is given by
\hat{\bold{H}} = \hat{\bold{F}}(\hat{\bold{F}}^T\hat{\bold{F}})^{-1}\hat{\bold{F}}^T,
where \bold{F} = \bold{F}(\bold{\beta})
is the n\times p
local model matrix with ith
row \partial f_i(\bold{\beta})/\partial\bold{\beta}
and \hat{\bold{F}} = \bold{F}(\hat{\bold{\beta}})
.
Note
This function never creates the prediction matrix and only obtains its diagonal elements from
the singular value decomposition of \bold{X}
or \hat{\bold{F}}
.
Function hatvalues
only is a wrapper for function leverages
.
References
Chatterjee, S., Hadi, A.S. (1988). Sensivity Analysis in Linear Regression. Wiley, New York.
Cook, R.D., Weisberg, S. (1982). Residuals and Influence in Regression. Chapman and Hall, London.
Ross, W.H. (1987). The geometry of case deletion and the assessment of influence in nonlinear regression. The Canadian Journal of Statistics 15, 91-103. doi:10.2307/3315198
St. Laurent, R.T., Cook, R.D. (1992). Leverage and superleverage in nonlinear regression. Journal of the Amercian Statistical Association 87, 985-990. doi:10.1080/01621459.1992.10476253
Walker, E., Birch, J.B. (1988). Influence measures in ridge regression. Technometrics 30, 221-227. doi:10.1080/00401706.1988.10488370
Examples
# Leverages for linear regression
fm <- ols(stack.loss ~ ., data = stackloss)
lev <- leverages(fm)
cutoff <- 2 * mean(lev)
plot(lev, ylab = "Leverages", ylim = c(0,0.45))
abline(h = cutoff, lty = 2, lwd = 2, col = "red")
text(17, lev[17], label = as.character(17), pos = 3)
# Leverages for ridge regression
data(portland)
fm <- ridge(y ~ ., data = portland)
lev <- leverages(fm)
cutoff <- 2 * mean(lev)
plot(lev, ylab = "Leverages", ylim = c(0,0.7))
abline(h = cutoff, lty = 2, lwd = 2, col = "red")
text(10, lev[10], label = as.character(10), pos = 3)
# Leverages for nonlinear regression
data(skeena)
model <- recruits ~ b1 * spawners * exp(-b2 * spawners)
fm <- nls(model, data = skeena, start = list(b1 = 3, b2 = 0))
lev <- leverages(fm)
plot(lev, ylab = "Leverages", ylim = c(0,0.25))
obs <- c(1,9)
text(obs, lev[obs], label = as.character(obs), pos = 3)