Time Series of Macroinvertabrates Abundance in the Arkansas River.

Description

A time series of 16 years (5 replicates per year) of mayfly (Ephemeroptera:Heptageniidae) abundance in the fall at the monitoring station AR1 on the Arkansas River in Colorado, USA.

Usage

data(Arkansas, package='SiZer')

Format

A data frame with 90 observations on the following 2 variables.

year

The year of observation

sqrt.mayflies

The Square root of observed abundance.

Source

Sonderegger, D.L., Wang, H., Clements, W.H., and Noon, B.R. 2009. Using SiZer to detect thresholds in ecological data. Frontiers in Ecology and the Environment 7:190-195.

Examples

require(ggplot2)

data(Arkansas)
ggplot(Arkansas, aes(x=year, y=sqrt.mayflies)) + 
   geom_point()

Calculate SiZer Map

Description

Calculates the SiZer map from a given set of X and Y variables.

Usage

SiZer(
  x,
  y,
  h = NA,
  x.grid = NA,
  degree = NA,
  derv = 1,
  grid.length = 41,
  quiet = TRUE
)

Arguments

Details

SiZer stands for the Significant Zero crossings of the derivative. There are two dominate approaches in smoothing bivariate data: locally weighted regression or penalized splines. Both approaches require the use of a 'bandwidth' parameter that controls how much smoothing should be done. Unfortunately there is no uniformly best bandwidth selection procedure. SiZer (Chaudhuri and Marron, 1999) is a procedure that looks across a range of bandwidths and classifies the p-th derivative of the smoother into one of three states: significantly increasing (blue), possibly zero (purple), or significantly negative (red).

Value

Returns list object of type SiZer which has the following components:

x.grid

Vector of x-values at which the derivative was evaluated.

h.grid

Vector of bandwidth values for which a smoothing function was calculated.

slopes

Matrix of what category a particular x-value and bandwidth falls into (Increasing=1, Possibly Zero=0, Decreasing=-1, Not Enough Data=2).

Author(s)

Derek Sonderegger

References

Chaudhuri, P., and J. S. Marron. 1999. SiZer for exploration of structures in curves. Journal of the American Statistical Association 94:807-823.

Hannig, J., and J. S. Marron. 2006. Advanced distribution theory for SiZer. Journal of the American Statistical Association 101:484-499.

Sonderegger, D.L., Wang, H., Clements, W.H., and Noon, B.R. 2009. Using SiZer to detect thresholds in ecological data. Frontiers in Ecology and the Environment 7:190-195.

See Also

plot.SiZer, locally.weighted.polynomial

Examples

data('Arkansas')
x <- Arkansas$year
y <- Arkansas$sqrt.mayflies

plot(x,y)

# Calculate the SiZer map for the first derivative
SiZer.1 <- SiZer(x, y, h=c(.5,10), degree=1, derv=1, grid.length=21)
plot(SiZer.1)
plot(SiZer.1, ggplot2=TRUE)

# Calculate the SiZer map for the second derivative
SiZer.2 <- SiZer(x, y, h=c(.5,10), degree=2, derv=2, grid.length=21);
plot(SiZer.2)

# By setting the grid.length larger, we get a more detailed SiZer
# map but it takes longer to compute. 
#
# SiZer.3 <- SiZer(x, y, h=c(.5,10), grid.length=100, degree=1, derv=1)
# plot(SiZer.3)  
  

Coerce SiZer object to a Data Frame

Description

Coerce SiZer object to a Data Frame

Usage

## S3 method for class 'SiZer'
as.data.frame(x, row.names = NULL, optional = FALSE, ...)

Arguments

Examples

data('Arkansas')
x <- Arkansas$year
y <- Arkansas$sqrt.mayflies

plot(x,y)

# Calculate the SiZer map for the first derivative
SiZer.1 <- SiZer(x, y, h=c(.5,10), degree=1, derv=1, grid.length=21)
as.data.frame(SiZer.1)

Fits a bent-cable model to the given data Fits a bent-cable model to the given data by exhaustively searching the 2-dimensional parameter space to find the maximum likelihood estimators for \alpha and \gamma.

Description

Fits a bent-cable model to the given data Fits a bent-cable model to the given data by exhaustively searching the 2-dimensional parameter space to find the maximum likelihood estimators for \alpha and \gamma.

Usage

bent.cable(x, y, grid.size = 100)

Arguments

Details

Fit the model which is essentially a piecewise linear model with a quadratic curve of length 2\gamma connecting the two linear pieces.

The reason for searching the space exhaustively is because the bent-cable model often has a likelihood surface with a very flat ridge instead of definite peak. While the exhaustive search is slow, at least it is possible to examine the contour plot of the likelihood surface.

@return A list of 7 elements:

log.likelihood

A matrix of log-likelihood values.

SSE

A matrix of sum-of-square-error values.

alphas

A vector of alpha values examined.

gammas

A vector of gamma values examined.

alpha

The MLE estimate of alpha.

gamma

The MLE estimate of gamma.

model

The lm fit after alpha and gamma are known.

Author(s)

Derek Sonderegger

References

Chiu, G. S., R. Lockhart, and R. Routledge. 2006. Bent-cable regression theory and applications. Journal of the American Statistical Association 101:542-553.

Toms, J. D., and M. L. Lesperance. 2003. Piecewise regression: a tool for identifying ecological thresholds. Ecology 84:2034-2041.

See Also

piecewise.linear

Examples

data(Arkansas)
x <- Arkansas$year
y <- Arkansas$sqrt.mayflies

# For a more accurate estimate, increase grid.size
model <- bent.cable(x,y, grid.size=20)
plot(x,y)
x.grid <- seq(min(x), max(x), length=200)
lines(x.grid, predict(model, x.grid), col='red')

Plot a SiZer map using 'ggplot2'

Description

Plot a 'SiZer' object that was created using 'SiZer()'

Usage

ggplot_SiZer(x, colorlist = c("red", "purple", "blue", "grey"))

Arguments

Details

The white lines in the SiZer map give a graphical representation of the bandwidth. The horizontal distance between the lines is 2h.

Author(s)

Derek Sonderegger

References

Chaudhuri, P., and J. S. Marron. 1999. SiZer for exploration of structures in curves. Journal of the American Statistical Association 94:807-823.

Hannig, J., and J. S. Marron. 2006. Advanced distribution theory for SiZer. Journal of the American Statistical Association 101:484-499.

Sonderegger, D.L., Wang, H., Clements, W.H., and Noon, B.R. 2009. Using SiZer to detect thresholds in ecological data. Frontiers in Ecology and the Environment 7:190-195.

See Also

plot.SiZer, locally.weighted.polynomial

Examples

data('Arkansas')
x <- Arkansas$year
y <- Arkansas$sqrt.mayflies

plot(x,y)

# Calculate the SiZer map for the first derivative
SiZer.1 <- SiZer(x, y, h=c(.5,10), degree=1, derv=1, grid.length=21)
plot(SiZer.1)
plot(SiZer.1, ggplot2=TRUE)
ggplot_SiZer(SiZer.1)

# Calculate the SiZer map for the second derivative
SiZer.2 <- SiZer(x, y, h=c(.5,10), degree=2, derv=2, grid.length=21);
plot(SiZer.2)
plot(SiZer.2, ggplot2=TRUE)
ggplot_SiZer(SiZer.2)


# By setting the grid.length larger, we get a more detailed SiZer
# map but it takes longer to compute. 
#
# SiZer.3 <- SiZer(x, y, h=c(.5,10), grid.length=100, degree=1, derv=1)
# plot(SiZer.3)  
  

Smoothes the given bivariate data using kernel regression.

Description

Smoothes the given bivariate data using kernel regression.

Usage

locally.weighted.polynomial(
  x,
  y,
  h = NA,
  x.grid = NA,
  degree = 1,
  kernel.type = "Normal"
)

Arguments

Details

The confidence intervals are created using the row-wise method of Hannig and Marron (2006).

Notice that the derivative to be estimated must be less than or equal to the degree of the polynomial initially fit to the data.

If the bandwidth is not given, the Sheather-Jones bandwidth selection method is used.

Value

Returns a LocallyWeightedPolynomial object that has the following elements:

data

A structure of the data used to generate the smoothing curve

h

The bandwidth used to generate the smoothing curve.

x.grid

The grid of x-values that we have estimated function value and derivative(s) for.

degrees.freedom

The effective sample size at each grid point

Beta

A matrix of estimated beta values. The number of rows is degrees+1, while the number of columns is the same as the length of x.grid. Notice that

\hat{f}(x_i) = \beta[1,i]

\hat{f'}(x_i) = \beta[2,i]*1!

\hat{f''}(x_i) = \beta[3,i]*2!

and so on...

Beta.var

Matrix of estimated variances for Beta. Same structure as Beta.

Author(s)

Derek Sonderegger

References

Chaudhuri, P., and J. S. Marron. 1999. SiZer for exploration of structures in curves. Journal of the American Statistical Association 94 807-823.

Hannig, J., and J. S. Marron. 2006. Advanced distribution theory for SiZer. Journal of the American Statistical Association 101 484-499.

Sonderegger, D.L., Wang, H., Clements, W.H., and Noon, B.R. 2009. Using SiZer to detect thresholds in ecological data. Frontiers in Ecology and the Environment 7:190-195

See Also

SiZer, plot.LocallyWeightedPolynomial, spm in package 'SemiPar', loess, smooth.spline, interpSpline in the splines package.

Examples

data(Arkansas)
x <- Arkansas$year
y <- Arkansas$sqrt.mayflies
layout(cbind(1,2,3))
model <- locally.weighted.polynomial(x,y)
plot(model, main='Smoothed Function', xlab='Year', ylab='Sqrt.Mayflies')

model2 <- locally.weighted.polynomial(x,y,h=.5)
plot(model2, main='Smoothed Function', xlab='Year', ylab='Sqrt.Mayflies')

model3 <- locally.weighted.polynomial(x,y, degree=1)
plot(model3, derv=1, main='First Derivative', xlab='Year', ylab='1st Derivative')

Calculates the log-Likelihood value

Description

Calculates the log-Likelihood value

Usage

## S3 method for class 'PiecewiseLinear'
logLik(object, ...)

Arguments

Return the log-Likelihood value for a fitted bent-cable model.

Description

Return the log-Likelihood value for a fitted bent-cable model.

Usage

## S3 method for class 'bent_cable'
logLik(object, ...)

Arguments

Creates a piecewise linear model

Description

Fit a degree 1 spline with 1 knot point where the location of the knot point is unknown.

Usage

piecewise.linear(
  x,
  y,
  middle = 1,
  CI = FALSE,
  bootstrap.samples = 1000,
  sig.level = 0.05
)

Arguments

Details

The bootstrap samples are taken by resampling the raw data points. Sometimes a more appropriate bootstrap sample would be to calculate the residuals and then add a randomly selected residual to each y-value.

Value

A list of 5 elements is returned:

change.point

The estimate of \alpha.

model

The resulting lm object once \alpha is known.

x

The x-values used.

y

The y-values used.

CI

Whether or not the confidence interval was calculated.

intervals

If the CIs where calculated, this is a matrix of the upper and lower intervals.

References

Chiu, G. S., R. Lockhart, and R. Routledge. 2006. Bent-cable regression theory and applications. Journal of the American Statistical Association 101:542-553.

Toms, J. D., and M. L. Lesperance. 2003. Piecewise regression: a tool for identifying ecological thresholds. Ecology 84:2034-2041.

See Also

The package segmented has a much more general implementation of this analysis and users should preferentially use that package.

Examples

data(Arkansas)
x <- Arkansas$year
y <- Arkansas$sqrt.mayflies

model <- piecewise.linear(x,y, CI=FALSE)
plot(model)
print(model)
predict(model, 2001)

Creates a plot of an object created by locally.weighted.polynomial.

Description

Creates a plot of an object created by locally.weighted.polynomial.

Usage

## S3 method for class 'LocallyWeightedPolynomial'
plot(
  x,
  derv = 0,
  CI.method = 2,
  alpha = 0.05,
  use.ess = TRUE,
  draw.points = TRUE,
  ...
)

Arguments

Plots a piecewise linear model

Description

Plots a piecewise linear model

Usage

## S3 method for class 'PiecewiseLinear'
plot(x, xlab = "X", ylab = "Y", ...)

Arguments

Plot a SiZer map Plot a SiZer object that was created using SiZer()

Description

Plot a SiZer map Plot a SiZer object that was created using SiZer()

Usage

## S3 method for class 'SiZer'
plot(
  x,
  ylab = expression(log[10](h)),
  colorlist = c("red", "purple", "blue", "grey"),
  ggplot2 = FALSE,
  ...
)

Arguments

Details

The white lines in the SiZer map give a graphical representation of the bandwidth. The horizontal distance between the lines is 2h.

Author(s)

Derek Sonderegger

References

Chaudhuri, P., and J. S. Marron. 1999. SiZer for exploration of structures in curves. Journal of the American Statistical Association 94:807-823.

Hannig, J., and J. S. Marron. 2006. Advanced distribution theory for SiZer. Journal of the American Statistical Association 101:484-499.

Sonderegger, D.L., Wang, H., Clements, W.H., and Noon, B.R. 2009. Using SiZer to detect thresholds in ecological data. Frontiers in Ecology and the Environment 7:190-195.

See Also

plot.SiZer, locally.weighted.polynomial

Examples

data('Arkansas')
x <- Arkansas$year
y <- Arkansas$sqrt.mayflies

plot(x,y)

# Calculate the SiZer map for the first derivative
SiZer.1 <- SiZer(x, y, h=c(.5,10), degree=1, derv=1, grid.length=21)
plot(SiZer.1)
plot(SiZer.1, ggplot2=TRUE)

# Calculate the SiZer map for the second derivative
SiZer.2 <- SiZer(x, y, h=c(.5,10), degree=2, derv=2, grid.length=21);
plot(SiZer.2)

# By setting the grid.length larger, we get a more detailed SiZer
# map but it takes longer to compute. 
#
# SiZer.3 <- SiZer(x, y, h=c(.5,10), grid.length=100, degree=1, derv=1)
# plot(SiZer.3)  
  

Calculates predicted values from a piecewise linear object

Description

Calculates predicted values from a piecewise linear object

Usage

## S3 method for class 'PiecewiseLinear'
predict(object, x, ...)

Arguments

Return model predictions for fitted bent-cable model

Description

Return model predictions for fitted bent-cable model

Usage

## S3 method for class 'bent_cable'
predict(object, x, ...)

Arguments

Prints out the model form for a Piecewise linear model

Description

Prints out the model form for a Piecewise linear model

Usage

## S3 method for class 'PiecewiseLinear'
print(x, ...)

Arguments