Modern Analogue Technique (MAT) transfer function models for palaeoecology. The fitted values are the, possibly weighted, averages of the environment for the k-closest modern analogues. MAT is a k-NN method.
mat(x, …)# S3 method for default
mat(x, y,
method = c("euclidean", "SQeuclidean", "chord", "SQchord",
"bray", "chi.square", "SQchi.square",
"information", "chi.distance", "manhattan",
"kendall", "gower", "alt.gower", "mixed"),
kmax, …)
# S3 method for formula
mat(formula, data, subset, na.action,
method = c("euclidean", "SQeuclidean", "chord", "SQchord",
"bray", "chi.square", "SQchi.square",
"information", "chi.distance", "manhattan",
"kendall", "gower", "alt.gower", "mixed"),
model = FALSE, …)
# S3 method for mat
fitted(object, k, weighted = FALSE, …)
# S3 method for mat
residuals(object, k, weighted = FALSE, …)
a data frame containing the training set data, usually species data.
a vector containing the response variable, usually
environmental data to be predicted from x.
a symbolic description of the model to be fit. The details of model specification are given below.
an optional data frame, list or environment (or object
coercible by as.data.frame to a data frame) containing
the variables in the model. If not found in data, the
variables are taken from environment(formula), typically the
environment from which wa is called.
an optional vector specifying a subset of observations to be used in the fitting process.
a function which indicates what should happen when
the data contain NAs. The default is set by the
na.action setting of options, and is
na.fail if that is unset. The "factory-fresh" default
is na.omit. Another possible value is NULL, no
action. Value na.exclude can be useful.
a character string indicating the dissimilarity (distance) coefficient to be used to define modern analogues. See Details, below.
logical; If TRUE the model frame of the fit is
returned.
numeric; limit the maximum number of analogues considered
during fitting. By default, kmax is equal to \(n - 1\),
where \(n\) is the number of sites. For large data sets this is
just wasteful as we wouldn't expect to be averaging over the entire
training set. kmax can be used to restrict the upper limit on
the number of analogues considered.
an object of class mat.
numeric; the k-closest analogue models' for which fitted values and residuals are returned. Overides the default stored in the object.
logical; should weighted averages be used instead of simple averages?
arguments can be passed to distance to
provide additional optios required for some dissimilarities.
mat returns an object of class mat with the following
components:
list; the model statistics based on simple averages of k-closest analogues. See below.
list; the model statistics based on weighted of k-closest analogues. See below.
matrix of pairwise sample dissimilarities for the training
set x.
the original training set data.
the original environmental data or response, y.
the matched function call.
the dissimilarity coefficient used.
If model = TRUE then additional components "terms" and
"model" are returned containing the terms object
and model frame used.
fitted.mat returns a list with the following components:
estimatednumeric; a vector of fitted values.
knumeric; this is the k-closest analogue model with
lowest apparent RMSE.
weightedlogical; are the fitted values the weighted averages
of the environment for the k-closest analogues. If
FALSE, the fitted values are the average of the environment
for the k-closest analogues.
The Modern Analogue Technique (MAT) is perhaps the simplest of the transfer function models used in palaeoecology. An estimate of the environment, \(x\), for the response for a fossil sample, \(y\), is the, possibly weighted, mean of that variable across the k-closest modern analogues selected from a modern training set of samples. If used, weights are the reciprocal of the dissimilarity between the fossil sample and each modern analogue.
A typical model has the form response ~ terms where
response is the (numeric) response data frame and terms
is a series of terms which specifies a linear predictor for
response. A typical form for terms is .,
which is shorthand for "all variables" in data. If . is
used, data must also be provided. If specific species
(variables) are required then terms should take the form
spp1 + spp2 + spp3.
Pairwise sample dissimilarity is defined by dissimilarity or
distance coefficients. A variety of coefficients are supported --- see
distance for details of the supported coefficients.
k is chosen by the user. The simplest choice for k is to evaluate the RMSE of the difference between the predicted and observed values of the environmental variable of interest for the training set samples for a sequence of models with increasing k. The number of analogues chosen is the value of k that has lowest RMSE. However, it should be noted that this value is biased as the data used to build the model are also used to test the predictive power.
An alternative approach is to employ an optimisation data set on which to evaluate the size of \(k\) that provides the lowest RMSEP. This may be impractical with smaller sample sizes.
A third option is to bootstrap re-sample the training set many times. At
each bootstrap sample, predictions for samples in the bootstrap test
set can be made for \(k = 1, ..., n\), where \(n\) is the
number of samples in the training set. \(k\) can be chosen from the
model with the lowest RMSEP. See function bootstrap.mat for
further details on choosing \(k\).
The output from summary.mat can be used to choose
\(k\) in the first case above. For predictions on an optimsation or
test set see predict.mat. For bootstrap resampling of
mat models, see bootstrap.mat.
The fitted values are for the training set and are taken as the, possibly weighted, mean of the environmental variable in question across the k-closest analogues. The fitted value for each sample does not include a contribution from itself --- it is the closest analogue, having zero dissimilarity. This spurious distance is ignored and analogues are ordered in terms of the non-zero distances to other samples in the training set, with the k-closest contributing to the fitted value.
Typical usages for residuals.mat are:
resid(object, k, weighted = FALSE, \dots)
Gavin, D.G., Oswald, W.W., Wahl, E.R. and Williams, J.W. (2003) A statistical approach to evaluating distance metrics and analog assignments for pollen records. Quaternary Research 60, 356--367.
Overpeck, J.T., Webb III, T. and Prentice I.C. (1985) Quantitative interpretation of fossil pollen spectra: dissimilarity coefficients and the method of modern analogues. Quaternary Research 23, 87--108.
Prell, W.L. (1985) The stability of low-latitude sea-surface temperatures: an evaluation of the CLIMAP reconstruction with emphasis on the positive SST anomalies, Report TR 025. U.S. Department of Energy, Washington, D.C.
Sawada, M., Viau, A.E., Vettoretti, G., Peltier, W.R. and Gajewski, K. (2004) Comparison of North-American pollen-based temperature and global lake-status with CCCma AGCM2 output at 6 ka. Quaternary Science Reviews 23, 87--108.
summary.mat, bootstrap.mat for boostrap
resampling of MAT models, predict.mat for making
predictions from MAT models, fitted.mat and
resid.mat for extraction of fitted values and residuals
from MAT models respectively. plot.mat provides a
plot.lm-like plotting tool for MAT models.
# NOT RUN {
## Imbrie and Kipp Sea Surface Temperature
data(ImbrieKipp)
data(SumSST)
data(V12.122)
## merge training set and core samples
dat <- join(ImbrieKipp, V12.122, verbose = TRUE)
## extract the merged data sets and convert to proportions
ImbrieKipp <- dat[[1]] / 100
ImbrieKippCore <- dat[[2]] / 100
## fit the MAT model using the squared chord distance measure
ik.mat <- mat(ImbrieKipp, SumSST, method = "chord")
ik.mat
## model summary
summary(ik.mat)
## fitted values
fitted(ik.mat)
## model residuals
resid(ik.mat)
## draw summary plots of the model
par(mfrow = c(2,2))
plot(ik.mat)
par(mfrow = c(1,1))
## reconstruct for the V12.122 core data
coreV12.mat <- predict(ik.mat, V12.122, k = 3)
coreV12.mat
summary(coreV12.mat)
## draw the reconstruction
reconPlot(coreV12.mat, use.labels = TRUE, display.error = "bars",
xlab = "Depth", ylab = "SumSST")
## fit the MAT model using the squared chord distance measure
## and restrict the number of analogues we fit models for to 1:20
ik.mat2 <- mat(ImbrieKipp, SumSST, method = "chord", kmax = 20)
ik.mat2
# }
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