Up to this point we have generally been preforming ordination analyses according to the protocol of:

- calculate a dissimilarity/distance matrix that represents the pairwise relations of vegetation composition
- calculate a mapping from high-dimensional space to a space of only a few dimensions
- (optionally) test the quality of the mapping
- perform
*post-hoc*analyses of environmental variability on the configuration of points in the ordination.

With the exception of RDA (Lab 7), the configuration of the points in PCA, PCO, and NMDS ordinations has been completely determined by the vegetation compositional data. Alternatively, for CCA and DB-RDA the configuration of points is first calculated as a CA or PCO respectively, which is then subjected to weighted regression against environmental or experimental variables, keeping the fitted values of the regression as coordinates. Accordingly, CCA and DB-RDA are referred to as "constrained ordinations" as they constrain the values of the underlying ordination to achieve what is typically referred to as their "canonical" axes. FSO is the first technique that directly incorporates the environmental data into the calculation of the configuration.

The initial application of FSO to single variables bears more resemblance to the surface fitting routines we have employed so far than to the dimensional mapping routines. That is, when we calculate an FSO on elevation, for example, the result is an estimate of the relative elevation for each plot, based on its composition. If a variable is influential in determining or constraining the distribution of vegetation, then we ought to be able to estimate the value of that variable based on the composition. This is an approach called "calibration" by Ter Braak and colleagues (19xx). Whether or not the ordination achieves an efficient mapping from high-dimensional space to low dimensional space is a secondary consideration, and generally requires solving for multiple dimensions. All this is more easily understood by example.

The FSO is calculated with a function called `fso()`, which requires two
arguments: a variable (or multiple variables in a formula or data.frame) and a
similarity matrix. Because the convention in R is to work with `dist`
objects, `fso()` automatically converts the `dist` object from
`dist()`, ` dsvdis()` or `vegdist` to a full similarity
matrix as required. From our previous work with other methods, we have good
reason to expect that elevation and slope may be important, so let's start with
elevation.

First, let's get the fso package with the necessary functions. If you are
running Windows, simply use the "packages" drop-down menu feature to get the
most recent version, and then enter `library(fso)`.
If you are running linux/unix, got to CRAN
and select the "packages" link. Scroll down to fso_2.0-1 (unless there is a newer
version) and click on the link to fetch the package. As root (if possible),
enter `R CMD INSTALL fso_2.0-1.tar.gz` to install. Then inside of R, enter
`library(fso)`.

As we have throughout the previous labs, we'll analyze the Bryce Canyon data. So, if you haven't already

library(labdsv) data(bryceveg) data(brycesite) attach(brycesite) dis.bc <- dsvdis(bryceveg,'bray/curtis')

elev.fso <- fso(elev,dis.bc)

`fso()` does its work silently. to examine the results, use

plot(elev.fso,title="Bryce Canyon",xlab="Elevation",ylab="Apparent Elevation")

As you can see, the relationship between elevation and "apparent elevation" is pretty good, with a correlation of 0.83 (upper left corner). That means that on average we can predict the elevation of a sample from its composition fairly well. There are some interesting outliers, though, that bear looking into. How about slope?

slope.fso <- fso(slope,dis.bc) plot(slope.fso,title="Bryce Canyon")

Hmmm. While a correlation of 0.35 is not too bad, the distribution leaves something to be desired. The problem is that the distribution of sample plots is not remotely uniform for slope, with many more plots on gentle slopes than steep. As a possible fix, we can try

slope.log.fso <- fso(log(slope+1),dis.bc) plot(slope.log.fso)

Despite the emphasis on *a priori* concepts about the control of
vegetation composition by environment it has proven helpful to write routines
that facilitate rapid screening and analysis of a broad range of possible
variables. `fso()` will accept a formula or dataframe as its first
argument, and perform sequential FSOs on the variables. For example,

grads.fso <- fso(~elev+av+slope+annrad+grorad,dis.bc)

Notice that there is no dependent variable. The formula starts out with the
summary(grads.fso)

Fuzzy set statistics col variable r p d 1 1 elev 0.8303255 0.000000e+00 0.6119908 2 3 slope 0.3495267 5.891172e-06 0.3208229 3 4 annrad 0.3050769 8.758970e-05 0.2255343 4 5 grorad 0.3013482 1.078098e-04 0.5811717 5 2 av 0.1402331 7.694954e-02 0.1985226 Correlations of fuzzy sets elev av slope annrad grorad elev 1.0000000 0.5936768 -0.5018290 -0.7666193 -0.8255718 av 0.5936768 1.0000000 -0.6119128 -0.9184280 -0.5206253 slope -0.5018290 -0.6119128 1.0000000 0.6292742 0.1255797 annrad -0.7666193 -0.9184280 0.6292742 1.0000000 0.7297448 grorad -0.8255718 -0.5206253 0.1255797 0.7297448 1.0000000

Finally, in the bottom matrix it presents a correlation matrix of the derived fuzzy sets.

To plot one of the FSOs, specify it by name or column number.

plot(grads.fso,"annrad")

The estimated p-values are computed from a Normal approximation with the appropriate numbers of degrees of freedom. Nonetheless, in my experience they have proven somewhat problematic (slightly high type I error rate), and I have included a permutation-based statistic as an alternative. Essentially, the routine permutes the input vector (permute -1 times), calculates an FSO, and stores the correlation in a vector. The statistic reported is

sum(cors >= obs) + 1 / (number of reps)

This is a conservative estimate of the p-value. If you ask for 1000
permutations, and never in 999 permutations is the correlation equaled or
exceeded, then
(0 + 1) / 1000 = 0.001

The number of desired permutations is given with the grads.fso <- fso(~elev+av+slope+annrad+grorad,dis.bc,permute=1000) summary(grads.fso)

Fuzzy set statistics col variable r p d 1 1 elev 0.8303255 0.001 0.6119908 2 3 slope 0.3495267 0.001 0.3208229 3 4 annrad 0.3050769 0.001 0.2255343 4 5 grorad 0.3013482 0.001 0.5811717 5 2 av 0.1402331 0.064 0.1985226

grads.mfso <- mfso(~elev+slope+grorad,dis.bc)

There are several options to control the relative scaling of MFSOs. The
default (scaling=1, lm=TRUE) plots the actual mu values (memberships in fuzzy sets)
for the first variable, and then the regression residuals of subsequent mu values
on all previous axes, similar to a Gramm-Schmidt algorithm.
The last step guarantees a second axis uncorrelated with the first.
The alternative scalings are:

scaling | algorithm |

1 | no scaling, use actual mu values |

2 | relativize both fuzzy sets [0,1] |

3 | center and scale proportional to their correlation coefficient |

In either case, the regression step (and subsequent rescaling of the second axis
to the residuals of the regression) is controlled by `lm=`.

To see the result, simple use the `plot` function.

plot(grads.mfso)

Because in this case the two sets were not very strongly correlated, the second
set retains a fairly broad range of mu values. For sets that are more strongly
correlated, the Y axis may have a much more restricted range. How good a job
does this mapping do? The `dis=` argument allows you to specify a
dissimilarity/distance matrix to compare the ordination distances to. If
specified, the values show up in the last plot.

plot(grads.fso,dis=dis.bc)

In this case, the correlation between ordination distances and dissimilarities in the original matrix is 0.721. Looking back to Lab 8, the PCO two-dimensional projection of the Bray/Curtis dissimilarity matrix achieved 0.788, so our two-dimensional FSO is almost as good as a PCO in terms of faithful representation of dissimilarities, and has the added benefit of an obvious ecological interpretation.

The alternative, using `scaling=2` turns out slightly better.

grads2.mfso <- mfso(~elev+slope+grorad,dis.bc,scaling=2) plot(grads2.mfso,dis.bc)

Slope and grorad is also an interesting combination.

fso.2 <-mfso(~slope+grorad,dis.bc) plot(fso.2)

elev.fso.lm <- lm(elev.fso$mu~elev) summary(elev.fso.lm)

Call: lm(formula = elev.fso$mu ~ elev) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.819e-01 3.856e-02 -4.717 5.23e-06 *** elev 9.162e-05 4.892e-06 18.729 < 2e-16 *** --- Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: 0.03801 on 158 degrees of freedom Multiple R-Squared: 0.6894, Adjusted R-squared: 0.6875 F-statistic: 350.8 on 1 and 158 degrees of freedom, p-value: 0

elev.fso.res <- residuals(elev.fso.lm)

Positive residuals are plots that appear to be at a higher elevation than they
really are, and negative residuals are plots that appear to be at a lower
elevation than they really are. One possible explanation is that sites with high
radiation loads (e.g. south-facing slopes) would appear to be lower than sites with low
radiation loads (e.g. north-facing slopes). We'll compare growing season
radiation (summary(lm(elev.fso.res~grorad))

Call: lm(formula = elev.fso.res ~ grorad) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.401664 0.078248 5.133 8.27e-07 *** grorad -0.002460 0.000479 -5.136 8.15e-07 *** --- Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: 0.03519 on 158 degrees of freedom Multiple R-Squared: 0.1431, Adjusted R-squared: 0.1377 F-statistic: 26.38 on 1 and 158 degrees of freedom, p-value: 8.148e-07

(-0.002460 * (max(grorad)-min(grorad))) / (max(elev.fso.res)-min(elev.fso.res))

[1] -0.5161084

Growing season radiation is not uniformly distributed, so its total effect size
is obviously smaller than the maximum contribution calculated (51% of the range
of the residuals), but this gives us a basis for comparison.
How about topographic position? We'll add it to growing
season radiation to look for conditional significance.
The R `lm()` routine handles
categorical variables like `pos` without recoding, establishing contrasts
to the first category.

summary(lm(elev.fso.res~grorad+pos))

Call: lm(formula = elev.fso.res ~ grorad + pos) Residuals: Min 1Q Median 3Q Max -0.0665706 -0.0239119 -0.0006207 0.0232542 0.0939038 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.3714292 0.0816540 4.549 1.09e-05 *** grorad -0.0023628 0.0004907 -4.816 3.48e-06 *** poslow_slope 0.0233538 0.0099608 2.345 0.0203 * posmid_slope 0.0148793 0.0093715 1.588 0.1144 posridge 0.0165458 0.0113768 1.454 0.1479 posup_slope 0.0118876 0.0098836 1.203 0.2309 --- Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: 0.03499 on 154 degrees of freedom Multiple R-Squared: 0.174, Adjusted R-squared: 0.1472 F-statistic: 6.488 on 5 and 154 degrees of freedom, p-value: 1.675e-05

anova(lm(elev.fso.res~grorad+pos))

Analysis of Variance Table Response: elev.fso.res Df Sum Sq Mean Sq F value Pr(>F) grorad 1 0.032666 0.032666 26.6776 7.337e-07 *** pos 4 0.007056 0.001764 1.4406 0.2232 Residuals 154 0.188566 0.001224 --- Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

To look at the effects of soil depth, remembering from Lab 2 that soil depth is correlated with elevation, we need to correct for elevation first. We'll run the summary of the regression first to get the signs and coefficients and then the anova to check for sequential significance.

summary(lm(elev.fso.res~elev+depth))

Call: lm(formula = elev.fso.res ~ elev + depth) Residuals: Min 1Q Median 3Q Max -0.075054 -0.024821 -0.002617 0.020813 0.092883 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.120e-02 3.683e-02 -0.304 0.761 elev -1.016e-06 4.729e-06 -0.215 0.830 depthshallow 2.829e-02 5.839e-03 4.845 3.27e-06 *** --- Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: 0.03325 on 142 degrees of freedom Multiple R-Squared: 0.1467, Adjusted R-squared: 0.1347 F-statistic: 12.2 on 2 and 142 degrees of freedom, p-value: 1.285e-05

anova(lm(elev.fso.res~elev+depth))

Analysis of Variance Table Response: elev.fso.res Df Sum Sq Mean Sq F value Pr(>F) elev 1 0.001033 0.001033 0.9341 0.3354 depth 1 0.025951 0.025951 23.4743 3.274e-06 *** Residuals 142 0.156981 0.001105 --- Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

range(elev.fso.res)

[1] -0.06672459 0.10010083

0.0289 / (0.10010083 - -0.06672459)

[1] 0.173235

it's about 17% of the range.

dis | the dissimilarity/distance object | no default |

start | the variables in the initial model as a data.frame | no default, can be NULL |

add | the variables to consider for addition to the model as a data.frame | no default |

numitr | the number of random permutations to use to calculate p-values | default = 100 |

scaling | the scaling to use on MFSO (see above) | default = 1 |

Here's an example of `step.mfso()` in use

attach(brycesite) step.mfso(dis.bc,start=NULL,add=data.frame(elev,slope,av,grorad,annrad))

Baseline = 0 variable delta_cor p_val 1 elev 0.6119908 0.08 2 slope 0.3208229 0.66 3 av 0.1985226 0.90 4 grorad 0.5811717 0.08 5 annrad 0.2255343 0.82

step.mfso(dis.bc,start=data.frame(elev),add=data.frame(slope,av,grorad,annrad))

Baseline = 0.6119908 variable delta_cor p_val 1 slope 0.101305309 0.01 2 av -0.001551554 0.98 3 grorad 0.050204388 0.01 4 annrad -0.002196274 1.00

step.mfso(dis.bc,start=data.frame(elev,slope),add=data.frame(av,grorad,annrad))

Baseline = 0.7132961 variable delta_cor p_val 1 av -0.0014125421 0.96 2 grorad 0.0078175173 0.04 3 annrad -0.0009994769 0.99

step.mfso(dis.bc,start=data.frame(elev,slope),add=data.frame(av,grorad,annrad), numitr=1000)

Baseline = 0.7132961 variable delta_cor p_val 1 av -0.0014125421 0.906 2 grorad 0.0078175173 0.035 3 annrad -0.0009994769 0.993

The outcome of the step-wise function in this case gives us the same model we
had analyzed above (`elev` then `slope`) with arguably a third
dimension for `grorad`. To screen large numbers of variables it is
sometimes advisable to set `numitr=1` during the variable selection
stage, and then put just the selected variable in the `add` data.frame
for permutation testing.

We also have a primary
gradient for slope which is nearly independent of the elevation gradient.
Regressing the residuals of the slope FSO (not shown) only showed one
statistically significant effect (`annrad`), such that sites with high
annual radiation appeared to be slightly steeper than they really are. This
appears to be because radiation is an asymmetric function of aspect, where
level sites have radiation budgets more similar to south
slopes than to north slopes.

We don't have sufficient climate data to tease apart the elevation effect; it combines both temperature and presumably precipitation, but we can't partition the two. It's an example of what Austin and Smith (19xx) would call a "complex gradient." The slope gradient is also probably best viewed as a complex gradient. The ecological characteristics of slope (apart from aspect effects ON slopes) is likely soil erosion and consequent soil texture effects. Steep slopes are likely to suffer from removal of fine-textured soil particles and run-off of precipitation, leading to drier sites than level sites at equivalent elevations.

FSO is thus useful for analysis of primary gradients, as well as nested secondary effects within primary gradients. The multi-FSO plots are nearly as efficient as PCO, with much more immediate ecological interpretation possible.

Roberts, D.W. 2008. Statistical analysis of multidimensional fuzzy set ordination. Ecology 89:1256-1260.

Roberts, D.W. 2009. Comparison of multidimensional fuzzy set ordination with CCA and DB-RDA. Ecology 90:2622-2634.

plot.fso <- function (x, which = "all", xlab = x$var, ylab = "mu(x)", title = "", r = TRUE, pch = 1, ...) { if (class(x) != "fso") { stop("You must specify n object of class fso from fso()") } if (is.matrix(x$mu)) { if (which == "all") { for (i in 1:ncol(x$mu)) { if (is.na(x$r[i])) { cat(paste("variable ", x$var[i], " has missing values \n")) } else { plot(x$data[, i], x$mu[, i], xlab = xlab[i], ylab = ylab, main = title) if (r) { ax <- min(x$data[, i]) ay <- max(x$mu[, i]) text(ax, ay, paste("r = ", format(x$r[i], digits = 3)), pos = 4) } } readline("\nHit Return for Next Plot\n") } } else if (is.numeric(which)) { for (i in which) { plot(x$data[, i], x$mu[, i], xlab = xlab[i], ylab = ylab, main = title) if (r) { ax <- min(x$data[, i]) ay <- max(x$mu[, i]) text(ax, ay, paste("r = ", format(x$r[i], digits = 3)), pos = 4) } readline("\nHit Return for Next Plot\n") } } else { for (j in 1:ncol(x$mu)) { if (which == x$var[j]) plot(x$data[, j], x$mu[, j], xlab = xlab[j], ylab = ylab, main = title) if (r) { ax <- min(x$data[, j]) ay <- max(x$mu[, j]) text(ax, ay, paste("r = ", format(x$r[j], digits = 3)), pos = 4) } } } } else { plot(x$data, x$mu, xlab = xlab, ylab = ylab, main = title) if (r) { ax <- min(x$data) ay <- max(x$mu) text(ax, ay, paste("r = ", format(x$r, digits = 3)), pos = 4) } } } dummify <- function (var) { if (!is.factor(var)) stop('You must pass a factor variable') cols <- length(levels(var)) out <- matrix(0,nrow=length(var),ncol=cols) for (i in 1:cols) { out[,i] <- var == levels(var)[i] } out <- data.frame(out) names(out) <- levels(var) out }

ellip.mfso <- function (ord, overlay, ax = 1, ay = 2, cols = c(2, 3, 4, 5, 6, 7), ltys = c(1, 2, 3), ...) { if (class(ord) != "mfso") stop("You must pass an object of class mfso") require(cluster) if (inherits(overlay, c("partana", "pam", "clustering"))) { overlay <- overlay$clustering if (min(overlay) < 0 || (length(table(overlay)) != max(overlay))) { cat("WARNING: renumbering clusters to consecutive integers\n") overlay <- match(overlay, sort(unique(overlay))) } } else if (is.logical(overlay)) overlay <- as.numeric(overlay) else if (is.factor(overlay)) overlay <- as.numeric(overlay) pass <- 1 layer <- 0 lty <- ltys[pass] for (i in 1:max(overlay, na.rm = TRUE)) { x <- ord$mu[, ax][overlay == i & !is.na(overlay)] y <- ord$mu[, ay][overlay == i & !is.na(overlay)] pts <- chull(x, y) layer <- layer + 1 if (layer > length(cols)) { layer <- 1 pass <- min(pass + 1, length(ltys)) } col <- cols[layer] lty = ltys[pass] x <- as.matrix(cbind(x[pts], y[pts])) elp <- ellipsoidhull(x) lines(predict(elp),col=col) } }