graph.fanova | R Documentation |

One-way ANOVA tests for functional data with graphical interpretation

```
graph.fanova(
nsim,
curve_set,
groups,
typeone = c("fwer", "fdr"),
variances = "equal",
contrasts = FALSE,
n.aver = 1L,
mirror = FALSE,
savefuns = FALSE,
test.equality = c("mean", "var", "cov"),
cov.lag = 1,
...
)
```

`nsim` |
The number of random permutations. |

`curve_set` |
The original data (an array of functions) provided as a |

`groups` |
The original groups (a factor vector representing the assignment to groups). |

`typeone` |
Character string indicating which type I error rate to control,
either the familywise error rate ('fwer') or false discovery rate ('fdr').
Further arguments to the FWER or FDR envelope can be passed in argument |

`variances` |
Either "equal" or "unequal". If "unequal", then correction for unequal variances
as explained in details will be done. Only relevant for the case |

`contrasts` |
Logical. FALSE and TRUE specify the two test functions as described in description part of this help file. |

`n.aver` |
If variances = "unequal", there is a possibility to use variances smoothed by appying moving average to the estimated sample variances. n.aver determines how many values on each side do contribute (incl. value itself). |

`mirror` |
The complement of the argument circular of |

`savefuns` |
Logical. If TRUE, then the functions from permutations are saved to the attribute simfuns. |

`test.equality` |
A character with possible values
If
where
See Mrkvicka et al. (2020) for more details. |

`cov.lag` |
The lag of the covariance for testing the equality of covariances,
see |

`...` |
Additional parameters to be passed to |

This function can be used to perform one-way graphical functional ANOVA tests described in Mrkvička et al. (2020). Both 1d and 2d functions are allowed in curve sets.

The tests assume that there are `J`

groups which contain
`n_1,\dots,n_J`

functions
`T_{ij}, i=\dots,J, j=1,\dots,n_j`

.
The functions should be given in the argument `curve_set`

,
and the groups in the argument `groups`

.
The tests assume that `T_{ij}, i=1,...,n_j`

is an iid sample from
a stochastic process with mean function `\mu_j`

and
covariance function `\gamma_j(s,t)`

for s,t in R and j = 1,..., J.

To test the hypothesis

`H_0 : \mu_1(r) \equiv \mu_2(r)\equiv \dots \equiv \mu_J(r),`

you can use the test function

`\mathbf{T} = (\overline{T}_1({\bf r}), \overline{T}_2({\bf r}), \dots , \overline{T}_J({\bf r}))`

where `\overline{T}_i({\bf r})`

is a vector of mean values of functions in the group j.
This test function is used when `contrasts = FALSE`

(default).

The hypothesis can equivalently be written as

`H_0 : \mu_i(r) - \mu_j(r) = 0, i=1,\dots,J-1, j=1,\dots,J.`

and, alternatively, one can use the test function (vector) taken to consist of the differences of the group averages,

```
\mathbf{T'} = (\overline{T}_1({\bf r})-\overline{T}_2({\bf r}),
\overline{T}_1({\bf r})-\overline{T}_3({\bf r}), \dots , \overline{T}_{J-1}({\bf r})-\overline{T}_J({\bf r})).
```

The choice is available with the option `contrasts = TRUE`

.
This test corresponds to the post-hoc test done usually after an ANOVA test is significant, but
it can be directed tested by means of the combined rank test (Mrkvička et al., 2017) with this test vector.

The test as such assumes that the variances are equal across the groups of functions. To deal with unequal variances, the differences are rescaled as the first step as follows

`S_{ij}(r) = \frac{T_{ij}(r) - \overline{T}(r))}{\sqrt{Var(T_j(r))}} \sqrt{Var(T(r))} + \overline{T}(r))`

where `\overline{T}({\bf r})`

is the overall sample mean and
`\sqrt{Var(T(r))}`

is the overall sample standard deviation.
This scaling of the test functions can be obtained by giving the argument `variances = "unequal"`

.

Mrkvička, T., Myllymäki, M., Jilek, M. and Hahn, U. (2020) A one-way ANOVA test for functional data with graphical interpretation. Kybernetika 56 (3), 432-458. doi: 10.14736/kyb-2020-3-0432

Mrkvička, T., Myllymäki, M., and Hahn, U. (2017). Multiple Monte Carlo testing, with applications in spatial point processes. Statistics and Computing 27 (5): 1239-1255. doi:10.1007/s11222-016-9683-9

Myllymäki, M and Mrkvička, T. (2020). GET: Global envelopes in R. arXiv:1911.06583 [stat.ME]

`frank.fanova`

```
#-- NOx levels example (see for details Myllymaki and Mrkvicka, 2020)
if(require("fda.usc", quietly=TRUE)) {
# Prepare data
data("poblenou")
fest <- poblenou$df$day.festive; week <- as.integer(poblenou$df$day.week)
Type <- vector(length=length(fest))
Type[fest == 1 | week >= 6] <- "Free"
Type[fest == 0 & week %in% 1:4] <- "MonThu"
Type[fest == 0 & week == 5] <- "Fri"
Type <- factor(Type, levels = c("MonThu", "Fri", "Free"))
# (log) Data as a curve_set
cset <- create_curve_set(list(r = 0:23,
obs = t(log(poblenou[['nox']][['data']]))))
# Graphical functional ANOVA
nsim <- 2999
res.c <- graph.fanova(nsim = nsim, curve_set = cset,
groups = Type, variances = "unequal",
contrasts = TRUE)
plot(res.c) + ggplot2::labs(x = "Hour", y = "Diff.")
}
#-- Centred government expenditure centralization ratios example
# This is an example analysis of the centred GEC in Mrkvicka et al.
data("cgec")
# Number of simulations
nsim <- 2499 # increase to reduce Monte Carlo error
# Test for unequal lag 1 covariances
res.cov1 <- graph.fanova(nsim = nsim, curve_set = cgec$cgec,
groups = cgec$group,
test.equality = "cov", cov.lag = 1)
plot(res.cov1)
# Add labels
plot(res.cov1, labels = paste("Group ", 1:3, sep="")) +
ggplot2::xlab(substitute(paste(italic(i), " (", j, ")", sep=""), list(i="r", j="Year")))
# Test for equality of variances among groups
res.var <- graph.fanova(nsim = nsim, curve_set = cgec$cgec,
groups = cgec$group,
test.equality = "var")
plot(res.var)
# Test for equality of means assuming equality of variances
# a) using 'means'
res <- graph.fanova(nsim = nsim, curve_set = cgec$cgec,
groups = cgec$group,
variances = "equal", contrasts = FALSE)
plot(res)
# b) using 'contrasts'
res2 <- graph.fanova(nsim = nsim, curve_set = cgec$cgec,
groups = cgec$group,
variances = "equal", contrasts = TRUE)
plot(res2)
# Image set examples
data("imageset3")
res <- graph.fanova(nsim = 19, # Increase nsim for serious analysis!
curve_set = imageset3$image_set,
groups = imageset3$Group)
plot(res)
# Contrasts
res.c <- graph.fanova(nsim = 19, # Increase nsim for serious analysis!
curve_set = imageset3$image_set, groups = imageset3$Group,
contrasts = TRUE)
plot(res.c)
```

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