`mfd`

classLet us show how the `funcharts`

package works through an example with the dataset `air`

, which has been included from the R package `FRegSigCom`

and is used in the paper of Qi and Luo (2019).

NOTE: since the objective of this vignette is only to illustrate how the package works, in the following we will use only 5 basis functions and a fixed smoothing parameter to reduce the computational time.

`mfd`

classWe provide the `mfd`

class for multivariate functional data. It inherits from the `fd`

class but provides some additional features:

- It forces the
`coef`

argument to be an array even when the number of functional observations and/or the number of functional variables are one - It provides a better subset function
`[`

that never drops dimensions, then it always returns a`mfd`

object with three-dimensional array argument`coef`

; moreover it allows extracting observations/variables also by name - When possible, it stores the original raw data in the long data frame format

The first thing is to get the `mfd`

object from discrete data. We currently allow two types of input with the two functions:

`get_mfd_data.frame`

: first input must be a data.frame in the long format, with:- one
`arg`

column giving the argument (`x`

) values, - one
`id`

column indicating the functional observation, - one column per each functional variable indicating the corresponding
`y`

values

- one
`get_mfd_list`

: first input must be a list of matrices for the case all functional data are observed on the same grid, which:- must have all the same dimension,
- have the variable names as name of the list,
- are such that, for each matrix:
- each row corresponds to a functional observation
- each column corresponds to a point on the grid

In this example, the dataset `air`

is in the second format (list of matrices, with data observed on the same grid)

```
library(funcharts)
data("air")
<- names(air)[names(air) != "NO2"]
fun_covariates <- get_mfd_list(air[fun_covariates],
mfdobj_x grid = 1:24,
n_basis = 5,
lambda = 1e-2)
```

In order to perform the statistical process monitoring analysis, we divide the dataset into a phase I and a phase II dataset.

```
<- 1:300
rows1 <- 301:355
rows2 <- mfdobj_x[rows1]
mfdobj_x1 <- mfdobj_x[rows2] mfdobj_x2
```

Now we extract the scalar response variable, i.e. the mean of `NO2`

at each observation:

```
<- rowMeans(air$NO2)
y <- y[rows1]
y1 <- y[rows2] y2
```

We also provide plotting functions using ggplot2.

`plot_mfd(mfdobj_x1)`

`plot_mfd(mfdobj_x1[1:10, c("CO", "C6H6")])`

This functions provides a layer `geom_mfd`

, which is basically a `geom_line`

that is added to `ggplot()`

to plot functional data. It also allows to plot the original raw data by adding the argument `type_mfd = "raw"`

. `geom_mfd`

accepts the argument `data`

as input, which must be a data frame with two columns, `id`

and `var`

, in order to use aesthetic mappings that allow for example to colour different functions according to some columns in this data frame.

```
<- data.frame(id = unique(mfdobj_x1$raw$id)) %>%
dat mutate(id_greater_than_100 = as.numeric(id) > 100)
ggplot() +
geom_mfd(mapping = aes(col = id_greater_than_100),
mfdobj = mfdobj_x1,
data = dat,
alpha = .2,
lwd = .3)
```

For class `mfd`

we provide a function `pca_mfd`

, which is a wrapper to `pca.fd`

. It returns multivariate functional principal component scores summed over variables (`fda::pca.fd`

returns an array of scores when providing a multivariate functional data object). Moreover, the eigenfunctions or multivariate functional principal components given in `harmonics`

argument are converted to the `mfd`

class. We also provide a plot function for the eigenfunctions (the argument `harm`

selects which components to plot).

```
<- pca_mfd(mfdobj_x1)
pca plot_pca_mfd(pca, harm = 1:3)
```