# FuncNN v1.0

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## Functional Neural Networks

A collection of functions which fit functional neural network models. In other words, this package will allow users to build deep learning models that have either functional or scalar responses paired with functional and scalar covariates. We implement the theoretical discussion found in Thind, Multani and Cao (2020) <arXiv:2006.09590> through the help of a main fitting and prediction function as well as a number of helper functions to assist with cross-validation, tuning, and the display of estimated functional weights.

# FuncNN

This package allows the user to build models of the form: f(z, g(x) | θ) where f() is a neural network, z is a vector of scalar covariates, and g(x) is a vector of functional covariates. The package is built on top of the Keras/Tensorflow architecture.

## Installation

You can install the released version of FuncNN from CRAN with:

#install.packages("FuncNN")

You can install FuncNN from GitHub with the following commands:

library(devtools)
install_github("b-thi/FuncNN")

## Example

The package functions can be as simple (or complicated) as you want them to be! To illustrate, we'll use the classic Tecator meat sample dataset for a classification problem.

# Library
library(FuncNN)

tecator = FuncNN::tecator

We'll do a classification example using meat samples with fat contents > 25 as "high fat" and < 25 as "low fat" as the dependent variable.

# Making classification bins
tecator_resp = as.factor(ifelse(tecator$y$Fat > 25, 1, 0))

We have our response, what about our predictors? Well, to keep it simple, let's first consider our scalar covariates; we'll use the water contents of the meat samples as a scalar covariate

# Non functional covariate
tecator_scalar = data.frame(water = tecator$y$Water)

Let's now add some functional covariates:

# Splitting data
ind = sample(1:length(tecator_resp), round(0.75*length(tecator_resp)))
train_x = tecator$absorp.fdata$data[ind,]
test_x = tecator$absorp.fdata$data[-ind,]
scalar_train = data.frame(tecator_scalar[ind,1])
scalar_test = data.frame(tecator_scalar[-ind,1])
train_y = tecator_resp[ind]
test_y = tecator_resp[-ind]

In the chunk of code above, I split the absorbance curves into a test and train set with a 25/75 split. Since we are doing this the "easy" way, we won't need to do any pre-processing on the raw absorbance points (of each meat sample). I also split the previously defined scalar covariates and response in the same way.

Before fitting the model, we need to get the functional covariates in the proper format (okay, so there is a little bit of processing). In the case where we are passing in the raw curve points, we need to pass them in as a list of K dimensions where K is the number of functional covariates. In the situation at hand, we only have one functional covariate so our list will have one element:

# Making list element to pass in
func_covs_train = list(train_x)
func_covs_test = list(test_x)

Now we can fit the model!

# Now running model
fit_class = fnn.fit(resp = train_y,
func_cov = func_covs_train,
scalar_cov = scalar_train,
hidden_layers = 6,
neurons_per_layer = c(24, 24, 24, 24, 24, 58),
activations_in_layers = c("relu", "relu", "relu", "relu", "relu", "linear"),
domain_range = list(c(850, 1050)),
learn_rate = 0.001,
epochs = 100,
raw_data = T,
early_stopping = T)
#> [1] "Evaluating Integrals:"
#> Model
#> Model: "sequential"
#> ___________________________________________________________________________
#> Layer (type)                     Output Shape                  Param #
#> ===========================================================================
#> dense (Dense)                    (None, 24)                    216
#> ___________________________________________________________________________
#> dense_1 (Dense)                  (None, 24)                    600
#> ___________________________________________________________________________
#> dense_2 (Dense)                  (None, 24)                    600
#> ___________________________________________________________________________
#> dense_3 (Dense)                  (None, 24)                    600
#> ___________________________________________________________________________
#> dense_4 (Dense)                  (None, 24)                    600
#> ___________________________________________________________________________
#> dense_5 (Dense)                  (None, 58)                    1450
#> ___________________________________________________________________________
#> dense_6 (Dense)                  (None, 2)                     118
#> ===========================================================================
#> Total params: 4,184
#> Trainable params: 4,184
#> Non-trainable params: 0
#> ___________________________________________________________________________
#>
#>
#>
#> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
#> xxxxxxxxxxxxxxxxxxxx

#>
#> Trained on 128 samples (batch_size=32, epochs=100)
#> Final epoch (plot to see history):
#>                   loss: 0.008045
#>     mean_squared_error: 0.008045
#>               val_loss: 0.000001007
#> val_mean_squared_error: 0.000001007

Here, we fit a 6 layer model. There are a number of warnings and error checks in place to make sure that all the dimensionality is consistent. The first three inputs are all objects we defined above! Also, observe that raw_data is true here; this is important to indicate as it tells the model function to do the pre-processing.

Now that we have our model, we can make some predictions:

# Running prediction
predict_class = fnn.predict(fit_class,
func_cov = func_covs_test,
scalar_cov = scalar_test,
domain_range = list(c(850, 1050)),
raw_data = T)

# Rounding predictions (they are probabilities)
rounded_preds = ifelse(round(predict_class)[,2] == 1, 1, 0)

# Confusion matrix
caret::confusionMatrix(as.factor(rounded_preds), as.factor(test_y))
#> Confusion Matrix and Statistics
#>
#>           Reference
#> Prediction  0  1
#>          0 39  0
#>          1  0 15
#>
#>                Accuracy : 1
#>                  95% CI : (0.934, 1)
#>     No Information Rate : 0.7222
#>     P-Value [Acc > NIR] : 2.335e-08
#>
#>                   Kappa : 1
#>
#>  Mcnemar's Test P-Value : NA
#>
#>             Sensitivity : 1.0000
#>             Specificity : 1.0000
#>          Pos Pred Value : 1.0000
#>          Neg Pred Value : 1.0000
#>              Prevalence : 0.7222
#>          Detection Rate : 0.7222
#>    Detection Prevalence : 0.7222
#>       Balanced Accuracy : 1.0000
#>
#>        'Positive' Class : 0
#>

Okay that's all I have for now, have fun!

## Functions in FuncNN

 Name Description fnn.fit Fitting Functional Neural Networks fnn.tune Tuning Functional Neural Networks daily Classic Canadian weather data set. fnn.plot Plotting Functional Response Predictions fnn.cv Functional Neural Networks with Cross-validation fnn.fnc Output of Estimated Functional Weights fnn.predict Prediction using Functional Neural Networks FuncNN-package FuncNN: Functional Neural Networks tecator Classic Tecator data set. No Results!