# Equipment Health Assessment: Time Series Analysis for Wind Turbine Performance
## 2 Data and Methods
### Methods
#### 2.2.2 Functional Neural Networks (FNN)  
On the other hand, Functional data analysis (FDA) [35; 36; 37] is a branch of statistics that deals with data that are functions or curves, rather than simple numeric or categorical values. In FDA, the data is treated as a continuous function, and the goal is to analyze and model the behavior of the function over time or space. FDA methods allow researchers to extract meaningful information from functional data, such as trends, patterns, and underlying structures. This information can then be used to develop models and make predictions or forecasts. A Functional Linear Model (FLM) [38; 39] is a statistical model used  
Figure 3: LSTM Cell ([[https://towardsdatascience.com/lstm-recurrent-neural-networks-how-to-teach-a-network-to-remember-the-past-55e54c2](https://towardsdatascience.com/lstm-recurrent-neural-networks-how-to-teach-a-network-to-remember-the-past-55e54c2) ff22e]([https://towardsdatascience.com/lstm-recurrent-neural-networks-how-to-teach-a-network-to-remember-the-past-55e54c2](https://towardsdatascience.com/lstm-recurrent-neural-networks-how-to-teach-a-network-to-remember-the-past-55e54c2) ff22e), accessed on on 15 January 2024). in FDA that extends the linear regression model to functional data. It is designed to handle data that consists of a collection of curves or functions, rather than discrete data points. The FLM assumes that the relationship between the response variable and the predictor variables can be represented by a linear function, but with a functional rather than scalar form. The predictor variables can be either functional or scalar and can be continuous or categorical. In FLM, the functional form of the predictor variable is expressed using a basis expansion of a finite set of basis functions. These basis functions can be any set of orthogonal functions that span the space of functions under consideration. For example, Fourier, wavelet, or B-spline basis functions can be used. Traditional time series techniques rely on statistical methods, such as ARIMA, exponential smoothing, and FLM. These methods have been widely used for decades and have proven to be effective in many applications. However, they have some limitations, such as being sensitive to the assumptions made about the data and the need for manual feature engineering. One of the more popular deep learning models for time series prediction is Functional Neural Networks (FNN) [30; 40], that have been shown to outperform traditional time series models in many applications due to their ability to handle complex and nonlinear relationships in the data. This is particularly useful in applications where the relationships between variables are not well understood or there are many interacting factors. A Functional Neural Network (FNN) [16; 17] as shown in Figure 4, is composed of a series of interconnected continuous neurons that are designed to process functional data. The input layer of the network takes in functional data, and each continuous neuron given by Equation (1) in the continuous hidden layer performs a non linear transformation on the input data or values coming in from the previous layer. The output layer of the network produces a functional output that can be used for prediction or forecasting. \[H_{(l)}^{(r)}(s)=\sigma\Big{(}b_{(l)}^{(r)}(s)+\sum_{j=1}^{l}\int w_{(l)}^{(r, j)}(s,t)H_{(l-1)}^{(r,j)}(t)dt\Big{)}, \tag{1}\]  
where \(l\) indicates the layer number and \(r,j\) are the neuron number, \(l=1,2,3,\ldots,L\), \(H_{(0)}^{(r)}(s)=X_{r}(s)\) (input time series), \(H_{(L)}^{(r)}(s)=\widehat{Y_{r}(s)}\) (output time series), \(\sigma(\cdot)\) is a non-linear activation function, \(b_{(l)}^{(r)}\in\mathcal{L}^{2}(\mathcal{T})\) is the unknown intercept function, \(w_{(l)}^{(r,j)}\in\mathcal{L}^{2}(\mathcal{T}\times\mathcal{T})\) is the bivariate parameter function for the \(r^{th}\) continuous neuron in the \(l^{th}\) hidden layer coming from the \(j^{th}\) continuous neuron of the \((l-1)^{th}\) hidden layer.