## 2 Development of a Servo Valve Prediction Model
### Rationale for Expectancy Projections  
#### 2.2.1 Theoretical Foundations of Exponential Smoothing Forecasting and Fusion Prediction  
Exponential smoothing is a time series forecasting method for univariate data with a wide range of applications for the predictive analysis of data with systematic genera and with seasonal components and is one of the oldest and most studied time series forecasting methods. It is most effective when the time series values follow an asymptotic trend and show seasonal behavior (i.e., when the values follow a repeating periodic pattern over a given number of time steps). Exponential smoothing uses an exponential window function to smooth time series data. In a simple moving average, past observations have the same weight, whereas the exponential function is used to specify weights that are exponentially decreasing over time. In the exponential smoothing process, past observations are proportionally weighted to the data obtained geometrically decreasingly. The prediction result is based on the weighted average produced by the exponential smoothing method, with an exponentially decaying trend in the weights obtained based on the difference in time between the data obtained immediately and the current value. That is, the closer the observation time, the higher the relevant weight. Exponential smoothing methods are equivalent to peer-to-peer methods and are applied interchangeably with ARIMA (autoregressive integrated moving average model) methods for time series forecasting. These forecasting methods provide error, trend, and seasonal characteristics to construct models. There are three main types of exponential smoothing time series forecasting methods: simple methods that assume no systematic structure, extensions that deal explicitly with trends, and state-of-the-art methods that support seasonality. The primary use of single exponential smoothing (SES), also known as simple exponential smoothing, is a method for time series forecasting of univariate data. Set the smoothing factor to \(a\). This smoothing factor constrains the observations for the previous time step and can have an effect on the rate of exponential decay. \(a\) is usually set to a value between [0, 1]. The range of values implies whether the model is primarily concerned with recent past observations or whether it will take more history into account when making predictions. Exponential smoothing is a simple but powerful method for predicting time series. Exponential smoothing is also used as a building block by many other models. Compared to simpler forecasting models such as simple or moving averages, exponential smoothing models have two advantages: one is that the weight of each observation decreases exponentially over time. This compares favorably with a moving average model that assigns the same weight to all relevant historical months. In addition, the effect of outliers and noise is less than that of using the plain method. The basic idea of the exponential smoothing model is that in each period, the model learns a little from the most recent demand observation and remembers the last forecast it made. The final forecast populated by the model already contains part of the previous demand observation and part of the previous forecast. Therefore, this previous forecast includes everything the model has learned so far based on the demand history. The smoothing parameter (or learning rate) \(a\) will determine the importance of the most recent demand observation. It is expressed as follows:  
\[f_{t}=ad_{t-1}+(1-a)f_{t-1}\ 0\leq a\leq 1 \tag{2}\]  
where \(a\) is the ratio (or percentage) of importance that the model will assign to the most recent observation of the historical importance of demand. \(ad_{t-1}\) denotes the previous demand observation multiplied by the learning rate. It can also be expressed that the model attaches \(a\) certain weight a to the last demand occurrence. \((1-a)f_{t-1}\) indicates how much the model has remembered from previous predictions.