SANY Sensors Anywhere

Temporal fusion can be used to predict the target variable directly from past observations of the target variable itself. The essential difference between temporal fusion and causal fusion is that temporal fusion takes the internal structure of data into account. In the SensorSA time-series data from in-situ sensors are obtained from SOS instances. The resultant predictions from the temporal fusion service are supplied to an OGC compliant SOS instance via a ‘virtual sensor’ controlled by an SPS instance.

Methods for time series analysis are often divided into two domains: timedomain and frequency domain. The frequency domain approach is more suited to exploratory analysis. The time-domain approach is discussed here. Time series usually contains some typical patterns:

• Trend: represents long-run movements in the series.
  
• Seasonal cycle: seasonality pattern repeats itself more or less around a fixed period.
  
• Autoregressive component: represents data as a function of the past history plus a white noise.
  
• Moving average component: assumes data model is a linear combination of a prior random process.

Apart from the above regular patterns, an irregular component in the time series reflects non-systematic movements in the process.

The regular patterns can be identified through exploratory analysis or empirical knowledge of the process. At this stage, one must decide the order of trend, i.e., whether it is a random walk or a local linear trend, the existence
of seasonal component and its period, the order of autoregressive and moving average components.

Two temporal fusion algorithems have been investigated in SANY: state-space modelling and Kalman filters

State-space Modelling

Once data patterns are identified, models for time series can be formed using an autoregressive integrated moving average model or state-space form.

The state-space form has enormous power to handle a wide range of time series models.

The basic structures such as trend and seasonal cycles are expressed explicitly in the model and are easy to interpret. The state-space form consists of a measurement equation and a transition equation. The transition equation contains the dynamics of the system under investigation and generates state variables. The measurement equation relates observable variables to state variables.

Kalman filters

After time series are modelled and put in state-space form, the Kalman filter algorithm may be used to produce predictions and smoothing of the statespace vector.

The Kalman filter is an important algorithm in many applications since it facilitates online estimation and enables the estimation and prediction of the state vector to be continually updated as new observations become available.

The Kalman filter is derived on the assumption that the disturbance and initial state vector are normally distributed. It gives optimal estimation of the state vector in the sense that it minimizes the mean square error within the class of linear estimators. It consists of two steps: prediction and update. The prediction step predicts the state variable and the prediction error to the next time step using the transition equation. The update step modifies the prediction once the observation at the current time step becomes available.

The Kalman filter also facilitates maximum likelihood estimation of the unknown parameters in the model. It enables the likelihood function to be calculated via prediction error decomposition. The maximum likelihood estimation can be carried out numerically or by an Expectation Maximization (EM) algorithm. The EM algorithm takes on a simple form comparing to the numerical solution and it always increases the likelihood during the iteration. The EM algorithm also tolerates missing observations and has a natural procedure to adjust the estimators.