A Forecasting Framework Based on Kalman Filter Integrated Multivariate Local Polynomial Regression: Application to Urban Water Demand

Abstract

In this study, a forecasting framework for daily urban water demand has been proposed. It was developed based on the extended Kalman filter (EKF) which consists of state estimation, forecasting and error correction. The forecasting and error correction models can be substituted. As an example, a multivariate local polynomial regression (MLPR) was used to linearize the complex system which is essential for EKF. A correctional prediction of residual based on relevance vector regression was employed to update and substitute error estimation value in the EKF. To improve the precision of the forecasts, the historical data series was decomposed into low- and high-frequency subseries using discrete wavelet transformation. Five category forecasts with the lead time of 1-day were assessed in comparison of the proposed model: MLPR, multi-scale relevance vector regression, autoregressive moving average, Back Propagation neural network and multiple linear regression. According to the performance criteria, the MLPR is slightly beneficial in capturing the basic dynamics of the daily urban water demand in the short term, but the state estimation and error correction can greatly improve the results. The proposed model obtains better forecasting performances than existing models, which is attributed to good state estimation from the Kalman transmission gain and favorable feature learning performance using MLPR.

Appendices

Appendix A

1.1 Wavelet Transform for Decomposition

The wavelet transform is a series mathematic function that decomposes a non-stationary time series into multiple subseries with diverse frequency domains by a mother wavelet (Daubechies ‘db’). In this study, Discrete wavelet transform (DWT) is utilized because it is easy to be implemented and more less amount of calculation. The DWT function of a time series x is simplified as following [14]

$$ W_{f} (p,q) = 2^{ - p/2} \sum\limits_{i = 1}^{N} {x_{i} \psi (2^{ - p} i - q)} $$

(A.1)

where p and q are integers that determine the magnitude of wavelet dilation and translation, respectively. W_f shows the specified wavelet coefficient of the subseries. ψ is the transforming function.

In Eq. (A.1), W_f applies down sampling to obtain an approximation coefficient (${\mathbf{x}}^{\text{aL}} = \{ x_{i}^{{^{\text{aL}} }} \}$) at level L with a lowpass LP(ψ(i)), and detail coefficients (${\mathbf{x}}^{{{\text{d}}k}} = \{ x_{i}^{{{\text{d}}k}} \}$) at levels k = 1, 2,…, K with a high pass filter HP(ψ(i)). Then the raw signal could be defined as

$$ x_{i} = x_{{_{i} }}^{aL} LP(\psi (i)) + \sum\limits_{z = 1}^{L} {x_{i}^{dz} HP(\psi (i))} $$

(A.2)

Appendix B

2.1 C–C Method for m and τ

The C–C method will be undertaken based on:

• (1) Compute the $\overline{S} (t),\;\Delta \overline{S} (t)$ and $S_{cor} (t)$ of each time subseries

  $$ \overline{S} (t) = \frac{1}{4 \cdot Z}\sum\limits_{j = 1}^{4} {\sum\limits_{m \ge 2} {\Delta S(m,N,r_{j} ,t)} } $$

  (B.1)
  $$ \Delta \overline{S} (t) = \frac{1}{4}\sum\limits_{m \ge 2} {\Delta S(m,t)} $$

  (B.2)
  $$ S_{cor} (t) = \Delta \overline{S} (t) + |\overline{S} (t)| $$

  (B.3)

  where $\Delta S(m,N,r_{j} ,t)$ and $\Delta S(m,t)$ are statistic functions varied with m, N, time and characteristic values, see as [27, 28].

• (2) Select the index lag t for the first local minimum of $\Delta \overline{S} (t)$, the global minimum of $S_{cor} (t)$ and $\overline{S} (t) = 0$. Determine m and τ based on Eq. (2).