A nonlinear time-series prediction methodology based on neural networks and tracking signals

Figure 1
Multilayer feedforward ANN structure. Source: Balestrassi et al. (2009)Balestrassi, P., Popova, E., Paiva, A., & Lima, J. (2009). Design of experiments on neural network’s training for nonlinear time series forecasting. Neurocomputing, 72(4-6), 1160-1178. http://dx.doi.org/10.1016/j.neucom.2008.02.002.
http://dx.doi.org/10.1016/j.neucom.2008.... .

Thumbnail

Figure 2
Prediction Methodology Flowchart. Source: Authors.

Thumbnail

Figure 3
DOE for simulation. Source: Bianchesi et al. (2019)Bianchesi, N., Romão, E., Lopes, M., Balestrassi, P., & Paiva, A. (2019). A design of experiments comparative study on clustering methods. IEEE Access: Practical Innovations, Open Solutions, 7, 167726-167738. http://dx.doi.org/10.1109/ACCESS.2019.2953528.
http://dx.doi.org/10.1109/ACCESS.2019.29... .

Thumbnail

Figure 4
Cohen’s d representation. Source: Bianchesi et al. (2019)Bianchesi, N., Romão, E., Lopes, M., Balestrassi, P., & Paiva, A. (2019). A design of experiments comparative study on clustering methods. IEEE Access: Practical Innovations, Open Solutions, 7, 167726-167738. http://dx.doi.org/10.1109/ACCESS.2019.2953528.
http://dx.doi.org/10.1109/ACCESS.2019.29... .

Thumbnail

Figure 5
Relationship of Control Chart to Normal Curve. Source: Montgomery (2009)Montgomery, C. D. (2009). Introduction to statistical quality control (6th ed.). New York: John Wiley & Sons..

Thumbnail

Figure 6
STAR1 - Time Series Forecasting. Source: Authors.

Thumbnail

Figure 7
BL1 - Time Series Forecasting. Source: Authors.

Thumbnail

Figure 8
STAR1 – Training samples. Source: Authors.

Thumbnail

Figure 9
STAR1 - Original series. Source: Authors.

Thumbnail

Figure 10
STAR1 – Modified series. Source: Authors.

Thumbnail

Figure 11
Pareto Chart of the Standardized Effects for ARL, α=0.05 – STAR1. Source: Authors.

Thumbnail

Figure 12
Pareto Chart of the Standardized Effects for ARL, α=0.05 – BL1. Source: Authors.

Thumbnail

Figure 13
Pareto Chart of the Standardized Effects for ARL, α=0.05 – NMA. Source: Authors.

Thumbnail

Figure 14
Main Effects Plot for ARL – STAR1. Source: Authors.

Thumbnail

Figure 15
Main Effects Plot for ARL – BL1. Source: Authors.

Thumbnail

Figure 16
Main Effects Plot for ARL – NMA. Source: Authors.

Thumbnail

Figure 17
Interaction Plot for ARL – STAR1. Source: Authors.

Thumbnail

Figure 18
Interaction Plot for ARL – BL1. Source: Authors.

Thumbnail

Figure 19
Interaction Plot for ARL – NMA. Source: Authors.

Thumbnail

Figure 20
Surface Graph for ARL – STAR1. Source: Authors.

Thumbnail

Figure 21
Surface Graph for ARL – BL1. Source: Authors.

Thumbnail

Figure 22
Surface Graph for ARL – NMA. Source: Authors.

Thumbnail

Figure 23
Response Optimization for ARL (a). Source: Authors.

Thumbnail

Figure 24
TOG Time Series Forecasting – Samples 1 to 54. Source: Authors.

Thumbnail

Figure 25
TOG Time Series Forecasting – Samples 55 to 84. Source: Authors.

Thumbnail

Figure 26
TOG Time Series Forecasting – Samples 85 to 100. Source: Authors.

Thumbnail

Table 1
Nonlinear Time Series Models.

Thumbnail

Table 2
Pseudocode.

Thumbnail

Table 3
Experimental Factors.

Thumbnail

Table 4
Design Matrix.

Thumbnail

Table 5
ARL Results.

Thumbnail

Table 6
Desirability Results.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

Model	Equation
Bilinear (BL1)	$y_{t} = 0.7 y_{t - 1} ε_{t - 2} + ε_{t}$
Nonlinear moving average (NMA)	$y_{t} = ε_{t} - 0.3 ε_{t - 1} + 0.2 ε_{t - 2} + 0.4 ε_{t - 1} ε_{t - 2} - 0.25 ε_{t - 2}^{2}$
Smooth transition autoregressive (STAR1)	$y_{t} = 0.8 y_{t - 1} - 0.8 y_{t - 1} {[1 + exp (- 10 y_{t - 1})]}^{- 1} + ε_{t}$

Step		Pseudocode
1	1:	$y \leftarrow$ dataset
	2:	$t \leftarrow$ time series size
	3:	end $\leftarrow$ $y (l a s t P o s i t i o n)$
	4:	$n_{p r e d} \leftarrow$ number of predictions
2	5:	for t $\leftarrow 1$ to $50$
	6:	$m l p_{n e t} \leftarrow$ RNA_MLP_train( $t, y$ )
	7:	$y_{p r e d_{1}} \leftarrow$ RNA_MLP_test( $m l p_{n e t}, y, n_{p r e d}$ )
3	8:	$T S_{1} \leftarrow$ cts( $y_{p r e d_{1}}, y$ )
	9:	end for
	10:	$l_{s} \leftarrow$ upper limit
	11:	$l_{l} \leftarrow$ lower limit
4	12:	for t $\leftarrow 51$ to $e n d$
4	13:	$y_{p r e d_{t}} \leftarrow$ RNA_MLP_test( $m l p_{n e t}, y, 1$ )
5	14:	$T S_{t} \leftarrow$ cts( $y_{p r e d_{t}}, y$ )
	15:	if ( $c t s (t) > l_{s}$ or $c t s (t) < l_{i})$
	16:	$m l p_{n e t} \leftarrow$ RNA_MLP_train( $t, y$ )
	17:	else
	18:	end for
	19:	// Functions
	20:	function $c \leftarrow cts$ ( $y p r e d, y$ )
	21:	$N \leftarrow$ $y p r e d$ length
	22:	$e n d \leftarrow$ $y (l a s t P o s i t i o n)$
	23:	$y \leftarrow$ $y (e n d - N + 1 : e n d)$
	24:	$e r r o r \leftarrow$ $y p r e d - y$
	25:	$e r r o r_{a b s} \leftarrow$ absolute value of error
	26:	$m a d (1) \leftarrow$ $e r r o r_{a b s} (1)$
	27:	$c u s u m (1) \leftarrow$ 0
	28:	$t s b (1) \leftarrow$ $0$
	29:	$a d (1) \leftarrow$ $e r r o r_{a b s} (1)$
	30:	for $t \leftarrow 1$ to $N$
	31:	$a d (1) \leftarrow$ $e r r o r_{a b s} (t) + a d (t - 1)$
	32:	$m a d (1) \leftarrow$ $a d (t) / t$
	33:	$c u s u m (1) \leftarrow$ $e r r o r (t) + c u s u m (t - 1)$
	34:	$c t s (1) \leftarrow$ $c u s u m (t) / m a d (t)$
	35:	end for
	36:	end function
	37:	function $n e t \leftarrow RNA_MLP_train (N, y)$
	38:	$t \leftarrow$ $1 : (N - 1$ )
	39:	$x (t) \leftarrow$ $y (t)$
	40:	$y_{p} (t) \leftarrow$ $y (t + 1)$
	41:	$n e t_{p a r a m_{t r a i n}} \leftarrow$ 80/100
	42:	$n e t_{p a r a m_{v a l i d}} \leftarrow$ 10/100
	43:	$n e t_{p a r a m_{t e s t}} \leftarrow$ 10/100
	44:	$n e t_{t y p e} \leftarrow$ RNA MLP Backpropagation
	45:	$h i d d e n L a y e r_{s i z e} \leftarrow$ 10
	46:	$n e t \leftarrow$ $f e e d F o r w a r d N e t (h i d d e n L a y e r_{s i z e},^{'} t r a i n l m^{'})$
	47:	$n e t_{p a r a m_{s h o w}} \leftarrow$ 50
	48:	$n e t_{p a r a m_{l r}} \leftarrow$ 0.01
	49:	$n e t_{p a r a m_{e p o c h s}} \leftarrow$ 1000
	50:	$n e t_{p a r a m_{g o a l}} \leftarrow$ 1e-06
	51:	$[n e t, t r] \leftarrow$ $t r a i n (n e t, x, y_p)$
	52:	end function
	53:	function $y_{p r e d} \leftarrow RNA_MLP_test (N, y, n_{p r e d})$
	54:	$y_{p r e d} \leftarrow$ $n e t (y$ )
	55:	end function

	Mean (X)	Standard Deviation (SD)
(-1)	0.2	0.5
(+1)	0.8	3.0

Run Order	Replication	Mean	SD
1	1	0.20	0.50
2	1	0.80	0.50
3	1	0.20	3.00
4	1	0.80	3.00
5	1	0.08	1.75
6	1	0.92	1.75
7	1	0.50	0.02
8	1	0.50	3.52
9	1	0.50	1.75
10	1	0.50	1.75
11	1	0.50	1.75
12	1	0.50	1.75
13	1	0.50	1.75
14	2	0.20	0.50
15	2	0.80	0.50
16	2	0.20	3.00
17	2	0.80	3.00
18	2	0.08	1.75
19	2	0.92	1.75
20	2	0.50	0.02
21	2	0.50	3.52
22	2	0.50	1.75
23	2	0.50	1.75
24	2	0.50	1.75
25	2	0.50	1.75
26	2	0.50	1.75

Run Order	ARL
Run Order	STAR1	BL1	NMA
1	35.76	32.26	25.59
2	18.15	16.11	20.56
3	17.76	16.00	17.12
4	16.94	16.00	29.65
5	23.84	16.95	18.21
6	17.04	16.00	17.17
7	17.66	18.74	17.83
8	17.07	16.00	19.90
9	17.60	16.01	19.90
10	17.60	16.01	19.90
11	17.60	16.01	19.90
12	17.60	16.01	19.90
13	17.60	16.01	19.90
14	47.94	31.25	26.38
15	28.55	16.20	20.63
16	18.63	16.00	17.46
17	16.94	16.00	30.34
18	41.83	17.82	18.49
19	17.61	16.00	16.97
20	16.23	18.53	18.03
21	16.83	16.00	20.05
22	18.27	16.00	20.05
23	18.27	16.00	20.05
24	18.27	16.00	20.05
25	18.27	16.00	20.05
26	18.27	16.00	20.05

Factors	STAR1	BL1	NMA
Mean	0.89	0.59	0.80
SD	0.02	2.41	3.52
ARL	11.79	15.62	15.45

[1] *natalia.bianchesi@unifei.edu.br, naty_bianchesi@hotmail.com.br

Brasil

Brasil

A nonlinear time-series prediction methodology based on neural networks and tracking signals

Abstract

Paper aims

Originality

Research method

Main findings

Implications for theory and practice