Improving artificial neural network model predictions of daily average PM₁₀ concentrations by applying principle component analysis and implementing seasonal models

Figures & data

Table 1. Descriptive statistics of seven yearly data sets (2007–2014) used for investigation

Variable	Range	Min.	Max.	Mean	SD	Variance
PM10 (µg/m^3)	601	12	613	87.81	72.29	5226.84
Air temperature (AT) (°C)	39.10	−7.30	31.80	15.86	8.34	69.56
Wind direction (WD) (deg.)	291	12	303	123.01	72.47	5651.69
Wind speed (WS) (m/sec)	10.40	0.10	10.50	1.38	0.79	0.631
Relative humidity (RH) (%)	116	23	139	73.51	15.15	229.55

Figure 1. Temporal changes in the daily average concentrations of PM₁₀ and AT for the years 2007–2014.

Figure 2. A schematic diagram of the use of PCA to determine most significant lags to feed ANN.

Figure 3. Template structure of ANN models for daily PM₁₀ estimates.

Table 2. Performance measures used in the model evaluations

Measure	Equation	Description
Index of agreement		Closer to 1.0 indicates the better agreement with the selected model
Fractional mean bias		FMB is nonlinear and bounded by ±2, which has an ideal value of zero for an ideal model
Root mean square error		The smaller values of RMSE denote the better model performance
Coefficient of determination		Close to 1.0 indicates the greater explained variance

Notes: n = total number of annual measurements; P_i = predicted values; O_i = observed values; = mean of the predicted values; Ō = Mean of the observed values; σ_P = standard deviation of the predicted values; σ_O = standard deviation of the observed values.

Table 3. PCA results of PM10_t against the lagged variables in the data set

PCA-1PM10 with PM10ts		PCA-2PM10 with ATts		PCA-3PM10 with WSts		PCA-4PM10 with WDts
PC1 (77.26%)		PC1 (71.52%)		PC1 (60.73%)		PC1 (71.35%)
PM10t	0.748	PM10t	−0.527	PM10t	−0.596	PM10t	−0.726
PM10t−1	0.862	ATt−1	0.905	WSt−1	0.712	WDt−1	0.755
PM10t−2	0.862	ATt−2	0.890	WSt−2	0.681	WDt−2	0.691
PM10t−3	0.776	ATt−3	0.878	WSt−3	0.685	WDt−3	0.671
PM10t−4	0.640	ATt−4	0.678	WSt−4	0.645	WDt−4	0.659
PM10t−5	0.636	ATt−5	0.679	WSt−5	0.621	WDt−5	0.618
PM10t−6	0.633	ATt−6	0.668	WSt−6	0.586	WDt−6	0.592
PM10t−7	0.620	ATt−7	0.654	WSt−7	0.540	WDt−7	0.583

Table 4. Structures of identified seasonal and nonseasonal ANN models

ANN Models	Variables in Input Vector
Seasonal models
NN_W1 for winter  (8-8-1)	PM10t−1, PM10t−2, PM10t−3, ATt−1, ATt−2, ATt−3, WSt−1, WDt−1
NN_S1 for summer  (8-7-1)	PM10t−1, PM10t−2, PM10t−3, ATt−1, ATt−2, ATt−3, WSt−1, WDt−1
NN_W2 for winter  (6-6-1)	PM10t−1, PM10t−2, ATt−1, ATt−2, WSt−1, WDt−1
NN_S2 for summer  (6-6-1)	PM10t−1, PM10t−2, ATt−1, ATt−2, WSt−1, WDt−1
NN_W3 for winter  (4-5-1)	PM10t−1, ATt−1, WSt−1, WDt−1
NN_S3 for summer  (4-5-1)	PM10t−1, ATt−1, WSt−1, WDt−1
Nonseasonal models
NSNN1 (8-9-1)	PM10t−1, PM10t−2, PM10t−3, ATt−1, ATt−2, ATt−3, WSt−1, WDt−1
NSNN2 (6-8-1)	PM10t−1, PM10t−2, ATt−1, ATt−2, WSt−1, WDt−1
NSNN3 (4-6-1)	PM10t−1, ATt−1, WSt−1, WDt−1

Table 5. Performance measures obtained from seasonal and nonseasonal models

NN Models	LR	LM	Training				Testing
			RMSE	R^2	FMB	IA	RMSE	R^2	FMB	IA
Seasonal ANN models
NN_W1	0.05	—	0.587	0.722	0.10	0.83	0.615	0.711	0.09	0.82
NN_S1	0.05	—	0.602	0.727	0.13	0.81	0.682	0.715	−0.11	0.79
NN_W2	0.10	0.2	0.610	0.683	0.11	0.79	0.657	0.670	−0.09	0.80
NN_S2	0.10	0.2	0.606	0.681	−0.12	0.81	0.693	0.671	0.11	0.78
NN_W3	0.10	—	0.655	0.693	0.16	0.78	0.721	0.688	−0.14	0.73
NN_S3	0.05	—	0.637	0.719	−0.19	0.82	0.669	0.709	−0.21	0.79
Singular (nonseasonal) ANN models
NSNN1	0.01	—	0.636	0.654	0.11	0.75	0.653	0.633	−0.09	0.71
NSNN2	0.10	—	0.685	0.667	0.09	0.79	0.703	0.656	0.22	0.67
NSNN3	0.10	—	0.711	0.689	0.10	0.78	0.739	0.668	−0.24	0.68

Figure 4. Training and testing performance plots of ANN models: (a) NN_W1, (b) NN_S1, (c) NSNN1.

Figure 5. Estimates in training and testing stage of the models NN_W1 and NN_S1 against to actual data.

Figure 6. Testing estimates of NN_W1 and NN_S1 models for seasons in sequence.