1 Introduction Table 1. 3 Benchmark data specification for multiclass problems Classes Training data Test data Iris 75 Thyroid Blood cell 3,097 Hiragana-50 Hiragana-13 8.356 Satimage USPS 032983600 4,435 2,000 2,007 MNIST 784 60,000 0.000 where t and T denote time and time delay, respectively By integrating(1. 22), we can obtain the time series data z(0), r(1), a(2), a(t),.... Using z prior to time t, we predict r after time t. Setting T=17 and using four inputs a(t-18), r(t-12), 1(t-6), I(t), we estimate r(t+6) The first 500 data from the time series data, r(118),., I(1117), are used to train function approximators, and the remaining 500 data are used to test performance. This data set is often used as the benchmark data for function pproximation and the normalized root-mean-square error (NRMSe),i.e the root-mean-square error divided by the standard deviation of the time series data is used to measure the performance. In a water purification plant, to eliminate small particles floating in the water taken from a river, coagulant is added and the water is stirred while these small particles begin sticking to each other. As more particles stick together they form flocs, which fall to the bottom of a holding tank. Potable water is obtained by removing the precipitated flocs and adding chlorine Careful implementation of the coagulant injection is very important to obtain high-quality water. Usually an operator determines the amount of coagulant needed according to an analysis of the water qualities, observation of floc formation, and prior experience. To automate this operation, as inputs for water quality, (1) turbidity, (2)temperature, (3) alkalinity, (4)pH, and (5)flow rate were used, and to replace the operator's observation of floc properties by image processing, (1) doc diameter,(2)number of flocs, 3)floc volume, (4)floc density, and(5 illumination intensity were used 44] The 563 input-output data, which were gathered over a 1-year period were divided into 478 stationary data and 95 nonstationary data according o whether turbidity values were smaller or larger than a specified value. Then each type of data was further divided into two groups to form a training data set and a test data set; division was done in such a way that both sets had similar distributions in the output space12 1 Introduction Table 1.3 Benchmark data specification for multiclass problems Data Inputs Classes Training data Test data Iris 4 3 75 75 Numeral 12 10 810 820 Thyroid 21 3 3,772 3,428 Blood cell 13 12 3,097 3,100 Hiragana-50 50 39 4,610 4,610 Hiragana-105 105 38 8,375 8,356 Hiragana-13 13 38 8,375 8,356 Satimage 36 6 4,435 2,000 USPS 256 10 7,291 2,007 MNIST 784 10 60,000 10,000 where t and τ denote time and time delay, respectively. By integrating (1.22), we can obtain the time series data x(0), x(1), x(2), ...,x(t),.... Using x prior to time t, we predict x after time t. Setting τ = 17 and using four inputs x(t − 18), x(t − 12), x(t − 6), x(t), we estimate x(t + 6). The first 500 data from the time series data, x(118),...,x(1117), are used to train function approximators, and the remaining 500 data are used to test performance. This data set is often used as the benchmark data for function approximation and the normalized root-mean-square error (NRMSE), i.e., the root-mean-square error divided by the standard deviation of the time series data is used to measure the performance. In a water purification plant, to eliminate small particles floating in the water taken from a river, coagulant is added and the water is stirred while these small particles begin sticking to each other. As more particles stick together they form flocs, which fall to the bottom of a holding tank. Potable water is obtained by removing the precipitated flocs and adding chlorine. Careful implementation of the coagulant injection is very important to obtain high-quality water. Usually an operator determines the amount of coagulant needed according to an analysis of the water qualities, observation of floc formation, and prior experience. To automate this operation, as inputs for water quality, (1) turbidity, (2) temperature, (3) alkalinity, (4) pH, and (5) flow rate were used, and to replace the operator’s observation of floc properties by image processing, (1) floc diameter, (2) number of flocs, (3) floc volume, (4) floc density, and (5) illumination intensity were used [44]. The 563 input–output data, which were gathered over a 1-year period, were divided into 478 stationary data and 95 nonstationary data according to whether turbidity values were smaller or larger than a specified value. Then each type of data was further divided into two groups to form a training data set and a test data set; division was done in such a way that both sets had similar distributions in the output space