Application of an artificial neural network to improve short-term road ice forecasts

Expert Systems with Applications PERGAMON

Expert Systems With Applications 14 (1998) 471–482

Application of an artificial neural network to improve short-term road ice forecasts J. Shao* Vaisala TMI Limited, Vaisala House, 349 Bristol Road, Birmingham B5 7SW, UK

Abstract This paper describes how a three-layer artificial neural network (NN) can be used to improve the accuracy of short-term (3–12 hours) automatic numerical prediction of road surface temperature, in order to cut winter road maintenance costs, reduce environmental damage from oversalting and provide safer roads for road users. In this paper, the training of the network is based on historical and preliminary meteorological parameters measured at an automatic roadside weather station, and the target of the training is hourly error of original numerical forecasts. The generalization of the trained network is then used to adjust the original model forecast. The effectiveness of the network in improving the accuracy of numerical model forecasts was tested at 39 sites in eight countries. Results of the tests show that the NN technique is able to reduce absolute error and root-mean-square error of temperature forecasts by 9.9–29%, and increase the accuracy of frost/ice prediction. q 1998 Elsevier Science Ltd. All rights reserved.

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1. Introduction During the period from late autumn to early spring, vast areas in North America, western Europe and many other countries (e.g., Japan) suffer from frequent snow, sleet, ice and frost. Such adverse weather conditions lead to dangerous driving conditions for road users. In recent years, accurate numerical prediction of road surface temperature and surface state, as a part of an expert system, has been widely accepted by both highway engineers and meteorologists as an appropriate and valuable technique for cutting winter road maintenance costs (e.g., by 20%), protecting the environment, and keeping and raising road safety standards (Shao and Lister, 1996). A recent survey commissioned by the UK Meteorological Office shows that about £170 million ($272 million) and 25–50 lives have been saved each year with the road ice prediction system in the UK (Thornes, 1994). In Wisconsin, USA, the use of a winter weather system which includes a numerical road ice prediction model and remote automatic roadside weather stations can save $75 500 and reduce salt usage by 2500 tons during a single winter storm (Stephenson, 1988). It is obvious that the higher the accuracy of road ice prediction, the higher the potential of economic savings. * Corresponding author. Tel.: 44 121 683 1217; fax: +44 121 683 1299; e-mail: [email protected].

0957-4174/98/$19.00 q 1998 Elsevier Science Ltd. All rights reserved. PII: S0 95 7 -4 1 74 ( 98 ) 00 0 06 - 2

However, uncertainty in initial and boundary conditions of a numerical model, complex topography, frequent snow cover, heavy traffic and some unknown mechanisms cause significant deterioration of model performance at some forecast sites and when the forecast period becomes longer. In order for highway engineers and meteorologists to achieve more savings on the costs of winter road maintenance, such error needs to be reduced. However, due to complex physical processes embedded in the numerical model and varying operating environments of the model, the error is neither linear nor systematic. As a result, conventional error-correction methods (such as a statistic template; see Thornes and Shao, 1992) have a limited effect on removing the error. A new method is therefore needed to improve the model forecast. In the last decade or so, the results of research on artificial neural networks (NNs) and their applications have reemerged in many published papers, and NN techniques have been shown to be a useful tool in solving problems of control, prediction and classification in industry, environmental sciences and meteorology (e.g., McCann, 1992; Boznar et al., 1993; Jin et al., 1994; Aussem et al., 1995; Bankert and Aha, 1996; Eckert et al., 1996; Marzban and Stumpf, 1996). Therefore, it is natural to think of combining the NN and numerical road ice prediction techniques to yield better forecasts for highway engineers. In this paper, an attempt is made to investigate the possibility of using a simulated multilayer NN to correct or reduce the error of

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J. Shao/Expert Systems With Applications 14 (1998) 471–482

forecasts (3–12 hours ahead) of road surface temperature for winter road maintenance. In this paper, an automatic numerical road ice nowcasting model (called the Icebreak model; see Shao and Lister, 1996) is employed to test the applicability of a NN for a possible improvement of road ice prediction. It has been found that the error of the Icebreak model is neither linear nor systematic, and may be caused, in part, by some unknown and non-linear mechanisms such as a sudden perturbation in the atmospheric boundary layer and/or the surface layer due to the atmosphere itself, traffic and other environmental disturbances (e.g., underground hot water pipes in an urban area). In that one of the most important and valuable features of an NN with non-linear activation functions is being able to emulate complex and non-linear time series or patterns, it is expected that a properly trained NN is able to identify the magnitude and pattern of forecast error under certain weather conditions, and thus improve the prediction of road ice. As an investigation, a three-layer and errorbackpropagated NN was provided with preliminary meteorological parameters and original model prediction errors in the previous 3 hours. After a learning process, the network then outputs a set of temperature adjustments to the initial model forecasts generated from the Icebreak model for the next 3–12 hours. In Section 2, the design and algorithm of the NN are presented. The input features of the network are described in Section 3. Verification of the NN is described in Section 4 for 39 forecast sites in Austria, Belgium, Holland, Italy, Japan, Norway, Switzerland and the United Kingdom to verify the effectiveness of the NN technique in improving the forecasts. The results are discussed and concluding remarks are given in Section 5.

2. The network and algorithm An NN technique is a relatively new method of statistical mapping (in the sense that the mapping is neither unique nor determinative, and depends on learning samples). It differs from traditional statistical techniques in its ability to successfully generalize patterns that have not been presented previously, and to learn and emulate unanticipated and nonlinear features of a time series without requiring an insight into the underlying mechanisms. This gives the NN technique an advantage in dealing with complex problems. Although four or more layers of an NN have been used by some researchers, in fact, a three-layer NN is able to represent an arbitrary function (no matter how complex) according to the Kolmogorov representation theorem (Fausett, 1994). Therefore, the NN used in this paper consists of three layers (an input layer, a middle hidden layer and an output layer), as shown in Fig. 1. In the figure, N 1, N 2 and N 3 are numbers of neurons at the input, hidden and output layers respectively. The neurons in the input layer serve to

Fig. 1. A three-layer neural network.

distribute the value they receive to the next layer and do not perform a weighted sum. However, each neuron in the hidden layer and the output layer has a connection to all neurons in its previous layer by a weight, denoted as w ij(t), which represents the weight from neuron i to neuron j at time t. The weighted sum at neuron j is expressed as net j and its output as out j. Each such neuron performs a simple computation: it receives signals from the neurons of its previous layer and computes a new activation level that it sends along each of its output connections. The computation of the activation level is based on the values of each input signal received from a previous neuron, and the weights on each input connection. The first part, or the linear component, of the computation is to compute the weighted sum of the neuron’s input values. The second part, or the non-linear component, of the computation is to transform the weighted sum into the final value, that serves as the neuron’s output value to other neurons of its next layer. The weighted sum of each neuron j at the hidden layer or output layer can be written as X wij · outi , (1) netj ¼ i

where out i is the output of neuron i at a previous layer. The output from each neuron j (out j) is then derived from an activation function or a sigmoid transfer function, defined as f (netj ) ¼

1 , 1 þ exp( ¹ k · netj )

(2)

which has the range 0 , f(net j) , 1. Here, k is a positive constant that controls the ‘spread’ of the function or the gain of the neurons, and has been assigned a value of 2 in this paper. Before making a generalization, the network is first trained by providing pairs of known input and output (i.e. supervised learning). The input goes directly into the neurons at the first layer. Each neuron in the hidden layer then accepts the values from input neurons and sums them according to its connecting weights. Then the output at the hidden layer becomes the input for the output layer. At the output layer, NN output is compared to actual sensor measurements (targets) and the error obtained is used to adjust backwards connection weights between layers of neurons. The network is designed to approach the desired known output or targets (T j, j ¼ 1, 2,…, N 3) by dynamically

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J. Shao/Expert Systems With Applications 14 (1998) 471–482

adjusting the weights on the links between the neurons, i.e. to minimize the error function 1X (Tj ¹ outj )2 : (3) Err ¼ 2 j The object of the minimization of Eq. (3), which is a function of the weights and inputs, is to find a minimum (which ideally should be a global minimum) in the error space. The sequence described above is called a learning process. The learning rule for the multilayer NN is called the ‘generalized delta rule’ or the ‘backpropagation rule’ (Beale and Jackson, 1994), by which the error is first calculated in the output neurons and then passed back through the network to the hidden neurons. This allows the neurons to change their connection weights proportionally to both the error in the neurons and output of the neurons. As a result, the weight change moves the network in the direction of maximum error reduction or in the steepest downward direction on the error surface. A general expression of weight adjustment is:

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2. All weights become stable, i.e. the change of all weights is less than 0.0001. 3. t $ t max. After successful training using historical data, the network is fed with the latest real-time input. It then compares this input with the trained input patterns, searches for the most similar input pattern and produces output through the connections in the network.

3. Neural network architecture

where t max is the maximum allowed learning time (5000 iterations). The learning process is terminated when any of the following conditions is met:

It is understood that good generalization based on learned examples cannot be achieved without a good deal of background knowledge of physical processes in the near-ground atmosphere and on the road surface. Therefore, it is very valuable to incorporate such knowledge into the NN design and learning processes. Early results of sensitivity analysis (Thornes and Shao, 1991) have shown that the Icebreak model is sensitive to air temperature, cloud amount, cloud type, wind speed and humidity. Experience gained in operational use and research of the model indicate that (a) the error of the model forecast is usually (though not always) larger during daytime and smaller at night, and (b) the error is non-linear and is highly correlated with small-scale temporal variations of the near-ground (at a 2-m height) atmospheric environment. This knowledge and experience is helpful in choosing the right input variables, formats and features for the network. However, as such a short-term forecast by the Icebreak model uses only basic sensor measurements (such as road surface temperature, road surface state, air temperature, dew point and wind speed) at an automatic roadside weather station, selection and design of input features for the neurons of the network has to be simple and must be based on these limited variables in order to maintain the model’s important automation property. As a result, cloud data, which are not available at a roadside weather station, are unfortunately excluded from being used in this study. Although there are generally two time-consuming approaches (pruning and leave-one-out) to finding the optimal network architecture (Gallant, 1993), there is currently not a general rule to decide the optimum number of neurons at each individual layer without a number of tedious trials. Too few neurons may cause the network to be incapable of representing the desired mapping between input and output and thus to fail. On the other hand, if too many neurons are chosen, the network will be able to memorize all the learned examples but will not generalize well with inputs that have not been seen; overfitting will cause the network to fail in an operational stage. In designing the architecture of the network, the following specifications and constraints must be taken into account:

1. The error of fit becomes less than the accuracy (0.18C) of sensor measurements.

1. The short-term forecast model needs to remain automatic and speedy.

wij (t þ 1) ¼ wij (t) þ h · dj · outi ,

(4)

where h is a gain term or learning rate and d j is an error term on neuron j. For output neurons, the error term is represented as: dj ¼ k · outj · (1 ¹ outj )(Tj ¹ outj ):

(5)

For hidden neurons, the expression is: X dj ¼ k · outj · (1 ¹ outj ) dk wjk ,

(6)

where the sum is over the k neurons in the output layer. Initially, all connections of neurons in hidden and output layers are assigned with a computer-generated small random weight. These weights are then modified according to the learning rule along the deepest downward direction to minimize the error function. In order to reduce the possibility of being ‘trapped’ in a local minimum where a learning process likely fails, the learning rate is progressively decreased. As the rate decreases, the network takes smaller downhill steps and its weights settle into a minimum configuration without overshooting the stable position. Therefore, the network is hopefully able to bypass local minima and then find some deeper minima or a better solution without oscillating wildly. For this purpose, the learning rate is taken as a function of time (t, or number of iterations in the learning process):   t , (7) h ¼ 1 ¹ ln 1 þ tmax

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Table 1 List of input neurons (note: all temperatures are in 8C, wind speed in m/s and wind direction in 0–360 degrees) Number

Description

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Time (hour) of day: sin(t) Time (hour) of day: cos(t) Forecast error 1 hour earlier: Tf1 ¹ Ts1 Forecast error 2 hours earlier: Tf2 ¹ Ts2 Forecast error 3 hours earlier: Tf3 ¹ Ts3 Variation of surface temperature 1 hour earlier: Ts1 ¹ Ts2 2 3 Variation of surface temperature 2 hours earlier: ÿ Ts ¹
Ts ÿ
Derivative of surface temperature variation: Ts1 ¹ Ts2 ¹ Ts2 ¹ Ts3 1 2 Variation of air temperature 1 hour earlier: Ta ¹ Ta 2 3 Variation of air temperature 2 hours earlier: ÿ 1 Ta ¹
Ta ÿ 2
2 Derivative of air temperature variation: Ta ¹ Ta ¹ Ta ¹ Ta3 1 2 Variation of dew point 1 hour earlier: Td ¹ Td 2 3 Variation of dew point 2 hours earlier: ÿ Td ¹
Td ÿ
Derivative of dew point variation: Td1 ¹ Td2 ¹ Td2 ¹ Td3 Difference of surface and air temperatures 1 hour earlier: Ts1 ¹ Ta1 1 1 Difference of air temperature and dew ÿpoint 1 hour earlier:
Ta ¹ Td 1 2 3 Mean wind speed for the last 3 hours: W þ ÿW þ W =3
Mean wind direction for the last 3 hours: sin[ ÿD1 þ D2 þ D3
=3] Mean wind direction for the last 3 hours: cos[ D1 þ D2 þ D3 =3] Road surface state (dry, wet, icy, etc.) 1 hour earlier: S 1

q 2. Only a limited number of meteorological variables are available. 3. The learning period of the network should be as short as possible for its practical usage. 4. The architecture of the network should be sufficiently general to produce acceptable forecasts wherever it is applied operationally. After a brief trial and based on the above considerations and knowledge of physical mechanisms governing the variation of road surface temperature, a set of 20 variables was assigned to individual neurons in the input layer (see Table 1). All of the variables in the table were inputs to the network at an interval of 1 hour and were normalized in terms of their extremes that may occur in the atmospheric boundary layer. To determine the number of hidden neurons (N 2), an estimate is made based on the relationship described by Baum and Haussler (1989): N2 ¼

« · Nt , N1 þ N3

for up to 1 week, i.e. its learning process was restricted to the previous 7 days of data. The training data for the network were automatically updated every time the network generated its output. In other words, the memory of the network was limited to a 168-hour history. This training feature was specifically designed for the purpose of potential application of the technique in a real-time environment (usually on a PC), where high speed and less computing space are desired. Having been trained by all known input–output patterns, the network produced a mapping from a set of new inputs to a new set of hourly outputs for the Icebreak model. The outputs (i.e. projected adjustments to errors of original model prediction) from the network were then used to adjust

(8)

where « is the error to be tolerated in future generalization, N t is the number of training examples (168), N 1 and N 3 are numbers of neurons at the input and output layers respectively. For the 3- to 12-hour forecasts with a tolerance error of 0.2, two neurons in the single hidden layer were enough to represent and generalize the desired input–output relations. The number of neurons in the output layer is the number of hours of forecasts. Each neuron in the output layer is associated with an hourly forecast error. For computing efficiency, the network was given a limited ‘memory’

Fig. 2. A numerical ice prediction system (NIPS) with an artificial neural network (NN). Note: X is a meteorological parameter, T is temperature and t is time.

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J. Shao/Expert Systems With Applications 14 (1998) 471–482 Table 2 Brief description of test site and test data Site code

Country

Property a

Test period

BL001 NO001 SM001 SM002 SM003 SM004 SM005 SM006 SM008 SM009 SM010 SM011 SM012 SM014 GM004 GN001 NM002 SMi01 CA001 CA002 CA003 MR001 SA001 SA002 RL002 RL004 RL005 RL006 LZ013 LZ015 LZ018 LZ023 LZ028 LZ030 LZ033 LZ035 WD001 LM001 ST009

Austria Austria Belgium Belgium Belgium Belgium Belgium Belgium Belgium Belgium Belgium Belgium Belgium Belgium Holland Holland Holland Italy Japan Japan Japan Japan Japan Japan Norway Norway Norway Norway Switzerland Switzerland Switzerland Switzerland Switzerland Switzerland Switzerland Switzerland Switzerland UK UK

DA, mountain DA, mountain PA, plain DA, plain DA, plain DA, plain DA, plain DA, plain PA, plain PA, plain DA, plain DA, plain PA, plain PA, plain DA, plain DA, plain DA, plain DA, valley DA, airport DA, airport DA, airport Bridge, near tunnel DA, urban centre DA, urban centre DA, mountain DA, mountain DA, mountain DA, mountain DA, mountain DA, mountain DA, mountain DA, mountain DA, mountain DA, mountain DA, mountain DA, mountain DA, mountain PA, airfield PA, plain

10/02/96–20/04/96 18/01/96–15/04/96 15/12/96–11/01/97 25/12/96–13/01/97 25/12/96–13/01/97 25/12/96–12/01/97 25/12/96–13/01/97 25/12/96–08/01/97 25/12/96–01/01/97 25/12/96–11/01/97 26/12/96–13/01/97 25/12/96–13/01/97 26/12/96–13/01/97 26/12/96–13/01/97 28/02/94–13/03/94 05/03/94–14/03/94 20/02/94–09/03/94 28/01/94–15/03/94 12/10/93–09/01/94 19/10/93–03/11/93 12/10/93–16/12/93 29/10/96–05/05/97 26/01/94–08/02/94 23/01/94–02/02/94 20/03/94–19/04/94 19/03/94–04/04/94 07/04/94–16/04/94 27/03/94–18/04/94 12/11/94–30/01/95 11/11/94–23/01/95 08/11/94–27/01/95 03/11/94–30/01/95 05/11/94–29/01/95 15/11/94–30/01/95 15/11/94–26/01/95 07/11/94–28/01/95 20/12/93–20/01/94 10/11/92–30/04/93 03/11/95–30/12/95

a

DA, dense asphalt; PA, porous asphalt.

original model forecasts [(T(t þ Dt), Dt ¼ 1, 2,…, 12 hours] for the next forecast period. The function of the NN in the ice prediction system is demonstrated schematically in Fig. 2.

4. The test In order to verify the application of the NN technique in road surface temperature forecasts, 39 forecast sites in Austria, Belgium, Holland, Italy, Japan, Norway, Switzerland and the United Kingdom were used for the test (see Table 2). These test sites cover a variety of forecast environments (e.g., flat plains, low lands, high mountains, airport, urban centres, bridge decks, porous asphalt, etc.). These various environments provide a sound platform for the verification of the NN technique.

In the test, historical data of sensor measurements at the sites were collected. The data cover different testing periods, as shown in Table 2. The test period was mainly from late winter to early spring, when model prediction error is usually large due to higher solar elevation and more variable weather conditions. The predictions of road surface temperature with and without the network were compared with sensor measurements of surface temperature at each of the test sites on an hourly basis (except maximum and minimum temperatures, which are on a daily basis). The comparison was carried out for overall temperature (every hour available throughout each testing period), as well as maximum and minimum temperatures. Performance of the model was measured using absolute error (AE), bias and root-mean-square (rms) error. A negative bias means that the forecast is colder than actual,

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Table 3 Comparison of 3-hour temperature forecast with an error back-propagation neural network to those without the network (in parentheses) (8C) Site code

Days

AE

Overall

Maximum

Minimum

Bias

rms

Bias

rms

Bias

rms

BL001 NO001 SM001 SM002 SM003 SM004 SM005 SM006 SM008 SM009 SM010 SM011 SM012 SM014 GM004 GN001 NM002 SMi01 CA001 CA002 CA003 MR001 SA001 SA002 RL002 RL004 RL005 RL006 LZ013 LZ015 LZ018 LZ023 LZ028 LZ030 LZ033 LZ035 WD001 LM001 ST009

61 24 16 10 18 17 18 13 6 14 17 18 17 17 10 8 3 45 38 13 16 179 14 10 20 3 4 9 35 33 37 44 42 28 25 44 25 163 54

1.11 (1.26) 0.98 (1.12) 0.70 (0.91) 0.62 (0.84) 0.56 (0.73) 0.52 (0.73) 0.62 (0.87) 0.49 (0.84) 0.82 (1.12) 0.75 (0.81) 0.60 (0.81) 0.54 (0.80) 0.57 (0.96) 0.61 (0.82) 0.63 (0.92) 0.87 (1.16) 0.76 (1.11) 1.16 (1.22) 1.11 (1.25) 1.21 (1.35) 1.39 (1.70) 1.09 (1.29) 0.70 (0.94) 1.28 (1.67) 1.16 (1.33) 1.12 (1.39) 1.16 (1.12) 1.14 (1.31) 0.99 (1.23) 0.83 (0.93) 0.96 (1.19) 1.03 (1.19) 0.95 (1.21) 0.97 (1.21) 1.01 (1.22) 0.85 (1.06) 0.71 (0.84) 0.91 (1.19) 0.58 (0.69)

0.1 (0.2) 0.1 (0.3) 0.1 (0.2) 0.0 (0.0) 0.0 (¹0.2) 0.1 (0.1) ¹0.1 (¹0.4) ¹0.1 (¹0.6) 0.2 (0.6) 0.0 (¹0.1) 0.0 (¹0.3) 0.0 (¹0.3) ¹0.1 (¹0.4) 0.0 (0.3) ¹0.1 (¹0.2) 0.0 (¹0.3) ¹0.3 (¹0.6) 0.0 (0.0) 0.2 (0.5) 0.1 (0.5) 0.1 (0.4) 0.2 (0.5) 0.1 (0.5) ¹0.2 (0.0) 0.0 (0.1) 0.4 (0.7) 0.3 (0.5) 0.4 (0.7) 0.1 (0.1) 0.1 (0.3) 0.0 (0.2) 0.0 (0.0) 0.0 (0.1) 0.0 (0.1) 0.1 (0.2) 0.1 (0.2) 0.2 (0.4) 0.0 (0.2) 0.0 (0.1)

1.83 (2.06) 1.61 (1.79) 0.99 (1.35) 0.82 (1.11) 0.73 (0.96) 0.69 (0.98) 0.85 (1.13) 0.65 (1.03) 1.08 (1.48) 1.01 (1.32) 0.81 (1.08) 0.71 (1.01) 0.76 (1.28) 0.82 (1.11) 1.03 (1.40) 1.52 (1.73) 1.16 (1.61) 1.97 (2.13) 1.54 (1.81) 1.74 (1.93) 2.13 (2.46) 1.72 (2.00) 0.90 (1.24) 2.13 (2.52) 1.75 (1.97) 1.86 (2.18) 1.84 (1.84) 1.89 (2.19) 1.59 (1.89) 1.23 (1.38) 1.41 (1.69) 1.61 (1.79) 1.49 (1.79) 1.41 (1.68) 1.41 (1.68) 1.24 (1.63) 1.00 (1.26) 1.50 (1.81) 0.82 (0.94)

1.4 (1.8) 0.6 (0.6) 0.2 (0.1) 0.6 (0.0) 0.7 (¹0.1) 0.0 (0.0) 0.5 (¹0.2) 1.0 (0.0) 0.2 (0.0) 0.2 (0.0) 0.0 (¹0.3) 0.4 (¹0.2) 0.3 (0.0) ¹0.1 (0.2) 0.4 (0.2) ¹0.7 (¹1.4) 0.2 (¹0.8) ¹0.1 (¹0.6) 0.8 (0.6) 1.7 (1.8) 2.0 (2.1) 0.0 (0.0) 0.4 (0.7) 0.4 (1.5) ¹0.3 (¹0.5) 1.2 (0.6) 0.6 (0.7) 0.2 (¹0.6) 0.7 (0.4) 0.9 (0.6) 0.9 (0.8) 0.8 (0.5) 0.8 (0.4) 1.0 (0.7) 1.0 (0.8) 1.1 (0.9) 0.5 (0.5) ¹0.3 (¹0.5) 0.4 (0.4)

3.45 (4.12) 1.70 (2.11) 0.42 (0.39) 0.99 (0.95) 0.83 (0.48) 0.52 (0.59) 0.98 (0.62) 1.12 (0.57) 0.55 (0.58) 0.60 (0.72) 0.55 (0.55) 0.68 (0.38) 0.56 (0.44) 0.40 (0.62) 0.90 (1.32) 0.86 (3.19) 0.41 (1.07) 1.75 (2.06) 2.37 (2.99) 3.69 (3.94) 4.32 (4.70) 2.29 (2.56) 0.55 (0.92) 0.57 (1.68) 2.03 (2.52) 1.52 (2.26) 2.19 (2.22) 2.17 (2.48) 1.71 (2.19) 1.71 (1.90) 1.75 (2.20) 2.49 (2.56) 2.08 (2.34) 2.03 (2.35) 1.66 (2.19) 2.00 (2.34) 1.06 (1.21) 1.81 (2.13) 0.94 (1.39)

¹0.3 (¹0.2) ¹0.4 (0.0) ¹0.6 (¹0.4) ¹0.5 (¹0.6) ¹0.4 (¹0.5) ¹0.6 (¹0.4) ¹0.6 (¹0.6) ¹0.5 (¹0.7) ¹0.4 (¹0.3) ¹1.0 (¹1.2) ¹0.6 (¹0.8) ¹0.5 (¹0.6) ¹0.8 (¹1.0) ¹0.7 (¹0.1) ¹0.3 (¹0.2) ¹0.4 (¹0.3) ¹1.2 (¹1.2) ¹0.2 (0.1) ¹0.4 (0.4) ¹0.2 (0.5) ¹0.1 (0.4) ¹0.3 (0.3) ¹0.4 (0.6) ¹0.2 (0.8) ¹0.2 (0.2) ¹0.1 (¹0.2) 0.0 (0.4) ¹0.5 (0.0) ¹0.5 (¹0.3) ¹0.4 (0.0) ¹0.7 (¹0.2) ¹0.6 (¹0.2) ¹0.4 (0.1) ¹0.8 (¹0.2) ¹0.6 (¹0.1) ¹0.5 (¹0.3) ¹0.4 (0.2) ¹0.5 (0.0) ¹0.4 (0.0)

0.72 (1.13) 0.79 (0.75) 0.79 (0.97) 0.64 (0.88) 0.58 (0.84) 0.76 (0.78) 0.71 (1.01) 0.76 (0.97) 0.66 (1.12) 1.37 (1.83) 0.80 (0.99) 0.65 (0.88) 0.99 (1.47) 0.92 (0.61) 0.39 (0.76) 0.54 (1.12) 1.45 (2.06) 0.64 (0.55) 0.63 (0.93) 0.42 (0.73) 0.46 (1.05) 0.74 (0.95) 0.51 (0.89) 0.73 (0.93) 0.73 (0.90) 0.32 (0.78) 0.36 (1.10) 0.59 (0.28) 0.68 (0.67) 0.58 (0.51) 0.87 (0.63) 0.90 (0.53) 0.58 (0.56) 1.02 (0.66) 0.77 (0.57) 0.59 (0.63) 0.79 (0.84) 0.75 (0.79) 0.77 (0.85)

Average Improvement (%)

— —

0.92 (1.12) 17.9

0.1 (0.2) —

1.41 (1.68) 16.1

0.4 (0.2) —

1.76 (2.04) 13.7

¹0.4 (0.0) —

0.73 (0.81) 9.9

and a smaller rms error or AE shows better performance of the model. Because the number of testing days varies from site to site, AE, bias and rms error of model forecasts at individual sites were normalized according to the formula: X ni xi i X , (9) WM ¼ ni i

where WM stands for weighed mean, n i is number of days at site i, and x i is either the bias, rms error or absolute error at the site.

The results of the test are summarized in Tables 3–6 for 3-, 6-, 9- and 12-hour forecasts respectively. In the tables, the error of model prediction with the network is compared with that of model prediction without the network (shown in parentheses in the tables). The second last row in the tables shows mean bias and rms error normalized according to Eq. (9). It is apparent from first glance at Tables 3–6 that the effectiveness of the NN technique varied significantly from site to site, with generally significant improvements using the NN technique. From a summary analysis of the tables, it is seen that: •

Both forecasts with and without the network were little biased. However, the network tended to slightly

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J. Shao/Expert Systems With Applications 14 (1998) 471–482 Table 4 Same as Table 3, but for 6-hour forecast Site code

Days

AE

Overall

Maximum

Minimum

Bias

rms

Bias

rms

Bias

rms

BL001 NO001 SM001 SM002 SM003 SM004 SM005 SM006 SM008 SM009 SM010 SM011 SM012 SM014 GM004 GN001 NM002 SMi01 CA001 CA002 CA003 MR001 SA001 SA002 RL002 RL004 RL005 RL006 LZ013 LZ015 LZ018 LZ023 LZ028 LZ030 LZ033 LZ035 WD001 LM001 ST009

60 24 16 10 18 17 18 13 6 13 18 17 15 17 10 7 3 44 37 13 18 187 14 10 19 3 3 9 34 32 37 43 42 27 25 44 24 159 53

1.16 (1.46) 1.16 (1.48) 0.95 (1.36) 0.76 (1.20) 0.62 (0.95) 0.60 (0.92) 0.71 (1.07) 0.52 (0.97) 1.17 (1.56) 0.80 (1.21) 0.62 (1.06) 0.61 (0.97) 0.70 (1.12) 0.67 (1.08) 0.75 (1.07) 0.94 (1.29) 0.74 (1.24) 1.26 (1.59) 1.60 (2.01) 1.39 (1.88) 1.71 (2.22) 1.35 (1.83) 0.89 (1.40) 1.79 (2.39) 1.28 (1.65) 1.43 (1.71) 1.19 (1.40) 1.25 (1.59) 1.18 (1.54) 0.90 (1.16) 1.10 (1.50) 1.10 (1.39) 1.10 (1.49) 1.13 (1.44) 1.07 (1.38) 1.00 (1.36) 0.89 (1.25) 1.04 (1.48) 0.69 (0.98)

0.0 (0.2) 0.0 (0.3) 0.1 (0.3) 0.0 (0.1) 0.0 (¹0.2) 0.1 (0.2) ¹0.1 (¹0.3) ¹0.1 (¹0.7) 0.4 (0.7) 0.0 (¹0.1) 0.0 (¹0.2) 0.0 (¹0.3) ¹0.1 (¹0.3) 0.1 (0.4) ¹0.1 (0.0) ¹0.2 (¹0.2) ¹0.2 (¹0.7) 0.1 (0.4) 0.4 (0.8) 0.3 (0.9) 0.1 (0.3) 0.3 (0.7) 0.2 (0.7) ¹0.3 (0.0) 0.2 (0.5) 0.5 (0.7) 0.6 (0.8) 0.5 (0.9) 0.1 (0.1) 0.2 (0.2) 0.1 (0.2) ¹0.1 (0.0) 0.1 (0.2) 0.0 (0..0) 0.1 (0.2) 0.1 (0.2) 0.4 (0.7) 0.1 (0.3) 0.0 (0.1)

1.97 (2.35) 1.95 (2.27) 1.32 (1.95) 1.03 (1.60) 0.83 (1.29) 0.81 (1.29) 0.99 (1.48) 0.71 (1.24) 1.65 (2.10) 1.17 (1.73) 0.84 (1.40) 0.82 (1.32) 0.98 (1.53) 0.97 (1.54) 1.27 (1.62) 1.40 (1.84) 1.20 (1.79) 1.86 (2.26) 2.25 (2.79) 1.97 (2.57) 2.51 (3.10) 2.19 (2.71) 1.22 (1.82) 2.59 (3.24) 2.01 (2.45) 2.10 (2.56) 1.73 (2.09) 2.00 (2.48) 1.88 (2.30) 1.35 (1.70) 1.58 (2.10) 1.71 (2.09) 1.71 (2.17) 1.59 (2.00) 1.51 (1.88) 1.47 (1.89) 1.29 (1.78) 1.63 (2.15) 0.98 (1.35)

0.7 (1.0) ¹0.6 (¹0.9) 0.2 (0.1) 0.2 (¹0.3) 0.9 (¹0.1) 0.0 (0.0) 0.6 (¹0.2) 0.9 (0.0) 0.5 (0.1) 0.1 (¹0.2) 0.1 (¹0.3) 0.4 (¹0.2) 0.4 (¹0.1) 0.1 (0.3) 0.0 (0.0) ¹0.8 (¹1.3) ¹0.6 (¹1.7) ¹0.9 (¹1.3) 1.0 (1.0) 1.7 (1.9) 1.3 (1.3) ¹0.1 (¹0.3) 0.5 (0.9) 3.1 (4.5) ¹1.4 (¹1.8) ¹0.1 (0.3) 0.2 (0.5) 0.1 (0.1) 1.0 (0.5) 0.8 (0.8) 0.3 (0.3) 0.5 (0.3) 0.6 (0.3) 0.6 (0.2) 0.7 (0.7) 0.9 (1.0) 0.9 (1.0) ¹0.3 (¹0.4) 0.2 (0.1)

3.38 (4.08) 2.80 (3.58) 0.43 (0.39) 0.71 (1.00) 1.04 (0.48) 0.67 (0.59) 1.16 (0.62) 1.18 (0.57) 0.78 (0.80) 0.42 (0.61) 0.53 (0.55) 0.90 (0.39) 0.61 (0.46) 0.74 (0.73) 1.06 (1.46) 2.14 (3.01) 1.11 (2.27) 2.16 (2.89) 4.19 (5.00) 4.02 (5.08) 5.17 (6.20) 4.13 (5.16) 0.71 (1.13) 3.45 (4.77) 3.59 (4.18) 1.09 (0.76) 0.83 (0.98) 1.65 (2.20) 3.28 (3.91) 2.62 (3.30) 1.78 (2.65) 1.75 (2.31) 2.27 (3.00) 1.86 (2.37) 1.78 (2.39) 1.75 (2.59) 1.54 (1.85) 2.02 (2.28) 1.35 (2.05)

¹0.2 (¹0.2) ¹0.5 (0.0) ¹0.7 (¹0.3) ¹0.6 (¹0.5) ¹0.5 (¹0.5) ¹0.7 (¹0.5) ¹0.6 (¹0.6) ¹0.7 (¹0.9) ¹0.4 (¹0.8) ¹0.8 (¹1.1) ¹0.7 (¹0.8) ¹0.4 (¹0.5) ¹0.9 (¹1.1) ¹0.5 (0.0) ¹0.5 (¹0.4) ¹0.7 (¹0.5) ¹0.6 (¹1.1) ¹0.1 (0.2) 0.0 (0.7) 0.3 (1.2) ¹0.1 (0.1) ¹0.1 (0.5) ¹0.2 (0.7) ¹0.1 (0.7) 0.0 (0.4) ¹0.4 (¹0.4) ¹0.2 (¹0.4) ¹0.1 (¹0.1) ¹0.4 (¹0.4) ¹0.2 (¹0.2) ¹0.5 (¹0.2) ¹0.3 (¹0.1) ¹0.1 (0.2) ¹0.5 (¹0.2) ¹0.3 (0.1) ¹0.6 (¹0.4) ¹0.2 (0.3) ¹0.4 (0.0) ¹0.2 (0.0)

0.90 (1.42) 0.79 (0.84) 0.95 (1.25) 0.71 (1.15) 0.81 (1.15) 0.87 (0.95) 0.80 (1.32) 0.89 (1.33) 0.93 (1.66) 1.22 (2.05) 0.94 (1.41) 0.64 (0.97) 1.05 (1.61) 0.88 (1.01) 0.88 (1.20) 1.17 (1.69) 0.68 (1.71) 0.75 (0.93) 1.13 (1.90) 0.95 (1.76) 1.44 (2.25) 0.97 (1.39) 0.51 (0.96) 0.68 (1.06) 0.84 (1.17) 0.45 (1.00) 0.29 (0.62) 0.54 (0.77) 0.76 (1.20) 0.55 (0.79) 1.03 (1.28) 0.78 (0.88) 0.79 (1.12) 0.84 (0.98) 0.86 (0.97) 0.90 (1.01) 0.76 (1.02) 0.88 (1.31) 0.71 (0.96)

Average Improvement (%)

— —

1.08 (1.47) 26.5

0.1 (0.3) —

1.66 (2.13) 22.1

0.2 (0.0) —

2.37 (2.93) 19.1

¹0.3 (0.0) —

0.87 (1.22) 28.7

•

•

enlarge the bias of the minimum temperature forecast, although it reduced the bias of the overall temperature forecast. Absolute error and rms error of overall, maximum and minimum temperature forecasts were significantly reduced by the network. The reduction of the errors or the improvement of the forecasts was 9.9–29.0%. The improvements became more significant for the forecasts longer than 3 hours and for the minimum temperature forecast. With the aid of the network, the rms error of the minimum temperature forecast was well below 18C in a 3- to 6-hour period, and well below 1.58C for up to 12 hours

•

ahead. The mean absolute error was around 18C for 3and 6-hour forecasts, and around 1.48C for 9- to 12-hour forecasts. Although the forecasts with the network were generally and significantly better than those without the network, there were a few cases when rms error of either maximum or minimum temperature forecast was slightly enlarged by the network (e.g., 3-, 6- and 12-hour maximum temperature forecast at some of the Belgian sites, and 3-hour minimum temperature forecast at some sites in Switzerland). Four examples are chosen to show the effectiveness of the

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Table 5 Same as Table 3, but for 9-hour forecast Site code

Days

AE

Overall

Maximum

Minimum

Bias

rms

Bias

rms

Bias

rms

BL001 NO001 SM001 SM002 SM003 SM004 SM005 SM006 SM008 SM009 SM010 SM011 SM012 SM014 GM004 GN001 NM002 SMi01 CA001 CA002 CA003 MR001 SA001 SA002 RL002 RL004 RL005 RL006 LZ013 LZ015 LZ018 LZ023 LZ028 LZ030 LZ033 LZ035 WD001 LM001 ST009

59 23 16 10 18 17 18 13 6 13 17 17 15 16 10 7 2 44 38 14 18 184 14 10 19 3 3 9 34 32 37 43 42 27 25 44 24 160 53

1.48 (1.92) 1.24 (1.67) 1.18 (1.69) 0.80 (1.36) 0.85 (1.39) 0.83 (1.36) 0.92 (1.39) 0.77 (1.37) 1.49 (2.38) 1.32 (1.89) 0.84 (1.39) 0.80 (1.34) 0.91 (1.62) 0.90 (1.59) 1.15 (1.56) 0.94 (1.35) 1.76 (2.58) 1.48 (1.94) 1.83 (2.40) 2.05 (2.61) 1.92 (2.46) 1.57 (2.13) 1.00 (1.60) 1.99 (2.59) 1.70 (2.20) 1.35 (1.90) 1.20 (1.62) 1.88 (2.49) 1.24 (1.71) 1.00 (1.44) 1.28 (1.72) 1.27 (1.68) 1.21 (1.73) 1.20 (1.77) 1.08 (1.53) 1.17 (1.70) 1.14 (1.68) 1.36 (1.86) 0.79 (1.15)

¹0.1 (0.0) 0.2 (0.5) 0.2 (0.5) 0.1 (0.1) ¹0.3 (¹0.6) 0.1 (0.1) ¹0.2 (¹0.5) ¹0.3 (¹0.9) 0.1 (0.5) 0.0 (¹0.1) ¹0.1 (¹0.3) ¹0.2 (¹0.4) ¹0.2 (¹0.5) 0.2 (0.5) 0.0 (0.0) ¹0.2 (¹0.5) ¹1.3 (¹2.5) 0.1 (0.3) 0.3 (0.8) 0.5 (1.0) 0.1 (0.3) 0.4 (0.9) 0.2 (0.6) ¹0.4 (0.0) 0.3 (0.6) 0.4 (0.7) 0.7 (1.3) 1.2 (1.8) 0.1 (0.1) 0.2 (0.3) 0.0 (0.1) ¹0.2 (¹0.1) 0.1 (0.2) 0.0 (0.0) 0.0 (0.2) 0.1 (0.2) 0.4 (0.7) 0.1 (0.3) 0.1 (0.1)

2.40 (2.86) 1.97 (2.49) 1.78 (2.42) 1.21 (1.87) 1.18 (1.86) 1.13 (1.77) 1.41 (1.92) 1.09 (1.84) 1.95 (2.85) 1.94 (2.67) 1.22 (1.88) 1.16 (1.81) 1.35 (2.16) 1.29 (2.06) 1.89 (2.36) 1.45 (2.01) 2.29 (3.31) 2.36 (2.89) 2.59 (3.28) 2.94 (3.63) 2.94 (3.54) 2.57 (3.19) 1.34 (2.02) 3.18 (3.73) 2.51 (3.12) 2.01 (2.62) 1.90 (2.49) 3.37 (4.03) 1.99 (2.51) 1.47 (1.96) 1.98 (2.45) 1.97 (2.43) 1.86 (2.44) 1.79 (2.44) 1.52 (2.03) 1.71 (2.33) 1.68 (2.37) 2.19 (2.71) 1.13 (1.57)

0.2 (0.1) 0.0 (0.0) 0.3 (0.3) ¹0.3 (¹0.9) ¹0.6 (¹1.4) 0.0 (0.0) ¹0.6 (¹1.2) ¹0.4 (¹1.3) 0.2 (0.7) 0.0 (¹0.3) ¹0.3 (¹0.6) ¹0.4 (¹0.7) ¹0.1 (¹0.5) ¹0.1 (0.3) 0.0 (¹0.4) ¹1.2 (¹1.9) ¹1.3 (¹2.9) ¹1.6 (¹2.1) 0.4 (0.6) 1.6 (1.9) 1.6 (2.0) ¹0.2 (¹0.3) 0.9 (1.3) 2.0 (3.4) ¹1.2 (¹1.3) 0.8 (0.9) ¹0.6 (0.7) 1.5 (2.1) 0.4 (0.2) 0.3 (0.2) 0.4 (0.1) ¹0.1 (¹0.4) 0.1 (¹0.3) 0.1 (¹0.4) 0.8 (0.7) 0.9 (1.0) 0.9 (1.2) ¹0.7 (¹0.8) 0.4 (0.4)

4.40 (4.69) 3.15 (4.11) 1.05 (1.28) 0.85 (1.61) 1.68 (2.42) 0.81 (1.44) 1.98 (2.39) 1.37 (2.02) 1.89 (1.50) 0.82 (1.10) 1.45 (2.13) 1.33 (1.99) 1.60 (2.37) 0.83 (1.67) 3.67 (4.21) 2.71 (3.56) 2.62 (3.83) 4.26 (5.14) 4.24 (5.11) 5.47 (6.48) 6.25 (6.83) 4.47 (5.30) 1.28 (1.99) 2.73 (4.07) 3.61 (4.50) 1.12 (1.99) 1.15 (1.37) 5.24 (6.07) 3.11 (3.83) 2.56 (3.21) 2.01 (2.16) 3.13 (3.62) 2.40 (3.05) 2.83 (3.70) 2.26 (2.85) 2.60 (3.50) 1.66 (2.44) 3.22 (3.77) 1.58 (2.21)

¹0.6 (¹0.6) ¹0.4 (0.0) ¹0.5 (¹0.3) ¹0.4 (¹0.6) ¹1.0 (¹1.1) ¹0.5 (¹0.4) ¹0.4 (¹0.5) ¹0.9 (¹1.2) ¹1.4 (¹1.5) ¹1.3 (¹1.7) ¹0.9 (¹0.9) ¹0.6 (¹0.7) ¹1.0 (¹1.2) ¹0.4 (0.2) ¹0.4 (¹0.3) ¹0.7 (¹0.6) ¹1.9 (¹3.4) ¹0.1 (0.1) ¹0.1 (0.4) 0.4 (1.1) ¹0.3 (¹0.2) 0.0 (0.7) 0.4 (1.1) ¹0.1 (0.7) ¹0.5 (0.1) ¹0.3 (0.1) ¹0.4 (0.1) ¹0.5 (0.0) 0.0 (¹0.5) ¹0.4 (¹0.3) ¹0.6 (¹0.6) ¹0.4 (¹0.1) ¹0.2 (0.2) ¹0.4 (¹0.2) ¹0.2 (0.1) ¹0.6 (¹0.8) ¹0.2 (0.1) ¹0.3 (0.0) ¹0.3 (¹0.1)

1.56 (2.05) 1.00 (1.30) 1.51 (2.07) 1.20 (1.83) 1.28 (1.88) 1.05 (1.52) 0.97 (1.41) 1.14 (1.87) 1.95 (2.93) 2.18 (3.15) 1.33 (1.87) 0.81 (1.38) 1.37 (2.05) 1.02 (1.33) 0.68 (1.04) 1.12 (1.75) 1.91 (3.45) 0.80 (1.14) 1.73 (2.57) 1.84 (2.56) 2.16 (2.88) 1.05 (1.63) 0.86 (1.54) 0.94 (1.44) 1.01 (1.19) 0.94 (1.53) 0.38 (0.59) 0.86 (0.82) 1.33 (1.87) 0.84 (1.12) 1.18 (1.67) 1.02 (1.04) 0.86 (1.19) 0.86 (1.03) 0.94 (1.39) 1.05 (1.47) 0.86 (1.26) 1.00 (1.45) 0.91 (1.15)

Average Improvement (%)

— —

1.30 (1.81) 28.2

0.1 (0.3) —

2.01 (2.59) 22.4

0.0 (¹0.1) —

3.02 (3.71) 18.6

¹0.3 (0.0) —

1.10 (1.55) 29.0

network. The first is an urban site (SA001), where surrounding buildings, traffic and underlying water pipes seriously affect the heat balance of the road surface and its sublayer, causing a largely distorted diurnal curve of road surface temperature. The second is a typical normal (i.e. dense asphalt) site (GM004) on a flat terrain. The third is porous asphalt (SA009) and the last one is a mountain site (LZ015). The four examples are shown in Fig. 3(a)–(d). These figures clearly show how the network was able to make the forecast curve closer to the curve of sensor measurements. Although the network over-adjusted the original forecast curve (i.e. departed further from the actual) at a few times, the overall gain from its adjustment was dominant.

The minimum temperature forecast plays an important role in winter road maintenance. Two examples of minimum temperature forecast are shown in Fig. 4(a)–(d) for site LM001 and in Fig. 5(a)–(d) for site MR001. Both sites are characterized by long and little broken historical data records. Site LM001 is a porous asphalt site. Site MR001 is a bridge site and is regarded as a poor site because it is near a tunnel, its roadside automatic weather station is far away from a road surface sensor and the weather station is sheltered by a nearby high roadside fence. Figs. 4 and 5 show that the minimum temperature forecast with the network in 3-, 6- and 9-hour forecasts was largely in line with the variation of the actual minimum temperature. The 12-hour

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J. Shao/Expert Systems With Applications 14 (1998) 471–482 Table 6 Same as Table 3, but for 12-hour forecast Site code

Days

AE

Overall

Maximum

Minimum

Bias

rms

Bias

rms

Bias

rms

BL001 NO001 SM001 SM002 SM003 SM004 SM005 SM006 SM008 SM009 SM010 SM011 SM012 SM014 GM004 GN001 NM002 SMi01 CA001 CA002 CA003 MR001 SA001 SA002 RL002 RL004 RL005 RL006 LZ013 LZ015 LZ018 LZ023 LZ028 LZ030 LZ033 LZ035 WD001 LM001 ST009

59 23 17 10 18 17 18 13 6 13 18 17 15 17 10 7 2 44 38 14 19 187 14 10 19 3 3 9 34 32 37 43 42 27 25 44 24 159 53

1.60 (2.06) 1.35 (1.90) 1.51 (2.23) 1.11 (1.82) 0.85 (1.43) 0.88 (1.42) 0.89 (1.48) 0.66 (1.34) 1.52 (2.22) 1.29 (1.96) 0.96 (1.58) 0.75 (1.27) 0.80 (1.41) 1.04 (1.66) 0.90 (1.46) 1.17 (1.77) 0.84 (1.19) 1.53 (1.92) 2.23 (2.80) 2.29 (2.86) 2.30 (2.84) 1.74 (2.34) 1.31 (1.90) 2.64 (3.22) 1.48 (2.03) 2.08 (2.75) 1.36 (1.94) 1.43 (1.85) 1.55 (2.00) 1.04 (1.46) 1.38 (1.86) 1.42 (1.88) 1.19 (1.70) 1.32 (1.89) 1.23 (1.66) 1.30 (1.71) 1.22 (1.68) 1.38 (1.97) 0.83 (1.30)

0.0 (0.2) 0.4 (0.8) 0.4 (0.7) 0.3 (0.4) 0.0 (¹0.2) 0.2 (0.2) ¹0.2 (¹0.4) ¹0.2 (0.7) 0.8 (1.2) 0.1 (0.0) 0.0 (¹0.1) ¹0.1 (¹0.3) ¹0.1 (¹0.2) 0.3 (0.8) ¹0.1 (¹0.1) ¹0.1 (¹0.1) ¹0.2 (¹0.5) 0.2 (0.3) 0.6 (1.0) 0.5 (0.9) 0.2 (0.5) 0.6 (1.1) 0.6 (1.2) ¹0.5 (¹0.2) 0.3 (0.7) 1.2 (1.6) 0.6 (0.9) 0.8 (1.3) 0.0 (0.0) 0.2 (0.2) 0.0 (0.0) ¹0.2 (¹0.3) 0.0 (0.1) 0.0 (¹0.2) 0.0 (0.0) 0.1 (0.1) 0.2 (0.4) 0.3 (0.5) 0.2 (0.4)

2.51 (2.98) 2.12 (2.76) 2.21 (3.02) 1.45 (2.36) 1.20 (1.93) 1.25 (1.89) 1.33 (2.07) 0.94 (1.72) 2.22 (3.03) 1.88 (2.68) 1.35 (2.14) 1.08 (1.76) 1.30 (2.00) 1.47 (2.27) 1.45 (2.05) 1.73 (2.38) 1.15 (1.93) 2.16 (2.58) 3.04 (3.68) 3.11 (3.69) 3.18 (3.76) 2.71 (3.34) 1.93 (2.52) 3.78 (4.37) 2.38 (2.92) 3.24 (4.02) 2.03 (2.72) 2.09 (2.60) 2.39 (2.91) 1.49 (2.02) 2.01 (2.57) 2.10 (2.67) 1.82 (2.40) 1.88 (2.56) 1.69 (2.22) 2.02 (2.43) 1.76 (2.27) 2.15 (2.80) 1.22 (1.78)

0.0 (¹0.1) 1.0 (1.2) 0.4 (¹0.1) 0.2 (¹0.3) 0.8 (¹0.1) 0.1 (0.0) 0.5 (¹0.2) 0.9 (0.0) 0.2 (0.2) 0.2 (¹0.2) 0.1 (0.3) 0.4 (¹0.2) 0.3 (¹0.1) 0.4 (0.3) 0.1 (0.0) ¹1.0 (¹1.3) ¹1.9 (¹3.1) ¹1.3 (¹1.5) 0.6 (0.9) 1.8 (2.3) 1.1 (1.3) 0.2 (0.3) 1.3 (1.9) 3.5 (4.5) ¹1.3 (¹1.4) 2.7 (3.5) ¹1.7 (¹1.4) 0.0 (0.5) 0.3 (¹0.1) 1.0 (1.1) 0.1 (¹0.1) ¹0.9 (¹1.4) 0.0 (¹0.3) ¹0.2 (¹0.7) 0.2 (¹0.2) 1.2 (1.2) 0.5 (0.8) ¹0.4 (¹0.5) 0.1 (0.1)

4.98 (5.48) 3.97 (4.92) 1.05 (0.82) 0.52 (0.76) 1.18 (0.48) 0.82 (0.59) 1.33 (0.62) 1.21 (0.57) 0.69 (1.07) 0.85 (0.61) 0.98 (0.55) 1.01 (0.39) 0.70 (0.47) 0.89 (0.78) 0.99 (1.46) 2.33 (3.02) 2.43 (3.65) 2.78 (3.18) 4.29 (5.07) 4.56 (5.44) 5.24 (6.14) 4.95 (5.77) 2.25 (2.84) 3.80 (4.77) 4.16 (4.67) 5.44 (6.25) 1.83 (2.21) 1.63 (2.37) 4.56 (5.38) 2.53 (3.15) 2.36 (3.16) 3.66 (4.57) 2.99 (3.67) 2.33 (3.22) 2.49 (3.24) 3.11 (3.68) 1.77 (2.42) 2.14 (2.42) 1.35 (1.94)

¹0.5 (¹0.4) ¹0.6 (0.0) ¹0.8 (¹0.3) ¹0.8 (¹0.7) ¹0.4 (¹0.6) ¹0.3 (¹0.4) ¹0.6 (¹0.8) ¹0.3 (¹0.9) ¹0.6 (¹0.6) ¹1.0 (¹1.4) ¹1.1 (¹1.2) ¹0.4 (¹0.6) ¹0.5 (¹0.6) ¹0.1 (0.4) ¹0.3 (¹0.2) 0.3 (0.2) ¹0.6 (¹1.1) 0.5 (0.6) 0.2 (0.8) 0.7 (0.9) 0.0 (0.1) 0.3 (1.0) 0.9 (1.5) 0.0 (0.6) ¹0.1 (0.3) ¹0.6 (¹0.6) ¹1.0 (¹0.4) ¹0.1 (0.1) ¹0.6 (¹0.8) ¹0.3 (¹0.5) ¹0.5 (¹0.4) ¹0.5 (¹0.4) ¹0.2 (0.1) ¹0.4 (¹0.3) ¹0.1 (0.1) ¹0.4 (¹0.5) ¹0.7 (¹0.5) ¹0.1 (0.2) 0.0 (0.4)

1.73 (2.11) 1.18 (1.39) 1.89 (2.04) 1.35 (1.95) 1.12 (2.06) 0.95 (1.40) 1.17 (1.91) 0.79 (1.84) 1.61 (2.31) 1.95 (2.93) 1.48 (2.17) 0.93 (1.47) 1.19 (1.98) 1.23 (1.71) 0.90 (1.48) 2.02 (2.25) 0.67 (1.10) 1.37 (1.80) 2.47 (3.00) 2.33 (2.65) 2.96 (3.25) 1.50 (2.18) 1.49 (2.13) 0.67 (1.18) 1.14 (1.66) 0.62 (1.22) 1.10 (0.64) 0.78 (0.92) 1.33 (1.70) 0.95 (1.52) 1.78 (2.00) 1.04 (1.49) 0.80 (1.36) 1.15 (1.71) 0.94 (1.30) 1.20 (1.48) 1.53 (1.92) 1.27 (1.90) 0.84 (1.36)

Average Improvement (%)

— —

1.40 (1.94) 27.8

0.2 (0.4) —

2.10 (2.71) 22.5

0.2 (0.1) —

2.96 (3.47) 14.7

¹0.2 (0.1) —

1.35 (1.87) 27.8

forecast, however, showed a recognizable mismatching to the actual on some days. To gain a picture about error distribution, overall and minimum temperature forecasts were compared with actual in 6 0.58C and 6 1.08C error bands. The results are listed in Table 7. It is seen from the table that, on average, 81.6% of the minimum temperature forecast and 63.8% of the overall temperature forecast were within a 6 1.08C error band in the 3-hour forecast. The percentage remained above 65% and 50% respectively for the forecasts up to 12 hours ahead. The improvement by the network was generally greater as forecast period was prolonged. The study also shows that the accuracy [weighted by Eq.

(9)] of forecast of frost/ice nights (when surface temperature falls at or below 08C) increased from 91.7% to 94.9%, 90.7% to 93.5%, 89.8% to 93.7% and 88.0% to 91.0% for 3-, 6-, 9- and 12-hour forecasts respectively (see Table 8). The error of predicting the time when road surface temperature falls at or below 08C is also compared. Without the network, the predicted time of icing was 38–60 min later than actual in the 3- to 12-hour forecasts. With the network, the error of the prediction was reduced to 1 min earlier (3hour forecast) to 41 min later (12-hour forecast). The improvement of accuracy of the prediction is more significant for a shorter term forecast. The results of the test show that the NN technique pro-

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Table 7 Comparison of weighted frequency distribution (%) of error of overall temperature (T all) and minimum temperature (T min) predictions with and without a neural network in 6 0.58C and 6 18C error bands Forecast period (hours) 3

6

9

12

6 0.58C 1.08C 6 0.58C 6 1.08C

36.1 63.8 49.8 81.6

32.9 58.4 47.8 76.4

29.9 53.8 42.1 70.4

28.1 50.7 42.2 64.8

Without NN T all: 6 0.58C 6 1.08C 6 0.58C T min: 6 1.08C

31.0 55.4 52.3 80.9

25.2 46.6 39.4 63.8

21.6 39.9 32.3 57.6

19.5 37.4 24.9 46.8

Improvement in 6 1.08C error band 8.4 11.8 T all 0.7 12.6 T min

13.9 12.8

13.3 18.0

With NN T all: T min:

q posed in this paper is generally able to improve upon objective forecasts of road surface temperature. At some sites, the improvement is much greater than at other sites. The results shown in the paper demonstrate that an NN technique can be a very helpful and highly valuable supplementary tool to improve the accuracy of road ice forecasts, especially at sites where a numerical model is unable to deliver satisfactory results due to some known or unknown factors and mechanisms (e.g., sheltering and heavy traffic).

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5. Discussion and summary It has been shown that the backpropagation neural network described in this paper is able to reduce both bias and rms error of road surface temperature forecasts. The reduction, however, varies from site to site and from time to time. At some sites, the network successfully and significantly reduced the error of all temperature forecasts. At some other sites, however, it reduced the error of overall and maximum temperature forecasts at the expense of increased error in the prediction of minimum temperature. One such example is given in Fig. 6. It is noticed from the figure that, while the daytime temperature forecast with the network was better than that without the network, its forecast from 1900 to 2400 was worse. Further analysis indicated that, during that period before midnight, the road was treated with salt and there was snowfall. The combination of salting and snowfall was not seen in the network’s previous 7-day learning period and was probably too much for the network to learn. This example of failure shows the limit of the power of the kind of network described in this paper. A closer look at Tables 3–6 reveals that the rms error of maximum and minimum temperature forecasts was

Fig. 3. (a) Comparison of hourly road surface temperature forecast to actual sensor measurement for 3-hour forecast at SA001 (2 February 1994). (b) Same as (a), but for 6-hour forecast at GM004 (4 March 1994). (c) Same as (a), but for 9-hour forecast at ST009 (10 December 1995). (d) Same as (a), but for 12-hour forecast at LZ015 (7 January 1995).

increased by the network only when (though not always) the rms error of the original forecasts was small. This suggests that the network may not be suitable when the original model forecast is good enough (e.g., rms error below or around 0.78C). In general, the inability of the technique to significantly improve temperature forecasts in these cases could be caused by: 1. A non-optimal selection of input features. 2. Learning difficulties of the network in finding a global minimum or deeper minima in its error space. 3. Insufficiency of input–output examples for learning (as shown in Fig. 6).

J. Shao/Expert Systems With Applications 14 (1998) 471–482

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Fig. 4. (a) Comparison of daily minimum temperature forecast with a neural network to the actual for 3-hour forecast at LM001 (10 November 1992–30 April 1993). (b) Same as (a), but for 6-hour forecast. (c) Same as (a), but for 9-hour forecast. (d) Same as (a), but for 12-hour forecast.

The author has noticed that greater improvements can be achieved in some cases by choosing different numbers and format of inputs for the network at individual sites. Therefore, a possible solution for selecting better input features could be an algorithm of optimum input feature selection, as applied by Bankert (1994). As for solving the second

Fig. 5. (a) Comparison of daily minimum temperature forecast with a neural network to the actual for 3-hour forecast at MR001 (27 October 1996–5 May 1997). (b) Same as (a), but for 6-hour forecast. (c) Same as (a), but for 9-hour forecast. (d) Same as (a), but for 12-hour forecast.

problem, although there is neither a general guiding rule nor a guarantee of finding a global minimum, some additional approaches, such as addition of internal neurons, momentum term and noise perturbation, may help to locate deeper minima. As the success of prediction or generalization for a neural network largely depends on the range of domain covered by the learning input–output examples, it is

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Table 8 Comparison of the accuracy of ice state forecast (%) and ice time forecast (min) with and without the neural network for all test sites Forecast period (hours) 3

6

9

12

Ice state forecast With NN 94.9 Without NN 91.7 Improvement 3.2

93.5 90.7 2.8

93.7 89.8 3.9

91.0 88.0 3.0

Ice time forecast With NN ¹1 Without NN 38 Improvement 37

21 53 32

29 55 26

41 60 21

likely that the network fails to make a good prediction if its current input pattern is too far away from its past and learned examples. This problem may be cured by expanding the size of the examples (e.g., by increasing the memory of the NN from 7 days to 2 weeks or 1 month). However, the difficulty we face in reality is that these approaches need to be tested at individual forecast sites. One approach may be better at one site but worse at another. Application of these approaches also requires extra computing power, which may not be easily available in operational forecasts. In summary, the study shows that a three-layer neural network is generally able to significantly improve the accuracy of 3-, 6-, 9- and 12-hour numerical model forecasts of road surface ice by learning from historical data of roadside weather stations, especially at sites with complex environments and unknown underlying mechanisms. The neural network technique can be a useful tool to improve road ice prediction for better winter road maintenance.

Fig. 6. Comparison of 3-hour road surface temperature forecasts with and without the neural network at LZ023 (18 January 1995).

Acknowledgements I wish to thank the staff (especially William D. Fairmaner) of the Vaisala group for providing the data used in this research.

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