# On the Search of Models for Early Cost Estimates of Bridges: An SVM-Based Approach

## Abstract

**:**

_{SVR}2, were as follows: root mean square error: 1.111; correlation coefficient of real-life bridge construction costs and costs predicted by the model: 0.980; and mean absolute percentage error: 10.94%. The research resulted in the development and introduction of an original model capable of providing early estimates of bridge construction costs with satisfactory accuracy.

## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Research Objectives

## 2. Methodology and Concept of a Model

**x**, represent information such as the features, characteristics, and specificity of bridges. The sought-for model was intended to provide multidimensional mapping from the set of cost predictors to the set of values representing total construction costs. Formally, the implicit regression function f, which is supposed to provide the mapping

**x**→ y denoted as:

**x**),

#### 2.1. Support Vector Machines Method in Regression Analysis

**x**, y] ∈ R

^{m}× R } and Φ is a nonlinear transformation used to determine a new feature space H for the inputs: Φ: R

^{m}→ H, Φ(

**x**) ∈ H, y ∈ R, then the function f can be given as follows:

**x**) =

**w**

^{T}Φ(

**x**) + w

_{0}

**x**) is supposed to increase the expressive power of the representation, and the approximation function is computed in the higher dimensional, linear feature space H. Support vectors (sv) are the training data points that lie closest to the hyperplane and thus they affect its optimal location.

**x**),y) = |y − f(

**x**)|

_{ε},

**x**)|

_{ε}= 0 for |y − f(

**x**)| ≤ ε and |y − f(

**x**)|

_{ε}= |y − f(

**x**) | − ε for |y − f(

**x**)| > ε,

**w**ǁ

^{2}+ CΣ(ξ − ξ*) → min,

**w**

^{T}Φ(

**x**) + w

_{0}− y ≤ ε + ξ and y − (

**w**

^{T}Φ(

**x**) + w

_{0}) ≤ ε + ξ* and ξ, ξ* ≥ 0

**x**) = Σ

_{nsv}(α − α*)Φ(x)

^{T}Φ(x′) + w

_{0},

^{T}Φ(x′) is difficult and computationally complex. To simplify the computations, the kernel functions K(x, x′) are introduced instead:

^{T}Φ(x′),

^{2}),

^{d},

**x**) = Σ

_{sv}(α − α*)K(x, x′) + w

_{0},

#### 2.2. Variables of the Model and the Concept of Model Development

_{14}, x

_{22}, and x

_{27}were introduced to represent more than one nominal value that were ARCHED/BOX, COLUMNS/PILES, and k/C/D/E, respectively (see also the footnotes under Table 1). This was done due to the fact that some nominal values were not numerous enough in the dataset to be represented alone by one binary variable. It is important to note that for each of the characteristics listed in Table 1, only one nominal value was allowed, so only one of the binary variables belonging to this characteristic could take value 1. For example, for the type of a structure of which the nominal value was VIADUCT, the values x

_{1}− x

_{3}equaled x

_{1}= 0, x

_{2}= 1, x

_{3}= 0.

**x**and y as used for model development, and p stands for pattern number.

- The choice is made on the basis of the a priori knowledge of the problem and/or users’ expertise;
- Values are selected on the basis of the grid search;
- Determination of the parameters directly from the data;
- Assuming C equal to the range of output values;
- Tuning ε parameter to the training data noise density.

^{p}):

_{y}σ

_{ŷ}),

^{2})

^{0.5},

^{p}= 100%· (|y

^{p}− ŷ

^{p}|)/y

^{p},

_{y}and σ

_{ŷ}standard deviations of real values of the bridges total construction costs and values predicted by a model, respectively; n—cardinality of either L or T subset, y − ŷ—prediction errors, computed after completion of the machine learning process for either L or T subset; and p—pattern index. The SVM machine learning process was made with the use of STATISTICA

^{TM}software suite.

^{p}are considered, the above rule can be reformulated into the expectation about the desired range of APE

^{p}between 0% and +25%/+30%. What is obvious is that the predictions of the bridges’ total construction costs are still required to be provided by the models with errors as small as possible. However, the rule can be used for the purposes of the models’ performance comparison and assessment.

## 3. Results

_{y}|; |E(y) − 3σ

_{y}|},

_{y}= 4.22 computed for y

^{p}belonging to subset L resulted in C = 19.27. After this, it was assumed that 20 will constitute the upper boundary of C. Values of C were sought for with the use of grid search; the values of ε (threshold of the loss function) were also sought for with the use of grid search. The considered ranges of C and ε, as well as the grid search details, are given in Table 3.

_{SVR}) are introduced in Table 4. Characteristics of the models include values of meta-parameters C and ε, number of support vectors (sv), and number of bounded support vectors and values of the constants w

_{0}. The support vectors are the data patterns belonging to subset L that determine the position of the regression hyperplane for a certain model. Furthermore, errors of 10-fold cross-validation are also presented. General error and performance measures RMSE, R, and MAPE for the five BCCPM

_{SVR}models, computed for L and T subsets, are set together in Table 5.

^{p}errors and the rule, (presented in Section 2.2) that refers to the desired range of APE

^{p}values for bridge construction early cost estimates.

^{p}errors of predictions of total bridge construction costs both for L and T subsets under the conditions that APE

^{p}≤ 25% or APE

^{p}≤ 30%. In light of the analysis of the values in Table 6, model BCCPM

_{SVR}2 was proven to perform better than the others—the model reached the highest shares of APE

^{p}≤ 25% for L and T subsets and the same shares of APE

^{p}≤ 30% for L and T subsets as BCCPM

_{SVR}1.

_{SVR}2, the scatter plots of values of y (actual bridge construction costs, presented on the horizontal axes) and ŷ (bridge construction cost predictions by model BCCPM

_{SVR}2, presented on the vertical axes) are depicted in Figure 4 and Figure 5. The former shows the scatter plot of y and ŷ values for subset L, the latter for subset T. The charts include also the cones of errors ±25% and ±30%.

^{p}errors of bridge construction cost predictions provided by the model BCCPM

_{SVR}2 (both for L and T subsets) divided into intervals of a range equal to 5%. Additionally, distributions (cumulated shares) of APE

^{p}errors are given in the Table.

^{p}; ŷ

^{p}) in the scatter plots (in Figure 4 and Figure 5) is even along the line of a perfect fit. Moreover, for both of the subsets L and T, the vast majority of bridge construction cost predictions are located within the ±25% cone of errors; almost all of the predictions are located within the ±30% cone of errors.

^{p}, (in Table 6), as complementary information, confirm that most of the bridge construction cost predictions made by BCCPM

_{SVR}2 meet the condition of early cost estimates.

## 4. Discussion

_{SVR}2, and its measures are presented in Section 3. The predictions of the bridge construction costs provided by the model can also be analyzed in a way that focuses on selected characteristics and features of bridges as the model’s input.

^{p}, computed for the machine learning subset, belonging to certain intervals (compare Table 7) with regard to variables of a nominal type (coded as binary values for machine learning). The relative percentage shares of APE

^{p}for variables of nominal type were computed as follows:

- For each of the variables x
_{j}for j = 1 − 8 or j = 12 − 27, the number of predictions that fulfilled the condition of having corresponding APE^{p}that fell into the certain interval were counted and divided by the number of occurrences of x_{j}= 1.

^{p}, computed for the machine learning subset, belonging to certain intervals (compare Table 6) with regard to variables of a numerical type.

^{p}for these variables were computed as follows: for each of the variables x

_{j}for j = 9 − 11:

- Predictions for variables values that fulfilled the conditions of falling into certain range of values and having corresponding APE
^{p}from a certain error’s interval were counted and divided by the number of occurrences.

## 5. Conclusions

- RMSE: 1.058 and 1.111;
- Pearson’s correlation coefficient R of real-life bridge construction costs and costs predicted by the model: 0.974 and 0.980;
- MAPE: 13.85% and 10.94%.

## Funding

## Acknowledgments

^{TM}software suite.

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the investigated support vector machine (SVM)-based regression models.

Characteristic | Nominal Values | Coding | Symbol |
---|---|---|---|

Type of a structure | BRIDGE | binary | x_{1} |

VIADUCT | binary | x_{2} | |

WHARF | binary | x_{3} | |

Type of a bridge | ROAD BRIDGE | binary | x_{4} |

RAIL BRIDGE | binary | x_{5} | |

ANIMAL BRIDGE | binary | x_{6} | |

Type of a project | BUILD | binary | x_{7} |

DESIGN&BUILD | binary | x_{8} | |

Total length | LENGTH [m] | numerical | x_{9} |

Width of a structure | WIDTH [m] | numerical | x_{10} |

Number of spans | SPANS | numerical | x_{11} |

Structural solution | BEAM | binary | x_{12} |

FRAME | binary | x_{13} | |

ARCHED/BOX | binary | x_{14} | |

Material solution | REINFORCED CONCRETE | binary | x_{15} |

PRESTRESSED CONCRETE | binary | x_{16} | |

STEEL | binary | x_{17} | |

Bridgehead supports | SOLID-WALLED | binary | x_{18} |

COLUMNS | binary | x_{19} | |

Intermediate supports | NONE | binary | x_{20} |

SOLID-WALLED | binary | x_{21} | |

COLUMNS/PILES | binary | x_{22} | |

Supports’ foundations | SHALLOW | binary | x_{23} |

DEEP | binary | x_{24} | |

Load class * | A | binary | x_{25} |

B | binary | x_{26} | |

k/C/D/E ^{1} | binary | x_{27} |

^{1}k for rail bridges or C, D, E for other bridges; * according to standards applied in Poland.

p | 10 | 77 | 83 | 104 | 109 | 111 | 119 | 150 | 166 |
---|---|---|---|---|---|---|---|---|---|

x_{1} | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |

x_{2} | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 |

x_{3} | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 |

x_{4} | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |

x_{5} | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |

x_{6} | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |

x_{7} | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |

x_{8} | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |

x_{9} | 0.069 | 0.151 | 0.150 | 0.219 | 0.197 | 0.180 | 0.456 | 0.095 | 0.715 |

x_{10} | 0.114 | 0.057 | 0.027 | 0.114 | 0.114 | 0.092 | 0.097 | 0.426 | 0.049 |

x_{11} | 0.000 | 0.000 | 0.143 | 0.143 | 0.143 | 0.000 | 0.286 | 0.071 | 0.500 |

x_{12} | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |

x_{13} | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |

x_{14} | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |

x_{15} | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |

x_{16} | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |

x_{17} | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |

x_{18} | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

x_{19} | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

x_{20} | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |

x_{21} | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |

x_{22} | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

x_{23} | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

x_{24} | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |

x_{25} | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |

x_{26} | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |

x_{27} | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |

y [PLN] | 3.02 | 5.49 | 6.11 | 9.84 | 10.82 | 12.54 | 14.15 | 6.83 | 19.85 |

y [EUR] ^{1} | 0.70 | 1.28 | 1.42 | 2.29 | 2.52 | 2.92 | 3.29 | 1.59 | 4.62 |

^{1}training and testing of the model was done with the use of costs given in millions of PLN.

Parameter | Lower Boundary | Step | Upper Boundary |
---|---|---|---|

C | 5 | 1 | 20 |

ε | 0.05 | 0.05 | 0.20 |

Model | C | ε | sv | Bounded sv | w_{0} | Cross-Validation Error |
---|---|---|---|---|---|---|

BCCPM_{SVR}1 | 7 | 0.050 | 91 | 50 | −0.108761 | 0.038 |

BCCPM_{SVR}2 | 8 | 0.050 | 85 | 47 | −0.118497 | 0.037 |

BCCPM_{SVR}3 | 8 | 0.100 | 59 | 24 | −0.132814 | 0.037 |

BCCPM_{SVR}4 | 9 | 0.100 | 58 | 23 | −0.137849 | 0.036 |

BCCPM_{SVR}5 | 10 | 0.100 | 56 | 22 | −0.130312 | 0.035 |

Model | RMSE_{L} | RMSE_{T} | R_{L} | R_{T} | MAPE_{L} | MAPE_{T} |
---|---|---|---|---|---|---|

BCCPM_{SVR}1 | 1.115 | 1.112 | 0.971 | 0.979 | 14.64% | 11.33% |

BCCPM_{SVR}2 | 1.058 | 1.111 | 0.974 | 0.980 | 13.85% | 10.94% |

BCCPM_{SVR}3 | 1.175 | 1.141 | 0.968 | 0.978 | 17.03% | 11.44% |

BCCPM_{SVR}4 | 1.139 | 1.152 | 0.970 | 0.978 | 16.69% | 11.28% |

BCCPM_{SVR}5 | 1.115 | 1.161 | 0.971 | 0.978 | 16.56% | 11.30% |

**Table 6.**Comparison of absolute percentage error for p-th case (APE

^{p}) errors for the five selected models.

Subset L | Subset T | |||
---|---|---|---|---|

Model | APE^{p} ≤ 25% | APE^{p} ≤ 30% | APE^{p} ≤ 25% | APE^{p} ≤ 30% |

BCCPM_{SVR}1 | 85.38% | 92.31% | 81.08% | 91.89% |

BCCPM_{SVR}2 | 86.92% | 92.31% | 83.78% | 91.89% |

BCCPM_{SVR}3 | 72.31% | 80.00% | 81.08% | 89.19% |

BCCPM_{SVR}4 | 73.08% | 80.77% | 81.08% | 89.19% |

BCCPM_{SVR}5 | 73.85% | 82.31% | 83.78% | 89.19% |

Subset | Subset | ||||
---|---|---|---|---|---|

Share | L | T | Distribution | L | T |

APE^{p} ≤ 5% | 20.77% | 27.03% | APE^{p} ≤ 5% | 20.77% | 27.03% |

5% < APE^{p} ≤ 10% | 20.77% | 37.84% | APE^{p} ≤ 10% | 41.54% | 64.86% |

10% < APE^{p} ≤ 15% | 24.62% | 10.81% | APE^{p} ≤ 15% | 66.15% | 75.68% |

15% < APE^{p} ≤ 20% | 10.77% | 5.41% | APE^{p} ≤ 20% | 76.92% | 81.08% |

20% < APE^{p} ≤ 25% | 10.00% | 2.70% | APE^{p} ≤ 25% | 86.92% | 83.78% |

25% < APE^{p} ≤ 30% | 5.38% | 8.11% | APE^{p} ≤ 30% | 92.31% | 91.89% |

APE^{p} > 30% | 7.69% | 8.11% | APE^{p} > 30% | 100.00% | 100.00% |

**Table 8.**APE

^{p}predictions’ errors for machine learning with regard to the type of bridge its structure and type of a project.

Relative Percentage Share of APE^{p} | |||||||
---|---|---|---|---|---|---|---|

0–5% | 5–10% | 10–15% | 15–20% | 20–25% | 25–30% | >30% | |

BRIDGE (x_{1}) | 35.71% | 14.29% | 14.29% | 7.14% | 7.14% | 14.29% | 7.14% |

VIADUCT (x_{2}) | 15.31% | 21.43% | 28.57% | 12.24% | 11.22% | 3.06% | 8.16% |

WHARF (x_{3}) | 60.00% | 40.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |

ROAD BRIDGE (x_{4}) | 17.65% | 11.76% | 26.47% | 8.82% | 14.71% | 5.88% | 14.71% |

RAIL BRIDGE (x_{5}) | 18.39% | 25.29% | 25.29% | 10.34% | 9.20% | 5.75% | 5.75% |

ANIMAL BRIDGE (x_{6}) | 60.00% | 10.00% | 10.00% | 20.00% | 0.00% | 0.00% | 0.00% |

BUILD (x_{7}) | 15.65% | 22.61% | 26.96% | 11.30% | 10.43% | 6.09% | 6.96% |

DESIGN&BUILD (x_{8}) | 62.50% | 6.25% | 6.25% | 6.25% | 6.25% | 0.00% | 12.50% |

**Table 9.**APE

^{p}predictions’ errors for machine learning with regard to the structural and material solutions.

Relative Percentage Share of APE^{p} | |||||||
---|---|---|---|---|---|---|---|

0–5% | 5–10% | 10–15% | 15–20% | 20–25% | 25–30% | >30% | |

BEAM (x_{12}) | 19.64% | 20.54% | 26.79% | 9.82% | 8.93% | 5.36% | 8.93% |

FRAME (x_{13}) | 15.31% | 21.43% | 28.57% | 12.24% | 11.22% | 3.06% | 8.16% |

ARCHED/BOX (x_{14}) | 54.55% | 9.09% | 9.09% | 9.09% | 9.09% | 9.09% | 0.00% |

REINFORCED CONCRETE (x_{15}) | 18.33% | 15.00% | 26.67% | 8.33% | 15.00% | 5.00% | 11.67% |

PRESTRESSED CONCRETE (x_{16}) | 20.83% | 29.17% | 25.00% | 8.33% | 6.25% | 6.25% | 4.17% |

STEEL (x_{17}) | 30.43% | 17.39% | 17.39% | 21.74% | 4.35% | 4.35% | 4.35% |

**Table 10.**APE

^{p}predictions’ errors for machine learning with regard to the types of bridgehead and intermediate supports and supports’ foundations.

Relative Percentage Share of APE^{p} | |||||||
---|---|---|---|---|---|---|---|

0–5% | 5–10% | 10–15% | 15–20% | 20–25% | 25–30% | >30% | |

SOLLID-WALLED (x_{18}) | 21.77% | 20.16% | 23.39% | 10.48% | 10.48% | 5.65% | 8.06% |

COLUMNS (x_{19}) | 14.29% | 28.57% | 42.86% | 14.29% | 0.00% | 0.00% | 0.00% |

NONE (x_{20}) | 11.11% | 14.29% | 34.92% | 12.70% | 11.11% | 6.35% | 9.52% |

SOLLID-WALLED (x_{21}) | 36.36% | 18.18% | 22.73% | 9.09% | 9.09% | 0.00% | 4.55% |

COLUMNS/PILES (x_{22}) | 28.26% | 30.43% | 10.87% | 8.70% | 8.70% | 6.52% | 6.52% |

SHALLOW (x_{23}) | 12.68% | 22.54% | 30.99% | 7.04% | 14.08% | 5.63% | 7.04% |

DEEP (x_{24}) | 31.67% | 18.33% | 16.67% | 15.00% | 5.00% | 5.00% | 8.33% |

Relative Percentage Share of APE^{p} | |||||||
---|---|---|---|---|---|---|---|

0–5% | 5–10% | 10–15% | 15–20% | 20–25% | 25–30% | >30% | |

A (x_{25}) | 24.71% | 22.35% | 21.18% | 11.76% | 8.24% | 5.88% | 5.88% |

B (x_{26}) | 0.00% | 36.36% | 45.45% | 9.09% | 9.09% | 0.00% | 0.00% |

k/C/D/E ^{1} (x_{27}) | 20.00% | 11.43% | 25.71% | 8.57% | 14.29% | 5.71% | 14.29% |

^{1}(compare with Table 1).

**Table 12.**APE

^{p}predictions’ errors for machine learning with regard to the total length of bridge (x

_{9}).

LENGTH (x_{9}) | Relative Percentage Share of APE^{p} | ||||||
---|---|---|---|---|---|---|---|

0–5% | 5–10% | 10–15% | 15–20% | 20–25% | 25–30% | >30% | |

up to 25 m | 0.00% | 16.67% | 36.67% | 6.67% | 16.67% | 6.67% | 16.67% |

25–50 m | 14.63% | 19.51% | 26.83% | 21.95% | 9.76% | 2.44% | 4.88% |

50–75 m | 18.18% | 22.73% | 31.82% | 4.55% | 13.64% | 9.09% | 0.00% |

75–100 m | 36.36% | 31.82% | 13.64% | 4.55% | 4.55% | 0.00% | 9.09% |

more than 100 m | 45.45% | 9.09% | 0.00% | 4.55% | 0.00% | 9.09% | 4.55% |

**Table 13.**APE

^{p}predictions’ errors for machine learning with regard to the width of bridge (x

_{10}).

WIDTH (x_{10}) | Relative Percentage Share of APE^{p} | ||||||
---|---|---|---|---|---|---|---|

0–5% | 5–10% | 10–15% | 15–20% | 20–25% | 25–30% | >30% | |

up to 11 m | 11.76% | 29.41% | 35.29% | 5.88% | 11.76% | 0.00% | 5.88% |

11–14 m | 15.87% | 12.70% | 26.98% | 9.52% | 11.11% | 9.52% | 14.29% |

14–17 m | 29.73% | 29.73% | 21.62% | 13.51% | 5.41% | 0.00% | 0.00% |

17–20 m | 5.41% | 5.41% | 2.70% | 5.41% | 2.70% | 2.70% | 0.00% |

more than 20 m | 8.11% | 2.70% | 0.00% | 0.00% | 2.70% | 0.00% | 0.00% |

**Table 14.**APE

^{p}predictions’ errors for machine learning with regard to the of number of spans (x

_{11}).

NUMBER OF SPANS (x _{11}) | Relative Percentage Share of APE^{p} | ||||||
---|---|---|---|---|---|---|---|

0–5% | 5–10% | 10–15% | 15–20% | 20–25% | 25–30% | >30% | |

1 | 9.09% | 13.64% | 33.33% | 15.15% | 13.64% | 6.06% | 9.09% |

2 | 15.00% | 40.00% | 10.00% | 10.00% | 15.00% | 5.00% | 5.00% |

3 | 34.48% | 31.03% | 20.69% | 3.45% | 3.45% | 0.00% | 6.90% |

4 | 3.45% | 0.00% | 6.90% | 3.45% | 0.00% | 6.90% | 0.00% |

5 and more | 27.59% | 3.45% | 0.00% | 0.00% | 0.00% | 0.00% | 3.45% |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Juszczyk, M.
On the Search of Models for Early Cost Estimates of Bridges: An SVM-Based Approach. *Buildings* **2020**, *10*, 2.
https://doi.org/10.3390/buildings10010002

**AMA Style**

Juszczyk M.
On the Search of Models for Early Cost Estimates of Bridges: An SVM-Based Approach. *Buildings*. 2020; 10(1):2.
https://doi.org/10.3390/buildings10010002

**Chicago/Turabian Style**

Juszczyk, Michał.
2020. "On the Search of Models for Early Cost Estimates of Bridges: An SVM-Based Approach" *Buildings* 10, no. 1: 2.
https://doi.org/10.3390/buildings10010002