## 3 Advancements and Alternatives to Modal-Based Damage Identification
### Advancements to Environmental & Operational Challenges
#### 3.1.1 Regression Models  
To accurately estimate temperature influence, it must be recorded with the intention to utilise both its spatial and temporal evolution. Bridges subjected to large daily temperature variations will exhibit greater spatial thermal gradients, while larger bridges will take longer to respond to temperature fluctuations and will thus have greater temporal thermal gradients.  
While spatial thermal gradients can be determined with additional thermocouples as shown by [PERSON] et al. (2017), temporal thermal gradients require the implementation of a regression model, such as least squares, that fits a linear equation to the variable relationship. Multi-linear regression models provide a suitable option for most bridge types and environments, as shift changes in frequency due to sub-zero temperatures can be modelled. Generally, a regression error value of circa 5% is used to reduce the effect of erroneous data, as seen in ([PERSON] et al., 2017). [PERSON] et al. (2018) employed a multivariate linear regression method called the Least Trimmed Square (LTS) estimator, which is fashioned upon the popular least squares approach, but incorporates an initial screening procedure called a Concentration step (C-step) ([PERSON] et al., 2017). The C-step is an iterative process that finds the minimum determinate of a number of data subsets, with the aim of identifying the sub-scatter of highest density that is most representative of the data. This allows for erroneous data points or outliers to be removed before the linear regression model is fitted. In this way, the LTS is less sensitive to local variations and dubious data. [PERSON] et al. (2010) also employed the same technique to increase their regression model's robustness when used in conjunction with Single Value Decomposition (SVD) for damage detection. Their subsequent comparison with standard linear regression techniques showed that the inclusion of a screening stage such as the C-step improves the accuracy of the model and subsequent damage detection.  
More complex regression models may incorporate a polynomial curved fit as per [PERSON] and [PERSON] (2010), whose polynomial regression model exhibited an excellent capability in regard to mapping daily averaged output-only modal frequency variability using daily averaged temperature fluctuations in a long-span suspension bridge. The authors used daily averaged values to reduce the effect of random modal variations that arise from the modal identification algorithm being subjected to non-stationary loading. Their success could also be partly attributed to their utilisation of the Iterative Windowed Curve-fitting Method (IWCM), instead of the standard peak peaking method for modal frequency identification. The IWCM is a frequency domain curve fitting technique that overcomes spectral bias errors that occur in standard frequency domain analyses, caused by the attenuation of values around spectral peaks. It achieves this by applying the same window function used on the time domain data to the resulting frequency domain data, as per [PERSON] (2010).