# Mixed nonlinear regression for modelling historical temperatures in Central–Southern Italy

## Authors

## Abstract

This paper has exploited, for Central and Southern Italy (Mediterranean Sub-regional Area), an unprecedented historical dataset as an attempt to model seasonal (winter and summer) air temperatures in pre-instrumental time (back to 1500). Combining information derived from proxy–documentary data and large-scale simulation, a statistical downscaling approach in the form of mixed regression model was developed to adapt larger-scale estimations (regional component) to the sub-regional temperature pattern (local component). It interprets local temperature anomalies by means of monthly based Temperature Anomaly Scaled Index in the range −5 (very cold conditions in June) to 2 (very warm conditions). The modelled response agrees well with the independent data from the validation sample (Nash–Sutcliffe efficiency coefficient, >0.60). The advantage of the approach is not merely increased accuracy in estimation. Rather, it relies on the ability to extract (and exploit) the right information to replicate coherent temperature series in historical times.

## 1 Introduction

Modelling can be described as an art because it involves experience and intuition as well as the development of a set of mathematical skills.

Mark Mulligan and John Wainwright (eds.), 2004.

Environmental Modelling, Wiley, Chichester, p. 8.

The Mediterranean is one of the few regions in the world holding a large volume of weather documentary proxies for the past 500–1,000 years (Camuffo and Enzi 1992; Jones et al. 2009; Rodrigo et al. 2011). However, such large amounts of documents and archives have not yet been fully explored to reproduce with high spatio-temporal resolution the different climates of the Mediterranean (García-Herrera et al. 2007). Determining the climatic history in these unrepresented places of the world is a challenging and complex issue.

In the recent decades, considerable progress has been made in pre-instrumental temperature modelling at both hemispheric and regional scales (e.g. Mitchell and Jones 2005; Rutherford et al. 2005). Luterbacher et al. (2004) and Xoplaki et al. (2005) were able to map seasonally resolved temperature reconstructions across European land areas back to 1500. In particular, Luterbacher et al. (2004) developed separate multiple regression equations between each principal component (PC) of the instrumental data and all leading PC of the proxy records. In this way, they assimilated proxy records into reconstructions of the underlying spatial patterns of past climate changes. In hemispheric, continental and regional reconstructions, proxy coverage (e.g. tree-ring width data) is often irregular and heterogeneous (Esper et al. 2002). Thus, temperature and precipitation reconstructions may become poorly accurate at sub-regional and local scales, or over particular periods (Mann et al. 2008; Mann 2007; Ogilvie and Jonsson 2001; Diodato et al. 2008). On the other hand, high-resolution climate information is increasingly needed for the study of past, present and future climate changes (Vrac et al. 2007). Then, the issues of sub-regional reconstructions and downscaling of large-scale (e.g. the regional scale) climate data to smaller spatial and temporal scales should be made a priority in order to achieve a better understanding of sub-regional climates (Riedwyl et al. 2009). Documentary proxies’ investigation remains a reliable approach to trace back the temperature extremes before the advent of instrumental recording of meteorological data (e.g. Brázdil et al. 2005; Brewer et al. 2007; Jones et al. 2009). However, to the best of the authors’ knowledge, temperature series have not been modelled for the Mediterranean areas so far. Moreover, for this region, continuous and homogeneous instrumental series cannot be extended before the nineteenth century (Camuffo et al. 2010).

Several authors such as Luterbacher and Xoplaki (2003), Pauling et al. (2003) and Ge et al. (2005) suggested that pre-modern instrumental weather indices might be promising to enrich climate reconstructions. Different sets of proxy variables have indeed been used to find relationships for high-resolution climate reconstructions (e.g. Briffa et al. 2002; Oberhuber and Kofler 2002; Larocque and Smith 2005; Moberg et al. 2005, 2009; Diodato 2007; Davi et al. 2008; Garnier et al. 2011). Many of these reconstructions depend on empirical relationships between proxy records and climate data, such as linear regression-based algorithms and neural networks (Helama et al. 2009). These relationships are seldom based on a training process capable to capture all the possible data combinations that occur when extrapolation is performed (i.e. reconstruction period) and can engender bias in the estimates if not employed with care (Robertson et al. 1999; Moberg et al. 2005; von Storch et al. 2005). Observing and documenting changes in the phenologies of various plant species may support efforts to reconstruct past climates but challenges still remain with respect to spanning scales of observation, integrating observations across taxa and modelling phenological sequences (Morisette et al. 2009). With reference to dendroclimatological studies, correlation between tree-ring proxies and temperature data was found to only explain about 50 % of the variance (Liang et al. 2008; Helama et al. 2009; Tan et al. 2009). Documentary data series are expected to better correlate with temperature, the overall explained variance being of about 70 % (Leijonhufvud et al. 2008; Dobrovolný et al. 2010).

We have approached the statistical modelling of temperature variability, based on both documentary records and previous large-scale reconstructions. A documentary-based technique was developed based on multiscale temperature regression (MTR) model at sub-regional level. An area covering Central and Southern Italy, and named in this paper Mediterranean Sub-regional Area (MSA), is the focus of the investigation. The selected sub-region, centrally located in the Mediterranean region, is an interesting test area rich in proxy–documentary data and modern weather records useful to improve the spatial resolution of past climate reconstructions. The goal was to produce a relatively simplified MTR model has a multiscale spatial structure because it combines documentary proxy-based local-scale weather anomalies with large-scale temperature data to adapt regional temperature data to specific sites and seasons. In a previous paper (Diodato and Bellocchi 2011), the structural form of the model was only partially described for winter season and its basic assumptions not discussed in detail. In this paper, we provide a general formulation of the model for both winter and summer seasons and the rationale behind the notion that legitimates its structure. The next section describes the geographical environment, the datasets and the developed methods. Section 3 illustrates the novel mixed-model approach in detail and evaluates its results. Conclusions (Section 4) point out the main findings and look at horizons for future research.

## 2 Environmental setting, data and methods

### 2.1 Study area and datasets

Regional temperature data (*T*_{R}) were derived from Luterbacher et al. (2004) for Europe over 1500–2002. The data, upscaled at about 0.5° grid resolution (∼50 km) from historical instrumental series and multi-proxy data (http://www.ncdc.noaa.gov/cgi-bin/paleo/eurotemp.pl), covers an area extending from 25° West to 40° East and from 35° to 70° North (Fig. 1a). It is possible to observe the lack of temperature records and documentary evidence (used for regional reconstruction of temperatures) over Southern Europe (including the MSA), as suggested by both data density (Fig. 1a) and correlation pattern (Fig. 1b).

In order to fill this deficiency in the data available, a new documentary dataset was derived from chronicles found in two main sources, *Moio and Susanna Manuscript* (Ferrari 1977) and *Corradi’s Annals* (Corradi 1972). It is in *Moio and Susanna Manuscript* (Ferrari 1977) that a continuous temperature series is supplied for Catanzaro (38° 54′ North, 16° 36′ East) from 1461 to 1768. More recent weather information is available from different sources at sparse sites, which are suitable for reconstruction of primitive series (Camuffo et al. 2010). The Italian scientist Alfonso Corradi (1833–1892) collected the historical documents from 5 to 1850 ac, related to meteorology and epidemics (Corradi 1972). More recently, the historian Umberto Ferrari published the chronicles of Giovanni Battista Moio and Gregorio Susanna quoting climate extremes, famines from 1710 to 1769 and weather information over the sixteenth and seventeenth centuries for the Calabria region in Southern Italy (Ferrari 1977). A data bank (Catalogue EVA-Environmental Events of the ENEA-Italian National Agency of for New Technologies, Energy and the Environment, Clemente and Margottini 1991) was also referred to and used when necessary.

### 2.2 Monthly temperature anomaly scaled index

Information held in the written documentary sources was extracted to derive temperature related indices. As a general rule (after Pfister 1999, 2001; Brázdil et al. 2005), a 7-point scale is employed, ranging from −3 for ‘extreme coldness’ to +3 for ‘extreme hotness’, with 0 indicating ‘normal’ conditions. However, this ordinal scale bears the limitation of concise discrimination across the full range of extremes since all events above a certain magnitude are likely assigned to the same extreme class (Glaser and Riemann 2009). To obtain a more realistic degree of variability (and therefore higher differentiating power), we used an asymmetric index to ensure a more accurate detection of extreme anomalies. Examples of such events include those recorded during the Little Ice Age (e.g. when rivers froze over). Temporal shifts between proxy and actual anomalies in different seasons of the year were also taken into account. In fact, as an example, a river freezing on March or April is a more negative anomaly than a frozen river on January.

Monthly scaled index for decoding temperature anomalies from proxy-documentary data

Month | Temperature anomaly classes | |||||
---|---|---|---|---|---|---|

Freezing | Very cold | Cold | Normal | Warm | Vary warm | |

Dec | −4 | −2 | −1 | 0 | 1 | 2 |

Jan | −4 | −2 | −1 | 0 | 1 | 2 |

Feb | −4 | −2 | −1 | 0 | 1 | 2 |

Jun | N/A | −4 | −2 | 0 | 1 | 2 |

Jul | N/A | −5 | −2 | 0 | 1 | 2 |

Aug | N/A | −5 | −2 | 0 | 1 | 2 |

Temperature anomalies reconstruction for a selected number of years

Year | Monthly TASI | ΣTASI | Source | ||||||
---|---|---|---|---|---|---|---|---|---|

Dec | Jan | Feb | Jun | Jul | Aug | Winter | Summer | ||

1752 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | A and M |

1753 | 1 | 0 | 0 | −2 | 0 | 0 | 1 | −2 | A and M |

1754 | 0 | −1 | 0 | 0 | 0 | −1 | −1 | −1 | A and M |

1755 | 0 | −3 | −1 | −2 | −2 | −1 | −4 | −5 | A and M |

1756 | 0 | 0 | −1 | 0 | 0 | 0 | −1 | 0 | A and M |

1757 | 0 | 0 | −1 | −2 | −2 | 0 | −1 | −4 | A, M and EVA |

### 2.3 Model parameterisation and evaluation

The split-samples approach was used to segregate the available temperature data into a calibration set and a validation set. Particular attention was paid to the calibration procedure in order to provide at reliable model for time series reconstruction. Two distinct climate periods (1867–1903 and 1972–2002) were included in the calibration dataset (68 years in total) for two main reasons. The first was to ensure model calibration accuracy and stability by accounting for both cold and warm intervals, and the second to ensure that the model was able to simulate air temperature on periods with either accurate (as in recent times) or inaccurate data (as in historical times). The validation dataset contained instrumental temperature reconstruction for the MSA (as performed by Camuffo et al. 2010). In particular, the periods 1742–1754 and 1792–1818 were selected for model validation, because measured data are available for these two intervals in Central–Southern Italy.

The entire workflow was executed using a spreadsheet for data collection, model development and graphical assembling, with the support of STATGRAPHICS online statistical package (http://www.statgraphics.com) and Statistics Library-R modules (Wessa 2009) for statistics performance (e.g. autocorrelation) and graphical outputs (e.g. confidence intervals of regression lines), respectively. The agreement between estimates and observations was evaluated using a set of statistics, including the modelling efficiency by Nash and Sutcliffe (1970), ranging from negative infinity to positive unity (the latter being the optimum value). In order to have a visual inspection of the quality of results, a set of comparative scatterplots and histograms is also presented.

## 3 Modelling of sub-regional winter temperatures

In this study, regional temperatures (*case*) from Luterbacher et al. (2004) are the basis for modelling sub-regional temperatures (*response*). In this situation, it is possible to have more than one response for each case, and a central problem is to find a function that combines several responses to arrive at realistic estimates for the temperature of the sub-region of interest. Multiscale predictors are generally needed to model temperatures over different spatial and temporal scales (after Bates and Watts 2007). Based on this understanding, the information collected (regional temperature data) was downscaled to reasonably approximate the behaviour of the disturbance terms (or stimulus variables) driving the temperature measurements at sub-regional scale. These approximations lie on the general assumption that sub-regional air temperature depends on two disturbance terms: synoptic regional forcing and local weather conditions. The regional scale can drive the general temperature trend while area-specific temperatures are met by local conditions. Both weather variables and climate indices were used as predictors, which are the basis for the multiscale regression model.

### 3.1 Inferences for multiscale temperature estimation

_{R}) and sub-regional ((.)

_{S}) scales. To extend the procedure for extrapolations outside the range represented by the calibration sample, the equation was iteratively rearranged towards a robust solution whereby two additive components (nonlinear, regional; linear, local) are used:

The first term, *y*(*T*_{MTR}), is the seasonal mean temperature output (in degree Celsius) of the MTR model. *T*_{R} is the regional component of temperature (in degree Celsius), supplied as a boundary condition source for the sub-regional scale model. The part in brackets is the sub-regional component of temperature (in degree Celsius), supplied as a local constraint.

*Y*= model estimates and

*X*= actual data, according to the following criteria:

*a*), the second is to approximate the unit slope (

*b*) of the straight line that would minimise the bias and the third is to maximise the goodness-of-fit (

*R*

^{2}) of the linear function. A knowledge-driven approach to bias correction was used to estimate the parameters of Eq. 1 using an iterative procedure (after Box et al. 1978). For instance, after a first run, it was found that regional temperatures (

*T*

_{R}) led to increasingly biased and imprecise estimates over historical times (purely lieanr solution). Likewise, earliest regional inferences in Mann et al. (2008) tended to be associated with decreased performance. To account for this noninvariance over the historical time scale and to rebalance internally the quality of calibration, a power law was assigned to

*T*

_{R}with the exponent forced to be lower than one (and finally set equal to 0.5). Such iterative fitting of the data allowed for correcting most of the bias initially observed and capturing the full range of sub-regional scale variability.

The scale parameter *k* (°C^{0.5}) was initially set equal to one and, for reasons of parsimony as given in Grace (2004), not treated as a free parameter because the initial value resulted in a fit that satisfied the criteria outlined above (Eq. 2). *T*_{R} appears in both the square root (power of 0.5) and linear terms. In the first case, it returns a direct, nonlinear effect while in the brackets it crosses the sub-regional anomalies identified by the TASI to correct the bias observed in the historical times. The square root of *T*_{R} and parameter *β* are mainly to define the order of magnitude of the process used to downscale the MTR model to the sub-regional scale. These are empirical terms that represent a trade-off between parsimony and plausibility in the specific situation depicted by Eq. 1. The other two terms into the brackets are seasonally varying (index *S*) shift parameters (in degree Celsius) of *T*_{R}. They force the model with meteorological (sum of monthly values of the Temperature Anomaly Scaled Index (ΣTASI_{S}) defined above) and climatological (*Ω*_{S}; for winter (*Ω*_{w}) and summer (*Ω*_{s})) boundary conditions.

### 3.2 Model parameterisation and evaluation

*β*= 0.268,

*Ω*

_{w}= 11.0 °C,

*Ω*

_{s}= 43.5 °C. Using the estimated parameter values, the nonlinear response to

*T*

_{R}is depicted in Fig. 3, as translated into Eq. 1 for different values of ΣTASI

_{S}.

In the temperature series supplied by Luterbacher et al. (2004), standard deviation (SD) for winter increases in more recent years, i.e. after the LIA (SD = 0.96 °C against 0.74 °C for 1739–1783). This contrasts with the instrumental observations, for instance those performed by Domenico Cirillo in the eighteenth century (SD = 1.1 °C) and documented by the Meteorological Diaries of the Royal Society of London for the Kingdom of Naples (Derham 1733). The reconstructed series based on Eq. 1 gives SD∼1.0 °C for both recent and historical times. For summertime, SD∼0.6 °C was registered for 1783 in the regional dataset, also approached by the reconstructed series.

_{1}, b

_{1})) in both seasons.

Performance and autocorrelation statistics for MTR model (Eq. 1) at the calibration and validation stages

Scale of the estimation | Dataset | Performance statistics | Autocorrelation statistics | |||
---|---|---|---|---|---|---|

Nash–Sutcliffe efficiency coefficient | Correlation coefficient | Mean absolute error (°C) | Lag-1 residual correlation | Durbin–Watson statistic | ||

Sub-regional (Eq. 1) | Calibration | |||||

Winter | 0.88 | 0.94 | 0.24 | 0.27 | 1.45 ( | |

Summer | 0.87 | 0.88 | 0.24 | 0.04 | 1.83 ( | |

Validation | ||||||

Winter | 0.66 | 0.82 | 0.33 | 0.19 | 1.59 ( | |

Summer | 0.63 | 0.74 | 0.24 | 0.03 | 1.92 ( | |

Regional (Luterbacher et al. 2004) | Validation | |||||

Winter | −0.43 | 0.26 | – | – | – | |

Summer | −0.30 | 0.50 | – | – | – |

*d*statistic was calculated to verify the presence of autocorrelation in the residuals

*e*(the index

*t*indicating the

*t*th year):

Two critical values, *d*_{L, α} and *d*_{U, α}, vary depending on the level of significance (*p*), the number of observations and the number of predictors in the regression function. In the calibration dataset, indication of possible correlation is produced at 0.01 < *p* < 0.05 significance level for the winter season only (Table 3). The existence of the autocorrelation can be understood as the result of a functional misspecification (e.g. Green 2003). This aspect is similar to the multicollinearity problem in linear regression, usually dealt with separately from autocorrelation, but also examined by its autocorrelation effect in the error term (e.g. Ramsey et al. 2001). In our case, autocorrelation may be due to some internal constraints at the calibration stage, probably related to the fact that winter temperatures in the regional dataset and model outputs are more similar in recent times (the period of years used for calibration) than it was in historical times. The calibration dataset is from recent times (covering periods around the twentieth century), when estimates from Luterbacher et al. (2004) better approach observed temperatures. Under such conditions, the model likely represents some redundancy in the explanatory variables that means, other predictors than the regional temperature component might not be effective in improving upon the sub-regional estimates. However, both calibration results in summer and the results of validation in both seasons assume statistical independence of the residuals, with type-I error probability of 0.09 and 0.36 of Durbin–Watson test statistic (Table 3).

The mean absolute error (0.24–0.33 °C), similar between calibration and validation and between seasons, and the other statistics of Table 3 indicate for the validation set a satisfactory performance.

### 3.3 Limitations of the study

The scope of our modelling approach and model parameterisation was restricted to capturing the temporal variability of seasonal temperature data in the study area, and some limitations of the methodology should be acknowledged.

Knowledge of the historical climate in the MSA and the modelling background of the authors represented the basis of this study. Indeed, there may be other ways to assess temperature series at sub-regional scale. For instance, structural equation modelling accounts for correlations among predictors and can estimate indirect effects of predictors on other predictor variables that taken together affect the outcome (Hair et al. 1998). Other solutions were indeed explored, but this model had the best fit, was the most parsimonious and was consistent with the climatology of the study area. To determine an appropriate balance between computation, complexity and uncertainty, we have relied on ad hoc model development and trial-and-error assignment of model parameter values via spreadsheet utility. The use of spreadsheet solver to minimise the square error of estimation is indeed a common solution, as in previous papers (e.g. Diodato and Bellocchi 2007, 2010), although algorithmic improvements are available (e.g. Menascé 2008).

Parameter estimation achieved in more steps makes confidence bounds for model parameters not easily quantifiable. The model error (mismatch between the observed and the modelled values) is however an indication of total model uncertainty (e.g. Shrestha and Solomatine 2008), and Nash–Sutcliffe efficiency values of 0.6 can discriminate between bad and good performances (e.g. Lim et al. 2006). The efficiency values obtained in the validation stage (>0.8) thus indicate limited model uncertainty (likely associated with narrow parameter uncertainty). Since the results of model calibration were satisfactory, the robustness of the solution was relied on and sensitivity analysis was not added to the study. The reconstruction of temperatures series has thus used generic optimised parameters, which are crude estimates over multiple years. This ensures a generic representation for the MSA, with evidence of improved performance compared with previous estimates. Since geographical locations have characteristics that require specific model structures and local optimisation, then the application of the model to other sub-regions may be limited by the ability to provide representative drivers and parameter values.

## 4 Conclusions

This paper addresses two main issues: it generates a composite of documentary and proxy sources relating to temperatures in Central and Southern Italy, and it devises a new statistical model relating this proxy series and the Luterbacher et al. (2004) European temperature reconstructions to the temperature of the study area. The main novelty of this paper is the introduction of a relatively simple model to reconstruct past seasonal (winter and summer) temperature variability at sub-regional scale, based on documentary data and existing reconstructions. In general, the use of data deriving from diverse spotted sources is not straightforward to reconstruct climate in Southern Europe. Data used in the previous seasonal temperature reconstruction over Europe, especially over the Mediterranean areas, are from few and early instrumental series (data before 1850) that, for their nature, are difficult to find, evaluate, correct and convert or present in a Celsius scale in terms of temperature anomalies. With the extension to summertime, as carried out in this study, we have gone beyond a previous application of the model (Diodato and Bellocchi 2011) and confirmed that the use of proxy–documentary data was appropriate to correct regional estimates for the effect of microclimatic conditions (especially evident in summer season).

The multiscale regression approached here corrects the loss of variance inherent in both early instrumental records and univariate least-squares calibration equations. In general, multiscale, process-based climate models can be accurate. However, the authors argue that improvements in model sophistication may not be as profitable as the ability to reconstruct confidently the overall picture of temperature-related events (and therefore temperature data) over historical times and in different geographical places. Validation, from this point of view, is a major statistical instrument to develop a reliable model to add robustness to past temperature reconstructions. Furthermore, in this paper, we took advantage of the MTR model versatility to evaluate, through proxy–documentary data, how the sub-regional temperatures signal is driven by local and boundary conditions. The accuracy of these signals depends not only on the intrinsic properties of the model itself. It also depends on the possibility to recover homogeneous documentary records able to maintain unchanged the climate information and to replicate, through the model application, the actual temperature series. The new proxies and the model offer us the possibility to make statements about the historical climate of Central and Southern Italy (and its relationship to the wider regional climate) that are unprecedented in their power, detail, and confidence. However, bringing it up and discussing it here would need to accommodate too many aims and objectives in only one paper. We have already applied the model to the analysis of climate signals in historical times, i.e. the winters of the years 1675 to 1715 coinciding with a period of reduced solar activity (along with high volcanic activity) and known as ‘Maunder minimum’ (Diodato and Bellocchi 2011). The modelling work can be extended to other historical periods, and this could be the subject of a future paper. The approach is also potentially suitable for applications elsewhere in the Mediterranean basin, provided that model parameters will be documented for other sub-regions than the one investigated here. Further research extending the modelling approach developed here towards other sub-regions of the Mediterranean area would provide additional insight into the implications for the production of valuable knowledge from proxy–documentary data and can be considered the natural evolution of this study.