Selection of global climate models for India using cluster analysis

Assessment and analysis of maximum and minimum temperatures have become important due to their increasing impact on climate. Surface temperature has a significant effect on most of the parameters of the hydrological cycle. However, climate impact assessment studies based on temperature are few when compared to precipitation over India (Kothawale et al. 2010; Sonali & Nagesh Kumar 2013). Global climate models (GCMs), which are simulation tools for the Earth's climate, are gaining importance due to their capability to predict future climate variables that will be useful to develop long, medium and short term water resources planning strategies. This raises the question of choosing suitable GCMs from the large number available, making the problem more complex. The problem becomes aggravated as most of the GCMs are similar, i.e.: (a) some GCMs developed by various agencies share part of the same code by way of exploring similar numerical schemes, parameterisations, and resolutions, and some are even developed by the same agency; (b) some institutions provide simulations from more than one variation of the same GCM; (c) employing many GCMs results in similar forecasts due to sharing of similar oceanic and atmospheric components (Pennell & Reichler 2011). In the present study clusters and groups are used synonymously.

In this regard, cluster analysis is becoming prominent due to its capability of reducing the number of GCMs, eliminating those of a similar nature, to a manageable homogeneous subset. Complimentarily, cluster validation methods have proved to be efficient and can be explored as the basis to determine the optimal number of clusters of GCMs (Wang et al. 2009). Thirty-six GCMs are grouped homogeneously by using K-means cluster analysis. Skill score is the indicator employed for evaluation. Cluster validation methods, namely the Davies–Bouldin Index (DBI) and F-statistic, are applied to determine an optimal number of clusters of GCMs. Outputs from the F-statistic and DBI are used for formulating an ensemble of GCMs, or in other words, a group of GCMs. The following sections present a literature review, objectives of the study, K-means cluster analysis, cluster validation methods, performance indicator, data collection, results, discussion and conclusions.

A brief but relevant literature review related to the evaluation and clustering of GCMs is presented below.

Cai et al. (2009) assessed the performance of 17 GCMs for simulation of precipitation and temperature using a skill score for the periods of 1931–1960 and 1961–1990. No single GCM was found to be advantageous in predicting precipitation or temperature for the whole world, even though some GCMs performed better for particular regions. Chen et al. (2012) evaluated the ability of nine GCMs of the Coupled Model Intercomparison Project 5 (CMIP5) category to simulate the near surface wind over China. Spatial fields of wind speed at the end of the 21st century are almost similar to those of the last 35 years. Gómez-Navarro et al. (2012) studied the ranking of an ensemble for predicting the regional climate for Spain. They discussed the weighted ensemble average quality with reference to the uncertainties in the observational dataset.

Fu et al. (2013) evaluated the performance of 25 GCMs using the multicriterion method at the regional scale for air temperature, monthly mean sea level pressure and rainfall, and annual rainfall over south eastern Australia for 1960–1961 to 1999–2000. They concluded that simulated monthly rainfall and monthly mean sea level pressures are not satisfactory compared to monthly temperature. Su et al. (2013) assessed the performance of 24 GCMs of the CMIP5 category for the Eastern Tibetan Plateau by comparing the GCM outputs with field data for the period 1961–2005. They concluded that most of the GCMs satisfactorily assessed the spatial variations and climatological patterns of temperature. Tiwari et al. (2014) analysed the skill of five GCMs in predicting precipitation in winter over Northern India for the period 1982–2009. The climatology, correlation coefficients, and inter annual standard deviation were computed and compared with the observed values. It was noticed that the GCMs were able to replicate the climatology and inter annual standard deviation to varying degrees, whereas prediction skill is too low. Barfus & Bernhofer (2015) assessed the capabilities of 18 GCMs using indices such as Vertical Totals, Total Totals, Cross Totals, Severe WEAther Threat (SWEAT), K, and Showalter to the case study of the Arabian Peninsula. No overall best performing GCM was identified. Grose et al. (2014) evaluated the 27 GCMs' performance for the Western tropical Pacific, and their differences from the Coupled Model Intercomparison Project 3 (CMIP3) were estimated. The CMIP5-based GCMs showed some improvements over CMIP3 in the late 20th century. Perez et al. (2014) evaluated the skill of GCMs from CMIP3 and CMIP5 databases for the North-East Atlantic Ocean region. The skill of GCMs to replicate the synoptic situations, the historical inter-annual variability, and the consistency of GCM experiments for 21st century projections were considered.

Hidalgo & Alfaro (2015) evaluated 107 climate runs from 48 GCMs for Central America. A research basis to ascertain the performance of the GCMs is suggested. Similar studies were reported by Palazzi et al. (2015) for Karakoram-Himalaya for the precipitation rate with 32 GCMs and Kumar et al. (2015) for 22 regions across the world for wind extremes with 15 GCMs.

Pennell & Reichler (2011) mentioned the limitations about GCMs as ‘the limitations of these models tend to be fairly similar, contributing to the well-known problem of common model biases and possibly to an unrealistically small spread in the outcomes of model predictions’. They evaluated 24 GCMs for their capability to simulate 20th century climate. The effective ensemble size of GCMs was observed to be less than the actual number. They also used hierarchical clustering to assess the similarities among different GCMs and concluded that the present approach of interpreting multi-model ensembles may induce over prediction of climate. Yokoi et al. (2011) applied cluster analysis to 43 model performance metrics, which were evaluated using 22 GCMs. They employed K-means and Ward's method and found that both methods provided a similar dynamical and physical result pattern. Yuan & Wood (2012) studied issues such as overconfidence of the ensemble of GCMs and the necessity of an effective sub set of GCMs that can maintain skill and predictability. They stressed the necessity for clustering of models, and applied hierarchical cluster analysis for grouping 12 seasonal forecast models at different lead times for surface air temperature and precipitation. It was concluded that cluster analysis was found to be useful for grouping climate models. Similar studies were reported by Knutti (2008), Knutti et al. (2013), Reifen & Toumi (2009), Eden et al. (2012), and Jurya (2015).

It is observed from the aforementioned literature that no significant work was conducted in the clustering of GCMs in association with cluster validation methods for Indian climate conditions. The formulated objectives are as follows:

1. Identification of a performance indicator for evaluating GCMs.

2. Examining the suitability of K-Means cluster analysis for grouping 36 GCMs for maximum temperature (MAXT), minimum temperature (MINT) and a combination of maximum and minimum temperature (COMBT).

3. Examining the suitability of DBI and F-statistic to achieve optimal clusters of GCMs.

4. Suggesting ensembles of GCMs for MAXT, MINT, and COMBT.

K-means is an iterative algorithm used for clustering GCMs. The algorithmic steps consist of: (a) assignment of each GCM initially to any one group (g) and obtaining the average of the assigned group(s); (b) identification of the objective function; (c) identification of termination criteria for stopping the algorithm, which provides the basis for choosing final groups and corresponding GCMs; and (d) computation of total squared error, as the error in cluster analysis significantly represents the deviation of individual data set from the group average ⁠. Computation of total squared error value for cluster K is done using where is error value (Jain & Dubes 1988; Raju & Nagesh Kumar 2014a). K-means cluster analysis can be tried a good number of times with possible minimum and maximum groups for effective appraisal of the problem.

For clustering algorithms, the number of groups is an important input. However, it is difficult to assess optimal clusters manually for a set of GCMs. In this regard, cluster validation methods inherently indicate the quality of the chosen group. Brief descriptions of the DBI and F-statistic are presented below.

Thirty-six GCMs belonging to CMIP5 presented in Table 1 provide both monthly maximum and minimum temperatures and are chosen for evaluation. It may be noted that these 36 GCMs were chosen based on data availability from among the CMIP5 models. CMIP5 has selected these models based on their performance amongst all the existing GCMs. In addition, more than one GCM from the same institution is explored to observe the similarities/differences between them in the classification perspective, instead of only one model from the same institution (Knutti 2008). Accordingly, outputs from GCMs are compared with the Climate Research Unit 2.1 datasets for the model evaluation study, and the period considered is 1961–1999. Forty grid points of 2.5⁰ × 2.5⁰ covering India are considered.

A skill score of 40 grid points corresponding to each GCM is computed using MATLAB (Equation (3)). A data matrix of 36 GCMs and skill scores for each grid point are developed for three variables, i.e. MAXT, MINT, COMBT, and are used as the basis for K-means cluster analysis. The Cluster Validity Analysis Platform (CVAP) developed by Wang (2007) is used for K-means cluster analysis and performed for 2–9 clusters and even beyond. CVAP is run for 2–30 clusters (five times for each cluster) to assess the occupancy of GCMs in each cluster, totaling 145 runs. It is observed that as the number of clusters is increased beyond nine, many clusters are not occupied. Therefore, the analysis is restricted to nine clusters. For each chosen group, the DBI and F-statistic is evaluated (Equations (1) and (2)) and used as the basis for determining the optimal cluster. The following sections present the results relating to MAXT, MINT, and COMBT.

Studies on three variables, namely, MAXT, MINT and COMBT, using the skill score as indicator, show the existence of similarity for most of the GCMs (Figures 1, 5 and 6). It is observed from Figure 4 that the F-statistic values decrease with an increase in the number of clusters for MAXT, MINT and COMBT. F-statistic values are almost the same for cluster 7 and very nominal difference for clusters 6 and 8 is observed for MAXT, MINT and COMBT, mainly due to the similar ratio values (Equation (2)). It is observed that HadCM3 is the common GCM in the three ensembles of MAXT, MINT, and COMBT. Numerous authors suggested the necessity for an ensemble of GCMs (or more than one GCM) (Mujumdar & Nagesh Kumar 2012; Raju & Nagesh Kumar 2014b; Hidalgo & Alfaro 2015; Palazzi et al. 2015; Raju et al. 2016). In this context, heterogeneous GCMs obtained through cluster analysis can be used as support for formulating the ensemble, which in our opinion is innovative as compared to an ensemble of GCMs of homogeneous nature. In addition, the F-statistic and DBI values are found to be complimentary to each other and provide an opportunity to find the optimum ensemble size with more confidence, as more GCMs than required burden the process whereas a smaller ensemble size may not serve the intended purpose.

K-means cluster analysis along with F-statistic and DBI is applied to a case study of India. It is observed that the optimum cluster is 2 for both MINT and COMBT whereas it is 3 in the case of MAXT on the basis of the F-statistic test and DBI. Ensembles of {HadCM3, IPSL-CM5A-LR, GFDL-ESM2M}, {ACCESS1.3, HadCM3} and {MPI-ESM-MR, HadCM3} are suggested for MAXT, MINT and COMBT, respectively.

In our opinion, this is the first application of K-means cluster analysis and optimal clustering of GCMs, and it can be used as the basis for choosing an ensemble of GCMs for Indian climate conditions.

It is relevant to note that the inferences emanating from the present paper depend on the chosen grid points as well as on the chosen clustering and validation method. However, the thrust of the present paper is on a methodology that can be applied and replicated with ease.

Future work is targeted to employ fuzzy cluster analysis to handle impreciseness in the skill score values and skewness in the data. There is a possibility that fuzzy cluster analysis may give better results, but it needs to be explored in detail.

This work is supported by the Council of Scientific and Industrial Research, New Delhi, through project no. 23(0023)/12/EMR-II dated 15.10.2012. The authors are grateful to Dr Sonali P, Indian Institute of Science, Bangalore, for providing the skill scores of different GCMs. The authors acknowledge the modelling groups for making their simulations available for analysis, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) for collecting and archiving the CMIP5 model output, and the WCRP's Working Group on Coupled Modelling (WGCM) for organizing the model data analysis activity. The authors also thank the Indian Meteorological Department (IMD) for providing the gridded temperature dataset.