Where t1 and t2 are the averages of the time series of climate variables in the first and second time slice respectively, and St1 is the standard deviation of the interannual variability from the first time slice (t1) for climate variable k .
Changes in probability of local climate extremes
Extreme value theory provides a statistical framework for making informed assessments regarding the likelihood of exceptionally rare or extreme events. It aims to predict the probability of such events, by measuring the deviations from the central tendency in the frequency distribution of the events. Building upon the foundations of extreme value theory, specifically the Generalized Extreme Value (GEV) distribution, we can analyze a sample of extreme events and derive the parameters that best characterize the underlying probability distribution of these extremes. The GEV has three parameters: shape, scale, and location (Gaines & Denny, 1993; Katz et al., 2005). The shape parameter delineates three possible distributions: i) light-tailed (Gumbel), heavy-tailed (Frerchet), and iii) bounded (Weubull) (see Katz et al., 2005). The location parameter specifies where the distribution is “centred”, while the scale parameter quantifies its “spread”. The climetrics R package leverages the statistical method presented by Katz et al., (2005) to assess extremes. In this approach, the probability of extreme events is defined based on climate variables and is calculated as percentiles (e.g., the 95th and 5th percentiles of the distributions of the temperature and precipitation, respectively), for each grid cell within the baseline period (t1). Subsequently, the percentiles of the future distributions in the second time period (t2), corresponding to the extreme values within the baseline, are computed. These percentiles representing the probability of exceeding historical extreme values.
To quantify the probability of extreme events for any two variables at each grid cell, the algorithm combines the two probabilities by summing them and then subtracting the product of the two probabilities to prevent double-counting. The probability of the second time period (t2 ) is then subtracted from the probability of the first time period (t1 ). Positive values indicate an increased probability of extremes in the second time period while negative values suggest a decrease. These calculations provide insights into specific aspects of extreme climate events (e.g., hot and dry spells). The same methodology can be applied to incorporate other factors like wind speed.
The formula for calculating the probability of exceeding a threshold (x) in given by:
\begin{equation} \mathbf{\text{P\ }}\left(\mathbf{A\ \cap B}\right)\mathbf{=P(A)\times P(B)}\nonumber \\ \end{equation}
P (A ∩ B) is the probability of the joint occurrence of both events (A&B), which calculates by multiplying the individual probabilities of each event: P(A) × P(B).
Changes in areas of analogous climates:
The algorithm measures different aspects of risk arising from climate change by quantifying changes in the spatial distribution of future climate conditions relative to those in the recent past. To this end, it begins by using as input a map of climate zones for each time period. Such climate zones can be provided by users. Else, the climetricsR package generates them using the updated version of Köppen-Geiger climate classification (Peel et al., 2007) for both the baseline (t1) and the second time period (t2) using temperature (minimum, mean, and maximum) and precipitation. The Köppen-Geiger climate classification relies on annual temperature and precipitation, which are subjected to a sufficiently large time or ensemble averaging. The climetrics R package employs the method developed by Ohlemüller et al., (2006) to quantify the percentage of changes in the areas of similar classes between the baseline (t1) and the second time period (t2). In this context, positive values indicate expansions or gain in these areas, whereas negative values indicate signify contraction or losses. This analysis provides valuable insights into the shifting patterns of analogous climate regions over time.
\begin{equation} \Delta C_{\text{ij}}\ =\ \sqrt{\left(C_{t2j}-C_{t1i}\right)^{2}}\nonumber \\ \end{equation}
Ct1i = Climate Zones int1Ct2j = Climate Zones int2ΔCij = Changes in areas of Analogues climates
Novel climates:
Novel climates are characterized by environmental conditions that lack analogues in the recorded past (Saxon et al., 2005; Ackerly et al., 2010). Within the climetrics R package, the method proposed by Williams et al., (2007) is employed to quantify dissimilarities between the baseline (t1) and either future orpast time slices (t2) using the Standardized Euclidean Distance (SED) as described in the “Standardized Local Anomalies” section. The algorithm measures the SED for each grid cell between the first (t1) and second time periods (t2). To identify a novel climate at each cell, it assesses the climate realization in the second time slice (t2) for the grid cell against climate realizations from the baseline (t1) across all grid cells, ultimately retaining the minimum SED (SEDmin). SEDmin represents the upper limit of local climate change indices. Since the pool of potential climatic analogues is global, a high value of SEDmin indicates that the climate conditions second time period (t2) have no close analogues anywhere in the baseline (t1). In essence, The larger the SEDmin score, the greater dissimilarity of the future climate relative to the global pool of potential climate analogues. This metric effectively highlights the presence of novel and unprecedented climate conditions.
Change in distance to analogous climates:
The climetrics R package implemented the method developed by Ohlemüller et al., (2006) to quantify similarities between climate zones. For a grid cell i in the baseline (t1), the algorithm calculates the geographic distance to all other cellsj that belong to the same climate classification as grid celli. This computation employs the great-circle distance (Zar, 1989), also known as orthodromic distance, which measures the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere. Then for each cell i, the algorithm computes the median of these great-circle distances below the 10th percentile of the distribution of all such distances for both the baseline (t1) and the second time periods (t2). Subsequently, it maps the changes in these distances over time. To illustrate the changes in distances between the baseline (t1) and the second time period (t2), with regard to a specific dimension of climate change, the algorithm calculates the difference in distances between these two time points (Δkm = kmt2 – kmt1). A negative value indicates a temporal decrease in distance over time, signifying a closer similarity between the climate zones. Conversely, a positive value indicates an increase in distance, suggesting greater dissimilarity between the climate zones over time.
Climate change velocity
Climate change velocity is a measure of the local rate of movement or displacement of climatic conditions across Earth’s surface (Loarie et al., 2009; Sandel et al., 2011). The climetrics R package employs three distinct algorithms to assess climate change velocity. The first algorithm (“dVe”) represents a distanced-based velocity measurement developed by Sandel et al., (2011). It calculates the changing velocity in terms of geographical distance per unit of time (km/year). The algorithm, firstly, calculates the temporal gradient by measuring the local difference between the baseline (t1) and the second time period (t2). Secondly, it calculates spatial gradients by taking the slope of the specific climate parameter for the baseline (t1) using a 3x3 grid-cell neighbourhood.
\begin{equation} dVe\ (km\ /year)=\frac{\text{Temporal}}{\text{Spatial}}=\frac{\left(t1-t2\ \right)\text{yea}r^{-1}}{\left(Slope\ 3*3\right)km^{-1}}\nonumber \\ \end{equation}
t1 = Climate variable for the baseline
t2 = Climate variable for the second time period (future/past)
The second algorithm for quantifying velocity (“ve”) inclimetrics , is implemented based on the method and code developed by Hamman et al., (2015). This algorithm accounts for the fact that no two grid cells poses identical climate values. Instead, it relies on a user-defined threshold to identify a climate match for the climate surface of the baseline in the climate surface representing the second period (t2 ). Then, it calculates the geographic distance of all matching t2 climate cells to the baseline cell and finds the shortest geographic distance. The distance is then divided by the number of years between t1and t2 , providing the measure of velocity (see Hamann et al., 2015).
The third algorithm of climate change velocity (“gVe”) adopts a gradient-based methodology inspired by the work of Burrows et al. (2011). This approach focuses on quantifying velocity by dividing the long-term trend of climate variables by the spatial gradient along the same direction. For an angle, θ, with 0° as North and 180° as South, the velocity of climate change along those angles, Vθ, is given by:
\begin{equation} V\theta=\frac{\text{Temporal\ trend}}{S_{\text{NS}}\ cos\theta\ \ +S_{\text{EW}}\text{\ \ sinθ}}\nonumber \\ \end{equation}
where SNS is the North-South spatial gradient andSEW is the East-West spatial gradient. When θ is perpendicular to the angle of the velocity of climate change, the velocity of climate change in that direction is infinite, since the denominator in equation 8 becomes zero.