The global warming causes changes to the Earth’s climate, or long-term weather patterns that vary from place to place. While we think of “Global warming” and “Climate change” as synonyms, scientists use the term climate change when describing the complex shifts affecting our planet’s weather and climate systems in different parts, because some areas actually get cooler in the short term, while the others become warmer.

Climate change encompasses not only rising average temperatures but also extreme weather events, shifting wildlife populations and habitats, rising seas and a range of other impacts. All of those changes are emerging as humans continue to add heat-trapping greenhouse gases to the atmosphere, changing the rhythms of climate that all living things have come to rely on. It has become clear that humans have caused most of the past century’s warming by releasing heat-trapping gases called “greenhouse gases”. Their levels are higher now than at any time in the last 800,000 years and, as a result, glaciers are melting, sea levels are rising and cloud forests are dying.

The warming that happens when certain gases in Earth’s atmosphere trap heat is considered as the greenhouse effect. These gases let in light but keep heat from escaping, just like the glass walls of a greenhouse, hence the name Greenhouse. Scientists have known about the greenhouse effect since 1824, when Joseph Fourier calculated that the Earth would be much colder if it had no atmosphere ( [1] [2] ). This natural green house effect is what keeps the Earth’s climate livable; and without it, the Earth’s surface would be an average of about 60˚F (33˚C) cooler.

In this section, we will review some existing literature on different models/methods used to measure the climate change.

[8] used recent advances in time series econometrics to estimate the relation among emissions of carbon dioxide and methane, the concentration of these gases, and global surface temperature. These models were estimated and specified to answer two questions; whether the human activity affects global surface temperature and whether the global surface temperature affects the atmospheric concentration of carbon dioxide and methane. In this study, regression results provided direct evidence for a statistically meaningful relation between radioactive forcing and global surface temperature. A simple model based on these results indicated that greenhouse gases and anthropogenic sulfur emissions were largely responsible for the change in temperature over the last 130 years.

[9] used statistical models consisting of a trend plus serially correlated noise fitted to observed climate data, for example global surface temperature, the trend and noise representing systematic change and other variations, respectively. When such a model was fitted, the estimated character of the noise determined the precision of the estimated trend. In this study, the results of fitting such models to global temperature implied that there was uncertainty in the amount of temperature change over the past century of up to 0.2˚C and that the change was significantly different from zero.

To characterize observed global and hemispheric temperatures, previous studies have proposed different types of data-generating processes (see e.g. [10] [11] [12] [13] ). The most common among them are random walk and trend-stationary, however, these approaches offering contrasting views regarding how the climate system works.

[14] presented an analysis of the time series properties of global and hemispheric temperatures using modern econometric techniques. Their results showed that the temperature series can be better described as trend-stationary processes with a one-time permanent shock. They suggested that the climate change has affected the mean of the processes but not their variability. During the last century, it has manifested in global and Northern Hemisphere temperatures, while a second stage is yet possible in the Southern Hemisphere. They argued that significant anthropogenic interference with the climate system has already occurred.

In [15] , the authors provided evidence of anthropogenic influence over the warming of the 20th century is presented and the debate regarding the time-series properties of global temperatures is addressed in depth. The 20th century global temperature simulations produced for the Intergovernmental Panel on Climate Change’s Fourth Assessment Report and a set of the radiative forcing series used to drive them are analyzed using modern econometric techniques. Results show that both temperatures and radiative forcing series share similar time-series properties and a common nonlinear secular movement.

We obtained the Combined Land-Surface Air and Sea-Surface Water Temperature Anomalies (Land-Ocean Temperature Index, LOTI) from Goddard Institute for Space Studies (GISS), NASA https://data.giss.nasa.gov/gistemp/. The data are available for Global mean, Northern Hemisphere mean and Southern Hemisphere means (monthly, quarterly and annual) since 1880 to present, updated through the most recent month [16] .

Functional Time Series (FTS) and Sliced Functional Time Series (SFTS)

[17] first introduced the functional time series (FTS) models. Using these models, the interest lies in forecasting a series of functional data observed over time. The functional curves are observed (with error) at time t = 1, … n, and we wish to forecast the functions for times t = n + 1, … n + h. Let [f_t(x_j)] denote the observed data, where j = 1, … p. We assume that there are underlying L₁ continuous and smooth functions [s_t(x)] such that:

f t ( x j ) = s t ( x j ) + δ t ( x j ) e i . j (1)

where [e_i.j] are independent and identically distributed variables with zero mean and unit variance, and δ t ( x j ) allows for heteroskedasticity.

The technique in [17] uses non parametric smoothing on each curve f_t(x) separately to obtain estimates of the smooth functions [s_t(x)]. Panelized regression splines are used for smoothing, and then a functional principal component approach [18] is used to decompose the time series of functional data into a number of principal components and their scores. The functional time series (FTS) model can be written as follows:

s t ( x ) = μ ( x ) + ∑ k = 1 K ϕ t , k Ψ k ( x ) + e t ( x ) (2)

where Ψ_k(x) is the k^th principal component, the set of coefficients [ ϕ 1 , k , ⋯ ϕ m , k ] are the corresponding scores, e_t(x) denote independent and identically distributed random functions with zero mean, and K is the number of principal components to be used.

To plot a functional time series, [19] proposed three new graphical methods. They include the rainbow plot, the “Functional Bagplot” and the functional highest density region “(HDR) Boxplot”. Their approach has a side benefit of identification of outliers, which may not be obvious from the plot of the original data. These outliers are two types, either 1) magnitude outliers (i.e. the curves lie outside the range of the vast majority of the data), or 2) they may be the shape outliers (the curves that are within the range of the rest of the data but they have different shape from other curves). It is also possible that the curves may exhibit a combination of these two features. The presence of the outliers may have serious effect on the modeling and forecasting series.

To detect the outliers from a functional time series, the first step is to obtain the functional curves and the data are transformed into sliced functional time series (SFTS). For this, the entire data are sliced for each year as a function of 12 months. These curves are plotted in rainbow order with red for the earlier years and violet for the most recent year. The functional curves are then projected into a finite dimensional subspace. The subspace R² is chosen for simplicity. Each of the functional data point in R² are ordered by 1) data depth and 2) data density, based on halfspace Bagplot in [20] and HDR Boxplot in [21] . Those curves with lowest depth and/or lowest density are considered to be the outliers (see [19] for details).

Functional Bagplot

The functional bagplot uses halfspace location depths described in [20] which is based on the bivariate bagplot of [22] , applied to the first two principal component scores. The depth region R_k is the set of all θ, with r(θ, z) ≥ k. Since the depth regions form a series of convex hulls, we have R k 1 ⊂ R k 2 for k₂ > k₁. The Tukey bivariate depth median is defined as the value of θ which minimizes r(θ, Z) if there is such a unique θ, otherwise it is defined as the center of gravity of the deepest region.

Functional HDR boxplot

The functional HDR boxplot is based on the bivarate HDR Boxplot [21] , which is applied to the first two principal component scores. The bivariate HDR boxplot is constructed using a bivariate kernel density estimate f(z), which is defined as

f ( z ) = 1 / n ∑ i = 1 n k h i ( z − Z i ) , (3)

where Z_i represents a set of bivariate points; K_hi(×) = K(×/hi)/hi; K is the kernel function; and h_i is the bandwidth for the ith dimension. The bandwidths were selected using smoothed cross validation. Using the kernel density estimates, a HDR is defined as

R α = { z : f ( z ) ≥ f α } , (4)

where f_α is such that ∫_Rαf(z)dz = 1 − α; that is, it is the region with probability of coverage 1 − α, where all points within the region have a higher density estimate than any of the points outside the region, hence the name highest density region.

Figure 2 represents the average monthly global temperature since 1880 till 2018, as compared to the long-term average of twentieth century. Though warming has not been uniform across the planet, the upward trend in the globally averaged temperature shows that more areas are warming than cooling, specifically after 1980. According to the international State of the Climate in 2017 report, it was observed that since 1901, the planet’s surface has warmed by 0.7˚ - 0.9˚ Celsius (1.3˚ - 1.6˚ Fahrenheit) per century, but the rate of warming has nearly doubled since 1975 to 1.5˚ - 1.8˚ Celsius (2.7˚ - 3.2˚ Fahrenheit) per century.

Figure 3 depicts the mean monthly temperature anomalies of Northern and Southern Hemispheres from 1880 to 2018. If we compare the two series, we can observe that the northern series is more volatile with increasing trend than the southern series. The trend is evident from 1980 in northern and from 1960 in the southern hemisphere. The larger values of temperature anomalies in the Northern hemisphere may be due to the fact that it comprises of more land areas (represented by green color), whereas the southern hemisphere has more ocean/sea areas (represented by blue color). The ocean temperatures increase more slowly

than land temperatures because the oceans lose more heat by evaporation and they have a larger heat capacity.

Next, the data are transformed into sliced functional time series. The first step is to obtain the functional curves. For this, the entire data are sliced for each year as a function of 12 months, as plotted in Figure 4 and Figure 5. These curves are plotted in rainbow order with red for the earlier years and violet for the most recent year. We use R package rainbow [19] to construct these plots.

Figure 4 shows the global temperature anomalies (1880-2018) as sliced functional time series. The corresponding series for northern and southern hemispheres are plotted in Figure 5. The curves are plotted in rainbow order, with earlier years as red and most recent years as violet. It confirms that the average temperature is continuously rising in recent years. Some of the anomalies in Northern hemisphere series are as high as 1.0 - 1.5 (considered to be as outliers).

Variability in Northern series is higher than the variability in southern series due to more land areas in Northern Hemisphere.

Outlier Detection in Temperature Series

Next, the functional curves are projected into a finite dimensional subspace, the subspace R² is chosen for simplicity. Based on halfspace bagplot [20] and HDR boxplot [21] , each of the functional data point in R² are ordered by data depth and data density. Those curves that have either lowest depth or lowest density are considered to be the outliers.

1) The Functional Bagplots

2) The Functional HDR Plots

3) The Functional Bivariate plots

Functional bagplots for Global, Northern and Southern Hemisphere series are plotted in Figures 6-8. Their respective functional HDR plots are shown in Figures 9-11; whereas, the functional bivariate plots based on the first two principle

components are constructed in Figures 12-14 respectively. The outliers depicted from these plots are shown in Table 1. In global series, 1916, 2015 and 2016 are confirmed outliers from all three methods. The corresponding outlier years in the Northern hemisphere are 2015, 2016 and 2017, whereas 1997, 1998 are consistently appeared to be the outliers in Southern series.

Application of FTS Model

Next, we apply the functional time series model of [17] and obtain the forecasts for next twenty years (2019-2038). The various components of FTS model (the mean function, the first two bases functions and corresponding time series coefficients) for global and the two hemisphere aeries are plotted in Figures 15-17 respectively. Figures 18-20 show the forecasts of average temperature in the three series; again the years are plotted in rainbow order with earlier year in red and most recent year in violet color.

The forecasts clearly show the warming in the three series. For global series, the forecasts values are relatively lower for the months of January and February, highest in March and then they are expected to be lower for April-July, slightly increase for August and October and relatively lower for the other months (Figure 18). The Northern Series shows the similar pattern with highest values in the month of March and lowest in July and relatively smaller in the other months (Figure 19).

The Southern hemisphere series forecasts depict a different pattern. The forecasts curves show increase in the average temperature in the next twenty years, with the maximum temperature in May and August and minimum in the months of November and December (Figure 20).

Forecast Comparison with the other Models

Finally the forecasting performance of Sliced Functional Time Series (SFTS) model will be measured by Mean Error (ME), Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE). These measures are described below:

*The years in bold are magnitude outliers with high values and beyond the outer region.

1) ME = ∑ i = 1 N ( Y i − F i ) / N

2) RMSE = ∑ i = 1 N ( Y i − F i ) 2 / N

3) MAE = ∑ i = 1 N | Y i − F i | / N

4) MAPE = ∑ i = 1 N | Y i − F i | Y i × 100

where Y_i denotes the observed value and F_i denotes the corresponding forecast value. These measures of forecast accuracy are also computed for ARIMA/SARIMA models of Box and Jenkins [23] , exponential smoothing state space models [24] and random walk with drift models. For this, the global temperature series was divided into two sets: the training set 1880-1988 (109 years) and the test set 1989-2018 (30-years). The out-of-sample forecasts accuracy is measured and the results are summarized in Table 2.

Forecasts from these models are plotted in Figures 21-23 respectively. From these figures and Table 2, it is clear that the forecasts obtained by SFTS models have smaller values of error measures with relatively narrow prediction intervals.

As part of the Paris Agreement on climate change [25] , the international community committed in 2015 to limit rising global temperatures to well below 2˚C by the end of the 21st century and to pursue efforts to limit the temperature increase even further to 1.5˚C. However, these global temperature targets mask a lot of regional variation that occurs as the Earth warms. For example, land warms faster than oceans, high-latitude areas faster than the tropics, and inland areas faster than coastal regions.

In this paper, the global temperature data are analyzed through sliced functional time series (SFTS) model, a relatively new method of forecasting, and the

monthly forecasts for the next twenty years (2019-2038) are obtained along with 80% prediction intervals. These forecasts are also compared with the forecasts

obtained from Autoregressive Integrated Moving Average (ARIMA), exponential smoothing state space (ETS) and random walk with drift (RWD) models. It is found that the Sliced Functional Time Series models performed better than standard ARIMA, ETS and RWD models and the forecasts obtained from SFTS models are not only more accurate and reliable, but also they have narrow prediction intervals as compared to other models.

By 2038, the SFTS model projects that the average global surface temperature is expected to be 1.05 degree Celsius warmer than 1901-2000 average in the month of March, 0.95 degrees warmer in the months of January and February and about 0.85 degrees warmer in other months (see Figure 18). This similarity in temperatures regardless of total emissions is a short-term phenomenon: it reflects the tremendous inertia of Earth’s vast oceans. The high heat capacity of water means that ocean temperature doesn’t react instantly to the increased heat being trapped by greenhouse gases.

Given the size and tremendous heat capacity of the global oceans, it takes a massive amount of accumulated heat energy to raise Earth’s average yearly surface temperature, even a small amount. Behind the seemingly small increase in global average surface temperature over the past century is a significant increase in accumulated heat. That extra heat is driving regional and seasonal temperature extremes, reducing snow cover and sea ice, intensifying heavy rainfall, and changing habitat ranges for plants and animals by expanding some and shrinking others.

The author declares no conflicts of interest regarding the publication of this paper.

Yasmeen, F. (2019) Measuring Global Warming: Global and Hemisphere Mean Temperature Anomalies Predictions Using Sliced Functional Time Series (SFTS) Model. Open Journal of Applied Sciences, 9, 316-334. https://doi.org/10.4236/ojapps.2019.95026