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In this study, the sliced functional time series (SFTS) model is applied to the Global, Northern and Southern temperature anomalies. We obtained the combined land-surface air and sea-surface water temperature from Goddard Institute for Space Studies (GISS), NASA. The data are available for Global mean, Northern Hemisphere mean and Southern Hemisphere means (monthly, quarterly and annual) since 1880 to present (updated through March 2019). We analyze the global surface temperature change, compare alternative analyses, and address the questions about the reality of global warming. We detected the outliers during the last century not only in global temperature series but also in northern and southern hemisphere series. The forecasts for the next twenty years are obtained using SFTS models. These forecasts are compared with ARIMA, Random Walk with drift and Exponential Smoothing State Space (ETS) models. The comparison is made on the basis of root mean square error (RMSE), mean absolute percentage error (MAPE) and the length of prediction intervals.

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 ( [

The concept of “global average temperature” is convenient for detecting and tracking changes in planet’s energy budget that is how much sunlight Earth absorbs minus how much it radiates to space as heat over time. The concept of an average temperature for the entire globe may sometimes seem odd, as the highest and lowest temperatures on Earth are about more than 55˚C or 100˚F apart. In the Northern and Southern Hemispheres, temperatures vary from night to day and between seasonal extremes, means that some parts of Earth are quite cold while other parts are downright hot.

In order to calculate a global average temperature, scientists begin with temperature measurements taken at various locations around the globe. Because the goal is to track changes in temperature, these measurements are converted from absolute temperature readings to “temperature anomalies”. These are the differences between the observed temperature readings and the long-term average temperature for each location and time. Multiple independent research groups across the world performed their own analysis of the surface temperature data, and they all showed a similar trend in upward direction [

From increasing greenhouse gas concentrations, different parts of the world respond in different ways to warming. For example, high-latitude regions including far north or south of the equator become warm faster than the global average due to positive feedbacks from the retreat of ice and snow, an increased transfer of heat from the tropics to the poles in a warmer world also enhances warming.

According to the American Meteorological Society’s State of the Climate in 2017, the year brought an end to new record temperatures that were set each year from 2014 to 2016. Depending on the data set used, 2017 came in second or third warmest, after 2016 (warmest) and 2015 (second or third warmest) [

twenty-first century, the annual global temperature record has been broken five times, The top 10 warmest years on record have all occurred since 1998, and the four warmest years on record have all occurred since 2014.

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

[

[

To characterize observed global and hemispheric temperatures, previous studies have proposed different types of data-generating processes (see e.g. [

[

In [

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 [

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

[_{t}(x_{j})] denote the observed data, where j = 1, … p. We assume that there are underlying L_{1} 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 [_{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 [

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, [

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 R^{2} is chosen for simplicity. Each of the functional data point in R^{2} are ordered by 1) data depth and 2) data density, based on halfspace Bagplot in [

Functional Bagplot

The functional bagplot uses halfspace location depths described in [_{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 k_{2} > k_{1}. 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 [

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 ∫_{R}_{α}f(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.

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

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 R^{2} is chosen for simplicity. Based on halfspace bagplot [^{2} 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

Application of FTS Model

Next, we apply the functional time series model of [

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 (

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 (

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:

Method | Global Series | Northern Hemisphere Series | Southern Hemisphere Series |
---|---|---|---|

Functional Bagplot | 1903, 1912, 1916, 2015, 2016 | 1903, 1912, 1916, 2015, 2016, 2017 | 1911, 1924, 1997, 1998 |

Functional HDR plot | 1911, 1916, 1997, 2015, 2016, 2017, 2018 | 1997, 2002, 2006, 2007, 2015, 2016, 2017 | 1911, 1976, 1977, 1997, 2015, 2016, 2017 |

Functional Bivariate plot | 1902, 1903, 1916, 2015, 2016, 2017 | 1889, 1903, 1935, 2000 2015, 2016, 2017 | 1911, 1924, 1997, 1998 |

*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 [

Forecasts from these models are plotted in Figures 21-23 respectively. From these figures and

As part of the Paris Agreement on climate change [

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

Models | ME | RMSE | MAE | MAPE |
---|---|---|---|---|

ARIMA/SARIMA | 0.2238 | 0.2954 | 0.2451 | 55.8238 |

ETS | 0.2616 | 0.3416 | 0.2835 | 62.3048 |

RW with Drift | 0.2348 | 0.3340 | 0.2863 | 52.3225 |

Sliced FTS | −0.1875 | 0.2834 | 0.2047 | 48.3212 |

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

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