1. Introduction
Identifying climate change through temperature measurements or their proxies continues to play a central role in understanding the dynamics of global climate over time. Changes in temperatures at both surface and upper-atmospheric layers add much clarity toward finding evidence for climate change and global warming. To this end, some recent articles are of interest: McShane and Wyner (2011), Seidel et al. (2011), Thorne and Vose (2010), Thorne et al. (2010), Zou and Wang (2010), Schleip et al. (2009), Randel et al. (2009), Karl et al. (2006), and Ramaswamy et al. (2001). Many of these articles have concluded that changes (cooling or warming) have occurred in the atmospheric temperatures of the earth. These articles take a univariate approach while identifying changes in the mean temperatures as trends on a decadal scale at various atmospheric layers of the globe. In this article, we consider the radiosonde temperature measurements of Angell (2012) at surface, troposphere, tropopause, and lower stratosphere layers of the earth at both the South and North Polar zones for the years 1958–2008. Deviating from the univariate approach, we study annual temperature anomalies at the four layers based on a multivariate model formulation. Mainly, our aim is to take advantage of the multivariate formulation and identify abrupt changes not only in the means but also in the covariances of temperature anomalies in these four layers.
Sustained changes in climatological factors can occur either gradually or in an abrupt manner. Real changes in climatological factors may be gradual in response to changes in one or more of the external forces known as forcings. However, changes in external forcings that reach beyond a threshold point end up inducing an abrupt change in the climate (Alley et al. 2002; Randall et al. 2007). Abrupt change in climate has been referred to as a large shift in climate that persists (Alley et al. 2002). Such changes may be manifested by changes in average temperature, altered patterns of storms, floods, or droughts over a widespread area. For example, abrupt change can occur as a result of a sudden massive volcanic eruption. Greenhouse gases such as carbon dioxide in the atmosphere can induce abrupt changes in Earth’s atmospheric temperatures when they reach beyond a threshold point. Randall et al. (2007) also discuss the possibility of unforced abrupt changes in the climate although such instances might be rare. Apart from such natural abrupt changes, one can also anticipate artificial abrupt changes in climatic data that are a result of changes in data measurement practices (Peterson et al. 1998; Gaffen et al. 2000). In their review article, Peterson et al. (1998) discuss a host of possible ways in which artificial abrupt changes can occur in climatic time series; these include changes in instruments, observing practices, station locations, and formulas used to calculate means. Usually, such artificial abrupt changes can be identified with meticulous documentation of changes in data measurement practices. However, documentation problems in data measurement practices are common and thus artificial abrupt changes in climatic data must also be identified along with real changes. Alley et al. (2002) point out that most statistical models are based on the assumption that climates are not changing but are stationary. Thus, such models have limited use for nonstationary climatic variables that change abruptly over time. They go on to call for statistical approaches that better reflect the properties of abrupt climate changes. This article may be viewed as an attempt to address this issue.
Given the large natural variability in climate time series, it can often be difficult to identify whether and when an abrupt change (real or artificial) may have occurred. From a statistical standpoint, this is often pursued by implementing change detection methods. In simple terms, change-point detection techniques are designed to detect the timing of abrupt changes and decide whether they are significant or not. Thus, there are two inferential problems associated with the identification of the unknown time points at which abrupt changes in the parameters of a statistical model might have occurred: detection and estimation. In the detection part, one aims to develop methods to test for the null hypothesis of no change point in the parameters against the alternative hypothesis of the presence of one or more unknown change points. Upon detecting an unknown change point, one follows it up by estimating its location. While both of these inferential problems are important, the detection part has received greater attention in the literature, including the climatological literature. In this regard, Reeves et al. (2007) give an overview of common change-point detection techniques used in climate literature. Chen and Gupta (2001) provide a survey of different change-point models and Beaulieu et al. (2012) present various applications of change-point detection to climate data. A Bayesian approach has been adapted by Schleip et al. (2009) and Berliner et al. (2000). Ducré-Robitaille et al. (2003) compared eight methods for the detection of shifts in time series data. Overall, change-point methodology has recently become an important tool for identifying changes in climatic factors (Alexandersson 1986; Easterling and Peterson 1995; Jaruskova 1996; Alexandersson and Moberg 1997; Vincent 1998; Horváth et al. 1999; Lund and Reeves 2002; Rodionov 2004; Fearly and Sweeny 2005; Zhao and Chu 2006; DeGaetano 2006; Lau and Wu 2007; Wang et al. 2007; Briggs 2008; Jandhyala et al. 2009; Jaruskova 2010; Fotopoulos et al. 2010).
Upon detecting change in a climatological series, it is important to identify the location of the change point. Knowledge about the location of a change point goes much further than a mere point estimate. One seeks an interval estimate for the change point and this can be constructed via the maximum likelihood estimate (MLE) provided its distribution is known. However, distribution of the change-point MLE is largely intractable under the classical abrupt change modeling (Jandhyala and Fotopoulos 1999; Borovkov 1999; Fotopoulos et al. 2010). Recently, Fotopoulos et al. (2010, 2011) derived computable expressions for the distribution of the change-point MLE when a change occurred in the mean vector of a multivariate Gaussian series and in the parameter of an exponential distribution. In pursuit of performing change-point analysis of radiosonde temperature series observed at four layers of the earth’s atmosphere, we first develop in this paper the large sample distribution of the change-point MLE when a change has occurred in both the mean vector and the covariance matrix of a multivariate Gaussian series. Then, we extend the Bayesian conjugate posterior derived by Perreault et al. (2000c) for change in the Gaussian mean vector only to the case of identifying change in both the mean vector as well as the covariance matrix. We also adapt the Bayesian noninformative prior of Son and Kim (2005) and the conditional solution of Cobb (1978) into the change-point analysis of the radiosonde temperature series. This article includes a simulation study for comparing the change-point MLE and three Bayesian solutions; it is also used to evaluate the robustness of the methodologies against deviations from Gaussianity.
The article is organized as follows. A description of the data and the main goal of the study are presented in section 2. Section 3 covers the MLE and different Bayesian approaches for estimating an unknown change point; it also contains a simulation study for evaluating the performance of various methodologies. All aspects of carrying out change-point analysis of the radiosonde series are included in section 4. Section 5 contains a discussion and concluding remarks.
2. Data
Angell (2012) reported radiosonde temperature deviations at various altitudes as a part of the output of 63 globally distributed stations. The data were measured 1) at the surface, 2) in the troposphere (850–300 hPa), 3) at the tropopause (300–100 hPa), and 4) in the lower stratosphere (100–50 hPa). The observations of atmospheric layers were made at both the South Polar (60°–90°S) and North Polar (60°–90°N) zones of the earth. Data on mean annual air temperature measurements for the period 1958–2008 were reported as deviations from the mean of 1958–1977. One may see Angell (2012) for a graphical illustration of the four layers as well as the full details of the sources of the recorded data. Here, we graphically present the data in Figs. 1 and 2 for the South and North Polar zones, respectively.
Annual radiosonde temperature measurement anomalies (°C) for the South Polar zone at the (a) surface, (b) troposphere, (c) tropopause, and (d) lower-stratosphere layers for the years 1958–2008.
Citation: Journal of Applied Meteorology and Climatology 53, 3; 10.1175/JAMC-D-13-084.1
As in Fig. 1, but for the North Polar zone.
Citation: Journal of Applied Meteorology and Climatology 53, 3; 10.1175/JAMC-D-13-084.1
Our goal in this article is to carry out change-point analyses of the above annual data in order to identify and understand the presence of abrupt changes in the temperature series during the period 1958–2008. While this can be carried out for data from each layer in a univariate manner, we model data from each zone as a multivariate set of dimension d = 4. The multivariate formulation enables us to analyze temperature changes at each of the four layers in a simultaneous manner. The ability to analyze temperature changes among the four layers simultaneously is a major advance over the usual univariate methods that detect changes in each layer one at a time. Moreover, the multivariate formulation allows us to identify changes not only in the mean vector but also in the covariance structure, which is otherwise not possible through any of the univariate methods. For detection of change points, we adapt likelihood ratio tests and their distribution theory, as presented by Csörgö and Horváth (1997). However, change-point estimation for changes in a multivariate model requires important developments without which we cannot proceed further. In what follows, we pursue this first for the MLE and then for the Bayesian posterior under the conjugate prior. Other Bayesian procedures are also a part of our study.
3. Change-point estimation in the mean and covariance of a multivariate Gaussian series
We begin by letting 
a. Estimation of change point under the MLE approach
b. Estimation of change point under the Bayesian approach
Developments under the Bayesian approach to change-point analysis are vast. Included among them are changes in the mean and/or variance of the univariate normal distribution (Perreault et al. 1999; Perreault et al. 2000a,b), changes in the mean vector of a multivariate normal distribution (Perreault et al. 2000c), changes in linear regression (Seidou et al. 2007), and multiple change points (Fearnhead 2005, 2006; Seidou and Ouarda 2007; Ruggieri 2012).
In this article, we consider three different ways of computing the Bayesian posterior distribution. First, we consider the conjugate priors method that Perreault et al. (2000c) adopted for the case of change in the mean vector only of a multivariate Gaussian sequence and we extend it to the case of change in both the mean vector as well as the covariance matrix. Conjugate priors are convenient in Bayesian analysis due to the fact that both prior and posterior distributions belong to the same family. Next, we state the posterior of the noninformative prior approach adopted by Son and Kim (2005). Finally, we present the conditional solution of Cobb (1978) since it can be applied to the situation of change in both the mean vector and covariance matrix. All three methods, including the derivation of the conjugate priors method, are presented in appendix B.
c. Simulation study
We carry out a simulation study with several goals in mind, all related to the performance of various methods of change-point estimation. The first goal is to assess the closeness of the asymptotic distribution of change-point MLE to the case of finite samples and then to evaluate the asymptotic equivalence of Hinkley (1972) for known and estimated parameters. Our next goal is to carry out a comparison among change-point MLE and the three Bayesian solutions. The final goal is to assess the robustness of the change-point MLE to deviations from normality. Details of the simulation study and its results are all presented in appendix C.
4. Change-point analysis of radiosonde temperatures from South and North Polar zones
a. Detection of change points in radiosonde temperature series
To begin the change detection analyses, let 
The p values and the corresponding change-point MLEs based on application of the likelihood ratio statistic under alternative hypotheses of change in mean only [
Applying the step-down procedure to p values in Table 1, we notice that our layer-3,4 bivariate model shows significance against all three alternatives. The corresponding univariate models also show significance, and, moreover, the change-point MLEs coincide (almost) at both univariate and bivariate levels. Thus, at the South Polar zone, we conclude that a change in both the mean and covariance occurred in layers 3 and 4, and we label the corresponding model S-34. Next, we note that the layer-1,2 model shows significance only in the mean. Moving down, the layer-1 and layer-2 models also show significance in their means only. However, the change-point MLE for the layer-1,2 model is 19 whereas the corresponding MLE for the layer-1 model is 8 and it is 19 for the layer-2 model. Because of this discrepancy, we pursue change for the layer-1 (model S-1) and layer-2 (model S-2) models on a univariate basis only. Similarly, applying the same step-down procedure to p values in Table 2 for the North Polar zone, we conclude that a significant change in the mean vector only has occurred in the layer-1,4 two-dimensional model (model N-14). Next, as we step down, layer-2 shows a change in the mean only (p value = 0.0406). The same test for layer-3 shows that the p value = 0.0637, which we conclude to be marginally significant. Thus, we pursue change in the mean only for the layer-2 (model N-2) and layer-3 (model N-3) models on a univariate basis.
b. Cross validation of the assumptions of Gaussianity and independence
The estimation methodologies presented in section 3 cannot proceed without cross validating the assumptions of independence and Gaussianity. We begin our cross-validation process with the data from the South Pole wherein the models to be pursued are S-34, S-1, and S-2. First, let 
c. Parameter estimates and distributions of change-point estimates
Upon estimating the parameters of any given model for the South or North Polar zones, one computes the asymptotic distribution of the MLE as well as the three Bayesian posteriors of the change point by following the details provided in the respective appendixes. The computed distributions for models S-34, S-1 , and S-2 of the South Polar zone are presented in Tables 3–5. We skip presenting similar tables for models from the North Polar zone to save space. One may then use these distributions to compute a confidence interval estimate of the change point through the distribution of the MLE, and credible regions for the change point through any of the three Bayesian solutions. For each model, the computed parameter estimates, the approximately 95% confidence intervals as well as the approximately 95% Bayesian credible regions, are presented in Tables 6 and 7 for the South and North Polar zones, respectively.
Distributions of change-point estimates obtained by the methods of MLE, Cobb’s conditional solution, Bayesian conjugate prior, and Bayesian noninformative prior for bivariate model S-34 involving layer 3 (tropopause) and layer 4 (lower stratosphere) from data on temperature measurements of the South Polar zone during the years 1958–2008.
Model parameter estimates and confidence set based on MLE and the three Bayesian credible regions for the model change point from models of the South Polar zone.
d. Summary of conclusions from change-point models
The models and the various estimates presented in Tables 6 and 7 enable us to draw important conclusions about the behavior of radiosonde temperature measurements over time at both the South and North Polar zones. First, it is clear from model S-34 and its parameter estimates that the amount of change in absolute (standardized) terms is −1.41°C (−2.01°C) at the tropopause layer and it is −2.47°C (−1.74°C) at the stratosphere layer. In addition, models S-1 and S-2 suggest that the amount of change at the surface layer is 0.81°C (1.65°C) and that at the troposphere layer is 0.50°C (1.55°C). Thus, a cooling effect has occurred around the year 1981 in both the tropopause and stratosphere layers of the South Polar zone whereas there is a warming effect in the surface and troposphere layers with the warming taking place at the surface layer much earlier (around 1965) compared to the year 1976 at the troposphere layer. Moreover, estimates of model S-34 suggest that a significant increase in the variability also took place at the tropopause and stratosphere layers of the South Polar zone. More importantly, the estimated correlation between the tropopause and stratosphere layers before 1981 is −0.07, whereas it has changed to 0.83 after 1981. Thus, a strong positive correlation has begun between these two layers that was nonexistent before 1981.
As for the North Polar zone, estimates for models N-14, N-2, and N-3 suggest that the amount of change at each layer and the year in which the change has occurred are 1.06°C (2.03°C) around the year 1988 for the surface layer, 0.52°C (0.94°C) around the year 2001 at the troposphere layer, −1.27°C (−1.31°C) around the year 2001 for the tropopause layer, and −1.34°C (1.22°C) around the year 1988 for the stratosphere layer. Thus, the changes at both the South and North Polar zones are similar in the sense that the warming effect at the surface and troposphere layers and the cooling effect at the tropopause and stratosphere layers found at the South Polar zone also persist at the North Polar zone. However, they also differ in the sense in that the change in correlation structure found at the tropopause and stratosphere layers of the South Polar zone is not observed at the North Polar zone. Even the variability in temperatures at various layers of the North Polar zone remained steady throughout the time period. Also, the warming and cooling effects began occurring much earlier at the South Polar zone (between the years 1965 and 1981) in comparison to the later year changes (between the years 1987 and 2001) found at the North Polar zone.
e. Change-point models versus linear-trend models
Climatologists often look for linear trends in atmospheric factors. Despite its significance against no change, a change-point model is preferred only when it is significantly better than a linear-trend model. For example, Seidel and Lanzante (2004) compared linear-trend models against change-point alternatives for modeling temperature series of the atmosphere. Here, we settle this issue by applying the celebrated Akaike information criterion (AIC) procedure that Akaike (1973) proposed for model selection. Omitting details of the computations, we present in Table 8 for the South Polar zone the results of applying the AIC to no change model, to the linear-trend model, and to the proposed change-point model. The same table also contains trend estimates and the corresponding p values that show the nature of their significance. Table 8 shows that the change-point model is the best model by virtue of its AIC value being minimum (as identified by the asterisk) in each case. Similar computations have also been performed for data from the North Polar zone. While we do not present the details of these computations, we have found that the change-point model was the best for all three North Polar zone models also. Overall, it is clear that change-point models explain temperature trends at both the South and North Polar zones better than the models of no change or models with a linear trend.
Slope coefficients and the corresponding p values, AIC number for mean only with no change, change-point model with change in mean, and simple linear model with no change for layers 1 and 2; and for bivariate mean vector model with no change, bivariate mean vector with change in mean vector as well as covariance, and bivariate simple linear regression model for layers 3 and 4; for data on temperature measurements from the South Polar zone for the years 1958–2008.
5. Discussion and concluding remarks
The observed changes at the South and North Polar zones could merely be a result of either instrument changes or changes introduced in the procedures for operating the instruments. In this regard, Gaffen (1994), Gaffen et al. (2000), Lanzante et al. (2003), and Angell (2003) looked into the contamination of data due to changes in radiosonde instruments and the corresponding operational procedures. Angell (2003) pointed out that there were nine stations that had anomalies in measurements and they were all located in the tropical region of the globe. Gaffen et al. (2000) noted changes in instrumentation at stations located in Tahiti, Japan, the former Soviet Union, Australia, and New Zealand. On balance, there is no evidence of documentation or instrumentation problems that may have caused the changes in the data series at both the South and North Polar zones.
The above being the case, what do climatological studies have to say regarding the cooling effect at the tropopause and stratosphere layers and the warming effect at the surface and troposphere layers? Moreover, what are the consequences of these temperature changes?
Climatologists have been investigating changes in atmospheric temperatures for some time (see Angell 1986, 1999; Randel and Wu 1999; Compagnucci et al. 2001; Ramaswamy et al. 2001; Karcher et al. 2003; Comiso 2003; Schleip et al. 2009; Randel et al. 2009; Thorne et al. 2010; Seidel et al. 2011). These studies are based on different datasets that include radiosonde, Microwave Sounding Unit (MSU), rocketsonde, and lidar observations. Yet, their conclusions are similar in that there has been a warming effect at the surface and troposphere layers and a cooling effect at the tropopause and stratosphere layers. Instead of dwelling on the details of these studies, we address below the issue of climatological consequences of the observed temperature changes.
Leaving aside the surface layer, we discuss below existing studies from climatological literature that focus on changes in the troposphere, tropopause, and stratospheric layers of the atmosphere. In this regard recent studies (see Santer et al. 2003; Seidel and Randel 2006; Son et al. 2011; Feng et al. 2012) have found that the tropopause layer plays a central role in how temperature trends vary over time in these three layers. While there may be several other sources in the literature, it seems to us that these four articles are representative of the current knowledge on the coupling effect of the tropopause with the troposphere and stratospheric layers. In this regard, Santer et al. (2003) note that the tropopause represents the boundary between the troposphere and stratosphere and is marked by large changes in the thermal, dynamical, and chemical structures of the atmosphere. Supporting this view, Son et al. (2011) observed that due to its promise as an indicator of climate change, the tropopause has often been examined to better understand stratosphere–troposphere exchange and coupling. Feng et al. (2012) stated that the tropopause has attracted much attention in the recent literature owing to its extreme sensitivity to climate variability and climate change. On the basis of reanalysis data and climate model simulations, recent studies have found that the global tropopause height may be a sensitive indicator of anthropogenic climate change (Seidel and Randel 2006).
The summary conclusion of all these articles has been that the warming of the troposphere and the cooling of the stratosphere contribute significantly to increases in the height of the tropopause. For example, based on results from Parallel Climate Model (PCM) experiments and the reanalysis of data for the period 1979–93 obtained from European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA) data, Santer et al. (2003) concluded that both stratospheric cooling and tropospheric warming lead to increases in tropopause height. Seidel and Randel (2006) analyzed radiosonde data from 100 stations archived at the National Climatic Data Center (NCDC) for the period 1980–2004. From their analysis they found that increases in tropopause height may be more directly affected by the cooling of the stratosphere and much less by the warming of the troposphere. In contrast, Feng et al. (2012) carried out an analysis of radiosonde data available in the Integrated Global Radiosonde Archive (IGRA) for the period 1965–2004 and found statistically significant thickening of the tropopause layer. Moreover, they found that the thickness of the tropopause is coupled primarily with the temperature of the lower stratosphere than the upper troposphere.
The conclusions of the present study support the findings of Seidel and Randel (2006) and Feng et al. (2012). It may be recalled that we found the presence of a strong correlation between the tropopause and lower stratosphere (of the South Polar zone) subsequent to 1981 while such a correlation was not present prior to 1981. Also, we have not found any association between the tropopause and troposphere throughout the period of our study (1958–2008). Thus, the warming of the troposphere may not have contributed to the thickness of the tropopause layer whereas the presence of a correlation between the lower stratosphere and tropopause and the cooling of the lower stratosphere appear to have contributed to the increase in the thickness of the tropopause layer. It is noteworthy that the data of Seidel and Randel (2006) begin with 1980, which closely matches with the year in which we found a strong correlation began between the tropopause and lower stratosphere layers. Even Santer et al. (2003) concluded that tropospheric height changes began during 1979.
In the final part of our discussion, we investigate why abrupt changes in the temperatures of atmospheric layers might have occurred. To this end, the anthropogenic and natural forcings studied in the PCM experiments of Santer et al. (2003) have substantial relevance for identifying the causes for tropospheric and stratospheric temperature changes. The anthropogenic forcings that Santer et al. considered were well-mixed greenhouse gases (G), direct scattering effects of surface aerosols (A), and tropospheric and stratospheric ozone (O), while the natural forcings were changes in solar irradiance (S) and volcanic aerosols (V). On the basis of their study, Santer et al. (2003) found that the largest contribution to tropopause height increases during the satellite era (1979–99) came from well-mixed greenhouse gasses and tropospheric and stratospheric ozone. The solar irradiance and volcanic aerosols together reduced the tropopause height, but only marginally.
There are other studies that have regarded ozone depletion since 1980s as the major factor for the cooling of the lower stratosphere (Angell 1986; Randel and Wu 1999; Ramaswamy et al. 2001; Steinbrecht et al. 2003; Cagnazzo et al. 2006). Forster et al. (2007) reasoned that the cooling was due to the decrease in absorption of longwave radiation from reduced ozone levels. Solomon et al. (2007) reported that ozone depletion in the Arctic was far less than that in the Antarctic. Consequently, the cooling at the South Pole was more prominent than in the North Polar zone. Besides ozone depletion, Ramaswamy et al. (2001) found that greenhouse gases such as carbon dioxide not only warmed the surface but also affected the temperatures of the lower stratosphere.
Acknowledgments
We are grateful to the editor and reviewers for constructive comments and suggestions that led to substantial improvements in the paper and its organization. The research of VKJ and SBF was supported by National Science Foundation Grant DMS-0806133. VKJ thanks the C. R. Rao Advanced Institute of Mathematics, Statistics and Computer Science (AIMSCS), Hyderabad, India, for their kind hospitality when he spent time there doing a part of this research work. The research work of IBM was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
APPENDIX A
Asymptotic Distribution of Change-Point MLE
a. Developments toward computational procedure
b. Implementation of algorithm
We shall now proceed to implement the algorithmic procedure of Jandhyala and Fotopoulos (1999) together with the modifications suggested in Fotopoulos et al. (2010). In implementing the algorithmic procedure of Jandhyala and Fotopoulos (1999), one should keep in mind that we implement here only their second approximation. Consequently, upon completing steps 
Steps 
Step 
Step 
APPENDIX B
Bayesian Approaches to Change-Point Estimation
a. Conjugate prior approach
b. Noninformative prior approach
c. Cobb’s conditional approach
APPENDIX C
Simulation Study and Results
a. The simulation study
First, we carried out some preliminary simulations and found that the change-point MLE and the three Bayesian solutions for the most part depended on 
b. Results of the simulation study
First, we present here results for the bivariate case only; the univariate case showed similar results. Next, results for the bias were negligible in all cases and hence we do not present them here. However, RMSE values were found to vary considerably together with the method of estimation, sample size, location of the change point, and amount of change, as well as departure from Gaussianity. Thus, we present them in Tables C1–C3. To better capture the behavior of RMSE values, we also presented them in figures for the following special cases: (a) comparing the behavior of MLE and Cobb’s methods, (b) effects of sample size and change-point location, (c) comparison among various methods, and (d) effects of degrees of freedom. However, due to limitation of space we omit all figures, and only present a summary of what we observed in them.
Comparing MLE and Cobb’s methods: RMSE values under the MLE method were all close to each other, except when
Effect of sample size and change-point location: The MLE and Cobb’s methods tended to yield higher values under ee and when the sample size was small. When
Comparison among various methods: All estimation methods showed similar results when the change in mean was at a higher level
Effect of degrees of freedom: RMSE values for the MLE were close to each other except at 5 degrees of freedom; other methods also exhibited similar behavior. When the degrees of freedom was 5, the noninformative prior method consistently showed higher RMSE values, particularly so when
RMSE for various change-point estimates in the bivariate case when 
Overall, it may be concluded that the MLE method was more robust to changing conditions than were the other methods. Generally, RMSE values for Cobb’s method were closer to the MLE method. The two Bayesian methods showed similar values only when 
APPENDIX D
Likelihood Ratio Change-Detection Statistics
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