# Robustness Evaluation of Regression Tasks with Skewed Domain Preferences

Nuno Costa
INESC TEC
Porto, Portugal  Nuno Moniz
Lucy Family Institute for Data & Society
University of Notre Dame
Indiana, USA

###### Abstract

In natural phenomena, data distributions often deviate from normality. One can
think of cataclysms as a self-explanatory example: events that occur almost
never, and at the same time are many standard deviations away from the common
outcome. In many scientific contexts it is exactly these tail events that
researchers are most interested in anticipating, so that adequate measures can
be taken to prevent or attenuate a major impact on society. Despite such
efforts, we have yet to provide definite answers to crucial issues in
evaluating predictive solutions in domains such as weather, pollution, health.
In this paper, we deal with two encapsulated problems simultaneously. First,
assessing the performance of regression models when non-uniform preferences
apply - not all values are equally relevant concerning the accuracy of their
prediction, and there’s a particular interest in the most extreme values.
Second, assessing the robustness of models when dealing with uncertainty
regarding the actual underlying distribution of values relevant for such
problems. We show how different levels of relevance associated with target
values may impact experimental conclusions, and demonstrate the practical
utility of the proposed methods.

##  1 INTRODUCTION

Among the most common assumptions in statistical-based learning is the premise
of normality. However, real-world domains commonly present departures (even if
slightly) from normality, raising two critical caveats for machine learning.
First, cases framed within the central tendency of the distributions are the
most well-represented in the data. Thus, resulting models will tend to be
better at recognising and anticipating such cases. Second, cases with
extreme/rare values are often associated with high-impact events, e.g. fraud.
Therefore, in certain situations, such cases should be prioritised and not
under-valued w.r.t the ability to anticipate them.

Imbalanced learning is a sub-topic of machine learning that has gathered
increasing attention for over 30 years. It mostly focuses on the ability to
correctly model and predict rare and extreme events. Well-known as a very
challenging machine learning task [5], it usually faces issues hampering the
learning process, including identifying small disjunct areas, lack of density
and information in training data, overlapping classes, noisy data and data set
shift (different distribution for training and test data) [9]. Since its
inception, most of the research focus in this area concerns classification
tasks, and work concerning regression is residual. In spite of this, the
problem of imbalanced regression has witnessed an increasing level of interest
in recent years. Such interest resulted in multiple contributions concerning
the formalisation of the task in different data scenarios [10, 13] and
multiple proposals for pre-processing strategies [20, 11]. However, the major
linchpin for the challenge of imbalanced regression still lies on the issue of
evaluation, and the need to adopt suitable metrics to correctly describe the
performance of models in predicting extreme values.

Work concerning the evaluation of imbalanced regression tasks is still
deserving of much more attention. In respect to this, two major milestones
include the work on utility-based metrics by Ribeiro [16] – based on
adaptations of the popular F-Score [18] metric – which were later reviewed by
Branco [2] and Moniz [10]. This work has two main shortcomings. First, it is a
threshold-based metric, in scenarios where a decision concerning which
threshold to use is very sensitive and difficult to pinpoint. Second, the use
of this threshold leads to the complete disregard of model performance on the
majority of cases – those deemed as non-extreme. More recently, Ribeiro and
Moniz [17] proposed the Squared-Error Relevance Area (SERA) measure, which
unlike previous work, disregards the use of any threshold while still focusing
its evaluation of model performance on the more extreme values of the domain.
Nonetheless, there is an issue that runs through both lines of previous work,
which is that of robustness and the practical limitations of the foundations
of imbalanced regression tasks.

The most crucial concept in imbalanced regression tasks today concerns
relevance functions [21]. For now, it should suffice to clarify that it serves
as a mapping between domain values, i.e. target values, and its importance
(usually measured between 0 and 1) to the user in terms of obtaining an
accurate prediction. Conceptually, this should be a user-defined process, at
least partially – it is unfeasible to associate a relevance judgement to all
(infinite) values of a continuous domain. In such a scenario, users in
possession of domain knowledge or experience would make judgements on the
relevance of certain target values, and the remaining would be inferred, i.e.
interpolated. However, in most situations, we lack such domain knowledge, and
thus rely on automatic methods that are distribution-based. Essentially, those
in the centre of the distribution would be associated to relevance judgements
near 0, and as those values separate from such centre, their relevance should
approximate (or assume the value of) 1. Naturally, this assumes that the
available data describes a good sample of the underlying data domain, which is
not always the case. Notwithstanding, even in the situation where some domain
knowledge is available, small deviations from a ground truth might have a
significant impact in model evaluation and selection, e.g. the lack of
consensus in the quantification of biomarker concentration for
characterisation of neuro-degenerative disease stages [3].

###  1.1 Contributions

In this paper, we present methods to ascertain the robustness of machine
learning algorithms under domain preferences uncertainty. They are named
convolution and elastic, and both consist in simulating a multitude of
relevance scenarios through simulation, with the aim to understand how
different algorithms perform under different relevance criteria. Each of these
parts are associated to significant contributions, detailed as follows.

  1. 1.

Metrics suitable for imbalanced numerical prediction. Given that we assume
there is greater relevance in identifying specific intervals of the target
value, we cannot formalise prediction tasks as standard regression. Instead,
we leverage recent work concerning imbalanced regression [17], and the use of
non-standard evaluation/optimisation metrics;

  2. 2.

Robustness Evaluation. Given the lack of consensus on what target values
describe each regime of the underlying phenomena, we must evaluate the
robustness of our models. We propose a robustness evaluation method that
explores different ground-assumptions concerning the relation between target
level regions and importance regimes.

The remainder of the paper is organised as follows. Section 2 describes
relevant previous work and further contextualises the contributions of
imbalanced regression in machine learning. The problem of imbalanced
regression is detailed in Section 3. The robustness evaluation techniques are
thoroughly explained in Section 4. Experimental results are thoroughly
presented and discussed in Section 5, followed by conclusions and future work
in Section 6.

##  2 BACKGROUND

The main assumption of standard metrics for evaluation of numerical prediction
tasks is that each value in the domain has equal importance, i.e. uniform
domain preferences. Several metrics have been proposed for learning settings
where non-uniform preferences apply, i.e. a different level of relevance is
associated to distinct ranges of the target variable. These range from
asymmetrical approaches, common in the scope of finance, e.g. LIN-LIN [4], to
adaptations of well-known metrics for classification tasks, such as the
utility-based F-Score [16, 2, 12]. More recently, Ribeiro and Moniz, [17]
provide a thorough discussion on the limitations of existing metrics in the
scope of learning with non-uniform preferences, and propose the Squared Error-
Relevance Area (SERA) metric, which tackles several shortcomings in previous
work – we provide further details in Section 3.

###  2.1 The biomarker concentration problem - use case in detail

Medical researchers have been working on building concentration curves for
different plasma biomarkers across the various stages of AD (Alzheimer’s), and
there are hints pointing to dynamic, nonlinear and nonparallel patterns over
the course of the disease’s development in an individual. Research has also
shown there is a significant correlation between blood plasma and CSF
biomarker concentration [6], a finding that reinforces the usefulness of
blood-based biomarker tests. Notwithstanding, this can also be seen as an
additional step of uncertainty in asserting the concentration values that
require clinical intervention. In other words, concentration distributions for
CSF and brain samples (where it’s possible to find a ground truth) are still
not well determined. Concentration distributions for blood plasma can only be
inferred from the former - and it is virtually impossible to know the ground
truth. In Figure 1, it is possible to observe hypothetical concentration
curves from 4 of the most popular AD biomarkers in plasma [3].

Figure 1: Peptide concentration across AD stages, from [3], concerning Healthy
(HC), Mild cognitive impairment (MCI) and Alzheimer’s (AD) stages.

This is a classic example where the underlying problem has a great scientific
value and its solution would present a meaningful impact for neurodegenerative
disease treatment. Unfortunately the scientific community is yet to unravel a
precise regime classification through biomarker concentration levels. It
would, however, be possible to build a tentative relevance function using the
information presented in Figure 1.

##  3 IMBALANCED REGRESSION

Standard regression tasks assume that a function $f()$ maps predictor
variables to a continuous target variable. Such function is formalised as
$Y=f(X_{1},X_{2},\cdots,X_{p})$, where $Y$ is a numerical target variable, and
features describing each case are denoted with $X_{1},X_{2},\cdots,X_{p}$. An
approximation $h()$ (a model) to this function is obtained by using a data set
with examples of the function mapping (known as a training set), i.e.
$\mathcal{D}=\\{\langle\mathbf{x}_{i},y_{i}\rangle\\}_{i=1}^{n}$. Here, our
objective is to optimise models using a certain loss function, $L(\hat{y},y)$,
such as the absolute or the squared error of estimations w.r.t. the true
values. When all target values are considered equally important, i.e. uniform
domain preferences, standard regression tasks assume that the utility $U$ of
the estimations is a function of the estimation error $L(\hat{y},y)$ and that
it is inversely proportional to the loss function, $U\propto L^{-1}$. However,
this is not the case for learning tasks that are confounded by varying user
preferences across range(s) of target values or in scenarios where estimation
error may not hold as a useful metric across all possible target values. The
aforementioned is the problem statement for imbalanced regression tasks.

###  3.1 Relevance Functions

Imbalanced regression faces the challenge of providing a formal approach
capable of describing non-uniform preferences over continuous domains. Figure
2 illustrates the difference between the standard assumption of uniform vs.
non-uniform preferences.

Ideally, the specification of domain preferences should be carried out by
users or domain experts. Nonetheless, as previously discussed, this is often
difficult (if not unfeasible). In such cases, we can rely of distribution-
based statistics, as to automatically infer a suitable relevance function.

Figure 2: Example of target values’ relevance for a specific domain
preference. The dashed magenta line represents the case where no preferences
are given, therefore all values are considered equally important.

##### Definition

A relevance function $\phi(Y):Y\to[0,1]$ is a continuous function that encodes
a user-defined domain preference Y by mapping it into a $[0,1]$ scale of
relevance, where 0 and 1 represent the minimum and maximum relevance,
respectively [17].

The relevance function construction starts with control points mapping to
regions where relevance is known, ideally provided by domain experts. These
are either specified by users or automatically derived from the data. When the
latter is carried out, such process relies on the use of the adjusted box-plot
method [19], as suggested by Ribeiro and Moniz, [17]. From the control points,
an interpolation method is used to approximate a continuous function ready to
use in potentially infinite domains. For this occasion we will use Piecewise
Cubic Hermite Interpolating Polynomials [7], also as suggested by Ribeiro,
[16].

###  3.2 Evaluation

Concerning the evaluation of solutions in imbalanced regression tasks, we
propose the use of a recent evaluation metric, SERA (Squared Error-Relevance
Area), proposed by Ribeiro and Moniz, [17]. Such a metric is able to enhance
the quantitative focus in ranges other than the most common values, leveraging
the previously presented concept of relevance functions.

##### Definition

For a given data set D and a relevance function $\phi$, consider the subset
$D^{t}$ where the relevance of the target value is above or equal to a cutoff
$t$. In such case, it’s possible to obtain an estimate of the Squared Error-
Relevance of a model as:

$SER_{t}=\sum_{i\in D^{t}}(\widehat{y}_{i}-y_{i})^{2}$ (1)

where $\widehat{y}_{i}$ and $y_{i}$ are the estimated and true values for data
point $i$, respectively. For this estimate, only the subset of estimations for
which the relevance of the true target value is above a known threshold t, are
considered. SERA is defined as:

$SERA=\intop\nolimits_{0}^{1}SER_{t}dt=\intop\nolimits_{0}^{1}\sum_{i\in
D^{t}}(\widehat{y}_{i}-y_{i})^{2}dt$ (2)

In general terms, the smaller the area under the curve (SERA), the better is
the model’s performance. In a situation where no domain preferences are
defined, i.e. $\phi(Y)=1$ , SERA converges to the sum of squared errors. For a
more in-depth analysis and presentation of this recent proposal, we recommend
the reading of [17].

##  4 EVALUATION OF ROBUSTNESS UNDER UNCERTAINTY

The formal definition for imbalanced regression tasks provided by Ribeiro and
Moniz, [17] provides the basis for tackling the first issue in this project:
how to evaluate and compare solutions for which the relevance of obtaining
correct estimations in different target values is different? Indeed, having a
more appropriate metric for the evaluation of regression models on extreme
target values, we are left with the second challenge to ponder over: in
situations of uncertainty concerning the exact value range of predictive
focus, what is the best model to ensure a robust solution?

In this section, we propose two methods that can be applied in tandem with a
SERA-based evaluation of predictive solutions. Such methods are based on the
simulation of neighbouring configurations of the relevance function based on
the initial hint. The overall intuition is that by evaluating models in a
considerable amount of near configurations (w.r.t the original “hint”) for
SERA, it is possible to understand the behaviour of different
algorithms/strategies under uncertainty. It is also possible to understand
what is the most robust method under such events. To derive functions in the
vicinity of the relevance function constructed from the original data, we
propose two methods to simulate two degrees of freedom: i) x-axis displacement
and ii) slope incline. For both methods, the variation range can be
established according to the problem needs.

  * •

Convolution. First (Algorithm 1), we can think of the situation in which the
hint regarding the shape is correct, but inaccurate concerning which are the
most relevant values. By shifting the relevance function across a certain
range – much like a convolution – it is possible to collect the performance of
models for an additional number of similarly derived relevance functions;

  * •

Elastic. Second (Algorithm 2), we consider a situation where the relevance
function derived from the original data is correct about which values are the
most relevant, but not correct about the shape. It consists in fixating the
point where the relevance is larger than 0, while varying the first point
where the relevance is equal to 1.

For both methods we start by determining the range of the target variable
$Y\in\mathcal{Y}$ and obtaining a relevance function $\phi(Y)$. The function
should be a mapping of the domain $\mathcal{Y}$ into a $[0,1]$ scale of
relevance and it should be a smooth function. Next, we define the peak
relevance $\pi_{max}$ for $\phi(Y)$, as the minimum value for which
$\phi(Y)=1$, i.e. the first maxima encountered. Similarly, we define the base
relevance, $\pi_{min}$ as the largest target value for which $\phi(Y)=0$, i.e.
the point immediately before the slope of $\phi(Y)$. Note that, within the
scope of our work, we are dealing with one-tail right distribution of values.
In case of a distribution skewed left, the peak and base relevance values are
exchanged.

Result: Performance for Convolution method

$Y=$ training set target variable;

$\delta=$ search step size;

$\phi()=$initial relevance function;

$\pi_{max}=$initial peak relevance;

$\pi_{min}=$initial base relevance;

$\textbf{n}=\frac{range(Y)}{\delta}$;

for  _i in n_ do

update relevance function $\phi()$ s.t. $\pi_{max}=\pi_{max}+\delta$ and
$\pi_{min}=\pi_{min}+\delta$;

evaluate the model using SERA with $\phi()$ as the criterion;

end for

Algorithm 1 Convolution simulation for relevance function

Result: Performance for Elastic method

$Y=$ training set target variable;

$\delta=$ search step size;

$\phi()=$initial relevance function;

$\pi_{max}=$initial peak relevance;

$\pi_{min}=$initial base relevance;

M = $\tilde{Y}$, median $Y$;

$\textbf{n}=\frac{range(M,M+\sigma(Y))}{\delta}$;

for  _i in n_ do

update relevance function $\phi()$ s.t. $\pi_{max}=\pi_{max}+\delta$ and
$\pi_{min}=\pi_{min}$;

evaluate the model using SERA with $\phi()$ as the criterion;

end for

Algorithm 2 Elastic simulation for relevance functions.

Figure 3 illustrates the transformations performed on a sigmoid relevance
function: a) refers to Convolution, while b) refers to Elastic.

Figure 3: Left - Original relevance function. Dashed line is the inflection
point. Right - Relevance methods: a) Convolution b) Elastic

In closing, we should note the following. For our experimental study, we use a
Convolution range from $\pi_{max}$-$\sigma_{target}$ up to
$\pi_{max}$+$\sigma_{target}$ of the target value, while the Elastic range we
define as $\pi_{1}$-$\sigma_{target}$ up to $\pi_{1}$+$\sigma_{target}$ for
the first extreme $\pi,$ and $\pi_{2}$+$\sigma_{target}$ up to
$\pi_{1}$-$\sigma_{target}$ for the second extreme $\pi$. Extremes can be
$\phi=1$ or $\phi=0$

##  5 EXPERIMENTAL STUDY

In this section, we present an experimental study concerning the usefulness of
our robustness assessment approaches using multiple data sets. We start by
presenting the main research questions that drive our study: how robust are
the conclusions concerning the performance of predictive models in the context
of relevance uncertainty?

###  5.1 Methods

In this subsection, we provide experimental details concerning data, including
adopted methodology, custom feature selection methods, learning algorithms and
estimation procedures.

####  5.1.1 Learning Algorithms

For this study, we consider the following algorithms: Multivariate Adaptive
Regression Splines (mars), Extreme Gradient Boosting (xgboost) and Random
Forest. These are listed in Table 1, with the respective hyperparameter grid
and implementation details. The hyperparameters pertaining to xgboost and
Random Forest parameters are chosen using a random search. From the high-
dimensional grid space, 15 random combinations are considered. On the other
hand, MARS evaluates 24 fixed combinations.

Algorithm | Grid size | Parameters | R Package
---|---|---|---
mars | 24 |  degree $\in\\{1,2,3\\}$ | earth
|  |  nprune $\in\\{1,2,3,4,6,7,8\\}$ |
xgboost | 50 |  nrounds $\in\\{1:1000\\}$ | xgboost
|  |  max depth $\in\\{1:10\\}$ |
|  |  eta $\in\\{0,001:0,6\\}$ |
|  |  gamma $\in\\{0:10\\}$ |
random forest | 50 |  mtry $\in\\{0:92\\}$ | ranger
|  |  min node size $\in\\{1,2,3,4,5\\}$ |
|  | n.trees = 500 |
Table 1: Parameters considered for grid search during model training

####  5.1.2 Estimation

##### Validation.

Concerning estimation methodologies for the purposes of validation, we adopt a
10-fold cross validation method [8] on the train set for hyperparameter
optimisation.

##### Relevance function.

The data sets used in this study are of unknown nature, apart from features’
names. We are not capable of introducing any domain expertise to build any
kind of guess to what the relevance profile should look like. Therefore, we
assume the relevance function automatically generated according to Ribeiro and
Moniz, [17] as our reference.

(a) Germany data set (b) Beijing data set (c) Norway data set

Figure 4: Pollution data sets. In each chart, the vertical dashed line
corresponds to the nearest approximation of the domain knowledge relevance
function.

For the Convolution technique, we propose to simulate neighbouring
configurations of relevance functions within the range of one standard
deviation to the left and right of the automatically generated points. For
example, for a function with peak relevance at 100, i.e. $\phi(100)=1$, we
will simulate scenarios where values between $100-\sigma$ and $100+\sigma$
will be reference points for peak relevance. In short, the relevance function
is incrementally shifted to both sides in its original form. For the Elastic
technique, we apply it in the same range, but this time moving the left
extremity further to the left while the right extremity moves to the right,
and vice-versa. A good analogy is as if one decides to stretch and shrink the
original relevance function by pulling/pushing its peaks (for 1 and 0) Of
course, this is an educated guess of different scenario assumptions, enabling
to explore the data. We again emphasise that our final goal is to find a
method that can better withstand such uncertainties regarding relevance bands.

##### Optimisation criteria.

As previously discussed, a particularity of our learning setting is that,
differently from the standard setting of regression tasks, our work is set in
a context where one assumes that different values have different relevance to
users. Previous work has extensively demonstrated how the use of traditional
optimisation criteria within this scope may be misleading [17]. Accordingly,
our analysis concerning which are the best hyper-parameter combinations within
the scope of this experimental study is based on the use of the SERA metric
(Section 3.2).

###  5.2 Results

In this section, our goal is to demonstrate that the techniques proposed in
this paper are capable of properly assessing the robustness of predictive
models’ performance in the context of uncertainty in imbalanced regression
tasks. Accordingly, we divide this section in two parts:

  1. 1.

Case-study analysis of the Convolution and Elastic techniques applied to three
air pollution data sets and the indicator $PM10$, to grasp the impact of using
such techniques with policy-informed control points;

  2. 2.

Evaluation of prediction models in all available data sets concerning the
robustness of their performance when applying our our proposed techniques;

Figure 5: Rank shift probability for all data sets according to the
application of the Convolution and Elastic techniques.

Concerning the first part, a note on the data sets used. The first data set
concerns hourly averages of concentration levels in Beijing (China) from
August 2012 to March 2013 [22]; the second, daily averages for rural
background stations in Germany from 1998 to 2009 [14]; and, finally, hourly
values for a station in Alnabru, Oslo (Norway), between October 2001 and
August 2003 [1]. The benefit of using these data sets is that we can base the
set of control points used on official recommendations per the World Health
Organisation [15] for denoting 24-hour averages as normal or dangerous:
$\phi(50\mu$g$/m)=0$ and $\phi(150\mu$g$/m)=1$, respectively. Figure 4
illustrates the outcomes for the best models of each learning algorithm used,
estimated according to the methodology described in Section 5.1.2. The
scenario corresponding to the x-axis value of $0$ concerns the relevance
function using policy-informed control points.

A careful analysis of the results provides interesting insights concerning
possible scenarios. For example, concerning the first data set (left), we
observe that, when applying both techniques, there is a considerable
performance overlap despite different underlying relevance functions. In the
second data set (centre), we clearly notice a consistent distinction between
each of these models throughout the entire set of neighbouring relevance
functions. As for the third (right), we gain insights into how conclusions
concerning a certain relevance function may be brittle. In this case, we find
that small shifts in the reference points for relevance judgements may have
great impact in model evaluation and comparison. It also illustrates how
different shapes of the underlying relevance function may have distinct
impacts – by comparing the impact of the Convolution and the Elastic
techniques.

Concerning the second part, we test the Convolution and Elastic techniques in
the 29 data sets, having as reference the relevance function defined by the
automatic method proposed in Ribeiro and Moniz, [17]. The experimental
resolution for neighbouring configurations when applying our proposed
techniques is 19 steps for Convolution and 19 steps for the Elastic technique.
Results are presented in two distinct manners. First, illustrating an overview
of the experimental results, Figure 5 describes the rank shift probability for
the prediction models estimated as the best (best rank) among all candidates.
The rank shift probability translates as the percentage of neighbouring
scenarios where the best model according to the automatic relevance function
did not remain as the best. In other words, a robust scenario would describe a
near 0% of rank shift probability, meaning that the best model according to
initial performance rank remained as such in all neighbouring scenarios
tested, and vice-versa. Second, by analysing concrete cases that demonstrate
the usefulness of using the proposed techniques, based on interesting
scenarios from the first set of results – see Figure 6.

As an overall assessment of the results – see Figure 5 – we observe the same
type of scenarios as in the first part of our experimental results. In almost
half of the data sets used (14), neither of the proposed techniques
demonstrated performance rank shifts for the best model w.r.t the reference
relevance function. Around a quarter of the data sets demonstrated varying
levels of rank shift probability when applying both the Convolution and
Elastic techniques. We should highlight cases at both extremes, such as with
the data sets a7 and violentcrimes. The former presents probabilities of 50%
and 89% for Convolution and Elastic techniques, respectively; and, for the
latter, 17% and 22%. Finally, we observe 6 cases where only the application of
the Elastic technique rendered a positive probability. No cases with such
behaviour were observed when applying the Convolution technique. In this
aspect, we should mention the case of the data set Richter, with 39% of rank
shift probability for the Elastic technique and 0% for the Convolution
technique.

(a) Data set: Output (b) Data set: Richter (c) Data set: a7

Figure 6: Performance of models in experimental data sets under the
application of the Convolution and Elastic techniques

To better grasp the usefulness of the proposed techniques, we propose to look
closer at the results (Figure 6) from three data sets and their illustrative
scenarios: the Output, Richter and a7 data sets. In the Output data set, we
find a scenario of robustness in terms of performance, where simulations based
on both techniques conclude that the best initial performer (in this case a
Random Forest model) stays as such in all neighbouring configurations. As for
the Richter data set, we observe a good example where, if the relevance
function is convoluted to the right, algorithms present a similar performance,
but when the relevance function is convoluted to the left, the best model
(from the mars algorithm) is no longer the best option for the particular
imbalanced regression task. Finally, regarding the a7 data set we observe a
considerable discrepancy between results for both the Convolution and Elastic
techniques when contracting/expanding the form of the relevance function or
when sliding it right or left. This impact raises concrete issues on the
particular sensitivity of some domains to small changes in assertions
concerning what is most important to be accurate in terms of predictive
performance.

Given the experimental results obtained concerning the proposed methods, we
highlight the following conclusions.

  * •

Regime shift. Using both techniques, we observe that some data sets are
sensible to shifts in the relevance criteria for the target value, leading to
the best predictive model losing its initial advantage.

  * •

Function form. We also note that the duality between the approach in the
Convolution and Elastic techniques is useful: in several cases, the Elastic
technique demonstrates differences in model dominance, while the Convolution
technique does not;

  * •

Variety of scenarios. We consider a valuable contribution the diversity that
both the Elastic and Convolution techniques introduce in trying to understand
algorithm dominance under different relevance scenarios.

##  6 CONCLUSIONS AND FUTURE WORK

In this paper, we propose two simulation-based techniques that allow key
insights into the robustness of prediction models in imbalanced regression
tasks. We note that this is a particularly common scenario where domain
knowledge regarding which values are more important is not available. The two
proposed methods, Convolution and Elastic, extend the more traditional
performance evaluation process with the use of multiple relevance functions
aiming at the simulation of different scenarios of target value’s relevance
regions. Altogether, these methods have allowed us to better evaluate each
model’s performance on the relevant target value band, and by extending the
analysis to multiple relevance configurations, we are able to identify the
most consistent and robust model, i.e. the ones that will consistently perform
better than others regardless of which target value range is deemed the most
relevant.

Based on the experimental analysis of these methods provided in this paper, we
demonstrate how they can be useful for scientists working with imbalanced
regression problems where domain knowledge is unavailable or even yet to
answer details about which values are more important to predict – a known
scenario is key challenges such as early detection of Alzheimer’s disease.

For the sake of reproducible science, all data sets and code necessary to
reproduce the results presented in this paper are available in (removed for
anonymous submission).

#### References

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Published in Nature Scientific Reports, 12, 7105, 2 May 2022.

§ ABSTRACT

An experimental effort was conducted to measure the change in internal energy of non-ideal carbon dioxide as its volume rapidly expanded with the sudden opening of a valve from one to two compressed gas cylinders.  This was achieved by measuring the mass heat capacity of the gas cylinders and the manifold-valve, and measuring the change in temperature from the sudden doubling of volume of the non-ideal carbon dioxide.  It was determined that an empirical equation for the change in internal energy of a non-ideal fluid was more accurate than previous methods used for estimating the change in internal energy by estimating the change in entropy.  With this empirical equation, a theoretical ideal Stirling cycle heat engine that exceeds the Carnot efficiency was realized by utilizing non-ideal carbon dioxide as a working fluid.

§ INTRODUCTION

In an earlier publication [1], a theoretical Stirling cycle heat engine utilizing a real working fluid with significant intermolecular attractive (and repulsive) Van der Waals forces [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] was proposed.  The intermolecular attractive Van der Waals forces would both decrease the required work input during the cold isothermal compression, as well as reduce the work output recovered during the hot isothermal expansion.  If one were to look at empirical equations of state for real fluids, such as Redlich-Kwong [23] and Peng-Robinson [24], it becomes clear that the intermolecular attractive force increases with decreasing temperatures.  Because of the temperature dependence of the Van der Waals forces, which increase in strength with decreasing temperature, the reduction in cold work input is greater than the loss of the hot work output; therefore, the ideal efficiency of this macroscopic heat engine could in theory exceed the Carnot efficiency ${\eta}_C$

\begin{eqnarray}
\label{eq:eqEffCarnot}
\end{eqnarray}

Originally there were measurements of the enthalpy of vaporization of water [25, 26]; when calculating for the change in internal energy during vaporization, which is isothermal expansion of a real fluid, it was observed that the change in specific internal energy ${{\delta}u}_{{\delta}T=0}$ (J/kg) during isothermal compression and expansion followed a distinct empirical equation

\begin{eqnarray}
\label{eq:eqDeltaU_isothermal_mytheory}
{{\delta}u}_{{\delta}T=0}&=&{{{a'}{\cdot}{{T}^{-0.25}}}{\cdot}{({\rho_1}-{\rho_2})}}, \\
\end{eqnarray}
where ${\rho}_1$ and ${\rho}_2$ (m$^3$/kg) represent the density, \emph{T} (K) represents the absolute temperature, $R_G$ (J/kg${\cdot}$K) represents the gas constant, $T_C$ (K) represents the critical temperature, and $P_C$ (Pa) represents the critical pressure.  This equation has been found to match well for numerous fluids [1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 25, 45, 46, 47, 26, 48, 49, 50].

For non-isothermal changes in specific internal energy of a fluid ${{\delta}u}$ (J/kg), one must include the specific intermolecular kinetic energy $u_{KE}={C_V}{\cdot}{R_G}{\cdot}{T}$ (J/kg) [51],
\begin{eqnarray}
\label{eq:eqDeltaU_mytheory}
{{\delta}u}&=&{{C_V}{\cdot}{R_G}{\cdot}{{\delta}{T}}}-{{{a'}{\cdot}{{T}^{-0.25}}}{\cdot}{{\delta}{\rho}}}, \\
\end{eqnarray}
where \emph{T} (K) represents the absolute temperature, $R_G$ (J/kg${\cdot}$K) represents the gas constant, and ${C_V}$ is equal to the number of degrees of freedom of the molecule plus one half (ex. monatomic fluids $C_V=1.5$, diatomic fluids $C_V=2.5$, etc).  By integrating equation \ref{eq:eqDeltaU_mytheory} from a given density $\rho$ (kg/m$^3$) to infinitely low density (a true ideal gas) to find the intermolecular potential energy, and the temperature from absolute zero to the current temperature \emph{T}, one can calculate the total specific internal energy \emph{u} (J/kg) with equation \ref{eq:eqU_mytheory}
\begin{eqnarray}
\label{eq:eqU_mytheory}
{u}&=&{{C_V}{\cdot}{R_G}{\cdot}T}-{{{a'}{\cdot}{\rho}}{\cdot}{{T}^{-0.25}}},\\ \nonumber
\end{eqnarray}

Clausius' Theorem for the second law [2]
\begin{eqnarray}
\label{eq:eqClausius}
\end{eqnarray}
states that any internally reversible thermodynamic cycle must generate a positive entropy ${\delta}{s_u}{\geq}0$ to the surrounding universe, where the change in specific entropy ${\delta}s$ (J/kg$\cdot$K) is defined as [3, 4, 5, 6, 7]
\begin{eqnarray}
\label{eq:eqSideal}
\end{eqnarray}
where \emph{T} (K) is the absolute temperature, and ${\delta}q$ (J/kg) represent the heat transferred per unit mass.  If a fluid were to consistently follow equation \ref{eq:eqClausius}, then the change in specific internal energy ${{\delta}u}$ (J/kg) would consistently follow equation \ref{eq:eqdU_ideal} [1, 3, 4, 6]
\begin{eqnarray}
\label{eq:eqdU_ideal}
\end{eqnarray}
It should be noted that in most of the experimental measurements of the enthalpy of vaporization [1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 25, 45, 46, 47, 26, 48, 49, 50] there is great similarity between equation \ref{eq:eqDeltaU_mytheory} and equation \ref{eq:eqdU_ideal}.

Clausius' equation \ref{eq:eqClausius} makes intuitive sense for a reversible thermodynamic process utilizing an ideal-gas as its working fluid.  With a real-fluid, however, intermolecular Van-der-Waals forces impact the molecular behavior and thermodynamic properties.  In most published references and tables, the specific internal energy \emph{u} (J/kg) is often set to zero at an arbitrary point (often the triple-point), and calculated assuming equation \ref{eq:eqdU_ideal}, which was formulated for the purpose of holding equation \ref{eq:eqClausius} applicable.

In contrast, equation \ref{eq:eqDeltaU_mytheory} is an empirical equation based on measurements of the change in internal energy for numerous different molecules during vaporization.  In addition, equation \ref{eq:eqDeltaU_mytheory} makes physical sense, as it includes both the internal kinetic energy of the molecules as defined with the Kinetic Gas Theory [51], as well as the intermolecular potential energy.  Lennard-Jones [6, 52, 53], a well-established approximation for the potential energy from intermolecular attractive and repulsive Van der Waals forces, assumes the attractive force is inverse proportional to the molecular distance to the sixth power $r^{-6}$ (m$^{-6}$); the attractive intermolecular force is thus inverse proportion to the specific volume squared $v^{-2}$ (m$^{-6}$), and this is observed in most empirical equations of state for a real fluid [23, 24].  Integrating a potential force inverse-proportional to the specific volume squared would yield a potential energy inverse proportional to the specific volume \emph{v} (m$^3$/kg), or proportional to the density $\rho$ (kg/m$^3$), as observed in equations \ref{eq:eqDeltaU_mytheory} and \ref{eq:eqU_mytheory}.
\begin{eqnarray}
\end{eqnarray}

\section{Non-Ideal Stirling Cycle Heat Engine}

A Stirling cycle heat engine [4, 1] has isothermal (constant temperature) compression at a cold sink (Stage 1-2), isochoric (constant volume) heating to a hot temperature (Stage 2-3), isothermal expansion at the hot temperature source (Stage 3-4), and isochoric cooling back to the cold temperature (Stage 4-1).  For a true, ideal-gas Stirling engine to reach the Carnot efficiency (equation \ref{eq:eqEffCarnot}), it is necessary for all of the heat output from the isochoric cooling to go to the isochoric heating.

As a demonstration, carbon dioxide will be the working fluid, with a cold temperature of 32$^{\circ}$C and a hot temperature of 82$^{\circ}$C.  The density will shift from 70 kg/m$^3$ to 700 kg/m$^3$.  Using equation \ref{eq:eqEffCarnot}, the Carnot efficiency ${\eta}_C$
\begin{eqnarray}
\end{eqnarray}
The thermodynamic properties of this Stirling cycle engine are tabulated in Table \ref{tb:tbStirlingEng}, using both equation \ref{eq:eqU_mytheory} for the specific internal energy \emph{u} (J/kg) and the National Institute of Standards and Technology (NIST) Chemistry WebBook [54].  The pressures used to calculate the specific work inputs and outputs $w={\int}P{\cdot}{\delta}v$ (J/kg) from NIST [54] are tabulated in Table \ref{tb:tbPnist}.  The efficiency is calculated as
\begin{eqnarray}
\label{eq:eqEffStirling}
\end{eqnarray}
If the Stirling engine is utilizing an ideal-gas, then inherently ${q_{23}}=-{q_{41}}$; with a real fluid ${q_{23}}>-{q_{41}}$, and thus these values need to be included in equation \ref{eq:eqEffStirling}.

The efficiency $\eta$ calculated with the values from NIST [54], derived with equation \ref{eq:eqdU_ideal}, is 13.92\%,
\begin{eqnarray}
\end{eqnarray}
The efficiency calculated with equation \ref{eq:eqU_mytheory} is 16.89\%,
\begin{eqnarray}
\end{eqnarray}
which actually exceeds the Carnot efficiency $\eta_C$ of 14.08\%!

If one calculates the change in entropy ${\delta}s$ (J/kg${\cdot}{^{\circ}}$C) to the ambient universe with equation \ref{eq:eqSideal} throughout the full cycle
\begin{eqnarray}
{{\delta}s_{NIST}}={{\oint}{\frac{{\delta}q}{T}}}=({\frac{{q_{23}}+{q_{34}}+{q_{41}}}{T_H}}+{\frac{q_{12}}{T_L}})=({\frac{261,305}{82+273.15}}+{\frac{-224,939}{32+273.15}})=-1.38,\nonumber \\
\end{eqnarray}
it is noticed that the net total change in entropy per cycle with the NIST [54] internal energies ${{\delta}s_{NIST}}$ (J/kg${\cdot}{^{\circ}}$C), derived with equation \ref{eq:eqdU_ideal}, obeys Clausius' equation \ref{eq:eqClausius}; equation \ref{eq:eqdU_ideal} was in fact originally derived to ensure a thermodynamic cycle obeys Clausius' equation \ref{eq:eqClausius}.  If one assumes the internal energy of a real fluid can be determined with equation \ref{eq:eqU_mytheory}, an empirical equation based on previous measurements of the enthalpy of vaporization [1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 25, 45, 46, 47, 26, 48, 49, 50]
then the net total change in entropy per cycle ${{\delta}s_{calc}}$ (J/kg${\cdot}{^{\circ}}$C) fails to obey Clausius' equation \ref{eq:eqClausius}.

If the theoretical macroscopic Stirling cycle heat engine utilizing real fluids described is to exceed the Carnot efficiency, then equation \ref{eq:eqDeltaU_mytheory} must be the most accurate description of the change in internal energy for a non-ideal fluid.  If Clausius' (equation \ref{eq:eqClausius}) remains applicable in the presence of intermolecular Van der Waals forces, however, then equation \ref{eq:eqdU_ideal} would apply; inherently equation \ref{eq:eqdU_ideal} would have greater changes in internal potential energy during isothermal compression and expansion of a real fluid (observed in Table \ref{tb:tbStirlingEng}), and ensuring the ideal Stirling efficiency $\eta$ is less than or equal to the Carnot efficiency ${\eta}_C$ defined in equation \ref{eq:eqEffCarnot}.  It is thus desired to perform an experiment to determine which, equation \ref{eq:eqDeltaU_mytheory} or equation \ref{eq:eqdU_ideal}, is the most accurate definition of the change in specific internal energy ${\delta}{u}$ (J/kg) of a non-ideal fluid.

\section{Experimental Description}

To determine if equation \ref{eq:eqDeltaU_mytheory} or equation \ref{eq:eqdU_ideal} is the most accurate definition of ${\delta}{u}$ (J/kg), a simple and easily repeatable experiment was performed.  Two compressed gas cylinders, manufactured by \emph{Luxfur}, with a volume of 3.4 liters each, and designed to hold 5 lbs of carbon dioxide (CO$_2$), were obtained.  They were connected by a manifold assembly (Figure \ref{fig:fig_Manifold}) which included adapters from the CGA-320 valve to NPT, a ball valve, a tee, and a separate ball valve (to bleed the small amount of CO$_2$ during disassembly).  Three calibrated thermocouples (Taylor \# 9940, Panel-Mount LCD Thermometer with Remote Probe; range  {-40}$^{\circ}$C to 150$^{\circ}$C) were used, with one attached via aluminum tape to each cylinder, and the third attached to the manifold.

Before the experiment could commence, it was necessary to characterize the mass heat capacity $mC_P$ (J/$^{\circ}$C) of both the individual cylinders, as well as the manifold assembly.  First, the cylinder or manifold was left in a freezer (maintained at a temperature of -20$^{\circ}$C) for at least 24 hours, and the temperature inside the freezer was first measured and recorded as $T_{0,Sample}$ ($^{\circ}$C).  A mass (kg) of water was first weighed, and this water was poured into a bag resting inside an insulated chest.  A thermocouple was left in the bottom of the bag, and the temperature was then collected as the initial temperature of the water $T_{0,Water}$ ($^{\circ}$C).  The sample, either the cylinder or the manifold, was quickly moved into the water-filled bag, and insulating material was then piled on top of the bag to the limit that the insulated chest could be securely closed.  The water temperature, measured by the thermocouple, would quickly drop and later settle.  It was observed that the temperature would often settle after 15 minutes, but the final temperature $T_F$ ($^{\circ}$C) was collected after 60 minutes (minimal difference between the 15 minute measurements).  By using the NIST Chemistry WebBook [54] to determine the heat $Q_{water}$ (J) out of the mass of water from the change in temperature, the heat into the sample could be determined, and the mass heat capacity $mC_P$ (J/$^{\circ}$C)  could be estimated
\begin{eqnarray}
{mC_P}&=&{\frac{Q_{water}}{{T_F} - {T_{0,Sample}}}}.
\end{eqnarray}
This was performed twice with both the cylinder and the manifold, and the results are tabulated in Table \ref{tb:tbExp_mCp}.  The averaged measured mass heat capacity of the cylinder is 2,524 J/${^{\circ}}$C, and the mass heat capacity of the manifold is 454 J/${^{\circ}}$C.

The process of the experiment was to have a mass of CO$_2$ in Cylinder 1, and leave Cylinder 2 empty.  The mass of CO$_2$ was determined by simply weighing Cylinder 1 before the experiment, and subtracting the measured mass of the empty cylinder (3,460.6 g).  The initial temperature on Cylinder 1, Cylinder 2, and the manifold (3) was recorded, and then the two-cylinder assembly was added to the insulated chest and thoroughly covered in insulation (Figure \ref{fig:fig_Test_Setup}).  To commence the experiment, the ball valve was suddenly opened, allowing for CO$_2$ to flow from Cylinder 1 to Cylinder 2, thus instantly doubling the volume and suddenly dropping the temperature due to the Joule–Thomson effect [55].  Over time, the temperature of the cylinders and manifold would drop, as heat would flow from the aluminum cylinders and steel manifold into the cooler CO$_2$ until they reached thermal equilibrium, and the final temperature on Cylinder 1, Cylinder 2, and the manifold (3) was recorded.  The results of these temperature measurements are tabulated in Table \ref{tb:tbExp_Numbers1}.

Afterwards, the remaining mass of CO$_2$ in Cylinder 1 and Cylinder 2 was recorded, and these results, along with the initial mass in Cylinder 1, are tabulated in Table \ref{tb:tbExp_Numbers2}.  In addition, with the known temperatures (Table \ref{tb:tbExp_Numbers1}) and densities (mass over the 3.4 liter volume), the specific internal energy \emph{u} (kJ/kg) values as determined by the NIST Chemistry WebBook [54] (which were derived by NIST with equation \ref{eq:eqdU_ideal}) were collected and tabulated in Table \ref{tb:tbExp_Numbers2}.  On average, less than 1\% of the CO$_2$ was lost in the disassembly of the manifold or due to leaking.  In addition, it was frequently observed that only a small portion of the mass (approximately 400g) would travel from Cylinder 1 to Cylinder 2; this is expected, as much of the liquid CO$_2$ (and thus most of the mass) would stay in the original Cylinder 1 rather than travel through the manifold.  It was also noticed that at times Cylinder 2 would experience a slight \emph{increase} in temperature; mainly due to the kinetic energy of expansion, and the low masses of CO$_2$ that made its way into Cylinder 2.

\section{Experimental Analysis}

Carbon dioxide (CO$_2$) [56] has a critical temperature $T_C$ of 304.128 K; a critical pressure $P_C$ of 7,377,300 Pa; a critical specific density $\rho_C$ of 467.6 (kg/m$^3$), 3 degrees of freedom, a molar mass of 44 g/mole, and a Pitzer eccentric factor [57] of 0.228.  The specific gas constant $R_G$ for CO$_2$ is 188.924 J/kg${\cdot}{^{\circ}}$C; the ideal-gas specific heat at a constant volume $C_V=3.5{\cdot}{R_G}$; and the specific heat ratio ${\kappa}={\frac{C_P}{C_V}}={\frac{4.5}{3.5}}=1.28$.  For CO$_2$, the value of \emph{a'} as defined in equation \ref{eq:eqU_mytheory} is 728.46 {Pa}${\cdot}${K$^{0.25}$}$\cdot${{m$^6$}${\cdot}$kg$^{-2}$}.

The density of saturated liquid CO$_2$ $\rho_L$ (kg/m$^3$) and saturated gas CO$_2$ $\rho_G$ (kg/m$^3$) is defined with equation \ref{eq:eqRhoLG} [56]
\begin{eqnarray}
\label{eq:eqRhoLG}
{ln(\frac{\rho_L}{\rho_C})}&=&{{\Sigma}_{i=1}^4}{a_i}{\cdot}{(1-{\frac{T}{T_C}})^{t_i}}, \\ \nonumber
\end{eqnarray}
where $T_C$ is 304.128 K, $\rho_C$ is 467.6 (kg/m$^3$), and the values of $a_i$, $t_i$, $b_i$, and $u_i$ are tabulated in Table \ref{tb:tbRhoLG}.

Table \ref{tb:tbExp_Dens} contains the tabulated densities $\rho$ (kg/m$^3$) of the CO$_2$, both before the experiment $\rho_0$ (kg/m$^3$), and after the experiment in Cylinder 1 $\rho_{F1}$ (kg/m$^3$) and Cylinder 2 $\rho_{F2}$ (kg/m$^3$).  In addition, the densities of a saturated liquid $\rho_L$ (kg/m$^3$) and a saturated gas $\rho_G$ (kg/m$^3$) for the experimentally measured CO$_2$ temperatures (Table \ref{tb:tbExp_Numbers1}) as determined with equation \ref{eq:eqRhoLG} is also tabulated in Table \ref{tb:tbExp_Dens}.  With the density $\rho$ (kg/m$^3$), saturated liquid density $\rho_L$ (kg/m$^3$), and saturated gas density $\rho_G$ (kg/m$^3$), the vapor quality \emph{X} was determined with equation \ref{eq:eqX}, and tabulated in Table \ref{tb:tbQual_Exp}.
\begin{eqnarray}
\label{eq:eqX}
\end{eqnarray}
Utilizing the density $\rho$ (kg/m$^3$), saturated liquid density $\rho_L$ (kg/m$^3$), and saturated gas density $\rho_G$ (kg/m$^3$) tabulated in Table \ref{tb:tbExp_Dens}, as well as the experimentally measured temperatures tabulated in Table \ref{tb:tbExp_Numbers1}, and the masses of CO$_2$ tabulated in Table \ref{tb:tbExp_Numbers2}, the internal energy \emph{U} (kJ) was calculated using the empirical equation \ref{eq:eqU_mytheory}, and tabulated in Table \ref{tb:tbU_CO2_xLG}.

For qualities \emph{X} greater than 1, the CO$_2$ is treated as a vapor, and the internal energy is estimated solely with the empirical equation \ref{eq:eqU_mytheory}, and the final internal energy \emph{U} (kJ) was tabulated in Table \ref{tb:tbU_final_theory}.  For qualities \emph{X} less than 1 (there were no measurements at a greater density than the saturated liquid density), the internal energy of the liquid-vapor mixture \emph{U} (kJ) was calculated with equation \ref{eq:eqUfctX} from the quality \emph{X} (Table \ref{eq:eqX}), saturated liquid internal energy $U_L$ (kJ), and the saturated gas internal energy $U_G$ (kJ), both tabulated in Table \ref{tb:tbU_CO2_xLG}.
\begin{eqnarray}
\label{eq:eqUfctX}
\end{eqnarray}
All of these final internal energies \emph{U} (kJ) are tabulated in Table \ref{tb:tbU_final_theory}.

Finally, the estimated heat inputs were determined with equation \ref{eq:eqFindQ}, plotted in Figure \ref{fig:fig_Q_in_figure}, and tabulated in Table \ref{tb:tbQin_combo}.  These include the heat estimates $Q_{theory}$ (kJ) utilizing the internal energies derived from the empirical equation \ref{eq:eqU_mytheory} and tabulated in Table \ref{tb:tbU_final_theory}; as well as the heat estimates $Q_{NIST}$ (kJ) utilizing the specific internal energies collected from the NIST Chemistry WebBook [54] and tabulated in Table \ref{tb:tbExp_Numbers2}.
\begin{eqnarray}
\label{eq:eqFindQ}
\end{eqnarray}
The values of $Q_{theory}$ (kJ) and $Q_{NIST}$ (kJ) are compared to the experimentally measured heat inputs $Q_{EXP}$ (kJ), determined by comparing the measured changes in temperature (Table \ref{tb:tbExp_Numbers1}) with the mass heat capacity tabulated in Table \ref{tb:tbExp_mCp}, as described in equation \ref{eq:eqFindQexp},
\begin{eqnarray}
\label{eq:eqFindQexp}
\end{eqnarray}

\section{Conclusion}

When analyzing the results of Table \ref{tb:tbQin_combo} and Figure \ref{fig:fig_Q_in_figure}, with 25 independent test results, the correlation between $Q_{Theory}$ (equation \ref{eq:eqDeltaU_mytheory}) and $Q_{EXP}$ is 0.9560; in excess of the correlation of 0.9229 between $Q_{NIST}$ (equation \ref{eq:eqdU_ideal}) and $Q_{EXP}$.  The average error between $Q_{Theory}$ and $Q_{EXP}$ is 29\%, less than the average error of 64\% between $Q_{NIST}$ and $Q_{EXP}$.  The median error between $Q_{Theory}$ and $Q_{EXP}$ is 17\%, less than the median error of 27\% between $Q_{NIST}$ and $Q_{EXP}$.  Finally, the standard deviation of the error between $Q_{Theory}$ and $Q_{EXP}$ is 29\%, less than the standard deviation of the error of 153\% between $Q_{NIST}$ and $Q_{EXP}$.  The experimental data suggests that equation \ref{eq:eqDeltaU_mytheory} is the most accurate definition of the change in internal energy of a real fluid ${\delta}{u}$ (J/kg), as compared to equation \ref{eq:eqdU_ideal}.  This effort provides an experimental justification to the possibility of the theoretical macroscopic Stirling cycle heat engine utilizing real fluids described earlier [1] exceeding the Carnot efficiency.

\section*{Data availability}
All data generated or analysed during this study are included in this published article [and its supplementary information files].

\clearpage

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c | c | c |}
\hline
Stage & P (Pa) & T (K) & $\rho$ (kg/m$^3$) & $u_{NIST}$ (J/kg) & $u_{calc}$ (J/kg)\\
\hline
\hline
1 & 3,334,500 & 305.15 & 70 & 430,810 & 189,575\\
\hline
2 & 8,650,400 & 305.15 & 700 & 275,020 & 79,771\\
\hline
3 & 26,745,000 & 355.15 & 700 & 322,020 & 117,374\\
\hline
4 & 4,132,200 & 355.15 & 70 & 468,660 & 223,091\\
\hline
\hline
Stages & w (J/kg) & ${\delta}u_{NIST}$ (J/kg) & $q_{NIST}$ (J/kg) & ${\delta}u_{calc}$ (J/kg) & $q_{calc}$ (J/kg)\\
\hline
\hline
12 & 69,149 & -155,790 & -224,939 & -109,804 & -178,953\\
\hline
23 & 0 & 47,000 & 47,000 & 37,603 & 37,603\\
\hline
34 & -105,515 & 146,640 & 252,155 & 105,717 & 211,232\\
\hline
41 & 0 & -37,850 & -37,850 & -33,516 & -33,516\\
\hline
\end{tabular}
\caption{The stages of the non-ideal Stirling cycle heat engine, as well as the specific work inputs and outputs \emph{w} (J/kg), specific heat inputs and outputs \emph{q} (J/kg), and specific internal energies \emph{u} (J/kg) from both \emph{NIST} [54] and calculated \emph{calc} with equation \ref{eq:eqU_mytheory}.}
\label{tb:tbStirlingEng}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c || c | c |}
\hline
$\rho$ (kg/m$^3$) & $P_{L}$ (MPa) & $P_{H}$ (MPa) & $u_{L}$ (kJ/kg) & $u_{H}$ (kJ/kg)\\
\hline
\hline
70 & 3.3345 & 4.1322 & 430.81 & 468.66\\
\hline
88.146 & 3.5471 & 5.0367 & 428.9 & 463.45\\
\hline
107.63 & 3.7598 & 5.9412 & 426.92 & 457.93\\
\hline
128.64 & 3.9724 & 6.8457 & 424.86 & 452.08\\
\hline
151.42 & 4.185 & 7.7502 & 422.73 & 445.85\\
\hline
176.22 & 4.3977 & 8.6548 & 420.5 & 439.22\\
\hline
203.31 & 4.6103 & 9.5593 & 418.18 & 432.14\\
\hline
232.93 & 4.823 & 10.464 & 415.74 & 424.59\\
\hline
265.24 & 5.0356 & 11.368 & 413.18 & 416.58\\
\hline
300.17 & 5.2482 & 12.273 & 410.47 & 408.18\\
\hline
337.28 & 5.4609 & 13.177 & 407.6 & 399.53\\
\hline
375.61 & 5.6735 & 14.082 & 404.52 & 390.85\\
\hline
413.83 & 5.8861 & 14.986 & 401.2 & 382.42\\
\hline
450.56 & 6.0988 & 15.891 & 397.59 & 374.49\\
\hline
484.73 & 6.3114 & 16.795 & 393.6 & 367.23\\
\hline
515.82 & 6.524 & 17.7 & 389.12 & 360.69\\
\hline
543.76 & 6.7367 & 18.604 & 383.94 & 354.85\\
\hline
568.77 & 6.9493 & 19.509 & 377.72 & 349.63\\
\hline
591.17 & 7.1619 & 20.413 & 369.68 & 344.96\\
\hline
611.31 & 7.3746 & 21.318 & 357.28 & 340.75\\
\hline
629.52 & 7.5872 & 22.222 & 303.52 & 336.94\\
\hline
646.07 & 7.7999 & 23.127 & 289.34 & 333.46\\
\hline
661.22 & 8.0125 & 24.031 & 283.9 & 330.26\\
\hline
675.15 & 8.2251 & 24.936 & 280.23 & 327.31\\
\hline
688.03 & 8.4378 & 25.84 & 277.38 & 324.57\\
\hline
700 & 8.6504 & 26.745 & 275.02 & 322.02\\
\hline
\end{tabular}
\caption{The pressures and specific internal energies versus the density $\rho$, taken from NIST [54], and used to solve the work and heat inputs and outputs listed in Table \ref{tb:tbStirlingEng}.  The pressure $P_L$ (MPa) and specific internal energy $u_L$ (kJ/kg) is at 32$^{\circ}$C, and the pressure $P_H$ (MPa) and specific internal energy $u_H$ (kJ/kg) is at 82$^{\circ}$C.}
\label{tb:tbPnist}
\end{center}
\end{table}

\clearpage

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c | c | c |}
\hline
Sample & Mass Water & $T_0$ \emph{Sample} & $T_0$ \emph{Water} & $T_F$ & $mC_P$\\
{} & (kg) & ($^{\circ}$C) & ($^{\circ}$C) & ($^{\circ}$C) & (J/$^{\circ}$C)\\
\hline
\hline
Cylinder & 4.01 & -16.1 & 23.7 & 18.4 & 2575.23\\
\hline
Cylinder & 3.9 & -16.3 & 23.2 & 18 & 2473.09\\
\hline
\hline
Manifold & 1.6 & -20 & 24.7 & 21.8 & 464.56\\
\hline
Manifold & 1.77 & -19.3 & 24.9 & 22.4 & 443.96\\
\hline
\end{tabular}
\caption{The results of the effort to find the mass heat capacity $mC_P$ (J/$^{\circ}$C) of the cylinder and the manifold.  }
\label{tb:tbExp_mCp}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c || c | c | c |}
\hline
Test & $T_{1,0}$ & $T_{2,0}$ & $T_{3,0}$ & $T_{1,F}$ & $T_{2,F}$ & $T_{F,3}$ \\
Number & ($^{\circ}$C) &  ($^{\circ}$C) &  ($^{\circ}$C) &  ($^{\circ}$C) &  ($^{\circ}$C) & ($^{\circ}$C)\\
\hline
\hline
1-1 & 18.4 & 19.1 & 19.4 & 10.7 & 11.7 & 13.1\\
\hline
1-2 & 18.2 & 18.7 & 18.3 & 7.9 & 19.9 & 11.8\\
\hline
1-3 & 17.9 & 19.3 & 19.9 & 6.9 & 20.3 & 12.9\\
\hline
1-4 & 18.4 & 19.2 & 18.7 & 8.4 & 20.3 & 12.7\\
\hline
1-5 & 17.9 & 18.1 & 18.6 & 13.6 & 19.3 & 12.4\\
\hline
\hline
2-1 & 15.1 & 15.6 & 15.6 & 7.5 & 8.2 & 9.8\\
\hline
2-2 & 14.7 & 15.2 & 16.0 & 4.8 & 16.1 & 8.9\\
\hline
2-3 & 10.6 & 14.5 & 12.9 & 1.1 & 15.6 & 6.9\\
\hline
2-4 & 14.5 & 15.5 & 16.2 & 3.4 & 16.7 & 8.3\\
\hline
2-5 & 13.5 & 15.3 & 16.3 & 10.2 & 16.3 & 12.1\\
\hline
2-6 & 13.1 & 13.3 & 13.2 & 11.4 & 14.2 & 11.6\\
\hline
\hline
3-1 & 10.9 & 10.0 & 10.3 & 3.1 & 6.6 & 4.9\\
\hline
3-2 & 10.2 & 9.0 & 9.7 & 0.7 & 10.5 & 4.2\\
\hline
3-3 & 7.6 & 8.9 & 10.2 & -1.7 & 10.2 & 3.8\\
\hline
3-4 & 4.7 & 9.4 & 8.5 & -4.5 & 10.4 & 1.7\\
\hline
3-5 & 11.2 & 11.5 & 11.4 & 0.4 & 12.7 & 5.2\\
\hline
3-6 & 7.2 & 11.4 & 9.9 & 2.2 & 11.7 & 4.1\\
\hline
3-7 & 8.2 & 11.2 & 11.0 & 7.0 & 12.0 & 8.6\\
\hline
\hline
4-1 & 10.9 & 11.8 & 12.2 & 4.5 & 4.8 & 6.4\\
\hline
4-2 & 8.0 & 9.0 & 9.2 & 0.8 & 5.3 & 3.9\\
\hline
4-3 & 9.8 & 11.1 & 11.0 & 0.5 & 11.6 & 5.1\\
\hline
4-4 & 9.4 & 11.2 & 10.4 & -0.1 & 12.2 & 4.9\\
\hline
4-5 & 8.0 & 11.2 & 10.9 & -1.6 & 12.0 & 3.8\\
\hline
4-6 & 11.3 & 11.8 & 12.3 & 6.2 & 12.6 & 8.5\\
\hline
4-7 & 11.6 & 12.1 & 12.2 & 9.9 & 13.0 & 10.6\\
\hline
\end{tabular}
\caption{The measured temperature ($^{\circ}$C) of Cylinder \emph{1} and Cylinder \emph{2}, as well as the manifold temperature \emph{3}, both before \emph{0} and after \emph{F} the manifold valve was opened.}
\label{tb:tbExp_Numbers1}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c || c | c | c |}
\hline
{Test} & $mass_0$ &  $mass_{F1}$ & $mass_{F2}$ & $u_{0}$ & $u_{F1}$ & $u_{F1}$ \\
{Number} & (g) & (g) & (g) & (kJ/kg) & (kJ/kg) & (kJ/kg) \\
\hline
\hline
1-1 & 2026.7 & 1565.6 & 459.8 & 256.95 & 249.92 & 391.15\\
\hline
1-2 & 1565.2 & 1200.2 & 364.1 & 272.42 & 258.99 & 408.49\\
\hline
1-3 & 1199.1 & 848.4 & 350.8 & 292.83 & 285.54 & 410.19\\
\hline
1-4 & 847.4 & 484.4 & 342.4 & 332.90 & 370.17 & 411.06\\
\hline
1-5 & 483.8 & 246.9 & 236.3 & 394.34 & 415.90 & 421.39\\
\hline
\hline
2-1 & 1880.4 & 1385.4 & 491.0 & 251.91 & 247.99 & 366.59\\
\hline
2-2 & 1385.4 & 1054.0 & 332.0 & 270.20 & 258.64 & 408.73\\
\hline
2-3 & 1053.8 & 764.2 & 288.0 & 278.59 & 273.67 & 413.02\\
\hline
2-4 & 764.0 & 444.7 & 305.1 & 329.22 & 357.58 & 412.07\\
\hline
2-5 & 444.6 & 225.9 & 216.4 & 394.46 & 415.61 & 421.24\\
\hline
2-6 & 226.3 & 110.9 & 109.5 & 417.76 & 258.16 & 431.34\\
\hline
\hline
3-1 & 1941.2 & 1582.5 & 349.8 & 238.88 & 227.90 & 398.84\\
\hline
3-2 & 1582.2 & 1270.0 & 299.9 & 247.85 & 233.03 & 407.66\\
\hline
3-3 & 1270.0 & 995.4 & 272.2 & 254.02 & 241.40 & 410.48\\
\hline
3-4 & 995.8 & 738.6 & 238.5 & 262.85 & 255.59 & 414.36\\
\hline
3-5 & 735.7 & 442.4 & 283.4 & 319.53 & 341.88 & 411.23\\
\hline
3-6 & 442.6 & 197.7 & 182.7 & 380.36 & 412.78 & 421.50\\
\hline
3-7 & 197.6 & 97.2 & 94.8 & 417.26 & 427.74 & 431.45\\
\hline
\hline
4-1 & 2233.5 & 1679.9 & 551.5 & 232.55 & 228.77 & 329.70\\
\hline
4-2 & 1679.9 & 1356.8 & 321.0 & 238.32 & 229.54 & 401.02\\
\hline
4-3 & 1356.8 & 1058.5 & 296.0 & 256.33 & 244.19 & 408.97\\
\hline
4-4 & 1058.4 & 773.8 & 283.2 & 274.00 & 267.86 & 410.85\\
\hline
4-5 & 773.5 & 503.8 & 268.1 & 299.86 & 311.51 & 412.35\\
\hline
4-6 & 503.6 & 257.5 & 244.1 & 379.36 & 408.98 & 415.44\\
\hline
4-7 & 257.2 & 128.7 & 126.6 & 413.23 & 426.21 & 428.62\\
\hline
\end{tabular}
\caption{The measured mass (g) of CO$_2$ in the 3.4 liter cylinders, as well as the specific internal energy \emph{u} (kJ/kg) collected from the NIST webbook [54].}
\label{tb:tbExp_Numbers2}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c | c | c |}
\hline
{ } & 1 & 2 & 3 & 4 & 5 \\
\hline
$a_i$ & 1.9245108 & -0.62385555 & -0.32731127 & 0.39245142 & {-} \\
\hline
$t_i$ & 0.340 & 0.5 & (10/6) & (11.6) & {-} \\
\hline
\hline
$b_i$ & -1.7074879 & -0.8227467 & -4.6008549 & -10.111178 & -29.742252\\
\hline
$u_i$ & 0.340 & 0.5 & 1 & (7/3) & (14/3)\\
\hline
\end{tabular}
\caption{Coefficient values for equation \ref{eq:eqRhoLG}.}
\label{tb:tbRhoLG}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c || c | c | c || c | c | c |}
\hline
{Test} & $\rho_{0}$ & $\rho_{0-L}$ & $\rho_{0-G}$ & $\rho_{F1}$ & $\rho_{F1-L}$ & $\rho_{F1-G}$ & $\rho_{F2}$ & $\rho_{F2-L}$ & $\rho_{F2-G}$ \\
{Num} & ($\frac{kg}{m^3}$) & ($\frac{kg}{m^3}$) & ($\frac{kg}{m^3}$) & ($\frac{kg}{m^3}$) & ($\frac{kg}{m^3}$) & ($\frac{kg}{m^3}$) & ($\frac{kg}{m^3}$) & ($\frac{kg}{m^3}$) & ($\frac{kg}{m^3}$)\\
\hline
\hline
1-1 & 596.09 & 789.83 & 182.35 & 460.47 & 855.86 & 138.39 & 135.22 & 848.19 & 143.19\\
\hline
1-2 & 460.35 & 791.80 & 180.96 & 353.00 & 876.26 & 126.02 & 107.09 & 774.46 & 193.42\\
\hline
1-3 & 352.68 & 794.72 & 178.89 & 249.53 & 883.22 & 121.94 & 103.18 & 770.16 & 196.58\\
\hline
1-4 & 249.24 & 789.83 & 182.35 & 142.47 & 872.72 & 128.12 & 100.71 & 770.16 & 196.58\\
\hline
1-5 & 142.29 & 794.72 & 178.89 & 72.62 & 832.99 & 152.94 & 69.50 & 780.74 & 188.86\\
\hline
\hline
2-1 & 553.06 & 820.32 & 161.31 & 407.47 & 879.06 & 124.37 & 144.41 & 874.14 & 127.28\\
\hline
2-2 & 407.47 & 823.76 & 159.02 & 310.00 & 897.32 & 113.88 & 97.65 & 811.50 & 167.27\\
\hline
2-3 & 309.94 & 856.61 & 137.92 & 224.76 & 920.75 & 101.12 & 84.71 & 815.95 & 164.25\\
\hline
2-4 & 224.71 & 825.47 & 157.89 & 130.79 & 906.39 & 108.85 & 89.74 & 806.04 & 171.01\\
\hline
2-5 & 130.76 & 833.81 & 152.40 & 66.44 & 859.61 & 136.07 & 63.65 & 809.69 & 168.51\\
\hline
2-6 & 66.56 & 837.08 & 150.29 & 32.62 & 850.51 & 141.72 & 32.21 & 828.00 & 156.21\\
\hline
\hline
3-1 & 570.94 & 854.34 & 139.33 & 465.44 & 908.30 & 107.80 & 102.88 & 885.27 & 120.75\\
\hline
3-2 & 465.35 & 859.61 & 136.07 & 373.53 & 923.18 & 99.84 & 88.21 & 857.36 & 137.46\\
\hline
3-3 & 373.53 & 878.37 & 124.78 & 292.76 & 937.45 & 92.52 & 80.06 & 859.61 & 136.07\\
\hline
3-4 & 292.88 & 897.98 & 113.51 & 217.24 & 953.40 & 84.68 & 70.15 & 858.11 & 136.99\\
\hline
3-5 & 216.38 & 852.05 & 140.76 & 130.12 & 925.00 & 98.89 & 83.35 & 840.30 & 148.21\\
\hline
3-6 & 130.18 & 881.15 & 123.15 & 58.15 & 913.96 & 104.74 & 53.74 & 848.19 & 143.19\\
\hline
3-7 & 58.12 & 874.14 & 127.28 & 28.59 & 882.53 & 122.34 & 27.88 & 845.85 & 144.67\\
\hline
\hline
4-1 & 656.91 & 854.34 & 139.33 & 494.09 & 899.29 & 112.78 & 162.21 & 897.32 & 113.88\\
\hline
4-2 & 494.09 & 875.56 & 126.44 & 399.06 & 922.58 & 100.16 & 94.41 & 894.02 & 115.74\\
\hline
4-3 & 399.06 & 862.58 & 134.25 & 311.32 & 924.40 & 99.21 & 87.06 & 848.97 & 142.70\\
\hline
4-4 & 311.29 & 865.51 & 132.46 & 227.59 & 928.01 & 97.33 & 83.29 & 844.28 & 145.67\\
\hline
4-5 & 227.50 & 875.56 & 126.44 & 148.18 & 936.87 & 92.81 & 78.85 & 845.85 & 144.67\\
\hline
4-6 & 148.12 & 851.28 & 141.24 & 75.74 & 887.99 & 119.18 & 71.79 & 841.10 & 147.70\\
\hline
4-7 & 75.65 & 848.97 & 142.70 & 37.85 & 861.84 & 134.70 & 37.24 & 837.89 & 149.76\\
\hline
\end{tabular}
\caption{The calculated density of CO$_2$ in the 3.4 liter cylinders, taken from the mass tabulated in Table \ref{tb:tbExp_Numbers2}.  The densities of a saturated liquid and a saturated gas are defined with equation \ref{eq:eqRhoLG}.}
\label{tb:tbExp_Dens}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c |}
\hline
{Test} & $X_{0}$ & $X_{F1}$ & $X_{F2}$\\
\hline
\hline
1-1 & 0.0976 & 0.1656 & 1.0709\\
\hline
1-2 & 0.2133 & 0.2490 & 2.0746\\
\hline
1-3 & 0.3641 & 0.4068 & 2.2155\\
\hline
1-4 & 0.6511 & 0.8820 & 2.2782\\
\hline
1-5 & 1.3319 & 2.3548 & 3.2654\\
\hline
\hline
2-1 & 0.1183 & 0.1907 & 0.8611\\
\hline
2-2 & 0.2444 & 0.2754 & 1.8982\\
\hline
2-3 & 0.3385 & 0.3820 & 2.1758\\
\hline
2-4 & 0.6323 & 0.8093 & 2.1497\\
\hline
2-5 & 1.2025 & 2.2451 & 3.0805\\
\hline
2-6 & 2.5332 & 5.0139 & 5.7458\\
\hline
\hline
3-1 & 0.0967 & 0.1281 & 1.2011\\
\hline
3-2 & 0.1593 & 0.1784 & 1.6650\\
\hline
3-3 & 0.2238 & 0.2411 & 1.8312\\
\hline
3-4 & 0.2989 & 0.3303 & 2.1340\\
\hline
3-5 & 0.5814 & 0.7313 & 1.9447\\
\hline
3-6 & 0.9372 & 1.9050 & 3.0028\\
\hline
3-7 & 2.3928 & 4.8072 & 6.0527\\
\hline
\hline
4-1 & 0.0586 & 0.1176 & 0.6588\\
\hline
4-2 & 0.1303 & 0.1598 & 1.2595\\
\hline
4-3 & 0.2141 & 0.2368 & 1.7682\\
\hline
4-4 & 0.3217 & 0.3606 & 1.9050\\
\hline
4-5 & 0.4808 & 0.5853 & 2.0069\\
\hline
4-6 & 0.9443 & 1.6626 & 2.2824\\
\hline
4-7 & 2.0654 & 4.0326 & 4.6798\\
\hline
\end{tabular}
\caption{The calculated vapor quality solved with equation \ref{eq:eqX}, utilizing the density $\rho$ (kg/m$^3$), saturated liquid density $\rho_L$ (kg/m$^3$), and saturated gas density $\rho_G$ (kg/m$^3$) tabulated in Table \ref{tb:tbExp_Dens}.}
\label{tb:tbQual_Exp}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c || c | c | c || c | c | c |}
\hline
{Test} & $U_{0x}$ & $U_{0-L}$ & $U_{0-G}$ & $U_{F1-x}$ & $U_{F1-L}$ & $U_{F1-G}$ & $U_{F2-x}$ & $U_{F2-L}$ & $U_{F2-G}$ \\
{Num} & (kJ) & (kJ) & (kJ) & (kJ) & (kJ) & (kJ) & (kJ) & (kJ) & (kJ)\\
\hline
\hline
1-1 & 177.74 & 108.52 & 325.56 & 165.91 & 56.05 & 255.40 & 75.57 & 17.45 & 74.92\\
\hline
1-2 & 174.49 & 83.02 & 251.60 & 147.67 & 35.93 & 196.14 & 63.69 & 20.91 & 58.15\\
\hline
1-3 & 156.19 & 62.70 & 192.94 & 119.41 & 23.67 & 138.68 & 61.70 & 20.52 & 55.93\\
\hline
1-4 & 126.13 & 45.37 & 136.12 & 77.91 & 15.00 & 79.14 & 60.37 & 20.03 & 54.59\\
\hline
1-5 & 80.97 & 25.30 & 77.84 & 43.64 & 10.41 & 40.13 & 42.80 & 13.20 & 37.83\\
\hline
\hline
2-1 & 174.55 & 85.70 & 304.78 & 156.63 & 40.34 & 226.43 & 78.73 & 15.00 & 80.23\\
\hline
2-2 & 163.86 & 61.86 & 224.73 & 135.42 & 24.98 & 172.30 & 57.77 & 15.91 & 53.69\\
\hline
2-3 & 139.75 & 37.50 & 171.92 & 107.84 & 12.63 & 124.75 & 50.68 & 13.46 & 46.63\\
\hline
2-4 & 114.95 & 33.76 & 123.98 & 70.93 & 9.32 & 72.67 & 53.64 & 15.06 & 49.26\\
\hline
2-5 & 73.98 & 18.64 & 72.27 & 39.66 & 7.85 & 36.87 & 38.99 & 10.47 & 34.98\\
\hline
2-6 & 40.17 & 9.29 & 36.81 & 20.22 & 4.14 & 18.08 & 20.18 & 4.76 & 17.78\\
\hline
\hline
3-1 & 167.94 & 70.32 & 316.61 & 157.46 & 32.23 & 258.59 & 58.30 & 9.55 & 57.18\\
\hline
3-2 & 165.71 & 54.96 & 258.22 & 145.02 & 20.02 & 207.26 & 51.55 & 10.61 & 48.93\\
\hline
3-3 & 151.34 & 37.24 & 207.56 & 126.37 & 11.20 & 162.14 & 47.13 & 9.45 & 44.42\\
\hline
3-4 & 130.91 & 23.40 & 162.78 & 102.34 & 4.50 & 119.95 & 41.75 & 8.39 & 38.92\\
\hline
3-5 & 110.09 & 27.13 & 119.96 & 69.71 & 6.72 & 72.18 & 49.38 & 11.38 & 46.13\\
\hline
3-6 & 71.79 & 12.62 & 72.34 & 33.94 & 3.68 & 32.29 & 32.67 & 6.93 & 29.77\\
\hline
3-7 & 34.72 & 6.04 & 32.29 & 17.51 & 2.73 & 15.89 & 17.41 & 3.66 & 15.44\\
\hline
\hline
4-1 & 159.16 & 80.91 & 364.28 & 160.29 & 38.82 & 274.61 & 85.40 & 13.07 & 90.16\\
\hline
4-2 & 164.64 & 50.64 & 274.52 & 148.83 & 21.64 & 221.44 & 53.70 & 7.93 & 52.48\\
\hline
4-3 & 157.68 & 45.98 & 221.50 & 132.51 & 16.28 & 172.72 & 51.16 & 11.17 & 48.24\\
\hline
4-4 & 139.20 & 34.98 & 172.83 & 108.15 & 11.02 & 126.21 & 49.25 & 11.06 & 46.12\\
\hline
4-5 & 112.49 & 23.32 & 126.40 & 77.07 & 5.76 & 82.07 & 46.80 & 10.35 & 43.67\\
\hline
4-6 & 81.49 & 18.68 & 82.10 & 44.09 & 6.82 & 42.10 & 43.02 & 9.75 & 39.73\\
\hline
4-7 & 44.98 & 9.71 & 41.92 & 23.22 & 4.39 & 21.01 & 23.12 & 5.17 & 20.60\\
\hline
\end{tabular}
\caption{The calculated internal energy \emph{U} (kJ), solved with equation \ref{eq:eqU_mytheory}, utilizing the measured temperature
in Table \ref{tb:tbExp_Numbers1}, and the densities tabulated in Table \ref{tb:tbExp_Dens}.}
\label{tb:tbU_CO2_xLG}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c |}
\hline
{Test} & $U_{0}$ & $U_{F1}$ & $U_{F2}$\\
{Num} & (kJ) & (kJ) & (kJ)\\
\hline
\hline
1-1 & 129.69 & 89.06 & 75.57\\
\hline
1-2 & 118.97 & 75.82 & 63.69\\
\hline
1-3 & 110.12 & 70.46 & 61.70\\
\hline
1-4 & 104.46 & 71.57 & 60.37\\
\hline
1-5 & 80.97 & 43.64 & 42.80\\
\hline
\hline
2-1 & 111.61 & 75.84 & 71.17\\
\hline
2-2 & 101.66 & 65.55 & 57.77\\
\hline
2-3 & 83.00 & 55.46 & 50.68\\
\hline
2-4 & 90.81 & 60.59 & 53.64\\
\hline
2-5 & 73.98 & 39.66 & 38.99\\
\hline
2-6 & 40.17 & 20.22 & 20.18\\
\hline
\hline
3-1 & 94.15 & 61.24 & 58.30\\
\hline
3-2 & 87.34 & 53.43 & 51.55\\
\hline
3-3 & 75.36 & 47.59 & 47.13\\
\hline
3-4 & 65.07 & 42.64 & 41.75\\
\hline
3-5 & 81.09 & 54.60 & 49.38\\
\hline
3-6 & 68.60 & 33.94 & 32.67\\
\hline
3-7 & 34.72 & 17.51 & 17.41\\
\hline
\hline
4-1 & 97.51 & 66.55 & 63.85\\
\hline
4-2 & 79.81 & 53.57 & 53.70\\
\hline
4-3 & 83.56 & 53.32 & 51.16\\
\hline
4-4 & 79.33 & 52.56 & 49.25\\
\hline
4-5 & 72.88 & 50.42 & 46.80\\
\hline
4-6 & 78.57 & 44.09 & 43.02\\
\hline
4-7 & 44.98 & 23.22 & 23.12\\
\hline
\end{tabular}
\caption{The calculated internal energies \emph{U} (kJ), solved with equation \ref{eq:eqU_mytheory}, adjusting for mixed liquid-vapor, using the internals energies \emph{U} (kJ) tabulated in Table \ref{tb:tbU_CO2_xLG}, with the qualities \emph{X} tabulated in Table \ref{tb:tbQual_Exp}, solved with equation \ref{eq:eqUfctX}.}
\label{tb:tbU_final_theory}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ | c || c | c | c |}
\hline
{Test} & $Q_{Theory}$ & $Q_{NIST}$ & $Q_{EXP}$\\
Num & (kJ) & (kJ) & (kJ)\\
\hline
\hline
1-1 & 34.9442 (14.71\%) & 50.3454 (22.88\%) & 40.9726\\
\hline
1-2 & 20.5379 (20.76\%) & 33.1792 (28.01\%) & 25.9194\\
\hline
1-3 & 22.0339 (22.47\%) & 35.0143 (23.21\%) & 28.4180\\
\hline
1-4 & 27.4844 (9.12\%) & 37.9578 (50.70\%) & 25.1876\\
\hline
1-5 & 5.4758 (48.53\%) & 11.4785 (7.89\%) & 10.6392\\
\hline
\hline
2-1 & 35.3962 (12.59\%) & 49.8695 (23.16\%) & 40.4932\\
\hline
2-2 & 21.6599 (16.50\%) & 33.9698 (30.96\%) & 25.9394\\
\hline
2-3 & 23.1390 (3.29\%) & 34.5102 (44.24\%) & 23.9256\\
\hline
2-4 & 23.4268 (18.01\%) & 33.2143 (16.24\%) & 28.5742\\
\hline
2-5 & 4.6671 (39.48\%) & 9.6657 (25.33\%) & 7.7120\\
\hline
2-6 & 0.2402 (91.25\%) & -18.6774 (780.27\%) & 2.7456\\
\hline
\hline
3-1 & 25.3884 (17.36\%) & 36.4521 (18.66\%) & 30.7204\\
\hline
3-2 & 17.6409 (22.25\%) & 26.0571 (14.84\%) & 22.6890\\
\hline
3-3 & 19.3648 (16.16\%) & 29.4168 (27.36\%) & 23.0976\\
\hline
3-4 & 19.3141 (18.79\%) & 25.8576 (8.72\%) & 23.7840\\
\hline
3-5 & 22.8826 (15.39\%) & 32.7121 (20.95\%) & 27.0452\\
\hline
3-6 & -1.9849 (113.69\%) & -9.7327 (167.14\%) & 14.4960\\
\hline
3-7 & 0.1985 (90.54\%) & 0.0272 (98.70\%) & 2.0992\\
\hline
\hline
4-1 & 32.8876 (9.79\%) & 46.7398 (28.21\%) & 36.4548\\
\hline
4-2 & 27.4498 (8.25\%) & 39.8135 (33.08\%) & 29.9178\\
\hline
4-3 & 20.9232 (15.94\%) & 31.7417 (27.53\%) & 24.8898\\
\hline
4-4 & 22.4886 (6.11\%) & 33.6212 (40.37\%) & 23.9510\\
\hline
4-5 & 24.3473 (4.28\%) & 35.5481 (39.76\%) & 25.4346\\
\hline
4-6 & 8.5325 (32.17\%) & 15.6756 (24.62\%) & 12.5784\\
\hline
4-7 & 1.3649 (50.29\%) & 2.8338 (3.21\%) & 2.7456\\
\hline
\end{tabular}
\caption{The combined energy input \emph{Q} (kJ) into the CO$_2$, defined with equation \ref{eq:eqFindQ} and equation \ref{eq:eqFindQexp}, using the theory defined in equation \ref{eq:eqU_mytheory}, from NIST in Table \ref{tb:tbExp_Numbers2} [54], and measured experimentally.  The parentheses represent the percent (\%) error with $Q_{EXP}$.  These tabulated results are plotted in Figure \ref{fig:fig_Q_in_figure}.}
\label{tb:tbQin_combo}
\end{center}
\end{table}

\clearpage

\begin{figure}
\centering
\includegraphics[width=\linewidth]{Manifold}
\caption{The manifold to connect the two 3.4 liter CO$_2$ cylinders.  }
\label{fig:fig_Manifold}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=\linewidth]{Test_Setup}
\caption{The full experimental apparatus, with two CO$_2$ cylinders connected by the manifold (Figure \ref{fig:fig_Manifold}), inside the insulated chest, with the three thermocouples attached.  }
\label{fig:fig_Test_Setup}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=\linewidth]{Q_in_figure}
\caption{The combined energy input \emph{Q} (kJ) into the CO$_2$, defined with equation \ref{eq:eqFindQ} and equation \ref{eq:eqFindQexp}, using the theory defined in equation \ref{eq:eqU_mytheory}, from NIST in Table \ref{tb:tbExp_Numbers2} [54], and measured experimentally.  These plotted results are tabulated in Table \ref{tb:tbQin_combo}.}
\label{fig:fig_Q_in_figure}
\end{figure}

\clearpage

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\end{document}