# Uncertainty analysis and composite hypothesis under the likelihood paradigm

André Chalom  Paulo Inácio de Knegt López de Prado,
Department of Ecology, Institute of Biosciences, University of São Paulo
<EMAIL_ADDRESS>
(07/28/2015)

###### Abstract

The correct use and interpretation of models depends on several steps, two of
which being the calibration by parameter estimation and the analysis of
uncertainty. In the biological literature, these steps are seldom discussed
together, but they can be seen as fitting pieces of the same puzzle. In
particular, analytical procedures for uncertainty estimation may be masking a
high degree of uncertainty coming from a model with a stable structure, but
insufficient data.

Under a likelihoodist approach, the problem of uncertainty estimation is
closely related to the problem of composite hypothesis. In this paper, we
present a brief historical background on the statistical school of
Likelihoodism, and examine the complex relations between the law of likelihood
and the problem of composite hypothesis, together with the existing proposals
for coping with it. Then, we propose a new integrative methodology for the
uncertainty estimation of models using the information in the collected data.
We argue that this methodology is intuitively appealing under a likelihood
paradigm.

“In an ideal world, all data would come from well-designed experiments and
would be sufficient to simultaneously estimate all parameters using rigorous
statistical procedures. The world is not ideal. One must often combine
estimates from different experiments, or supplement high-quality data (…) with
uncertain data, or even assumptions (…). Think carefully about whether your
conclusions may be artifacts of your assumptions or calculations, and document
those assumptions and calculations so that your reader can ask the same
question, and then carry on.”

(H. Caswell, Matrix Population Models)

###### Contents

  1. 1 Motivation
  2. 2 Historical background
  3. 3 The challenge for uncertainty estimation
  4. 4 GLL: A general law of likelihood
  5. 5 WLL: A weak law of likelihood
  6. 6 PLUE: a proposal for likelihood profiling
    1. 6.1 Intuition
    2. 6.2 Method
  7. 7 Case study
    1. 7.1 Mathematical details
    2. 7.2 R code for performing the analyses

##  1 Motivation

The use of mathematical and computational models is now widespread in several
fields of the biological sciences. However, several applications of these
models lack one or more steps that should be taken in order to correctly
understand and interpret its results [Bart,, 1995]. Though seldom discussed
together, five steps can be seen as fundamental in this process: verification,
validation, calibration, uncertainty analysis and sensitivity analysis.

The two first steps are very tightly correlated, both in historical terms and
in the common practice, so it’s common to refer to the set of validation and
verification as “V & V”. We can think of verification as trying to ensure that
the computer code is correctly implementing the desired model, while
validation is trying to ensure that the theoretical model is capable of
reproduce the actual phenomena of interest. Another way to see it is to think
that validation is determining that the model is solving the right equations,
while verification is determining that the model is solving the equations
right [Meisner,, 2010].

The third step, calibration, consists in determining the right values for the
input parameters of a model in order to enhance the correspondence between a
model run and one observed situation. An appropriate calibration of the model
is then essential for the model to be used in a predictive fashion [Meisner,,
2010].

The last steps consist in determining how much the variation of the input
parameters is translated into the total variation of the results, which is
called uncertainty analysis, and how much of the variation of the results can
be ascribed to the variation of each individual parameter, which is called
sensitivity analysis [Helton and Davis,, 2003; Helton et al.,, 2005]. These
two have been extensively reviewed in another text ([Chalom and Prado,,
2012]), so we will just highlight the fact that these analyses, together with
model calibration, strongly depend on previously and correctly executing the
model verification and validation.

During the development and analysis of mathematical models in ecology, it is
common to have completely separated procedures for calibration by parameter
estimation (which we will call CPE) and sensitivity and uncertainty analyses
(SUA) of a model. These steps are carried out in different sections of the
research papers, discussed in different chapters of books (ie: [Caswell,,
1989]), and sometimes are even performed by different people. In matrix models
for population projections, the results discussion is focused on a single
combination of parameters, considered to be an optimum for the CPE, and the
SUA is presented as a separate step, being secondary to the main results (see,
for example, [Silva Matos et al.,, 1999]). We have no guarantees that even the
statistical schools of though considered for both procedures will be coherent.

Moreover, the parameter set used in the model calibration may have few to no
relation to parameters that have biological relevance. In the case of matrix
models, the former may be parameters related to a time series, while the later
might be the projection matrix entries. This way, the researcher responsible
for the calibration may see the system as individual counts over time, while
the researcher responsible for the uncertainty analyses will only be
interested in the vital rates, which are real valued. When developed in such
disparate worlds, the SUA will be unable to point out directions for the
planning of new experiments or to single out which parameters should be
targeted for more sampling effort.

This lack of integration between these fundamental steps for model analysis
may have its roots in the development of the CPE and SUA theories. While the
parameter estimation has always been seen as belonging to a statistical
theory, the sensitivity of biological models was developed as an analytical
tool, based on the linear expansion of functions around a privileged point.
Depending on the nature of the experiments designed to measure vital rates,
there may be a plethora of methods that might be used to determine the set of
values that best represent the status of knowledge that we have about a given
natural system - and these methods may stem from frequentist, Bayesian or
likelihood-based approaches to the statistical theory. The analytical theory
of SUA, on the other hand, is in great part due to the work of Hal Caswell
[Caswell,, 1989], and proceeds by taking a model that is already optimally
parametrized, and studying the first order derivatives of the model answer in
relation to each input parameter. This formulation disregards completely the
quantity and quality of collected data, except for their average, and
sensitivity measures are taken exclusively over the variation of the model
answer in response to infinitesimal changes. Thus, models that are
parametrized by data with large uncertainties will not present larger
sensitivity indexes than models parametrized by data with great precision.
Increasing the sampling effort to gather data about the system will not affect
the uncertainty values. We can conclude that the analytical formulation of the
SUA, ultimately, gives information about the structure of the model, but is
silent about the more embracing questions from the collection of field data to
the formulation and execution of the model. This might lead to a false sense
of confidence being placed on studies that present a stable matrix structure,
leading to small sensitivity indexes, but use data with large uncertainties.

Other than this, questions about model validation are not usually addressed by
literature in this field, but they are also intimately connected to the
relevant questions about uncertainty. We can divide the uncertainty about a
model in three main sources: structural uncertainty, parameter uncertainty and
stochastic uncertainty (see [Marino et al.,, 2008] for a review about the
latter two). While most of the SUA techniques are focused on the second
component, the validation of a model may point out how much confidence we can
have about whether one model (or a set of models) is adequate at representing
the desired phenomena.

A stochastic formulation for a theory of global uncertainty and sensitivity
analyses, as described in [Chalom and Prado,, 2012], and interpreted under the
likelihood paradigm for statistics, is able to regard in the same way all of
the uncertainty sources described above, generating a complete picture of our
knowledge about a system. Although these formulations may converge for very
simple systems, this is not the general case.

The main advantages of using a stochastic formulation are:

  1. 1.

The analytical procedures are local, thus responding only to what happens
after infinitesimal perturbations, and depends on the model functions being
“well-behaved” in the chosen neighborhood. The stochastic procedures are
global and have no such limitation.

  2. 2.

Any stochastic approach allows for the information contained in the sample
variability to be used to represent the uncertainty about the parameters.

  3. 3.

By accepting the Likelihood Principle, we know that all relevant information
on the samples is contained in the associated likelihood function. This way,
we can use the likelihood function alone to proceed in the analysis, while the
analytical procedures at times will disregard information, and at times will
lead to the false sensation of having more detailed information than the
sample allows.

##  2 Historical background

The statistical use of likelihoods is very widespread, starting from Fisher’s
definition on 1922, and spanning several schools of thought. On the one hand,
the framework for Neyman-Pearson’s hypothesis testing is built upon it; on the
other hand, it is a bridge between the a priori and a posteriori probabilities
in Bayesian analysis. Starting on the decade of 1960, the use of likelihoods
as a basis for statistical inference started to be more widely advocated as an
alternative to both frequentism and Bayesianism. The 1965 book by Ian Hacking
about the logic of inference and the 1972 book by A.W.F. Edwards on
likelihoods can be seen as the starting points for this position to become an
independent school of thought, sometimes called Likelihoodism. More recently,
this position will be strongly supported by the works of Richard Royall and
Elliott Sober.

The likelihoodist school of thought is based on the conjunction of the
likelihood principle, as demonstrated by Allan Birnbaum, and the law of
likelihood, as enunciated by Ian Hacking.

Birnbaum derived the likelihood principle in 1962, from two generally accepted
principles111 This demonstration is controversial to this date. See [Mayo,,
2010] for a recent critique and [Gandenberger,, 2012] for a recent defense:
the sufficiency principle and the conditionality principle. Informally, we can
think about the two first affirming that “data that does not aggregate new
information is irrelevant to the inference”, and “experiments that could have
been realized but were not are irrelevant to the inference”. From this
principles, Birnbaum deduces the likelihood principle, which can informally be
stated as “experimental results which are possible but were not observed are
irrelevant to the inference”. Or, in Birnbaum’s more formal terms:

> “The likelihood principle: If $E$ and $E\prime$ are any two experiments with
> the same parameter space, represented respectively by density functions
> $f(x,\theta)$ and $g(y,\theta)$; and if $x$ and $y$ are any respective
> outcomes determining the same likelihood function; then
> $Ev(E,x)=Ev(E\prime,y)$. That is, the evidential meaning of any outcome $x$
> of any experiment $E$ is characterized fully by giving the likelihood
> function $cf(x,\theta)$ (which need be described only up to an arbitrary
> positive constant factor), without other reference to the structure of $E$.”
> [Birnbaum,, 1962]

The law of likelihood was formulated by Hacking as:

> “If hypothesis $A$ implies that the probability that a random variable $X$
> takes the value $x$ is $p_{A}(x)$, while hypothesis $B$ implies that the
> probability is $p_{B}(x)$, then the observation $X=x$ is evidence supporting
> $A$ over $B$ if and only if $p_{A}(x)>p_{B}(x)$, and the likelihood ratio,
> $p_{A}(x)/p_{B}(x)$, measures the strength of that evidence.” [Hacking,,
> 1965]222Emphasis added

The divergence between the likelihood school and frequentist school is due to
the later rejecting the likelihood principle (although it accepts the
principles of sufficiency and conditionality). On the other hand, the Bayesian
school generally disagrees with the Likelihoodists by the interpretation of
the law of likelihood.

The likelihood school of thought can be seen as an intellectual heir to the
Neyman-Pearson hypothesis testing framework, as its breaking point from
Fisherian significance testing happens because of the logical necessity of
comparing competing hypothesis. Fisher’s tests give a special privileged
position to the so-called null hypothesis, which Neyman and Pearson will see
as incomplete333 For a more complete and extremely didactic insight into their
view, see the original paper from [Neyman and Pearson,, 1933]. The data
gathered by one experiment may indicate a small or a large support for one
given hypothesis, but this alone should not be considered as evidence in favor
of this hypothesis before the other alternatives have been also scrutinized.
In other words, the probability or improbability of a single hypothesis should
not be used to conclude about its veracity. As the fictional detective
Sherlock Holmes, by Sir Arthur Conan Doyle famously said, “How often have I
said to you that when you have eliminated the impossible, whatever remains,
however improbable, must be the truth?”.

However, while Neyman and Pearson will use the likelihood ratio from several
hypothesis to guide the behavior of the scientist in accepting or rejecting
hypothesis, Royall and Edwards will separate the concepts of (1) the degree of
certainty that we have about one hypothesis; (2) the strength of evidence that
some data confers to one hypothesis over the others and (3) the course of
action that should be taken after examining such evidences. While Neyman and
Pearson conflate the strength of evidence (2) with the course of action (3),
and Bayesian analysis conflate the degree of certainty (1) with the strength
of evidence (2), the proposers of Likelihoodism propose that the role of
statistical inference is to provide only the strength of evidence [Royall,,
1997]. The degree of certainty and course of action should take into account a
multiplicity of other factors, and should not be part of the inference. One of
the greatest precursors of the statistical thought, Marquis Pierre-Simon de
Laplace, had already foreshadowed this position with the problem of the number
of judges necessary to convict or acquit a prisoner: this decision, to
Laplace, needs to take into account not only the probability of convicting an
innocent of acquitting a guilty prisoner, but also whether the punishment will
be a fine or the death sentence[Laplace,, 1814].

Moreover, by dissociating the strength of evidence from the decision making,
Royall and Edwards are asking the scientist to reveal the untransformed
likelihood ratios derived from its data, instead of transforming it into
p-values or a posterioris, so that her peers are able to assess clearly what
is the presented evidence. By retaking the scientist role as a subjective
decision maker, this proposition converges to Fisher’s, who criticized Neyman-
Pearson’s plan as being useful for industrial processing, but useless to the
scientific community. The likelihoodist proposal thus echoes the words of
Fisher, and also the words of the Bayesian probabilist I. J. Good:

> “We have the duty of formulating, of summarizing, and of communicating our
> conclusions, in intelligible form, in recognition of the right of other free
> minds to utilize them in making their own decisions.” [Fisher,, 1955]

> “If a Bayesian is a subjectivist, he will know that the initial probability
> density varies from person to person and so he will see the value of
> graphing the likelihood function for communication.” [Good,, 1976]

However, there is a very important difference between the Bayesian and the
Likelihoodist paradigms: the a posteriori probabilities, constructed by the
multiplication of the likelihood by an a priori probability under the Bayesian
school, is a absolute quantity. It is thus possible to refer to the a
posteriori probability of event $A$, or the a posteriori probability of an
event $B$. In contrast, the Likelihoodist approach will see in the likelihood
ratio the end point for the statistical analysis. This is why the law of
likelihood is expressed over evidence supporting $A$ over $B$: because the
likelihoodist approach does not allow for the transformation of the relational
support of $A$ over $B$ to be translated into non-relational quantities of
support for each hypothesis444See the text by [Fitelson,, 2007] for more
details on this matter, including Carnap’s classification of confirmation.

##  3 The challenge for uncertainty estimation

From this section, we will consider problems where $\mathbf{x}$ represent a
vector of data obtained independently from one or more experiments from a
vector-valued random variable $\mathbf{X}$, such that
$P(\mbox{$\mathbf{X}$}\\!=\\!\mbox{$\mathbf{x}$})=f(\mbox{$\mathbf{x}$};\theta)$.
The parameter vector $\theta$ is unknown and may assume values in $\Theta$. We
will refer to the likelihood of $\theta$ given the observed data $\mathbf{x}$
as $\mathcal{L}(\theta|\mbox{$\mathbf{x}$})=f(\mbox{$\mathbf{x}$};\theta)$.
The likelihood ratio over the same data set can be abbreviated as
$L(\theta_{1},\theta_{2})=\frac{\mathcal{L}(\theta_{1}|\mbox{$\mathbf{x}$})}{\mathcal{L}(\theta_{2}|\mbox{$\mathbf{x}$})}$.

In order to perform an uncertainty analyses for a mathematical model under the
likelihood paradigm, the general question that we would like to answer can be
written as: “how much support does the collected data provide for the
concurrent hypothesis over the model results?”. For example, in a population
growth model, the question may become “how much support the collected data
provide for the hypothesis that the population is growing at the rate of 1
individual per year, versus the hypothesis that it is stable ?”555As we will
see, the question can become more general, such as “how much support the
collected data provide for the hypothesis that the population is growing or
stable versus the hypothesis that is declining?”

This question, formulated over the parameters of a probability distribution,
should be answered by means of the likelihood function. It can be demonstrated
that one-to-one functions over the parameters preserve the likelihood
function, such that if $\mathcal{L}(\theta|\mbox{$\mathbf{x}$})$ is the
likelihood of $\theta$ and $\phi=f(\theta)$ is given by a one-to-one function
$f(\cdot)$, the likelihood of $\phi$ is given by
$\mathcal{L}(\phi|\mbox{$\mathbf{x}$})=\mathcal{L}\left(f(\theta)|\mbox{$\mathbf{x}$}\right)$
[Edwards,, 1972].

However, the application of a mathematical model is analogous to the
application of a generic function, and the reasoning above does not hold for
arbitrary many-to-one functions. Thus, defining a methodology for uncertainty
analyses of arbitrary models is closely related to the problem of composite
statistical hypotheses. However, the law of likelihood is expressed in terms
of simple statistical hypothesis, and there is no way of combining these
values to form the “likelihood of a composite hypothesis”. As Edwards writes:

> “No special meaning attaches to any part of the area under a likelihood
> curve, or to the sum of the likelihoods of two or more hypotheses (…).
> Although the likelihood function, and hence the curve, has the mathematical
> form of a [known] distribution, it does not represent a statistical
> distribution in any sense.” [Edwards,, 1972]

How can we work with composite hypothesis then? The traditional answer from
the proposers of Likelihoodism is that we don’t. Edwards puts off composite
hypothesis as “uninteresting to science”, while Royall sees this restriction
to simple hypothesis as an advantage, not a limitation, to the likelihood
approach. However, this is necessary if we want to derive an uncertainty
estimation for model results. Here, we will present three paths that may be
pursued for this end:

  1. 1.

Formulate a new law of likelihood, which can be applied for composite
hypothesis;

  2. 2.

Present a methodology based on a logical extension of the law of likelihood;

  3. 3.

Address the issue using a model similar to the one applied for nuisance
parameters.

Proposal 1 is based on the axiomatic formulation of a generalized law of
likelihood (GLL) that is compatible with the law of likelihood (LL) for simple
hypothesis for simple hypothesis, but able to extend its domain for composite
hypothesis. And proposal 2 is based on the use of a weaker definition of
statistical evidence, based on a weak law of likelihood (WLL). We will review
some existing results related to these proposals in the following sections.

Proposal 3 is based on methods such as profiled likelihood, which are ad hoc
methods widely used to reduce the study of multiparametric statistical models
to visualizing one parameter at a time. A typical use for these methods is
exemplified by the problem of fitting a normal distribution over some data.
While, strictly, the comparable hypothesis must be given by pairs
$(\mu=\mu_{0},\sigma=\sigma_{0})$, representing one value for the mean and on
for the variance of the distribution, it is possible to compare the
approximate support for different values for the mean, while leaving the
variance free to assume any viable value - thus treating the composite
hypothesis $(\mu=\mu_{0},\sigma\geq 0)$. We will get back to this proposal in
section 6.

##  4 GLL: A general law of likelihood

The most straightforward way to pursue a general law of likelihood is by
considering the maximum666 Or more precisely, the supremum, as the set of
likelihoods does not need to have a maximum of the likelihoods of the simple
hypothesis as the likelihood of the composite hypothesis. This is precisely
the path followed by Zhang, [2009]; Zhang and Zhang, [2013] and Bickel,
[2010]. We will follow the more didactic approach by Zhang. Let us consider
two hypothesis, $H_{1}:\theta\in\Theta_{1}\subset\Theta$ versus
$H_{2}:\theta\in\Theta_{2}\subset\Theta$. Now we postulate the following two
axioms777The notation was sightly altered for coherency with the other
sections:

######  Axiom 4.1.

If
$\inf\mathcal{L}(\Theta_{1}|\mbox{$\mathbf{x}$})>\sup\mathcal{L}(\Theta_{2}|\mbox{$\mathbf{x}$})$,
then the observation $\mathbf{x}$ is evidence supporting $\Theta_{1}$ over
$\Theta_{2}$.

######  Axiom 4.2.

If $\mathbf{x}$ is evidence supporting $H_{1}^{*}$ over $H_{2}$ and
$H_{1}^{*}$ implies $H_{1}$, then $\mathbf{x}$ is evidence supporting $H_{1}$
over $H_{2}$.

The first axiom ensures that if the images of
$\mathcal{L}(\Theta_{1}|\mbox{$\mathbf{x}$})$ and
$\mathcal{L}(\Theta_{2}|\mbox{$\mathbf{x}$})$ are disjoint intervals, or in
other words, if every simple hypothesis that implies $H_{1}$ is supported over
every simple hypothesis that implies $H_{2}$, then $H_{1}$ is supported over
$H_{2}$. There seem to be no reason to reject this axiom, other than a
predisposition to ignore composite hypothesis. The second axiom is used to
ensure a form of logical coherence. It is important to stress that this
coherence axiom is not used to suppose that the logic structure of the
hypothesis is somehow preserved in the likelihood function. In particular, if
$H_{1}^{*}$ implies $H_{1}$, the axiom 4.2 does not warrant the assertion that
$H_{1}^{*}$ is better supported than $H_{1}$.

From these axioms, we can derive the following general law of likelihood
(GLL):

######  Theorem.

(GLL) “If
$\sup\mathcal{L}(\Theta_{1}|\mbox{$\mathbf{x}$})>\sup\mathcal{L}(\Theta_{2}|\mbox{$\mathbf{x}$})$,
then there $\mathbf{x}$ is evidence supporting $H_{1}$ over $H_{2}$.” [Zhang,,
2009]

###### Proof.

Consider
$\sup\mathcal{L}(\Theta_{1}|\mbox{$\mathbf{x}$})>\sup\mathcal{L}(\Theta_{2}|\mbox{$\mathbf{x}$})$.
Then there exists $\theta_{1}\in\Theta_{1}$ so that
$\mathcal{L}(\theta_{1}|\mbox{$\mathbf{x}$})>\sup\mathcal{L}(\Theta_{2}|\mbox{$\mathbf{x}$})$.
From the axiom 4.1, the hypothesis $H_{1}^{*}:\theta=\theta_{1}$ is supported
over $H_{2}$. But $H_{1}^{*}$ implies $H_{1}$, so the conclusion follows
directly from axiom 4.2. ∎

The use of a “generalized likelihood ratio” given by
$\sup\mathcal{L}(\Theta_{1}|\mbox{$\mathbf{x}$})/\sup\mathcal{L}(\Theta_{2}|\mbox{$\mathbf{x}$})$
seems natural to quantify the strength of evidence, just as the likelihood
ratio is used to quantify the strength of evidence under the law of likelihood
(LL). It is trivial to see that the GLL is compatible with LL in the case of
simple hypothesis, and with some particular cases proposed by [Royall,, 1997].
However, it is striking to see that this formulation of the GLL allows for an
absolute degree of hypothesis confirmations. One hypothesis
$H_{1}:\theta\in\Theta_{1}$ can be absolutely supported if
$\sup\mathcal{L}(\Theta_{1}|\mbox{$\mathbf{x}$})>\sup\mathcal{L}(\Theta_{1}^{c}|\mbox{$\mathbf{x}$}$,
where c represents the complimentary set. That is equivalent to
$L(\Theta_{1},\Theta_{1}^{c})>1$. This absolute likelihood will always be zero
for simple hypothesis concerning real values parameters, but this does not
happen then $\Theta$ is finite.

On the other hand, accepting the generalized likelihood ratio makes it
impossible for a researcher to combine evidence from multiple data sources, as
the generalized likelihood is not a multiplicative quantity. So, the greatest
problem here is to find a generalization that allows for a measure of strength
of evidence that can be combined across experiments.

##  5 WLL: A weak law of likelihood

A more general way of expressing statistical support is given by

WLL

“Evidence $E$ favors hypothesis $H_{1}$ over hypothesis $H_{2}$ if
$P(E|H_{1})>P(E|H_{2})$ and $P(E|\sim\\!\\!H_{1})\leq P(E|\sim\\!\\!H_{2})$.”
[Fitelson,, 2007]

Evidently, LL $\implies$ WLL. However, WLL is also implied by most modern
Bayesian proponents. Given a confirmation measure $c(H,E)$, the Bayesian
equivalent for the LL is:

$\dagger_{c}$

“Evidence $E$ favors hypothesis $H_{1}$ over hypothesis $H_{2}$ according to
measure $c$ if and only if $c(H_{1},E)>c(H_{2},E)$.” [Fitelson,, 2007]

There are three main formats for the confirmation measures:

Difference

$d(H,E)=P(H|E)-P(H)$

Ratio

$r(H,E)=\frac{P(H|E)}{P(H)}$

Likelihood ratio

$l(H,E)=\frac{P(E|H)}{P(E|\sim\\!\\!H)}$

It is important to remember that the Bayesian confirmation given by $c(H,E)$
has a non-relational nature, and the relation confirmation is constructed by
comparing two of those measures. Likelihoodists will see in the LL a primitive
relational relation. While the construction of the non-relational confirmation
measures depends on the specification of a prioris, the WLL makes no such
assumptions, using only catch-all probabilities (the $P(E|\sim\\!\\!H)$
terms). One theory based on the WLL without invoking a prioris could be used
as a base to an intermediate inference school of thought, one that is not
dependent on the specification of a prioris, but without the arbitrary
restriction on the use of composite hypothesis.

##  6 PLUE: a proposal for likelihood profiling

In this section, we will describe how the profiling techniques for likelihood
functions can be seen as a foundation for a methodology of uncertainty
estimation for mathematical models. We reason that this methodology is
intuitively appealing under a likelihood paradigm; and we will refer to it as
PLUE - Profiled Likelihood Uncertainty Estimation.

###  6.1 Intuition

Suppose that we have one mathematical model, for example a structured
population growth model like those exemplified in [Chalom and Prado,, 2012],
and one data set from which we can estimate the vital rates of the population,
such as survival, growth and fertility rates.

With this data in hands, we would like to ask the following questions:

“What is the support given by the data to the affirmation that the population
is stable versus that it is declining? What is the support given by the data
to the affirmation that the population will become extinct in up to 10 years
versus more than 10 years?”

These questions cannot be answered under a strict Likelihoodist point of view,
as they refer to composite hypothesis over the parameters. However, we can
take the maximum likelihood point as privileged and restrict the question to
the form:

“What is the support given by the data to point of maximum likelihood versus
any point in which the population growth is negative?”

Schematically, we can see in figure 1 that we are comparing the likelihood at
the global maximum ($x_{max}$) with the maximum value attained at a distinct
region ($x_{lim}$). This kind of question is now comparing one simple
hypothesis, corresponding to the maximum likelihood, with a composite
hypothesis. This kind of comparison avoids the problems and quirks that
afflict the general laws of likelihood due to the superposition of likelihood
intervals. In terms of the Zhang axioms (see section 4), we are accepting a
weak form of the first, but not the second.

This reasoning can be extended to the following:

“Which are the points in the parameter space for which the support given by
the data versus the point of maximum likelihood is greater than a certain
threshold $\delta$?”

By asking the same question for different values of $\delta$, we are
effectively profiling the likelihood surface for the parameter space. In this
point, we need to remember that profiling methods for likelihood are used in
inference since the decade of 1970 to reduce the dimensionality of parameters
spaces, in particular in the form of elimination of “nuisance parameters”.
Even if the status of this procedures is not fully understood under the
likelihood paradigm (see for example [Kalbfleisch and Sprott,, 1970] for the
relation between nuisance parameters and fiducial inference), these methods
are generally accepted as a valid procedure for exploratory analysis.

We have already discussed the typical example of fitting a normal distribution
to a data set. In this example, the likelihood is defined for pairs
$(\mu=\mu_{0},\sigma=\sigma_{0})$, but it is usual to make affirmations about
the average $\mu$ of the distribution, without referring to $\sigma$. However,
this is equivalent to consider the model $g(\mu,\sigma)=\mu$, that is, a model
that projects the bi-dimensional space formed by $\mu$ and $\sigma$ into an
unidimensional space. Our proposal is merely to use this same reasoning for
general non-invertible functions. Even if this is nothing new under the optics
of the likelihood profiling, this is the first time that this methodology is
proposed in the literature of uncertainty analyses, to the best of our
knowledge.

Figure 1: Schematic representation of a likelihood function, highlighting two
interest points: the global maximum $x_{max}$ and the maximum in a given
region $x_{lim}$. See details on the main text.

###  6.2 Method

Definitions

Consider first a mathematical model of interest. We will call this the
“biological model”888 But it could be a physical model, hydrological model,
geochemical model, etc. to differentiate between this and the statistical
model presented below. We will consider the simple case of a scalar response
$y$ for a model with a vector input $\mathbf{\theta}$:
$y=F(\mbox{$\mathbf{\theta}$})$.

Let’s represent by $\chi$ the set of data collected in field or laboratory
experiments. If we have $n$ parameters and $m$ observations, $\chi$ is a table
with $n$ columns and $m$ rows.

Now formulate different models to explain your data (ex: constant or grouped
fertility, constant or decreasing growth rate, models with 4 or 5 size
classes, etc.). These will be the statistical models. Write the likelihood
function $\mathcal{L}(\mbox{$\mathbf{\theta}$}|\chi)$ for each model, find the
best set of parameters that fits your data and determine the value of AIC for
each statistical model (see for example [Burnham and Anderson,, 2002] for
details in this step).

Sampling

Take the model with the best AIC, and use a Monte Carlo sampling to generate a
large number of samples with density proportional to
$\mathcal{L}(\mbox{$\mathbf{\theta}$}|\chi)$ (for example, use the Metropolis
algorithm [Tierney,, 1994]). We will call this sample $\mathbf{A}$. To each
element in $A_{i\cdot}$ (taken from
$\mathcal{L}(\mbox{$\mathbf{\theta}$}|\chi)$), we have associated the value of
$L_{i}=\mathcal{L}(A_{i\cdot}|\chi)$, the likelihood of this sample, and we
take $Y_{i}=F(A_{i\cdot})$, the result of the biological model over this
sample. Normalize $L_{i}$ to have the minimum log-likelihood in $0$.

Aggregation

Now we take the model results $Y_{i}$ and associated likelihoods $L_{i}$, and
create an upper profile in the following manner: for each increment $z$, find
the largest value $\bar{y}$ in $Y_{i}$ such that $L_{i}\leq z$. Note down this
value as $P_{sup}(z)=\bar{y}$, and repeat for a larger value of $z$.

Proceed in an analogous fashion to build the lower likelihood profile. Both
profiles, taken together, can be used to investigate the plausibility regions
of $y$ under $\mathcal{L}(\mbox{$\mathbf{\theta}$}|\chi)$.

##  7 Case study: a minimal model for population growth

Here, we will study a structured population model that be considered minimal:
the life stages are non-reproducing juveniles and reproductive adults. This
example is presented by [Caswell,, 2008] as a simple example of the analytical
sensitivity analysis. The model is described by the following matrix:

$A=\left[\begin{array}[]{ll}\sigma_{1}(1-\gamma)&f\\\
\sigma_{1}\gamma&\sigma_{2}\end{array}\right]$ (1)

Here, $\sigma_{1}$ is the probability of juvenile survival, $\sigma_{2}$ is
the probability of adult survival, $\gamma$ is the maturation probability and
$f$ is the adult fertility. We will represent by $\lambda$ the largest
eigenvalue of this matrix, and consider this as the model response.

To simplify some calculations, let’s presume that the survival rate does not
depend on the life stage ($\sigma_{1}=\sigma_{2}=\sigma$). We will also
suppose that it is possible to unequivocally mark the juveniles that were born
in the last time interval, and which passed through the maturation process to
become adults in the last time interval. The model becomes:

$A=\left[\begin{array}[]{ll}\sigma(1-\gamma)&f\\\
\sigma\gamma&\sigma\end{array}\right]$ (2)

For large animals with one offspring for reproductive seasons, such as whales,
$f$ can be modeled as the proportion of adults that generate offspring. The
parameter $\sigma$ is modeled as the proportion of individuals that survive
from one time interval to another, and $\gamma$ as the proportion of juveniles
that maturate from one time interval to another. This way, all of the three
parameters can be modeled as binomial distributions with unknown probabilities
$\theta_{i}$ and number of trials given by $n_{1}$, the number of juveniles,
$n_{2}$, the number of adults, and $n_{t}$, the total population size:

$\displaystyle\gamma$
$\displaystyle\sim\operatorname{binom}(\theta_{1},n_{1})$ (3) $\displaystyle
f$ $\displaystyle\sim\operatorname{binom}(\theta_{2},n_{2})$ (4)
$\displaystyle\sigma$
$\displaystyle\sim\operatorname{binom}(\theta_{3},n_{t}=n_{1}+n_{2})$ (5)

We will also assume that the parameters are independent, to arrive at the
likelihood functions graphed in 2 (mathematical details are presented in
section 7.1).

We will examine a numerical example with the initial population having 10
juveniles and15 adults. The population size is very small, which is important
to highlight the difference between the analytical and stochastic approaches.
After one time step, we can see 3 recent adults, 2 born juveniles and 23 total
survivors. It’s easy to see from the figure 2 that the best estimate for this
parameters is given by $\sigma=$ 0.92, $f=$ 0.13 and $\gamma=$ 0.3. Also, the
corresponding value for $\lambda$ is 1.02.

Figure 2: Likelihood function for each parameter in the model. Black =
$\sigma$, red = $f$ and green = $\gamma$.

The likelihood functions for each parameter were used to generate 50000
samples using Metropolis algorithm, from which we derive an empirical
distribution for $\lambda$, which is proportional to likelihood of the
parameters. This $\lambda$ distribution, together with the likelihood values
associated with each input, was used to construct a likelihood profile for the
model result. The minimum likelihood for $\lambda$ is attained in $\lambda=$
1.02.

Figures 4 and 5 show preliminary results of the application of sensitivity
techniques (such as described in [Chalom and Prado,, 2012]) to the generated
samples. It is important to highlight that this analyses have been carried out
in a neighborhood of the maximum likelihood point that is not infinitesimal,
nor arbitrary, which would be the case in analytical and other stochastic
methods.

Figure 3: Likelihood profile for the results of a minimal model of structured
population growth. Figure 4: Graph of the dispersion of parameter values (in
the x axis) and results of the minimal model of structured population growth,
generated by a likelihood profile method.  Figure 5: Partial Rank Correlation
Coefficient (PRCC) between the input and output values for a minimal model of
structured population growth.

The analysis shown indicates that the estimated value of $\lambda$ is very
unreliable, presenting a very open profile. It is important to notice that
this happens due to the small sample size. If we consider a sample in which
all the observations are 3 times larger, but keeping all the proportions, the
analysis results in a more closed profile (fig. 6).

Figure 6: Likelihood profile for the results of a minimal model of structured
population growth, but with larger sample size. In comparison to 3, this
profile is more closed and the plausibility regions are narrower.

###  7.1 Mathematical details

In this section, we will detail some mathematical points left out in the
example above.

For a given observed number of juveniles that have undergone maturation,
becoming adults in the last time interval, $x_{A}$, we can write the log-
likelihood function for $\gamma$ as:

$\mathcal{L}\left(\theta_{1}|x_{A}\right)=\log\left({n_{1}\choose
x_{A}}\theta_{1}^{x_{A}}(1-\theta_{1})^{n_{1}-x_{A}}\right)$ (6)

In the same way, the log-likelihood for $f$ is related to the number of
juveniles that were born in the last time interval, $x_{J}$, by the
expression:

$\mathcal{L}\left(\theta_{2}|x_{J}\right)=\log\left({n_{2}\choose
x_{J}}\theta_{2}^{x_{J}}(1-\theta_{2})^{n_{2}-x_{J}}\right)$ (7)

And last, the $\sigma$ log-likelihood function is given over the number of
surviving individuals $x_{S}$ (or the total population minus $x_{J}$):

$\mathcal{L}\left(\theta_{3}|x_{S}\right)=\log\left({n_{t}\choose
x_{S}}\theta_{3}^{x_{S}}(1-\theta_{3})^{n_{t}-x_{S}}\right)$ (8)

With the strong assumption that the three variables are independent, we can
write $n_{t}=n_{3}$ and $\\{x_{A},x_{J},x_{S}\\}=\\{x_{1},x_{2},x_{3}\\}$ in
order to arrive at the more economical notation999 It is important to remember
that $x_{1}$ is not the number of juveniles, and so on. for the likelihood
function over the entire parameter vector $\boldsymbol{\theta}$:

$\mathcal{L}\left(\boldsymbol{\theta}|\mathbf{x}\right)=\sum_{i}\log\left({n_{i}\choose
x_{i}}\theta_{i}^{x_{i}}(1-\theta_{i})^{n_{i}-x_{i}}\right)$ (9)

The result of the model is $\lambda$, the largest eigenvalue in $A$. For 2x2
matrices, this is given simply by:

$\det\left[\begin{array}[]{ll}\lambda-\sigma(1-\gamma)&-f\\\
-\sigma\gamma&\lambda-d\end{array}\right]=\lambda^{2}-\lambda\operatorname{tr}(A)+\operatorname{det}(A)=0$
(10)
$\displaystyle\lambda=\frac{1}{2}\left(\operatorname{tr}(A)+\sqrt{\operatorname{tr}^{2}(A)-4\operatorname{det}(A)}\right)$
(11)

###  7.2 R code for performing the analyses

In this section, we will present the R code used to generate the analyses and
graphs presented above. This can be seen as a tutorial in the usage of the
computational tools provided in the “pse” package. You should have installed R
101010This tutorial was written and tested with R version 3.0.1, but it should
work with newer versions along with an interface and text editor of your
liking, and the package “pse” (available on
http://cran.r-project.org/web/packages/pse).

####  7.2.1 Biological and statistical models

First, we should define our interest model. We will refer to this model as the
biological model111111 Because the models of interest in my research are
biological. It can also be a physical model, geochemical model, etc. to
distinguish this from the statistical model we will be using to estimate
likelihoods. This model must be formulated as an R function that receives a
data.frame, in which every column represent a different parameter, and every
line represents a different combination of values for those parameters. The
function must return an array with the same number of elements as there were
lines in the original data frame, and each entry in the array should
correspond to the result of running the model with the corresponding parameter
combination. For example, it can be this121212See the preceding sections for
the interpretation of the model, including parameters and results:

> tr <\- function (A) return(A[1,1]+A[2,2])> A.to.lambda <\- function(A)\+
> 1/2*(tr(A) + sqrt((tr(A)^2 - 4*det(A))))> getlambda = function (sigma, f,
> gamma) {\+ A.to.lambda (matrix(c(sigma*(1-gamma), f,\+ sigma*gamma,
> sigma),\+ ncol=2, byrow=TRUE) )\+ }> getlambda = Vectorize(getlambda)> model
> <\- function(x) getlambda(x[,1], x[,2], x[,3])

Following the definition of the model, we should define the likelihood
function for our parameters. To do this, we can formulate and test several
statistical models. In order to fit competing models to the data and select
the best of them, we recommend using the R package bbmle. Then, the best model
should be written as a function receiving a numeric vector representing one
realization of the parameter vector and returning the positive log-likelihood
of that vector.

For example, the best model may be that the parameters $\sigma$, $f$ and
$\gamma$ are all independent from each other, coming from the specified
binomial distributions:

> # Initial population: juveniles, adults, total> n <\- c(10, 15); n.t <\-
> sum(n)> # Observed quantities: matured, born, survival> obs <\- c(3, 2, 23)>
> # Best guess for the parameters> sigma <\- obs[3]/n.t> f <\- obs[2]/n[2]>
> gamma <\- obs[1]/n[1]> lambda <\- getlambda(sigma, f, gamma)

The likelihood function, in this case, should be:

> # NOTE: LL function uses GLOBAL obs and n!!!> LL <\- function (x)\+ {\+ t
> <\- dbinom(obs[3], n.t, as.numeric(x[1]), log=TRUE) +\+ dbinom(obs[2], n[2],
> as.numeric(x[2]), log=TRUE) +\+ dbinom(obs[1], n[1], as.numeric(x[3]),
> log=TRUE)\+ if (is.nan(t)) return (-Inf);\+ return(t);\+ }

Please note that this function uses the global variable obs, and that it
return minus infinity instead of not-a-number in cases where the likelihood is
not properly defined. This can happen, for instance, if any of the values of x
is negative.

####  7.2.2 Profiling: sampling and aggregating the results

After carefully constructing the model of interest and the likelihood
function, as described in the previous section, performing the PLUE analysis
is simply a matter of calling the PLUE function. This function performs three
steps. First, it performs a Monte Carlo sampling of the likelihood function in
order to generate a large sample from the likelihood distribution. Then, the
biological model is applied to this sample, and finally the model results are
combined by means of profiling the likelihood function associated with each
data point.

The pse package implements a simple Metropolis sampling function that can be
used by setting method=‘internal’ in the PLUE function call. For more
elaborate sampling schemes, and more control over the process, we recommend
using method=‘mcmc’, which uses the mcmc R package.

> library(pse)> factors = c("sigma", "f", "gamma")> N = 50000> # we set the
> random seed to ensure reproducibility:> set.seed(42)> # The starting point
> for the Monte Carlo sampling> start = c(sigma, f, gamma)> plue <\-
> PLUE(model, factors, N, LL, start,\+ method="mcmc", opts=list(blen=10),
> nboot=50)

In order to see the profiled likelihood of the model result, simply run:

> plot(plue, xlim=c(0.6, 1.3))

Notice that additional parameters may be specified for the underlying plotting
function, such as xlim, ylab or main. Additional plots may be seen by using
the plotscatter and plotprcc functions:

> # Sensitivity analyses over lambda> plotscatter(plue)> plotprcc(plue)

The interpretation of these graphs is analogous to those generated by the
Latin Hypercube Sampling, as described in [Chalom and Prado,, 2012]. However,
it is important to notice that, instead of the arbitrary region of the
parameter space that is sampled in the LHS scheme, the plots presented in this
document are representing a discretization of the likelihood surfaces of the
parameters, thus incorporating all the information about the data collected.

## Acknowledgments

This work was supported by a CAPES scholarship.

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# Hall field-induced magneto-oscillations near charge neutrality point in
graphene

Mrityunjay Pandey<EMAIL_ADDRESS>Centre for Nano Science and
Engineering, Indian Institute of Science, Bangalore 560012, India  Kenji
Watanabe  Research Center for Functional Materials, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Takashi Taniguchi
International Center for Materials Nanoarchitectonics, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Srinivasan Raghavan
Centre for Nano Science and Engineering, Indian Institute of Science,
Bangalore 560012, India  U. Chandni<EMAIL_ADDRESS>Department of
Instrumentation and Applied Physics, Indian Institute of Science, Bangalore
560012, India

###### Abstract

We explore the non-equilibrium transport regime in graphene using a large dc
current in combination with a perpendicular magnetic field. The strong in-
plane Hall field that is generated in the bulk of the graphene channel results
in Landau levels that are tilted spatially. The energy of cyclotron orbits in
the bulk varies as a function of the spatial position of the guiding center,
enabling us to observe a series of compelling features. While Shubnikov-de
Haas oscillations are predictably suppressed in the presence of the Hall
field, a set of fresh magnetoresistance oscillations emerge near the charge
neutrality point as a function of dc current. Two branches of oscillations
with linear dispersions are evident as we vary carrier density and dc current,
the velocity of which closely resembles the TA and LA phonon modes, suggestive
of phonon-assisted intra-Landau level transitions between adjacent cyclotron
orbits. Our results offer unique possibilities to explore non-equilibrium
phenomena in two-dimensional materials and van der Waals heterostructures.

Intra-Landau level transitions, non-equilibrium phenomena, drift velocity,
Hall field, acoustic phonons

In two dimensional electron systems, magneto oscillations underpin the Landau
quantization of electrons under an external magnetic field. In addition to the
conventional Shubnikov-de Haas (SdH) oscillations, a variety of novel low-
field oscillations have been investigated, primarily in GaAs-AlGaAs
heterostructures, where transition of electrons between Landau levels can lead
to oscillations in longitudinal resistance as the magnetic field ($B$) is
varied. These transitions are typically enabled by phonon-mediated resonances
Firsov et al. (1964); Tsui et al. (1980); Zudov et al. (2001); Brummell et al.
(1987); Hatke et al. (2009); Dmitriev et al. (2010), strong Hall field
Dmitriev et al. (2010); Yang et al. (2002); Zhang et al. (2007); Bykov et al.
(2005) or microwave illumination Yang et al. (2003); Zudov et al. (2014);
Dorozhkin et al. (2016); Dmitriev et al. (2012); Zudov et al. (2003). In
recent times, quantum oscillations have led to key discoveries in graphene and
other van der Waals heterostructures, including unconventional Berry’s phases
Novoselov et al. (2005); Zhang et al. (2005), the Hofstadter spectrum Hunt et
al. (2013) and magnetophonon resonances Kumaravadivel et al. (2019). These
studies were primarily done in the near-equilibrium limit with a small
electric field applied laterally. The fate of quantum Hall effect and magneto
oscillations in graphene when subjected to strong electric fields and high
currents has received much less attention Tan et al. (2011); Wang and Lei
(2013); Singh and Deshmukh (2009); Tian et al. (2016). Importantly, such non-
equilibrium effects are relevant in the context of electron interferometers
using edge channels for quantum information processing Neder et al. (2006);
Roulleau et al. (2008), generation of excitations such as magnetoplasmons in
the edge channels Kamata et al. (2010) and for the application of the quantum
Hall effect as a resistance standard Tian et al. (2016); Komiyama and
Kawaguchi (2000).

Figure 1: Electrical characterization: (a) Schematic of the non-equilibrium
magneto-transport measurement on a Si/SiO2 back gated device with a width of
18 $\mu$m and voltage probes separated by 3.8 $\mu$m where VBG is the back
gate voltage and $V_{xx}$ is the longitudinal voltage drop. The direction of
magnetic field ($B$), dc current ($I_{\textit{dc}}$), and Hall field ($E_{H}$)
are mutually perpendicular to each other. (b) Longitudinal resistivity $\rho$
as function of charge carrier density $n$ measured at 2.5 K for the graphene
Hall bar device. (c) Landau fan diagram of the longitudinal resistance
$R_{xx}$ as a function of $n$ and $B$, with the dashed lines denoting filling
factors. Scale bar extends from 0.25 $\mathrm{\SIUnitSymbolOhm}$ (red) 40
k$\mathrm{\SIUnitSymbolOhm}$ (yellow). (d) Differential resistance rxx as a
function of $B$ for I${}_{\textit{dc}}$ ranging between 0 to 10 $\mu$A in
steps of 2 $\mu$A shows suppression of SdH oscillations on increasing
$I_{dc}$. Data taken at a magnetic field between 0 to 3.75 T at $n$ = 3.5 x
1011 cm-2 Figure 2: Non-equilibrium measurements at constant B: (a)
Differential resistance $r_{xx}$ at a fixed magnetic field of 3 T as function
of $n$ and $I_{dc}$, at 2.5 K, with $I_{\textit{dc}}$ ranging from 0 $\mu$A to
58 $\mu$A. Color map is plotted in log scale. Black dashed line indicate the
resistance minima corresponding to LL indices $\nu$ = 18, 14, 10, 6 and 2.
Scale bar ranges from 0.25 $\mathrm{\SIUnitSymbolOhm}$ (yellow) to 6.4
k$\mathrm{\SIUnitSymbolOhm}$ (purple). (b) $r_{xx}$ at different
$I_{\textit{dc}}$ for the hole side showing the rise in the background
resistance, as well as the resistance minima corresponding to various LL
indices. (c) Normalised resistance $r_{norm}$ shown as line plots near the
charge neutrality point elucidating the appearance of a set of new
oscillations as a function of $I_{dc}$. Individual line plots are offset in
Y-axis for clarity. Peak positions are marked using dots.

In this letter, magnetotransport in the non-equilibrium regime is investigated
in a large area graphene Hall bar device, by introducing a high dc current
($I_{\textit{dc}}$) in addition to a small ac excitation (see Fig.1(a)). The
high quality of our hexagonal boron nitride (hBN)-encapsulated device allows
us to probe a series of phenomena over a wide range of charge carrier
densities. We observe signatures of archetypal charge carrier heating-induced
dissipation in differential resistance measurements, that are consistent with
the presence of a Hall field ($E_{H}$). In the presence of $I_{dc}$, to a good
approximation, the drift velocity $v_{d}$ due to $E_{H}$ can be expressed as
$v_{d}=I_{dc}/new$, where w is the width of the channel, n is the carrier
density and e is the electronic charge. In the regime vd $<$ vp,
$\omega_{c}/2k_{F}$ (where $v_{p}$ is the phonon velocity, $\omega_{c}$ is the
cyclotron frequency and $k_{F}$ is the Fermi wave vector), we observe a marked
suppression of SdH oscillations at constant $n$ when $I_{dc}$ is increased,
signifying carrier heating. On the other hand, quantum oscillations show
several interesting features when $n$ is varied at fixed $B$. The channel
shows dissipative transport as $I_{dc}$ is increased, with the quantum
oscillations disappearing in the low-$n$ regions at a lower $I_{dc}$ in
comparison to the high-$n$ regions. Most strikingly, near the charge
neutrality point we report a set of novel resistance oscillations, that
disperse linearly as a function of $I_{dc}$ and $n$. These oscillations are
observed in the density regime where $v_{d}\approx v_{p}$. The two sets of
oscillations observed in our data resemble the LA and TA modes of acoustic
phonons in graphene, providing evidence for Hall field-induced inelastic
intra-Landau level transitions near the charge neutrality point.

The experiments reported in this study were performed at 2.5 K on a hBN-
encapsulated graphene Hall bar device with a width of 18 $\mu$m. It was shown
in recent studies Kumaravadivel et al. (2019); Greenaway et al. (2021) that
devices with larger widths in comparison to the mean free path of carriers can
provide a platform to investigate the resistive behaviour of the bulk by
minimizing edge contributions. This regime is relevant to our devices as well,
where the electron-phonon interactions are found to be crucial. Fig. 1(b)-(c)
show the basic electrical characterization of our device at $I_{dc}=0$. Fig.
1(b) shows the four-probe resistivity as a function of the charge carrier
density $n$. Fig. 1(c) indicates the Landau fan diagram in the $(n,B)$
parameter space, with filling factors $\nu$ = $\pm$ 2, 6, 10 and 14 indicated
by dashed lines. SdH oscillations are investigated at several $I_{dc}$, as
shown in Fig. 1(d), where the differential resistance rxx as a function of $B$
for several $I_{dc}$ ranging from 0 to $10~{}\mu$A are shown for a carrier
density of 0.35 x 1012 cm-2. The high quality of our sample is apparent from
the observation of SdH oscillations at a low field of 0.5 T. The oscillations
were found to be suppressed with increase in $I_{dc}$. We have studied the
carrier energy loss to the lattice by measuring the SdH oscillations and
analyzing the damping of these oscillations (See supplementary information).
In graphene, the rate of energy loss is given as:

$P=I_{dc}^{2}r_{xx}/nA$ (1)

where $P$ is the dissipation power and A is the area of the channel,
indicating the power loss is expectedly large at low density and large current
Baker et al. (2012); Alexander-Webber et al. (2013). Consequently, carrier
temperature rises with increasing current and decreasing carrier density,
leading to dissipative transport in the channel, consistent with our
experimental observations. The estimated dissipation rates indicate that the
energy loss rate at $I_{dc}=10~{}\mu$A is of order 10-14 W however, at 2
$\mu$A it is of order 10-16 W, resulting in high carrier temperatures at large
dc currents as reported in previous studies Baker et al. (2012); Alexander-
Webber et al. (2013). This estimate assumes a very rapid thermalization of the
electrons on a time scale of few tens of femtoseconds, corroborated by recent
experiments in graphene and related materials Sun et al. (2008); Breusing et
al. (2009)

Figure 3: Intra-Landau level transitions near the charge neutrality point: (a)
Linear fitting of the resistance peaks denoted by dots in Fig.2(c) near the
charge neutrality point, as function of $n$ and $I_{dc}$. Velocities
calculated from the slopes are 14 and 8.5 km/s on the electron side and 13.8
and 7.3 km/s on the hole side, respectively, which match closely with the LA
and TA phonon modes in graphene. (b) Schematic showing the possible mechanism
of intra Landau level transitions in presence of a Hall field, assisted by
acoustic phonons. In presence of a Hall field, Landau levels are tilted
spatially as denoted by the red lines. A figure-of-8 configuration occurs for
a separation of $2R_{c}$, leading to maximum overlap of wavefunctions and
transitions between adjacent cyclotron orbits, favoured by phonons of energy
$\hbar qv_{p}$.

Our most significant results are illustrated in Fig. 2. In Fig. 2a, $r_{xx}$
is plotted as a color map for several $n$ and $I_{dc}$ values, at a fixed
magnetic field of 3 T. At $I_{dc}=0$, resistance minima corresponding to
filling factors $|\nu|=2$, 6, 10, 14 and 18 are clearly visible. As $I_{dc}$
is increased, the resistance value at the minima increases and eventually
merge with the higher resistance background. This feature is clearer on the
hole side denoted by the dashed lines. Remarkably, the resistance minima for
higher $\nu$ sustain till a larger $I_{dc}$. This observation is consistent
with $I_{dc}$-induced carrier heating. Electron temperature increases with the
increase in $I_{dc}$ resulting in disappearance of the resistance minima.
Energy loss rate as a function of $n$ at different $I_{dc}$ calculated using
eq. 1, reveals that dissipation rate is larger at low density. Consequently,
the quantum oscillations are suppressed at a lower $I_{dc}$ for lower $n$.
This also results in an increase in carrier temperature and the overall
resistance background as depicted in Fig. 2(b), which shows a sectional cut of
the color map in Fig. 2(a).

The most striking observation occurs near the charge neutrality point, where
we observe a new set of resistance oscillations to emerge, which become
increasingly stronger as $I_{dc}$ is increased. Fig. 2(c) shows the section of
magneto resistance (normalized as $r_{norm}=r_{xx}/r_{xx}^{o}$, where
$r_{xx}^{o}$ is the differential resistance at the charge neutrality point) in
the range of $\pm$ 0.4 x 1012 cm-2. It is evident from Fig. 2(c) that there
are four branches of resistance oscillations, two each on the electron and
hole sides, which show dispersive behavior over the ($n-I_{dc}$) subspace.
Notably, these oscillations were observed to be broader at a higher magnetic
field of 5 T (see supplementary information), while retaining the general
characteristics.

Magneto oscillations in two dimensional electron systems can have varied
origins depending on the experimental configurations. For instance, inter-
Landau level transitions are known to be favoured in microwave-induced
resistance oscillations Dmitriev et al. (2012); Zudov et al. (2003), Zener
tunneling at high currents Dmitriev et al. (2010); Yang et al. (2002); Zhang
et al. (2007) and magnetophonon oscillations involving acoustic or optical
phonons Tsui et al. (1980); Hatke et al. (2009). In all of these, the
transitions occur via a large momentum transfer of 2$k_{F}$, where $k_{F}$ is
the Fermi wave number, which is equivalent to a guiding center shift of twice
the cyclotron radius or 2$R_{c}$. Semiclassically, this implies that the
transitions between Landau levels are strongest when there is maximum spatial
overlap of wavefunctions in the figure-of-8 configuration for the orbits. This
momentum is provided by short range scatterers, phonons or defects in the
sample. Particularly in case of experiments involving non-equilibrium
transport, the Landau levels are tilted by the Hall field, resulting in
resistance oscillations when the energy and momentum conservation rules are
satisfied viz. $j\hbar\omega_{c}=2\hbar k_{F}v_{F}$, where $j$ is an integer.
However, such transitions across different Landau level orbits would need a
large Hall field to be present across a length scale of $2R_{c}$, particularly
at the lowest Landau levels in graphene which have the highest energy
seperation. In our experiment, the oscillations near the vicinity of the
charge neutrality point appear around the zeroth Landau level, where
$2R_{c}eE_{H}<j\hbar\omega_{c}$, restricting inter Landau level transitions.
Hence, in the Hall field regime that we have explored, the magneto
oscillations near charge neutrality point are rather surprising and cannot be
understood based on the above mechanisms.

We restrict our analysis to intra-LL transitions in light of the discussion
above. Since the Hall field tilts the Landau levels, it is imperative to have
an external excitation to aid the transitions we observe, since the energies
of the orbits at finite $E_{H}$ are not uniformly equal to the Landau level
energy at $E_{H}=0$. It is unlikely that the optical phonons contribute to
this process since we observe a dispersive feature in the ($n-I_{dc}$) space.
Magneto-excitons are also improbable candidates MacDonald et al. (1985); Chen
et al. (1990); Chui (1987) since they involve electron-hole pairs in adjacent
cyclotron orbits belonging to different Landau levels. To further understand
the nature of dispersion and the excitations underlying the resistance
oscillations, we have estimated the Hall field-induced drift velocity as a
function of $n$ and $I_{dc}$ (see supplementary figure S2). . It is apparent
that the novel oscillations observed in the magneto resistance at low
densities in Fig. 2 occur when the drift velocity becomes nearly equal to the
velocities of acoustic phonons in graphene ($\sim 10$ km/s), enabling phonon-
assisted transitions between adjacent cyclotron orbits belonging to the same
Landau level.

To extract the velocity of the relevant phonon excitations, the peak positions
as a function of $I_{\textit{dc}}$ and n are plotted in Fig. 3(a) and the
velocity is estimated as $v_{d}=I_{dc}/new$. The drift velocities associated
with the two branches were found to be $7.3\pm 0.8$ km/s and $13.8\pm 0.5$
km/s on the hole side and $8.5\pm 0.5$ km/s and $14\pm 1$ km/s on the electron
side, respectively, which match closely with the LA and TA phonon modes in
graphene Park et al. (2014); Michel and Verberck (2008). We speculate that
charge inhomogenieties near the neutrality point may lead to a marginal
decrease in the measured phonon velocities. Fig. 3(b) summarizes the mechanism
discussed above. In the presence of a Hall field $E_{H}$, cyclotron orbits of
an index N are no longer of equal energy due to the spatial tilting of Landau
levels given by the relation

$E_{N}(k_{y})=E_{N}-eE_{H}X(k_{y})$ (2)

where $E_{N}$ denotes the energy of the $N$th Landau level and
$X(k_{y})=-k_{y}l_{B}^{2}$; $k_{y}$ is the wavevector along $Y-$axis and
$l_{B}$ is the magnetic length. Consequently, the guiding centers of the
adjacent cyclotron orbits are at different energies. Transitions are possible
within a Landau level when the energy shift in the guiding centers of adjacent
orbits $eE_{H}2R_{c}$ matches the phonon energy $\hbar$qvp, with $q=2k_{F}$ as
shown in Fig. 3(b), which is equivalent to the condition $v_{d}=v_{p}$.

##  .1  Conclusions

In summary, our study of non-equilibrium transport in large area graphene
devices provide the framework to explore the strong Hall field regime. We
report pronounced Hall-field induced resistance oscillations near the charge
neutrality point in graphene, in addition to dissipative carrier heating
effects on Landau levels. It is apparent that these novel oscillations have a
linear dispersion with $I_{dc}$ and $n$, indicative of electron-phonon
interactions and intra-Landau level transitions. Remarkably, Hall-field
induced oscillations also provide a novel method to investigate low-energy
phonon modes in graphene and other two-dimensional layered materials and
heterostructures.

##  .2  Methods

Large area, high-quality heterostructures of hBN/Graphene/hBN were fabricated
using a dry pick up and transfer technique using a polymer stack of PPC/PDMS.
The stack was transferred on a Si/SiO2 wafer and annealed for 4 hours in argon
at 2 SLM flow rate at 350∘C. The geometrical extent of the large encapsulated
graphene was identified using electrostatic force microscopy Pandey et al.
(2019) to choose a clean region of interest. Hall bar was etched using ebeam
lithography and reactive ion etching, and Cr/Au (5/50 nm) contacts were
defined using standard device fabrication methods. All the electrical
measurements were performed in a Janis cryostat at 2.5 K. SR-830 lock-in
amplifier was used at 13 Hz to provide a small ac excitation current of 40 nA,
while a dc current was introduced using a Kepco programmer. An isolation
transformer circuit was used to pass both ac and dc currents through the Hall
bar channel. Keithley 2450 source meter was used to source the back gate
voltage.

##  .3 Acknowledgements

We thank S. Banerjee, J. P. Eisenstein, J. Jung, H. R. Krishnamurthy, J. Song
and J. Sutradhar for useful discussions and inputs. We gratefully acknowledge
the usage of the Micro and Nano Characterization Facility (MNCF) and National
Nanofabrication Centre at CeNSE, IISc. U.C. acknowledges support from Science
and Engineering Research Board (SERB) via grant SPG/2020/000164. KW and TT
acknowledge support from the Elemental Strategy Initiative conducted by the
MEXT, Japan (Grant Number JPMXP0112101001) and JSPS KAKENHI (Grant Numbers
JP19H05790 and JP20H00354).

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Supporting Information

Figure 4:  Fig S1. Dissipation rate as function of dc current: Dissipation
rate calculated using the equation $I_{dc}r_{xx}(0)/nA$, where $r_{xx}(0)$ is
the resistance of the Shubnikov de Haas (SdH) oscillations at $I_{dc}=0$.
Comparing the dissipation rate with previously reported values indicate the
carrier temperature falls in range of $\sim 10$ K at a current of 10 $\mu$A
and carrier density of 0.35 x 1012 cm-2[28, 29]. Figure 5:  Fig S2. Drift
velocity contour as function of dc current and carrier density: Drift velocity
induced due to the Hall field is calculated as a function of dc current and
carrier density in the hole doped region, close to the charge neutrality
point. Within the experimental conditions, drift velocity $v_{d}=I_{dc}/new$
has been evaluated for $I_{dc}$ ranging between 0 to 60 $\mu$A at 3 T,
indicating that in the region where the new set of oscillations are seen, the
drift velocity is of the same order of magnitude as the acoustic phonon
velocities in graphene. Dashed line indicates a velocity of 10 km/s. Figure 6:
Fig S3. Effect of Magnetic field on novel magneto-oscillations near the charge
neutrality point: (a) Differential longitudinal magnetoresistance as function
of carrier concentration and dc current at 5 T indicates the broadening of the
oscillations near charge neutrality point which is consistent with the Landau-
level broadening with increase in magnetic field. Consequently the two
branches of phonon modes are not well resolved, in comparison to Fig.2 in the
main text, although the general characteristics are retained. (b) Peak
position as function of dc current and carrier density in the current ranging
between 42 $\mu$A and 56 $\mu$A, as marked by the dots in (a). Velocities
calculated from the dispersion of the peaks are 12.2 $\pm$ 0.8 km/s (hole
side) and 11.8 $\pm$ 0.6 km/s and 6.5 $\pm$ 0.9 km/s (electron side)