text:
Bitter harvest
By Labonita Ghosh, Alka Dhupkar, Makarand GadgilLabonita Ghosh, Alka Dhupkar, Makarand Gadgil, Mumbai Mirror | Updated: Jul 7, 2019, 10:08 IST
Any debate around GM crops usually sparks polarised reactions. The answer to the biosafety of such products can only be found in science – and the promise of food security.
Vasant Vinayak Mule from Sindkhed Raja Taluka in Vidarbha, Maharashtra, will probably go down in history as the first farmer ever to be arrested for trying to cultivate herbicide-tolerant (HT) BT cotton. This trait of genetically modified (GM) crop is banned in India, so Mule was booked by the Vidarbha police under The Environment (Protection) Act, 1986, which regulates the manufacture, use, import, export and storage of hazardous, geneticallyengineered (GE) organisms.
On June 10, some 30,000 farmers across Maharashtra — according to the farmers' association, the Shetkari Sanghatana, which led the move — decided to defy the government's ban and cultivate HT cotton. Theirs was a token protest — many of them had sown these seeds only in a portion of their holdings — for an actual cause. The farmers insist that HT cotton is more productive, at almost double the yield than the permitted BT version, and therefore more lucrative for a community gripped by mounting debts and suicides.
"If I use HTBT cotton, I save on the pesticide cost," says Mahadu Khamkar, who planted these seeds in a portion of his four-acre plot in Ahmednagar. "The labour cost for weeding is so high that the use of HTBT is a huge relief. This was my first time sowing GM seeds, but I am ready to face any action the government might take against me."
Cultivators like Mule, Mahadu or Lalit Bahale of Akola district have never met farmers from the American Midwest, or they might have a different opinion.
Till as recently as 2013 — almost two decades after agriculturists in the US opted for HT variants of soy, corn and cotton — a report published in Nature stated that millions of hectares across America have been overrun by a killer 'superweed' that has developed a resistance to the herbicide used to treat crops. Some 24 different kinds of this superweed have now spread to 18 countries globally, the report says.
Indian farmers haven't encountered this problem yet, but the recent stir has reignited a debate — going on since 2002, when BT cotton (the only approved GM product) was first introduced in India — about whether other genetically-engineered crops should be allowed. "This debate resurfaces every time there is a new technology on the horizon," says geneticist and radiobiologist Dr PC Kesavan, who has been on several government-appointed committees on GM. "But it always ends in the same way: Inconclusively, with both proponents and opponents pulling out the same studies and data that they have, for many years. I don't expect this round to be any different."
Dr GV Ramanjaneyulu, executive director for the Centre for Sustainable Agriculture in Secunderabad, says: "Decisions regarding GM crops should be based on existing knowledge. Unfortunately, it always turns into an emotional discussion, getting completely delinked from science."
(Left) A farmer displays HTBT cotton seeds. In Vidarbha, the movement was led by the farmers' association, Shetkari Sanghatana
Safe or not?
So what exactly does science say about the biosafety of GM crops? Do they, for instance, carry health risks, both in the immediate or even long-term?
"The science on GM crops over the last 30 years says, unequivocally, that the crops grown worldwide today are completely safe," says Devang Mehta, geneticist with the department of Biological Sciences at the University of Alberta in Canada. "Genetically modified organisms (GMOs) haven't been found to result in any adverse outcomes in humans or animals. More than 500 scientific studies have looked at the question of GMO safety without finding a single negative outcome."
Adds Dr TM Manjunath, agricultural scientist and biotechnologist: "Science is 100 per cent in favour of GM crops. About 190 million hectares, in 24 countries across the world, have been given over to GMO, which 17 million farmers cultivate. In India, BT cotton is grown over 11.4 million hectares — or 93 per cent of the total cotton acreage in the country — by about seven million farmers. Would this have been allowed unless the technology was proven to be safe?" He argues that BT cotton has been farmed globally for almost 23 years before it was approved in India. "Today, its year-on-year adoption rate has gone up consistently, and no adverse effects have been reported by any farmer or scientifically-validated institutions," he says.
Studies show that four primary crops make for 99 per cent of all GMO cultivated around the world: soybean (approximately 50 per cent, in terms of global acreage); corn or maize (31 per cent); cotton (13 per cent) and canola (five per cent).
Trait-wise, about 89 per cent is herbicide tolerant, and HT corn and soy are also used for livestock feed in many parts of the world, besides finding their way into our processed food. However, officials recently discovered that BT brinjal was being illegally grown in Haryana, and scientists are puzzled about where those seeds came from. Some believe Bangladesh, which has recently allowed the cultivation of BT brinjal on its soil. But since it imports all GM varieties from India, it's possible there is an underground trade in GM seeds, based on samples leaked from research labs. "The Indian market has been flooded with illegal seeds," says Dr Ramanjaneyulu. "We don't even know if the vegetables that we currently consume are GM."
The farmers carried out a token protest earlier this month.
Realm of the unknown
This lack of monitoring is what fuels the naysayers' stand against GM. "BT and GM crops are strictly regulated," says Dr Manjunath. "Field tests are preceded by extensive lab tests and confined field trials. BT brinjal, for instance, has undergone safety tests for seven or eight years — and was found to be safe by various committees appointed by the government. Still, it has not been approved by the GEAC [Genetic Engineering Appraisal Committee, of the Ministry of Environment and Forests]."
Dr Suman Sahai, founder of the research and advocacy organisation Gene Campaign, says: "Tests are notoriously compromised because we, the ordinary folk, often don't understand the basic logic of what is happening. A lot of scientists say that the proof of safety has to be presented extremely carefully. But the government has been known to withhold data so as to not to reveal the downside."
With government funding for public sector research shrinking, scientists in the private sector are taking over GM trials — with not always reliable disclosures. "They are inevitably attached to big agricultural or seed lobbies, or companies like Monsanto [which has now been taken over by the German multinational company Bayer], whose interests they are trying to forward," says agricultural scientist Dr S Bala Ravi.
In a paper co-written with Dr Kesavan, Prof MS Swaminathan writes: "…several uncertainties and unscientific assumptions render genetic engineering an imprecise technology. And multinational corporations are running ahead of the science to drive this technology of GMO, and colluding with regulators."
Even if that were not the case, today the field is overcrowded. "The science is deeply conflicted because there are so many different kinds of scientists involved — ecological and biodiversity; agricultural; plant biotechnologists; nutritionists and many others," says an environmental studies professor who is working on a book on the socio-political impact of GM, and declined to be named. "The largest consensus, at the moment, is among ecologists, who say HT crops pose a genotxicity risk to biodiversity. With nutritionists, we are still in the realm of the unknown because Americans have been eating GM foods for many decades, and while there are enough studies to cause concern, there is nothing definitive."
Officials in Haryana collecting samples from a field where the banned BT brinjal was found to have been cultivated
The politics of it
GM technology was first introduced to India in the late 1990s. However, in 2010, acknowledging that there was not enough biosafety data, environment minister in the UPA government, Jairam Ramesh, placed a moratorium on all GM products except BT cotton — deemed safer since it was non-consumable, at least by humans. Questions about exposure had still not been framed then.
Over time, its opposers have pointed out that the benefit of BT cotton — its genetic design in turn making it naturally resistant to its most dreaded enemy, the bollworm — was that it didn't require harmful pesticides. But like the superweed, the bollworm, too, has begun to develop immunity. So farmers are now forced to use a higher grade herbicide than before. The most effective ones have been found to contain glyphosate, a compound the International Agency for Research on Cancer — an offshoot of the World Health Organisation — flagged off in 2015 as likely to be carcinogenic. Both advocates and critics have latched onto this ambivalence to strengthen their side of the argument: One lot points to the obvious risk; the other says it's merely perceived, not proven.
Over time, in India, the GM debate has also acquired a political tint. Today, the RSS affiliate Swadeshi Jagran Manch (SJM) is opposed to field trials of GM crops — which appears to be the sole reason the Narendra Modi government hasn't allowed such tests.
The moratorium, though, imposed under the UPA, continues by default under the present administration as well. Anil Gajke, national executive member of SJM, says: "We are only asking the government to be wary of science promoted by multinational companies, which is not for the greater good, but for the profits of a particular company." He adds: "In the initial stages, cotton farmers may see an increase in yield, but as the years pass, the input cost in terms of pesticides, fertilisers, water and such will go up, leaving the farmer in a debt trap. Farmers from Vidarbha and Marathwada now want to return to the traditional varieties of seed, but these are no longer available to them." When contacted, environment minister Prakash Javadekar refused to comment, saying it would be inappropriate for him to do so while Parliament is in session.
Bengaluru activists protest against the proposed introduction of BT brinjal
Food for the future?
The issue of GM crops has also gotten inextricably linked to questions of food security in the future.
"Even the handful of GMOs grown worldwide to date has had an enormous positive impact on food production," says Mehta. "Based on a peer-reviewed study published in 2011, GMOs so far have led to a 37 per cent decline in pesticide use, compared to non-GM alternatives, and increased the profits of farmers, growing them by 68 per cent, and improved yields by 22 per cent. Imagine the outcomes if more GMO crops were grown worldwide. Some researchers have also made impressive advances in improving the nutritional aspects of GMOs, like Dr Navreet Bhullar of the Swiss Federal Institute of Technology, who has produced GMO wheat and rice plants with increased iron, zinc and provitamin-A, which will address the deficiencies in diet of children in India."
GMOs are now grown in 24 countries including the US, Canada, Brazil and Argentina, says Mehta, and another 43 countries — including China — import GMO crops, even though they do not permit the planting of GMOs on their soil. "About 31 per cent of our rural population is small-holder farmers who're especially vulnerable to the threat of climate change," he says. "They could reap enormous benefits if they are allowed access to such revolutionary technology."
Maharashtra farmers know this. In Akola district — ground zero of the recent protests — the Shetkari Sanghatana has put up stalls selling banned HTBT seeds. During the recent protests, thousands of farmers were seen buying these, in open violation of the law. Sanghatana president Anil Ghanwat has an unarguable response when asked why the organisation has allowed farmers to risk their lives and livelihood with incarceration.
"Liquor is banned in Gujarat and gutkha is banned in our state. But you still get them, right?" he asks. "GM cotton and BT brinjal seeds are easily available from the black market. So instead of the black market making money, why can't the government allow scientific trials, regularise prices and monitor the adoption and cultivation of these seeds?" Adds Ajit Narde, head of the Sanghatna's technical committee: "The opposition to GM crops is unscientific and based on fear-mongering. Every year, India imports edible oil made from GM canola, but we don't allow the cultivation of GM crops. Where is the logic in that?"
In Aurangabad, farmers allegedly burnt the central government's June 12 notice to Maharashtra, reiterating the ban on GM seeds.
Clearly, a GM revolution is in the offing — and this one might bring a definite resolution.
More than 1,500 farmers in Vidarbha's Akola district protested, on June 10, demanding that they be allowed to cultivate herbicide-tolerant cotton, a variant of BT cotton, which is the only GM product permitted in the country at the moment
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Wow!  THANKS for the AMAZING progress.
Can you confirm that EME pull request 88 for closed ISSUE-52 is no longer needed.  If so can we close it without action?
While you continue to make progress on ISSUE-100's pull request there are still lots of unassigned open EME and MSE issues you could attack!
THANKS again for the progress today.
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York College is part of The City University of New York (CUNY), which is the largest city university system in the world. This public institution was founded in 1966 and is located in Queens County on a 50-acre campus.
Academic offerings include over 40 different majors dispersed among its three schools; the School of Arts and Sciences, the School of Business and Information Systems, and the School of Health and Behavioral Sciences.
There is also an Honors Program available for students with a minimum GPA of 3.5. In order for students to complete this program they must complete a minimum of 12-18 credit hours of Honors Program courses. Students enrolled in this program are also required to attend scholarly and cultural events. Transfer students who wish to be enrolled in the Honors Program must have a GPA of 3.25 or higher.
The Aviation Institute is located on campus and was established in 2003. Through this program students are able to receive a B.S. in Aviation Management and an Aviation Policy Minor. Courses can also be taken on a non-credit basis.
The North East Regional Office of the US Food and Drug Administration (USFDA) can be found on campus. Some of the tests performed by their staff of scientists include analysis for microbial pathogens, insanitation, quality and purity in pharmaceuticals, colors, pesticides, and decomposition and heavy metals in foods. Students can benefit from this facility by participating in internships at the FDA, attending workshops on Food Safety and proper Pharmaceutical techniques, and by attending lectures and laboratory sessions taught by the FDA scientific staff in the Departments of Chemistry and Biology.
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Transfer students must have a minimum GPA of 2.0 to be considered for admission. Students who wish to transfer in must complete an online application, pay a slightly higher application fee, and send official transcripts from all colleges or universities attended. If the student is transferring with less than 12 credit hours official high school transcripts or GED scores and diploma must be sent as well. Transfer students who wish to transfer into the SEEK program must have participated in SEEK, EOP, HEOP, or College Discovery at a prior institution.
All international students must be enrolled as full-time students and must also maintain a 2.0 GPA. International students must complete an application online, send in documentation of all courses taken and grades received, and are encouraged to send in SAT or ACT scores. After an international student has been accepted the student must prove financial ability by submitting a Certificate of Eligibility, also known as the SEVIS I-20. This form also contains the Affidavit of Free Room and Board. If the student was previously enrolled at a U.S. institution, a Transfer Enrollment Status Form must be forwarded from the student's International Student Advisor at their current institution, which will allow their SEVIS record to be electronically transferred. Applicants requiring a F-1 Visa must pay a one-time fee and this fee must be paid at least three business days before the student applies for a Visa. Also international students are encouraged to attend the International Student Symposium before beginning classes.
The federal and state grants available for students include: the Federal Pell Grant, the Tuition Assistance Program (TAP), the Aid for Part-Time Students (APTS), and the Supplemental Educational Opportunity Grant (SEOG). If students must borrow money to pay for their tuition they can do so through the William D. Ford Federal Direct Loans and the Federal Perkins Loans.
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Students participating in the FWS program must be enrolled in at least 6 credit hours. The maximum number of hours a student is allowed to work is 20 hours a week. This program offers both on-campus and off-campus worksites. The students' hourly wage for the on-campus sites depends on the amount of credits the student has received and the wages vary for students working at off-campus sites.
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Six months ago my neighbour applied for and was granted planning permission to develop a large complex that abuts my property. As I was sick at this time, I did not pay any attention as I had a lot to contend with. The other day I looked at the plans and I noticed that they sought to replace our mutual boundary hedge (full of beautiful old mature trees – oak, ash, beech, etc) with a 2.4m high concrete block wall. They did not consult with me in any way prior to their application. The planning permission states that "the development shall take place with the plans submitted". Does this mean that they must now remove our mutual boundary hedge? Can they do this despite the fact that I was not consulted on the matter?
You describe the hedge as a "mutual boundary hedge". This means that it is a party boundary and that you have joint ownership. Therefore your property (legal) boundary runs through the centre of the trunks/stems of the hedge. One party cannot substantially interfere with or remove a party wall, hedge or fence without the consent of the adjoining owner.


text:
\section{Introduction}

\setcounter{page}{1}

A theorem of McCann \cite{mcc:95} shows that for any two absolutely continuous probability measures on $\Bbb R^d$ there exists a monotone transformation sending one probability measure to the other.
A consequence of this theorem, relevant to statistics,  is that density estimation can be recast in terms of transformations. In particular, one can fix any absolutely continuous probability measure, call it $\Bbb P$, and then reparameterize the whole class of absolutely continuous probability measures as monotone transformations from $\Bbb P$.  The advantage of this new viewpoint is the flexibility in choosing the target measure $\Bbb P$ which can allow  prior information on the shape of true sampling distribution.
 For example, if it is known that the data is nearly Gaussian then choosing a  Gaussian $\Bbb P$  along with a strong penalty on transformations that are far from the identity allows one to construct penalized maximum likelihood estimates which effectively shrink the resulting nonparametric estimate in the direction of the Gaussian target $\Bbb P$. Moreover, when there is no knowledge about the true sampling measure, one can simply choose any absolutely continuous $\Bbb P$ and still construct a completely nonparametric density estimate.

In this paper we utilize this reparameterization of densities, as monotone transformations from some $\Bbb P$, to construct semiparametric and nonparametric density estimates. We focus our attention on classes transformations which are smooth,  invertible and which have attractive computational properties.  Formally, we model our data   $X_1,\ldots, X_n$ as being generated by some diffeomorphism $\phi\colon \Bbb R^d \rightarrow\Bbb R^d$ of some fixed but known probably distribution $\Bbb P$ on $\Bbb R^d$:
\begin{equation}
\label{model1}
X_1,\ldots,X_n\overset{iid}\sim \Bbb P\circ \phi .
\end{equation}
The notation $\Bbb P\circ \phi$ is taken to mean that the probability of $X\in A$ is given by   $ \Bbb P(\phi(A))$ where $\phi(A)=\{ \phi(x)\colon x\in A \}$. An important observation is that the model (\ref{model1}) implies that $\phi(X)\sim \Bbb P$. Therefore, one can imagine estimating $\phi$ by attempting to ``deform'' the data $X_1,\ldots,X_n$ by a transformation which satisfies
\[ \phi(X_1),\ldots,\phi(X_n)\overset{iid}\sim \Bbb P. \]

One of the main difficulties when working with such a model is the invertibility condition on the maps $\phi$. The nonlinearity of this condition in $\Bbb R^d$ when $d>1$ makes constructing rich classes and optimizing over such classes difficult.
One of the early attempts at circumventing such difficulties, found in \cite{and:11}, utilized  the class of quasi-conformal maps to generate penalized maximum likelihood estimates of $\phi$. However, these tools were only developed for $\Bbb R^2$ with no clear generalization for higher dimension.
In this paper we adapt the powerful tools developed by Grenander, Miller,  Younes, Trouv\'e and co-authors in the image processing and computational anatomy literature   (see \cite{you:10} and the references therein) and
apply them to the estimation of $\phi$ from data $X_1,\ldots, X_n$. Indeed, these techniques allow us to show that a penalized maximum likelihood estimate of $\phi$ exists and that the solution has a finite dimensional characterization, similar to those results found in the spline literature (see \cite{wahba:90}).

We start the paper in Section \ref{rich} with an overview of using the  dynamics of time varying vector field flows to generate rich classes of diffeomorphisms. Then in sections \ref{pmle} and \ref{EL} we define our penalized maximum likelihood estimate (PMLE) of $\phi$ and prove not  only existence, but also establish a  finite dimensional characterization which is key for numerically computing the resulting density estimate. In Section \ref{steinSection} we notice a surprising connection with the Euler-Lagrange equation for the PMLE of $\phi$ and a generalization of Stein's lemma for characterizing the normal distribution (see \cite{stein:04}). In sections \ref{npe} and \ref{spe} we give examples of our new density estimate, first as a nonparametric density estimate and second as a semiparametric density estimate where a finite dimensional model is used for the target probability measure $\Bbb P$. We finish the paper with an appendix which contains some technical details used for the proofs of the existence of the PMLE and for the finite dimensional characterization.

\section{A rich class of diffeomorphisms}
\label{rich}

 In this section we give a brief overview of the technique of using the  dynamics of time varying vector field flows to generate rich classes of diffeomorphisms. These time varying flows have been utilized  with spectacular success in the fields of computational anatomy and image processing
 (see  \cite{alla:07,
 Beg:2006ly,
 conf/miccai/BegMTY03,
  Beg:2005qf,
   cao:05,
   dup:98,
 Grenander:1998:CAE:309082.309089,
 Miller:1999bh,
 Miller:2006zr,
 Miller:2001ve,
 Trouve:1998ys,
 ty:dq,
 vaillant:04,
you:10,
Younes:2008}, and references therein).
  The tools developed in this body of work have been shown to be very powerful  for developing algorithms which optimize a diverse range of objective functions defined over classes of diffeomorphism.  It is these tools that we adapt for density estimation and statistics.

A map $\phi\colon \Omega \rightarrow\Bbb R^d$ is said to be a $C^k(\Omega,\Bbb R^d)$ diffeomorphism of the open set $\Omega \subset \Bbb R^d$ if  $\phi$ is one-to-one, maps onto $\Omega$, and  $\phi, \phi^{-1}\in C^{k}(\Omega,\Bbb R^d)$.  In what follows we generate classes of diffeomorphisms by time varying vector field flows. In particular, let  $\{ v_t\}_{t\in [0,1]}$  be a time varying vector field in $\Bbb R^d$, where $t$ denotes  `time' so that for each $t$, $v_t$ is a function mapping $\Omega$ into $\Bbb R^d$. Under mild smoothness conditions there exists a unique class of diffeomorphisms of $\Omega$, denoted $\{ \phi_t^v\}_{t\in [0,1]}$, which satisfy the following ordinary differential equation
\begin{align}
\label{ood}
 \partial_t \phi_t^v(x) &= v_t(\phi_t^v(x))
 \end{align}
 with boundary condition $\phi_0^v(x)=x$, for all $x\in \Omega$ (see Theorem \ref{ExistFlow} below). The interpretation of these flows is that $\phi_t(x)$ represents the position of a particle at time $t$, which originated from location $x$ at time $t=0$, and flowed according the the instantaneous velocity given by $v_t$. It will be convenient to consider the diffeomorphism that maps time $t$ to some other time $s$, this will be denoted $\phi_{ts}^{ v}(x)\equiv \phi_{s}^{v}( {\phi_{t}^{ v}}^{-1}(x)) $.

For the remainder of the paper we will assume that at each time $t$, $v_t$ will be a member of a  Hilbert space of vector fields mapping  $\Omega$ into $\Bbb R^d$ with inner product denoted by $\langle\cdot, \cdot \rangle_V$ and norm by $\|  \cdot \|_V$.
Indeed, how one chooses the Hilbert space $V$ will determine the smoothness properties of the resulting class of deformations $\{\phi_t\}_{t\in[0,1]}$.  Once the Hilbert space $V$ is fixed we can define the following set of time varying vector fields.

\begin{definition}
\label{defV01}
 Let  $V^{[0,1]}$ denote the space of measurable functions $v_t(x)\colon [0,1]\times \Omega\rightarrow \Bbb R^d$ such that $v_t\in V$ for all $t\in [0,1]$ and $\int_0^1 \| v_t\|^2_V dt <\infty $.
\end{definition}
One clear advantage of this class  is that it can be endowed with a Hilbert space inner product if $V$ is a Hilbert space. Indeed,
 $V^{[0,1]}$ is a Hilbert space with inner product  defined by $\langle v,h \rangle_{V^{[0,1]}} \equiv\int_0^1 \langle  v_t, h_t\rangle_V dt$ (see Proposition \ref{Hspace} in the Appendix or Proposition 8.17 in \cite{you:10}).
For the remainder of the paper we typically use $v$ or $w$ to denote  elements of  $ V^{[0,1]}$ and  $v_t$ or $w_t$ to denote the corresponding elements of $V$ at any fixed time $t$.
An important theorem found in \cite{you:10} relates the smoothness of $V$ to the smoothness of the resulting diffeomorphism.
Before we state the theorem, some definitions will be prudent. The Hilbert space $V$ is said to be continuously embedded in another normed space $H$ (denoted $V\hookrightarrow H$) if $V\subset H$ and there exists a constant $c$ such that
\[ \| v \|_H \leq c \| v\|_V \]
for all $v\in V$ where $\|\cdot \|_H$ denotes the norm in $H$.
 Also we let $C_0^k(\Omega,\Bbb R^d)$ denote the subset of $C^k(\Omega,\Bbb R^d)$ functions whose partial derivatives of order $k$ or less all have continuous extensions to zero at the boundary $\partial \Omega$.
\begin{theorem}[\cite{you:10}, \cite{dup:98}]
\label{ExistFlow}
If $V\hookrightarrow C_0^k(\Omega,\Bbb R^d)$, then for any $v\in V^{[0,1]}$ there exists a unique class of $C^k(\Omega,\Bbb R^d)$ diffeomorphisms $\{\phi^v_t\}_{t\in[0,1]}$ which satisfy (\ref{ood}) and  $\phi_0^v(x) = x$ for all $x\in \Omega$.
\end{theorem}

To derive our finite dimensional characterization we will make the additional assumption that $V$ is a reproducing kernel Hilbert space of vector fields. This will guarantee the existence of a reproducing kernel
 $K( x,y)\colon \Omega \times \Omega \rightarrow \Bbb R^{n\times n}$ which can be used to compute the evaluation functional. In particular, $K$ has the property that for any $x\in \Omega$ and $f\in V$ the following identity holds $\langle  K(x,\cdot)  p, f\rangle _V=  f(x)\cdot p$ for all $p\in \Bbb R^d$.
 To simplify the following computations we will only work with kernels of the form $K(x,y) = R(x,y)I_{d\times d}$ where $R:\Omega\times\Omega\rightarrow \Bbb R$ is a positive definite function and $I_{d\times d}$ is the $d$-by-$d$ identity matrix.

 Now the class of diffeomorphisms we consider in this paper  corresponds to the set of all time varying vector field flows evaluated at $t=1$: $\phi^v_1$, where $v$ ranges through $V^{[0,1]}$. The class $V^{[0,1]}$ will be completely specified by the reproducing kernel $R(x,y)I_{d\times d}$ which has the flexibility to control the smoothness of the resulting maps $\phi_1^v$ through Theorem \ref{ExistFlow}.

\section{Penalized maximum likelihood estimation}
\label{pmle}

In this section we construct a penalized maximum likelihood estimate (PMLE) of $\phi^v_1$ given the data $X_1,\ldots, X_n\overset{iid}\sim \Bbb P\circ \phi^v_1$ where $\{\phi^v_t\}_{t\in[0,1]}$ satisfies (\ref{ood}) and $v\in V^{[0,1]}$.  Under mild assumptions on $V$ and the density of $\Bbb P$ we prove the existence of a PMLE estimate of $\hat v$, whereby obtaining an estimate $ \Bbb P\circ \phi^{\hat v}_1$ of the true sampling  distribution.

The target probability measure $\Bbb P$ is assumed known with a bounded density with respect to Lebesque measure on $\Bbb R^d$.
Therefore, by writing the density of $\Bbb P$ as $\exp H$ for some  function $H\colon\mathbb R^d\rightarrow \mathbb R\cup \{-\infty\}$, the probability measure $\Bbb P\circ \phi^v_1$ has density given by
 \[ d\,\Bbb P\circ \phi_1^v(x)  =\det (D\phi_1^v(x))  \exp H \circ \phi_1^v(x) dx  \]
 where $\det (D\phi^v_1(x))$ is defined as the  determinant of the Jacobian of $\phi^v_1$ evaluated at $x\in \Omega$ (always positive by the orientation preserving nature of $\phi^v_1$).
Since $\phi_1^v$ ranges over an infinite dimensional space of diffeomorphisms, the  likelihood for $v$ given the data  will typically be unbounded as $v$ ranges in $V^{[0,1]}$. The natural solution is to regularize the log likelihood using the corresponding  Hilbert space norm on $V^{[0,1]}$ with a  multiplicative tuning factor $\lambda/2$.
The penalized log-likelihood (scaled by $1/n$) for the unknown vector field $v$ flow given data $X_1,\ldots, X_n\overset{iid}\sim \Bbb P\circ \phi^v_1$ is then given by
 \begin{equation}
 \label{energy1}
 E_\lambda(v)\equiv  \frac{1}{n} \sum_{k=1}^n \log\text{det}[D\phi(X_k) ]+H\circ\phi(X_k)  - \frac{\lambda}{2}  \int_0^1 \| v_t \|_V^2 dt.
 \end{equation}
 The estimated vector field $\hat v$ is chosen to be any element of $V^{[0,1]}$ which maximizes $E_\lambda$ over $V^{[0,1]}$.
 The following theorem establishes that such a $\hat v$ exists.

\begin{claim}
\label{claim1}
Let $V$ be a  Hilbert space which is  continuously  embedded in $C_0^2(\Omega, \Bbb R^d)$ where $\Omega$ is a bounded open subset of $\Bbb R^d$. Suppose $e^{H(\cdot)}$ is a bounded and continuous density  on $\Omega$. Then there exists a time varying vector field $\hat v \in V^{[0,1]}$ such that
\begin{equation}
\label{exist}
E_\lambda(\hat v)=\sup_{v\in V^{[0,1]}}E_\lambda(v).
\end{equation}
\end{claim}
\begin{proof}

We first establish $\sup_{v\in V^{[0,1]}}E_\lambda(v)<\infty$ by splitting the energy $E_\lambda$ into three parts
\begin{equation}
\label{decomp}
E_\lambda(v)=\underbrace{\frac{1}{n}\sum_{k=1}^n   \log\text{det} D\phi_1^v(X_k) }_{:=E_1(v)}+ \underbrace{\frac{1}{n}\sum_{k=1}^n H\circ\phi_1^v(X_k)}_{:=E_2(v)}  \underbrace{-\frac{\lambda}{2} \int_0^1 \| v_t \|_V^2 dt.}_{:=E_3(v)}
\end{equation}
Notice that each term is well defined and finite whenever $v\in V^{[0,1]}$, since
the assumption $V\hookrightarrow C_0^2(\Omega, \Bbb R^d)$ is sufficient for  Theorem 8.7   in \cite{you:10} to apply to the class  $ V^{[0,1]}$. In particular, for any  $v\in V^{[0,1]}$  there exists a unique class of $C^1$ diffeomorphisms of $\Omega$, $\{\phi_t^v\}_{t\in [0,1]}$, which satisfies (\ref{ood}) (also see Theorem 2.5 in \cite{dup:98}).
The term $E_2(v)$ is clearly bounded from above since  $\sup_{x\in\Omega}H(x)<\infty$ by assumption.
For the remaining two terms notice that  the determinant of the Jacobian is given by   $ \log\det D\phi_1^v(x)  = \int_0^1  \text{div}\,  v_t (\phi_t^v(x))dt$ (by equation (\ref{div}) in the Appendix).
Therefore
\begin{align}
E_1(v) + E_3(v) & = \frac{1}{n}\sum_{k=1}^n   \int_0^1 \left(  \text{div}\,  v_t (\phi_t^v(X_k))  - \frac{\lambda}{2}   \| v_t \|_V^2
\right)dt \nonumber\\
 & \leq \frac{1}{n}\sum_{k=1}^n   \int_0^1 \left( \sup_{x\in \Omega} |\text{div}\,  v_t (x)|  - \frac{\lambda}{2}   \| v_t \|_V^2
\right)dt \nonumber \\
 & \leq    \int_0^1 \left( c\| v_t \|_V  - \frac{\lambda}{2}   \| v_t \|_V^2
\right)dt, \,\,\text{by the assumption $V\hookrightarrow C_0^2(\Omega, \Bbb R^d)$} \nonumber \\
& \leq \frac{c^2}{2\lambda}<\infty. \nonumber
\end{align}

Now let $v^1,v^2,\ldots$ be any maximizing sequence that satisfies $\lim_{m\rightarrow \infty} E(v^m) = \sup_{v\in V^{[0,1]}}E_\lambda(v)$. Since $\sup_{v\in V^{[0,1]}}E_\lambda(v)<\infty $  we can construct the sequence $v^m$ so that there exists an $M<\infty$ such that $\| v^m \|_{V^{[0,1]}}\leq M$ for all $m$. Since $\Omega$ is bounded, closed finite balls in $V^{[0,1]}=L^2([0,1],V)$ are weakly compact (by \cite{dup:98}). Therefore we may  extract a subsequence from $v^m$ (relabeled by $m$) which weakly converges to  a $\hat v \in V^{[0,1]}$. In particular, $\langle v^m, w \rangle_{V^{[0,1]}}\rightarrow \langle \hat v, w \rangle_{V^{[0,1]}}$ for all $w\in V^{[0,1]}$. Furthermore we have lower semicontinuity of the norm
\begin{equation}
\label{liminf}
 \liminf_{m\rightarrow \infty} \| v^m \|^2_{V^{[0,1]}} \geq  \| \hat v \|^2_{V^{[0,1]}}.
\end{equation}
Now by Theorem 3.1 in \cite{dup:98}

\if10{
{\flushleft\textcolor{blue}{$\downarrow$---------begin long version---------}}\newline
\cite{dup:98}  applies since the assumption $[W_0^{3,2}(\Omega)]^3$ (where $\Omega\subset \Bbb R^3$ in their paper) is only used to establish that $[W_0^{3,2}(G)]^3\hookrightarrow C_0^1(G)$. Since we are assuming $V\hookrightarrow C_0^2(\Omega, \Bbb R^d)\hookrightarrow C_0^1(\Omega, \Bbb R^d)$ we are free to use the results in \cite{dup:98} .
{\flushleft\textcolor{blue}{$\uparrow$------------end long version---------}}\newline
} \fi
 we have that  $\phi^{v^m}_t(x) \rightarrow \phi^{\hat v}_t(x)$ uniformly in $t\in [0,1]$ as $m\rightarrow \infty$.
This allows us to show that $\log \det D\phi_1^{v^m}(x)\overset{m\rightarrow\infty}\longrightarrow \log \det D\phi_1^{\hat v}(x)$ for every $x\in \Omega$. To see why, one can use similar reasoning as in \cite{cao:05}. First write
\begin{align*}
|\log \det D\phi_1^{v^m}(x)- \log \det D\phi_1^{\hat v}(x)| &= \left| \int_{0}^1 \text{div}\, v_t^m(\phi_t^{v^m}(x)) -  \text{div}\, \hat v_t(\phi_t^{\hat v}(x)) dt \right|
=I + I\!I
\end{align*}
where the first term $I$ satisfies
\begin{align*}
I &\equiv \left| \int_{0}^1 \text{div}\, v^m_t(\phi_t^{v^m}(x)) -  \text{div}\, v^m_t(\phi_t^{\hat v}(x)) dt \right| \\
&\leq \int_{0}^1 \|\text{div}\, v^m_t\|_{1,\infty} \bigl|\phi_t^{v^m}(x) -  \phi_t^{\hat v}(x)\bigr| dt  \\
&\leq   \int_{0}^1 c\| v^m_t\|_{V} \bigl|\phi_t^{v^m}(x) -  \phi_t^{\hat v}(x)\bigr| dt,\,\,\text{since $V\hookrightarrow C_0^2(\Omega,\Bbb R^d)$} \\
&\leq  c\|  v^m\|_{V^{[0,1]}} \Bigl[\int_{0}^1 \underbrace{\bigl|\phi_t^{v^m}(x) -  \phi_t^{\hat v}(x)\bigr|^2}_\text{ $= o(1)$ uniformly in $t$} dt\Bigr]^{1/2},\,\,\text{by H\"older.} \\
&\rightarrow 0,\,\text{ since $\| v^m\|_{V^{[0,1]}}\leq M$ for all $m$.}
\end{align*}
For the second term $I\!I$ notice that  the map sending $v\mapsto \int_0^1 \text{div}\, v_t(y_t) dt$ is a bounded linear functional on $V^{[0,1]}$ (using the fact that $V\hookrightarrow C_0^1(\Omega, \Bbb R^d)$) where $y_t \equiv \phi_t^{\hat v}(x)$. By the Riesz representation theorem there exists a $w^{\hat v}\in V^{[0,1]}$ such that $\int_0^1 \text{div}\, v_t(y_t) dt = \langle v,w^{\hat v} \rangle_\text{\tiny $V^{[0,1]}$}$. Therefore
 \begin{align*}
I\!I
&\equiv \left| \int_{0}^1 \text{div}\, v_t^m(\phi_t^{\hat v}(x)) -  \text{div}\, \hat v_t(\phi_t^{\hat v}(x)) dt \right|\\
&=   \Bigl| \langle v^m -  \hat v, w^{\hat v}\rangle_{V^{[0,1]}}\Bigr|\rightarrow 0, \,\,\text{by weak convergence.}
\end{align*}
Combining the results for $I$ and $I\!I$ we can conclude that $\log \det D\phi_1^{v^m}(x)\overset{m\rightarrow\infty}\longrightarrow \log \det D\phi_1^{\hat v}(x)$ for every $x\in \Omega$.

To finish the proof notice that
\begin{align}
\sup_{v\in V^{[0,1]}} E_\lambda(v) &= \lim_{m\rightarrow \infty} E(v^m)= \limsup_{m\rightarrow \infty} E(v^m)\nonumber\\
&= \frac{1}{n}\sum_{k=1}^n   \log\text{det} D\phi_1^{\hat v}(X_k)+ \frac{1}{n}\sum_{k=1}^n H\circ\phi_1^{\hat v}(X_k) -\frac{\lambda}{2} \liminf_{m\rightarrow \infty} \int_0^1 \| v^m_t \|_V^2 dt \nonumber\\
&\leq \frac{1}{n}\sum_{k=1}^n   \log\text{det} D\phi_1^{\hat v}(X_k)+ \frac{1}{n}\sum_{k=1}^n H\circ\phi_1^{\hat v}(X_k) -\frac{\lambda}{2}  \int_0^1 \| \hat v_t \|_V^2 dt,\,\,\text{ by (\ref{liminf})}\nonumber \\
&= E_\lambda(\hat v) \nonumber
\end{align}

  \end{proof}

One of the important facts about any vector field flow $\hat v\in V^{[0,1]}$ which maximizes $E_\lambda$  is that the resulting estimated transformation  $\phi_1^{\hat v}$ is a geodesic (or minimum energy) flow with respect to the vector field norm $\int_0^1 \| \hat v_t\|^2_V dt$. To see this is first notice that the parameterization of time $t=1$ maps, $\phi_1^v$, by vector fields $v\in V^{[0,1]}$ is a many-to-one  parameterization.
In other words there exist multiple pairs of vector fields $v,w\in V^{[0,1]}$ such that $\phi_1^v =\phi^w_1$ but $v\neq w$. Notice, however, that the log-likelihood term in $E_\lambda$ only depends on   $\phi_1^{\hat v}$. This implies that any maximizer $\hat v$ of $E_\lambda$ must simultaneously minimize  the penalty $\int_0^1 \| \hat v_t\|^2_V dt$ over the class of all $w\in V^{[0,1]}$ which has the same terminal value, i.e.\! $\phi^{\hat v}_1=\phi^w_1$.
 Consequently, the PMLE estimate $\hat v$ must be a geodesic flow. An important consequence  is that  geodesic flows $\{ \hat v_t \}_{t\in [0,1]}$ are completely determined by the initial vector field $\hat v_0$.  This will become particularly important in the next section where the initial velocity field will be completely parameterized by $n$ coefficient vectors.

\section{Spline representation from Euler-Lagrange}
\label{EL}

In this section we work under the additional assumption that $V$ is a reproducing kernel Hilbert space. This assumption allows one to derive the Euler-Lagrange equation for any maximizer $\hat v$ of which satisfies (\ref{exist}). This leads to a finite dimensional characterization of  $\hat v$  which  parallel those results found in the spline literature for function estimation.

\begin{claim}
\label{claim2}
Let $V$ be a  reproducing kernel Hilbert space, with  kernel $R(x,y)I_{d\times d}$, continuously  embedded in $C_0^3(\Omega, \Bbb R^d)$  where $\Omega$ is bounded open subset of $\Bbb R^d$. Suppose $e^{H(\cdot)}$ is a $C^1(\bar \Omega,\Bbb R)$ density  on $\Omega$. Then any time varying vector field $\hat v \in V^{[0,1]}$ which satisfies (\ref{exist}) also satisfies the following
 Euler-Lagrange equation:
 \begin{align}
 \label{ELeq}
 \hat v_t(x)&=  \frac{1}{\lambda n}\sum_{k=1}^n \beta_{k,t}^T R(x,X_{k,t})  +  \frac{1}{\lambda n}\sum_{k=1}^n   \nabla_{y} R(x,y)\Bigr|_{y= X_{k,t}}
\end{align}
where $X_{k,t}\equiv \phi_t^{\hat v} (X_k)$ and
$ \beta_{k,t}\equiv   \nabla H(X_{k,1}) D\phi^{\hat v}_{t1}(X_{k,t})  +\nabla \log\det D\phi^{\hat v}_{t1} (X_{k,t})$.
\end{claim}
\begin{proof}
Let $E_1, E_2$ and $E_3$ decompose $E_\lambda$ as in (\ref{decomp}). Notice first that if  $h\in V^{[0,1]}$  and $\epsilon \in \Bbb R$ then $2 E_3(\hat v+\epsilon h) = {\lambda}\| \hat v \|^2_{V^{[0,1]}} + \epsilon 2 \lambda\langle v,h \rangle_{V^{[0,1]}}  +\epsilon^2{\lambda}\| h \|^2_{V^{[0,1]}} $. Therefore $E_3(\hat v+\epsilon h)$ is differentiable with respect to $\epsilon$ with derivative given by
\begin{equation}
 \label{dderE1}
{\partial_\epsilon}  E_3(\hat v+\epsilon h)\bigr|_{\epsilon=0}
=  \int_0^1 \langle h_t,\lambda \hat v_t \rangle_V dt.
\end{equation}
In addition, Theorem 8.10 of \cite{you:10}
 implies that  $\phi^{\hat v+\epsilon h}_1(x)$  is differentiable at $\epsilon = 0$.

Now, the assumption  $H\in C^1(\bar\Omega)$ combined with equation (\ref{111}), in the Appendix, gives
\begin{align}
\partial_\epsilon E_2(\hat v+\epsilon h)\bigr|_{\epsilon=0} \nonumber
&= -\frac{1}{n}\sum_{k=1}^n  \nabla H(\phi_1^{\hat v }(X_k)) \cdot {\partial_\epsilon}  \phi^{\hat v+\epsilon h}_{1}(X_k)\bigr|_{\epsilon = 0} \nonumber \\
&= -\frac{1}{n}\sum_{k=1}^n  \nabla H(\phi_1^{\hat v }(X_k)) \cdot \int_0^1  \bigl\{D\phi^{ \hat v}_{u1} h_u \bigr\}\circ{\phi^{ \hat v}_{u}(X_k)}\,   du \nonumber \\
&= -\frac{1}{n}\sum_{k=1}^n \int_0^1  \bigl\{  \nabla H( X_{k,1}) D\phi^{\hat v}_{u1}(X_{k,u}) \bigr\} \cdot  h_u (X_{k,u})\,   du \nonumber \\
&=  \int_0^1\Bigl\langle  h_u(\cdot),  -\frac{1}{n}\sum_{k=1}^n \bigl\{ \nabla H( X_{k,1}) D\phi^{\hat v}_{u1}(X_{k,u}) \bigr\}^T R(\cdot,X_{k,u})\Bigr\rangle_V \,du  \label{derE2}
\end{align}
Finally, Proposition \ref{proo}, from the Appendix, implies $E_3(\hat v+\epsilon h )$ is differentiable at $\epsilon = 0$ with derivative given by
\begin{align}
\partial_\epsilon  E_1(\hat v+\epsilon h)\bigr|_{\epsilon=0} \nonumber
&  =- \frac{1}{n} \sum_{k=1}^n  \partial_\epsilon  \log \det D\phi_{1}^{\hat v+\epsilon h }(X_k)\bigr|_{\epsilon = 0} \nonumber\\
&=- \frac{1}{n} \sum_{k=1}^n
 \int_0^1  \Bigl[ h_u \cdot \nabla \log\det D\phi_{u1}^{\hat v}  + \text{\rm div}\, h_u\Bigr] \circ  \phi^{\hat v}_{u}(X_k)  \, du \nonumber\\
&= \int_0^1 \Bigl\langle h_u(\cdot), - \frac{1}{n} \sum_{k=1}^n \left\{ \nabla \log\det D\phi_{u1}^{\hat v}(X_{k,u})\right\}^T  R(\cdot,X_{k,u})+\nabla_{y} R(\cdot,y)\bigr|_{y=X_{k,u}}\Bigr\rangle_V \,du \label{derE3}
\end{align}
{\em Remark:} the above equation requires $\partial_{x_i } ( e_i\cdot h_u(x))  = \partial_{x_i}   \langle e_i R(\cdot, x),  h_u\rangle_V=  \langle e_i  \partial_{x_i} R(\cdot, x),  h_u\rangle_V$ which follows since $\text{div}\, h_u \in V$ by the assumption $V\hookrightarrow C_0^2(\Omega,\Bbb R^d)$ (see \cite{aro:50}).
Now from (\ref{dderE1}),  (\ref{derE2}) and (\ref{derE3}), the energy $ E_\lambda(\hat v+\epsilon h)$ is differentiable with respect to $\epsilon$ at $0$ and
\begin{equation}
\label{EEEl}
0={\partial_\epsilon}  E_\lambda(\hat v+\epsilon h)\bigr|_{\epsilon=0}= \langle \mathcal E ^{\hat v},h \rangle_{V^{[0,1]}}
\end{equation}
where
\begin{equation}
\label{CalE}
 \mathcal E ^{\hat v}_t =\lambda \hat v_t   - \frac{1}{n}\sum_{k=1}^n \beta_{k,t}^T  R(\cdot,X_{k,t})  - \frac{1}{n}\sum_{k=1}^n \nabla_{y} R(\cdot,y)\bigr|_{y=X_{k,t}}
 \end{equation}
with $\beta_{k,t}\equiv  \nabla H( X_{k,1}) D\phi^{\hat v}_{t1}(X_{k,t})  + \nabla \log\det D\phi_{t1}^{\hat v}(X_{k,t}) $.
Since $h\in V^{[0,1]}$ was arbitrary, equation (\ref{EEEl}) implies $\mathcal E^{\hat v}=0$, which then gives (\ref{ELeq}). {\em Remark:} we are using the fact that the zero function in a reproducing kernel space is point-wise zero since the evaluation functionals are bounded.
\end{proof}

   There are a few things things to note here. First, the Euler-Lagrange equation  (\ref{ELeq}) only implicitly characterizes $\hat v$ since it appears on both sides of the equality ($\beta_{k,t}$ and $X_{k,t}$ also  depend on $\hat v$). Regardless,  (\ref{ELeq}) is useful since it implies that $\hat v$ must lie within a known $n\times d$ dimensional sub-space of $V^{[0,1]}$. In particular, as discussed at the end of Section \ref{pmle}, the estimate $\{ \hat v_t\}_{t\in [0,1]}$ is completely characterized by it's value at time $t=0$, i.e.\! $\hat v_0$ (by the geodesic nature of $\hat v$). Restricting equation (\ref{ELeq}) to  $t=0$ one obtains
   \begin{align}
 \label{ELeq0}
 \hat v_0(x)&=  \frac{1}{\lambda n}\sum_{k=1}^n \beta_{k,0}^T R(x,X_{k})  +  \frac{1}{\lambda n}\sum_{k=1}^n   \nabla_{y} R(x,y)\Bigr|_{y= X_{k}}.
\end{align}
  Simply stated, $\hat v_0$ has a finite dimensional spline characterization with spline knots set at the observations $X_1,\ldots, X_n$. Therefore to recover $\{ \hat v_t \}_{t\in[0,1]}$ one simply needs to find the $n$ vectors $\beta_{1,0},\dots, \beta_{n,0}$ which satisfy the following fixed point equation
        \begin{equation}
 \label{InitialMom}
  \beta_{k,0}= \nabla H(\phi^{\hat v}_1(X_k))  D\phi^{\hat v}_{1}(X_k)+ \nabla \log\det D\phi^{\hat v}_{1} (X_k)
  \end{equation}
 for all $k=1,\ldots,n$.

\section{Connection to Stein's Method}
\label{steinSection}

The Euler-Lagrange equation given in (\ref{ELeq}) has a surprising connection with a generalization of Stein's lemma for characterizing the normal distribution (see \cite{stein:04}). The main connection is that the Euler Lagrange equation for the PMLE estimate $\hat v_t$, simplified at initial time $t=0$ and terminal time $t=1$, can be reinterpreted as an empirical version of a generalization of Stein's lemma. This is interesting in it's own right, however, the connection may also bear theoretical fruit for deriving asymptotic estimation bounds on the nonparametric and semiparametric estimates derived from $\hat v$. In this section we make this connection explicit  with the goal of of motivating and explaining the Euler-Lagrange equation for $\hat v$ derived above.

To relate $\hat v_t$ at $t=0$ with Stein's lemma, and more generally Stein's method for distributional approximation,  first notice that (\ref{InitialMom}) implies the coefficients $\beta_{k,0}$,  from the implicit equation  (\ref{ELeq0}) for  $\hat v$, satisfy
$\beta_{k,0
 = \nabla \log \hat f(X_k)
$ where  $\hat f= e^{ H\circ \phi^{\hat v}_1}  |D\phi_1^{\hat v}| $ is the estimated density of $X$ using the pullback of the target measure with the estimated diffeomorphisms $\phi^{\hat v}_1$.
 Now by computing the inner product of  both sides of the Euler-Lagrange equation (\ref{ELeq0}) with any vector field $u\in V$ and applying the reproducing property of the kernel $R(\cdot,\cdot)$ one derives
 \begin{align}
\lambda \langle \hat v_0, u\rangle_
&=\Bbb E_n\bigl\{ \nabla \log \hat f(X)\cdot  u(X) + \text{div}\, u(X)\bigr\}. \label{stein2}
\end{align}
where $\Bbb E_n$ denotes  expectation with respect to the empirical measure generated by the data: $\frac{1}{n}\sum_{k=1}^n \delta_{X_k}$.
To relate with Stein first let $\Bbb E$ denote expectation with respect to the population density $f =    e^{ H\circ \phi} |D\phi| $ given in our basic model (\ref{model1}). Notice that a generalization of Stein's lemma shows that if the densities $f$ and $\hat f$ give rise to the same probability measure then
\begin{equation}
\label{ste}
0= \Bbb E\bigl\{ \nabla \log \hat f(X) \cdot u(X) + \text{div}\, u(X)\bigr\}
 \end{equation}
for all $u$ in a large class of test functions $\mathcal U$ (see Proposition 4 in \cite{stein:04}).
For example, a simple consequence of Lemma 2 in \cite{stein:81} implies that when $\hat f$ is the density of a  $d$ dimensional Gaussian  distribution $\mathcal N_d(\hat\mu,1)$  and $X\sim \mathcal N_d(\mu,1)$ then $\hat\mu = \mu$ implies $ \Bbb E\bigl\{ -  (X-\hat\mu) \cdot u(X) + \text{div}\, u(X) \bigr\} =0$
for any bounded function $u:\Bbb R^d \rightarrow \Bbb R^d$ with bounded gradient. Stein's method, on the other hand, generally refers to  a technique for  bounding the distance between two probability measures $f$ and $\hat f$ using bounds on departures from a characterizing equation, such as  (\ref{ste}) for example (see \cite{chen:05} for an exposition). The bounds typically take the form
\begin{align} \sup_{h\in \mathcal H} \left| \int ( h f - h \hat f ) \right| &\leq \sup_{u\in \mathcal U} \bigl| \Bbb E\bigl\{ \nabla \log \hat f(X)\cdot u(X) + \text{div}\, u(X)\bigr\} \bigr| \label{ssmethod}
 \end{align}
where $\mathcal H$ and $\mathcal U$ are two class of functions related through a set of differential equations.

  In our case, applying a H\"older's inequality to the Euler-Lagrange equation (\ref{stein2}) gives a bound on  right hand side of (\ref{ssmethod}) in terms of a regularization measurement on the PMLE $\hat v$ and an empirical process error:
    \begin{align}
    \label{reg1}
     \sup_{u\in \mathcal U} \bigl| \Bbb E\bigl\{ \nabla \log \hat f(X) \cdot u(X) + \text{div}\, u(X)\bigr\} \bigr|   &\leq \underbrace{ \lambda \| \hat v_0\|_V   \sup_{u\in \mathcal U} \|  u \|_V}_\text{regularization at $t=0$} + \underbrace{\sup_{u\in \mathcal U} \bigl| (\Bbb E -\Bbb E_n) \nu_{\hat f,u}\bigr|}_\text{ empirical process error}
 \end{align}
 where $\nu_{\hat f,u} = \nabla \log \hat f(X) u(X) + \text{div}\, u(X)$.
 This makes it clear that  theoretical control of the PMLE estimate $\hat v_t$ at time $t=0$, using the Euler-Lagrange equation characterization (\ref{ELeq0}), allows asymptotic control of the distance between the estimated density $\hat f$ and the true density $f$.

 At terminal time $t=1$, there is a similar connection with Stein's lemma. In contrast to time $t=0$, which quantifies the distance between the estimated  and population densities $\hat f$ and $f$,  time $t=1$ quantifies the distance between $\phi(X)$ (the target measure) with $\phi_1^{\hat v}(X)$ (the push forward of the true population distribution though the estimated map).  To make the connection, one follows  the same line of argument as above to  find that  for any $u\in V$
\begin{align}
\lambda \langle \hat v_1, u\rangle_V &=  {\Bbb E}^{\hat v}_n\bigl\{ \nabla H(X)\cdot u(X ) + \text{div}\, u(X)\bigr\} \label{stein}
\end{align}
where ${\Bbb E}^{\hat v}_n$ denotes  expectation with respect to the empirical measure $\frac{1}{n}\sum_{i=1}^n \delta_{\phi^{\hat v}_1(X_k)}$, which is simply the push forward of the empirical measure $\frac{1}{n}\sum_{i=1}^n \delta_{X_k}$ through the estimated map $\phi^{\hat v}_1$.
Now the analog to (\ref{reg1})  becomes
   \begin{align}
   \label{reg2}
    \sup_{u\in \mathcal U} \bigl| {\Bbb E^{\hat v}}\bigl\{ \nabla H(X)\cdot  u(X) + \text{div}\, u(X)\bigr\} \bigr|   &\leq \underbrace{ \lambda \| \hat v_1\|_V  \sup_{u\in \mathcal U} \|  u \|_V }_\text{regularization at $t=1$} +\, {\sup_{u\in \mathcal U} \bigl| (\Bbb E^{\hat v} -\Bbb E_n^{\hat v}) \gamma_{u}\bigr|}
 \end{align}
 where $\gamma_{u} = \nabla H(X) u(X) + \text{div}\, u(X)$ and $\Bbb E^{\hat v}$ denotes expectation with respect to the push forward of the population density $f =    e^{ H\circ \phi} |D\phi| $ though the estimated map $\phi_1^{\hat v}$.
Since the target measure $\Bbb P$ is assumed to have density $e^H$, this bounds the distributional distance between $\phi^{\hat v}_1(X)$ and $\Bbb P$ when $X\sim \Bbb P\circ \phi$.

\section{Nonparametric example}
\label{npe}

In this section we utilize the finite dimensional characterization of the PMLE $\hat v$ at time $t=0$, given in (\ref{ELeq0}), to construct nonparametric density estimates of the form $\hat f= e^{ H\circ \phi^{\hat v}_1}  |D\phi_1^{\hat v}| $ from {\em iid} samples $X_1,\ldots, X_n$.
As was discussed in the introduction, so long as the target measure $\Bbb P$ is absolutely continuous, the assumption that   $X_1,\ldots, X_n \overset{iid}\sim \Bbb  P\circ \phi$ encompasses all absolutely continuous measures. Since the class of diffeomorphisms $\{\phi_1^v\colon v\in V^{[0,1]}\}$ is nonparametric, the estimate $\hat f= e^{ H\circ \phi^{\hat v}_1}  |D\phi_1^{\hat v}| $ is inherently nonparametric regardless of the choice of target probability measure $\Bbb P$ (with density $e^H$). In effect, the choice of target $\Bbb P$ specifies a shrinkage direction for the nonparametric estimate: larger values of $\lambda$ shrink $\hat f$ further toward the target $\Bbb P$.
In this section we illustrate the nonparametric nature of the density estimate $\hat f$, whereas the next section explores semiparametric estimation with parametric models on the target $\Bbb P$.     One  key feature of our methodology is the use of the Euler-Lagrange equation (\ref{ELeq}) as a stopping criterion for a gradient based optimization algorithm for constructing $\hat v$. In fact, to avoid computational challenges associated with generating geodesics  with initial velocities given by (\ref{ELeq0}), we consider a finite dimensional subclass of $V^{[0,1]}$ which have geodesics that are amenable to computation (and for which gradients are easy to compute). The key is that we use the Euler-Lagrange identity (\ref{ELeq}) to measure of the richness of the subclass, within the larger infinite dimensional Hilbert space $V^{[0,1]}$, whereby allowing a dynamic choice  of the approximating dimension for a target resolution level.

Claim~\ref{claim2} shows that the PMLE vector field $\hat{v}\in V^{[0,1]}$ obeys a parametric form determined up to the identification of the $n$ functions $t\mapsto \beta_{k,t}$ as $t$ ranges in $[0,1]$. Moreover, the whole path of coefficients $\beta_{k,t}$ is determined from the initial values $\beta_{k,0}$, by the geodesic nature of $\hat v$.
In this way, we are free to optimize, over the vectors $\{\beta_{1,0}, \ldots, \beta_{n,0}\}\subset \Bbb R^d$ using equation (\ref{ELeq0})
 and are guaranteed that the global maximum, over the full infinite dimensional space $\{ \phi_1^v\colon v\in V^{[0,1]}\}$, has this form.  Unfortunately, deriving geodesic maps with this type of initial velocity field is challenging.
To circumvent this difficulty we choose an approximating  subclass of vector fields at time $t=0$ which are parametrized by the selection of $N$ knots $\{\kappa_{1},\ldots, \kappa_N\} \subset \Omega$ and $N$ initial momentum row vectors $\{\eta_{1},\ldots, \eta_N\} \subset  \mathbb{R}^d$ and have the form:
\begin{equation}
\label{knots}
 v_0(x)= \sum_{k=1}^N \eta_{k}^T R(x,\kappa_{k}).
 \end{equation}
The knots $\{ \kappa_1,\ldots, \kappa_N\}$ need not be located at the data points $\{X_1,\ldots, X_n \}$. Indeed, we will see that alternative configurations of knots can be numerically beneficial. The key point is that vector fields at time $t=0$, which satisfy (\ref{knots}), generate geodesics with respect to norm $\bigl[\int_0^1 \| v^t  \|^2_V dt \bigr]^{1/2}$ that are easy to compute. Moreover, the variational derivatives of the terminal map $\phi^v_1$ with respect to the initial $\eta$ coefficients and the knots $\kappa$ are easily computed when utilizing  similar techniques  as those developed in \cite{vaillant:04} and \cite{alla:07}. This enables efficient gradient based algorithms for optimizing the PMLE criterion over the class generated by (\ref{knots}).

\begin{figure}[t]
\centering
\includegraphics[height = 2.2in]{density_justn}
\includegraphics[height = 2.2in]{agreement_justn}\\
\includegraphics[height = 2.2in]{density_n_pm_h}
\includegraphics[height = 2.2in]{agreement_n_pm_h}\\
\caption{In this example we compare two different knot configurations, in (\ref{knots}), for generating nonparametric density estimates using approximate solutions to the  Euler-Lagrange equation (\ref{ELeq}). The left column of images shows two different density estimates (red), based on the same data set (blue), using two different knot configurations (top-left uses $10$ knots, bottom-left uses $30$ knots). The right column of images show the corresponding  diagnostic curves which characterize the richness of the approximating subclass generated by the knots. The fact that the two diagnostic curves shown bottom-right are similar suggests that the $30$ knots used generate the approximating subclass by (\ref{knots}) is sufficiently rich to reach the stationary points of the penalized log likelihood $E_\lambda$ given in (\ref{energy1}).
See Section \ref{npe} for details.
 \label{f2}
 }
\end{figure}

As a first illustration, we show that the na\"ive choice of initial knots obtained by setting $\{\kappa_1,\ldots, \kappa_N\}=\{ X_1,\ldots,X_n\}$ in (\ref{knots})  is {\em not} sufficient to solve  (\ref{ELeq}); then show how it can be easily fixed using the Euler-Lagrange methodology.
 Our data set, shown with blue sticks in Figure~\ref{f2}, consists of $n=10$ independent samples from a mixture of two normals, truncated so the support is $[0,1]$.  Our target probability measure $\Bbb P$ is set to the uniform distribution on  $[0,1]$ (smoothly tapering to zero $0$ outside of $[0,1]$ for numerical convenience).
  For simplicity we choose the Gaussian kernel $R(x,y)=\exp\bigl(-\frac{(x-y)^2}{2\sigma^2}\bigr)$, with $\sigma=0.1$, to generate the RKHS $V$ and use the  penalty parameter $\lambda$ set to $10$. The top left plot in Figure~\ref{f2} shows the non-parametric density estimate $\hat f= e^{ H\circ \phi^{\hat v}_1}  |D\phi_1^{\hat v}| $ in red, generated by applying a gradient based optimization algorithm applied to the subclass (\ref{knots}) where the knots $\{\kappa_1,\ldots, \kappa_N\}=\{X_1,\ldots, X_n\}$ are kept fixed and the coefficients $\eta_1,\ldots, \eta_N$ are optimized by minimizing the penalized log likelihood function $E_\lambda (v)$ given in (\ref{energy1}).
 To diagnose the richness of subclass (\ref{knots}) within the full Hilbert space we define the function $\mathcal D_t^v(x)$ for any $v\in V^{[0,1]}$ and any $t\in [0,1]$ as follows
 \begin{equation}
 \label{diag1}
 \mathcal D_t^v(x) \equiv    \frac{1}{ n}\sum_{k=1}^n \bigl[\beta_{k,t}^v \bigr]^T R(x, X_{k,t}^v )  +  \frac{1}{ n}\sum_{k=1}^n   \nabla_{y} R(x,y)\Bigr|_{y= X_{k,t}^v}
 \end{equation}
where $X_{k,t}^v\equiv \phi_t^{v} (X_k)$ and
$ \beta_{k,t}^v\equiv   \nabla H(X^v_{k,1}) D\phi^{v}_{t1}(X_{k,t}^v)  +\nabla \log\det D\phi^{ v}_{t1} (X^v_{k,t})$.
The function $\lambda v_t - \mathcal D_t^v$ serves as a diagnostic criterion in the sense that
  the Hilbert norm of $\lambda v_t - \mathcal D_t^v$ gives  the maximal rate of change of the penalized log-likelihood $E_\lambda(v)$, within the full infinite dimensional Hilbert space $V^{[0,1]}$. In particular,
 \[   \biggl[\int_0^1\|\underbrace{ \lambda v_t-\mathcal D_t^v}_\text{\scriptsize diagnostic } \|_{V}^2dt\biggr]^{1/2} = \sup_\text{\small $\{ u\colon \| u \|_{V^{[0,1]}}= 1 \}$} \left[\frac{d}{d\epsilon} E_\lambda (v+\epsilon u)\right]_{\epsilon = 0} .   \]
 Therefore if   $\lambda v_t(x)-\mathcal D_t^v(x) = 0$ for all $t\in [0,1]$ and $x\in \Bbb R^d$, then $v$ satisfies the Euler-Lagrange equation. Discrepancies between $\lambda v_t(x)$ and $\mathcal D_t^v$ when optimizing over the subclass (\ref{knots}) indicates the subclass that is insufficient rich to reach the stationary points of $E_\lambda(v)$.
 The diagnostic plots in this example, which correspond to our density estimate shown in the upper-left image of Figure~\ref{f2},  are shown in the upper-right plot of Figure~\ref{f2} where $\lambda v_0(x)$ is plotted in  black and $\mathcal D_0^v(x)$ is plotted as a dashed green line. The large amount of discrepancy between $\lambda v_0(x)$  and $\mathcal D_0^v(x)$ indicates that the knots $\{\kappa_1,\ldots, \kappa_N\}=\{ X_1,\ldots,X_n\}$  are insufficient.

\begin{figure}[t]
\centering
\includegraphics[height = 2.2in]{density_justn2}
\includegraphics[height = 2.2in]{agreement0_justn2}\\
\includegraphics[height = 2.2in]{density_subsample2}
\includegraphics[height = 2.2in]{agreement0_subsample2}
\caption{
 \label{f3}
 In this example we demonstrate that a small number of knots, in (\ref{knots}), can be enough to approximate solutions to the  Euler-Lagrange equation (\ref{ELeq}). The left column of images shows two different density estimates (red), based on the same data set  of size $n=240$ (grey histogram), using two different knot configurations (top-left uses $240$ knots, bottom-left uses $20$ knots). The right column of images show the corresponding  diagnostic curves which characterize the richness of the approximating subclass generated by the knots. The fact that the two diagnostic curves shown bottom-right are nearly identical suggests that the $20$ knots  used generate the approximating subclass by (\ref{knots}) is sufficiently rich to reach the stationary points of the penalized log-likelihood $E_\lambda$ given in (\ref{energy1}).
See Section \ref{npe} for details.
 }
 \end{figure}

To generate  knots which are sufficiently rich, in this first example,  we apply a discrete approximation at initial time $t=0$ to the gradient term $ \nabla_{y} R(x,y) $ appearing in  the Euler-Lagrange equation (\ref{ELeq}).
 For this approximation we use $N=3n$ knots in the pattern given by the following approximation
\begin{align}
 \hat v_0(x)&=  \frac{1}{\lambda n}\sum_{k=1}^n \beta_{k,0}^T R(x,X_{k})  +  \frac{1}{\lambda n}\sum_{k=1}^n   \nabla_{y} R(x,y)\Bigr|_{y= X_{k}} \\
 & \approx \frac{1}{\lambda n}\sum_{k=1}^n \beta_{k,0}^T R(x,\kappa_{k})  +  \frac{1}{\delta \lambda n}\sum_{k=1}^n   R(x,\kappa_{n+k})-R(x,\kappa_{2n+k}) \\
 & = \sum_{k=1}^N \eta_{k}^T R(x,\kappa_{k}) \label{eq:etakappa}
\end{align}
where
$\kappa_{k}\equiv\begin{cases}
X_{k} & \text{if $k \in 1 \dots n$} \\
X_{k}+\frac{\delta}{2} & \text{if $k \in n+1 \dots 2n$} \\
X_{k}-\frac{\delta}{2} & \text{if $k \in 2n+1\dots 3n$}
\end{cases}$ and $\eta_{k}\equiv\begin{cases}
\frac{1}{\lambda n} \beta_{k,0} & \text{if $k \in 1 \dots n$} \\
\frac{1}{\delta \lambda n} & \text{if $k \in n+1 \dots 2n$} \\
-\frac{1}{\delta \lambda n} & \text{if $k \in 2n+1 \dots 3n$} \\
\end{cases}$
with $\delta=10^{-4}$.
 With this new set of knots, the resulting PMLE over the new class is show at bottom left in Figure~\ref{f2}. Notice that now the diagnostic function $\lambda v_0 -\mathcal D_0^v$ (the difference between the black and green line in the bottom-right plot of Figure~\ref{f2}) is much closer to zero. Indeed, for every $t\in [0,1]$ the diagnostic function $\lambda v_t -\mathcal D_t^v$ is similarly close to zero (not pictured). This implies that the maximal rate of change  of the penalized log-likelihood within the infinite dimensional Hilbert space $V^{[0,1]}$, at our estimate, is very small and hence our knots are sufficiently rich.

In the previous example we used $N=3n$ knots in (\ref{knots}) to construct a sufficiently rich class for solving the Euler-Lagrange equation (\ref{ELeq}). Now we demonstrate that with larger data sets and smaller smoothness penalties one can actually use a smaller set of knots, $N\ll n$, to approximate the solutions to Euler-Lagrange equation (\ref{ELeq}). The histograms in the left column of Figure~\ref{f3} show $n=240$ {\em iid}  samples from the same truncated mixture of normals used in the previous example. The resulting density estimates, shown in red,  use a smoothness penalty set to $\lambda=1/4$. The estimate shown top-left  utilizes  $N=n=240$ knots set at the data points whereas the estimate shown bottom-left  uses $N=20$ knots randomly selected from the data.  The right column
shows the corresponding diagnostic plots ($\lambda v_0$ shown in black  and $\mathcal D_0^v$ shown in green). The relative agreement of the diagnostic curves in the bottom-right plot suggests that 20 knots are reasonably adequate for finding approximate solutions to the Euler-Lagrange equation. We expect this situation to improve as the number of data points increase. This has the potential to dramatically decrease the computational load when applying this estimate to extremely large data sets.

\section{Semiparametric example}
\label{spe}

In this section we demonstrate how the PMLE $\phi_1^{\hat v}$ can be used to generate semiparametric estimation procedures obtained by assuming a parametric model on the target distribution $\Bbb P$ then introduce a nonparametric diffeomorphism to the target model.
Indeed, any parametric model $\{\Bbb P_\theta\colon \theta\in \Theta\subset \Bbb R^m\}$ can  be extended to a   semiparametric class by considering diffeomorphisms of the data to the parametric target as follows: $\{\Bbb P_\theta \circ \phi^{v}_1 \colon \theta\in \Theta, v\in V^{[0,1]}\}$.
Since the  model $X_1,\ldots, X_n\overset{iid}\sim \Bbb P_\theta\circ \phi^v_1$ implies $\phi^v_1(X_1),\ldots, \phi_1^v(X_n)\overset{iid}\sim \Bbb P_\theta$ it is natural to alternate the optimization of $\theta$ and $\phi$ to compute the estimates $\hat \theta$ and $\hat \phi$ under this semiparametric model. This optimization routine is outlined explicitly in Algorithm \ref{alg2}.

\begin{algorithm}[h!]
\caption{Compute the semiparametric estimates $\hat\theta, \hat \phi$}
\label{alg2}
\begin{algorithmic}[1]
\STATE {{Set} $i=0$ and {initialize} $(\theta^0,\phi^0)$.}
\STATE { {Set}  $\phi^{i+1}$ to the PMLE of $\phi$ defined in Section \ref{pmle} under the model $X_1, \ldots, X_n \overset{iid}\sim \Bbb P_{\theta^i}\circ \phi$}.
\STATE {{Set} $\theta^{i+1}$ to the maximum likelihood estimate  of $\theta\in \Theta$ under the following model for the transformed data points:
\[ \phi^{i+1}(X_1), \ldots, \phi^{i+1}(X_n) \overset{iid}\sim \Bbb P_{\theta}\]}
\STATE {{If} $\theta^i\approx \theta^{i+1}$ and $\phi^i\approx \phi^{i+1}$ {then return} $(\hat\theta, \hat\phi)\leftarrow(\theta^{i+1},\phi^{i+1})$; {else
set} $i\leftarrow i+1$ and {return} to \mbox{step {\footnotesize 2}.}}
\end{algorithmic}
\end{algorithm}

\begin{figure}[t]
\centering
\includegraphics[height = 2.2in]{normal_target_density}
\includegraphics[height = 2.2in]{normal_target_agreement0}\\
\includegraphics[height = 2.2in]{mixture_target_density}
\includegraphics[height = 2.2in]{mixture_target_agreement0}
\caption{
In this example we demonstrate that by parametrically modeling the target distribution one can  produce flexible semiparametric density estimates.
The left column of histograms show the data (the same histogram plotted twice) sampled from the population density shown in black. Two semiparametric estimates are shown in red which correspond to different parametric targets. The estimated target distribution is shown in green on the left column of images. The right column of images show the corresponding  diagnostic curves which characterize the richness of the approximating subclass generated by the knots ($\lambda v_0$ is plotted in  black and $\mathcal D_0^v$ is plotted in dashed-green).  See Section \ref{spe} for details.
 \label{f4} }
 \end{figure}

To illustrate the semiparametric nature of our estimate we sample from a population density which is a mixture of a $\chi^2$ density (with $20$ degrees of freedom) and a Gaussian density ($\mu = 55$ and $\sigma = 3$) shown in black on the left column of plots in Figure \ref{f4}. The data  comprises $n=200$ independent samples from this mixture, the histogram of which is shown on the left column of plots in Figure \ref{f4}.
 We consider two different semiparametric estimates of the population density. The first uses a basic  location-scale Gaussian family to the parametric target model
 $\{ \Bbb P_\theta \colon \theta \in\Theta \} \equiv \{ \Bbb G_{\mu,\sigma} \colon   \mu \in \Bbb R,  \sigma \in \Bbb R^+ \}$ where $\Bbb G_{\mu,\sigma}$ denotes the Gaussian measure centered at $\mu$ with variance $\sigma^2$.
 The second example uses a
 mixture of two Gaussian measures $\{ \Bbb P_\theta \colon \theta \in\Theta \} \equiv \{ \alpha\,\Bbb G_{\mu_1,\sigma_1} +(1-\alpha)\,\Bbb G_{\mu_2,\sigma_2} \colon  \mu_1, \mu_2 \in \Bbb R,  \sigma_1,\sigma_2 \in \Bbb R^+,   0< \alpha < 1 \}$.
As in Section \ref{npe} we use a Gaussian reproducing kernel to generate the Hilbert space $V$. In this example, however, we use a wider kernel, with standard deviation set to half the sample standard deviation of the data.  This is done to illustrate the flexibility in the estimated density obtained by simply changing the kernel width and the penalty parameter $\lambda$ (which is decreased to $1/2500$ in this example).
Wider kernels tend to produce estimates which have restricted local variability but can still have sufficient flexibility to model large amplitude variations over large spatial scales.

The estimate obtained from the basic location-scale Gaussian family is shown in red in the top-left plot of Figure \ref{f4}. Conversely, the estimated density which uses the mixture target model is shown in red in the bottom-left plot of Figure \ref{f4}.   The corresponding estimated target density $d\Bbb P_{\hat \theta}/dx$ is shown in green on the left two plots.
To numerically approximate the PMLE initial velocity field $\hat v_0$, needed in step 2 of Algorithm \ref{alg2},  we used the approximating subclass for the initial velocity field given in the form (\ref{knots}) with $200$ knots located at the data values.
The corresponding time zero diagnostic plots are shown in the right column of Figure \ref{f4} (green for $\lambda v_0$ and black for $\mathcal D_0^v$).
 Notice that in both cases the semiparametric estimates do a good job at estimating the population density.
In the case of the location-scale Gaussian target the estimated target density does a poor job of explaining the true density. However, the presence of a nonparametric diffeomorphism allows this model to fit nearly as well as a fit from a mixture model. Notice also that the semiparametric estimate based on the location-scale Gaussian target overestimates the true sampling density between the two modes. This seems due to the fact that the estimation procedure prefers an overly dispersed target density which  allows the estimated diffeomorphism to effectively add  mass around the smaller mode. The situation seem to be corrected when using a mixture.

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'Parks and Recreation' to feature Indiana mansion
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VINCENNES, Ind. — The one-time Indiana home of the nation's ninth president will be getting widespread exposure in an upcoming episode of NBC's "Parks and Recreation."
The series starring Amy Poehler that's set in the fictional city of Pawnee, Indiana, will include in its Jan. 20 episode a visit to "Grouseland," the mansion William Henry Harrison called home when he was governor of the Indiana territory.
Grouseland executive director Lisa Ice-Jones tells the Vincennes Sun-Commercial she's thrilled the mansion built in 1800 will be featured in the show.
A film crew for "Parks and Recreation" filmed the mansion during a visit to Vincennes, but a set was built in Studio City, California, to resemble the mansion's interior.
Harrison lived at Grouseland from 1804 until 1812. He was elected U.S. president in 1840.


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You are here: Home1 / Art2 / Art Collective3 / History
The Plough History
A short overview of the history of the Plough will be provided in chronological order. It will focus on how the group came into existence, their stylistic developments, and their other endeavors. The Plough was a diverse group of artists, with different specialties and styles, and although these differences ultimately broke the original group apart, The Plough still exist to this day.
The Beginning (1918 – 1921)
During the summer of 1918 a group of young Groninger artists, consisting of Jan Altink, Toon Benes, Johan Dijkstra, Willem Reinders, and Jan Wiegers gathered to discuss the future of art in Groningen. The reason for this meeting was an exposition held the year before at the Kunstlievend Genootschap Pictura, which they felt didn't pay enough attention to young and upcoming artists. This meeting led to a new public meeting which was to be held on the 5th of June 1918.[1] Although the interest for this meeting was modest (only seven additional people showed), the meeting would be a great success. It was at this meeting that the 12 artists present founded the Groninger art circle The Plough.
Jan Altink had suggested the name The Plough, as it symbolized their goal to plough through the Groninger art-fields. The first months after its formation the group was immensely productive. Their first exhibit was held in February 1919 and displayed 111 works of art. Besides the works of the founders, the exhibition showed work of newer members such as Jan Jordens, George Martens, and, the first female member, Alida Pott. That same year the print-artist H.N. Werkman also joined The Plough. In the early years, the goal of The Plough was to search for new paths to take and to cleanse the artistic palettes. Besides these philosophical goals, they also set a number of practical goals, such as finding a shared atelier or clubhouse. Due to Jan Jordens' position as art teacher at the state's HBS, this goal would quickly be realized.[2]
Of great influence to the members of The Plough was F.H. Bach, who taught most of them during their time at the Minerva Art Academy in Groningen. Bach would occasionally take his students to Blauwborgje, a farm north of the town of Groningen, to sketch outside in the fresh air. Blauwborgje would prove to be one of the most important places of inspiration and gathering for The Plough. Besides this educational influence during those first years, The Plough were greatly influenced by the developments in art outside of Groningen. Jan Wiegers, for example, traveled through Europe and became acquainted with the developments of the 1920's. Another outside influence was the Bergen School, which Charley Toorop was a member of. The acquaintanceship with the Bergen School came through Jan Jordens, who enjoyed his artistic education in The Hague and Amsterdam, and who, during his studies, came into contact with the work of the Bergen School.
The Three Styles of The Plough (1921 – 1930)
It wasn't long until Jan Wiegers got diagnosed with consumption (TB). With financial aid from the other Plough members, Wiegers went to a rehabilitation center in Davos, where he would receive treatment. Wiegers' stay in Davos would later prove to be of crucial significance to the development of The Plough.
During his time in Davos, Wiegers met the German expressionist Ernst Ludwig Kirchner (1880 – 1938), a member of the German art group Die Brücke, who represented an expressionist movement now known for its sharp color contrasts, angular lines, and impulsive brushstrokes. Kirchner, also suffering from consumption, and Wiegers would regularly work together during their stay in Davos. Wiegers became an admirer of the style of Kirchner and brought his newfound enthusiasm for this particular style of expressionism back to Groningen when he returned in 1921. Within The Plough, an expressionist movement was formed that included Jan Altink, Jan Jordens, George Martens, Hendrik Werkman, Johan Dijkstra, and, of course, Jan Wiegers.
Wiegers also introduced a new painting technique that mixed paint with beeswax and petrol. This new technique was soon adopted by several other members of The Plough. Typical The Plough paintings were portraits, of each other and of friends. But also the Groninger landscape, which played a prominent role in many works. A relatively high placed horizon, with a perspective that was emphasized by a road or little waterway, combined with contrasting color blocks are signature stylistic aspects of The Plough's work during their heyday. Open-air painting was greatly favored by the members. The previously mentioned Blauwborgje was, besides being a place frequently used for meetings and discussions, a favored place for portrait and landscape painting.
It was also during this period that Wobbe Alkema and Jan van der Zee developed their own interpretation of constructivism. Van der Zee and Alkema became members of The Plough in 1923 and 1924 respectively but had already been working together on their geometric-abstract style well before that time. Together with John Faber, they founded the advertising agency 'Atelier Voor Artistieke Reclame' (AVAR).[3] In 1923 Hendrik Werkman published his 'druksels', in which the constructivist influence of Alkema and Van der Zee was very much apparent. In the magazine The Next Call, also published by Werkman, the same constructivist influence can be recognized again, alongside with traces of Dadaism. Jan Altink is seen as the catalyst of the more impressionistic movement that began to develop within The Plough around 1927. Before that time, Altink evidently was a member of The Plough's expressionist movement. A.M. Dommering wrote in Nieuwsblad van het Noorden that Altink's style had changed significantly over the course of a year, which was best seen in the change of use of color, from the deep purplish red to the lighter and more frivolous colors of impressionism.[4] Following Altink's example, Ekke Kleima and Jacob Gerard Hansen had also begun to adopt this style.
Peaks and Valleys (1931 – 1941)
The international character of The Plough was greatly exemplified by their 15-year anniversary exhibition. This exhibition consisted of a great overview of modern art (137 pieces) from all across Europe. Besides works of The Plough members themselves, big European artists such as Giorgio de Chirico, Paul Klee, James Ensor, Wassily Kandinsky, Constant Permeke, and Pablo Picasso were also shown.
After this great peak of international recognition, The Plough's development would stagnate in the 1930s. Whilst The Plough kept setting up collective exhibitions, the diversity among them grew. Individually, the members kept growing, but the cooperation and sense of collectiveness from the previous period were lost. As a result, the drive to be experimental and novel became less and less strong, and most artists became more moderate.[5] Werkman warned his fellow members against complacency in his pamphlet Preludium, but to no avail.
In 1941 The Plough would halt all their meetings and labors, as they refused to work under the Reichs Chamber of Culture (Reichskulturkammer).
The Post World War Period (1945– 2005)
In September 1945, the members of The Plough reassembled to prepare for their first major post-war exhibition. In January 1946, the usual annual exhibition of artworks made by The Plough was held in the Pictura gallery. During the first 15 years after the war, the image that surrounded The Plough was still dominated by the older generation. By that time, Jan Wiegers had become a Professor at the State Academy for Arts in Amsterdam, while Jan Altink, Jan Jordens, Johan Dijkstra and Hendrik de Vries received the cultural award of the municipality of Groningen. In response to the moderate attitude of The Plough's older members, a countermovement called 'Het Narrenschip' was established.
A great incentive for the artistic endeavors of The Plough was when Jos de Gruyter was appointed as director of the Groninger Museum; he sought to promote the local art scene again. This resulted in the resuming of the yearly tradition of 'Koppermaandag' in 1963, a tradition where the members of The Plough would jointly present the mayor of Groningen with an engraving.
This boost, however, would prove short-lived, as criticism on the work of The Plough was increasing. The main point of critique was that The Plough were ignoring the developments in modern art, largely due to the fact that there were almost no young members. Around the 1970s almost no members of The Plough were younger than 55. In 1978 a successful recruitment of a large number of younger artists took place. This, however, would lead to a very fractured number of styles and movements, which resulted in fellow exhibitions being replaced by thematic exhibitions instead. At the start of the 80s, The Plough became further divided as some members' wish for expansion clashed with other members' wish to remain a close group of friends. This led to a new countermovement called 'Het Groninger Kunstenaars Kollektief', founded on the 21st of January 1981.
After this fracture, The Plough was saved by painter Jan ten Hoope, who in 1996 provided The Plough with their own gallery called 'Het Ploeghuis'. This Ploeghuis, which was settled in the former Pictura, would become the center for all The Plough activities (such as monthly exhibitions), but it also functioned as living quarters and workshop. The Plough became more active again and housed some of the bigger modern Dutch artists such as Lizette Veenhuizen, Petra van Kalker, and Peter Wortel. This period of renewal and growth came to a halt in 2005 when – due to financial problems, internal discord, and arguments with the landlord – the Plough gallery had to close its doors on the 4th of May 2005.
The Plough still exists to this day and organizes yearly exhibitions in the province of Groningen. Though the group still exists, it is nowhere near its former glory. The heydays of The Plough lie back in the 20s of the previous century.
The Plough as authors
It is important to remember that The Plough was an art circle, and not just a collective of painters, printmakers, and drawers. Besides their pictorial significance, the members of The Plough also published regularly.
In their prime, The Plough and her members published numerous monthly magazines, the first of which was published in October 1921, titled: Blad Voor Kunst. It was a magazine about modern art, printed and published by Hendrik Werkman. A total of 6 issues were published. The editorial office consisted of Jan Wiegers and Jan Jordens for the visual arts, Auguste Defresne for literature, and Daniël Ruyneman for music. The cover of the first magazine was designed by Jan Wiegers.
Although many members of The Plough contributed to Blad Voor Kunst, it wasn't their official magazine. This would be published in 1924 under the name Het Kouter, which literally means plough blade, or plough iron. It was meant to create more exposure for the ideas of The Plough, containing lectures, comments on cultural events, and wood prints. Only 5 issues were published. The fourth issue, of October 1925, was also the exhibition-catalog.
Besides these monthly magazines, The Plough also published a number of books. The first official 'The Plough book' was published in 1923 and was called Schemer, Verzen, and was written by Jozef Cohen, and illustrated by Johan Dijkstra. The Second publication was Teekeningen: J. Wiegers, J. Altink, H.N. Werkman met bijschrift van J. Hansen, which was published in 1924.
In 1926, a The Plough calendar with individual graphical work was published. In 1927 they published another book called De Ploeg Groningen Holland 1927.
In 1936 Werkman created Zwart-Wit boek (Black-White book), with additions from Jan Jordens, Jannes de Vries, Jan Altink, Ben Walrecht, Ekke Kleima, and George Martens. This would be the first publication of a small series.
[1] J. van der Spek, De Schilders van De Ploeg, p. 9.
[2] Idem.
[3] Steenbruggen, 1999, p. 16.
[4] C. Hofsteenge, 1993, p. 84.
[5] Ten Bruggencate en Buschman, 1993, p. 33.


So you've saved up some money and landed a good job. Now you're ready to become a first time renter, well there are things that you need to know. Keep in mind that moving out of the Valdosta State dorms or your parents house means learning to pay bills on time!
Make yourself a budget checklist with your overhead costs for each rental you're considering. Carry this worksheet whenever you're apartment or house hunting. Make notes about deposits, policies, and which utilities each landlord pays. Remind yourself of the amenities and drawbacks for each unit before making a commitment.
Always get a receipt for any fees, deposits, or rent you pay.
One-time fees such as security, pet, and utilities deposits can add up. Your first month in a new apartment can easily reach triple your normal monthly budget. The good news: If you take care of the property, and abide by the lease, you should get some, or all, of your deposits back when you move.
As a tenant, it's your right to live in a safe and habitable environment. Both you and your landlord have certain maintenance responsibilities for the apartment. It's always good idea to put your tenant repair requests in writing or when provided an online property management request system, these usually log an track everything, as well as send you email you can store and track.


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France: North & Central: (18 days) Aug 8- Sept 14 2021
Limited to 16 people, we see a broad variety of the top attractions in the north and centre of the country while gett... more
Patagonia: + Optional Easter Island & Iguazu Falls: January 9 - 27 2021
"Patagonia" is famous for wild its untamed wildness...our tour begins in Buenos Aries- the Paris of the South. Escort... more
from $9,495.00 CAD per person (based on double occupancy)
Colombia, Guyana, Suriname & French Guiana: March 6-27 2021
Colombia, Guyana, Suriname & French Guiana (22 Days) March 6th – 27th Land Only: $8,186 Colombia Only (15 Days) March... more
Ukraine, Belarus & Moldova: June 2022
Please let us know if you are interested! Details will be available shortly, Highlights include Kiev, Odessa on the B... more
Northern India + Sri Lanka: March/April 2022
Our journey through Northern India will include Darjeeling, Calcutta, and the Punjab with Sri Lanka. There will also ... more
New! Norway July/August 2021
We love Norway and look forward to returning!! Please let us know if you are interested to join us! Details will be p... more
Our tour guide and driver made an amazing tour extraordinary. His knowledge of the animals, birds, customs , people skills and driving skills were exceptional. Overall the tour far exceeded our expectations in what we saw, food, accommodations and tour guides
By IE Vancouver BC trip to Kenya


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Excessive pesticides usage hazardous to human health, soil fertility
F.P. Report
MULTAN: Excessive use of pesticides and insecticides in agriculture poses serious hazards to human health, soil fertility, and the overall environment of the country's food basket and hub of agriculture in South Punjab.
Both male and female field workers are usually exposed to this phenomenon with 70% population of South Punjab living in rural areas and the majority among them directly associated with the agriculture sector.
According to South Punjab Agriculture Secretariat, around five million farmers, the majority among them women, are used to working in agriculture fields to lend a hand to their men and eke out earnings for fighting abject poverty.
Being frail in comparison with lower immunity levels than men, these women workers are more exposed to pesticide hazards than men due to direct contact with plants of different crops especially cotton as they have to stay at cotton fields for hours during picking season.
The Agriculture Secretariat of South Punjab, a few months back, conducted a survey and then lab analysis to determine the presence of chemical pesticide residue in cotton fields and the results were alarming.
Three parts of cotton bolls (peel, lint, and seed), collected from the pesticides usage fields after analysis at Chemical Pesticide Residue Lab, Kala Shah Kaku, revealed the presence of an abundant amount of Triazophos and Gamma-cyhalothrin in samples.
According to European standards, the permissible residue limit of Triazophos (pesticide) should be 0.01 mg/kg but in Multan's sample, it was 20.34 mg/kg besides the abundant presence of Gamma-cyhalothrin as well.
Medical practitioners see the lethal impacts on human beings exposed to pesticides and insecticides fearing these people suffer from unconsciousness, lung and liver complications, and cancer if the disease gets chronic.
"There are two complications emerging from excessive use of chemical pesticides – acute and chronic," said Dr Waqas Arqam. "In the acute form, the farmers become unconscious while spraying or visiting farms soon after spray. In the chronic form, they suffer from lethal diseases lung and liver problems, and finally cancer."
He said a good number of pesticide-affected patients are brought to emergencies during the cotton season. "Excessive chemical pesticides also cause contact-dermatitis leading to itching, bleeding, infection and many other complications."
He said, after continuous contact with pesticides the residues start accumulating in their bodies, and a particular level of accumulation, causes cancer, liver complications, and infertility, especially in women farmers. "Sometimes, this poison travels to their stomachs when they eat food with unwashed hands."
"Women are more vulnerable to this phenomenon that results in stunting growth, liver and breast cancer, infertility, and skin complications," Dr Waqas said.
He called upon the agricultural department to raise awareness among farmers on the safer use of pesticides and if possible to shift to bio-pesticides to secure crops and human beings from poisonous pesticides.
Although the Central Cotton Research Institute CCRI prepared a cotton picker machine and numbers of farmers are shifting to bio-pesticides under Integrated Pest Management (IPM) plan yet these practices are not very common and scores of poor laborers are still exposed to health hazards.
"Relentless use of chemical pesticide was damaging human health, land fertility, and environment as a whole," commented Malik Arif, a progressive farmer from Nawabpur. "The cotton picker can be used only on thick cotton crops and we have not yet this culture in our agriculture."
Therefore, he suggested promoting bio-pesticides on a larger scale and curtailing the use of chemical pesticides for the safety of people and land fertility.
"Promoting the use of bio-pesticides and encouraging farmers by the South Punjab Agriculture Secretariat to opt for it under the Integrated Pest Management model is an excellent attempt," he remarked.
He said the IPM model remained successful in health security, cutting the input cost and enhancing productivity when it was applied on nearly 500 plots. "But, it can be more beneficial if reaches all the farmers."
He said farmers can prepare bio-pesticides by soaking "neem tree, aak, kortuma, tobacco, and moringa in specific amount of water for 10 hours and then boiling it. After boiling, more water can be mixed in it and the spray is ready for crops.
Another progressive farmer Muhammad Afzal cultivates chemical pesticide-free crops at 70 acres and was getting handsome production.
"Bio-pesticides and bio-fertilizers had improved the fertility of my land. I get as much produce by using these products as others get by using chemical pesticides," he said. "Some farmers are in haste for results. But, if they are consistent, they can improve production gradually at low cost."
He also suggested an aggressive campaign to promote the use of bio-pesticides and to promote organic crops.
Dr Ameen, an official of the agriculture department mentioned efforts being made to promote the use of organic pesticides. "This is a cheaper mode to kill pests. But, as the farmers desire quick results and enhanced production, they hesitate to opt for it."
He regretted that excessive use of pesticides and fertilizers leaves residues in soil, water, and air and has adverse effects on human health and non-target organisms like pollinators, parasitoids, predators, and wild animals also.
In this scenario, it is direly needed to promote the culture of healthy farming to secure our coming generations and protect our environment. (APP)
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We are thrilled to announce, we are now working with Dancers Against Cancer (DAC) in order to raise money for anyone in the dance community who has been affected by cancer. DAC's mission is to create an alliance with the dance community that provides financial support and inspiration to dance educators, choreographers, dancers, and families who have been impacted by cancer. This organization hits home for us, as we've recently lost one of our former students/instructors to the disease, Monica Struck Kuechler. We hope that with our efforts we can help put an end to this vicious disease and make a difference in the lives of dancers fighting cancer.
Any and all donations are greatly appreciated and can be made out to "Dancers Against Cancer". We are a HOPE studio and can accept donations.