text:
An Optimal Treatment Assignment Strategy to Evaluate Demand Response Effect

Demand response is designed to motivate electricity customers to modify their loads at critical time periods. The accurate estimation of impact of demand response signals to customers' consumption is central to any successful program. In practice, learning these response is nontrivial because operators can only send a limited number of signals. In addition, customer behavior also depends on a large number of exogenous covariates. These two features lead to a high dimensional inference problem with limited number of observations. In this paper, we formulate this problem by using a multivariate linear model and adopt an experimental design approach to estimate the impact of demand response signals. We show that randomized assignment, which is widely used to estimate the average treatment effect, is not efficient in reducing the variance of the estimator when a large number of covariates is present. In contrast, we present a tractable algorithm that strategically assigns demand response signals to customers. This algorithm achieves the optimal reduction in estimation variance, independent of the number of covariates. The results are validated from simulations on synthetic data.

Authors

• 35 publications
• 15 publications
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• Measuring Average Treatment Effect from Heavy-tailed Data

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• Retailer response to wholesale stockouts

The purpose of this paper is to identify the immediate and future retail...
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• Errors-in-variables Modeling of Personalized Treatment-Response Trajectories

Estimating the effect of a treatment on a given outcome, conditioned on ...
06/10/2019 ∙ by Guangyi Zhang, et al. ∙ 0

• Principal Stratification for Advertising Experiments

Advertising experiments often suffer from noisy responses making precise...
11/19/2019 ∙ by Ron Berman, et al. ∙ 0

• Two paradoxical results in linear models: the variance inflation factor and the analysis of covariance

A result from a standard linear model course is that the variance of the...
03/09/2019 ∙ by Peng Ding, et al. ∙ 0

• High-dimensional semi-supervised learning: in search for optimal inference of the mean

We provide a high-dimensional semi-supervised inference framework focuse...
02/02/2019 ∙ by Yuqian Zhang, et al. ∙ 0

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I Introduction

As more uncertain and intermittent renewable resources are integrated into the power system, operators are increasingly exploring flexibility in customers’ consumptions to balance supply and demand. This operation is commonly known as demand response (DR). In a typical implementation of DR programs, customers receive a DR signal to elicit a change in their consumptions. This signal can be a modification of electricity prices or simply a message requesting a change in consumption [1]. An effective DR program improves the efficiency and sustainability of power systems and is a central pillar of the envisioned smartgrid [2, 3, 4, 5].

A natural question about demand response is quantifying the impact of a DR signal. That is, if a DR signal is sent to a subset of the users, what is the change in these users’ consumptions because of that signal? An accurate estimate of this change is central to the operation of demand response programs: if not enough change in demand is elicited, other measures need to be taken; if too much change is elicited, the program is inefficient.

Most of existing work in this area of demand response have approached the problem from a market optimization point of view. For example, authors in [5, 6] considered how to optimize the social welfare; and authors in [7] have considered how to create an efficient market for demand response. In all these settings, customers’ responses are captured by well-defined utility functions. These functions are assumed to be known to the operators, or at least to the users themselves.

In practice, these market based approaches can be difficult to implement because customers often do not have a clear model of their own utilities. For example, consider a household with a smart energy management system (e.g., a NEST thermostat). This household will respond to a DR signal, but the response can be a complicated function of the current conditions in the household–e.g., temperature, appliances that are on, number of people at home and so on– and the user may not be consciously aware of the households’ utility function. Therefore the operator needs to learn customers’ responses from past history. Furthermore, because of the advancement of household sensors, this response need to be learned under a possibly high dimensional setting.

By performing enough experiments with enough customers, that is, sending enough DR signals, the operator will eventually learn the users’ response with accuracy. Repeated experimentation with large group of customers, however, is impractical for two reasons. The first is that operators only sends out DR signals if the demand need to be modified appreciably, and this event does not occur all that often in the power system. The second is that because of use fatigue [8], most utilities have agreements with their customers that a single household will only receive a limited number of DR requests [9]. Under these limited data regimes, accurately learning the response of users is nontrivial.

In this paper, we adopt an experimental design approach to the problem estimating the response of users to DR signals. We design user selection algorithms where by carefully choosing the users that receive DR signals, the maximum information about the system response can be learned.

In this paper we consider the linear setting, where a user’s consumption is a linear function of a set of variables and the DR signal. We refer to the former abstractly as a user’s covariates, where they could represent measurements such as temperature, appliance status, building type, behavior patterns and etc. Because of the explosive growth in sensing devices, we are particularly interested in the high dimensional case, that is, where users have a large number of covariates. We assume the impact of DR signals is additive to the original consumption behavior. Using the language of experimental design, we regard DR signal as a treatment and a user receiving a DR signal as being assigned this treatment. Then learning the DR response of the users is equivalent to learning the average treatment effect, which is the average response of the customers to the treatment [10]. The metric with respect to the estimation of the treatment effect is the variance of the estimator as more experiments are performed. Randomized trial is usually thought as the “gold standard” in these types of models mainly due to the fact that randomly assigning treatments to users removes the effect of confounding factors and provides a consistent estimate of the treatment effect. In the presence of many covariates, however, random assignment can be extremely inefficient. In fact, as we show in this paper, in the high dimensional setting, random assignment does not reduce the variance of the estimate of the average treatment effect, even as the number of treatment grows without bound. Instead, following the outline in [11], we design a selection scheme users are picked based on their covariates to be treated.

Suppose there are total users in the system. Under the linear models considered in this paper, the best possible lower bound on the rate of variance reduction is ,111no estimator can reduce variance faster than given by considering the Fisher information [12]. As discussed above, under high dimensional settings, a randomized algorithm can only achieve , even when are large number of fraction of users are assigned DR signals. The main contributions of this paper are:

1. We show if the number of users selected is a constant fraction of the total number of users, there exists user assignments that achieve a variance reduction rate of . This rate is independent of the dimension of the covariates, as long as it is less than .

2. We develop a tractable user assignment algorithm. This algorithm is obtained by converting the variance reduction problem in a densest-cut problem on a graph [13, 14, 15, 16].

Our approach differs from previous effort in learning demand response in one important regard. In previous studies, the focus was on training the best predictive model and subtracting out the predicted consumption from the measured consumption [17, 18]. In our approach, we do not ever learn a predictive model, in the sense that we do not learn the relationship between the covariates and the consumption. Rather, focus on learning a single parameter: the response to the DR signal.

The results in [11] act as an impetus to this paper. The main difference is that in [11], the users are assigned a treatment of , therefore some information is always conveyed by this assignment. In our model, the users are assigned either (receives DR signal) or (no signal). Therefore, for the users assigned , we do not obtain any information about the impact of DR signals. This makes the problem much more technically challenging, and consequently we only consider the offline assignment problem whereas [11] also considers the online assignment problem. There are extensive literature on average treatment effect estimation, and the interested reader can refer to [19, 20] and references within.

The rest of the paper is organized as follows. Section II introduces the preliminaries and the problem throughout this paper. Section III presents the variance of the estimator obtained by random assignment. Section IV further presents the variance of the estimator by optimal assignment, followed by a tractable algorithm presented in Section V. Section VI details the simulation results obtained by either random assignment and optimal assignment. Section VII concludes the paper.

Ii Preliminaries and Problem Formulation

In this paper we assume that a user’s consumption is given by a linear model. Let denotes a binary DR signal, where represents that a signal is sent to user and

presents that no signal is sent. A covariate vector

is also associated with a user, representing available side information. For example, side information may include local temperature, user’s household size, and number of electrical vehicles and so on. We denote the dimension of the covariate vector by , and assume the last component is , which is the intercept. Let denote the consumption of user , which is given as

 yi=βxi+γTzi+ϵi, (1)

where

is white noise with variance

(for convenience). The coefficient is the impact of the DR signal and estimating it efficiently is the goal of the paper. The coefficient represents the effect of the covariate vectors. The main technical challenge is to accurately estimate the coefficient of interest , even when is high dimensional. For analytical simplicity, we assume that the entries of

are drawn as i.i.d. Gaussian random variables (possibly after centering and rescaling). In simulations (Section

VI), we show that the results holds for other types of distributions as well.

We assume there are total users. In this model, a single user that receives two demand response signals at two different times is equivalent to two users each receiving a demand response signal. Therefore, we suppress the time dimension and label all users by . Note that in (1), all users share a common response to DR signals.

We denote the estimate of by . The value of is a function of the DR assignments, that is, the value of the ’s. Under the linear setting in (1

), the ordinary least square (ols) estimator

of is unbiased for all possible allocations of DR assignments, is centered at the true value . The natural measure of performance is then the variance: . With some simple linear algebra, the variance of is given by [21]:

 Var^β=σ2xTPZ⊥x, (2)

where . The ’th row of the data matrix is given by . We adopt the notation that denotes a matrix that has rows and columns, while denotes the to column of a matrix , where .

We are primarily interested in the setting where an operator can assign a limited number of ’s to be . This setting reflects the limit in budget of an operator in sending DR signals. Specifically, let be the total number of DR signal that can be sent. The goal of the operator is to strategically assign ’s to be such that the variance of is minimized. In particular, we are interested in the rate of reduction of as increases and in settings where is a constant.

From (2), minimizing the variance of is equivalent to maximizing the quantity , and we focus on the latter quantity in the rest of the paper due to notational convenience. Two types of algorithms are of interest: i) the standard random assignment where each is chosen to be or

with probability

, and ii) an optimal assignment procedure where ’s are chosen to maximize . Both algorithms face the constrain that only out of ’s can be assigned to be . We characterize growth rate of quantity in terms of and , or equivalently, the decay rate of .

We show that when is relatively small compared to , the two strategies yield similar rates of . In a high dimensional setting where is comparable to , e.g., , however, the random assignment is essentially useless in estimating , in the sense that remains a constant in expectation as grows. Our proposed strategy, on the other hand, improves the rate to in this case, as long as is a constant. In Section III, we discuss the randomized strategy. The optimal assignment algorithm is then considered in Section IV.

Iii Random Assignment

Random assignment has been extensively studied in literature, mainly because it balances the covariates in two groups and eliminate the influence of confounders [19]. For our model in (1), random assignment means that a subset of ’s are chosen at random and assigned a value . Theorem 1 quantifies the rate of the increase of .

Theorem 1.

Random assignment achieves a rate of (). If is a constant, then this rate is ().

Before proving Theorem 1, we discuss the scaling rate under the setting when is a constant. In practice, this is the regime of interest since it is reasonable to suppose that a fraction (e.g. 10%) of users receives DR signals. In this case, the rate achieved by random assignment is . This rate is () when is relatively small compared to . However, when is large, e.g., , then this rate becomes (). This rate is not desirable as it indicates that the variance of the estimator is not decaying with even when is large. Thus we would like to design an assignment strategy which yields an estimator that still possesses a relatively good performance even when is very close to . In the next section, we show that with optimal assignment, we achieve the optimal rate () when . The proof of Theorem 1 follows.

Proof.

We consider a random assignment where . Then the rate becomes:

 Etr{xT(I−Z(ZTZ)−1ZT)x} (3) = k−Etr{xTZ(ZTZ)−1ZTx} (a)= k−tr{Z(ZTZ)−1Z%TExxT} (b)= k−tr{Z(ZTZ)−1Z%TE(~x+kn)(~x+kn)T} (c)= k−tr{Z(ZTZ)−1Z%T(k(n−k)n2I+0+k2n211T)} (d)= k−pk(n−k)n2−k2n2n = (n−p)k(n−k)/n2,

where () follows from linearity and cyclic permutation of the trace operator; () follows from defining ; () follows from multiplying out each terms inside and using the fact that each element in has a zero mean and a variance as ; () follows from being a projection matrix onto

. Using the fact that the eigenvalues of a projection matrix are either 0 or 1 and

has rank with probability one, then the trace of is with probability one. In addition, from Lemma 1, it is shown that if contains one column as intercept, so that the trace of , which completes the equality in . ∎

Lemma 1.

If is a by matrix (where ) with one column which contains all ones, then .

Proof.

Note that is the projection matrix which is orthogonal to , we then have the following:

 ZT(I−Z(ZTZ)−1ZT)=0, (4)

where is a zero vector that has length .

Note that has one column as the intercept, which suggests that has one row where each element takes value one. Since the equality in (4) holds for every row, we then have:

 1T(I−Z(ZTZ)−1ZT)=0, (5)

which indicates that .

Iv Optimal Assignment

Instead of being randomly assigned into DR programs, users can be optimally allocated to either the treatment group or the control group depending on their covariate information, in order to obtain the best estimator of . Mathematically speaking, we optimally assign each to be 0 or 1, in order to minimize the variance of the estimator . This optimization problem is:

 maximize^x xTPZ⊥x (6) subject to n∑i=1xi=k xi∈{1,0}.

We first discuss the upper bound on the quantity (which signifies the lower bound for ). We show that it is O(). Then we establish that under the regime of , there exist algorithms that achieve a rate that meets the upper bound of .

Iv-a Optimal Rate

Before proceeding on analyzing the rate obtained by the proposed strategy, we first discuss the upper bound on the rate of .

Proposition 1.

No assignment can achieve a better rate that O [11].

Proof.

The basic idea is to derive the Fisher information with the linear regression model in (

1). The inverse of the Fisher information provides a lower bound for the variance of the estimator obtained by least squares and thus an upper bound for the quantity . For more details, please refer to Proposition 1 in [11]. ∎

In the next subsection we will show that when which is a constant, we achieve this upper bound.

Iv-B Achievability of Optimal Rate

We first present the main result of this section. We assume that each element of

(excluding the intercept column) is drawn independently from a standard Gaussian distribution. This assumption will facilitate the calculation of the main result shown in Theorem

2. The algorithm associated with Theorem 2 is presented in Algorithm 1.

Theorem 2.

Recall that the rate is the growing rate of the inverse of the variance introduced in (2). This rate from optimal assignment is of (), which is independent of the dimension of covariates. More specifically, when is a constant, then this rate is linear rate, i.e., ().

Before proving Theorem 2, we first show in Lemma 2 that the worst case scenario for the rate is when . This scenario provides a minimum on the quantity for every where , which provides a maximum for for every . Thus if we can show in Theorem 2 that in the worst case scenario where , the growing rate of quantity is when is a constant, then this rate holds for all where .

Lemma 2.

is increasing in . Consequently, if = , the estimator yields the worse case performance [11]:

 inf1≤p

Proof.

This is a general result about linear estimation and the interested reader can refer to Lemma 5 in [11]. ∎

When , the rank of is one with probability one, thus we write , where is in the null space of [11], i.e., . Based on this observation, is written into a simpler form as . The problem is then to maximize under the constraint that we only get to assign ’s to be 1’s and the rest to be 0’s. The optimization problem is:

 maximize^x (yTx)2||y||2 (8) subject to n∑i=1xi=k xi∈{1,0}.

To solve the optimization problem in (8), we need to find ’s in such that their sum is maximized, where is the element of the vector . We observe that it actually suffices to provide a lower bound on this maximum sum to prove the rate.

To provide this lower bound we need to know the structure of . We then show in Lemma 3 that if is in the null space of , then each can be constructed to be drawn from an i.i.d. standard Gaussian distribution. Based on this observation, the problem is further reduced to find the lower bound on the largest , assuming that is smaller than to ensure that with overwhelming probability the largest is non negative. Let us refer to this statistic as the order statistic of and denote it by such that . We present a lower bound for in Lemma 4 when is smaller than . This lower bound facilitates the final proof for Theorem 2. The proof of Lemma 3 and Lemma 4 is left in the appendix.

Lemma 3.

The basis of the null space of can be constructed as an i.i.d. standard Gaussian vector with length , when each element of is independently drawn from a standard Gaussian distribution and the last column of is an all one column.

Lemma 4.

Let satisfies . If each is independent and follows standard Gaussian distribution, then , where C is a positive constant and .

Now we can use the introduced lemmas to prove Theorem 2. A summary is presented in Algorithm 1, illustrating the procedures to obtaining the rate stated in Theorem 2 using the proposed lemmas. This algorithm also provides the optimal assignment strategy when .

Proof of Theorem 2.

We will focus on the case when since it provides the worst case rate for every , as stated in Lemma 2.

From lemma 3, we know that , we then obtain the following results:

 maxx,xi∈{1,0},∑xi=kE(yTx)2||y||22 (9) ≥E{(y(n)+...+y(n−k+1))2}||y||22 (a)=E{(y(n)+...+y(n−k+1))2}E||y||22+O(1n) ≥E{(ky(n−k+1))2}n+O(1n) =k2E{y2(n−k+1)}n+O(1n) (b)≥k2(Ey(n−k+1))2n+O(1n) (c)≥k2C2lognkn+O(1n),

where () is based on the multivariate delta method [22], () comes from Jensen’s inequality and () is based on Lemma 3 and Lemma 4.

Specifically, if , then (9) can be written as:

 k2C2lognkn=C2ρ2log(ρ−1)n=Θ(n). (10)

Another interesting case is when , (9) can be written as:

 k2C2lognkn=C2(logn−loglogn)=Θ(logn). (11)

As can be seen from Theorem 2, we will obtain the optimal rate by replacing as a constant. This indicates that with optimal assignment, the estimation variance will indeed decay with instead of being a constant as shown in Theorem 1, when the dimension of the covariates is comparable to . This is very interesting as it indicates that even in a high dimensional setting, the variance of the estimator will decay optimally by solving the variance minimization problem.

V A Tractable Alternative

Algorithm 1 provides a simple way to find the optimal assignment when . When is less than , this algorithm cannot be applied. In fact, the optimization problem in (6

) is a nonconvex quadratic optimization problem which can be NP-hard. In this section, we present a tractable approximate algorithm by relaxing the original combinatorial optimization problem into and semidefinite program. We then demonstrate that this SDP problem approximates the original problem with a performance ratio that is better than

when is in the range of . This procedure follows the results established in [15].

We first revisit the original variance minimization problem in (6) and transform into . Denote each element in as , then each takes value in . Therefore the variance minimization problem is written as:

 maximize^x 14(^x+1)TPZ⊥(^x+1) (12) subject to n∑i=1^xi=2k−n ^xi∈{1,−1}.

This is a Dense--Subgraph (DSP) problem with existence of self edges. To illustrate this, let element at row and column in matrix be denoted as edge weight (except that is half of the value on the diagonals) associated with vertex and vertex , then (12) is trying to find a set of vertices such that the sum of edge weights induced by these vertices are maximized. Since the problem presented in (12

) contains binary variables, we relax this problem into a SDP formulation:

 maximizeX,^x 14∑i∑jwij(1+^xi+^xj+Xij) (13) subject to ∑i^xi=2k−n Xii=1 ∑i∑jXi,j=(2k−n)2 [1^xT^xX]⪰0.

The original problem in (12) is hard, we will only obtain a surrogate solution in polynomial time. We thus adopt Algorithm 2 to obtain an approximate solution from SDP formulation. Let us denote this solution by based on Algorithm 2. The performance of the approximation from SDP is evaluated by the performance ratio which satisfies:

 E14(^x∗+1)TPZ⊥(^x∗+1)≥rw∗. (14)

Here the randomness in is introduced by the random rounding procedure shown in Algorithm 2 and is the optimal value of the objective function shown in (12). Performance ratio can be used to quantify how close the solution from Algorithm 2 is to the optimal solution by solving the original hard problem.

There exists a fruitful line of work on the approximation algorithms using either greedy algorithm or LP/SDP relaxation for DSP problems [23, 13, 24, 14, 25, 26, 16, 15, 27]. Most recent research has improved the performance ratio to with LP relaxation in [27]. However, if is not decaying with , i.e., a constant, then this ratio is not desirable since it is decreasing in . The authors in [15] propose an improved performance ratio that is better than for a wide range of . We will adopt the approximation procedure in [15] and argue that the performance ratio is valid in our case here as well. In the following, we will first present the general algorithm and then show that the performance ratio in [15] is still applicable in our case.

The approximation procedure in [15] is presented in Algorithm 2, including three main procedures:

• Solve SDP problem in (13) (step 1). After this procedure we can obtain the optimal continuous solution for (13) and the optimal value of the objective function is denote by .

• Construct initial , where represents the initial subgraph and is a set of indices (step 2 through step 4). The ’s take value 1 such that and the rest -1. Let us denote them by . The value of the objective function in (13) is written as . Here is the total weights of edges in the subgraph induced by . At this point the cardinality of is not necessarily .

• Resize to such that contains exactly vertices (step 5 through step 16). The final assignment of ’s is that . Let us denote them by . The value of the objective function is and is the total weights of the edges induced by .

We now demonstrate that Algorithm 2 achieves the performance ratio shown in Proposition 2.

Proposition 2.

The performance ratio from Algorithm 2 defined as:

 Ew(~S)≥rw∗ (15)

satisfies the conditions presented in Proposition 2 in [15] and is plotted in Fig. 1, where is the optimal value of the objective function in problem (12). When is large, this ratio is better than either O() or O() obtained by LP relaxation.

As can be seen from Proposition 2, the performance ratio quantifies the gap between the approximated solution obtained by Algorithm 2 and the optimal solution from the original problem shown in (12). This ratio is the direct result from the random rounding procedure and the resizing procedure shown in Algorithm 2. The ratios associated with these two procedures are presented in (16) and (17):

 Ew(S)≥αw∗, (16)

and

 w(~S)≥ξw(S). (17)

In [15], the authors well define the parameter (which depends on and ) and (depends on and obtained from the random rounding procedure) so that the performance ratio satisfies Proposition 2 where there are no self edges, i.e., . However, in our case, the are the diagonal elements of and they are not necessarily zero, so the graph in our case contains non trivial self edges. If we show that the presence of self edges does not change the values of and , then Proposition 2 naturally holds in our case as well.

Let us first discuss (16). From [15], parameter does not depend on whether there are self edges in the graph, so (16) directly applies.

Next we need to check if the same applies at the presence of self edges. When there are no self edges, i.e., , the authors in [15] show that when and otherwise. We show that the same condition for holds even with the presence of non negative self edges, as stated in Lemma 5.

Lemma 5.

Let and be obtained from random rounding procedure and resizing procedure in Algorithm 2 from a graph with non negative self edges, i.e., . Then we have:

 w(~S)={k(k−1)|S|(|S|−1)w(S),if |S|>k1,otherwise

Proof of Lemma 5 is given in the appendix. Lemma 5 validates the and with the presence of non negative self edges, thus the performance ratio stated in Proposition 2 is valid. A list of performance ratio ’s for different values of is shown in [15] and is plotted in Fig. 1. This rate is satisfactory since it has a rate of but strictly larger than [15]. It is better than O() when is not decaying faster than . In fact, as long as lies within some constant range, then can be seen as a varying constant thus is not decaying as a function of .

Vi Simulation

In this section we show the comparison results between random assignment and optimal assignment. We simulate the covariates from two different distributions, i.e., Gaussian distribution and uniform distribution. We also validate the claim by simulated building data from

[28].

Vi-a Gaussian Ensemble

We first generate the covariates as they are drawn from i.i.d. Gaussian ensemble, i.e., . We compare two cases where and in Fig. 2. Note that Fig. 2 is shown in semilogarithmic plot where the -axis has a logarithmic scale and the -axis has a linear scale. In addition, we adopt the value of in [15], i.e., 0.9 for and 0.94 for . We let to obtain the worst case performance.

Besides SDP relaxation, we also simulate a greedy based assignment to maximize the weighted edges induced by vertices. The greedy assignment sequentially eliminates vertices and works as follows: we start with the original graph and a set containing all vertex. At each elimination step, the vertex with the least weighted edges are eliminated from the set until this set contains exactly vertices. This greedy algorithm is introduced in [24].

In addition, we use the result from branch and bound (upper bound) to serve as the reference in order to compare the random assignment and the proposed optimal assignment. The duality gap for branch and bound is set to be 0.05 for all when . Due to computational complexity and time constraints, we set this gap to be around 0.25 when is big in the case of .

In Fig. 2, we see that the semilog plot on () is growing with and similar to . This suggests that is linear in , as we stated in Section IV. In addition, It is very close to the solution obtained by branch and bound, meaning that the result from SDP relaxation is close to the optimal solution in (12). The empirical performance ratio from SDP relaxation is shown in Table I.

From Table I, we see that the performance ratio for between and is actually greater than , which is even better than the theoretical bound in Proposition 2. In addition, if we decrease with respect to , i.e., change from to , then the performance ratio is reduced. This is due to the fact we need to do more eliminations during the resizing procedure in Algorithm 2 and the deterioration increases. However, if is well defined within a range, then this deterioration is controlled and does not change the statement in Proposition 2.

On the other hand, the semilog plot on from random assignment is a constant on average across different values of , whether is small or large. This validates Theorem 1 as it states that the rate is () when . In this case, we cannot obtain an efficient estimator since the variance is not decaying even when is big.

What is more, the greedy assignment does not provide a relatively good performance as well. It yields a constant as random assignment. This suggests although both the greedy algorithm and SDP relaxation are aimed to solve an optimization problem, the solution from SDP relaxation is much better and reliable.

Vi-B Uniform Ensemble

Although we discuss the rate of quantity with respect to Gaussian ensemble, we also simulate the covariates where the elements are drawn from a uniform distribution in an interval [-1,1]. The results are shown in Fig. 3. Fig. 3 is again a semilogarithmic graph. We again take to be 0.9 when and 0.94 when . The duality gap is 0.05 when . When , the duality gap is around 0.1 to 0.2 for greater than 100 and is 0.25 for . The comparison result is shown in Fig. 3 and the performance ratio by SDP relaxation is shown in Table II.

The observation from Fig. 3 and Table II is similar to the analysis in the case of Gaussian ensemble, that the solution obtained from SDP relaxation is still within a constant of the branch and bound solution for . Again, greedy algorithm fails to find a solution close to that by branch and bound and performs as poorly as the random assignment. The performance ratio is again decreased when we decrease , which indicates the similar deterioration occurred during the resizing procedure.

Vi-C Building data

We also validate the claim in the paper by simulated building data obtained from [28]. Using this software, we can generate building covariates such as environment temperature, number of occupants, appliances scheduling, etc. The software outputs the energy consumption based on the covariates that we generate. The buildings vary from a small postal office, to a large commercial hotel, with different number of covariates involved in the modeling.

For the purpose of this paper, we include significant covariates into the linear regression model and the number of those covariates are comparable to the number of users in the simulation. The values of the covariates are whitened by the covariance matrix so that the covariates have zero mean and unit variance [29]. The details of this procedure is given in the appendix. In addition, we fix in the simulation.

The simulation results are shown in Fig. 4 and Fig. 5.

In Fig. 4, we observe that is decaying very fast with the optimal assignment strategy, whereas the variance stays unchanged if adopting a random assignment strategy.

The variance is a varying quantity in terms of the dimension of the covariates, i.e., . Assume that the number of users are fixed, i.e., is fixed, and that is a constant. According to Theorem 1 and in Theorem 2, with random assignment the variance decays with whereas with optimal assignment the worse case scenario with yields a decay rate of .

In Fig. 5, we show how varies with an increasing and a fixed . As can be seen from Fig. 5, the increasing deteriorates the variance much severely from random assignment than that from the proposed optimal assignment. This again validates the efficiency of the proposed strategy in improving estimation accuracy, especially in a high dimensional setting where is comparable to .

Vii Conclusion

In this paper, we estimate the average treatment effect of demand response (DR) signals. We adopt an additive linear regression model and discuss two different strategies to assign DR signals to users under limited assignment budgets. The first strategy randomly picks users and sends DR signals to them. The second strategy optimally assigns DR signals to users by minimizing the variance to estimate treatment effect. We show that in a high dimensional setting, the second strategy achieves order optimal rates in variance reduction, whereas random assignment does not reduce variance even as the number of users grows. We formulate the general assignment as a combinatorial optimization problem and present a tractable SDP relaxation. We show that this relaxation obtains a solution that is within bounded gap of the original optimal solution. The simulation results validate this proposition with the synthetic data on both i.i.d. Gaussian covariates and uniform covariates. This work provides a framework for further research in applying causal inference in analyzing consumption data and DR interventions.

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Appendix

Vii-a Proof of Lemma 3

Proof.

This proof follows the intuition in [30]. In [30]

, to prove the null space of a Gaussian random matrix, the authors use the fact that a standard multivariate Gaussian distribution is invariant to any orthogonal transform. The difference here is that the matrix

contains an extra column of 1’s as intercept.

Denote the null space of by . It is one dimensional and satisfies:

 [ZT1:n−21T]y=0. (18)

Now we multiply

by an orthogonal matrix

on the left hand side and multiply by on the right hand side. For simplicity let us write as . Since then the following holds:

 [ZT1:n−21T]UUTy=[ZT1:n−2U1TU]UTy=0. (19)

Because has i.i.d. standard Gaussian entries and is an orthogonal matrix, then each row of must follow a Gaussian distribution , which is still a standard Gaussian distribution. Thus has i.i.d. standard Gaussian entries. Denote , then and are drawn from the same distribution and each of their entries follows i.i.d. standard Gaussian distribution.

From (19), we also require that the orthogonal matrix satisfies . If such orthogonal matrix exists (which is easy to find), then we can rewrite (19) as:

 [~ZT1T]UTy=0. (20)

Comparing (20) and (18), we see that and

must be identically distributed. One distribution that satisfies this property is the standard normal distribution

. It is easy to show that identity matrix is the only covariance matrix that satisfies such condition whereas zero mean is based on the fact that

from (18). This observation concludes the final proof. ∎

Vii-B Proof of Lemma 4

Proof.

The proof is based on the fact that if in expectation, at least ’s is greater than a constant, then the th largest must be greater than this constant [31]. Mathematically speaking, we want , where is a constant. We write in the following form:

 E{|{i:yi≥C√lognk}|} =nPr{yi≥C√lognk} (21) =nQ(C√lognk).

function of Gaussian distribution does not have a close form expression, so we are interested in a tight lower bound for it in order to obtain a lower bound for (21). Many lower bounds are obtained in literature [32, 33, 34, 35]. In particular, [34] provides a lower bound that is valid when the argument is small, and [32] provides a lower bound that is the tightest when the argument becomes relatively large. The lower bounds (bound 1 through bound 4, from [32, 33, 34, 35] respectively) are presented in (22a) through (22d).

 Q(x) ≥112e−x2+1√2π(x+1)e−x22, (22a) Q(x) ≥e1π(κ−1)+22κ√1π(κ−1)(π(κ−1)+2)e−κx22,κ≥1, (22b) Q(x) ≥1−√1−e−2x2π2, (22c) Q(x) ≥√e(β−1)2πβ2e−βx22,β≥1. (22d)

Note that bound 3 (in (22c)) is only valid when is small. A comparison of these four bounds are shown in Fig.6. It is shown in semilogarithmic plot where -axis is in a logarithmic scale and -axis is in a linear scale.

Assume that is small, i.e., when which is slightly smaller than but greater than . In this range the lower bound provided in (22c) is the tightest. Use this lower bound we obtain the constant that is universally applicable for that (21) holds when is small. After some simple calculation, we obtain the constant .

Although (22a) provides the tightest bound when gets bigger, this lower bound includes two exponential terms which complicate the calculation. Actually when , we can obtain a fairly good but conservative constant for the lower bound provided in (22d) with only one exponential term. The constant in this case when is relatively large is calculated as .

Vii-C Proof of Lemma 5

Before proving Lemma 5, let us first introduce Lemma 6 that bounds the values of the diagonals of .

Lemma 6.

The diagonals of is non negative.

Proof of Lemma 6.

Recall that , and is a projection matrix, then we have the following:

 (Z(ZTZ)−1ZT)2 =Z(ZTZ)−1ZT. (23)

Denote the diagonals of the by , then we have , which implies that .

This suggests that the diagonals of , denoted by , is non-negative. ∎

Now we proceed to prove Lemma 5.

Proof of Lemma 5.

To prove that Lemma 5 holds, we add up a constant to each to make sure that there are no negative weights in the graph. This does not change the optimization problem since it only adds a constant to the objective function and the solution remains the same.

If , then we arbitrarily add vertices to until it contains exactly vertices. Since the weight on each edge is non negative and we keep adding more edges into the subgraph induced by , is at least 1 in this case.

Now suppose that


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# Evaluate an analysis in a virtual world (both virtual plant and imager)Link

Quantitative evaluation of a 3D reconstruction and/or automated measure from a phenotyping experiment is critical, from both developer and end-user perspectives. However, obtaining ground truth reference is often tedious (e.g. manual measurements, it must be anticipated (synchronous measures with image acquisition), and some type of data are just inacessible with available technologies (e.g. having a reference point cloud).

Virtual plants makes data acquisition inexpensive and allows to parametrize the type of data. By design, ground truth data can be easily extracted from these virtual datasets. We thus designed the Virtual Plant Imager to take images of any 3D object, as a digital twin of our real plant imager.

After reading this tutorial, you should be able to generate a single virtual plant dataset (including several ground truth reference) in order to evaluate the phenotyping results generated through an analysis pipeline made with our plant-3d-vision tool suite.

We highly recommend the use of docker containers to run ROMI software, if you wish to use the docker images we provide, have a look here.

If it is not already done, you must be able to build and run the docker images of:

• the (Virtual) Plant Imager by following these instructions. This is required to generate the virtual data (initial plant 3D model, ground truth and RGB images).
• the plant-3d-vision by following these instructions. This is required to reconstruct a 3D model from the virtual 2D images, as if they were images of real plants. This docker will also allow you to evaluate this reconstruction using the available virtual ground truth data.

Principle: You want to evaluate the results generated by an analysis pipeline made with our plant-3d-vision tool suite. Let's say that this pipeline is defined by a typical configuration file, test_pipe.toml.

The idea is to generate images of a virtual 3D plant and provide these picture as input to the tested analysis pipeline. Technically, the Virtual Plant Imager relies on Blender v2.81a to generate at set of (2D) RGB images from the plant 3d model, mimicking what a real camera would do on a real plant. Any virtual camera pose can be generated (ie. distance, angle), but virtual poses similar to the real robot (plant imager) are preferred. An HTTP server acts as an interface to drive Blender generation scripts.

The virtual plant 3D model (with some of its ground truth references) can be imported and given as an input. However, we provide an integrated procedure to generate a virtual 3D plant directly "on the fly" with Lpy, using a Lpy model and customizable parameters. Some ground truth references will also be automatically generated.

Once this virtual plant has been virtually imaged, there are all data and metadata required to run an analysis with the tested pipeline. The results of this analysis will be compared to the virtual ground truth. Four type of evaluations are currently implemented :

• evaluation of a 2D segmentation
• evaluation of a 3D segmentation of the point cloud
• comparison of point cloud similarity
• evaluation of phyllotaxis measures (angles and internodes)

### 1. Prepare data into a proper databaseLink

First, create a working directory on your host machine, let's say home/host/path/my_virtual_db. You can find an example of such a directory here.

This working directory is a proper "romi" database which contains additional data for the virtual plant generation and/or imaging grouped in a so-called vscan_data folder:

Legend:

• (*) the name is fixed and cannot be changed
• (!) the folder/file must exist (no tag means that the folder is not required for the program to run)
my_virtual_db/
│   romidb (!*) # a (empty) marker file for recognition by the plantdb module
└───vscan_data/ (!*)
│   └───hdri/ (*)
│       │   hdri_file1.hdr
│       │   hdri_file2.hdr
│       │   etc...
│   └───lpy/ (*)
│       │   my_plant_species_model.lpy
│   └───obj/ (*)
│       │   VirtualPlant.obj
│       │   VirtualPlant_mtl
│       │   hdri.json
│       │   lpy.json
│       │   obj.json
│       │   palette.json
│       │   scenes.json
│   └───palette/ (*)
│       │   my_plant_species_model.png
│   └───scenes/ (*)
│   files.json

Recommended if you are not familiar with the virtual plant imager. You can directly obtain a functional working directory from the repository of the plant-imager you cloned in your host machine

So if your working directory is named my_virtual_db, execute in a terminal:

cd plant-imager # enter the cloned repository in your host machine
cp -r database_example home/host/path/my_virtual_db

To skip details and directly run the virtual plant imager, go now to section [2.](#2-generate-virtual-images-from-a-lpy-virtual-plant-in-a-virtual-scene)

#### 1.2 Customize data of the virtual plant and/or of the virtual imagesLink

Warning

For advanced users. If you modify data, you most likely need to modify the configuration .toml file downstream.

You can modify and enrich the virtual dataset in several manner (modifying the LPy model and parameters, importing your own model and avoiding Lpy-generation, change background scenes, etc...). For all these options, please refer to the specifications of the virtual plant imager.

### 2. Generate virtual images from a (Lpy) virtual plant in a virtual sceneLink

Start the docker container of the plant-imager with your database mounted:

cd plant-imager/docker
./run.sh -db /home/host/path/my_virtual_db  # This will map your working databse to the db directory located in the docker's user home

Then, in this docker container, generate the virtual dataset by running the following command:

(lpyEnv) user@5c9e389f223d  romi_run_task --config plant-imager/config/vscan_lpy_blender.toml VirtualScan db/my_virtual_plant # Run VirtualScan by specifying the output folder 'my_virtual_plant'

The computation can take a few minutes, depending on your system capacities. if it works, the terminal should display something like that:

===== Luigi Execution Summary =====

* 2 complete ones were encountered:
- 1 LpyFileset(scan_id=vscan_data)
- 1 PaletteFileset(scan_id=vscan_data)
* 2 ran successfully:
- 1 VirtualPlant(...)
- 1 VirtualScan(...)

This progress looks :) because there were no failed tasks or missing dependencies

Results: in your database, a new folder (here called my_virtual_plant) should have been created, that contain data and metadata related to the virtual image acquisition of this virtual plant !

my_virtual_db
│   romidb
└───vscan_data/
└───my_virtual_plant/
│   ├── images/
│   │   └── images/
│   │   └── images.json
│   └── files.json
│   └── scan.toml

With the default parameters provided with this example (Lpy model and configuration file), there is only one generated plant, which has the following main characteristics

• It is a model of an Arabidopsis thaliana plant
• It has only a main stem and no lateral branches (simplified architecture)
• It is a mature plant, that has grown an elongated inflorescence stem bearing several mature fruit (called a 'silique', the typical pod of the Brassicaceae family) and still has some flowers at the very tip.

In the next two sections, we point to simple paramaters of the configuration file used for this task to modify either the virtual plant or the virtual imaging.

#### 2.1 (optional) how to modify the virtual plant with LPy parametersLink

Note

Detailed description can be found here

Below are the main lpy parameters that can be customized to change how the virtual plant looks like (age, size, branching, etc...).

[VirtualPlant.lpy_globals]
BRANCHON = false
MEAN_NB_DAYS = 70
STDEV_NB_DAYS = 5
BETA = 51
INTERNODE_LENGTH = 1.3
STEM_DIAMETER = 0.09

#### 2.2 (optional) how to modify the virtual imaging performed by the virtual imagerLink

Note

Detailed description can be found here

Below are the main lpy parameters that can be customized to change how the virtual images are taken (path, background scenes, resolution, etc...)

virtual camera path [ScanPath] class_name = "Circle" [ScanPath.kwargs] center_x = -2 center_y = 3 z = 65 tilt = 8 radius = 75 n_points = 18

### 3. Running a reconstruction pipeline on the virtual datasetLink

Once you have a virtual dataset of images that all look like a real one, you can analyze it like a real one with romi pipelines from our plant-3d-vision tool suite !

Remember that the pipeline you want to evaluate is defined by the following configuration file: test_pipe.toml.

To adapt to the virtual imaging and focus the evaluation to the downstream image analysis and 3D reconstruction, you can adapt the configuration file to include ground truth from virtual imaging to use ground truth poses. Create a new configuration file for the evaluation and modify it as follows:

cp test_pipe.toml test_pipe_veval.toml #copy and rename the configuration file of the pipeline you want to test

In the newly created test_pipe_veval.toml, deactivate use of colmap poses for the volume carving algorithm ([Voxel] Task of the pipeline).
[Voxels]
use_colmap_poses = false

Then the analysis pipeline can be run as usual except that colmap will not be run :

romi_run_task \ #romi pipeline scheduler
--config path/to/test_pipe_veval.toml \ # configuration of the pipeline to evaluate
AnglesAndInternodes \ #Last task to execute
/path/to/my_virtual_plant #folder inside the database on which the analysis will be run

This run should process all dependencies and generates notably a segmented point cloud and measures of the phyllotaxis (angles and internodes) !

Note

any available romi Tasks for image analysis can be runned here. Please refer to the list of Tasks implemented in our romi software suite.

Note

The command line can be executed in docker container or in a terminal if you have activated the correct virtual environments and proceeded to local installation of the software.

Please refer to this tutorial if you encounter problems to run pipeline from our plant-3d-vision tool suite.

After execution, the terminal should display luigi execution summary, as in this example:

===== Luigi Execution Summary =====

* 2 complete ones were encountered:
- 1 ImagesFilesetExists(scan_id=, fileset_id=images)
- 1 ModelFileset(scan_id=models)
* 6 ran successfully:
- 1 AnglesAndInternodes(...)
- 1 OrganSegmentation(scan_id=, upstream_task=SegmentedPointCloud, eps=2.0, min_points=5)
- 1 PointCloud(...)
- 1 Segmentation2D(...)
- 1 SegmentedPointCloud(scan_id=, upstream_task=PointCloud, upstream_segmentation=Segmentation2D, use_colmap_poses=False)
...

This progress looks :) because there were no failed tasks or missing dependencies

===== Luigi Execution Summary =====

Results: new subfolders and metadata resulting from the analysis should have been created in the folder of the analyzed plant dataset (called my_virtual_plant in this example).

The particular data generated depends on the pipeline you called. We provide an example here with a pipeline involving machine-learning based segmentation of 2D images and proceeding up to phyllotaxis measures (angles & internodes.)

legend: * (.) indicates data that were already present before the run of the pipeline (but the data content may have been modified) * folder names generated by the analysis generally start with the corresponding Task name end with a hashcode to keep track of task execution by the Luigi scheduler (e.g. _1_0_2_0_0_1_5f7aad388e). Such code is replaced by '_hashcode' suffix in the example below

my_virtual_db
│   romidb (.)
└───vscan_data/ (.)
└───my_virtual_plant/ (.)
│   ├── images/ (.)
│   ├── AnglesAndInternodes_hashcode/
│   ├── OrganSegmentation_hashcode/
│   ├── PointCloud_hashcode/
│   ├── Segmentation2D_hashcode/
│   ├── PointCloudGroundTruth_100000__VirtualPlantObj_hashcode/
│   ├── SegmentedPointCloud__Segmentation2D_PointCloud_3a1e8e0010/
│   ├── VirtualPlant/
│   ├── VirtualPlant_arabidopsis_note___BRANCHON___fal___angles____inte_hashcode/
│   ├── Voxels_False___background_____False_hashcode/
│   └── files.json (.)
│   └── scan.toml (.)
│   └── pipeline.toml

### 4. Evaluate the quality of the construction by comparing to the virtual ground truth dataLink

(work in progress)

Once the analysis results are generated, you can now compare this results to the expected ground truth reference of the virtual plant.

Several Evaluation Tasks have been developed by romi: check the list to know which results are evaluating each of them.

In the below example, we would like to evaluate the point-cloud reconstruction, so we run:

romi_run_task \ #romi pipeline scheduler
--config path/to/test_pipe_veval.toml \ # configuration of the pipeline to evaluate
PointCloudEvaluation \ #evaluation task of to run
/path/to/my_virtual_plant #analyzed data folder of the database that you want to evaluate

Note

Please refer to this [tutorial] if you encounter problems to run pipeline from our plant-3d-vision` tool suite.

### 5. View and scrutinize in 3D all data generated (images, reconstruction and evaluation)Link

(work in progress) Use of the plant-3d-explorer

format: plain


text:
Fano varieties with $\mathrm{Nef}(X)=\mathrm{Psef}(X)$ and $\rho(X)=\mathrm{dim}\, X-1$. (arXiv:1508.00182v2 [math.AG] UPDATED)

We classify mildly singular Fano varieties $X$ such that $\mathrm{Nef}(X)=\mathrm{Psef}(X)$ and that the Picard number of $X$ is equal to the dimension of $X$ minus $1$. 查看全文>>


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Home > Standard Error > Standard Error Vs Standard Deviation Formula

# Standard Error Vs Standard Deviation Formula

## Contents

This estimate may be compared with the formula for the true standard deviation of the sample mean: SD x ¯   = σ n {\displaystyle {\text{SD}}_{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}} For the purpose of hypothesis testing or estimating confidence intervals, the standard error is primarily of use when the sampling distribution is normally distributed, or approximately normally distributed. If one survey has a standard error of $10,000 and the other has a standard error of$5,000, then the relative standard errors are 20% and 10% respectively. The standard error falls as the sample size increases, as the extent of chance variation is reduced—this idea underlies the sample size calculation for a controlled trial, for example. have a peek at this web-site

We usually collect data in order to generalise from them and so use the sample mean as an estimate of the mean for the whole population. The standard error is also used to calculate P values in many circumstances.The principle of a sampling distribution applies to other quantities that we may estimate from a sample, such as All such quantities have uncertainty due to sampling variation, and for all such estimates a standard error can be calculated to indicate the degree of uncertainty.In many publications a ± sign The standard deviation of the age was 3.56 years. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1255808/

## Standard Error Vs Standard Deviation Formula

The standard deviation of all possible sample means of size 16 is the standard error. Review of the use of statistics in Infection and Immunity. The SEM can be added and subtracted to a students score to estimate what the students true score would be.

The SEM can be looked at in the same way as Standard Deviations. Similarly, the sample standard deviation will very rarely be equal to the population standard deviation. It remains that standard deviation can still be used as a measure of dispersion even for non-normally distributed data. Standard Error Of Measurement Example All journals should follow this practice.NotesCompeting interests: None declared.References1.

For the purpose of this example, the 9,732 runners who completed the 2012 run are the entire population of interest. Standard Error And Standard Deviation Difference If we want to indicate the uncertainty around the estimate of the mean measurement, we quote the standard error of the mean. All Rights Reserved. BMJ 1995;310: 298. [PMC free article] [PubMed]3.

It is useful to compare the standard error of the mean for the age of the runners versus the age at first marriage, as in the graph. Standard Error Of Measurement Calculator JSTOR2340569. (Equation 1) ^ James R. Secondly, the standard error of the mean can refer to an estimate of that standard deviation, computed from the sample of data being analyzed at the time. Bootstrapping is an option to derive confidence intervals in cases when you are doubting the normality of your data. Related To leave a comment for the author, please

## Standard Error And Standard Deviation Difference

The sample standard deviation, s, is a random quantity -- it varies from sample to sample -- but it stays the same on average when the sample size increases. https://en.wikipedia.org/wiki/Standard_error Roman letters indicate that these are sample values. Standard Error Vs Standard Deviation Formula This would be the amount of consistency in the test and therefore .12 amount of inconsistency or error. Standard Error Of Measurement Formula Larger the population standard deviation, large the standard error of statistic. 1:20 PM Post a Comment Newer Post Older Post Home Subscribe to: Post Comments (Atom) About Me Web blog from

James Brown at University of Hawai'i at Manoa gave an good comparison of these two concepts. Check This Out Contrary to popular misconception, the standard deviation is a valid measure of variability regardless of the distribution. As the SDo gets larger the SEM gets larger. A critical evaluation of four anaesthesia journals. When To Use Standard Deviation Vs Standard Error

The standard error of all common estimators decreases as the sample size, n, increases. As will be shown, the mean of all possible sample means is equal to the population mean. The standard error falls as the sample size increases, as the extent of chance variation is reduced—this idea underlies the sample size calculation for a controlled trial, for example. Source Student B has an observed score of 109.

Perspect Clin Res. 3 (3): 113–116. Standard Error Of Measurement And Confidence Interval Hyattsville, MD: U.S. doi:  10.1136/bmj.331.7521.903PMCID: PMC1255808Statistics NotesStandard deviations and standard errorsDouglas G Altman, professor of statistics in medicine1 and J Martin Bland, professor of health statistics21 Cancer Research UK/NHS Centre for Statistics in Medicine,

## The sample mean x ¯ {\displaystyle {\bar {x}}} = 37.25 is greater than the true population mean μ {\displaystyle \mu } = 33.88 years.

As you collect more data, you'll assess the SD of the population with more precision. These assumptions may be approximately met when the population from which samples are taken is normally distributed, or when the sample size is sufficiently large to rely on the Central Limit Theory (again) To illustrate the distinction between the standard deviation and standard error, the diagram below shows a normal population with mean =1000 and standard deviation =200. Use the slider Standard Error Of Measurement Reliability This can also be extended to test (in terms of null hypothesis testing) differences between means.

When we calculate the standard deviation of a sample, we are using it as an estimate of the variability of the population from which the sample was drawn. more... Warning Be particularly careful when reading journal articles. have a peek here If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean


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# Tag Archives: physics

I’m now about to finish my first year as a PhD student. Along the way I’ve done a lot of physics! Some of the concepts are very hard. I’ve sure spent my fair share of hours battling with abstract maths! But I’ve learnt something much tougher and infinitely more valuable in the past 12 months – how to do research.

The blessing and curse of research is it’s very hard to teach. You need just the right combination of perserverance, creativity and inspiration. Unlike most forms of employment, science is wonderfully, frustratingly unpredictable!

There’s one principle that stands out through every success and failure this year. Ask the obvious question! Whether in a seminar, a conversation with colleagues or in front of your desk, never be afraid to say something stupid. Often it’s the most basic idea which leads to the richest consequences.

At the end of the day, research is something of a confidence game. It’s a bit similar to my limited experience on a snooker table. If I think I’m going to win, I usually do. But when those doubts creep in, it’s much harder to keep the break going!

That’s why it’s so important that scientists communicate. Sadly the human brain doesn’t seem to be wired up to think deeply and laterally simultaneously. Regular breaks for discussion, evaluation and presentation of your work are vital!

I’ve had my clearest thoughts on walks to the tube, after chatting over coffee or writing a blog post. Although the life of a scientist might appear relaxed, ours is not a job where you can just clock in and out!

Asking the obvious question is not just important for researchers. Students, journalists, politicians, civil servants, lawyers, managers, even executives pose simple questions every day. In fact, it’s when public figures disguise their questions and answers with complex language that we struggle to relate.

A stupid, obvious question can do no harm. And more often than not, it’s exactly what you need to say.

# Physics Through the Looking Glass

I recently took part in the popular Three Minute Thesis competition. Each contestant gets just 3 minutes to explain their research to a panel of laymen. Although I didn’t make it to the national finals, it was nevertheless great fun.

Here’s an audio recording of my speech, taken live at the QMUL finals. For the experts among you, I’m giving an account of my attempts to use twistor techniques to investigate subleading soft theorems in gauge theory and gravity!

# Maths in Physics – Three Big Ideas

Let me take you back 2400 years to ancient Greece. The philosopher Plato is thinking about the universe. Fundamentally it has to be built of simple objects, he muses. The simplest 3D objects are the most symmetrical ones. So he associates each basic element (fire, water, air, earth and aether) with a regular solid.

Of course, we don’t believe in Plato’s theory today. Nevertheless it’s a fascinating example of the interplay between maths and physics. This dialogue has continued down the ages. For more than two millenia we’ve been striving to arrive at a mathematical understanding of reality.

But what’s the point? On the one hand, there’s the philosophical argument of Plato. Somehow mathematics seems like the fundamental language of the universe. We’d better speak it if we’re to hold a meaningful conversation about physics. But there’s a more prosaic reason too.

Maths is simply the most efficient way of encoding our predictions about experiments. Centuries of science have shown that expressing models mathematically enables swifter and more precise verification of our ideas. It seems then that maths and physics are inextricably linked.

For PhD students and layman alike, this realization is a terrifying one! Pretty much everyone finds mathematics hard. Symbols and equations are easy to muddle up. Calculations are opaque at best, and impossible at worst. Even the basic formula for the current understanding of particle physics is a daunting thicket of letters.

But now I’m going to make a bold assertion. Actually the maths is quite easy. It’s the physics that’s hard. Really maths is just a collection of interesting abstract ideas. Most of these are motivated by some natural intuition. This makes them easy to “get” on a gut-instinct level. The problems arise when you have to select the right mathematical ideas and apply them to the real world.

I’m going to give you three brief examples of big mathematical ideas in modern physics. You’ll see that the concepts themselves are beautifully simple. The scary mess of symbols only rears it’s ugly head once you bolt these ideas together with physics.

## 1. Symmetry

Most of us have an intuitive idea about symmetry. You’d say something’s symmetric if it’s the same after you rotate or reflect it. In many cultures objects and patterns with lots of symmetry are considered beautiful. That’s principally because humans are lazy.

We like symmetrical things because they’re simple. A snowflake is easy on the eye because it’s very regular. Look at just a small part of it and you can predict the rest. So symmetry provides us with useful information for free.

You can see why mathematicians and physicists like symmetry – it makes their job easier! The more symmetry you claim to have, the more constrained your answers must be. That’s all very well as an abstract principle perhaps, but where does the physics come in?

Fortunately the real world has lots of symmetry. Look to your left. Have the laws of physics changed? Hopefully not! This tells us that physics has a rotational symmetry. When you walk to the shops reality doesn’t suddenly shift. So there’s a so-called translational symmetry too.

From our experience with the snowflake we should be able to extract some information about the world. We need to get a feel for what that should be.

Imagine that walk to the shops again. You’re pretty meticulous so you always walk at the same speed. And you like some variety so you’ve got a choice of two routes. Each route is exactly a mile long.

It’s obvious that your journey will take the same time whichever route you take. That is providing there’s nothing special to distract you on the way. Perhaps one route has a travelator, so you go faster even though you aren’t pacing quicker. Maybe on the other route you bump into an old friend and stop for a chat.

But in both of these scenarios you’re changing your overall speed. In the absence of such distractions the speed you walk is conserved, precisely because there’s nothing special about either route you take. In other words the translational symmetry gives rise to a speed conservation law.

So spatial symmetry gives rise to speed conservation. More precisely physicists talk about momentum conservation, which takes into account mass as well. What about rotations? Well they lead to conservation of angular momentum. You’ll have felt its effects if you’ve ever been on a rollercoaster!

So we’ve got a double whammy – symmetry exists in the real world, and it makes doing physics easier. It’s no wonder that our best theory of particles (the boringly-named “Standard Model”) is based entirely on symmetries.

Modern physics has all kinds of more general symmetries. Supersymmetry, gauge symmetry, colour symmetry, flavour symmetry – the list goes on! These mathematical principles guide us towards particular models of reality. In fact the terms in the daunting formula above are really just a consequence of choosing appropriate physical symmetries.

Now for a quick tour of the current battle-lines. Lately people have discovered new symmetries related to particle scattering. Others have been able to exploit these to get a better perspective on the theoretical foundations.

The upshot is a symmetrical jewel which encodes a huge amount of information. This amplituhedron is something like Plato’s solids. So maybe the Greek geezer wasn’t so wrong after all!

## 2. Integrability

Unlike symmetry, you’ve probably never heard of integrability. At heart it’s just a posh word for something extremely simple. We say a physical model is integrable if you can get exact answers from it.

This seems a little arcane, until you realise that we only have approximate solutions to most physical problems. Unfortunately the equations that describe reality are pretty hard. So physicists have to estimate the answer using various techniques. Only in exceptional cases can we do a full mathematical derivation.

This malarkey started off with Newton. In his famous Second Law he claimed that force was proportional to acceleration. This is immediately obvious during takeoff in an aeroplane, for example.

Many of you will know that acceleration measures how fast your speed is changing. So if you know the force being applied, Newton is telling you how the speed changes. Mathematicians call this a differential equation since difference is a synonym for change.

The usual way of solving these equations is called integration. Suppose you know the speed at the start of the motion. Newton gives you the change in speed from one moment to the next. Adding up all the changes over these tiny moments gives you the speed at any point in the future!

Sadly, integration gets really tricky for complicated systems. That’s when the physicists have to approximate things. The most famous example dates back to Newton’s time. Suppose you want to understand the motion of three planets interacting gravitationally. Nobody’s ever been able to find an exact solution for this, and there’s strong evidence that one doesn’t exist at all!

The idea of integrability carries over into quantum mechanics too. Basically a quantum theory separates reality out into a spectrum of chunks. In particular, energy comes in discrete lumps called quanta. But finding out what energies are allowed is difficult. Again you’ve got to solve a differential equation!

In general physicists make approximations, usually by perturbing a system slightly from some ideal configuration. This works remarkably well, but can’t give us complete knowledge of a theory. This ignorance is one of the difficulties associated with forming a quantum theory of gravity.

It seems that our mathematical accounts of nature are just not integrable. Neither in Newtonian physics, nor on the quantum level do the complicated interactions admit a pretty solution. In fact, integrability is a rare and magical property.

Typically integrability is associated with high symmetry situations. We saw in the last part that symmetry gives us useful constraints on allowed physics. By adding a healthy tablespoon of symmetry to the mix we raise our chances of finding a neat solution. Such souped-up theories are an active area of research.

But what’s the point? If integrable theories aren’t very realistic, then why should we be looking at them. Quite apart from their mathematical elegance, they give us new insights into the structure of our model. Methods that start off in the highfalutin realms of an integrable theory can percolate down to more realistic descriptions.

That’s just what some people hope will happen for our modern day Platonic amplituhedron. Many regard it’s beauty as a remnant of the integrability of a relevant souped-up theory. But its lessons could well inform our view of the universe.

## 3. Duality

Racing into the final furlong, we’re confronted with another unfamiliar word. But you might have an inkling about what it means. Think of your favourite dynamic duo – two individuals that present different perspectives yet work well together. From Bush and Blair to Ant and Dec, these relationships are common in the celebrity world.

Curiously this kind of friendship crops up mathematically too! Here’s an easy example. Suppose you draw two random lines on a piece of paper. If your paper is big enough they’ll meet at a point. Similarly if you put two dots on your paper there’s a single line that joins them.

Writing this down as a formula we can roughly say

$\displaystyle \textrm{two lines} = \textrm{one point}\\ \textrm{two points} = \textrm{one line}$

So if you prove something about points, you can immediately translate it into a statement about lines and vice versa. This is a simple example of a mathematical duality. You’ve got two different perspectives on geometry that work together to enrich your understanding.

You’d be forgiven for finding that example unremarkable. There’s nothing very exciting about lines on bits of paper. So let me convince you that duality is important in the real world.

Suppose you’re sitting at your computer, listening to your favourite song. Chances are that it’s been compressed into MP3 format. Have you ever wondered how that’s done? You’ve got to take the wave encoding the song and smush it together somehow.

It’s hard to work out a procedure at first. Just putting all the wave peaks closer together won’t work because that’ll just raise the pitch of the whole song! The key is to come at the issue from a different perspective. You decompose the song into different frequencies and then remove pitches that a human can’t hear.

The trick we used was to exploit a duality between time and frequency. Mathematically the act of decomposing the song is called a Fourier transform. This provides a new perspective on the original wave. You can transform things back and forth as you wish depending on which viewpoint is most convenient.

Duality is convenient because it gives us several mathematical models reproducing the same physics. You’re free to analyse the theory from whatever perspective is the most appealing. We’ve seen that our models are generally very hard to solve. Duality can help to simplify the situation.

Physicists exploit duality in two main ways. Many dualities allow you to improve your approximations of a system. Others give you further details of hidden solutions or symmetries. Those which do both are especially highly prized.

We’ll finish with a modern duality which remains a hot topic today. It has been instrumental in improving our knowledge of several physical models. More specifically it relates a souped-up quantum theory to a stringy theory of gravity. We call it the AdS/CFT duality.

This duality is a mathematical chocolate ganache – decadent, moreish and extremely rich. Many of these qualities arise through large amounts of symmetry. Each of the AdS and CFT duo are exquisitely symmetrical and perhaps even integrable.

The heady mixture of duality, symmetry and integrability has inspired much research over the past 15 years. In many ways, the amplituhedron is the grandchild of this cross-fertilisation. And the story is similar elsewhere in physics. From cosmologists to materials scientists, everyone’s using this powerful brew.

So get out there and impress your friends! I’ve given you a toolbox of ideas from the forefront of theoretical research. In a sense, you are right at the cutting edge. Though you might need a bigger instruction set before you try to build anything.

# New Year’s Resolution

Happy New Year everyone! I’ll be starting my PhD at Queen Mary, University of London next week and I’ve decided to pick up my blog activity. I’ll try to post a quick article every couple of days detailing something I’ve learnt, or any interesting thoughts I’ve had. Over time I might build up to longer reviews.

If there are any specific requests for theoretical physics material you’d like to see covered, either at a layman or a research level then please do comment them below!


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Volume 301 - 35th International Cosmic Ray Conference (ICRC2017) - Session Gamma-Ray Astronomy. GA-galactic
Observing the Galactic Plane with the Cherenkov Telescope Array
R. Zanin,* J. Holder on behalf of the CTA Consortium
*corresponding author
Full text: pdf
Pre-published on: August 16, 2017
Published on: August 03, 2018
Abstract
The Cherenkov Telescope Array is a next generation ground-based
gamma-ray observatory designed to detect photons in the 20\,GeV to
300\,TeV energy range. With a sensitivity improvement of up to one order of
magnitude on the entire energy range with respect to currently operating facilities, coupled with
significantly better angular resolution, the array will be used to
CTA will explore the ultra-high energy (E $>$50 TeV) window with great
sensitivity for the first time.

CTA is expected to reveal a detailed picture of the Galactic plane at
the highest energies, and to discover around one hundred
new supernova remnants and many hundreds of pulsar wind nebulae,
according to current population estimates. The ability of the
observatory to resolve such a large number of Galactic sources is one
of the challenges to be faced. In this paper, we will present the first
simulated scan of the Galactic plane with a realistic observation
strategy, with particular attention to the potential source
confusion. We will also present prospects for morphological studies of extended
sources, such as the young SNR RX J1713.7-39.
DOI: https://doi.org/10.22323/1.301.0740
How to cite

Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.

Open Access


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# Pattern-based models and a cooperative parallel metaheuristic for high school timetabling problems

Our paper “Pattern-based models and a cooperative parallel metaheuristic for high school timetabling problems” by Landir Saviniec, Maristela Santos, Alysson M. Costa and Lana Santos has been accepted for publication at the European Journal of Operational Research. The paper has part of the results from Landir’s PhD thesis.

[doi] [preprint]
Pattern-based models and a cooperative parallel metaheuristic for high school timetabling problems.  Saviniec, L., Santos, M. O., Costa, A.M., Santos, L. M.  European Journal of Operational Research (Accepted, 09/Aug/2019).

Abstract

High school timetabling problems consist in building periodic timetables for class-teacher meetings considering compulsory and non-compulsory requirements. This family of prob- lems has been widely studied since the 1950s, mostly via mixed-integer programming and metaheuristic techniques. However, the efficient search of optimal or near-optimal solutions is still a challenge for many problems of practical size. In this paper, we investigate mixed- integer programming formulations and a parallel metaheuristic based algorithm for solving high school timetabling problems with compactness and balancing requirements. We pro- pose two pattern-based formulations and a solution algorithm that simultaneously exploits column generation and a team of metaheuristics to build and improve solutions. Extensive computational experiments conducted with real-world instances demonstrate that our formulations are competitive with the best existing high school timetabling formulations, while our parallel algorithm presents superior performance to alternative methods available in the literature.

# Examining Trade-Offs in Piggybacking Flow Events while Making Environmental Release Decisions in a River System

Our paper “Examining Trade-Offs in Piggybacking Flow Events while Making Environmental Release Decisions in a River System” by Kaur, S., Horne, A. C., Nathan, R., Szemis, J. M., Gibson, L., Costa, A. M., Webb, J.A., Stewardson, M. has been published at the Journal of Water Resources Planning and Management.

Abstract:

High flow pulses (or spells or freshes) play a crucial role in maintaining the ecological health of a river system. Impoundment of water in a reservoir and release or diversion of water for human water needs has significantly altered the magnitude and frequency of flow pulses in many river systems, often reducing river ecological health. A limited volume of water is sometimes available for release into the river to reintroduce pulses specifically aimed at meeting ecological requirements (environmental water). If aiming to achieve maximum envi- ronmental benefit, such releases from the reservoir should be timed to augment or piggyback natural unregulated catchment flow events. These decisions must be made in presence of uncertainty of near-future unregulated catchment inflows entering the river. Making flow release decisions under this uncertainty poses the risk of either not achieving the benefit of the environmental flow release because too little envi- ronmental water is released, or of causing flood damage because too much is released. To date, assessment of risks associated with piggy- backing environmental flows have focused solely on the flooding risks. This paper considers assessment of trade-offs between environmental risks and flooding risks while making piggybacking decisions. The key contribution of the paper is a risk framework that allows for the assessment of both flooding and environmental risks when piggybacking of natural flow pulses occurs. The risk framework is used to assess rules or rules with varying levels of piggybacking on the trade-offs between environmental outcomes and flooding risks when releasing piggybacking flows under these rules for flow events under near-future forecast uncertainty. Spawning flows for a key fish species in the Yarra River in southeast Australia is used as a case study to compare three piggybacking rules. DOI: 10.1061/(ASCE)WR.1943- 5452.0001048. © 2019 American Society of Civil Engineers.

# Call for papers: III International Conference on Agro Big Data and Decision Support Systems in Agriculture (BigDSSAgro 2019)

The Organizers of the III International Conference on Agro Big Data and Decision Support Systems in Agriculture (BigDSSAgro 2019) would like to invite you to participate in the conference by submitting a contributed paper (extended abstract).

BigDSSAgro 2019 will be held at Universidad Técnica Federico Santa María in Valparaíso, Chile, September 25-27, 2019. http://www.bigdssagro2019.usm.cl

The BigDSSAgro2019 conference is the third conference in the series devoted to decision support systems in agriculture, which combine Operational Research methods and Big Data techniques to support many real-world decision-making problems in small scale farming, industrial agriculture, smart farms, agri-business supply chains, land use and environmental protection, among others. In BigDSSAgro 2019, the scope will be extended to focus on decisions, technologies and innovation for the 2030 agriculture.

Authors are invited to submit extended abstracts of their recent work by May 20, 2019. See the submissions guidelines for more information in the conference web page. The Program Committee will select the papers to be presented on the basis of the submitted extended abstracts. It is expected that revised and extended versions will subsequently be submitted for publication in a post-conference special issue to be announced. The list of topics in which we invite submissions include, but are not limited to the following main areas:

-Operational research methods relevant to agriculture, forestry and environment.
-Sustainability indicators in agriculture and environment.
-Modelling sensor based data to get useful information.
-Big Data techniques with potential applications in agriculture.
-Optimization and simulation models, including agent based simulation.
-Decision support tools based on Big Data and Operational research techniques.
-Decision support tools based on GIS and their implications with Big Data.
-Economic aspects of adoption of Big Data analytics.
-Machine learning.
-Support Vector machine.
-Precision agriculture, digital agriculture.
-Industrial agriculture.
-Small scale farming.
-Forestry.
-Land Use.
-Environment studies and environmental protection.

Important dates.
Submission start: April 1, 2019.
Submission deadline: May 20, 2019.
Author acceptance/rejection notification: July 15, 2019.
Camera-ready Extended Abstract deadlines: August 15, 2019.

We look forward to seeing you in Valparaíso.

Kind regards,
Víctor M. Albornoz (Universidad Técnica Federico Santa María)
Alejandro Mac Cawley (Pontificia Universidad Católica de Chile)

Conference General Chairs

# A co-evolutionary matheuristic for the car rental capacity-pricing stochastic problem

Our paper “A co-evolutionary matheuristic for the car rental capacity-pricing stochastic problem” by Beatriz Oliveira, Maria Antónia Carravilla, José Fernando Oliveira and Alysson M Costa has been accepted for publication at the European Journal of Operational Research. This paper contains part of the results from Beatriz’s PhD thesis.

Abstract

When planning a selling season, a car rental company must decide on the number and type of vehicles in the fleet to meet demand. The demand for the rental products is uncertain and highly price-sensitive, and thus capacity and pricing decisions are interconnected. Moreover, since the products are rentals, capacity “returns”. This creates a link between capacity with fleet deployment and other tools that allow the company to meet demand, such as upgrades, transferring vehicles between locations or temporarily leasing additional vehicles.

We propose a methodology that aims to support decision-makers with different risk profiles plan a season, providing good solutions and outlining their ability to deal with uncertainty when little information about it is available. This matheuristic is based on a co-evolutionary genetic algorithm, where parallel populations of solutions and scenarios co-evolve. The fitness of a solution depends on the risk profile of the decision-maker and its performance against the scenarios, which is obtained by solving a mathematical programming model. The fitness of a scenario is based on its contribution in making the scenario population representative and diverse. This is measured by the impact the scenarios have on the solutions.

Computational experiments show the potential of this methodology regarding the quality of the solutions obtained and the diversity and representativeness of the set of scenarios generated. Its main advantages are that no information regarding probability distributions is required, it supports different decision-making risk profiles, and it provides a set of good solutions for an innovative complex application.

# Algorithms for the power-p Steiner tree problem in the Euclidean plane

Our paper “Algorithms for the power-p Steiner tree problem in the Euclidean plane” by Christina Burt, Alysson Costa and Charl Ras has been published on a special issue on Exact and Heuristic Solutions for Optimization Problems from RITA.

Abstract

We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a “beaded” minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case.

# Active management of environmental water to improve ecological outcomes

Our paper “Active management of environmental water to improve ecological outcomes” has been published at the Journal of Water Resources Planning and Management.

[doi] [cited by] [preprint] Active management of environmental water to improve ecological outcomes.  Horne, A., Kaur S., Szemis, J., Costa, A.M., Nathan, R., Angus Webb, J., Stewardson, M., Boland, N. , Journal of Water Resources Planning and Management 144 (2018)

Abstract:

Environmental water is being embraced by governments around the world as a means to partly restore rivers impacted by excessive river regulation and to protect those that are not yet overregulated. In cases where environmental water is provided as a water right, managers can make ongoing active management decisions regarding the timing and magnitude of environmental flows in response to changing conditions and understanding of environmental demands. Such active management should lead to improved environmental outcomes but also comes with a significant ongoing management cost when compared to rules-based environmental water releases. This study uses an optimization tool to examine the environmental gain achievable with the active management of environmental water compared to rules-based releases. A case study of the Yarra River, Victoria, Australia, demonstrates the potential for substantial environmental gains achieved through active management of environmental water.

# Congratulations to Dr. Cheng Cheng

Cheng Cheng has had her PhD thesis approved at the University of Melbourne. Her thesis is titled “Optimisation of Disaster Waste Management Systems” and proposes models and algorithms for planning waste removal after disasters. Among other results, Cheng analysed the effect of using intermediate waste processing centers in the efficiency of the operations.

Congratulations to Cheng!

# Congratulations to Dr. Beatriz Oliveira

Beatriz Brito Oliveira has defended her PhD thesis at the University of Porto. Her thesis is titled “Fleet and revenue management in car rental: quantitative approaches for optimization under uncertainty” and was supervised by Prof. Maria Antònia Carravilla and Prof. José Fernando Oliveira.

Beatriz spent three months with our research group in Melbourne. During this time she worked in the last chapter of her thesis, finalising the development of a matheuristic for the stochastic version of the car rental planning problem she tackled.

Congratulations to Beatriz, Maria Antònia and José Fernando for this strong piece of research!

# Optimising elective surgery schedules

Our paper A Sequential Stochastic Mixed Integer Programming Model For Tactical Master Surgery Scheduling, by Ashwani Kumar, Mark Fackrell, Peter Taylor and Alysson M. Costa is now available online.

Abstract:

In this paper, we develop a stochastic mixed integer programming model to optimise the tactical master surgery schedule (MSS) in order to achieve a better patient flow under downstream capacity constraints. We optimise the process over several scheduling periods and we use various sequences of randomly generated patients’ length of stay scenario realisations to model the uncertainty in the process. This model has the particularity that the scenarios are chronologically sequential, not parallel. We use a very simple approach to enhance the non-anticipative feature of the model, and we empirically demonstrate that our approach is useful in achieving the desired objective. We use simulation to show that the most frequently optimal schedule is the best schedule for implementation. Furthermore, we analyse the effect of varying the penalty factor, an input parameter that decides the trade-off between the number of cancellations and occupancy level, on the patient flow process. Finally, we develop a robust MSS to maximise the utilisation level while keeping the number of cancellations within acceptable limits.

# AMSI OPTIMISE 2018

Returning in 2018, AMSI Optimise is an annual networking and research-training event that aims to strengthen mathematical optimisation research engagement and its applications across industry.

This event will comprise a three-day industry-focused conference, followed by a two-day research workshop. The symposium features expert and end-user talks, international guest speakers, collaboration showcases, industry challenge sessions and tutorials. The themes of the 2018 conference are Decision Making Under Uncertainty and Humanitarian Applications.

AMSI Optimise is aimed at:

• anyone using optimisation, with opportunities to learn more about the current state of the art and to connect with others who have similar interests
• industry practitioners interested in exploring the benefits of engagement with optimisation research
• academics and postgraduate students wanting to better understand drivers and needs in this area

The event provides a forum to bring together people from the diverse companies and research disciplines that use and develop optimisation models and software. It provides a platform for research training and developing new collaborations in this vital area through PhD internships, research partnerships and postgraduate research projects.